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Radiation driven instabilities in stellar winds Carlberg, Raymond G. 1978

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RADIATION DRIVEN INSTABILITIES IN STELLAR WINDS by RAYMOND GARY CARLBERG M.Sc. , University of British Columbia, 1975 B.Sc., University of Saskatchewan, 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOB THE DEGREE OF DOCTOE OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES* Department of Geophysics and Astronomy We accept this thesis as conforming to the reguired standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1978 (c) Baymond Gary Carlberg, 1978 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be al1 owed without my written permission. Department of The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date Odf *? jl% ii ABSTRACT This thesis investigates the quantitative nature of the variability which is present in the stellar winds of high lu minosity early type stars. A program of optical observations with high time and spectral resolution was designed to provide quantitative information on the nature of the fluctuations. These observations found no optical variability over a time period of six hours and hence restrict the variability over this pericd to size scales of less than 5x1011 cm, but do confirm the variations on time scales exceeding one day. A class of X-ray sources comprised of a neutron star orbiting a star with a strong stellar wind provides another source of information on the variability of stellar winds. A theory of accretion onto a neutron star was developed which is used with X-ray intensity data to derive estimates of the density and velocity of the stellar wind. This analysis performed on Cen X-3 suggests that the velocity in the stellar wind increases as the wind density increases. A theoretical analysis of the stability of a stellar wind is made to determine whether the variability may originate in the wind itself. Two types of instability are founds those that amplify pre-existing disturbances, and absolute instabilities which can grow from random motions within the gas. It is found that short wavelength disturbances <<10* cm) are always strongly damped by conduction, and leng wavelength ones (>1011 cm) are damped by radiation if the gas is thermally stable, that is if the net radiative energy loss increases with temperature. In termediate wavelengths of about 108~9 cm are usually subject to iii an amplification due to the density gradient of the wind. The radiation acceleration amplifies disturbances of scales 107 to 10*1 ca. Absolute instabilities are present if the gas is ther mally unstable, if the flow is deccelerating, or if the gas has a temperature of several million degrees. On the basis of the information derived on stellar wind stability it is proposed that a complete theory should be based on the assumption that the wind is a nonstationary flow. CONTENTS Chapter 1: Introduction .................................... Chapter 2: Optical Observations Chapter 3: Supersonic Accretion Chapter 4: Physical Description of The Gas ................. Chapter 5: The Stability analysis .......................... Chapter 6: Conclusions ..................................... Bibliography Appendix 1: Radiative Effects In Supersonic Accretion ...... Appendix 2: Gas Physics .................................... Appendix 3: The Dispersion Relation ........................ Appendix 4: The Major Computer Prcgrams .................... V FIGURES 1. Lambda Cephei: The Effect Of Resolution ................ 10 2. lambda Cephei Time Series .............................. 14 3. lambda Cephei Day Tc Bay ...15 4. Alpha Camelopardalis Day To Day ........................ 16 5. Delta Orionis Day To Day ...17 6. Supersonic Accretion Schematic .........................22 7. Schematic X-ray Intensity Variation of Cen X-3 .........27 8. The Density Velocity Variation of Can X-3 .............. 28 9. Ionization 8ala nc € ..........»« .... ..........«•........ .35 10. Standard Heating And Cooling Rate ..................... 36 11. CNO Abundances 10 Times Solar Heating And Coding .....38 12. Losses In An Optically Thick Medium ................... 40 13. Radiation Force As A function Of Temperature ..........42 14. Momentum Balance ...................................... 46 15. Roots For A Static Pseudo Iscthermal Atmosphere ....... 55 16. Hith Heating And Cooling, V, dv/dz Nonzero ........... .60 17. 8ith The Radiation Force ..............................62 18 • No CooXxncj • • • • • • • • • * • * • • • • * ••••*••••*•«•••• * • • • • 63 1. Thermal Instability T=5x10s K ......................... .65 19. High Temperature Instability .......................... 66 20. Decceleration Instability .............................67 vi TABLES 1. Catalogue Of Observations 9 2. Scales In Supersonic Accretion ........................ .23 3. Atomic Abundances ......................................32 4. Maximum Velocity For An Accelerating Solution ..........47 5. The Dispersion Relations Plotted .......................68 6. Photoionization Cross Section Parameters ............... 90 7. Gaunt Factor For Hydrogen 92 8., The Recombination Hate Constants .......................93 9. Ionization Potentials And Subshell Populations .........97 10. Gaunt Factor Constants ................................ 102 11. The Resonance Lines ................................... 103 vii ACKNOWLEDGEMENTS The completion of this thesis owes a great deal to the teaching, assistance, and encouragement that I have received. My supervisor. Dr. Greg Fahlman, always provided an interested ear and a careful criticism of my ideas. He was also a scurce of support when needed. Drs. Gordon Walker and Jason Auman made a number of suggestions which substantially improved the presentation of the thesis. My induction into the world of observational astronomy came with three very pleasant summers spent in Victoria at the Dominion Astrophysical Observatory. . The frustrations of doing spectrophotometry with photographic plates were largely elimin ated by Dr. Gordon Walker's encouragement to use his fieticon detector system. The successful observations obtained with this system owe a great deal to the advice and technical support of Dr. Walker, Dr. Bruce Campbell, Dr. Chris Pritchett, Tim Lester, and Mike Creswell. The whcle data reduction process has been reduced to a straightforward and enjoyable activity by the use of Chris Pritchett*s data handling package, RETICENT. A large part of this thesis relied on the computer for its completion. The OBC Computing Centre has been a great assis tance and deserves much praise for its facilities, and the pro vision of a large number of well documented utility programs. Particular thanks go to A. C. Hearn, originally at the University of Utah, who wrote the program REDUCE. Financial support was provided by the National Research Council of Canada and by a MacMillan Family Fellowship. 1 CHAPT IB 1. INTE0DUC1I0N The optical spectra of many hot stars were noted to have emission components on the Harvard objective prism plates {see the Henry Draper Catalogue, Cannon and Pickering 1918). Many of these stars were studied in detail, leading to Beal's (1929) proposal that the line profiles could be explained by emission from gas being ejected from the star. A great deal of informa tion was accumulated on the optical spectra of emission line 0 and E stars in the following years, which is summarized in Eeals (1951) and Dnderhill (1960). A new observational window^ was opened by Morton (1967) using a rocket borne spectrograph. He found that 4 of the Orion stars, S, 6, and £., had absorption lines blue shifted to velocities of order 2000 Km s_l. These velocities exceeded the typical escape speed of 300 Km s~1 by such a large factor that there was no question that the stars were losing mass at a large rate. The Snow and Morton ultraviolet survey (1976) showed that all stars with an effective temperature greater than about 3x10* K, and luminosities greater than a bolometric magnitude of -6, have a detectable supersonic wind which carries away a signifi cant amount of the star's mass during its lifetime. Reliable mass loss rates have been obtained from optical observations (see Hutchings 1976) and OV observations (Snow and Morton 1976) which now have been extended to infrared (Barlow and Cohen 1977) and radio wavelengths (Bright and Barlow 1975), all of which provide confirming and complementary data on the magnitude of the mass loss. The derived mass loss rates for OB stars lie in the range of 10~9 to 10~s M0/year, with terminal velocities 2 ranging from 1000 to 3000 km s-». The stellar wind phenomenon poses several questions: what is the physical mechanism driving the mass loss? how does the mass loss effect the star's evolution? and how do these hot, luminous stars - %f feet the interstellar medium and the evolution of a galaxy? The answers to all cf these questions hinge on a thorough understanding of the physics of the stellar wind. This thesis is a contribution to the understanding of the basic nature of the stellar wind. The physical problem is to describe the dynamics of a qas moving in an intense radiation field; a situation which occurs in a number of astrophysical situations, including quasars and active galactic nuclei, (Mushotzky, et al. 1972 and Kippenhahn et al. 1975). The line profiles cf stellar wind stars, especially ones with high mass loss rates have long been known to show some variability over one day (see for instance Beals 1951, Underbill 1960, Conti and Frost 1974, Leep and Conti 1978, Brucato 1971, Snow 1S77, and Hosendahl 1S73). For ground based observation this time period makes it difficult to resolve the time evolu tion of the variation. An observational program was initiated to more closely define the nature of these reported variations, using a modern detector capable of measuring very small changes. This will be discussed in Chapter 2. Besides optical evidence of variability, there are a number of X-ray binary stars where the X-ray source is a neutron star accreting mass from the stellar wind fCcnti 1978). These sources show a number of scales of variation of their intensity which can be ascribed to variations of the stellar wind. A 3 theoretical analysis of the accretion process and how it effects the ctserved intensity was made in order that the X-ray data could be used to derive the prevailing density and velocity of the stellar wind at the location of the neutron star. This an alysis performed on X-ray data for the source Cen X-3 indicates a correlation between the wind velocity and density. This will be described in Chapter 3. The basic formulation of the theory of a stellar wind from a hot, luminous star was initially put forth by Lucy and Solomon (1970), who proposed that the acceleration was produced by the scattering of photons with wavelengths that fell within a few resonance lines. This was a generalization of Milne*s (1926) idea that momentum transfer from photons could selectively acce lerate certain ions. This was later extended by Castor, Abbott, and Klein (1975, referred to as CAK) to include the force on many lines of many ions. The theory provided an encouraqinq agreement with the limited data available on the velocity as a function of radius and mass loss rates. Recently the ultraviolet satellite observations have re vealed that some highly ionized species, in particular 0 VI and N V, are present in the wind. These ions would not be expected to be ionized in any observable quantity by the radiation field appropriate to these stars. York, jt al. (1977) have observed variations in the 0 VI line in three stars over a time periods as short as six hours. This observation suggests a "slab" moving outwards at an increasing velocity.. The presence of 0 VI in the stellar wind presents a puzzle as to the source of its excitation. At the present time there are three proposals. 4 First, Castor (1978) has modified his radiation driven wind to an arbitrarily specified temperature higher than radiative equi librium, which provides a suitable abundance of C VI. Second, Lamers and Snow (1978) have an empirical "warm radiation pressure" model, in which they show that the ions can be provi ded if the stellar wind is at a temperature of aJsout 2x10s K* Neither of these models specify the source of the additional heating. Third, Hearn (1975) has proposed that stellar winds are initially accelerated in a hot corona with a temperature cf several million degrees. Pursuing this idea Clson and Cassinelli (1978) have shown that a small corona, about 10% of a stellar radius, generates enough thermal X-rays to produce the reguired ionization ratios. To provide a heating mechanism for a corona, Hearn (1972) showed that radiation driven sound waves could be amplified while propogating outward in the atmosphere. The waves grow to a saturated amplitude sufficient to provide enough shock heating to maintain a corona (Hearn 1973). There are two difficulties with this analysis. Berthomieu et al. (1975) have pointed out that Hearn*s simplifying assumptions result in a scale length for the wave amplification which is the same as the atmospheric scale length. Therefore significant amplification only occurs over lengths which invalidate the as sumption cf small variations of the zero order quantities over the length for amplification. In addition, the unstable waves that he finds are amplifying instabilities (Castor 1977), and require some oscillator to initiate the wave motion. Motivated by theoretical arguments and the observations of fluctuations I have performed a stability analysis on the egua-5 tions governing the moving gas in the stellar radiation field. The complete set of eguaticns governing the motion with no a ££ipri simplifications were used. An accurate description of the gas physics was developed using an approximate treatment of radiation transfer dependent ccly on local guantities. As a result the state of the gas can be completely specified by the local radiation field, the gas velocity and its gradient, and the density and temperature, as described in chapter 4. The stability of the gas against vertical disturbances was investi gated with the aid of a computer to provide the numerical solu tions to the dispersion relation. Chapter 5 is comprised of this discussion. The purpose of this thesis is to investigate the quantita tive nature of instabilities in stellar winds and relate it to the observational and theoretical problems which have been out lined. This is not an attempt to create a unified theory of a stellar wind. Bather it is a detailed investigation of certain areas of the question in order to illuminate some of the physi cal mechanisms which are important in a stellar wind. This is required because not much is known about the basic physical pro cesses which dominate the observed variability of the stellar wind. The investigation is confined to the stellar wind itself, which is loosely defined as the region where the optical depth in tbe continuum is less than one and the gas is moving with greater than scnic velocities. As has been emphasized by Cannon and Thomas (1978), it is possible that some of the driving force for the the wind and hence seme of the wind instabilities may 6 originate within deeper layers of the star. It is assumed that there is no magnetic field. This is done mostly because of the tremendous simplification of the problem which results. But there is no observational evidence for a magnetic field, although if the wind is as chaotic as this thesis suggests, a magnetic field would be difficult tc detect. In summary this thesis is motivated by observations of stellar wind variability, and suggestions by ether authors that instabilities do exist which may be responsible for the creation of a high temperature corona. The investigations described are carried out in twe parts. observational evidence of the varia bility is acquired which suggests length and times scales cf the fluctuations which are present, and a correlation between the wind velocity and density. The theoretical analysis provides physical sources of several instabilities which can exist in the stellar wind. From this information I suggest that the stellar wind is an extremely chaotic medium in which the instabilities not only provide the source for the observed variability, but also can be used to provide an ionization source for the 0 VI ion and the corona as postulated by Hearn. The presence of these instabilities means that a model for a stellar wind should be in the form of mean flow guantities and associated fluctu ating guantities. 7 CHAPTER 2. OPTICAL OBSERVATIONS For many years several of the lines in the optical spectrum of several early type stars have been reported as varying (see references cited in the Introduction). Particular attention has been paid to the star Lambda Cephei, because it is a bright 06f star in the northern sky. The H ©cline has been reported to vary on time scales of one day (Leep and Conti 1978) and longer, with no apparent systematic variation. The amplitude of the varia tion is typically 10% of the intensity. This behaviour is fairly typical of the more luminous mass loss stars. The shor test period variations with a high confidence level are the UV observations made by the satellite Copernicus of Sori A, TjQri, and £ Pup (York et al. 1977), where a small feature of width about 150 Km s-1 was seen to "move" in the 0 VI line between two observations spaced about 6 hours apart. Host of the observations at optical wavelengths have been made with photographic plates, which have a photometric accuracy barely able to reveal the presence of the variation, let alone reveal much information as to its character. In fact Lacy (1977) made scanner observations of some of the lines in stars that were reported as varying and concluded on the basis cf a statistical analysis of the errors present in the eguivalent width that any variability present was less than the expected randcm error. However the eguivalent width of a line averages together all material emitting at that line freguency,, Observa tions which resolve the line can provide much more information, but at the cost of longer exposure times. The classical description of line formation in a stellar 8 wind was given by Beals (1951). The observed line can be consi dered to be made up of three almost independent parts; an under lying abscrpti.cn line formed in the photosphere of the star, a superposed emission line with its centroid at zero velocity pro duced by emission of photons in the stellar wind, and a blue shifted absorption line which is formed in the portion of the stellar wind which is silhouetted against the star. The analysis of line formation within the wind is simpli fied by the Sobolev approximation (Sobolev 1960), which says that the emission and absorption of photons in a given narrow wavelength interval, outside of the doppler core, is determined by the amount of gas moving at a velocity such that the line of sight velocity of the gas falls within the wavelength interval. This approximation is valid if the gas speed is supersonic. The assumption is supported by the observations of Hutchings (1976 and references therein) who has shown that the wind has a velo city exceeding the sound speed for distances greater than 10$ of the stellar radius. The observations were undertaken to confirm the reported variability and were to be made with sufficiently high signal tc noise, spectral resolution and time resolution to clearly re solve the variations, as they developed. In particular it was thought that there might be evidence for the nature of the mech anism of the variation, for instance, a spot rotating with the star cr a "blob" moving cut through the wind. fill observations were carried out with the 1.2 meter teles cope of the Dominion Astrophysical Observatory, Victoria, E.C. The 2.4 meter camera in the ccude spectrograph was used with a 9 red coated image slicer giving a projected slit width cf 6C mi crons. The spectrum was detected with a 10 24 element array of 25.4 micron diodes (a Beticon RL1024/C17) cooled to a tempera ture of -80° C (Walker et al. 1S76). The image slicer and de tector pixel size combination were chosen to give a properly oversampled spectrum. , All observations were centred on the Ho< line in the first order resulting in a dispersion of .125A/diode. Observations were made in September and October of 1977, and are tabulated below and shown in the accompanying fig ures. TABLE 1: CATALOGUE OF OESIBVATIONS # Star Date Time Expos 1977 PSI secon 1 lambda Cep Sept 11/12 22: 13 2250 2 ii •i 22:55 u 3 « it 23:41 tt 4 ti ti 00:23 ii 5 n •i 01:05 ii 6 ti n 01:48 ti 7 II ii 03:10 ti 8 n tt 03:54 ii 9 » II 04: 36 ii 10 it Oct 11/12 23:00 3000 11 ti n 01:08 30 00 12 ti Oct 12/13 22:15 3000 13 H Oct 16/17 21:05 3000 14 •t II 23:38 30 00 15 ti tt 04:28 3000 16 Alpha Cam Oct 11/12 00:27 1500 17 ii Oct 12/13 01:35 2002 18 II Oct 16/17 22:58 1800 19 Delta Ori Oct 12/13 02:31 600 20 •i tt 03:01 600 21 ti Oct 16/17 23:00 2641 22 it it 03:41 2830 The lines present in the 100 A region examined are identi fied in Figure 1., They include the stellar and He II 6527, the interstellar 6614 A feature, and a multitude of narrow. 10 Fig 1: Lambda Cephei: The Effect of Resolution 11 weak, telluric water vapour lines. The telluric water lines and their relative equivalent widths are indicated above the spec tra. This data was taken from Moore et al. (1966), and may not give the exact relative intensities for these observations. All cf the spectra have been filtered by a Fourier transform techni que to U0% of the Nyquist frequency, which is roughly the true resolution of the spectra,, All spectra had considerable (10$) response changes along the array, due mostly to a light frost on the window of the detector. This was removed by dividing by the spectrum of a lamp which was taken immediately before cr after the observation. For the time series spectra the underlyinq shape of the spectrum did not vary within the error (0. }%) of the lamp calibration. All spectra were rectified to a linear continuum, Fiqure 1 shows the absolute necessity to resolve the tel luric water lines. As the Fiqure 2 time series of Lamda Cep over 6 hours shows, the water lines vary significantly over one hour. In Figure 1 the top spectrum shows the mean of the time series of high resolution spectra. Below it are two spectra recorded at KPNO in June, 1978 (cocrtesy of G. G, Fahlman and G. A. H. Salker) using a lower resolution spectrograph. The bottom spectrum in Figure 1, is the top spectrum but convolved with a Gaussian to give approximately the same instrumental re solution as the KfHO spectra. It is evident that the variation in an H*, profile can be entirely due to telluric water line variations, if the instrumental resolution is inadequate to clearly separate these variations out. The time series of Lambda Cephei (spectral type 06f) shown 12 in Figure 2 covers 6.5 hours. The average of this time series of observations is shown at the top of Figure 1. The lines below are the individual spectra divided by the mean, then nor malized. Although there are suggestions cf underlying broad (say about 10 A) changes, these are less than the noise level. In the day tc day observations shown in Figure 3, there is clear evidence of a variation at the Hotline of the emission feature at velocities near 200 Km s_l, and on the absorption side at velocities near -300 Km s~*. The time series difference spectra, number 1 to 9 of Figure 2, can be analyzed to determine the statistical significance of any variations. The report by York et al. (1977) of a feature of FwHM 150 Km s~1 changing over a period of 6 hours is only slightly wider than seme of the telluric water features, and leads to some difficultly in interpreting changes. The standard deviation of the spectra is in the range of 0.6 to 0.8% of the mean, other than for spectrum 9. Assuming the noise to have a normal distribution with this variance, the fluctuations must have an amplitude exceeding 2.57 standard deviations to have a less than 1% probability of chance occurence. This amplitude is indicated in Figure 2., The smoothed series of plots in Figure 2 are the same spectra as those on the left but averaged over 11 diodes. This reduces the variance by a factor of the square root of 11. The lines for a statistical significance cf 99% are again drawn on the plot. It can be seen that there are many features which do vary significantly. But the features that are varying all correspond to the wavelengths of telluric lines, except for the feature at a velocity with respect to the Hot, line 13 of +200 Km s-1 (left dotted line). This exceeds the 99% signi ficance level in records 1,2,3,6,7, and 8, going from an excess to a deficiency with respect to the mean. The variation occurs in the line (see Figure 1) near the top of the emission feature. The subtraction would be very sensitive to very small shifts of the line in this region. There are two reasons to think that this feature may not be stellar in origin. First, its variation correlates very well with the water lines at a velocity with respect to H of -700 Km s~1 (dotted line on right). And sec ond, the feature shows no velocity shift over this time period, which might be expected in a wind. I conclude that real varia tions are present in the time series, but they are most likely due to telluric features. Alpha Cam (spectral type 09.51a) was chosen because cf spe ctral type, and the presence of the emission line at H$(. Of all the stars examined it seems to have the most significant varia tions, see Figure 4. Delta Orionis (spectral type 09.511) was observed because of the variation reported by York, et al. , (1977). Observations made within one night, Oct 16/17, have no real indication cf a change. There is only weak evidence for a profile change in five days, because of the confusion created by the different strength of the telluric lines. This star is a spectroscopic binary of period of 5 days, which produces velocity shifts, but probably not profile changes. This is shown in Figure 5. The Sobclev approximation allows an estimate of the size of the region producing the photons in a given wavelength interval. Although the thickness of the shells of equal line of sight ve-JiJ_JJ.j_ 1000 4?0 -1000 I A ' I I I Difference Spectra Smoothed over 11 points 10 Oct 11/12 23:00 11 Oct 11/12 01:08 21:05 1000 500 0 -500 -1000 Km s' 15 12 Oct 12/13 '\Jty&l 14 Oct 16/17 23:38 ytfl 15 Oct 16/17 04:28 1 ni fiq. 3: Lambda Cephei day tc day 16 Fig. 4: Alpha Cam day to day 17 Fig. 5: Delta Ori day to day 18 locity will vary with the velocity and distance from the star, the contribution to the profile will be weighted towards regions of higher density. To estimate the total intensity in a wave length interval, , all integrals of the emission over the shell will be replaced by average quantities. If the shell has an average line of sight thickness. As, then the total volume emitting in the wavelength interval is approximately 4trr^2Asf, where f is a factor containing the difference between the true emitting area and the assumed disk of two stellar radii, 2z$. The constant f will be assumed to be 1. The thickness of the constant line of sight velocity shells is approximately constant over the region of dominant emission. These assumptions are justified by the actual calculations of line profiles as done by Cassinelli, et al. (1978). : The error of these approximations may be as large as a factor of five, but depends on the line considered. The average shell thickness, As, will be estimated frcm the line cf sight velocity gradient, dv/ds (=cos2# dv/dr +sin2^ v/r, where $ is the angle between the line of sight and the star) as A fluctuation in the wind which changes the emission rate will be observed as some fractional change of the flux obtained by integrating over all regions emitting in that wavelength range. Assuming that the emission rate changes by 100%, in the follow ing section an estimate will be made of the size of the fluctua tion causing a given fractional intensity change. If the fluc tuation is a region of size jf, cnly that part of the fluctuation (D 19 which is moving at an appropriate velocity to effect the inten sity in the wavelength interval contributes to the intensity change. If the fluctuation is moving at a uniform velocity with an internal velocity dispersion less than the thermal speed, then the fractional change in intensity in one wavelength interval would be (2) where Xn is the size of the fluctuation moving with a ccamcn velocity, in units of 1011 cm, r ^ is the size of the star in units of 1012 cm, A> the spectral resolution in units of 0.3A, which was the spectrograph resolution used. A typical velocity gradient is found by taking a terminal velocity of 1000 Km s—1 reached over a distance of 1012 cm. For a fluctuation which has a velocity gradient which is the same as the wind, the minimum volume emitting in a given wavelength interval would be just the velocity shell thickness cubed. In this case the intensity fluctuation is A more realistic situation might be if a fluctuation of size X has cnly a thin slab of thickness As moving at the appro priate velocity to be in the desired wavelength interval. In this case AI> - **** - 8*-°- A2 rn-*. Htr AS (4) 20 The X Cep time series restricts the magnitude of an inten sity fluctuation to less than 2% over the six hour span. Equation 4 then limits the size cf the largest region to change in this time to 5x1011 cm. These observations have confirmed the variability of the stellar Hoc line profile over times longer than one day, and con clusively show that the variation is due to the change in the profile, net changing telluric lines. The amplitude of the in tensity change in any one pixel is only slightly greater than what might be due to noise, but considering that groups of more than 10 pixels show the same change gives considerable confi dence to the physical reality of the change. The one time series of > Cep has no convincing evidence for any short term variation, or evolution of the profile. The signal to noise in the time series spectra is only about 50, which was a constraint imposed on the maximum integration time by the detector ceding system. 21 CHAPTER 3. SUPERSONIC ACCRETION Optical observations of variability are averages ever the entire volume of emission at that particular wavelength. If the fluctuations in the wind contain components cn a small scale compared tc the scale of the wind, the detection of fluctuations by way of technigues in which the integrated light is observed, are limited by the signal to noise which can be acquired. The discovery of two X-ray binaries imbedded in stellar winds, namely Cen X-3 and 30170G-37 (=HD153919) allows the possibility of using the X-ray source as a probe of the stellar wind. Since the X-ray luminosity is directly related to the rate of accre tion of a small fraction of the stellar wind onto the neutron star, the intensity of the X-ray source can be used with the aid of a sufficiently detailed understanding of the accretion pro cess to derive estimates of the density and velocity in the wind. This was the subject of the published paper which has been attached as Appendix 1. A summary of the principle results of the paper which support the conclusions of this thesis is given below. A schematic drawing cf the supersonic accretion process is shown in Fiqure 5 and the regions referred to are numbered in the figure. The incoming gas, region 1, is moving at a speed V with respect to the neutron star of mass M. The streamlines are bent in by the neutron star»s gravitational field. The mass and the velocity define the accretion radius, RA=2GM/VZ, which gives (apart from an efficiency factor which is close to one) the cross section for accretion of material. The incoming gas strikes a shock cone trailing the neutron star, called the Fiq. 6: Supersonic accretion schematic. 23 sheath, region 2. The gas is shock heated to a temperature of 3x10* (r/10l4cm)-* K, If the density is sufficiently high the gas cools. In the sheath the gas loses its component of veloci ty away from the neutron star, joins the accretion column and starts falling down, region 3. Wear the accreting neutron star the X-ray luminosity may be large enough to raise the tempera ture by Compton heating, The column will then expand cut tc an almost spherical inflow, region 4, Eventually the flow encoun ters the magnetosphere of the neutron star, region 5, below which the dynamics of the flow are regulated by the magnetic field. The gas strikes another shock a short distance above the surface of the neutron star, region 6, where the kinetic energy of infall is converted into thermal energy which is mostly ra diated away as X-rays. The table below gives length and time scales characteristic of the different regions. TABLE 2: SCALES IH SUPEBSONIC ACCBETION region size scale time scale star 1012 cm 1 day accretion column 10*0-11 Cm 500 seconds magnetosphere 108—9 cm 1 second neutron star 106 cm 1 millisecond An analysis cf this model yields several quantities which are directly related to the major parameters of interest in the stellar wind, the stellar wind density, n , and velocity. The luminosity of the unobscured source is L =4.7x103* n1,'vft-3(fl/fl0)3(Hl(/1O« ci)-»|S erg s~», (5) where p is a factor usually of order one giving the efficiency 24 of the accretion, nn=n6/(1011 cnr3) , and Vg=V/108 cm s~l. The angle that the shock cone makes with the axis of the accretion cclunn is 0=2.7° <TCO|/106 K) Vs-2 . (6) The temperature in the column, TCoi, is not directly observable, but an upper limit can be obtained by considering the heating and cooling processes, (Tto|/106 K) < 1.9 n„V*s V0*/»s. (7) This can be used to estimate the optical depth up the centre of the accretion column, "Ccol , due to electron scattering TcoJ>2.2 n,,-*/ls VQSZ/IS. (8) The electron scattering optical depth up the sheath is less than that up the column if nuV8-2<3.5, which is independent of the estimate of the column temperature. With these simple relations in hand and some X-ray data of an object that is clearly fueled by a stellar wind it is possible to confirm the model of the accretion process outlined above. , More importantly the observa tions can be used to derive the density and velocity in the wind. Two fairly good sets of published data exist for the source Cen X-3, which appears to be the clearest case of accretion from a spherical supersonic wind. The source 3U1700-37 <=HD153919) would appear to be a very strong stellar wind source from its optical spectrum (see Fahlman, Carlberg, and Walker 1977) al though there are significant effects in the spectrum associated with the period of the neutron star orbiting the 06f primary. be These effects may^due to a wake of disturbed gas trailing the neutrcn star (see Appendix 1). Or they may represent a signifi-25 cant distortion of the stellar wind itself. In any case, the X-ray data from 3U1700-37 has a lower count rate than Cen X-3, hence greater statistical errors. In Cen X-3 the observational situation is almost the exact reverse; the X-ray source is one of the brighter sources in the sky, but its optical companion is a 14th magnitude OB star which has been poorly studied (Conti 1S78) . The X-ray data for Cen X-3 shows several scales of variabi lity; a 4.8 second pulsation period, ascribed to the rotation of the neutron star and its magnetic field, a 2.1 day orbital period sometimes superposed with "anomalous dips", and an aperiodic change in the mean intensity level with a time scale of order one month. A particularly exciting observation was made by Pounds, et al. (1S75), who observed regular dips occur ring every orbit during a transition from X-ray low to high state. Jackson (1975) proposed that the two distinct dips were due to the reduction of the received flux by scattering in the two sides of the sheath of the accretion column. He deduced a velocity of the wind with respect to the neutron star of between 375 and 620 km s-1, and a column semi-angle of 20°. From egua-tion (6) and the velocities guoted by Jackson, the implied column temperature is in the range 3,5-9. 6x10s K. Schreier j§t al. (1S76) estimate the density in the wind as 1-5x10** cm-3. These two estimates are consistent with the limiting temperature of 1.5x106 K from Eguation (7), Accepting Jackson's proposal that the double dips are due to scattering in the sheath, but using the theory developed in Appendix 1, more informaticr can be derived from the observations. Pounds, et al. (1S15) note 26 that the relative depths of the dips decrease as the source turns on. Also, from inspection of their published data one can see that the dips appear to become single as the source turns on. A schematic tracing of the X-ray intensity is shewn in Figure 7. From these observations and the model cf the long term variations proposed by Schreier, et al, A rough trajectory of the variation of the stellar wind density and velocity car be plotted, which is shown in Figure 8. At point A the source is in the X-ray lew state and at B the high state. Adopting the estimate of Schreier gt al. for the low state density as 5x10*1 cm-3, and high state density of 10*1 cm~3, fixes the densities at point A and B, but not the velocity.The data shows that as the wind density decreases allowing the source to become vi sible, the velocity must be such that the optical depth up the sheath exceeds the column optical depth, and be close to one in order to provide the deep dips. This puts point A near the Tc=1 line. As the wind density drops the dips have a decreasing fractional depth, which means that the velocity must be dropping fast enough that the density and velocity are moving further below the ^,= 1 line. Eventually the dips become single as the density and velocity cross the TT5 > fc line. The combined den sity and velocity variation is such that the accretion rate, and hence the intrinsic luminosity only increase slightly while going from low to high state. The source settles down at the high state, point B, with a density of 1011 cm-3 and a wind ve locity with respect to the neutron star of about 500 Km s~l. The cptical depth up the column is so small that dips are not 28 Fig. 8: Density Velocity Variation of Cen X-3 2 9 seen regularly. When the source starts to turn off the data suggests that a different density velocity trajectory is fol lowed, such that the optical depth up the column and sheath always remains small. As the source approachs the point A again it is obscured by the increasing density of the stellar wind. If the trajectory of the variation of the wind velocity and density is schematically correct it is possible to draw a con clusion as to the driving force for the wind. The trajectory in Figure 8 suggests a correlation between the wind velocity and the wind density which would imply the acceleration of the gas up to the location of the neutron star increases as the density in the wind increases. In the radiatively driven wind of CAK the acceleration of the wind varies as n- , where is a con stant slightly less than one., This implies that the radiation acceleration should drop as the density increases. On the other hand Hearn (1975) suggests that the wind is initially acce lerated in a hot corona. The corona would heated by shock waves which grow from instabilities within the atmosphere. As will be discussed in Chapter 5, one of the dominant instabilities pre sent is the thermal instability which grows cn a time scale which varies as n-2. This instability may provide the correla tion between wind velocity and density. The one month scale cf the high low state variability is an enigua. There appears to be no natural scale in the wind to explain it, so it may be connected to the subatmosphere of the star (Cannon and Thomas 1977 and Thomas 1973). The observations of Schreier et al. (1976) contain evi dence that there are small scale (cf order 1011 cm) fluctuations 30 in the wind. The count rate clearly varies with an amplitude greater that the statistical error on time scales cf about one hour. This time scale, which has been set by the spacecraft e^*^ orbit and pointing mode, is much longer than the natural res ponse time of the accretion process, which is about ten minutes, from Table 2. It would be extremely interesting to have data with a time resolution of a few minutes to see if the fluctua tions in the wind become time resolved., In summary the theory of supersonic accretion that was de veloped and applied to a limited amount of data on Cen X-3 shows that there is a positive correlation between the wind density and vind velocity during a period of transition from X-ray low tc high state..,• 31 CHAPTER 4. PHYSICAL DESCRIPTION OF THE GAS The stability of the Hind will be investigated with a linearized stability analysis. The analysis requires that the prevailing physical conditions be specified. This chapter is devoted tc the derivation of the reguired quantities. The dif ficult physical quantities are those describing the interaction of the gas and the stellar radiation field, which are the rate of energy gain and less, and the radiation acceleration. Since this interaction is probably the key to the stellar wind, an accurate physical description must be used. In order to derive the cooling rate, heating rate, and ra diation force it is necessary to know the distribution cf atcms over the various stages cf ionization, and the rate of absorp tion and emission of radiation by the ions. These calculations reguire atomic constants tc describe the radiation processes, which then become functions of the local density, temperature, and radiation field. The radiation field is influenced by the flow of the gas, so that a good approximation to the radiation field would re quire knowing the details of the flow. As an approximation I have taken the unattenuated, but geometrically diluted stellar radiation field. .. This offers the advantage of retaining a com pletely local analysis at the cost of oversimplifying the radia tive transfer. The approximation of the unattenuated field will have the effect of somewhat over estimating the radiation force because overlapping lines are ignored. As discussed later the effect is likely to be at most a factor of two. The gas is assumed tc be in ionization equilibrium, which 32 is valid for time scales longer than the recombination time scale, 30 (T/10* K) 1 /2 nn-1 seconds. Equilibrium implies that the rate of transitions out of an ionization state is balanced by the rate in. For element i the rate out of ionization state j is determined by the rate of ionization to the next higher ion and the recombination rate to the next lower ion. The rate into the ionization state is determined by recombinations from above and ionizations from below. Algebraically, n c,j-i (nec j-1 + "5 hJ-i > *n1 ne °S j# where in^,T) is the collisicnal ionization rate out of j ^ CJ is the photoionization rate rt£j{ng,T) is the recombination rate from level j to j-1. These ionization balance equations were solved for as many atoms cf significant stellar abundance for which gccd atomic data was available. The elements used are shown with their as sumed abundances in the accompanying table. , It would have been desirable to have included Nickel and Iron with their fairly high cosmic abundance and great number of spectral lines, but no reliable and consistent set of data for a wide temperature range could be found. TABLE 3: ATOMIC ABUNDANCES ELEMENT Z ABUNDANCE Hydrogen 1 1.0 Helium 2 8.5x10-2 Carbon 6 3.3x10-* Nitrogen 7 9.1x10-5 33 Oxygen 8 6.6x10-* Neon 10 8. 3x 10-s Magnesium 12 2.6x10-s Silicon 14 3.3x10-s Sulfur 16 1.6x10-5 These abundances were taken from Alien (1973). Standard rates were used for all the photoionization cross sections, recombination rates, and collisional ionzation rates. But since the gas has a fairly high density (order 10*1 ca-3) and is in an intense radiation field it is necessary to make some corrections. The density effects are allowed for by adding corrections to the recombination rate for three body recombina tions, and recombination to upper levels. A small correction for ionization out of upper levels is also included. The great est difficulty is allowing for the effect of both the radiation field and the density effects on the dielectronic recombination rate. This process depends upcn captures to levels of large guantum number, and it is possible that these levels may be reionized before they can stabilize by cascading down to lcwer levels. These effects have been crudely allowed for by calcula ting a multiplicative correction factor, based on a fit to the guantum mechanical calculations cf Summers (1574), All these rates and corrections are discussed in Appendix 2. The Ionization Balance The solution to the ionization balance equations is very simple since the lowest level only interacts with the second level, and then the second level is linked to the first and 34 third, and sc on., This gives the ratio of the population in a ionization state to the population in the next lower level. A normalization completes the solution. , The equations are weakly nonlinear through their dependence on the electron density, but usually two or three iterations suffices for an accuracy of about 1 part in 106. The results are given in terms of the ionization fraction X ij for ion j of atom i, where xt*y summed over j is unity. To get the number of atoms of type i,j we take the product Xtj-A^n, where At' is the abundance of atom 1. In Figure 9 the ionization balance for a gas of density 1011 cm-3, in the undiluted radiation field of the star, is shown for a range of temperatures. It is found that for the range of densi ties of interest the reduction of the dielectronic recombination rate by the density and radiation field effects is significant and tends to shift the ionization slightly to higher stages of ionization. At very high densities the distribution approaches to the .distribution expected for ITE. The heating and cooling rate for a gas of density 10-*1 cm-3 in a undiluted radiation field are shown in Figure 10. , The plotted quantities are the ceding and heating rates, -A and T , respectively. The plotted guantities are to be multiplied by the density squared to obtain the rates per cm-3. The genera lized cooling rate is taken as Jt* =n2 {A - r ). The quantity *€/n2 is plotted. The radiative equilibrium between the photoionization cea-tinq and radiative losses holds at temperatures of about 2x10* K for densities around 10*4 cm-3.; This is shown in Fiqure 10 for zero velocity of the qas with respect to the star. Of interest Sulfur Silicon Magnesium He on Oxygen Nitrogen Carbon at Helium ,—i—,——~- j—, .—• » 1 5 6 7 Log T Fiq. 9: Ionization Balance 36 o Fig. 10: Heating and Ceding Bates for Solar Abundances 37 to the "warm radiation acceleration" model is that near 2x10s K the loss rate in an optically thin medium is at a maximum. Such a temperature would be very difficult to maintain in the gas, reguiring an immense input of energy from seme other heat source. Between 10* and 107 K the loss rate drops to a minimum where the radiative losses would be more easily balanced. The radiation losses from such a hot gas would consist largely of X-rays, which would be suitable for producing the 0 VI ion, as has been suggested by Cassinelli and Olson (1978). The gas is thermally unstable (see Field 1965) to both iso-choric and isobaric disturbances when the temperature derivative of the generalized cooling rate at constant density is negative. The temperature gradient is not quite steep enough (logarithmic derivative of the cooling rate less than about -3) at any pcint to admit isentropic instability, wherein ordinary sound waves gain energy in the rarefactions and lose it in compressions. Even if there is a slight inaccuracy in the calculations such that this isentropic instability condition could be met, it would appear in a very narrowly defined temperature interval. Calculations by Raymond et al. (1S78) indicate that with the inclusion of the iron group elements the slope becomes even less steep, and the gas is further away from isentropic instability. The loss rate and its derivative turns out to be critical to the stability cf an accelerating atmosphere, so it has been plotted it for the CNQ elements enhanced by a factor of 10 in Figure 11. Obviously the abundance has a strong effect on the coding rate, since the CNO elements are responsible for the coding in the range 10s to 10* K. 39 The stellar wind is usually optically thin at optical and longer wavelengths for continuum emission, but can become opti cally thick in the resonance lines, which provide the line coo ling as well as most of the radiation acceleration. Using the alteration to the loss rate of Hybicki and Hummer (1978) the reduced cooling rate is shown in Figure 12 for a velocity gra dient of dv/dz=10-3. Note that the specific cooling rate (units of erg cm-3 s~l) will still increase approximately linearly with density, since the losses vary with the cooling rate in erg cm+3 s-1 times the density squared, over the optical depth., This is a very rough calculation, since no allowance has been made for the change of the local intensity due to the optically thick lines. Radiation Force The radiation force is defined as ij C J (9) where ffi = J A-iiii , and m t- is the atomic weight of the various ions. If the unattenuated radiation field is used it provides an upper limit to the radiation force. A more realistic esti mate is supplied by the method used by Castor, Abbott, and Klein (1975) , which is based on an analysis of the radiative transfer in one spectral line originally done by Lucy (1971). With the aid cf the Sobolev approximation the problem can be solved and it is found that the force due to lines is ^ rad - ^ rad ——-* (10) Fig. 12: Cooling with optically thick lines. 41 where T= Tte2/(mc) f ;J (1) hL X;J nc[ 1/2 (I*/*2) {dv/dz-v/r) +v/r ]-» and Tfe2/(mc)=. 02654 9 "rad is acceleration in an optically thin gas f/j(/)is the oscillator strength for line 1 of atom i ionization state j, c is the speed of light JJL is cosine of the angle subtended by the stellar radius from the point in the gas. In addition to the force on the lines there is the force on the electrons, - ft* * rr* ^c-q e - —— e — J C Tl TYV <12) where F is the flux integrated over all freguencies, and is the Thomson cross section. The force on the electrons in the undiluted radiation field in a ecu tietely ionized gas is 194.7 cm s-2. There also is the force on the continuum, which is usually guite small, with the undiluted radiation field at a density of 10** cm-3 it is 63.48 cm s-2. The line acceleration is dominated by optically thick lines, and increases almost linearly with the velocity gradient. A schematic of the acceleration as a function of dv/dz is shown in Figure 11 below. In Figure 13, the acceleration is a weak function of temperature in the range 10* <T< 3x10s K, but for temperatures larger than 2x10s the force on the lines rapidly decreases. The slight hump at 2x10s K is due to the cue ele ments changing ionization state and the entry cf some new strong lines. The rapid fall off is due to the removal of ions that 42 vo Temperature Fig. 13: Badiation Force as a Function of Temperature 43 have resonance lines near the maximum of the stellar radiation field. The only force left beyond 107 K is the force on the electrons. The acceleration found here can be compared with the result found by CAK. The acceleration can be represented in the same fcrm as they have, 9>«d= 9e 8 ft), <13> where t-crer,evtK (dv/dz)-*, ge is the radiation force on the elec trons, and v is the thermal velocity. I find that K(t) = .067 t-Q-9* for n=10io, and M(t) = .022 t~° •** for n=1013 cm-3, whereas CAK find .033 tr0,7, which is good agreement. There are two reasons for the density dependence of the acce leration. First, a few of the lines go from optically thick to thin as the density gees down, and secondly, the ionization bal ance is density dependent in this calculation, through the allo wance for ccllisicnal ionization, and through the density depen dence of the rate coefficients. The deficiencies in this calculation of the radiation force are due to a somewhat limited line list, mostly due to the lack of any iron group elements, and mere seriously a very simple treatment of radiation transfer. Within the approximation used these two deficiencies cancel each other out to a certain ex tent. The radiation force has been over estimated by not taking account cf overlapping lines, which would involve formulating a model cf the atmosphere intervening between the point in the gas and the star. The radiation force increases with the number of lines present, but the flux available decreases as the number of lines goes up., Klein and Castor (1978) have reported on new 44 calculations made by Abbott of the radiation force. He finds that the original CAK law is bracketted by twc alternative tran sfer schemes, and probably the CAK law represents a good approx imation to the force. The calculations here are in good agree ment with the CAK law. The line acceleration varies approximately as (ntJ/n) ((dv/dz) /n,-j )* where is in the range 0.7 to 0.9. As the velo city gradient increases all lines become optically thin, and the force levels off at the maximum value. This means that the force depends on the abundances roughly to a power in the range of .1 to .3, which is a very weak function. Therefore, the ra diation force is insensitive to the assumed abundances for flows in radiative equilibrium because most of the lines are optically thick. One aspect of the radiation transfer which is important to the analysis of the stability of the flow is the shape of the lines, which can provide an immediate source of instability, as has been reported by Nelson and Hearn (1978). The instability they find only acts in subsonic flow. This has been left out because it is dependent on the details of the radiation trans fer. H2fSUium Balance The number of free zero order quantities can be reduced by reguirinq that the eguations of mass and momentum conservation be satisfied. In the case of radiative equilibrium the tempera ture is determined by the balance of heatinq and coclinq, other wise the temperature is just arbitrarily specified., 45 lo illustrate the solutions cf the mass and momentum equa tions the one dimensional equation of mass conservation is subs tituted into the momentum balance equation (see CAK and in Chapter 5, below) with zero temperature derivatives. where the form of the mass conservation equation for a spheri cally symmetric system has been used. Spherical qecmetry has been used partly because the density qradient remains negative even if the velocity gradient acquires a small neqative value. The perturbations are in the form of plane naves, so all deriva tives will be made with respect to the heiqht z, instead of r. The independent variable is chosen to be dv/dz. In Figure 14 the two sides of the momentum equation are shown as functions of dv/dz. For supersonic flow, v2>2RT, there is always one dec-celerating solution, where the radiation force is less than the qravitational field. As can be seen from Fiqure 11, if the ve locity isn*t too larqe there are two solutions in which the qas is accelerated outwards. For typical stellar wind conditions the two solutions have dv/dz approximately equal to 10-* and 1, The two solutions are acceptable locally, but boundary and con tinuity conditions may rule out the hiqh qradient solution. By imposinq continuity of velocity from subsonic to supersonic flow CAK restrict themselves tc the low qradient solution. The solu tion with the larqe velocity qradient is acceleratinq so rapidly that the wind becomes optically thin in the resonance lines. This means that if the acceleration could be maintained over a distance of 0. 151 of the stellar radius, the qas would be acvinq acceleration g + g ,, low density - — -rad g + 8rad> hi§h density dv/dz < 0 dv/dz > 0 dv/dz > 0, high gradient solution 2RT 2RT Fig. 14; The Momentum Equation Solution 47 at the terminal velocity, although this solution is physically acceptable, observational evidence suggests that it may not be realized. As can be seen from Figure 14, if the velocity becomes too large no accelerating solution can be found. The maximum velo city fcr which accelerating solutions exist varies with the gra vity and the density of the gas, A table is given below which indicates the maximum velocity giving an outward acceleration. In Table 4 the Vmax column gives the maximum velocity at which a positive dv/dz can be found, the value of which is given in the next column. Two values for the gravity are used to show that the maximum velocity with dv/dz>0 is mostly effected by the gas density. The gas was chosen to be in radiative equilibrium, which gives a temperature of 2x10* K. , TABLE 4: LIMITING VELOCITY FOR ACCELERATING SOLUTIONS g=10*cm s-2 g=4000cm s~2 Vmax dv/dz Vmax dv/dz n=10»o 4.1x108 .16x10-3 4.45x10« .84x10-* n=10»> 4.7x107 .16x10-2 5.2x10* .16x10-2 n=10»2 5.7x10* .15x10-2 6.0x10* .64x10-2 Table 4 shows that the maximum velocity is approximately inversely proportional to the density. The maximum velocity for acceleration decreases nearly to the sound speed at a density of IO12 cm-3. Below this velocity the Sobolev approximation used for deriving the radiation acceleration is invalid. If a large portion of the flow, thicker than one Scbclev shell, (the sound speed divided by the velocity gradient) ac-48 quires a velocity which is greater than the maximum for a posi tive velocity gradient in momentum balance, then the gas will deccelerate. This situation could arise if the flow is a chao tic medium in which elements of the fluid are propelled to velo cities in excess cf the maximum for acceleration, or have a den sity increase which makes the velocity greater than the maxiaum. The wind might consist of many, quite larqe patches, which are beinq accelerated and deccelerated with respect to one another. Hhere these reqions collide their supersonic velocities vculd ensure shock heatinq which would produce temperatures appropri ate to 0 VI and like ions. This shock heated qas would cnly comprise a small portion cf the total qas in the flow, and after forminq would be blown away from the star., 49 CHAPTER 5. THE STAEILITY ANALYSIS The observations suggest that a stellar wind is an extreme ly variable, inhomogenous flow. On scales of a day to years there are large general variations, which may originate within the star. The X-ray observations suggest small scale fluctua tions cf order 1011 cm. This observed variability could have two causes: the flow may start out in the lower atmosphere as smooth, and then enter a region of instability where it breaks up; or the existence of the flow may be depend upon some insta bility. In this section the local stability of the flow will be investigated. This will be done by considering the propogation of infinitely small disturbances, i.e. a linearized analysis, with wavelengths short compared to the scale of variation within the stellar wind. This analysis is directed towards finding instabilities that are rapid amplifiers, i.e. the growth time scale is shorter than the time to move one scale length in the atmosphere; and absolute instabilities, which can actually gen erate oscillations cr lead to "clumps" within the wind, . One major limitation of this analysis is that it has only bee* done for cne dimensional wave propogation, that is the waves can only have a velocity component which is oriented along the direction of propagation. Eor instance, this immediately rules out the possibility cf the Rayleiqh Taylor instability,, (Krolik 1577, Nelson and Hearn 1978). Similar analyses, but with more approx imations in the linearization, have been performed by Hearn (1972) and for guasars by Hestel et al. , (1976). The basic equations that apply are the conservation cf mass 50 115) where -f is the mass density and v is the gas velocity. The con servation of momentum neglecting the viscosity is given by, T (16) where 9(which will be sometimes abbreviated as g r) is the acceleration due to radiation, P is the gas pressure, and g is the gravitational acceleration. The conservation of energy is expressed, 0* (17) where e and h are are respectively the specific internal energy and enthalpy. , ~£ is the generalized cooling rate in the frame of the gas, defined as / = L-(1-v/c) G, where v is the velocity of the gas relative to the star and L and G are the local specific coding and heating rates, respectively. , The thermodynamic re lations reguired are an eguation cf state P = kT(n+.ne) « where ne = (j-1)nt*;. (18) The sum i,j is over the ionization states and the elements, re spectively. The number density of atoms and electrons are n and n The internal energy is e = 3/2 kT(n+n ) + Zn £; f:.t , (19) where "X;.- is the ionization energy of ion i,j with density nty. J J The enthalpy is defined as, h = e *P/^ . (20) For the conductivity •% the standard value cf Spitzer (1962) has 51 been used. The above equations are linearized in order to obtain a dispersion relation, which is a polynomial describing the prcpar gaticn of waves of infinitesmal amplitude. . The linearized egua-tions are obtained by imposing a perturbation on the tempera ture, density and velocity cf the form Q(z,t) = QD(z,t)+£gt <e4k)expfi{kz-cdt) ], (21) and substituting into the conservation equations. It has been assumed that the scale of variation of Q0(z,t) and the radius of the star are much larger than the wavelength of the perturba tion. Equating terms of first order in results in the system of linearized equations. This can be written as a coefficient matrix, consisting of zero order guantities, their zero order derivatives, and powers of cD and k. The determinant of the matrix gives a polynomial cf third order in Ld, which is the dis persion relation. although this process could have been carried through by hand and the roots cf the cubic polynomial derived analytically, it was far easier and less prone to error to do it with the aid of a computer. Besides, this analysis is eventual ly to be extended to more complex motions, in which case the computer would have to be used, sc the experience obtained in this simpler case will be usefully applied there. The method of generating the algebraic fera of the dispersion relation is out lined in appendix 4. To define the coefficients of the polynomial, it is neces sary to knew the density, velocity and temperature, their first derivatives, the second derivative of the temperature, the ra diation force, cooling and heating rates, and the electron den-52 sity with their temperature and density derivatives. These guantities were derived in Chapter 4. The toots of the dispersion equation are found usinq a com puter program which finds the roots of complex polynomials./ The root found is improved in accuracy by substitutinq it back into the polynomial and doinq a Newton's method iteration until the fractional chanqe is less than 1 part in 10ls. Since the roots are computed for a sequence of k, the root for the next value of k is then estimated from the root just found by extrapolation, and the same iterative improvement performed. . The limits to the accuracy of the numerical solutions means that when the roots have real and imaqinary parts different by 15 orders of maqni-tude or the different roots themselves are widely separted, the smallest quantities may not be very accurate. The method of solution chosen was designed to suppress "numerical noise", but the resulting smoothness of the plotted roots usually overesti mate the accuracy of the numbers in the cases mentioned above. The perturbations are of the form exp[ i (kz-tJt) ], consequen tly if the imaginary part of the frequency is positive for a qiven real k, then there is an instability at that wave number. This instability can act as an amplifier of a preexisting wave, in which case it is called a convective or amplifyinq instabili ty, or it can qrow away from the startinq value, either in a monotonic qrowth or in ever increasinq oscillations, which is called an absolute instability. A mathematical method of dis-tinquishinq between the two types of instability based on deter-mininq the behaviour of the wave as t-^oo, has been developed by Dysthe (1966), Bers (1975), and Akhiezer and Polovin (1971). 53 They find several criteria for determining the type of stabili ty, the easiest of which to apply is that if the simultaneous solution to D( ,k) = 0 and dD/dk=0, where D is the dispersion re lation polynomial, exists, and has an imaginary freguency greater than zero, then the instability is absolute. This is a necessa ry and sufficient condition in the approximation of t->©«» in an infinite atmosphere. The criterion means that in the neighbour hood of the solution (oe,ko) to the two equations the root varies as u) = u0 + A(k-ke)2, where a is a constant. This imp lies that an absolute instability is a saddle point of the ima-qinary part of the frequency as a function of k. The imaqinary part cf the frequency will be at a maximum with respect to real k at the solution, and this freguency will dominate the qicwth rate. These two nonlinear equations, D=0 and dD/dk=0, were solved simultaneously with the aid of a computer routine, using the local maximum of the imaqinary part of the freguency for real wavenumbers as a startinq point. An attempt was made to find common roots to the two equations by ccnstructinq the dis criminant of the coefficients of the two equations. This was unsuccessful because of the impossibility of retaining suffi cient numerical accuracy. In order to understand the dispersion relation and the physical origin of the roots, analytic expressions for the roots will be derived for a number of simple limiting cases. The roots in a complex situation can be understood as superposition of these several simple cases. These limiting solutions have been derived with the aid cf numerical solutions, and unless noted the calculated roots plotted came from a dispersion rela-ticn with coefficients calculated from a gas in an undiluted radiation field, with a density of 1011 cm-3, and a velocity of 100 Km s—*. The resulting equilibrium guantities are in cgs units: T = 1.87x10* K; =0, d /'/dT = .46x10-*, d^/dn = 2.x10-i3, dv/dz = .2x10~3, dn/dz = -2.1, g rad = 1.18x10*, dgr/dT = -0.3x10-1, and dgr/dn = -.15x10-*. It is found that the char acter of the roots changes little with a variation of the physi cal parameters around these values for typical stellar wind con ditions. Case 1: Sound Haves In An Atmosphere The simplest case which has a non zero growth rate is a wave propogating vertically in a static, isothermal atmosphere, with no conduction or radiation present. In this case the dis persion relation as given in the Appendix 3 reduces to Defining H=n/(dn/dz) , this has solutions o>=0 and in the limit of large and small k the nontrivial roots become la.-5* 00 to —» Cv (22) This is essentially the well known solution of Lamb (1945) to the problem of wave propogation in an isothermal, exponential atmosphere. But note that the value of H, the scale height of Fig. 15: Pseudc Isothermal Static atmosphere Boots 56 the density gradient, used in the numerical calculations was not the isothermal scale height, but that the scale height was de termined by the velocity gradient through the mass conservation equation. In the short wavelength limit (k-»a>) the waves move at a phase and qroup velocity equal to the ordinary sound velo city. Outward movinq waves are amplified and inward moving waves are damped at a rate such that the momentum carried in the wave is kept constant. These waves are not absolute instabili ties. At lonq wavelengths (k—>0) the real part cf the freguency goes to a finite limit, called the acoustic cutoff frequency, and the damping goes to zero. . This means that these waves have a phase velocity going to infinity, but the group velocity goes to zero and no energy is prcpcgated. Physically this cutoff results from the atmosphere as a whole moving with the wave mo tion, rather than a wave propogating away from the source. The change over between the two limiting solutions occurs for k of order H_1. The solution is illustrated in the accompanying Figure 15. In Figure 15, and all other graphs of the roots of the dispersion relation, the logarithm (base 10) of the real and imaginary parts of the wave freguency are separately plotted against the logarithm of the wave number. , On the graph of the real part a symbol ^or O) on the line means that the real part is negative. On the graph of the imaginary part the same symbols indicates that the wave is unstable at that wave number, that is, the imaginary part is positive. Note that fre quently tfee two acoustic roots have an identical magnitude, but opposite sign, so that in the plot the two lines lie on top of each other. 57 The plots are done for k ranging from 10-15 to 1 cm-1, which is an unrealistically large range for the physical situa tion, but is done to illustrate the asymptotic limits of the roots. The physically acceptable range of wave numbers is for wave numbers less than the a wavelength of a stellar radius, 10-n cm—i, to a wavecumber corresponding to one mean free path, about 10-2 cm—i. There is a maximum freguency for which the solutions are valid, set by the longer time scale, recombination or the elec tron ion thermal equilibrium. The recombination freguency is o>t&c = .188 n lk (T/10* K)-i/2 s-i, (23) and the electron ion equilibrium frequency is UeL = 7x10* n „ • (T/10* K)~3/2 s-»... <2<4) The maximum frequency for which the calculations are valid then is the minimum of (Jr<.c and tJei. . The minimum freguency of in terest would be determined by the time for the complete replace ment of the star's stellar wind envelope. This freguency is about 6x10-5 s-i. Case 2: The Effect Of Conduction Allowing conduction affects mostly the short wavelength roots. Taking the dominant terms in the dispersion relation gives. 58 For this case the dominant terms cf the roots for k are, f tvd (25) which is a heavily damped non propagating disturbance. The sound waves are given as * * 1 / (26) which are isothermal sound waves, and always damped independent of direction cf propagation. The numerical solution shows that the analytic solutions only apply for k>10-3, and that the slow root has a small real part at short wavelengths. Case 3: Radiation Effects In the long wavelength limit we expect radiation effects to be dominant. The dominant terms of the dispersion relation becc me w* {.,f c.R]t u>» + u {c2 %4£] + <b I i* r ^ - ii: 1. r i*I + ^7 dt 7 + (27) This dispersion relation has been derived under the assumption 59 that 4"*- Tf ? ~HX Cu C ' 128) The dominant term of one root is for k->0 • c* R. ct T * <29) which essentially is the thermal stability condition. A parcel of qas with d J^/dT <0 would probably tend to collapse. In a general case if d J^dT were negative, part of the gas may cool and ccllapse, and other parts may rise in temperature. The ex istence of the hot, low density component depends on an appro priate heat source to maintain a temperature of order 10* to 10? K, where the gas is stable. If this bistable mode is possible within a stellar wind, it may lead to a two component outflow with a cool (T around 2x10* K) and hot (T around 107 K) com ponent. The hot component may be able to supply sufficient C VI atoms that there wculd be no need for a coronal region. The dominant terms cf the other two roots are sound waves, civ , l-Z RT  + h rl-r - J U = ' ' ' PI (30) "The roots are shown in Figure 16, for a gas with a nonzero velo city and acceleration. From Equation 30 we make the discovery that deccelerating flows are unstable, and the numerical calcu lation finds that it is an absolute instability. An example of this instability is discussed later and illustrated in Figure 20. Defining some basic time scales as t(dynamic) = Vjdv/dzj, t (cool) = f c> R. / \ c| c|T I , Pig. 16: Isotropic Badiation Field 61 t (acoustic) = H/Cp where cx= Jk2BTi The condition for the instability of deccelerating flews (Eg. 28) is, t (dynamic) x t(cool) << (t (acoustic) ) 2 Note that the conductive damping dominates the roots for k > 10-*. Allowing a radiation acceleration, gives the roots as plotted in Figure 17. The asymptotic limits are not changed by the radiation acceleration, but the inward propagating acoustic wave is unstable in the range cf wavenumber 10_11< k < 10~ 7. The resulting growth rate is close to 1200 seconds, but the instabi lity is only amplifying. The cooling due tc collisicnally excited lines may be di minished when the gas becomes optically thick in the resonance lines. The effect of this has been approximated by turning the loss rate off, but leaving the heating on. The roots of the dispersicn relation in this case are shown in Figure 18. Be sides the amplifying instability from the radiation force there is an additional range of instability for both inward and cut-ward acoustic waves for 10~7< k <10-s. This behaviour results from the term d jf/dn becoming signficant. Figure 19 shows the effect of a thermal instability, d^/dt<0. The "slow" root has a rapid growth rate, which is an absolute instability. The acoustic roots are changed only slightly, the amplification acting over a narrower range of wa venumber, and not guite as rapidly. Figure 20 shows the pressure dominated thermal instability which is present at 107 K. In this case the flow speed is sub-LO G- K Fig. 17: Radiation Force Cn 6<4 sonic, and the radiation force is less than 5% cf gravity. The instability results when Eguation 28 is violated. The acoustic waves have an absolute instability at long wavelengths. The growth rate is proportional to the element abundance through the cooling rate. An atmosphere of density 1012 cm-3 and a velocity of 100 Km s~1 exceeds the maximum velocity for an accelerating solution. The roots when dv/dz is negative are shown in Figure 21. Shis is an absolute instability at long wavelengths. In summary the roots of the dispersion relation can be un derstood in terms cf combinations of the roots which occur in simple physical cases. For k> 10~* cm-1, the conduction always provides a strong damping, especially tc the slow wave. Thus it can be concluded that in the long wavelength liuit, k< 10-11, the behaviour depends on the shortest time set by: the acoustic time scale, egual to the scale height divided by the sound speed; the dynamic time scale (dv/dz)-1; $s.<aii the cooling time scale, ^<^u R/(d>?/«(T|. There is always a "thermal wave", that is, a slow moving wave, compared to the scund speed, with a growth rate set by the thermal time scale. The slow wave is an absolute instability if the derivative d f/dT is negative. If t(acoustic) » t (dynamic) then the acoustic waves have growth rates given by the dynamic time scale. These acoustic waves will be absolutely unstable if the velocity gradient is nega tive, A hot gas, with t(acoustic) < t (dynamic) will have an absolute instability arising from the acoustic waves. At T=107 the growth rate is about one hour, (one hour at 3x10* K where d//dt<0) which increases as n2, until t (acoustic) exceeds LOG * Fig. 20: High Temperature Instability Fig. 21: Decceleraticn Instability at n=1012 cur3 68 TABLE 5: THE DISPEBSICfr BELATION £ PLOTTED Fig. n T V dv/dz d /dT g Bemarks 15 10*i 2x10* 0 0 0 0 pseudo isothermal 16 10*i 2x10* 100 2x10-* 5x10-s 0 isotropic 17 10i» 2x10* 100 2x10-* 5x10-s 1x10* standard case 18 10i* 2x10* 100 2x10-* 5x10-s 1x10* no cooling 19 10*» 5x10s 100 7x10-* -3x10-* 1x10* thermal 20 10** 1x107 100 5x10-5 8x10-8 270 high temperature 21 10*2 2x10* 100 -6x10-* -3x10-3 5x103 decceleration t(dynamic). The radiation acceleration only acts to provide an amplifying instability for inward acoustic waves. This amplifi cation acts for an observaticnally interesting range of wave lengths, from 5x107 to 5x10** cm, with growth times cf order 1200 seconds. on the basis of this analysis the original CAK "cool" atmo sphere is stable only if no waves are sent into the accelerating wind from lower layers. Otherwise the radiation force acts tc provide an amplification of the inward moving (with respect to the gas, but outwards with respect to the star) acoustic wave. A corona with a temperature of several million degrees will always have an absolute instability, either the ordinary thermal instability of the radiation losses, or the "high temperature" instability outlined above, which has a growth time cf order an hour. The semi empirical model of the wind proposed by Cassitelli et al. (1978) has the wind heated with an initial acceleration in a hot corona, and then cooling in the outer radiatively acce lerated 2one. The stability analysis leaves no doubt that this situation would be expected to show fluctuations. The hot corona is subject to instabilities which may be responsible for creating the shock waves to heat it. Bemnants of these fluctua tions would be carried out into the wind where the length scales of 107 to 10** cm would be amplified. 69 CHAPTER 6. , CONCLUSIONS The program of optical observations conclusively shows that stellar winds do vary on time scales of one day and more. , These observations were taken at sufficiently high resolution that any variations cf the telluric lines could be separated from varia tions of the stellar lines. A star which has often been re ported as varying. Lambda Cephei, was monitored with a time re solution of one hour over a period of six hours but no signifi cant variation was seen in the H ^line. This null observation puts an upper limit of about 5x1011 cm on the size of any "blobs" in the wind. Day to day variability was confirmed in X Cep and •x'Cam, but not conclusively for & Ori. These variations may not be due to fluctuations within the wind itself since this time scale is long enough to allow complete replacement of the material in the line formation region. Causes of the long time scale variation include rotation of the star, internal oscilla tions of the star, or a variation of the the emergent flux and hence the driving force for the wind. The analysis of the X-ray observations cf Cen X-3 provides confirming evidence for the suggested mechanism causing the long term X-ray intensity variation reported by Schreier et al. (1976)• That is, the wind density varies sufficiently that the source is occasionally smothered by the opacity of the stellar wind. In addition I have found that as the density in the wind changes, it must be correlated with the wind velocity in order to explain the changing character of the anomalous dips in the intensity at non-eclipse phases. Besides these semi-regular dips the X-ray intensity shows intensity fluctuations on a time 70 scalei. of less than one hour, which is probably due to changes in the amount of mass being accreted by the neutron star, The natural source for the variation in the accretion rate is the variation of the density and velocity in the stellar wind with size scales of 1Q*o tc 10*1 cm. The theoretical analysis of the stability of a wind finds a number of sources cf instability in the flow. In the long wave length limit the highest growth rate, of order 10 seconds, usually holds for the thermal instability which arises whenever the cooling rate minus the heating rate has a negative deriva tive with respect to temperature. This situation arises in the temperature ranges of 3x10s tc 107 K. The growth rate of this instability is directly related to the cooling rate, which is proportional to the abundances of the elements present. If this instability operates, one would expect significant differences between stars of different composition. That is, stars with higher CMO or metal abundances would have a greater cooling rate for temperatures exceeding 10s K in an optically thin gas., As a result the thermal instability would grow on a shorter time scale. This may have some bearing on wolf-Hayet stars, which appear to have higher CNO abundances than OE stars, and defini tely have higher tass loss rates. An amplifying instability for acocstic waves which is usually present is the simple growth of wave amplitude due to the density gradient in the atmosphere. In a moving atmosphere this occurs on a time scale of the gas speed divided by the scale height, about 2x103 seconds. The decceleration instability of acoustic waves is an absolute ins tability, . The growth rate is jdv/dz)-1, usually cf order 1000 71 to 10* seconds. If the gas is very hot, greater than 10s K, there is an absolute instability which operates cn the time scale cf an hour. The radiation force provides an amplification for wavelengths in the range 108-1011 cm on time scales of 1200 seconds. As a result of this work, a number of suggestions can be made for further investigation. Line variability should be pur sued in order to unravel the nature of the variation. High re solution observations, with good signal to noise must be ob tained. These observations should either be made in spectral regions free of telluric lines, or at a sufficiently high reso lution to niniffiize blending with the stellar line. A longer segment of the X-ray data should be analyzed to confirm the model given for the accretion process, and sore im portantly to estimate the density and velocity as a function of time. The theoretical analysis of instabilities can be extended to allow a vertical and horizontal wave vector, and allow the wave motion tc have a horizontal as well as a vertical component and then to more general waves, such as allowing vcrticity. Besides dynamical generalizations, different source spectra should be allowed, particularly X-rays. This would allow the analysis to be extended to guasars and nuclei of galaxies. The purpose of this whole study is to acguire information on the fundamental physical nature cf the mass loss in the pre sence of a strong radiation field. My thesis is that the wind is observed to be variable, and that the variability on time scales cf a day or less can be attributed tc instabilities which 72 exist in the wind. It is suggested that the presence of these instabilities changes the fundamental dynamics of the solutions to the flow of the stellar wind. The length and time scales established in the analysis will allow the nonlinear equations of the flow to be attacked with a knowledge of their local be haviour. 73 BIELICGBAPHY 2. Akhiezer, A. I. ana Polovin, B. V., 1971, SovA PJjs^ Ospekii , 14, 278. 3. Aldrovandi, M. V. and Pequignot, D., 1973, Astr. Ag., 25, 137. 4. Aldrovandi, M. V. and Pequignot, D., 1976, Astr. Aj., 47, 321. 5. Earlow, M. J. and Cohen, M. , 1977, A£.J.., 213, 737. 6. Eeals, C. S., 1929, J.N., 90, 202.. 7. Eeals, C. S., 1951, Puh^DjjO, ix, 1. 8. Eers, A., 1975, Plasma Physics. Ed. Deiitt, C. and Peyraud, J. . (Gordon and Ereach: New York), p. 121., 9. Eerthcmieu, G., Provost, J. and Bocca, A., 1S75, A str. Jj3., 47, 413. 10. Eethe, H. A. and Salpeter, E. E. 1957. The Quantum Mechanics of One And Two Electron Atoms, (Sprinqer-verlag: New York). 11. Brucato, B. J. , 1971, M.N., 153, 435. 12., Burgess, A. and Summers, H. P., 1976, M.J., 174, 345. 13. Burgess, A. and Summers, H.P., 1969, A£.J., 157, 1007. 14. Cannon, A. J. and Pickering, E. C., 1918, Ann.t of ObsA Q£ Harvard Coll.M vols, 91-99. 15. Cannon, C, J. and Thomas, B. N•, 1977, k£.J., 211, 910. 16. Cassinelli, J. P. and Castor, J. I., 1973, AD.J., 179, 189, 17. Cassinelli, J. P., Olson, G. I., and Stalio, B., 1978, AJE.J., 220, 573. 18. Castor, J. I., Abbott, D. C., and Klein, B. I. , 1975, JE.J., 195, 157. 19. Castor, J. I., 1977, in Colloquium Internationaux Eu CJBS NOJS., 250. Mouvements Dans Les Atmospheres Stellaires. ed. Cayrel, B. and Steinberg, M•, (CNBS: Paris). 20. Castor, J. I., 1978, at IAU Symposium No. 8 3, Mass Loss And 74 21. Chapman, E. E. and Henry, R. J. 8., 1971 Ap..J., 168, 169. 22. Chapman, E..D. and Henry, R. J. B., 1972 Ap.*?., 173, 243. 23. Conti, P. S. and Frost, S. A., 1974, Ap.J. Lett., 190, L137. 24. Conti, P. S., 1978, Astr. Ap. , 63, 225. 25. Cox, 0. E. and Tucker, B. H. , 1969, Ap,J., 157, 1157. 26. Dupree, A. K., 1968, Astrophvs. Lett.. 1, 125., 27. Dysthe, K. B., 1966, Nuclear Fusion. 6, 215. 28. Hearn, A. G., 1973, Astr. .Ap., 23, 97. 29.. Fahlman, G. G., Carlberg, fi. G., and Balker, G. A. H., 1977, Ap.J. Lett., 217, L35. 30. Field, G. B. and Steigman, G., 1S71, Aj.J., 166, 59. 31. Field, G. B. , 1965, JLp.J., 1«2, 531. 32. Flower, D.,E. 1968, in IAD Symposium NOj. 34., Planetary Nebulae, ed. Osterbrock, D. E. and O'Dell, C. R. (Reidel: Dordrecht), p. 205. 33. Hearn, A. G., 1972, Astr. Ap.., 19, 417., 34. Hearn, A. G., 1975, Astr. Ap.. , 40, 355., 35. Henry, B. J. W. 1970, Ap.J., 161, 1153. 36. Hutchings, J. B. , 1976, Aja. J., 203 , 438. 37. Jackson, J. C. , 1S75, JS.J. , 172, 483. 38. Johnson, L. C. , 1972, Ap* J., 174, 227. 39. Jordan, C, 1969, M.J. , 142, 501. 40. Kato, T., 1S76, Ajp.. J. Supji. , 30, 397. 41. Kippenhahn, E., Mestel, L., And Perry, J. J., 1975, Astr. Ap., 44, 123. 42. Klein, R. I. and Castor, J. I., 1978, Ap.J., 220, 902. 43. Krolik, J. H., 1977, Phys. Fluids 20, 364. 44. lacy, C. H., 1977, Ap.J., 212, 132. 75 45. Iamb, H, , 1945, Hydrody namies. (Dover: New York). 46. Earners, H. J, G..L. M. and Morton, D. C, 19*76, A£.J. Su££., 32, 715, 47. lamers, H.J. G, L. M., and Snow, 1. P., Jr., 1978, A£.J., 219, 504. 48. Lctz, W. , 1967, Jfi.J. SupjS., 14, 207. 49. Lucy, L. B. And Solomon, P. M., 1970, j£.J., 159, 879., 50. Lucy, L. B. , 1971, Ac.. J., 163, 95. 51. McWhirtler, E. 8. P., 1975, in Atomic .And, Molecular £Efi£J5se,s In Astrophysics, ed. Hubbard, M. C. E, and Nussbaumer, H (Geneva Observatory: Sauverny, Switzerland), p. 205. 52. flestel, I., Moore, D. W., and Perry, J. J., 1976 Astr* lj», 52, 203. 53. Mewe, fi., 1972, Astr. Ap_. , 20,215. 54. Mihalas, D., 1972, Non-LTE Model Atmospheres for B and 0 Stars, NCAB-TN/STB-76, (National Center for Atmospheric Research: Bculder). 55., Milne, E. A., 1926, M. JJ., 85, 813. 56. Moore, C. E,, Minnaert, M. G. J., and Houtgast, J., 1966, The Solar Spectrum. NBS Monograph 6 1. 57. Moore, C. E,, 1949, Atomic Energy levels, NBS Circular No. 467. 58. Morton, D.C and Smith, ». H., 1973, Ajg.J. Su££. , 26, 333. 59. Morton, D.C, 1967, Aj.J., 147, 1017. 60. Mushotzky, R. F., Solomon, P. M., And Strittmatter, P. A., 1972, Ap_. J. , 174, 7. 61. Nelson, G. D., and Hearn, A. G., 1978, Astr. An.. , 65, 223. 62. Pounds, K. A., Cooke, 6. A., Ricketts, M. J., Turner, M. J., and Elvis, M., 1975, M.N., 172, 473. 63. Raymond, J.C, Cox, D. P., and Smith, B. H., 1976, J. • 204, 290. 64. Bcsendahl, J..D., 1973, Afi.J., 182, 523., 76 65. Rybicki, G. B. and Hammer, D. G., 1978, Ap.J., 219, 654. 66. Schreier, E. J., Schwartz, K., Giacconi, R. , Fabbianc, G., and Morin, J., 1976, Aj.J., 204, 539. 67. Seatcn, M. J. 1958, EgVi JSod[A Phj^s., 30, 979. 68. Seatcn, M. J., 1959, M.N., 119, 81. 69. Silk, J. And Brown, B. L., 1971, Aj.J., 163, 495. 70. Snow, T. P. Jr. and Morton, E. C, 1976, Ap.J. Supja., 32, 429. 71. Snow, T. P. Jr., 1977, Ap.J., 217, 760. 72. Sobolev, V, V., 1960. Moving Envelopes of Stars* (Harvard University Press: Cambridge), 73. Spitzer, L, Jr., 1962, Physics oJ Fully. Ionized Gases. 2nd ed., (Interscience: New York) • 74. Steigman, G., Werner, M. 8., and Geldon, F.... B. , 1971, Ap.J. , 168, 37 3. 75. Summers, H. P., 1974, M.N., 169, 663. 76. Sunyaev, B. A. and Vainstein, L, A., 1968, Astrophvs. lett ., 1, 193. 77. Thomas, B. N., 1973, Astr. Ajg., 29, 297. 78. Underbill, A. B. , 1960, in JcJU of SJbars A^d S^ellajc Systems, Stellar Atmospheres. ed.,J. Greenstein, (ti. .of Chicago: Chicago), p. 411. Jr 79. Walker, G, A. H., Bucholz, V., Fahlman, G. G., Glaspey, J., Lane-Wright, D., flochnacki, S., and Condal, A., 1S76, Proc, IAU Colloquium 40, ed. M. Duchesne, (reidel: Dordrecht). 80. Hiese, W. L., Smith, M. W., and Glennon, B. M., 1966, NSBDS-NBS4. 81. Wiese, W. L., Smith, M. «., and Miles, B. M., 1969, NSBDS-NBS22. / 82. Wilson, B., 1962, J.O.S.B.T. 2. 477. 83. Wright, A. E., and Barlow, M. J., 1975, M.N., 170, 41. 84. Ycrk, D. G. , Vidal-Madjar. A., Laurent, C, Bonnet, fi. , 1S77, Ap.J. Lett., 213, 161.. 77 APEENDIX 1. SUPEBSGNIC ACCBETTON This appendix is a reprint of the paper entitled "Badiative Effects in Supersonic Accretion", which appeared in the A§*rcjhxsical Journal Volume 220, p. 1041. The theory developed was used in Chapter III tc deduce the correlation between the wind velocity and wind density in the observed intensity transi tion cf Cen X-3. / THE ASTROPHYSICAL JOURNAL, 220:1041-1050, 1978 March 15 © 1978. The American Astronomical Society. All rights reserved. Printed in U.S.A. RADIATIVE EFFECTS IN SUPERSONIC ACCRETION R. G. CARLBERG Department of Geophysics and Astronomy, University of British Columbia Received 1977 March 21; accepted 1977 September 22 ABSTRACT Supersonic gas flow onto a neutron star is investigated. There are two regimes of accretion flow, differentiated by whether the gas can cool significantly before it falls to the magnetosphere. If radiative losses are negligible, the captured gas falls inward adiabatically in a wide accretion column. If the radiative energy-loss time scale is less than the fall time, the gas will cool to some equilibrium temperature which determines the width of the wake. An accreting neutron star generates sufficient luminosity that radiation heating may determine the temperature of the accretion column, provided the accretion column is optically thin. Gas crossing the shock beyond the critical radius forms an extended turbulent wake which gradually merges into the surrounding medium. As a specific example, the flow for the range of parameters suggested for the stellar wind X-ray binaries is considered. Subject headings: shock waves — stars: accretion — X-rays: binaries I. INTRODUCTION Recent observations of X-ray binaries, at both optical (Conti and Cowley 1975; Dachs 1976) and X-ray (Jones et al. 1973; Pounds et al. 1975; Eadie et al. 1975) wavelengths, show phase-dependent absorption of radiation. It has been suggested that this is caused by a wake trailing the compact object which emits the X-rays. Models of the wake based on the X-ray observations were put forward by Jackson (1975) and Eadie et al. (1975). The general problem of a gravitating body moving through a gas at a velocity much greater than the sound speed was first discussed by Hoyle and Lyttleton (1939). More recently, wakes were discussed by Davidson and Ostriker (1973), Illarionov and Sunyaev (1975), and McCray and Hatchett (1975). These models are incomplete in that they lack a description of the gravitationally perturbed gas which is unbound, i.e., the far wake. Although most of these papers emphasize the importance of radiative effects, no clear analysis has been made of the variations in the flow of gas caused by radiative gains and losses. In this paper supersonic accretion onto a neutron star is considered. There are three basic physical parameters: the mass of the accreting body and the free stream velocity and density of the gas. The dynamics of the flow are essentially determined by the free stream velocity and the mass. The angular width of the accretion column depends on its temperature, which in turn is regulated by radiative cooling and heating and is sensitive to the gas density. The proposed description is worked out for linear motion, which is a good approximation for an accretion radius much smaller than the system dimen sions. A schematic of the model is shown in Figure 1. The important regions are labeled: (1) the incoming supersonic gas; pressure forces can be neglected and streamlines are taken to be coincident with particle trajectories in a gravitational field; (2) the shock-heated sheath where the incoming gas impinges on the accretion column; the transverse component of the velocity is rapidly halted, providing pressure to contain the accretion column; (3) the accretion column, in which gas falls inward, toward the accreting body; (4) a region of spherically symmetric flow which may exist near the accreting body; (5) the base of the accretion column; beyond this the flow is regulated by the physics of the magnetosphere around the accreting object; (6) the accreting body, where the kinetic energy of the gas is liberated at a surface shock; and (7) the far wake, several hundred times the length of the accretion column. The density contrast between the far wake and the surrounding medium gradually goes to zero. One major qualitative aspect of this model is that there is no bow shock standing off from the front of the body which is distinct from the tail shock. Un doubtedly there will be a preceding shock, but pressure waves generated there will not propagate very far in a transverse direction because the streamlines of the flow are bent in by gravitation. Consequently, the bow shock merges into the tail shock. In general, this will be the case for any body whose size is less than the "accretion radius" RA = 2GM/VQ2, where V0 is the free stream velocity. Calculations by Hunt (Eadie et al. 1975) indicate that part of the infalling column may "miss" the accreting body and force the leading shock forward. This occurs because small nonradial velocities increase toward the body, by conservation of angular momentum. This effect will be ignored. This model is to be applied to a neutron star orbiting a massive star with a strong stellar wind. For convenience, scaled variables will be used for the distance rv = rjRA: free stream density, wu = HQ/1011 cm"3; free stream velocity, VB = V0j 103 cm s~1; and 1041 1042 CARLBERG Vol. 220 FIG. 1.—A schematic of supersonic accretion gas flow showing: (1) the incoming supersonic gas; (2) the shock-heated sheath; (3) the accretion column; (4) the possible spherically symmetric inflow at the bottom of the column; (5) the Alfven surface at the bottom of the column; (6) the accreting body; and (7) the far wake. mass of the accreting body, m — Mx/M0. Similarly, in later calculations, the temperature of the column will be represented as T6 = T/IO6 K. This form of notation will be used throughout. II. DYNAMICS OF THE GAS FLOW A gravitating body is placed in a uniform stream of gas moving at some velocity V0. To the point where the gas crosses the tail shock, we assume that the stream lines of the flow can be found from particle dynamics, i.e., the flow is dominated by inertial forces. The velocity can be obtained from the equations of conservation of energy and angular momentum, Wr2 + V,2) GM Wo2 and V<t> — V0 • 0) where Vr and V# are, respectively, the radial and tangential components of the gas velocity relative to the accreting object. The trajectories are given by (Ruderman and Spiegel 1971), I = 5* 0 + C0S & + ISin * ' (2) where s is the impact parameter and <f> is the angle measured from the accretion axis. From these equa tions, Danby and Camm (1957) obtain the density n = n(r, </>) as n = 2 sin 0/2 2 + sin-+ sin2 - + sin 2 r A + sin2 (3) As a simplifying approximation, we take the case of (f> small and <j>2 « RA/r, to obtain the relations, and V* « V0(RJry*, 20 (4) d) The Sheath The gas crossing the shock has a discontinuity in its motion described by the equations for the conserva tion of the total energy and of the normal components of mass flux and momentum. Assuming that a strong shock occurs (good for Mach numbers greater than ~3), the postshock density and temperature are (for a ratio of specific heats y = 5/3) P2 = 4Pl, and ra = ^-^. (5) where R is the gas constant and 1 and 2 refer to the pre- and postshock conditions, respectively. Vn (x V^) is the component of velocity normal to the shock. An important point is that specific energy is conserved across a shock. If the gas is energetically unbound ahead of the shock, it will remain unbound behind the shock in the absence of cooling. On the other hand, if all the thermal energy is immediately lost, one finds that the gas is energetically bound for all radii less than RA. The sheath is bordered by the shock on the outside and the inward-flowing gas of the accretion column on the inside. The sheath is a dynamically defined region where the gas slows to a stop, changes direction, and joins the accretion column. 80 No. 3, 1978 RADIATIVE EFFECTS 1043 The semiangle to the shock cone will be approxi mated by the semiangle of the accretion column. To this end we demonstrate that the thickness of the sheath is small for the case of a narrow shock cone. The radial-velocity component is approximately parallel to the shock and is continuous across the shock. In the limit of Vr = V0, one easily finds that gas entering the shock sheath at r0 < RA will travel to a maximum distance r given by r0 (6) before the radial velocity is brought to zero. The gas would then join the accretion column. An estimate of the sheath thickness can be obtained by equating the mass influx between r0 and r, (r x r0, r « RA) -nn0V0r02, to the mass flux through a cross section at distance r, HSK02T7T</>SH', where vt' is the thick ness of the sheath, ns the postshock density at r, and <f>s the angle to the shock. This gives the semiangular width of the sheath as w r0 ar (7) Thus the maximum width of the sheath is only a function of distance from the accreting object, and for r « RA the sheath width will be negligible. The above calculation assumed laminar flow and no premature mixing of sheath gas into the column, whereas it is quite likely that the sheath is turbulent. The Reynolds number in the sheath is 4.1 x 104/j11r„2$~1K8-6, indicating the possibility of tur bulence. A turbulent sheath would come into equi librium with the column more rapidly than laminar flow through mixing. As a consequence, a turbulent sheath would be even thinner than the limit set in equation (7). b) The Accretion Column The mass flux in the accretion column is simply dM/dt = irp0V0sc2, where sc is the critical impact parameter, taken as the impact parameter of the streamline which would have a total energy of zero on the accretion axis. To allow for the thermal energy, a parameter (Z is introduced, such that the " true accre tion radius" is equal to fiRA. In principle, jS is deter mined once the physical parameters, the density, velocity, and accreting mass, are specified. The param eter jS will be taken as the ratio of the specific kinetic energy (%V02) to the specific enthalpy (5RT0) if (•JrKp2 < 5RT0), otherwise 0=1, where T0 is the equilibrium temperature of the column at r„ = 1. The accretion rate is then dM/dt = 3.65 x 1016/JH/M2K8-3 g s"1 . (8) For a column in equilibrium, the transverse momen tum of the incoming gas must be balanced by thermal pressure in the column, From this one obtains a relation between the central temperature of the column and the opening angle, V02 <f>c 4(2Y'2Rp 2.13 x lO7^"1 Krad"1 . (10) Pressure forces are unable to support the gas, and it falls inward toward the accreting object down the accretion column at a velocity v = (GM/r)112. Using equation (10) and mass conservation, we find that the equilibrium accretion column density is, for fiRA » r, nc = 6.40 x 1013r6-art,-3'a«11K8*/3-1 cm"3 (11) The assumption that the width of the column is maintained by gas pressure is justified by the required default of any stronger forces, turbulence in particular. One can do a pressure confinement calculation similar to the one above by assuming a fully turbulent accre tion column. The internal pressure in the column would be generated by the turbulent velocity, which can be taken to be a fraction/ of the velocity of fall. Requiring that the opening angle of the column be less than, say, 1 radian, we find that f is restricted by 2V2 r (12) Pi i (9) This implies that the turbulent velocity must become a smaller fraction of the fall velocity as it nears the neutron star; otherwise the turbulent pressure is im possible to contain. But the Reynolds number of the gas increases inward (except for adiabatic infall), and we would expect the turbulence, if present, to increase and disrupt the column. Therefore, if the column exists, it must be in laminar flow. There are several reasons to think that laminar flow can obtain in the column. The turbulence would probably originate in the "shear layer" between the sheath and the column, but the Reynolds number in the sheath decreases down the column. In addition, the gas is being strongly accelerated only in the radial direction, which does not provide a driving force for turbulence. III. RADIATIVE EFFECTS If the gas is unable to cool before joining the accretion column, the column gas will fall adiabatically and will resemble the accretion scenario found by Hunt (1971), i.e., a very wide accretion column trailing the accreting body. Note that Hunt's solutions were obtained with essentially zero pressure at the boundary of the accreting body, and that the accretion rate would probably be diminished by the nonzero base pressures of a magnetospheric shock above the neutron star. On the other hand, if the gas cools much faster than any time scale for movement, a cold, narrow, high-density column will be formed. In order to determine which regime prevails, we compare the time scales for radiative energy-loss mechanisms with the time scale for infall of the gas, which is the basic and only uniquely identifiable dynamic time scale of the 1044 CARLBERG Vol. 220 problem. The fall time from the accretion radius is approximately, '^ = 3V(GM) = 250RU3,2KR3/MS- °3) a) Cooling Time Scales In the absence of any heating, the temperature of the gas is entirely dependent upon whether or not a significant amount of cooling can take place in the gas before it reaches the surface of the accreting body. In this section an estimate is made of the cooling time scale, which divides the density-velocity parameter space into regions of cooling and no cooling. In the following, all radiative time scales will be defined as 3kT divided by the appropriate heating or cooling rate, where k is Boltzmann's constant. When the gas crosses the shock, the ions get most of the thermal energy, since they have a much shorter mean free path than the electrons. The electron-ion equilibrium time (Spitzer 1962) in the sheath is, with n = 4«0, a maximum of req = 50.6^-^-^ s. (14) This time is compared with the fall time and is plotted in Figure 2. For the postshock gas, the equilibrium time decreases with density at the same rate as the cooling time and is always shorter than it. Therefore the postshock gas comes into collisional equilibrium and the electron and ion temperatures are assumed equal. The cooling from the postshock temperature can be taken from Figure 1 of Cox and Daltabuit (1971), omitting the cooling due to forbidden and semi-forbidden lines. The postshock cooling is assumed to be unaffected by any radiation present. Two assump tions are made for the density of the postshock gas in the sheath. The rightmost line is drawn for the mini mum possible postshock density, 4n0. This is an underestimate, since the density increases from its free stream value toward the accretion axis, by approxima tion (4). The angle to the shock decreases with the temperature by equation (10), and choosing the minimum temperature in the column as 104 K results in the cooling line on the left. Gas flows with densities and velocities in between these two cooling lines may be subject to an instability from the cooling to noncooling state and vice versa. If hot, uncooled gas mixes into the accretion column and expands it such that the shock moves outward, it will decrease the density of the incoming gas, by approximation (4). If the density drops sufficiently, the incoming gas may no longer cool and the column will expand to its uncooled state. Consequently we take the rightmost cooling line (labeled 4n0) as the effective cooling line. A point of interest is that, for gas crossing the shock at a distance of less than 3 x 1010 cm, the postshock temperature is greater than 107 K, for which brems-strahlung is the dominant cooling mechanism, until the gas is close enough (see eq. [25]) to be Compton cooled. The time scale for bremsstrahlung losses varies with «-lr1'2, which remains constant with distance in the sheath, whereas the dynamic time scale is decreasing as r3'2. Consequently, even though gas entering the column at large radii may cool, lower down the gas in the sheath may remain hot. As shown in the Appendix, the sum of the pressure force for the postshock gas FIG. 2.—The cooling diagram constructed with the free stream density and velocity. Solid lines, regions of cooling for the maximum (IO4 K) and minimum (4n0) densities. Cooling occurs to the right, i.e., higher densities, of these lines. Below the dashed lines (teq < '/) the electron and ion temperatures are equal. The hatched region is heated to the Compton equilibrium temperature and is certainly subsonic, whereas below the dot-dashed line (/Wc < 1) the Mach number of the incoming gas based on the Compton heating rate is less than 1. 82 No. 3, 1978 RADIATIVE EFFECTS 1045 and the gravitational force is still directed downward, but eventually the excess energy must be lost if the gas is to be gravitationally bound to the neutron star. One way to do this would be through turbulent mixing at the boundary between the upward-flowing sheath gas and the downward-flowing accretion column. This would decrease the effective cooling time for the lower sheath by diluting the hot gas with the cooler, denser gas of the column. If a mixing process is required in order to capture gas entering the column at small radii, it would imply that, if gas at the top of the column is unable to cool, then the accretion may become very inefficient. b) Heating When the accreting body is a neutron star, the accretion luminosity may be sufficient to cause significant heating of the infalling gas. (An additional problem is that the incoming gas stream may be heated to a sufficiently high temperature that the assumption of supersonic flow is invalidated. This is considered in the Appendix.) Approximate rates for photoionization and Compton heating are derived and used to con struct a heating diagram similar to the previous cooling diagram. Optical depths must also be considered, because radiative heating will be impossible if the optical depth up the column becomes too large. The accreting object is assumed to be a neutron star of radius 106cm. The entire kinetic energy of the infalling gas is converted into radiation at the surface shock. The resulting luminosity is L = 4.70 x 1036/;11K8-3w3(^/106cm)-1ergss-1. (15) An upper limit to the gas temperature in the column due to photoionization heating is required. The calcu lations of Hatchett, Buff, and McCray (1976). show that, for log £ greater than 2, where £ = L/nr2, the CNO elements are completely ionized. The heating will then be limited by the total recombination rate to the ground state, so the limiting photoionization heating rate is neafx/3, where ne will be taken to be the density in the column from equation (11), a is the recombina tion rate to all levels for completely ionized oxygen, / (=10-3) is the fractional abundance of oxygen, increased slightly to allow for some carbon and nitrogen, and x/3 is the average energy deposited per ionization for a v~l spectrum. (The spectrum may not be but all spectra deposit an average energy of order x-) The recombination rate used is the expression given by Allen (1973) for the total recombination rate to the ground level, « = 3 x 10-loZ2r-3'4. The resulting heating time is fph = 22.9T615'V'2«ir1>V4j81 x (//10-3)-HZ/8)-4s. (16) Comparing this with the fall time, we find that the temperature is limited by T6 < 1.89«n4/15V15(//10-3)4'15 x (Z/8)I6/15. The expression used for photoionization heating applies only if heating in a static gas would be able to attain this temperature and the absorption and scattering optical depth up the column are less than 1. The static condition, log £ > 2, is equivalent to VB < \.02Te2l3rv~116, which is indicated in Figure 3, and is always satisfied in the cooling region. The photo ionization absorption can be estimated from the equations of ionization balance and of optical depth, Le-and w(D = ^p^G dr dr (18) where ne, nt, and na are the number densities of electrons, ions, and ground-state absorbers, respec tively. The integrals in the exact formulae have been approximated by quantities integrated over frequency, and it is assumed that one ionic species is doing the major part of the absorbing at any given temperature. In the neighborhood of 106 K, the major absorber is O VIII. Setting / as the fraction of atoms that are absorbing X-rays, we find a solution to the above equations similar to Mestel's (1954), = i*4 Ufa(T) f ne (19) The bottom of the column, rb, has been assumed to be the magnetosphere at a distance near 108 cm from the center of the neutron star, and it is assumed that there is no significant opacity between the source of radiation and this lower boundary. This is consistent with the magnetospheric model of Arons and Lea (1976). Using the total recombination rate and 5 keV as an average photon energy, we find that the above integral becomes 1 0.9897V19'4 ln (r/r^-3nuF85 x (Rx/106 cm)(//10-3)(Z/8)4 . (20) For numerical estimates, the distance dependence of the optical depth will be ignored [In (r/r„) ~ 1]. This optical depth is meant to be useful only as an indica tion of how radiative heating is attenuated. As it turns out, this estimate of the photoionization optical depth is always less than the electron scattering optical depth for the range of parameters plotted. A more precise calculation is required to estimate the trans mitted spectrum as a function of frequency. The other major source of heating in an X-ray illuminated gas is Compton scattering. The Compton heating rate is given by Buff and McCray (1974) as - a-^kT L <7r me<r 4-rrr 2 (21) (17) where L is the source luminosity, e is a parameter which describes the effective temperature of the spectrum, 1046 CARLBERG Vol. 220 6 7 LOG DENSITY FIG. 3.—The density-velocity diagram for the photoionization heated accretion column, with a Compton heated base. The cooling line (solid) and the subsonic line (Mc < 1) are repeated from Fig. 2. For log £ > 2 the approximation used for the heating rate would be valid for a static gas. As the flow parameters cross the TS > TC (dotted) line, the optical depth up the sheath exceeds that of the column. and a describes the shape of the spectrum (a = 1.04 for a blackbody and i for an exponential spectrum). In principle, a and e are determined by the physical parameters n0, V0, Mx, and the radius of the accreting object. In view of the complexity of the calculation of the emitted spectrum, we chose to leave them as parameters. As typical values we chose a = \ and <r = 5 keV. This corresponds to a Compton equi librium temperature {kT = ae) of 2.9 x 107 K. With this choice of parameters, the Compton heating time scale is tc = 1.20 x 109r6n11-V*V1'"~1/3""1 x CRJC/10°cm)-1s, (22) or tc < tf for "iiVs-2 > 0.4767'6rI)1'a)3-1m-2(i?x/106)-1 . (23) If the base of the column becomes optically thick, the radiation will be thermalized, so that Compton heat ing becomes negligible as a result of the e parameter's being reduced. As discussed by Felten and Rees (1972), spectrum alteration begins when the optical depth T* = (3TFFTCS)1/2 exceeds 1, where rff and Tcs are the free-free and electron scattering optical depths. The calculation indicates that the optical depth at a photon energy of 5 keV is always much less than 1 as long as the flow is supersonic. As one gets closer to the source of radiation, the time scale for Compton heating decreases faster than the fall time. The bottom of the accretion column may be heated to Compton equilibrium, even though the regions higher up may be in photoionization equi librium, or optically thick and cold. From equation (10) we see that the column is impossible to contain for VB < 0.9, and the column will widen and may even become spherical at the bottom. For electron scattering the new effective base of the column is at a distance where the Compton heating and the fall time scales are equal, r8 = 1.39/;112K8-6(«/0.5)-2(£/5 keV)~202. (24) The spherically infalling material below this has a negligible contribution to the optical depth, because the density is reduced by the much greater column angle. The electron scattering optical depth along the column is rc = 9.18«11K87V2(/-6/108 cm)"1'2. (25) The actual line plotted on Figure 3 for the electron scattering opacity assumes that the effective base of the column is at the Compton heated distance of equation (25) and that the column temperature is determined by the photoionization heating temperature of equation (17). The range of photoionization heating is thus extended by the reduction of column opacity. If one computes the Alfven radius from B2/8n = l/2py2, one finds that in some cases the Alfven radius exceeds the Compton heated radius and will determine the effective base of the flow. But these cases turn out to be in the region of the density-velocity diagram above the cooling line and hence of little interest to the heating calculation. The column is effectively optically thin to the side ways loss of radiation because the sideways optical No. 3, 1978 RADIATIVE EFFECTS 1047 depth of the column is dominated by electron scatter ing and is always less than 1 for the range of parameters plotted. The higher-energy X-rays will be attenuated by K shell absorption by elements with ionization potentials greater than the CNO elements. For a spectrum with a typical photon energy of 5 keV, the K shell cross sections of Daltabuit and Cox (1972) and the abun dances of Allen (1973) suggest that the dominant K shell absorber will be silicon. The calculations of Hatchett et al. indicate that a typical silicon atom, for log £ > 2, will have several electrons left, and therefore will have a cross section of order 10"19 cm2, which is relatively independent of temperature and radiation flux. Combining this with a fractional abundance of 3 x 10~5, the effective cross section at the absorption edge is only 4.5 times the electron scattering cross section. Photons below the edge will be less affected, primarily interacting with only lower-abundance magnesium, and for those above, the cross section decreases approximately as E~3, until another edge, due to low-abundance sulfur, is encountered. In general we expect that K absorption will be of the same magnitude as the electron scattering. Similarly, the K shell photoionization heating rate is different from Compton heating only by a multiplicative factor of order 1, and will be ignored. c) Accretion Scenarios In Figure 3 a box has been drawn which encloses the suggested range of wind densities and velocities for stellar wind X-ray sources Cen X-3 and 3U 1700 — 37. Only a small part of this region is subsonic and beyond the description given here. For a given density and velocity, it is possible to qualitatively describe the flow. If the density and velocity parameters of the free stream lie above the cooling line, the accretion may be less efficient as pressure forces in the hot gas of the sheath become more important. This would be re flected in a diminished luminosity. In general, flows with parameters above the cooling line would broadly resemble the scenario found by Hunt (1971). Captured gas falls inward with its temperature rising adia-batically. Below the cooling line, the gas temperature drops to some equilibrium value and falls down the accretion column. Although the K absorption edges will alter the spectrum somewhat, the line above which the electron scattering opacity exceeds 1 is almost co incident with the cooling line, so that heating of the most distant parts of the accretion column will be possible below the cooling line. Typical maximum temperatures for Ve — 1 are Te = 1 at nn — 0.1 and Te — 3.5 at /in = 10. The column semiangles are 2° and 9°, respectively. The gas will remain at the equilibrium temperature specified by the local radi ation field, since all radiation time scales are more rapid than the fall time. Near the source Compton heating dominates, the temperature rises to the Compton equilibrium value, and the column expands so that the infall becomes almost spherical. It is of particular interest to compare the electron scattering opacity up the column with that up the edge of the sheath in the postshock gas. The opacity up the sheath is Evaluating this integral, and choosing (somewhat arbitrarily) the maximum extent of the column to be the distance at which the density has dropped to 4/3n0, i.e., rm = RAl(2<j>2), gives rs = 2.26«nK82r6-2, (27) whereas the electron scattering opacity up a column with a Compton heated base is 7.79V^TQ'2. A.S a result we find that the opacity up the sheath is greater than up the column if «UKB"S > 3.45, a result which is independent of the column temperature. This line has been included in Figure 3. If the stellar wind in which, the neutron star is em bedded has velocity and density variations, this analysis predicts potentially observable effects. The most obvious is that, if the line-of-sight optical depth is constant, the X-ray luminosity responds to variations in n0V0~3 (eq. [16]) on times of variation longer than about two fall times, or 500VB~3 s. This variation reflects the local structure of the wind for regions of size greater than 2RA (5 x 1010K8~2 cm). Another expected effect is that, as n0V0~2 increases, the optical depth up the sheath will exceed that up the column. Thus, if the X-ray source is occulted by the accretion column, the X-ray absorption would change from a single dip (Tc > rs) to a double dip (TC < Ts). In addition, Jackson's calculations (1975) indicate that, if the gas fails to cool, the absorption up the sheath always dominates. IV. THE FAR WAKE The Reynolds number of the gas flow is extremely high (V0RAjv = lO12^-1/?!1^-5'2/;!!.), and the far wake is expected to be turbulent. Turbulence in super sonic flows is not well understood, but experimental studies of supersonic wakes (Demetriades 1968) indicate that a phenomenological theory as outlined by Townsend (1976) provides a reasonable description of supersonic far wakes. Unfortunately, the dynamics of laboratory wakes are not dominated by a gravita tional field, and therefore the applicability of the description to this case must be carefully considered. The subsonic theory is based on the observation that the wake remains self-similar with respect to a characteristic velocity and length scale. This is com bined with the momentum equation from which all small terms have been dropped. The axisymmetric far wake is found to be self-similar with respect to the 1048 CARLBERG Vol. 220 half-width, /, and the turbulent velocity scale, u, denned by I \ (±\113(_r_Y'3 where the so-called momentum radius RM (the radius such that the drag force is 1 /2pV02TrRM2) has been taken to be RA. RT is the turbulent Reynolds number, observed by Demetriades to be 12.8. The width scale of the wake implies that the small angle approximations for the density and transverse velocity apply for the exterior supersonic flow. Hence the effective exterior pressure will be the transverse momentum flux, which varies with distance along the wake. But the equations of momentum used to derive the length and velocity scale assumed that there was no pressure gradient in the free stream. For the gravitational wake, an order-of-magnitude estimate of the pressure gradient term in the small angle approxi mation-finds that it is of order u2/l(RA/l), whereas the retained terms in the momentum equation are of order u2jl. Consequently, for RA « I, the pressure gradient term can again be dropped. Using the half-width of equation (28), we find that RA/1 is of order (RJr)113. Thus the equations are consistent for r » RA, but the crucial observation that a gravitational wake is self-similar is unavailable. Experiments also indicate that the flow may not be self-similar for distances of several tens of the momentum radius, but the deviation of the turbulent velocity scale from the self-similar value is a factor of 2 or less. For sufficiently low Reynolds numbers, part of the wake may be in laminar flow, ln this case the velocity defect on the axis is u/V0 = 3/2RA/r, and the half-width varies as (rRA)xlz (Lamb 1924). The Reynolds number grows with distance, and the flow will even tually become turbulent. With the extreme Reynolds numbers present here, the wake is expected to become turbulent within the sheath of the accretion column. If the gas in the wake has no energy losses, i.e., 5RT + \v2 constant, the temperature on the axis is found to be T-Tn = ^ = 3.55 x }0eVe2l3m2l3rl2-2'3 K , (29) where TK is the temperature of the gas external to the wake. This temperature implies that the turbulent velocity scale is subsonic. The temperature becomes equal to 2Tm at R^Volc^, where RB is the Bondi radius, 10!4(104 K./T) cm, and cx is the sound speed at Tm. Experimentally it is observed (McCarthy and Kubota 1964) that the pressure is approximately constant across the wake. Equating pv'2 at the wake boundary to the gas pressure at the center gives the central density »„ = 1.06 x 1011n11ro1"%-1'3'-i2~1'6. (30) Note that this density is greater than that which would be found by using the external static pressure by a factor of ^f- = 376.07-4-1 JV'3ria-5/6 • (31) The temperature estimate and density estimate of equations (30) and (31) assume that the wake is iso-energetic, but at these temperatures and densities, radiation cooling can be significant. Time scales of interest are the cooling time (where A is the cooling coefficient) 12.4«1X"1K85/3/-12-1/2(10-22 ergs cm3 s"7A) s, (32) turbulent dissipation of kinetic energy time scale, 3RT/(u3/l) = 3.04 x 104K81/3m-2'3r125'3 s, (33) and the turbulent time scale, 2.40 x 103 K8-1/-12m2'3s. (34) These time scales imply that, for large distances, cooling removes most of the thermal energy from the wake. If the sound speed within the wake drops below the turbulent velocity scale, the turbulence would then become supersonic, leading to shocks which rapidly heat the gas, but the shocks would occur on the basic turbulence time scale, and would not be able to reheat the bulk of the gas. One might speculate that the temperature would decline to the minimum of 104 K, but with an extremely clumpy-distribution. If cooling is complete, the simple model used, which does not consider the energy budget, may break down com pletely. Its value lies in the fact that, as the gas cools, the Reynolds number becomes even greater, and the dynamics of the gas flow in the far wake are almost certainly dominated by turbulence. One can combine the density and width to show a column density across the wake of sufficient size to produce optical absorption of radiation from the primary. That is n22l = 2 x 1033 cm-5 for an optical depth in the wake of one at Ha, assuming the lower level is populated by recombinations at 104 K and depopulated by radiative transitions. This would be possible whenever the wake was silhouetted against the primary star. But these simple considerations fall well short of the ability to reproduce line profiles as seen by Conti and Cowley (1975). The wake will remain cold in the presence of an X-ray source for L/nr2 < 10, or distances from the X-ray source of r12 > 3L371'2/;!!112. The absorption cross section for X-rays by a cold gas of cosmic abundances is about 10-22(£'/keV)-3 cm2. Again the column densities are adequate for X-ray absorption, but the absorption would be very sensitive to the inclination of the wake 86 No. 3, 1978 with respect to the X-ray star, since the far wake is very narrow. V. CONCLUSIONS The supersonic accretion of gas onto a neutron star has been described, working from the basic model as shown in Figure 1; the main features are the sheath and the accretion column. The angular width of the column, a measurable quantity in the X-ray light curve, is found to depend on the ratio of V2 to the column temperature, and therefore yields information about the local wind velocity provided the column tem perature can be specified. An accurate estimate of the temperature would require a hydrodynamic calculation including radiation transfer, but upper limits to the temperature can be obtained by estimating the relevant heating and cooling rates. !The most important consideration in determining the thermal state of the gas is whether or not the gas can cool before it falls all the way down the accretion column. The cooling line of Figure 2 separates the flow into two main regimes. If the postshock gas in the sheath is unable to cool, it will fall inward adi-abatically in a wide accretion column, with the accretion efficiency (the /? factor) reduced by the thermal pressure. Below the cooling line of Figure 2, the gas will cool to an equilibrium value determined by the radiation field. In the region of the density and velocity parameters which apply to the stellar wind X-ray binaries, this means that the upper part of the accretion column will be photoionization heated to temperatures of order 10° K. The base of the column will be heated to the Compton equilibrium tempera-1049 ture, causing the pressure to rise sufficiently that the base of the column will spread to a broad inflow. It is predicted that the electron scattering will cause the X-ray light curve absorptions to change from single dips to double dips as n0V0~2 increases, if the param eters are in the cooling region. The gas that is gravitationally perturbed but does not become bound to the neutron star forms the far wake. The high velocity and low viscosity indicate that the far wake is almost certainly turbulent. An exten sion of the similarity description of supersonic wakes experimentally studied provides the temperature and density in the wake. But the cooling time of the gas in the wake is then found to be less than the basic turbulence time scale, which may mean that whole description is invalid. In spite of this, we suggest that the far wake is composed of a hot gas entering the wake and denser clumps of cold gas, a description which is marginally consistent with the "wake" observations of Conti and Cowley (1975). The model outlined here is intended to be useful for providing qualitative insight into the physics of super sonic accretion. The numerical quantities employed are expected to be accurate to a factor of 3 or so, and should provide basic regimes which can be further explored with a numerical model. G. G. Fahlman provided invaluable advice and criticism, and read several rough drafts of this paper during the course of this research. General encourage ment and useful discussions were provided by members of the UBC Institute of Astronomy and Space Science and the Dominion Astrophysical Observatory. RADIATIVE EFFECTS APPENDIX In the case of a heated gas, the thermal pressure forces may become large enough to destroy the assumption that the flow is dominated by inertial forces. In this Appendix the region of validity of the supersonic description of accretion is examined. The incoming free stream may be heated such that the Mach number, M = V0/{2yRT)1'2, becomes less than I. The maximum temperature which can be produced by Compton heating is 2.9 x 107(a/0.5) x (e/5 keV). This temperature can be attained for «UF8"Z > 13.8, which is shown as the crosshatched area in Figure 2. This gives only the area for which subsonic flow is guaranteed in the presence of Compton heating, but what is really wanted is a line on which the Mach number is equal to 1. If only Compton heating is considered, we find that (non-equilibrium) temperatures are produced such that the Mach number is less than 1 for '?n-/8~4 > 17.2. This line (MC < 1) is shown in Figure 2. If photoionization heating is included, the subsonic region is increased very slightly at low velocities. We conclude that most of the box of Figure 3 is indeed in supersonic flow. The deviation of streamlines from particle trajec tories will depend on the ratio of pressure forces to inertial forces. As a worst case we assume that the gas is fully Compton heated. The ratio of radial pressure force to gravitational force for gas outside the shock is 1^ dp /GM _ RT0 /GM p 8r SRT (Al) This implies that the net force is outward for r > 5RT, where the thermal radius is RT = GM/SRT0 = 3.19 x 10107'7_1 cm. Similarly, in the transverse direction, the ratio of pressure to the momentum flux is r 5%. (A2) In the shock-heated sheath, the ratio of radial pressure force to gravitational force is 9/16, which will act only to reduce the effective mass of the gravitating object in the sheath. In the column the ratio of pressure forces to gravitation is, for a constant temperature, 3r/5RT. In general the pressure forces can be safely ignored, even in the presence of strong heating, provided that we remain in the area of validity of the supersonic flow assumption. 1050 CARLBERG REFERENCES Allen, C. W. 1973, Astrophysical Quantities (3d ed.; London: Athlone Press). Arons, J., and Lea, S. M. 1976, Ap. J., 210, 792. Bondi, H., and Hoyle, F. 1944, M.N.R.A.S., 104, 273. Buff, J., and McCray, R. 1974, Ap. J., 189, 147. Conti, P., and Cowley, A. P. 1975. Ap. J., 200, 133. Cox, D. P., and Daltabuit, E. 197i, Ap. J., 167, 113. Dachs, J. 1976, Astr. Ap., 47, 19. Daltabuit, E., and Cox, D. P. 1972, Ap. J., Ill, 855. Danby, J. M. A., and Camm, G. L., 1957, M.N.R.A.S., 111, 50. Davidson, K., and Ostriker, J. P. 1973, Ap. J., 179, 585. Demetriades, A. 1968, AIAA J., 6, 432. Eadie, G., Peacock, A., Pounds, K. A., Watson, M., Jackson, J. C, and Hunt, R. 1975, M.N.R.A.S. Short Comm., Ill, 35p. Felten, J. E., and Rees, M. J. 1972, Astr. Ap., 17, 226. Hatchett, S., Buff, J., and McCray, R. 1976, Ap. J., 206, 847. Hoyle, F., and Lyttleton, R. A. 1939, Proc. Cambridge Phil. Soc., 35, 405. j Hunt, R. 1971, M.N.R.A.S., 154, 141. Illarionov, A. F., and Sunyaev, R. A. 1975, Astr. Ap., 39, 185. Jackson, J. C. 1975, M.N.R.A.S., 112, 483. Jones, C, Forman, W., Tananbaum, H., Schreier, E., Gursky, H., Kellogg, E., and Giacconi, R. 1973, Ap. J. (Letters), 181, L43. Lamb, H. 1924, Hydrodynamics (5th ed.; Cambridge: Cam bridge University Press). McCarthy, J. F., and Kubota, T. 1964, AIAA J., 2, 629. McCray, R., and Hatchett, S. 1975, Ap. J., 199, 196. Mestel, L. 1954, M.N.R.A.S., 114, 437. Pounds, K. A., Cooke, B. A., Ricketts, M. J., Turner, M. J., and Elvis, M. 1975, M.N.R.A.S., 172, 473. Ruderman, M. A., and Spiegel, E. A. 1971, Ap. J., 165, 1. Spitzer, L. 1962, Physics of Fully Ionized Gases (New York: Interscience). Townsend, A. A. 1976, 77;e Structure of Turbulent Shear Flow (2d ed.; Cambridge: Cambridge University Press). R. G. CARLBERG: Department of Geophysics and Astronomy, University of British Columbia, Vancouver, B.C., Canada, V6T 1W5 88 APPENDIX 2. GAS PHYSICS P hot c i o ni z a t ion The simplest process to describe is the photoionization rate which is independent of the gas temperature and density, and is simply given by where tf"cj is the photoionization cross-section of atom i, ioni zation level j. For these calculations the mean radiation field J was taken simply as the flux energing from a model atmosphere allowing for geometrical dilution. The numerical computations used a model computed by Hihalas (1972), specifically the Non-LTE 50,000 K, log g = 4 model. Mihalas gives the radiation field in terms of the emergent flux, F , whereas the mean intensity 4 J is needed for the ionization and heating calculations. To make this change, the mean intensity was assumed to vary as the dilution, H= 1/2 (1-(1-(r^/r) 2) l/2) , and the flux as (r*/r)2, where r is the stellar radius. For the computations illustrated here, all done at a radiation field corresponding to the surface of the star, 4<J>/ = 2 (fTy). Since the 0 VI ion is of particular importance the pho toionization rates of all ions up to comparable ionization po tentials (IP of 0 V is 113.9eV) were included. The photoionization cross sections used in the calculations are given below. The cross sections used are given in below. The Hydrogen cross section was taken from Bethe and Salpeter (1957). hi/ (All. 1) 89 2 * rrl* AO* 77c) ^ e^p (- ^ <*><c°*r\) (All. 2) where «C is the fine structure constant a.a is the Bohr radius CH. is the Hydberg Z is the ion charge The Helium I cross section was obtained from Brown (1971). The formula quoted by him was multiplied by 16 to agree with his numerical values, and a factor of 2 was included in the ex ponential factor to reduce to the Hydrogen formula. The cross section is (All.3) where * ^ k W/> and l/j= 1^ with all ^'s replaced by ji's. The constants are *=2. 182846 0 = 1.188914 Zr2 Zb=1 k=( (hV-24. 587e¥)/13.59 8eV) V2 The Helium II cross section is the same as the phctcicriza-ticn cross section of Hydrogen but with Z=2 everywhere. 90 For the remaining ions the cross sections have been calcu lated by various authors using the principles of guantum me chanics, and then making a fit to a standard polynomial to rep resent the data as a function of incident photon energy. Two forms for the polynomial are used here, one due to Seatcn (1958) (All. 4) where i^,is the threshold freguency and the other constants are fitting parameters. The other form is a slightly more general polynomial due to Chapman and Henry (1971) *•<.»)* </»-*<)fej"*"V(i*-v»^pi7. (All.5) TABLE 6: PHOTOICNIZiTICN Icr s ev 10-l8cm* C I 11.26 12. 19 2.0 C II 24.383 4.6 3.0 C III 47.887 1.84 2. 6 C IV 64.492 0.713 2.2 N I 14.534 11.42 2.0 N II 29.601 6.65 3.0 N III 47.448 2.06 1.62 N IV 77.472 1.08 3.0 N V 97.89 0.48 2.0 0 I 13.618 2.94 1.0 16.943 3.85 1.5 18.635 2.26 1.5 0 II 35.117 7.32 2.5 0 III 54.943 3.65 3.0 0 IV 77.413 1.27 3.0 0 V 113.90 0.78 3.0 0 VI 138,2 0.36 2. 1 Ne I 21.564 5. 35 1.0 Ne II 40.962 4, 16 1. 5 CBOSS SECTION PARAMETERS ot- ft Beference 3. 17 -- H70 1.95 H70 3. 0 — SB 71 2.7 - - SE71 4.287 — H70 2.86 H70 3.0 — H70 2.6 — F68 1.0 — F6 8 2.661 — H70 4.378 --4.311 3.837 -- H70 2.014 — H70 0.831 H70 2.6 F68 1.0 — F68 3.769 — — H70 2.717 — H70 91 44.166 2.71 1.5 47.874 0.52 1.5 Ne III 63.45 1.89 2.0 68.53 2.50 2.5 71. 16 1.48 2.5 Me IV 97.11 3.11 3.0 Ne V 126.21 1.40 3.0 Ne VI 157.93 0.49 3.0 Mg I 7.646 9.92 1.8 Mg II 15.035 3.416 1.0 Mg III 80.143 5.2 2.65 Mg IV 109.31 3,83 2.0 Mg V 141.27 2.53 2.3 Si I 7.37 •12.32 3 8. 151 •25.17 5 Si II 16.345 2.65 3.0 Si III 33.492 2.48 1.8 Si IV 45.141 0.854 1.0 S I 10;360 12.62 3.0 12.206 19.08 2. 5 13.40 8 12.70 3.0 S II 23.33 8.2 1.5 S III 33.46 • .35 2.0 34.83 • .24 2.0 S IV 47.30 0.29 2.0 S V 72.68 0.62 1.8 S VI 88.05 0.214 1.0 2.148 2.126 2.277 — H70 2.346 2.225 1.963 -- H70 1.471 -- H70 1.145 -- H70 2.3 — Iso To Si III 2.0 — Iso Io Si IV 2.65 S58 1.0 — S58 1.0 — S58 6.459 5.142 CH71 4,420 8.943 0.6 -- SB71 2.3 — SB71 2.0 — SB71 21.595 3.062 CH72 0.135 5.635 1.159 4.734 1.6-95 -2. 236 CH72 10.056 -3.278 CH72 16.427 0.592 6.837 4.459 CH72 2.3 -- Iso To Si III 2.0 — Iso To Si IV * means that the cross section weighted by the statistical weights of the fine structure transitions. The abbreviation Iso means extrapolation along an isoelectronic seguence. The references coded above are: H70: Henry 1970 S58: Seatcn 1958 SB71: Silk and Brown 1971 CH71 Chapman and Henry 1971 CH71 Chapman and Henry 1972 F68: Flower 1968. 92 The Jecombination Bate The recombination rate for Hydrogen was calculated usirg an expression given by Johnson (1972) which has a correction factor built in allowing for finite density. The rate to level n is •z S (c»,n) = D<I*/kT)3/2exp(WkT) £ g^(n)x-^' c-o E <x0I*/kT) (All. 6) Above the level nw the populations can be assumed to be in egui-librium with the continuum, that is the populations are as in Saha eguilibriuo. The value of n is calculated from an expres sion given by Jordan (1969) y\b Z. n Vie 17 (All. 7) The value of x, is defined to be 1-(n/n„)2. The constant is 5. 197x10-** cm2. The functions E «"are tne exponential integrals, and the g (n) are Gaunt factor coefficients, determined as shown in the following table. TABLE 7: GAONI FACTOBS n=1 n=2 n>2 g0(n) 1. 1330 1.0785 0. 9935 + . 25 28n~ , 1296n~^ g, (JJ) -0.4059 -0.2319 -n~» {.6282-. 5598n~*• . 52 99n-*) g^(n) .07014 . 02947 n~* (.3887-1. 181n-*+1.470n-z) The values in this table were taken from Johnson (1972). In order to obtain the total recombination rate, recombina tions to the levels n=1 to 9 summed together. Computations of the recombination rate for all of the other ions cf interest have been made by Aldrovandi and Peguignot 93 (1973), with errata in Aldrovandi and Peguignot (1976). The data is provided in the form of fits to simple functions. The radia tive rate is given by *r= *r»el <T/10* K)~\. (All.8) and the dielectronic recombination rate by oL^-= A^T-3/2exp (-Tc/T)|; l4Bc(i€Xp (-T,/T) 1. (AIT.9) The various constants used are given in the accompanying table. The range of validity of the fits are Tmax/1000 < T < Tmax. Tcrit gives the temperature above which dielectronic recombina tion is important. TABLE 8: RECOMBINATION FIT CONSTANTS ATOM ABAC ETA TMAX TCBIT ADI TO BDI T1 HE I 4. 3E-13 .672 1E5 5. 0E4 1. 9E-3 4.7E5 0.3 9.414 C I 4.7E- 13 . 624 514 1.2E4 6. 9E-4 1. 1E5 3.0 4.914 C II 2.3E- 12 ,645 1E5 1.2E4 7.0E- 3 1.515 0. 5 2. 315 C III 3.21- 12 .770 3E5 1. 1E4 3. 8E-3 9. 1E4 2.0 3.7E5 C IV 7.5E- 12 .817 1E6 4.4E5 4. 8E-2 3.4E6 0.2 5. 115 C V 1.71-11 .721 3E6 7. 0E5 4, 8E-2 4. 1E6 0.2 7.6E5 N I 4. 1E-13 .608 1E5 1.8E4 5. 2E-4 1.3E5 3. 8 4.814 N II 2.2E- 12 .639 115 1. 8E4 1. 7E-3 1.415 4.1 6.8E4 N III 5.0E- 12 .676 3E5 2.4E4 1. 2E-2 1.8E5 1.4 3. 8E5 N IV 6.5E-12 .743 315 1. 5E4 5. 5E-3 1. 1E5 3.0 5.915 N V 1.5E- 11 .8 50 3E6 6.8E5 7.6E- 2 4.7E6 0.2 7. 215 N VI 2.91- 11 .750 117 1.0E6 6. 6E-2 5.4E6 0.2 9.8E5 0 I 3. 1E-13 .678 5E4 2.7E4 1.4E- 3 1.7E5 2.5 1.3E5 C II 2.0E- 12 .646 215 2. 2E4 1. 4E-3 1.7E5 3.3 5.8E4 0 III 5. 1E-12 .666 5E5 2.4E4 2. 8E-3 1.815 6. 0 9. 114 0 IV 9.6E-12 .670 116 2. 5E4 1.7E- 2 2.2E5 2.0 5.9E5 0 V 1.2E- 11 .779 6E5 1.6E4 7. 1E-3 1.3E5 3.2 8.0E5 G VI 2. 3E-11 . 802 3E6 1.0E6 1. 1E-1 6.2E6 C.2 9.5E5 0 VII 4. 1E-11 .742 1E7 1.5E6 8. 6E-2 7.016 0. 2 1.316 NE I 2.2E- 13 .759 115 3. 0E4 1. 3E-3 3.115 1.9 1.5E5 NE II 1. 5E-12 .693 1E5 3. 3E4 3. 1E-3 2.915 0.6 1.715 NE III 4.41- 12 .675 215 3. 3E4 7. 5E-3 2.6ES 0.7 4.515 NE IV 9. 1E-12 .668 3E5 3.5E4 5.7E- 3 2.415 4. 3 1.715 NE V 1.51-11 .684 615 3. 6E4 1.0E- 2 2.4E5 4.8 3.515 NE VI 2.3E- 11 .704 1E6 3.6E4 4. 0E-2 2.9E5 1.6 1. 116 NE VII 2.8E- 11 .771 1E6 2. 9E4 i. 1E-2 1.7E5 5.C 1 .3E6 NE VIII 5.0E- 11 .832 6E6 1. 5E6 1.8E-1 9. 8 16 0. 2 1.416 NE IX 8.6E- 11 .769 317 3. 816 1. 3E-1 1.1 E7 0.2 2.6E6 MG I 1.4E- 13 .855 3E4 4.0E3 1. 7E-•3 5.114 0.0 0.0 MG II 8. 8E-13 .838 115 7. 4E4 3. 5E-3 6.1E5 0.0 0.0 MG III 3.5E- 12 .734 3E5 6.6E4 3. 9E-•3 4.415 3.0 4. 1E5 9a MG IV 7.71- 12 .718 515 5. 5E4 9. 3E-3 3.9E5 3.2 8.715 MG V 1. 4E-11 .716 1E6 4.4E4 1.5E-2 3.4E5 3. 2 1.016 MG VI 2.3E-11 .695 1E6 4. 5E4 1.2E- 2 3. 1E5 6.7 5.4E5 MG VII 3.2E- 11 .6 91 1E6 4.5E4 1. 4E-2 3.115 4.4 3.615 MG VIII 4.6E- 11 .711 2E6 5. 014 3. 8E-2 3.6E5 3.5 1.616 MG IX 5.8E- 11 .804 3E6 3.4E4 1.4E-2 2.115 10.0 2. 116 MG X 9. 1E-11 . 830 117 2. 4E6 2. 6E-1 1.4E7 0.2 2.416 MG XI 1. SB 10 .779 5E7 4.0E6 1.7E- 1 1.517 0.2 3.5E6 SI I 'S. 9 E-13 .601 3E4 1. 114 6. 2E-3 1.1E5 CG 0.0 SI II 1. OE-12 .786 1E5 1. 1E4 1.4E-2 1.215 0.0 0.0 SI III 3.7E-12 .693 2E5 1. 1E4 1. 1E-2 1.0 E5 0.0 0.0 SI IV 5. 5E-12 .821 3E5 1.7E5 1. 4E-2 1.216 0.0 0.0 SI V 1.21-11 .735 615 9.514 7. 8E-3 5.5E5 10.0 1.0E6 SI VI 2. 1E-11 .716 1E6 8. 0E4 1.6E-2 4.915 4.0 1.3E6 SI VII 3.0E- 11 .702 1E6 7. 4E4 2. 3E-2 4.2E5 8. 0 1.7E6 SI V III 4.3E- 11 .6 88 2E6 6. 8E4 1..1E-2 3.8E5 6. 3 6.015 SI IX 5.8E- 11 .703 2E6 6.6E4 1. 1E-2 3.7E5 6.0 1.116 SI X 7.7E- 11 .714 3E6 6.5E4 4. 8E-2 4.2E5 5.0 2.516 SI XI 1.2E- 10 .855 117 4. 514 1. 8E-2 2.5E5 10.5 2.816 SI XII 1. 5E-10 .831 3E7 3.7E6 3.4E- 1 1.917 0. 2 3. 1E6 SI XIII 2. 1E-10 .765 517 6. 316 2. 1E-1 2.0E7 0.2 4.4E6 S I 4.1E- 13 .630 3E4 2.2E4 7.3E-5 1. 115 0. 0 0.0 s II 1.8E- 12 .6 86 115 1. 2E4 4. 9E-3 1.215 2.5 8. 8E4 s III 2.7E- 12 .745 2E5 1.4E4 9. 1E-3 1.315 6.0 1.515 s IV 5.7E- 12 .755 3E5 1.5E4 4. 3E-2 1.8E5 0,0 0.0 s V 1.2E- 11 .701 5E5 1.4E4 2. 5E-2 1.515 0.0 0.0 s VI 1.7E- 11 .849 1E6 2.9E5 3. 1E-2 1.9E6 0.0 0.0 s VII 2.7E- 11 .733 1E6 1.3E5 1. 3E-2 6.7E5 22.0 1. 8 E6 s VIII 4.0E- 11 .696 216 1. 115 2. 1E-2 5.9E5 6.4 2.0E6 s IX 5.5E- 11 .711 2E6 9. 0E4 3. 5E-2 5.515 13.0 2. 316 s X 7. 4E-11 .716 316 9.0E4 3. OE-2 4.7E5 6. 8 1.216 s XI 9.2E-11 .714 5E6 9.0E4 3. 1E-2 4.215 6. 3 1.316 s XII 1. 4E-10 .7 55 6E6 8. 3E4 6. 3E-2 5.0E5 4.1 3.416 s XIII 1.7E- 10 .832 1E7 6.0E4 2. 3E-2 3.015 12.0 3.616 s XIV 2. 5E-10 .852 118 5. 0E6 4.2E- 1 2.4E7 0.2 4.616 s XV 3.3E- 10 .783 2E8 9. 0E6 2. 5E-1 2.517 0. 2 5.516 A number of small changes have been made in the limits of the approximations in order to smooth the turn on transition for dielectronic recombination. The dielectronic recombination rates computed above were based on the assumption of a low density gas with no radiation field present, whereas the envelope of a stellar wind star is an environment of moderately high density and strong radiation field which will effect the rate. Dielectronic recombination occurs when a free electron excites a bound electron tc a higher 95 energy level, thereby allowing the free electron to lose enough energy to become bound into a very high guantum level. The atom then can stabilize by a series of cascades of the two electrons to the ground state. Schematically this process is S(n)+!+e-5*J+l-'in',n") —> A**-* (n,n«») + h ^ * A**-i (n,n 1»•) +h v t where in general n* = n+1 and n'*>>n, nIf the gas becomes sufficiently dense or if the radiation field strong enough, the electron in the high lying guantum level n*1 can be either ccl-lisicnally or radiatively ionized out of the atom before it has time to stabilize by photoemission. A rough empirical correction factcr was devised to allow for this decrease in the dielec tronic recombination rate. The principal guantum number of the state at which half the captured electrons are stabilized by cascades to ground and half are reionized is given by, 1(collisions)=<1.4x10ls26Tl/2/ne)l/7 Dupree (1968) 1 (radiative) -Z (3 flUn (1) / (HkT r<wj)) V2 Sunyaev and Vainstein (1968) where W is the geometrical dilution factor of the radiation field approximated by a blackbcdy of temperature Trad* These numbers can be calibrated against the depression of the recom bination rate calculated by Summers (1974). It was found that data is roughly fitted by the multiplicative factor f, such that ^=t *^(n=0,W=0), where f is f = exp[-2. 303* (. C15*a2+,092*a) ] where a = 12.55-7*log10(1). 96 That is, the adjusted recombination rate is found by multiplying the value found from the fits given by Aldrovandi and Peguignot times the f factor given above. In addition to this correction to the dielectronic rate, the semicoronal approximation of Wilson (1962) has been used to add tc the radiative rate. This allows for some recombinaticn to upper levels, = 1.8X10-A» "&j(kT)-a/* Ycj (All. 10) where Xcjl1) ~ ^/l2 (collisions) . In addition three body recom bination makes significant contributions at low temperatures, and is simply approximated by (Burgess and Summers 1976) ^^=1,16xl0-«J3fV2fle, (All.11) where J is the charge of the ion. , Collisional Ionization The rate of collisional ionization for Hydrogen was also taker from Johnson (1972) as (All.12) where ^ Zfftr "to i*3 V* vL = %I^/kT z'v = x0 (I^/kT+r^) | (t) = E0 (t)-2E, (t)*E^(t) , 97 and the Gaunt factors and x are as for the recombination rate in Hydrogen. Only ionization from the ground state n=1 will be considered, so r|=0.45 and b(= -0.603. All other ions have ccllisional ionization rates based upon an approximation investigated in detail by lotz (1967). A slight modification to the original formula has been made by McWhirtier (1975) to allow for the decrease of the ionization rate at high temperatures. The rate is given by where s goes from 1 to 2 in the calculations here, n (s) is the number of electrons in the subshell, and %••{ 1) is the normal J ionization potential as given by Allen (1973), yt'(2) is ")(t''(1) plus the excitation energy of the lowest excited level in the new ion with one of the inner shell electrons removed. For in stance, the ionization of C II which has an electronic confi guration of 1s22s22p can proceed with the addition cf 1S6659 cm-1 of energy to C III 1s22s2 by removing the one outer shell electron, or the ionization can take 196659*52315 car-1 and ionize to C III 1s22s2p, by removing one of the two s shell ele ctrons. The energy 52315 cm-* is simply the energy to go from C III 2s2 to C III 2s2p. The attached table gives ionization po tentials (in eV) and the number of subshell electrons. , The values were obtained from the tables of lotz (1967) and Moore (1949) * TABLE 9: IONIZATION POTENTIAL AND SHELL ELECTBON POPULATIONS ATOM IP1 N1 IP2 N2 H I 13.598 1 HE I 2 4.587 2 z. (All.13) 98 HE II 54.416 1 C I 11.260 2 16.6 2 C II 24.383 1 30.9 2 c III 47.887 2 323. 2 c IV 64.492 1 342. 2 c V 392.08 2 c VI 489.98 1 N I 14.5 34 3 20.3 2 N II 29.601 2 36.7 2 N III 47.448 1 55.8 2 N IV 77.472 2 469. 2 N V S7.89 1 492. 2 N VI 552.06 2 N VII 667.03 1 0 I 13.618 4 28.5 2 0 II 35. 117 3 42. 6 2 0 III 54.934 2 63.8 2 0 IV 77.413 1 87.6 2 0 V 113.90 2 642. 2 0 VI 138.12 1 66 9. 2 0 VII 739.32 2 0 VIII 871.39 1 NE I 21. 564 6 48.5 2 NE II 40.962 5 66.4 2 NE III 63.45 4 86. 2 2 NE IV 97.11 3 108. 2 NE V 126.21 2 139. 2 NE VI 157.93 1 172. 2 NE VII 207.26 21072. 2 ME VIII 239.09 11106. 2 NE IX 1195.8 2 NE X 1362.2 1 MG I 7.646 2 60.420 6 MG II 15.035 1 67.809 6 MG III 80.143 6 118.768 2 MG IV 109.31 C -/ 14 4.42 2 MG V 141.27 4 17 2. 0 1 2 MG VI 186.51 3 201.22 2 MG VII 224. 95 2 241.14 2 MG VIII 265.92 1 283.38 2 MG IX 328.0 21680.4 2 MG X 367.5 11719.8 2 MG XI 176 1.8 2 MG XII 196 3. 1 SI I 8. 151 2 13.616 2 SI II 16.345 1 22.870 2 SI III 33.492 2 137,709 6 SI IV 45. 141 1 149. 358 6 SI V 166.77 6 217. 170 2 SI VI 20 5.08 5 250.48 2 SI VII 246.49 4 285.26 2 SI VIII 303. 16 3 321.76 2 SI IX 351. 1 2 371. 2 2 SI X 401. 4 1 422.4 2 SI XI 476. 1 22340.8 2 SI XII 523. 12388. 2 99 SI XIII 2438. 2 SI XIV 2673. 1 S I 10.360 4 20.204 2 s II 23. 33 3 33.747 2 s III 34.83 2 43.737 2 s IV 47.30 1 57.60 2 S V 72.68 2 24 3.31 6 s VI 68.05 1 258.68 6 s VII 280.01 6 342.45 2 s VIII 3 28.33 5 352.24 2 s IX 379. 1 4 402.8 2 s X 447. 1 3 46S.0 2 S XI 504.7 2 551. 2 2 s XII 565. 1 621. 2 s XIII 652. 2 s XIV 707. 1 s XV 3224. , 2 s XVI 3494. 1 Again, following Silson (1962) we make a small addition to the ionization rate allowing for high density effects cf ioniza tions out of upper levels, = 4.8x10-« I-*/2exp {-Vy/kTJ/OCiyiz {collisions)). (All. 14) Charge Exchange In order to increase the general usefulness of this program the charge exchange rates of H+ • 0 ^ 0+ + H H+ + N ^ N+ + H were included using expressions exactly as given by Field and Steigman (1971) and Steigman, et al. (1971). Since the tempera tures here are usually in excess of 10* K, the charge exchange rate is at almost constant and at its maximum. , 100 The Heating Bate All heating is due tc energy gain by photoionization, which is simply given by (All.15) The total gain is "J J (All. 16) Cjgcl in.g Bates The emission of radiation is calculated under the assump tion the medium is optically thin. The cooling due to bremsstra-hlung is (Cox and Tucker 1969). , -A8= 2.29x10-2* 1-1/2 n^/n2 (All. 17) where n» is the number density of Hydrogen. This loss mechanism dominates for temperatures in excess of 107 K. The radiative recombination energy loss rate is -A'Jj = * rCj ( V,;j., + kT) X t-j A c r- o. 07/3 + ± JUV *• O.bH Lt~''* 1 (All. 18) where 0=/X^/kT. The correction factor in brackets was derived frcm the analysis of Seatcn (19 59) for the recombination process in Hydrogen. It represents the correction to the radiative re combination rate reguired to convert it to the energy rate, ac counting for the preferential capture of slow electrons. The loss rate due to dielectronic recombinations was esti-101 mated as, -A^ =^7}<{^yjH+ 6E^)XtJ- At- . (All.19) The recombination radiation is the dominant loss mechanism for temperatures of 2x10* K and less. .The energy difference AErj is taken as the lowest energy permitted transition tc the ground state. Between 2x10* and 107 K the dominant loss mechanism is ccl-lisicnal excitation of lines. In principle a calculation of this rate reguires the collisional cross section for excitation of a particular transition as a function of incident electron energy. With the aid of the Milne relation, which relates the collision cross section tc the inverse process of photcabscrpticn, the loss rate can be approximated as (Mewe 1972) -A* = 1.7x10-3T~V2 f. "IgUrf t] (Uexpi-Altj (l)/kT)Ac X(J' (All. 20) where g is a gaunt factor, ftJ (1) is the f value for the transi tion, and 4Et'j (1) is the energy of the emitted photon. The g factor has been calculated by Mewe (1972) for many transitions and given a simple extension by Kate (1976) to cover all transi tions. They both use the same fitting function for the Gaunt factor, g= A* (By-Cy2+D) exp (y) E, <y)*Cy (All. 21) where y=AE tj'(l) /kT, and A, E, c, D are constants given by Mewe and Rato, which are listed in the accompanying table, E , is the first exponential integral. The constants are identified by a transition number (G ID), which is matched tc a transition number of all the lines used in the calculation. For the actual computation the complete line list given was reduced by taking a multiplet average over fine structure levels. In the Table 10 102 the A, B, C, and D correspond tc the constants for the fitting function. When a value of 99.0 is entered the constant becomes a simple function as given by Mewe.. TABLE 10: THE CONSTANTS FOE THE LINE GAD NT FACTCJB A E C 0 G 0. 13 -0.12 0.13 0.28 1 0.04 0.04 0.02 0. 28 2 0.20 0.06 0. 0.28 3 0.25 0.04 0. , 0. 28 4 0.27 0.03 0. 0.28 5 0.28 0.02 0. 0. 28 6 0.2 9 0.02 0. 0.28 7 0.05 -0.04 0. 0. . 8 0.05 0.01 0. 0. 9 0.02 0.02 0. 0. 28 10 0.2 0.05 0. 0.28 11 0.02 0. 0. 0. 12 0.3 0.05 0. 0.28 13 0. 0. 0. 07 0. 14 0. 0. 0. 1 0. 15 0. 0. 0.2 0. 16 0. 0. 0.2 0. 17 0, 0. 0.04 0. 18 0. 0. 0. 3 0. 19 99. 99. 99. 0. 28 20 99. 99. 99. 0.28 21 99. 99. -0. 2 0. 28 22 99. 0. 0. 0. 23 0. 13 0. 0. 0. , 24 0. 11 0. 0. 0. 25 0. 1 0. 0. 0. 26 0.09 0. 0. 0. 27 99. 99. 0. 0. 28 0.54 -0.25 0. 0. , 29 0.43 -0. 19 0. 0. 30 0.35 -0.15 0. 0. 31 0.3 -0.12 0. 0. 32 0.0 5 0.2 0. 0.28 33 0.2 0. 15 0. 0. 28 34 -0.17 0.25 0. 0.28 35 -0.04 0.2 0. 0. 28 36 -0.3 0.4 0. , 0.28 37 -0.3 0.5 0. 0.28 38 -0.2 0.3 0. 0.28 39 -0.2 0.5 0. . 0. 28 40 0. 15 0. 0. 0.28 41 0.6 0. 0. 0.28 42 0.59 0.21 0. 04 0.28 43 44 0.27 0.08 0. 0.28 45 0.33 0.05 o. 0. 28 46 10 3 0.36 0.04 0. 0. 28 47 0.37 0.03 0. 0. 28 48 0.38 0.03 0. 0. 28 49 The following table gives the line list used in the calcu lation of the radiation acceleration and the cooling rate. The lines were taken from tables compiled by Morton and Smith (1973), Morton (private communication, but mentioned in Lamers and Morton 1976), Kate (1976), Miese, et a£-. (1966), and Siese, Si £2•. (1969). In the table the line is identified by atom and ionization species, usually with a remark about the multiplet of origin, the wavelength is given in Angstroms, the f value of the transition, a number identifying which set cf constants are to be used to calculate the Gaunt factor, the atomic number and the ion species are given. TABLE 11: LINES USED FOB THE CALCULATIONS ATOM LAMDA F VALUE G ID 2 J HI 9 920.960 0. 16C5E-02 7 1 1 HI 8 923.150 0.2216E-02 7 1 1 HI 7 926.220 0. 3 183E-02 7 1 1 HI 6 930.740 0.4814E-02 7 1 1 H I 5 937.803 0.7800E-02 6 1 1 H I 4 949.743 0. 1394E-01 5 1 1 H I 3 972.537 0. 2899E-01 4 1 1 H I 2 1025.722 0.7910E-01 3 1 1 H I 1 1215.670 0.4162E+C0 1 1 1 HE I 10 507.058 0.2093E-02 13 2 1 HE I 9 507.718 0. 2748E-02 13 2 1 HE I 8 508.643 0.3991E-02 13 2 1 HE I 7 509.998 0. 5931E-02 13 2 1 HE I 6 512.098 0.8480E-02 13 2 1 HE I 5 515.617 0. 1531E-01 13 2 1 HE I 4 522.213 0.3017E-01 13 2 1 HE I 3 537.030 0. 7342E-01 11 2 1 HE I 2 584.334 0.2763E+00 10 2 1 HE II 10 229.736 0. 1201E-02 7 2 2 HE II 9 230,139 0. 1605E-02 7 2 2 HE II 8 230.686 0.2216E-02 7 2 2 BE II 7 231.454 0.3183E-02 7 2 2 HE II 6 232.584 0. 4814E-02 7 2 2 HE II 5 234.34 7 C.7799E-02 6 2 2 HE II 4 237.331 0. 1394E-01 c ~j 2 2 HE II 3 243,027 0.2899E-01 4 2 2 104 BE II 2 256.317 C.7 912E-01 3 2 2 HE II 1 303.786 0.4162E+00 1 2 2 C I 31AUTO 945. 191 0.2730E+00 42 6 1 C I 31AUTO 945.338 0.2730E+00 42 6 1 C I 31 AUTO 945.579 0.2720E+00 42 6 1 C I 9 1260.736 0.3790E-01 41 6 1 C I 9 1260.927 0. 1260E-01 41 6 1 C I 9 1260.SS6 0.9480E-02 41 6 1 C I 9 1261. 122 0. 1580E-01 41 6 1 C I 9 1261.426 0.9480E-02 41 6 1 C I 9 1261.552 0. 2840E-01 41 6 1 C I 7 1277.245 C.8S70E-01 41 6 1 C I 7 1277.282 0.6730E-01 41 6 1 C I 7 1277.513 0.2240E-01 41 6 1 C I 1277.550 0.7S30E-01 41 6 1 C I 7 1277.723 0.1350E-01 41 6 1 C I 7 1277.95 4 0. 8S70E-C3 41 6 1 C I 6 1279.229 0.3810E-02 41 6 1 C I 5 1279.890 C. 6400E-02 41 6 1 C I 5 1280.135 0.2020E-01 41 6 1 C I 5 1280.333 0. 1510E-01 41 6 1 C I 5 1280.404 0.5040E-02 41 6 1 C I 5 1280.597 0.6720E-02 41 6 1 C I 5 1280.847 C.5040E-02 41 6 1 C I 4 1328.833 0. 3920E-01 42 6 1 C I 4 1329. C86 0. 1310E-01 42 6 1 C I 4 1329. 100 0. 1630E-01 42 6 1 C I 4 1329.123 C.S800E-02 4 2 6 1 C I 4 1329.578 0. 2940E-G1 42 6 1 C I 4 1329.600 C.9800E-02 42 6 1 C I 3 1560.310 C. 6 100E-01 42 6 1 C I 3 1560.683 C.6C80E-01 4 2 6 1 C I 3 1560.708 0. 2020E-01 42 6 1 C I 3 1561.341 0.1210E-01 42 6 1 C I 3 1561.367 0.8100E-03 42 6 1 C I 3 1561. 438 C.6800E-01 42 6 1 C I 2 1656.266 0.5660E-01 41 6 1 C I 2 1656.928 0.1360E+00 41 6 1 C I 2 1657.008 0.1020E+00 41 6 1 C I 2 1657.380 C.3400E-01 41 6 1 C I 2 1657.907 0. 45 30E-C1 41 6 1 C I 2 1658.122 0.3390E-01 41 6 1 CII B1 43. 200 0.3800E+00 42 6 2 CII 10 687^050 0.2700E+00 41 6 2 CII 10 687.350 0.2300E+00 41 6 2 CII 9 858.090 0.4600E-01 41 6 2 CII 9 858.550 0. 4600E-G1 41 6 2 CII 3 903.62 0 0.1700E+00 42 6 2 CII 3 903.960 0.3400E+00 42 6 2 CII 3 904.140 0.43COE+00 42 6 2 CII 3 904.480 0.8400E-01 42 6 2 C II 2 1036.337 0.1250E+00 42 6 2 C II 2 1037.018 0. 1250E+C0 42 6 2 C II 1 1334.532 0.1180E+00 4 2 6 2 C II 1 1335.662 0. 1180E-01 42 6 2 C II 1 1335.708 0.1060E+00 4 2 6 2 105 CI 11 BE1 42.510 0.5660E+00 42 6 3 C III 3.09 270.324 0.3287E-02 41 6 3 C III 3.08 274.051 0.3378E-02 41 6 3 C III 3.07 280.043 0.3527E-02 41 6 3 C III 3.03 291.326 0.3817E-02 41 6 3 C III 3 310.170 0.16C1E-01 41 6 3 C III 2.03 322. 574 0. 4680E-02 41 6 3 C III 2 386.203 0.2549E+00 41 6 3 C III 1 977.026 0.6740E+00 42 6 3 CIV 4 222.790 C.2630E-01 41 6 4 C IV 3 244. 907 0. 1S87E-01 22 6 4 C IV 3 244.907 0.3975E-01 22 6 4 C IV 2 312.422 0. 1335E + 00 21 6 4 C IV 2 312.453 0.6673E-01 2 1 6 4 C IV 1 1548.202 0.1940E+00 20 6 4 C IV 1 1550.774 0.S700E-01 20 6 4 CV HE1 32.800 C.2800E-01 13 6 5 CV HE2 33.430 C.5600E-01 13 6 5 CV HE3 34.970 0.1460E+00 11 6 5 CV HE4 40.270 0.6940E+00 9 6 5 CVI H5 26,000 0. 8000E-02 48 6 6 CVI H4 26.400 0.14 00E-01 47 6 6 CVI H3 27.000 0.2900E-01 46 6 6 CVI H2 28.500 0.7900E-01 45 6 6 CVI H1 33.700 0.4160E+00 43 6 6 N I 2 1134.165 0.1340E-01 42 7 1 N I 2 1134.415 0. 2680E-G1 42 7 1 N I 2 1134.980 0.4020E-01 42 7 1 N I 1 1199.549 0. 1330E + 00 41 7 1 N I 1 1200.224 0.8850E-01 41 7 1 N I 1 1200.711 0. 4420E-01 41 7 1 Nil 1310 529.680 0.820OE-01 41 7 2 Nil 9 533.500 0.2600E+00 41 7 2 Nil 9 533.570 0.1900E+00 4 1 7 2 Nil 9 533.640 0. 6500E-C1 41 7 2 Nil 9 533.720 0.2200E+00 41 7 2 Nil 9 533.880 0. 3900E-01 41 7 2 Nil 3 644.620 0.2300E+00 42 7 2 Nil 3 644. 820 0.2300E+00 42 7 2 Nil 3 645.160 0.2300E+00 42 7 2 Nil 7 671.010 0. 3700E-01 41 7 2 Nil 7 671.390 C.6700E-01 41 7 2 Nil 7 671.390 0.8900E-G1 41 7 2 Nil 7 671.620 0.2200E-01 41 7 2 Nil 7 671.770 0. 3000E-01 41 7 2 Nil 7 671.990 0.2200E-01 41 7 2 N II 2 915.612 0. 1490E+00 42 7 2 N II 2 915.962 0.4950E-01 4 2 7 2 N II 2 916.012 0.6190E-01 42 7 2 N II 2 916.020 0.3710E-01 42 7 2 N II 2 916.701 0. 1110E + 00 42 7 2 N II 2 916.710 0.3710E-01 42 7 2 N II 1 1083.990 0.1010E+00 42 7 2 N II 1 1084.562 0.2520E-01 42 7 2 N II 1 1084.580 0.7550E-01 42 7 2 N II 1 1085.529 0.1010E-02 42 7 2 106 H II 1 1085.546 0. 1510E-01 4 2 7 2 N II 1 1085.701 0. 8450E-01 42 7 2 N III AUTO 246.206 0. 1515E-02 41 7 3 N III AUTO 246.249 0. 1363E-02 41 7 3 N III AUTO 246.311 0. 15 15E-03 41 7 3 N III 7.25 262.184 0.1718E-02 41 7 3 N III 7.25 262. 233 0. 15 46E-02 41 7 3 N III 7. 25 262.289 0.1718E-03 41 7 3 N III 7. 15 268.347 0. 1801E-02 41 7 3 N III 7.15 268.473 0.1800E-03 41 7 3 N III 7. 15 268.473 0. 1620E-02 41 7 3 N III 7.12 270.073 0.1824E-02 41 7 3 M III 7. 12 270.200 0. 1823E-03 41 7 3 H III 7. 12 270.201 0. 1641E-02 41 7 3 K III 7. 10 272.523 0. 18 57E-02 41 7 3 N III 7.10 272.653 0.1857E-03 41 7 3 N III 7. 10 272.654 0.1671E-02 41 7 3 N III 7.08 276.193 0.1908E-02 41 7 3 K III 7.08 276.326 0. 19C7E-03 41 7 3 13 III 7.08 276.326 0. 1716E-02 41 7 3 N III 7.07 278.436 0.3878E-03 41 7 3 N III 7.07 278.572 0. 3876E-03 41 7 3 8 III 7.06 282.070 0. 2481E-01 41 7 3 N III 7.06 282.209 0.2480E-02 41 7 3 » III 7.06 282.209 0. 2232E-01 41 7 3 N III 7.05 285.855 0.4088E-03 41 7 3 N III 7.05 286.000 0.4086E-03 41 7 3 N III 7.04 292.447 0.4666E-01 41 7 3 M III 7.04 292.595 0. 4149E-G1 41 7 3 N III 7.04 292.596 0.4655E-02 41 7 3 N III 7.03 299.661 0.4492E-03 41 7 3 N III 7. 03 299.818 0.4490E-03 41 7 3 N III 7.02 305.761 0. 4677E-C3 41 7 3 N III 7.02 305.920 0.4674E-03 41 7 3 K III 7.01 311.550 0.2427E-02 41 7 3 N III 7.01 311.636 0.2183E-02 41 7 3 N III 7.01 311.721 0. 2426E-G3 41 7 3 N III 7 314.877 0. 1091E-01 41 7 3 III 6 323.436 0. 5235E-03 41 7 3 N III 6 323.493 0.1047E-02 41 7 3 S III 6 323.6 20 0. 13C8E-02 41 7 3 N III 6 323.675 0.2615E-03 41 7 3 N III 5.01 332.140 0.7063E-02 41 7 3 N III 5.01 332.333 0.7G46E-02 41 7 3 N III 5 374.204 0.2918E+00 41 7 3 N III 5 374.441 0.2625E*00 41 7 3 N III 5 374.449 0.2916E-01 41 7 3 N III 4 451.869 0.2381E-01 41 7 3 N III 4 452.226 0. 2379E-01 41 7 3 N III 3 684.996 0. 1207E+00 42 7 3 N III 3 685.513 0.2412E+00 42 7 3 N III 3 685.816 0.3013E*00 42 7 3 N III 3 686.335 0.6022E-01 42 7 3 N III 2 763.340 0.5664E-01 42 7 3 N III 2 764.357 0.5657E-01 42 7 3 N III 1 989.790 0.1C70E+00 41 7 3 107 N III 1 991.514 0. 1060E-01 41 7 3 N III 1 991.579 0.S580E-01 41 7 3 N IV 2 247.205 0.5497E+00 41 7 4 N IV 1 765.148 0.5451E+00 42 7 4 NV LI1 148.000 0. 3000E-01 22 7 5 NV 3 162.560 G.6690E-01 22 7 5 KV LI2 162.560 0. 67G0E-01 22 7 5 NV 2 209.270 0.1570E+00 21 7 5 KV LI5 209. 280 0.2360E+00 21 7 5 NV 2 209.330 0.7840E-01 21 7 5 N V 1 1238.821 0.1520E+00 41 7 5 N V 1 1242.804 0.7570E-01 41 7 5 NVI KE1 23.300 0. 2800E-01 13 7 6 NVI HE2 23.770 0.5600E-01 13 7 6 NVI HE3 24.900 0. 1460E+00 11 7 6 NVI HE4 28.79 0 0.6940E+00 9 7 6 NVII H5 19. 100 0. 8000E-02 48 7 7 NVII H4 19.400 0. 1400E-01 47 7 7 NVII H3 19.800 0. 2900E-01 46 7 7 NVII H2 20.900 C.7900E-01 45 7 7 NVII H1 24.800 0.4160E+00 4 3 7 7 OI J39 811.370 C.7700E-02 41 8 1 CI H5 878.450 0.3700E-01 41 8 1 0 I 5 988.581 0.5100E-03 42 8 1 0 I 5 988.655 0. 7640E-02 42 8 1 0 I 5 988.773 0.4280E-01 42 8 1 0 I 5 990.127 0. 1270E-01 42 8 1 0 I 5 990.204 0.3810E-01 42 8 1 0 I 5 990.801 0.5C80E-01 42 8 1 0 I 4 1025.762 0.62 00E-01 42 8 1 0 I 4 1025.762 0. 1110E-01 42 8 1 0 I 4 1025.762 0.7380E-03 42 8 1 0 I 4 1027.431 0.5530E-01 42 8 1 0 I 4 1027.431 0.1840E-01 42 8 1 0 I 4 1028. 157 0. 7360E-01 42 8 1 0 I 2 1302.169 0.4860E-01 42 8 1 0 I 2 1304.858 0. 4850E-C1 42 8 1 0 I 2 1306.02 9 0.4850E-01 42 8 1 CII 10 429.910 0. 5400E-01 41 8 2 on 10 430.040 0. 1100E+00 41 8 2 on 10 430.170 0.16C0E+00 41 8 2 on 2 539.080 0.5600E-01 41 8 2 CII 2 539.540 0. 3700E-01 41 8 2 on 2 539.850 0.1900E-01 41 8 2 on 1 832.750 0.7000E-01 42 8 2 on 1 833.320 0.15G0E+00 42 8 2 CII 1 834.460 0.2100E+00 42 8 2 0 III 228.834 0.8128E-02 41 8 3 0 III 228.893 0.7SS6E-02 41 8 3 0 III 228. S88 0.7962E-02 41 8 3 0 III 240.979 0. 2523E-01 41 8 3 0 III 241.000 0.3366E-03 41 8 3 0 III 241.000 0. 84C9E-02 41 8 3 0 III 241.000 0.3364E-01 41 8 3 0 III 241.037 0. 2825E-01 41 8 3 0 III 248.468 0.1533E-01 41 8 3 108 0 III 248.538 0.6129E-01 41 8 3 0 III 248.574 0.4596E-01 41 8 3 0 III 248.618 0. 5147E-01 41 8 3 0 III 248.693 0.9188E-02 41 8 3 G III 255.000 0. 15 46E-G2 41 8 3 0 III 255.044 0. 1932E-02 41 8 3 0 III 255. 113 0. 4636E-02 41 8 3 0 III 255.158 0.3476E-02 41 8 3 G III 255. 188 0. 1159E-02 41 8 3 0 III 255.302 0. 1158E-02 41 8 3 c III 262.000 0. 5868E-01 41 8 3 0 III 262.700 0.1951E-01 41 8 3 G III 262.700 0. 1463E-01 41 8 3 0 III 262.729 0. 2438E-01 41 8 3 0 III 262.882 0.4386E-01 41 8 3 0 III 262.900 0.1462E-01 41 8 3 G III 263.692 0. 1549E+G0 41 8 3 0 III 263.728 0. 10C7E+00 41 8 3 0 III 263.768 0. 2254E-01 41 8 3 0 III 263.818 0.5768E-02 41 8 3 c III 263.818 0.1002E+00 41 8 3 0 III 263.903 0.1384E-03 41 8 3 c III 264. 257 0.2291E-01 41 8 3 0 III 264.317 C.7636E-02 41 8 3 G III 264. 329 0. 5793E-02 41 8 3 0 III 264.338 0.9613E-02 41 8 3 0 III 264.471 0.5768E-02 41 8 3 0 III 264.480 0. 1742E-01 41 8 3 0 III 266.843 0.S355E-02 41 8 3 0 III 266. S67 0.6310E-01 41 8 3 c III 266.985 0. 47C8E-01 41 8 3 0 III 267.030 0.5141E-01 4 1 8 3 G III 267.050 0. 1559E-G1 41 8 3 0 III 267. 188 0.6325E-03 41 8 3 0 III 275.281 0. 3020E-01 41 8 3 0 III 275.336 0.2971E-01 41 8 3 0 III 275.513 0. 2958E-C1 41 8 3 0 III 280.116 0.5406E-02 41 8 3 C III 280.234 0. 13 18E-01 41 8 3 0 III 280.265 0.9573E-02 41 8 3 c III 280.328 0. 3257E-02 41 8 3 0 III 280.412 0. 4394E-02 41 8 3 0 III 280.483 0.3 244E-02 41 8 3 0 III 6 303.411 0.1383E+00 41 8 3 C III 6 303.460 0.4611E-01 41 8 3 0 III 6 303.515 0.3457E-01 41 8 3 G III 6 303.621 0. 5761E-01 41 8 3 0 III 6 303.693 0.3455E-01 41 8 3 0 III 5 303.769 0. 3521E + 00 41 8 3 0 III 6 303.799 0.1036E+00 41 8 3 G III 5 305.596 0.4167E+00 41 8 3 0 III 5 305.656 0.3125E+00 4 1 8 3 0 III 5 305.703 0.1041E+00 41 8 3 0 III 5 305.836 0.6245E-01 41 8 3 G III 5 305.879 0.4166E-02 41 8 3 0 III 308.306 0.1995E-02 41 8 3 109 0 III 4 373.605 0. 2573E-01 41 8 3 0 III 4 374.005 G.6171E-01 41 8 3 0 III 4 374.075 0. 4627E-01 41 8 3 0 III 4 374.165 0. 1542E-01 41 8 3 0 III 4 374.331 0. 2055E-01 41 8 3 0 III 4 374.436 0.1541E-01 41 8 3 0 III 3 507. 391 0.1387E+00 42 8 3 0 III 3 507.683 0.1387E+00 42 8 3 0 III 3 508.182 0. 1385E+00 42 8 3 0 III 2 702.332 0. 14C4E+00 42 8 3 0 III 2 70 2. 822 0. 4676E-01 42 8 3 0 III 2 702.891 0.3507E-01 42 8 3 0 III 2 702.899 0.5844E-01 42 8 3 0 III 2 703.645 0.3502E-01 42 8 3 C III 2 703.850 0. 1051E+00 42 8 3 0 III 1 832.927 0. 1049E+00 42 8 3 G III 1 833.701 0.2621E-01 42 8 3 0 III 1 833.742 0.7863E-01 42 8 3 C III 1 835.055 0. 1048E-02 42 8 3 0 III 1 835.096 0.1570E-01 42 8 3 C III 1 835.292 0. 8791E-01 42 8 3 01V B2 195.860 C.96 00E-01 41 8 4 01V E3 203.000 0. 1730E+00 41 8 4 0 IV 5 238.360 0.4977E+01 41 8 4 G IV 5 238.571 0. 4 47 6E + 01 41 8 4 0 IV 5 238.580 G.4973E+00 41 8 4 0 IV 4 279.631 0.3560E-01 41 8 4 0 IV 4 279.933 0.3556E-01 41 8 4 0 IV 3 553.330 0.94 32E-01 42 8 4 0 IV 3 554.075 0.1884E+00 42 8 4 0 IV 3 554.514 0.2353E+G0 42 8 4 0 IV 3 555.261 0.4700E-01 42 8 4 0 IV 2 608.398 0. 7062E-01 42 8 4 0 IV 2 609.829 0.7046E-01 42 8 4 0 IV 1 787.711 0. S345E-01 42 8 4 0 IV 1 7 90.109 0.9317E-02 42 8 4 0 IV 1 790. 199 0. 8384E-01 42 8 4 ov 2 172.160 G.59C0E+00 42 8 5 0 V 1 629.730 0.4405E+00 42 8 5 OVI LI1 104.810 0.3200E-01 22 8 6 OVI LI2 115. 800 0.7300E-01 22 8 6 OVI 2 150.080 0. 1750E+00 21 8 6 OVI 2 150. 120 0. 6740E-01 21 8 6 0 VI 1 1031.945 0. 1300E+00 20 8 6 0 VI 1 10 37.627 0. 6480E-01 20 8 6 OVI I HE 1 17. 420 0.2800E-01 13 8 7 OVI I HE2 17.770 0. 5600E-01 13 8 7 OVII HE3 18.630 0.1460E+00 11 8 7 OVII HE4 21.600 0.6940E+00 c 8 7 OVIII H5 14.600 0.80C0E-02 48 8 8 OVII I H4 14.820 0. 1400E-01 47 8 8 OVIII H3 15.200 0.2900E-01 46 8 8 OVIII H2 16.000 0. 7900 E-01 45 8 8 OVIII H1 19.000 0.4160E+00 43 8 8 Nil 2 735.890 0. 1620E+C0 35 10 1 NEI 1 743.700 0.1180E-01 35 10 1 110 NE II 324. 567 0. 1066E-02 41 10 2 HE II 324.570 0.1091E-01 41 10 2 NE II 325.393 0. 1256E-01 41 10 2 NE II 326. 519 0.4988E-02 41 10 2 NE II 326.542 0. 3962E-01 41 10 2 NE II 326.787 0.2443E-01 41 10 2 NE II 327.250 0. 1018E-01 41 10 2 NE II 327.262 0.5104E-01 41 10 2 NE II 327.355 0. 4666E-01 41 10 2 NE II 327.626 0.2393E-Q1 41 10 2 NE II 328.090 0. 4355E-C1 41 10 2 NE II 328. 102 0.5OC0E+0O 41 10 2 NE II 329.773 0. 5177E-02 41 10 2 NE II 330.214 0.2177E-02 41 10 2 NE II 330.626 0.2685E-01 41 10 2 NE II 330.658 0.9726E-02 41 10 2 NE II 330.790 0. 3784E-G1 41 10 2 NE II 331.069 0.8491E-02 41 10 2 NE II 331.515 0.2393E-01 41 10 2 NE II 352.24 7 0.2805E-02 41 10 2 NE II 352.956 0. 1374E-G1 41 10 2 NE II 353.215 0.1094E-01 4 1 10 2 NE II 353.935 0. 5 236E-02 41 10 2 NE II 354.S62 Q.1066E-01 41 10 2 NE II 355.454 0.5469E-02 41 10 2 NE II 355.948 0. 4775E-01 41 10 2 NE II 356.441 0. 2084E-01 41 10 2 NE II 356.541 0.7726E-02 41 10 2 NE II 356.800 0. 30C6E-01 41 10 2 NE II 357.536 0.2685E-01 41 10 2 NE II 361.433 0. 1577E-01 41 10 2 NE II 362.455 0.1694E-01 41 10 2 NE II 4 405.846 0. 1251E-01 41 10 2 NE II 4 405.854 0.1126E+00 41 10 2 NE II 4 407. 138 0.1247E+00 41 10 2 NE II 3 445.040 0.1723E-01 4 1 10 2 NE II 3 446.226 0. 8590E-01 41 10 2 NE II 3 446.590 0.6867E-01 41 10 2 NE II 3 447.815 0. 3424E-01 41 10 2 NE II 1 460.728 0.3300E+00 42 10 2 NE II 1 462.391 0.3288E+G0 42 10 2 NEIIIM13 227.400 0.5500E-01 41 10 3 NEIIIM11 227.620 0.1200E+00 41 10 3 NEIIIM11 229.060 0.9600E-01 41 10 3 NE III 5 251.120 0.1858E+01 41 10 3 NE III 5 251.129 0.3317E+00 41 10 3 NE III 5 251. 134 0.2213E-01 41 10 3 NE III 5 251.540 0. 1656E+01 41 10 3 NE III 5 251.549 0.5519E+00 41 10 3 NE III 5 251.720 0.2206E+01 41 10 3 NE III 4 267.047 0. 8576E-02 41 10 3 NE III 4 267.070 0.2573E-01 4 1 10 3 NE III 4 267.500 0. 1142E-01 41 10 3 NE III 4 267.512 0.8561E-02 41 10 3 NE III 4 267.530 0. 1427E-01 41 10 3 NE III 4 267.710 0. 3422E-01 41 10 3 111 NE III 3 283. 125 0. 5632E-03 41 10 3 NE III 3 283.150 C.8440E-02 41 10 3 NE III 3 283.170 0. 4726E-01 41 10 3 NE III 3 283.647 0.1404E-01 41 10 3 NE III 3 283.660 0.4212E-01 41 10 3 NE III 3 283.870 0.5612E-01 41 10 3 NE III 2 313.050 0. 3977E-01 41 10 3 NE III 2 313.680 0.3969E-01 41 10 3 NE III 2 313.920 0.3966E-01 41 10 3 NE III 1 488.100 0.4108E-01 42 10 3 NE III 1 488.870 0. 5469E-01 42 10 3 NE III 1 489.500 0.1229E*00 42 10 3 NE III 1 489.640 0. 4095E-01 42 10 3 NE III 1 490.310 0.1636E+00 42 10 3 NE III 1 491.050 0.6806E-01 42 10 3 NEIV N7 148.800 0.6100E+00 41 10 4 NEIV Ni 172.600 0.5400E+00 41 10 4 NEIV M7 208.€3 0 C.95C0E-01 41 10 4 NE IV 1 541.127 0. 2S58E-G1 42 10 4 NE IV 1 54 2.073 0.59C5E-01 42 10 4 NE IV 1 543.891 0. 8829E-01 42 10 4 NEV C9 118.80 0 G.2500E+00 41 10 5 KEV M8 142.610 0. 2000E+00 41 10 5 NEV m 143.320 0.61G0E+0O 4 1 10 5 NEV C5 173.900 0.8600E-01 41 10 5 NE V 3 357.S50 0. 1795E-01 42 10 5 NE V 3 358.480 0. 1792E-01 42 10 5 NE V 3 359.390 0.1788E-01 4 2 10 5 NE V 2 480.410 0. 1522E*00 42 10 5 NE V 2 4 81.280 0.5G63E-01 42 10 5 NE V 2 481.360 0. 63 27E-01 42 10 5 NE V 2 481.367 0.3796E-01 42 10 5 NE V 2 482.990 0. 1135E + 00 42 10 5 NE V 2 482.990 0.3784E-01 42 10 5 NE V 1 568.420 0. 9 259E-C1 42 10 5 NE V 1 569.760 0. 2309E-01 42 10 5 NE V 1 56 9.830 0.6927E-01 42 10 5 NE V 1 572.03 0 C.9994E-03 42 10 5 NE V 1 572.110 0. 1380E-01 42 10 5 NE V 1 572.340 C.7724E-01 42 10 5 NEVI E1 14.100 C.49C0E+00 41 10 6 N EVI B2 98.000 0.1020E+00 41 10 6 SEVI E3 111. 100 0. 1750E+C0 41 10 6 NEVI M9 122.620 C.5400E+00 4 1 10 6 NEVI M8 138.550 0.2 900E-01 41 10 6 NE VI 399. 820 0.4909E-01 42 10 6 NE VI 403.260 0. 2434E-01 42 10 6 NE VI 410.140 0.S787E-01 42 10 6 NE VI 410.930 0.1194E+00 42 10 6 NE VI 433.180 0.5273E-01 42 10 6 NE VI 435.650 0.5243E-01 42 10 6 NE VI 558.590 0.8388E-01 42 10 6 NE VI 562.710 0. 6 3 28E-02 42 10 6 NE VI 562.800 0.7493E-01 42 10 6 NEVII BE1 13.920 0.67C0E+00 41 10 7 NE VII 465.221 0. 3748E+00 42 10 7 112 NEVIII LI1 60.810 0. 3300E-01 22 10 8 NEVIII 2 88.130 0. 2980E+00 4 1 10 8 NEVIII LI2 98.000 0. 8000E-01 22 10 8 NEVIII 1 770.400 0.1020E+00 42 10 8 NEVIII 1 780.320 0. 5020E-01 42 10 8 NE IX HE1 10.800 0.2800E-01 13 10 9 NE IX HE2 11.000 0. 5600E-01 13 10 9 NE IX HE3 11.560 0.1490E+00 1 1 10 9 NE IX HE4 13.440 0.7230E+00 9 10 9 NE X 6 9.370 0.8000E-02 48 10 10 NE X 5 9. 490 0. 1400E-01 47 10 10 NE X 4 9.720 0.2900E-01 46 10 10 NE X 3 10.250 0.7900E-01 45 10 10 NE X 2 12.150 0.4160E+00 43 10 10 MG I 1827.940 0.5260E-01 41 12 1 MG I 2 2025.824 0.1610E+00 4 1 12 1 MG I 1 2852.127 0.1900E+G1 42 12 1 MG II 1025.968 0. 1480E-02 41 12 2 MG II 1026. 113 0.7 400E-03 41 12 2 MG II 1239.925 0.9680E-03 41 12 2 MG II 1240.395 0.4840E-03 41 12 2 MG II 1 2795.528 0.5920E+00 42 12 2 MG II 1 2802.704 0.2950E+00 42 12 2 MGIII NE1 171.500 0.1000E+00 34 12 3 MGIII NE2 182. 500 0.8G00E-02 36 12 3 MGIII 5 186.510 0.27COE+00 33 12 3 MGIII 4 187.190 0. 1600E + 00 33 12 3 MGIII 3 188.530 0.4000E-02 33 12 3 MG III 231.730 0.2101E+00 35 12 3 MG III 1 234.258 0.1111E-01 35 12 3 MGIV F1 120.000 0.2500E+00 41 12 4 MGIV F5 130.000 0.1340E+00 41 12 4 MGIV F2 147.000 0. 1500E + 01 41 12 4 MGIV F3 181.000 0.3200E*00 41 12 4 MG IV 320.994 0.1348E+00 42 12 4 MG IV 323.307 0.1339E+00 42 12 4 MGV 07 103.900 0.1200E+00 41 12 5 MGV 02 114.030 0.1800E+00 41 12 5 MGV 01 121.600 0.3000E+00 41 12 5 MGV 03 132.500 0.1340E+00 41 12 5 MGV 05 137.800 0. 4800E-01 41 12 5 MGV 04 146.500 0.2900E-01 41 12 5 MG V 351.089 0. 5643E-01 42 12 5 MG V 352.202 C.7500E-01 42 12 5 MG V 353.094 0. 1683E+00 42 12 5 MG V 353.300 0.56C7E-01 42 12 5 MG V 354.223 0.2237E+00 42 12 5 MG V 355.326 0.9293E-01 4 2 12 5 MGVI N7 80.100 0.27C0E+0O 41 12 6 MGVI N1 95.500 0.5000E«-00 41 12 6 MGVI N3 111.600 0.7200E-01 41 12 6 MG VI 399.289 0.4383E-01 42 12 6 MG VI 400.676 0. 87 35E-01 42 12 6 MG VI 403.315 0. 1302E+00 42 12 6 MGVII C9 68.100 0.2400E+00 41 12 7 AG VII C1 77.100 0.1000E+00 41 12 7 113 MGVII C2 78.400 0.4500E-01 41 12 7 MGVII C3 83.96 0 0.2100E+00 41 12 7 MGVII C4 84.020 0.61G0E+00 41 12 7 MGVII C5 95.300 0.5900E-01 41 12 7 MG VII 276.145 0.1201E+00 4 2 12 7 MG VII 277.007 0.1197E+00 42 12 7 MG VII 278.406 0. 1191E+00 42 12 7 MG VII 363.770 0. 1115E+00 42 12 7 MG VII 365.230 0. 4628E-01 42 12 7 MG VII 365.267 0.2777E-01 42 12 7 MG VII 365.270 0. 37C2E-01 42 12 7 MG VII 367.679 0.8276E-01 42 12 7 MG VII 367.701 0. 27 58E-01 42 12 7 MG VII 429.134 0.1040E+00 4 2 12 7 MG VII 431.220 0.2588E-C1 42 12 7 MG VII 431.318 0.7762E-01 42 12 7 MG VII 434.615 0. 1028E-02 42 12 7 MG VII 434.710 0.1540E-01 42 12 7 MG VII 434.923 0. 8621E-01 42 12 7 MGVIII B1 9.470 0.5100E+00 41 12 8 MGVIII B3 64. 500 0. 1670E+00 41 12 8 MGVIII B2 69.000 0.1070E+00 41 12 8 MGVIII11 74.850 0.6100E+00 41 12 8 MGVIII11 75.030 0.5500E+00 41 12 8 MGVIII1 1 75.040 0.6100E-01 41 12 8 MGVIII10 82.590 0.2420E-01 41 12 8 MGVIII10 82.820 0. 2400E-01 41 12 8 MGVIII 3 311.780 0.6800E-01 42 12 8 MGVIII 3 313.730 0.1400E+00 42 12 8 MGVIII 3 315.020 0.1700E+00 42 12 8 MGVIII 3 317.010 0. 340OE-01 4 2 12 • 8 MGVIII 2 335.250 0.4500E-01 42 12 8 MGVIII 2 339.010 0. 4500E-01 42 12 8 MGVIII 1 430.470 0.8800E-01 42 12 8 MGVIII 1 436.680 0.87C0E-02 42 12 8 MGVIII 1 436.730 0.7800E-01 42 12 8 MGIX BE1 9. 380 0.7000E+00 42 12 9 MGIX 6 62.750 G.5800E+00 41 12 9 MGIX 2 368.070 0.3 140E+00 4 2 12 9 MGX LI1 41.000 0.3500E-01 22 12 10 MGX 112 44. 050 0.8500E-01 22 12 10 MGX LI5 57.89C 0.3200E+00 21 12 10 MG XI HE4 7.310 0. 2770E-01 13 12 11 MG XI HE3 7.470 0.5690E-01 13 12 11 MG XI HE2 7.8 50 0.1520E+00 13 12 11 MG XI HE 1 9.160 C.7450E+00 13 12 11 MGXII 6 6. 510 0.8000E-02 48 12 12 MGXII 5 6. 590 0.1400E-01 47 12 12 MGXII 4 6.750 0. 2900E-01 46 12 12 MGXII 3 7. 120 G.7900E-01 45 12 12 MGXII 2 8. 440 0.4160E+00 43 12 12 SI I 41.12AO 1255.276 0.2200E+00 41 14 1 SI I 41.12AU 1256.490 0.2200E+00 41 14 1 SI I 41. 12AU 1258.795 0.2200E+00 41 14 1 SI I 10 1845.520 0. 1520E+G0 41 14 1 SI I 10 1847.473 0. 1140E+00 41 14 1 114 SI I 10 1848.150 0. 3800E-01 41 14 1 SI I 10 1850.672 0. 1280E+00 4 1 14 1 SI I 10 1852.472 0. 2 280E-01 41 14 1 SI I 10 1853.152 0.1520E-02 4 1 14 1 SI I 7 1977.579 0.2110E-G1 41 14 1 SI I 7 1979.206 0.1G40E-01 41 14 1 SI I 7 1980.618 0. 7770E-02 41 14 1 SI I 7 1983.232 0.1290E-01 41 14 1 SI I 7 1986.364 0.7750E-02 41 14 1 SI I 7 1988.994 0.2320E-01 41 14 1 SI I 3 2207.S78 0. 5890E-01 42 14 1 SI I 3 2210.894 0.4420E-01 42 14 1 SI I 3 2211. 744 0. 1470E-01 42 14 1 SI I 3 2216.669 0.4930E-01 42 14 1 SI I 3 2218.057 0.8800E-02 42 14 1 SI I 3 2218.915 0.5870E-03 42 14 1 SI I 1 2506.897 0. 6 520E-G1 41 14 1 SI I 1 2514.316 0.1560E*00 41 14 1 SI I 1 2516. 112 0.1170E+00 41 14 1 SI I 1 2519.202 G.3890E-01 41 14 1 SI I 1 2524. 108 O.E180E-01 41 14 1 SI I 1 2528.509 0.3880E-01 4 1 14 1 SI II 6 989.867 0.2440E+00 41 14 2 SI II 6 992,675 0.2190E+00 41 14 2 SI II 6 992.690 0. 24 3OE-01 41 14 2 SI II 5,01 1020.699 0.4820E-01 41 14 2 SI II 5.01 1023.693 0. 4800E-01 41 14 2 SI II 5 1190.418 C.6500E+00 42 14 2 SI II 5 1193.284 0. 1300E+01 42 14 2 SI II 5 1194.496 0.1620E+01 42 14 2 SI II 5 1197.389 0.3230E+00 42 14 2 SI II 4 1260.418 O.S590E+00 42 14 2 SI II 4 1264. 730 0.6600E+00 42 14 2 SI II 4 1265.023 0.9560E-01 42 14 2 SI II 3 1304,369 0. 1470E + 00 42 14 2 SI II 3 1309.274 0.1470E+00 42 14 2 SI II 2 1526.719 0. 7640E-01 41 14 2 SI II 2 1533.445 0.7600E-01 41 14 2 SI II 1 1808.003 0. 3710E-02 42 14 2 SI II 1 1816.921 0.3320E-02 42 14 2 SI II 1 1817.445 0.3690E-G3 42 14 2 SI III 11 566.610 0.4600E-01 41 14 3 SI III 2 1206.510 0. 1660E + 01 42 14 3 SI IV 2.02 327.13 7 0.4886E-02 41 14 4 SI IV 2.02 327. 181 0. 2449E-02 41 14 4 SI IV 2.01 361.560 0.9527E-02 41 14 4 SI IV 2.01 361.659 0. 47751-02 41 14 4 SI If 2 457.818 0.2201E-01 41 14 4 SI IV 2 458. 155 0.1100E-01 41 14 4 SI IV 1 1393.755 0.5280E+00 42 14 4 SI IV 1 1402.769 0.2620E+00 42 14 4 SI V NE 1 85.200 0.2700E+00 34 14 5 SI V NE2 90.500 0. 1000E-01 36 14 5 SI V 5 96.430 0. 2000E+00 35 14 5 SI V 4 97.140 C.8400E+00 35 14 5 SI V 3 98.200 0.3800E-02 35 14 5 SI V 2 117.860 0. 1900E+00 35 14 5 SI V 1 118.970 0.2100E-01 35 14 5 SI VI F5 69.200 0. 2 100E+00 41 14 6 SI VI F1 70.000 0.2500E+00 41 14 6 SI VI F2 83.000 0.1500E+01 41 14 6 SI VI F3 99.400 0.9000E+00 41 14 6 SI VI 246.001 0. 1133E+00 42 14 6 SI VI 249.125 0.1119E+00 42 14 6 SI VII 07 60.800 0.1400E+00 41 14 7 SI VII 01 68.000 G.44C0E+00 41 14 7 SI VII 02 69.660 0.2100E+00 41 14 7 SI VII 03 79.500 0.2600E-01 41 14 7 SI VII 05 81.900 0. 4 300E-01 41 14 7 SI VII 04 85.600 0.2600E-01 41 14 7 SI VII 272. 641 0.3448E-01 42 14 7 SI VII 274.175 0.4571E-01 42 14 7 SI VII 275.352 0. 1024E + 00 42 14 7 SI VII 27 5.665 0.3410E-01 42 14 7 SI VII 276.839 0.1358E+00 42 14 7 SI VII 278.445 0.5627E-01 42 14 7 SI VII 1 314,310 0. 3900E-01 42 14 7 SI VII 1 316.200 0.7400E-01 42 14 7 SI VII 1 319,830 0. 1100E+00 42 14 7 SI VIII N7 50.000 0.3100E+00 41 14 8 SI VIII U3 69.600 0. 5500E- 01 41 14 8 SI IX C9 44.200 0.2300E+00 4 1 14 9 SI IX C1 52.800 0. 2000E-01 41 14 9 SI IX C3 55.100 0.23 00E+00 41 14 9 SI IX C4 55.300 G.6300E+00 41 14 9 SI IX C5 61.600 Q.5700E-01 41 14 9 SI IX 3 223.720 0. 1000E+00 42 14 9 SI IX 3 225.03 0 0.9900E-01 42 14 9 SI IX 3 227.000 0.9900E-01 42 14 9 SI IX 2 290.630 0.9200E-01 42 14 9 SI IX 2 292. 830 0. 2300E-01 42 14 9 SI IX 2 292.830 0.3000E-01 42 14 9 SI IX 2 292.830 0. 3800 E-01 42 14 9 SI IX 2 296,190 C.6800E-01 42 14 9 SI IX 2 296.190 0. 2300E-01 42 14 9 SI IX 1 341.950 0.6500E-01 42 14 9 SI IX 1 345.010 0.2100E-01 42 14 9 SI IX 1 345.100 0.6200E-01 42 14 9 SI IX 1 349.670 0. 6300E-03 42 14 9 SI IX 1 349.770 0. 1300E-01 42 14 9 SI IX 1 439.960 0.6900E-01 42 14 9 SI X B1 6. 850 0.5400E+00 42 14 10 SI X E2 39.000 0. 1100E+00 41 14 10 SI X B3 47.540 0. 1430E+00 41 14 10 SI X B9 54.900 0. 2400E-01 41 14 10 SI X 4 253.810 C.6000E-01 42 14 10 SI X 4 256.580 0. 12COE+00 42 14 10 SI X 4 258.390 0.1500E*00 42 14 10 SI X 4 261.270 0. 2900E-01 42 14 10 SI X 2 272.000 0.3700E-01 42 14 10 SI X 2 277.270 0. 3600E-01 42 14 10 SI X 1 347.430 0.7400E-01 42 14 10 SI X 1 356.070 0. 6600E-01 42 14 10 SI X 1 356.070 0.7300E-02 42 14 10 SI XI BE1 6.780 0.7200E+00 42 14 11 SI XI 2 303.580 0.2640E+00 42 14 11 SI XII LI1 28.500 0. 3700E-01 22 14 12 SI XII LI 2 31.000 C.8800E-01 22 14 12 SI XII 499.399 •0. 7 2941-01 42 14 12 SI XII 520.684 0.3498E-01 42 14 12 SI XIII HE1 5. 290 0.28COE-01 13 14 13 SI XIII HE2 5.410 0.5700E-01 13 14 13 SI XIIJ HE3 5. 680 0. 15C0E + 00 11 14 13 SI XIII HE4 6.650 C.7500E+00 9 14 13 SI XIV 6 4.780 0. 8000E-02 48 14 14 SI XIV 5 4. 840 0.1400E-01 47 14 14 SI XIV 4 4. 9 60 0. 2900E-01 46 14 14 SI XIV 3 5.230 0.7900E-01 45 14 14 SI XIV 2 6.200 0.4160E+00 43 14 14 S I 9 1295.661 0.1C80E+00 41 16 1 S I 9 1296. 174 0.3610E-01 41 16 1 s I 9 1302.344 0.6000E-01 41 16 1 s I 9 1302. £65 0. 3600E-01 41 16 1 s I 9 1303.114 0.4790E-01 41 16 1 s I 1303.420 0. 1630E-01 41 16 1 s I 9 1305.885 0.1440E+00 41 16 1 s I 1310.210 0. 1620E-01 41 16 1 s I 1313.250 0.1610E-01 41 16 s I 8 1316.570 0. 3450E-01 41 16 1 s I 8 1316.610 0.6150E-02 41 16 1 s I 8 1316.620 0.4110E-03 41 16 1 s I 8 1323.521 0. 3060E-01 41 16 1 s I 8 1323.530 0. 1020E-01 41 16 1 £ I 8 1326.635 0.4C70E-01 4 1 16 s I 6 1401. 541 0.1580E-01 41 16 1 s I 6 1409.368 0.1570E-01 41 16 1 s I 6 1412.899 0. 1570E-01 41 16 1 <; I 5 1425.065 0.1810E+00 42 16 1 s I 5 1425.229 0.3220E-01 42 16 1 s I 5 1425.240 0.2150E-02 42 16 1 s I 5 1433.328 0. 1600E + 00 42 16 1 I 5 1433.328 0.5340E-01 42 16 1 s I 5 1437.005 0.2130E+00 42 16 1 s I 3 1474.005 0.7820E-01 41 16 1 s I 3 1474.390 0. 1400E-01 41 16 1 s I 3 1474.56 9 0.9320E-03 41 16 1 s I 3 1483.036 0.6940E-G1 41 16 1 s I 3 1483.232 0.2310E-01 41 16 1 s I 3 1487.149 0.S230E-01 41 16 1 s I 2 1807.341 0.1120E+00 41 16 1 s I 2 1820.361 0.1110E+00 41 16 1 s I 2 1826.261 0.1110E+00 41 16 1 s II 1 1250.586 0.5 350E-02 42 16 2 s II 1 1253.812 0.1C70E-01 42 16 2 s II 1 1259.520 0. 1590E-G1 42 16 2 s III 484.194 0.4074E-01 41 16 3 s III 484.580 0.3111E-01 41 16 3 s III 484.892 0.8568E-02 41 16 3 117 s III 485.220 0. 3 476E-C1 41 16 3 s III 485.640 0.4179E-02 41 16 3 s III 486. 154 0. 2296E-G3 41 16 3 s III 7 677.750 0. 9644E+00 41 16 3 s III 7 678.460 0.7225E+00 41 16 3 s III 7 679.110 0.24G6E+00 41 16 3 s III 7 680.690 0.8066E+00 41 16 3 s III 7 680.950 0.1440E*00 41 16 3 s III 6 680.979 0. 5593E-01 41 16 3 s III 6 681.500 0.1341E+00 41 16 3 s III 7 681.587 0.S5S7E-02 41 16 3 s III 6 682.883 0.3346E-01 41 16 3 s III 6 683.070 0, 4 460E-01 41 16 3 s III 6 683.470 0.1003E+00 41 16 3 s III 6 685.350 0. 3334E-01 41 16 3 s III 5 698.730 0.7406E-02 41 16 3 s III 5 700.150 0.3080E-02 41 16 3 s III 5 700.184 0.1848E-02 41 16 3 s III 5 700.290 0. 2463E-G2 41 16 3 s III 5 702.780 0.5523E-02 41 16 3 s III 5 702.820 0. 1841E-02 41 16 3 s III 4 724.290 0.4677E+00 41 16 3 s III 4 7 25.852 0.47C8E+00 41 16 3 s III 1 1190.206 0.2240E-01 42 16 3 s III 1 1194.061 0. 1670E-01 42 16 3 s III 1 1194.457 G.5570E-02 42 16 3 s III 1 1200.970 0. 1860E-01 42 16 3 s III 1 1201.730 0.3320E-02 42 16 3 s III 1 1202. 132 0. 2220E-03 4 2 16 3 s IV 5 551.170 0. S5C7E-01 41 16 4 s Iv 5 554.070 0. 9 4571-01 41 16 4 s IV 4 657.34 0 0.9106E+00 41 16 4 s IV 4 661.420 0.81451+00 41 16 4 c IV 4 661.471 0.9049E-01 41 16 4 s IV 3 744.920 0. 3155E+C0 41 16 4 £ IV 3 748.400 0.6295E+00 41 16 4 s IV 3 750.230 0.8278E+00 41 16 4 s IV 3 753.760 0. 1730E+00 4 1 16 4 s IV 2 809.690 0.1514E+00 41 16 4 s IV 2 815.970 0.1502E+00 41 16 4 s IV 1 9 33.382 0.4260E+00 41 16 4 s IV 1 944.517 0.2100E+00 41 16 4 s IV 1 1062.672 0. 3770E-01 42 16 4 s IV 1 1072.992 0.3360E-01 42 16 4 s IV 1 1073.522 0.3730E-02 42 16 4 s V 1 786.480 0.1263E+01 42 16 5 SVI 73 191.510 0. 2800E-01 41 16 6 SVI 2 248.980 0.4710E-01 41 16 6 s VI 2 249.270 0. 25C6E-01 42 16 6 SVI 2 249.270 0.2440E-01 41 16 6 s VII HE1 52.000 0. 4200E+00 34 16 7 s VII NE2 54.800 0. 1000E-01 36 16 7 SVII 5 60. 160 0.16C0E+0O 41 16 7 SVII 4 60.800 0.1400E+01 41 16 7 SVII 2 72. 020 0.1700E+00 41 16 7 SVII 1 72.660 0.3600E-01 41 16 7 S VIII F5 45. 300 0.26C0E+0O 41 16 8 S VIII F1 46.COO 0.2500E+00 41 16 8 S VIII F2 53.000 0. 1500E+01 41 16 8 S VIII F3 63.300 0.6000E-01 41 16 8 S VIII E4 199.900 C.9600E-01 41 16 8 S IX 07 41.000 0.2300E+00 41 16 9 S IX 02 47.400 0.2300E+00 41 16 9 S IX 01 49.200 0.8G00E+00 41 16 9 S IX 05 54. 100 0. 40 00E-01 41 16 9 S IX 03 54.200 0.2400E-01 41 16 9 S IX 04 56. 100 0.23 00 E-01 41 16 9 S IX 06 224.750 0.1600E+00 41 16 9 S X N7 35. 500 0.32OOE+0O 41 16 10 S X N1 42.500 0.1700E+00 41 16 10 S X N3 47.700 0. 4800E-01 41 16 10 S X N6 257.100 0.1900E+00 41 16 10 S XI C9 31.000 0.2100E+00 41 16 11 S XI C3 39.300 0.2100E+00 41 16 11 S XI C4 39. 300 C.6100E+00 41 16 11 S XI C5 41.000 0.3500E-01 41 16 11 S XI C6 188.600 0. 8600E-01 41 16 11 S XI C7 247.000 0. 8400E-01 41 16 11 S XII Bl 5. 180 0.55G0E+00 42 16 12 S XII B2 27.800 0.1120E+00 41 16 12 S XII B3 33.300 0.1190E+00 41 16 12 S XII B10 221.000 0.1600E+00 41 16 12 S XII B11 227.200 0. 2900E-G1 41 16 12 S XIII BE 1 5. 130 C.7300E+00 42 16 13 S XIII EE13 256.680 0.2500E+00 42 16 13 S XIV LI1 21.000 0.3800E-01 22 16 14 S XIV 112 23.050 0. 9000E-G1 22 16 14 £ XIV 115 30.430 0.3500E+00 21 16 14 S VI 2 248.990 0. 4775E-01 41 16 14 S XIV 417.640 0.5573E-02 42 16 14 S XIV 445.694 0. 2611E-02 42 16 14 SI XIII HE 1 4. 010 0.2800E-01 13 16 15 SI XIII HE2 4. 100 0.57C0E-01 13 16 15 SI XIII HE3 4.300 0.1500E+00 11 16 15 SI XIII a£4 5.040 0.7500E+00 9 16 15 SI XVI 6 9.3 70 C.8000E-02 48 16 16 SI XVI 5 9. 490 0. 1400E-G1 47 16 16 SI XVI 4 9.720 0.2900E-01 46 16 16 SI XVI 3 10.250 C.7900E-01 45 16 16 SI XVI 2 12.150 0.4160E+00 4 3 16 16 119 APPENDIX 3, THE LINEARIZED EQUATIONS AND THE DISPERSION BELATION The linearized equations and the dispersion relation used here were derived with the aid of a program called BEDUCE avail able from the UBC Computing Centre. Of particular interest here was its ability to allow the algebraic definition of functions cf the form Q(x,t)=QB+v{dQ/dz)+&q, exp[i <kz-<4fc) J. (AIII.22) The term v (dQ/dz) appears because the analysis is done in the frame moving along at the gas speed. These definitions are made for the density, temperature, and velocity, and other guantities such as the cooling rate have perturbations expressed in terms of their density and temperature derivatives., Then the various partial derivatives with respect to z and t in the conservation eguations are evaluated, and the terms of first order in S are collected. This gives the set of linearized eguations as given below. The results eventually must be expressed in the form of a matrix of coefficients times a vector of perturbation guanti ties. The determinant of this matrix will give the dispersion relation. In order to reduce the number of multiplications in volved in the evaluation of the determinant, the coefficients of and k have been combined together as much as possible. This is the motivation for the form of the eguations below. The result ing linearized eguations have coefficients which are labelled by their eguation of origin, m,p, and e, for mass momentum and energy; the term being multiplied labelled by the coefficient, w, k, and c; and the linearized guantity being multiplied la belled by n, T, and v. The equations have been written in the form of a series of terms which when summed together must be 120 equal to zero. The derivatives with respect to z are abbreviated as just the "numerator" cf the derivative, i.e. dv/dz goes to dv. The "*" is the multiplication sign and ** represents ex ponentiation. The results are presented in the form of EGBIBAN statements because this is essentially how they are output from BEEUCE , and it is how they are input to the program which does the numerical computations.. , The linearized eguations are: mass conservation, n1* <-i*w+dv) +t1*C + v1* <i*k*n0+dn)=0. Momentum conservation, n1* (i.*.k.*pkn*pcn) +t1* <i*k*pkt+pct) + v1* (-i*w+dvg)=0. find energy conservation, n1*<-i*w*ewn+i*k*ekn+ecn +t1* (-i*w*ewt+i*k*ekt-k**2*ccnkap*ect) + v1* (-i*w*ewv+i*k*efcv+ecv) =0. In the following vc1=1.-v0/c and rhocv is the mass density divided by the nuaber density. pkn=kboltz/ (n0*rhocv) * (dnedn*t0+t0) pcn=kbcltz/ (n0**2*rhocv) * (-dn*t0+nO*dnedn*dt -neO*dt-(dnedn*dn+dnedt*dt)*t0)-vc1*dgrdn pkt=kboltz/ (n0*rhocv) * (neO + dnedt*t0*n0) pct=kbcltz/<nO*rhocv)* (dn*dnedt*dt+dnedt*dt+dnedn*dn) -vc1*dgrdt dvg=dv+gradO/c 121 ewn=(dedn*nO+eO)*rhocv ekn=-dkdn*dt ecn=rhocv* (dv*hO+ dv*nG*dhdn+)-d2t*dkdn-vc1*dgdn+dldn •rhocv*(vC*(dedn*dn+dedt*dt)+dedn*vO*dn) ewt=dedt*nO*rhocv ekt=-dkdt*dt+ (-dkdt*dt-dkdn*dn) ect=dldt-vd *dgdt* (dhdt*dv*nO)*rhocv-dkdt*d2t +rhocv*dedt*vO*dn ew v=0 ekv=hO*nO*rhccv ecv=Gn*hO+nO*(dhdt*dt+dhdn*dn)) *rhocv+gO/c •rhocv* (nG*vO*dv) In order that the above set of algebraic eguations have a ncntrivial solution the matrix of the coefficients must have a zero determinant, which gives the dispersion relation as fol lows. The dispersion relation is computed fro a the constants by, D{w,k) =i £ ujm-ir*-i (crd (n,m)*i*cid (n,m) )•• (1) where the coefficients crd and cid are: crd (1,1) = -ecn*pct*dn-dvg*ect*dv+ pct*ecv*dv+ pcn*dn*ect cid (2,1) = -ecn*pct*nO-ecn*pkt*dn-ekn*pct*dn -dvg*ekt*dv+pct*ekv*dv+pkt*ecv*dv+pcn*dn*ekt +pcn*nO*ect+pkn*dn*ect crd(3,1) = ecn*pkt*nO 122 *€kn*pct*nO*-ekn*pkt*dn+dvg* (-conkap) *dv-pkt*ekv*dv •pcn*dn*ccnkap -pcn*nO*ekt-pkn*dn*ekt-pkn*nO*ect cid(4,1) = -pkt*ekv+(-conkap)*dv)+ ekn* pkt*nO-pcn*nO*(-conkap)-pkn*dn*(-conkap)-pkn*nO*ekt crd(5,1) = pkn*nO* (-conkap) cid(1,2) = ewn*pct*dn+dvg*ect*dvg*ewt*dv-pct*ecv-pct*ewv*dv -pcn*dn*e¥t*ect*dv crd (2,2) = -ewn*pct*nO-ewn*pkt*dn -dvg*€kt+pct*ekv*pkt*ecv+pkt*ew w*d v*pcn*nO*ewt + pkn *dn*ewt-ekt*dv cid (3 ,2) = -€wn*pkt*nO-dvg*(-ccnkap)+pkt*ekv •pkn*nC*e«t-(-conkap)*dv crd (1,3) = dvg*ewt-pct*ewv+ect«-ewt*d v cid (2,3) = -pkt*ewv+ekt crd(3,3)=conkap cid (1,4)=-ewt The form of these coefficients is such that if k is re placed by the negative of its complex conjugate the root found will be the negative of the complex conjugate of the original 123 root. This behaviour is demanded in order that the same physical solution be recovered independent cf the signs of and k. \ 124 APPENDIX 4. THE HAJOB COMPUTER PBCGBAMS This appendix describes the major computer programs for actually performing the numerical computations. They are all written in the FOBTBAN language. The photoionization cross sec tions and the resulting ionization and heating rates are calcu lated by the program PHOTION using the tables given in the ap pendix 2 as input. The ionization balance and heating and coo ling rates are calculated by, HCMAIN with subroutine S0BCHEAT. The zero order physical guantities and radiation acceleration are worked out by, COEF. The coefficients of the dispersion re lation are done in COCALC, and the roots of the dispersion rela tion in DISPEB. The flow cf the programs can be followed with the aid of the comments. 125 PEOGBAM PHOT ION C PHOTOIONIZATION AND HEATING BATES C DIMENSION INDEX (16,9) INTEGEB IZED{9) BEAT NJNU (100) , DELNU{100) ,PHOT ("76) ,PHEAT(76) INTEGEB NOO (76),NUF (76) BE AX. SIGMA(100,76) , INU (100) , FLUX {100) ,DELE (100) LOGICAL VEBBOS COMMON /A/ INDEX,IZED,DEN,T,VEBBOS,NFEEQ,N NOT COMMON /HELIDM/ ALP HA,BITA,A2,B2,ZF,ZB,BEN 1,ZF2 $ ,AEZ3,BZB31,AZB31 COMMON /PH/ PHOT,PHFAT,NJNU,DELNU,SIGMA EQUIVALENCE (DELNU (1) ,DELE (1) ) KAMELIST /PP/ VEBBOS DO 20 IJ=1,76 DO 20 IN= 1,100 20 SIGMA(IN,IJ) = 0. C C UNIT 1 HAS STELLAS RADIATION FLUXES AND FBEQUENCIES C UNIT TBC HAS PHOTOIONIZATION IDCES AND STELLAB C FLUX FEEQUENCIES C BEAD (1) NFBEQ BEAT (1) ENU,FLUX,NJNU,DELE BEAD (2,9774) (NUO (IJ),NUF(IJ),IJ=1,76) 9774 FOBMAT (214) C C CALCULATE TOTAL FLUX C FTGT=0. DO 22 IN=2,NFBEQ FTOT=FTOT + . 5* (FLUX (IN-1) +FLUX(IN) ) *DELE (IN) 22 CONTINUE VEBBOS=,TBUE. SBITE (6, 9775) 9775 FOBMAT ('1') REAL(5,PP) HBITE(6,PP) C C GO THBCUGH ALL ATOMS (I) C AN £ ALL IONS OF ATOMS (J) C DO 10000 1=1,9 II=IZED (I) DO 10001 J=1,II IJ=INDEX(J,I) NUNOT=NUO(IJ) NUI NF= NUF (I J) NUINF1=NUINF-1 NUNCT1=NUN0T+1 PHOT(IJ)=0.0 EHEAT (IJ)=0.0 C 126 C EEINCH TO COEBECT ATOM C C ATOMS ABE IDENTIFIED £Y EBANCH LABEL C COBBESPONDS TO 2 OF ATOM C GO TO (1,2,6,7,8,10,12,14,16) ,1 1 XIP=13.598 ZADJ=1.0 GO TO 9910 2 IF(J.EQ.2) GO TO 202 ALPHA= 2. 182846 BETA=1.188914 A2=4.7648166 B2=1. 4135164 DEN1=0.567759716 AB23=139.8332 ZF=1. ZB=2. ZF2=1. , BZB31=0.03083696 AZB31=0.01366421 XIP=24.587 GO TO 9920 202 ZADJ=0.25 XIP=54. 416 GO TO 9910 C C ALL FOLLOWING CALCULATIONS ABE IDENTIFIED BY THE C Z OF THE ATOM AND THE J OF TEE ION C EG 601 IS CI C EG 1204 IS MG III C 6 GO TO (601,602,603,604,605,606),J 601 SIGNOT=12. 19 FZEBC=11.26 A=3.317 S=2.0 GO TO 9930 602 SIGNOT=4.60 FZEBO=24.383 A=1.95 S=3.0 GC TC 9930 603 SIGNOT-1.84 FZERC=47.887 A=3.0 S=2.6 GO TO 9930 604 SIGNGT=0.713 FZEBO=64.492 A=2.7 S=2.2 GO TC 9930 605 GO TO 99 80 606 GO TO 9980 7 GOTO (701 ,702,703 ,704,7G5,706,707) ,J 127 701 SIGNOT=11.42 FZEHO=14.534 8=4.287 S=2.0 GO TO 9930 702 SIGNOT=6.65 S=3.0 A=2.86 IZEBO=29.601 GO TO 9930 703 SIGNCT=2.G6 A=3.0 S=1.626 FZEBO=47.448 GO TC 9930 704 SIGNOT-1.08 A=2.6 S=3.0 FZEBG=77.472 GO TO 9930 705 SIGNGT=0.48 S=2.0 A= 1.0 FZEB0=97.89 GO TO 9930 706 CONTINUE 707 GO TO 9980 8 GO TO (801,802,803,804,805,606,9980,9980),J 801 BO 811 IN= NUNOT,NUINF SIGMA (IN,IJ) =2.94*SEATGN (END (IN) , 13. 6 18,2. 661 ,1.0) IF (ENU (IN) . IT. 16.943) GO TO 611 SIGMA (IN,IJ)=SIGMA (IN,IJ)+3. 65*SEATON (ENU(IN) , $ 16.943,4.378,1.5) IF (ERU (IN) . LT. 18. 635) GO TO 811 SIGMA (IN,IJ)=SIGHA (IN,I J) +2.26*SEATON (ENU(IN) , $ 18.635,4. 311, 1. 5) 811 CONTINUE GO TO 999 802 SIGNOT=7.32 S=2.5 A=3.837 FZEEC=35.117 GO TO 9930 803 SIGNCT=3.65 S=3.0 A=2.014 FZEBO=5 4.943 GO TO 9930 804 SIGNOT=1.27 S=3.0 A=0,831 FZEB0=77,413 GO TO 9930 805 SIGNCT=0.78 S=3.0 A= 2.6 FZEB0=113.90 GO TC 9930 806 SIGNOT=0.36 S=2. 1 A=1.0 FZEB0=138. 12 GO TO 9930 10 GOTO (1001, 1002, 1003, 1004, 1005, 1006) #J GO TO 99 80 1001 SIGNOT=5.35 S-1.0 A=3.769 FZEEO=2 1.564 GG TO 9930 1002 DO 1012 IN=NUNOT,NUINF SIG MA(IN,IJ)=4. 16*SEATON(ENU (IN), 40.962,2. 717,1.5) IF (ENU (IN) . LT. 44. 1 66) GG TO 1012 SIGMA(IN,IJ) =SIGMA (1N,IJ) +2.71*SEATON (ERU (IN) , $ 44. 166,2. 148, 1.5) IF(ENU(IN) .LT. 47. 874) GG TO 1012 SIG8A(IN,IJ)=SIGMA(IN,IJ) +0.52*SEATON (ENU(IN) , $ 47.874,2.126,1.5) 1012 CCNTINUE GO TO 999 1003 DO 1013 IN=NUNOT,NUINF SIGMA(IN,IJ)=1.80*SEATGN(ENU (IN),63.45,2.277,2.0) IF (ENU (IN) . LT. 68.53) GO TO 1013 SIGMA (IN,IJ)=SIGMA (IN,IJ) + 2. 50*SEATON (ENU(IN) , $ 68.53,2.346,2.5) IF (ERU (IN) . LT. 71. 16) GO TO 1013 SIGMA(IN,IJ)=SIGMA (IN,IJ) +1.48*SEATON(ENU(IN) , $ ,71. 16,2.225,2. 5) 1013 CCNTINUE GO TO 999 1004 SIGNOT=3.11 JZEBC=97.11 A=1.963 S=3.0 GO TO 9930 1005 SIGNOT=1.40 FZERO=126.21 0 A= 1.471 S=3.0 GO TO 9930 1006 SIGNOT=0.49 FZE80=157.93 A=1.145 S=3.0 GO TO 9 930 12 GOTO (1201,1202,1203,1205),J GO TO 9980 1201 SIGNOT=9.92 A=2.3 S=1.8 FZERO=7.646 GG TO 9930 1202 SIGNOT=3.416 A=2.0 S=1.0 FZEEC=15.0 35 GO TO 9930 1203 SIGNOT=5.2 A=2.65 S=2.0 FZEBO=80.143 GO TO 9930 1204 SIGNOT=3.83 A= 1. 0 £=2.0 FZEEC=109. 31 GO TO 9930 1205 SIGNGT=2.53 A=1.0 S=2.3 FZEBO=141.27 GO TO 9930 14 GO TO (1401,1402,1403,1404),*] GO TO 9980 1401 DO 1411 IN=NUNOT,NUINF SIGMA(IN,IJ)=12.32*CHAHEN(ENU(IN),7.370,6.459, $ 5.142,3.) IF(ENU(IN) .LT.8. 151) GOTO 1411 SIGMA (IN,IJ)=SIGMA(IN,IJ)+25.18*CHAHEN(END(IN) $ 8. 151,4.420, $ 8.934,5.) 1411 CONTINUE GG TC 999 1402 SIGNOT=2.65 A=0.6 S=3.0 FZEBO=16.345 GO TO 9930 1403 SIGNGT=2.48 A=2.3 S=1.8 FZEBO=33.492 GG TO 9930 1404 SIGNOT=0.854 A=2.0 S-1.0 FZEIO=45. 141 GO TO 9930 16 GO TO(1601, 1602, 1603, 1604, 1605, 1606) ,J GO TO 99 80 1601 DO 1611 IN=NUNOT,NUINF SIGMA (IN ,IJ) = 12.62*CHAHEN (E NO (IN) ,10.360, $ 21. 595,3.062,3. 0) IF (ENU (IN) .LT. 12.206) GO TO 1611 SIGMA(IN,IJ)=SIGMA(IN,IJ)+19.08*CHAHEN (ENU(IN) $ 12.206,0. 135,5.635, $ 2.5) IF (ENU (IN) . LT. 13. 40 6) GO TO 1611 13 0 SIGMA (IN,IJ)=SIGMA (IN ,1 J) +1 2 .70*CHAHEN (END (IN) , % 13.408, 1. 159,4.7113, $ 3.0) 1611 CONTINUE GO TO 999 1602 SIGNOI=8.20 FZEBC-23.33 A=1.695 E=-2.236 S=-1. 5 GO TC 9940 1603 DO 1631 IN=NUNOT,NUINF SIGMA(IN,IJ)=. 350*CHABEN(ENU(IN) , 33.46, 10. 056, $ -3.276,2.0) IF (ENU(IN).LT.34.83) GO TO 1631 SIGMA (IN,IJ)=SIGMA (IN,IJ) •. 244*CHAHEN (ENU(IN) , $ 34.83,18.427, % 0.592,2.0) 1631 CONTINUE GO TG 999 1604 SIGNOT-0.29 FZEEC=47.30 A=6.837 E=4.459 S=2.0 GG TO 9940 1605 SIGNOI=0.62 A=2.3 S=1.8 FZEBC=72. 68 GO TO 99 30 1606 SIGNOT=0.214 A=2.0 S=1.0 FZEBO=88.0 5 GC TO 9930 C C NOW THAT CONSTANTS ABE SET UP C IN THE RELEVANT FORMULA C CALCULATE THE CBOSS SECTION AT TBE C INTEGFBEQ FREQUENCIES (UNIT 2) C 9910 DO 9911 IN=NUNOT,NUINF SIGMA (IN,IJ)=ZADJ*HS1G (ENU (IN) ,XIP) 9911 CONTINUE GC TC 999 9920 DO 9921 IN=NUNOT,NUINF 9921 SIGMA (IN,IJ)=HEISIG(ENU (IN) ,XIP) GO TO 999 9930 IF (NUNOT. GE, NUINF) GOTO 9960 DO 993 1 IN=NUNOT,NUINF 9931 SIGMA(IN,IJ)=SIGNOT*SIATON(ENU(IN) ,FZEBO,A,S) GO TO 999 9940 DO 9941 IN=NUNOT,NUINF 9941 SIGMA(IN,IJ)=SIGNOT*CHAHEN(ENU (IN) , FZEBC,SIGNOT , A , B,S ) GG TO 999 131 9980 SIGMA(NFREQ ,IJ)=-1 .0 GG TO 9981 999 DO 998 INU=NUNOT1,NUINF PHINT=. 5* (NJNU (INU-1) *SIGM A < INU- 1, IJ) *NJNU (INU) $ *SIGMA(INU,IJ)) fHOT(IJ) =PHINT*EEL E (INU) + PHCT (IJ) PHINT=.5*(FLUX(INU-1)*S1GMA(INU-1,IJ)• $ FLUX (INU) *SIGMA (INU,IJ) ) PHEAT (IJ) = PHINT*BELI (INU) + PEEAT(IJ) 998 CONTINUE C FLUX HAS UNITS ERG CM-2 S-1 (IV)-1 C NJNU HAS UNITS # CM-2 S-1 (EV)-1 C PHOT HAS UNITS # S-1 C P HEAT HAS UNITS EEG S-1 PHEAT (IJ) = PBEAT (IJ) 9981 BRIIE(6,9771) II,J,ENU(NUNOT),ENU(NUINE) $ , PHOT (IJ) , PHEAT (I J) 9771 FORMAT(* 0ION•,213,* FREQUENCIES*,2F12.3, $ • IONIZATION, HEATING RAT IS*,2E15.4) IF(VERBOS) WRITE (6,9773) II, J 9773 FORMAT{'OCRGSSECTIONS FOR ION (Z,N)=',2I3) IF(VEBBOS) flBITE(6,9772) (SIGMA (IN , IJ) , $ IN=1,NFREQ) 9772 FOBMAT(1X,10E12.3) 10G0 1 CONTINUE 10000 CCNTINUE WRITE (7) PHOT,PHEAT,SIGMA,FTOT C C FLUX HAS UNITS ERG CM-2 S-2 (EV)-1 C NJNU HAS UNITS # CM-2 S-1 (EV)-1 C PHOT HAS UNITS # S-1 C PHFET HAS UNITS ERG S-1 C STOP END ELGCK DATA COMMON /A/ INDEX,IZED ,DEN,T,VERBOS,LAST,NNOT INTEGER IZED (9) DATA IZED /1,2,6,7,8,10,12,14,16/ DIMENSION INDEX (16,9) DATA INDEX /1,15*0,2,3,14*0 ,4 ,5 ,6 ,7,8 ,9,10*0, $ 10,11,12,13,14,15,16,9*0, $ 17,18,19,20,21,22,23,24,8*0, $ 2 5,26,27,28,29,30,31,3 2,33,34,6*0, $ 35,36,37,38,39,40,41,4 2,43,44,45,46,4*0, $ 47,48,49,50,51,£2,53,54,55,56,57,58,59,60,2*0, $ 61,62,63,64,65,66,67,68,69, I 70,71,72,73,74,75,76/ END REAL FUNCTION HSIG(E,XIP) C C FOE CALCULATING THE HYDROGEN C CECSS SECTION C IF (ABS (E-XIP).LT. 0.0001) GG TO 1 ETA1=SQRT(E/XIP-1.0) ETA=1./ETA1 HSIG=3.442Q4E-16*(XIE/E)**4. $ *EXP(-4.*ETA*ATSN (E TA 1) )/ $ (1.-EXP (-6.238185 + ETA) ) BETUBN 1 HSIG=6.30432E-18 BETUBN END BEAL FUNCTION HEISIG(E,XIP) C C HELIUM I CBOSS SECTION C COMMON /HELIUM/ ALPHA,BETA,A2,B2,ZF,Z£,BEN1,ZF2 $ ,A EZ3,EZB31,AZ E31 BK2=(E-XIP) /13. 598 IF(BK2.LE.0.0) GO TO 1 BK=SQBT(BK2) FEXP=-6.283185*ZF/BK ALPHAI=(2.*ALPHA-ZF)*EXP <FEXP*ATAN(BK/ALPHA)) I *(BK2 + A2)**(-3. ) BETAI=(2.*BETA-ZF)*EXP(FEXP+ATAN(BK/BETA)) $ *(BK2*B2) **<-3.) DFE=2730.667*E*ZF*AEZ3* (BK2+ZF2) * $ (ALPHAI*BZB31+BETAI*AZB31)**2 $ (1. -EXP (FEXP) ) *DEN1 HEISIG=8.067291E-18*DFE BETUBN 1 IF (XIP. GT. 24. 587) GO TO 2 HEISIG=8.334E-18 BETUBN 2 IF (XIP.GT.392.08) GO TO 3 BEISIG=4.7113E-19 BETUBN 3 IF (XIP. GT. 552. 06) GO TO 4 HEISIG=3.316E-19 BETUBN 4 IF(XIP.GT.739.32) 60 TO 5 HEISIG=2.46E-19 BETUBN 5 8BITE (6,1000) E,XIP 1000 FOBMAT (' HEISIG PBOBLEMS • ,2F15.4) BETUBN END BEAT FUNCTION SEATON (F, IZEBO , A, S) C C SEATCN CBOSS SECTION FOBMULA C EN= FZEBC/F SEATON=1.0E-18*FN** (+S) *(A+ (1.-A) *FN) BETUBN END BEAL FUNCTION CHAHEN (F, FZEBO,A, B, S) C C CHAPMAN AND BENBY CBOSS SECTION FOBMULA C FN= FZEBO/F CHAHEN=A+{B-2.*A)*FK+ (1.+A-B)*FN*FN CHABEN=1.E-18*FN**S*CHAHFN RETURN END 134 PBGGBAM HCMAIN C C PASAKETEBS C DEN:TOTAL DENSITY C Ts TEMPEBATUBE C FJ: COVEBSION FBOM FIBST TO 2EBOTH MOMENT C BADIATION FIELD C =1 FOB UNIDIRECTIONAL =2 FOB A C HEM ISP HEBE C NIT: NUMBER OF ITERATIONS IN ION FRACTIO C N LOOP C VEBBGS: OUTPUT ALL CAICULATED QUATITIES C AT END OF NIT LOOP C ULTRA: OUTPUT DITTO EVERY CYCLE C TEBSE=.TRUE. C FABUND: MULTIPLY ALL ABUNDANCES Z>2 BY T C HIS NUMEEB C FE: GUESS AT ELECTRON DENSITY C WF: DILUTION FACTOR FOB B AEI AT ION FIELD C NLINE: NUMBER OF LINES IN CCCIING CALCUL C ATION C WLINE: WRITE INDIVIDUAL LINE CCCIING AND HEATING C TCL: TOLERANCE FCB CONVERGENCE OF BNOT,DENE,EQUILIBRIUM C TEMPERATURE C EGUIM: TRUE FOB FOBCING BALANCE OF HEATING AND COOLING C BATES C CHABGX TBUE FOB CHABGI EXCHANGE H-N, H-0 CALCULATIONS C NELMNT: # OF ELEMENTS STARTING WITH H IN IONIZATION CA C LCULATICN C USEFUL SOMETIMES IN EQUIM FOB PBELIMINAEY C ESTIMATE C DMAX: MAX FRACTIONAL CHANGE ALLOWED IN DELTA T, C PBEVENTS WILD OSCILLATIONS C DIELEC: FALSE TURNS ALL DIELECTBONIC RECOMBINATION OFF C MI-OOP: NUMEEB OF ITEBATIONS ALLOWED IN CONVEBGENCE TO C T IF EQOIM IS CN C TBAD: BACIATION TEMP EBATUBE OE PHOTON SOURCE C WFTBAD IS APPROXIMATELY A BRIGHTNESS TEMPEBATUBE C FUDGE: TBUE FOB BE DUCT ION OF DIELECTBONIC B ECO MB IN A HO N C WITH DENSITY C FUDGE FACTOR CALCULATION IS DESIBED C THEEEB: THREE BODY RECOMBINATION C DXENDT TRUE FOB COMPUTING EEBIVATIVES IN N AND T C SEBIES IS (T,N) , (T* { 1+-DERDEL) , N) ) . <T ,N* < 1 + -DEBDEL) ) C OUTPUT FALSE IF NO OUTPUT OF QUANTITIES TO UNIT 7 C CNGAB CHANGE OF CNO ELEMENTS FSCM SCLAE VALUES C TSERIE TBUE IF SERIES OF TEMPERATURES TO EE CALCULATED C DSEE1E TBUE FOB A DENSITY SEBIES C SEEINC SEBIES STARTS FROM INPUT DENSITY AND TEMFEBATUEE C AND INCBEASE LOGABITHMICALLY EY 10 TO SERINC C SERENC MAX VALUE OF N OR T C WTNE WEIGHT GIVEN TO OLD VALUE CF ELECTRON DENSITY C IN CONVERGENCE OF IONIZATION EQUATIONS. C DV FOR ESTIMATE OF EFFECTS OF OPACITY ON LINE COOLING 135 C TAUMAX GREATER THAN ZERO TO TURN ON CALCULATION C LOGICAL VERBOS,SEMICO,ULTRA,WLINE,EQUIM,CNVG,FIRST, $ CHABGX LOGICAL YERBO,WLIN,DIELEC,F UDGE,THREEB,TERSE,NOW AST $ ,QUIT,DXDNDT LOGICAL 0 UTPUT , TSERI E , DSERI E , BOTH DE,FSEE REAL SIGHA(100,76) ,PHOT (76) ,PEEAT(76) REAL PPHGT(76), PPHEAT(76), CPHEAT(76) REAL RATIO (16) , REL ( 16) ,X (17, 9) REAL TOPIN( 16) ,TOPOUT(16) REAL HLCCOL (9) REAL LOWLIN (76) INTEGER INDEX (16,9) INTEGER IZED (9) EEAL ABUND (9) REAL CHIT (76) REAL IP1 (76) ,IP2(76) ,CS(76) INTEGER NUM1 (76) ,NUM2(76) BEAL ARAC(76) , ETA(76) ,TMAX(76) , TCRIT (76) , ADI (76) , $ TO(76),BDI(76) , $ T1 (76) ,BREC(76) , BflEC (76) ,UREC(76) BEAL AREC{76) ,SLTE (76) REAL LINLOS,LRRAD,LBREMS,PHEET REAL AG(49) ,BG(49) ,CG(49) ,DG(a9) REAL LCOOL(76) ,ELINE(407) ,FI (407) BEAL LCLX (76) INTEGER IIND(407) , JIND (407) , I DENT (407) COMMON /A/ INDEX,IZID,DEN,DENE,T,TK,TKI,T4,TSQRT, $ VERBO,LAST COMMON /RECG/ RREC,IREC,UREC,ARAD,ETA,TMAX,TCRIT, $ ADI,TO,BDI,T1 COMMON /CICN/ IP2,NUM1,NUM2,CS,SLTE COMMON /COLREC/ IP1,CHIT,RN NOT COMMON /LINE/ LCOOL,ELINE,FL,IDENT,IIND,JIND,NLINE COMMON /GFACT/ AG,BG,CG,DG COMMON /CCNTRO/ SEMICO,ULTRA COMMON /CFUDJ/ FUDJ,RNOT,FUDGE COMMON /THICK/ X,ABUND,DV,TAUMAX NAMELIST /PARAM/ DEN,T,FJ ,NITP,VERBOS,FABUND,FE,WF, $ NLINE,SEMICO $ ,ULTRA,WLINE,TOL,EQUIM,CHARGX,NELMNT,DMAX,DIEIEC $ ,NLOCP,TRAD $ ,FUDGE,THREEB,TER SE,D XDNDT,DERDEL,OUTPUT,C NOAB $ ,TSEBIE, DSERIE, S EBINC, EOT H EE, SER END, WTN E $ ,DV,TAUMAX C C SET UP DEFAULTS C REWIND 1 REWIND 2 ABUND (1) =1.0 ABUND(2)=8.5E-2 ABUND(3)=3.3E-4 ABUND (4) = 9. 1E-5 ABUND (5) =6.6E-4 136 A BUND (6)=8.3E-5 ABUND(7)=2.6E-5 ABUNB (8)=3.3E-5 AB0ND(9) = 1.6E-5 NLOGf-15 SEMICO=.TRUE. ULTRA=. FALSE. FUDGE=.TRUE. THBEEB=. TSUE, TERSE=.FALSE. NGWAST=. FALSE. TBAB=50000. FJ=1. NITP=10 VEBEOS=.TBUE. WLINE=.TEUE. FABUND=1. CNOAB=1. FE= 1.002 WTNE= 1. WF=1.0 WFvJ0LD=-1. N1INE-407 EQUIM=.FALSE. EMAX=.25 BIELEC=.TRUE. NELEKT=9 CHARGX=.TBUE. T0L=1.E-03 DXDNDT=.FALSE. DEEDEL=.01 OUTPUT=.TRUE. ESEE=.TRUE. X (1, 1) =0. TAUEAX=0. DV=0. C TSEBIE=.FALSE. CSEE3E=. FALSE. BOTHDE=.FALSE. SEBINC=.1 SEBEND=0. C BEAD IN I AT A C UNIT 1 HAS PHOTOIONIZATION DATA CALCULATED BY PHOTION C ASSUMED TC EE GF THE ICBM: FIRST MOMENT OF RADIATION C FIELD*QDANTITIES C UNIT 2 HAS THE CGNSTANTS REQUIRE £ FOR RECOMBINATION, I C ONIZATION C AND IINECOOLING (LINES AND EXCITATION G FACTOR) C LOWLIN HAS LOWEST DELTA ENERGY LINES FCR IONS HITH D C IELECTBGNIC BECCM C BEAE (1) PHOT,PHEAT,SIGMA,FTGT BEAD (2) ABAD,ETA,TMAX, TCRIT ,ADI,10,BDI ,T1 BEAD (2) IP1,NUM1,IP2, NUM2 READ (2) ELINE,FL,IDENT,IIND,JIND READ (2) AG,BG,CG,DG BEAD (2) L01LIN C C SET DP OF INITIAL CONDITIONS FOB MULTIPLE LOOPS C ITDER=0 10000 CONTINUE QUIT-,FALSE. DIFOLD=0. DIFF=0. CNVG=.FALSE. ICLOOP=0 IF (IT DER. EQ, 5} DEN= DEN/ (1. - DEB DEL) IF{DXDNDT.AND. IIDEE.IE.4) GO TO 10100 ITDF B=0 IF(TSERIE.OR,DSERIE) GO TO 3001 BEAD (5,PARAM, END=10001) LAST-INDEX(IZED(NELMKT),NELMNT) IF(EQUIM) DXDNDT=. FALSE. IF(DXDNDT) EQDIM=,FALSE. IF {. NOT, EQUIM) CNVG=.TRUE, IF (ULTRA) VERBOS=.TBUE. IF (TEBSE) VEBBOS=. FALSE. IF (VEBBOS) «LINE=.TBUE. IF (TEBSE) SLINE=. FALSE. IF(NITP.LT.2) NITP=2 IE (X (1 , 1) . NE.O. ) GOTO 200U X (1, 1) =0. X (2,1) = 0. X(1,4)=0. X(2,4) = 0. X (1,5) =0. X (2,5)=0. EUO=0. EDC=0. BUN=0. EDN=0, C C TE P. PEE ATUBE OB DENSITY SERIES LOGIC C 2004 IF (TSEBIE. OB. DSEBIE) SEBINC= 10. **SEBINC IF(.NOT.(TSERIE.OR.DSEBIE)) GC TO 3004 IF(.NOT.EOTBDE) GO TO 3002 EQUIM=.FALSE. DXDNDT-. TRUE. , CUTEUT=.TRUE. 3001 IF(.NOT.BOTHDE) GO TO 3002 EQUIM=.NOT.EQUIM EXDNDT=.NOT.DXDNDT CUTPUT=.NOT.OUTPUT 3002 IF(DXDNDT.AND.ITDER.NE.O) GC TO 3004 IF (DSERIE) GO TO 3003 IF(.NOT.TSERIE) GO TO 3004 IF (. NOT. FS ER. AND. {. NOT. EOTH DE. OR. EQUIM) ) T=T*SERINC IF(T.GT.SEREND) GO TO 10001 GO TO 3004 138 3003 CONTINUE IF (. NOT. FSEB. AND. (. NGT. EOTH BE.OB. EQUIM) ) DEN=DEN* $ SEEINC IF (DEN. GT. SEBENB) GOTO 10001 3004 CONTINUE IF (EQUIM) GO TO 10 101 10100 CONTINUE IE (.NOT. DXDNDT) GO TO 10101 C C EEEIVATIVE CALCUIATICN LOGIC C ITDEB=ITDEB*1 IF(ITDER.EQ.1) GO TO 10101 IF (ITDEB. EQ. 2) T=T* (1. + DEB DEL) IF (ITDEB. EQ. 3) T=T* (1.-DEBDEL) / (1 . + DEBDEL) IF(ITDEB.EQ.4) GO TG 10111 IF (ITDEB. EQ. 5) EEN=EEN* ( 1. - DERDEL) / (1.+ DEBDEL) GO TO 10101 10111 T=T/ (1. -DEBDEL) DEN=DEN*(1. + DERDEL) 10101 CONTINUE IF (EQUIM) NIT=MIN0 (5,NITP) IF (. NOT, EQUIM) NIT=NITP FIBSI-.TBUE. TCLD=0. IF (.NOT,TERSE.OR..NOT,NGWAST) WRITE (6,1008) 1008 FOBHAT (11 *) IF (. NOT. TERSE. OB. . NOT. NGSAST) WRITE (6 , PA BAM) C C SET UP TEMPERATURE, DENSITY, AEUNDANCES, EIUX FACTOB C DENE=DEN*FE DEOID=DENE IF(.NOT.FSER) GO TO 2005 SUM=0. C C CALCULATION OF ABUNDANCES C DO 101 IEL=1,NELMNT IF (IEL.GT.2) ABUND (IEL) =FAEUND*ABUND (IEL) IF ( (IEL. GE. 3) . AND. (IEl.LE. 5) ) ABUND (IEL)=CNOAB* $ ABUND (IEL) SDM=SUM+ABUND(IEL) 101 CONTINUE DO 102 IEL=1,NELMNT ABUND (IEL) =ABUND (IEL) /SUM 102 CONTINUE FABUND=1. CNOAB=1. IF(.NOT.TERSE.OR. . NOT. NGH AST) WRITE (6 ,1002) FABUND, $ RELHNT,ABUND 1002 FORMAT (• RELATIVE ABUNDANCES WITH FABUND= » , F6 .3 , $ 5X,'NELMNT=»,I3,/1X,9E13.3) C C BAIIATICN DILUTION APPLIED C 139 SFJ=WF*EJ WFTRAD=WFJ*TBAD IF (WFJ. EQ. WFJQLD) GO TO 200G1 WFJ0LD=8FJ WMJ=»FJ*12.56637 C C ADJUSTMENT TO FLUX MADE INCLUDING A 4*PI MULTIPLICATION C EFTCT=FTOT*WF DO 103 IJ=1,76 PPHOT (IJ) = FHQT (IJ) *BMJ PPHEAT (IJ)=PHEAT (IJ) *WMJ 103 CONTINUE FSEB=. FALSE. 20001 ICLOOP=ICLOOP+1 IF (ICLCOP.GT.NLOOP) GO TO 30000 2005 VERBO=VERBOS.AND. (CN VG. OB-. , NOT. EQUIM) WLIN=WLINE. AND. (CNVG.OB. . NOT. EQUIM) TK=T/11604.8 TKI=1./TK T4=T*1.E-4 ETHBEE=0.0 TM4 5=T**(-4.5) TSQBT=SQBT (T) IF ( . NOT . FIB ST. AND. EQUIM) NII-MINO <3,NITP) 122 IF (, NOT. CHABGX) GO TO 10004 CALL CHGEX(BUO,BDO, BUN, BDN,T) C C CHAEGE EXCHANGE CALCULATION FOB NITROGEN AND OXYGEN C U IS U EE ATE FOB I TO II OF N AN E 0 C D IS DOWNBATE C IF((X(1,1) .NE.1.E-07) .OB. (X (1 ,1) . NE. 0. ) ) GOTO 10004 X(1,1) = 1.E-07 X(2,1)=1. X (1 ,5)=0.5 X(2,5)=X(1,5) X (1,4) = 0.5 X(2,4)=X(1,4) 10004 DC 1 IT=1,NIT DO 2 1=1# NELMNT 11= IZED (I) ZED=II ENUCLD=1.E4 IF (WFTRAD.LE.O.) GO TO 45 C C THIS IS A CALCULATION OF LOWEST LEVEL IN EQUILIBRIUM WITH C CONTINUUM DUE TO RADIATION FIELD C RNUCLD=2.72 NNIT=0 44 ENCTNU=ZED*SQRT (3. *ALOG (RNUOLD) * 1 57802./WETfiAD) NNIT=NNIT+1 DIE N= R NOT NU -BNUOLD R N UOL D=R NO T N U-DIF N/(1.-.5/AIOG(RNOTNU)) IF (BNUOLD. LE. 1. ) BNUOLD=2. IF(NNIT.GT.4) GO TO 45 IF (ABS (DIFN/BNOTNU) . GT. TOL) GO TO 44 45 111=11+1 IZ=II DO 3 J=1,II IJ=INDEX(J,I) CALL LEVEL (J ,1) BNOT=AMIN1(BNNOT,BNUGLD,2,)+.5 IF(I.EQ.1) BNNOTH=BNOT CALL CCLION <J,I) CALL BEC(J,I) IE (ULTEA, OB, ( (IT. EQ. WIT.OB. QUIT) .AND. VEBBO) ) WRITE ( $ 6,1029)BNOT, $ ENNOT,ENUOLD, FUDJ 1029 FOBMAT {» RNOT,RNNCT,RNUOLD,FUDGE FACTOB•,2F20.1, $ 2E15.3) BNNOT=BNGT DBEC(IJ)= DREC(IJ)* F DD J IF(.NOT.DIELEC) DBEC(IJ)= 0,0 C C TH BEE ECDY BECGMBINATION FBOM SUMMERS AND BDBGESS C WITH A DIFFERENT Z DEPENDENCE C IF(,NOT.THREEB) GO TC 46 ETHBEE=1. 16E-08* (J**3) *TM45*DENE IF(BBEC(IJ) .EQ.O. 0) BTHBEE=0. 0 46 ABFC(IJ)=RR£C(IJ)+EEEC(IJ)+ UB EC(IJ)+RTHREE C C BEC(J,I) IS RECOMBINATION BATE INTO J FROM J + 1 C COL(J,I) IS COLLISION RATE OUT OF J TO J + 1 C IF ( ULTRA .OR . ( (IT. EQ. NIT. OB. QUIT) • AND. VEBBO) ) WRITE ( $ 6,1001) $ II , *3,CS (I J) , SLTE (IJ) ,PPHOT (IJ) 1001 FORMAT(« COLLISIONS, UPPER LEVELS, PHOTO IONIZATION $ ,»BATE«, $ • ION (Z,N) ' ,2I3,3E15. 4) IE (ULTRA. OB, ( (IT. EQ, NIT.OB wQDIT) . AND. VEBBO) ) WBITE{ $ 6,1000) $ II,J,RREC(IJ) , DREC (IJ) ,UREC (IJ) 1000 FOBMAT (• BAEIATIVE, EIELECTEONIC, UPPER LEVELS, REC* $ »OMBINAIION $ *RATE* f 213, 3E15.4) C C TOPOUT (J) IS RATE J TO J + 1 C TOPIN(J) IS RATE J+1 TO J C 3 CONTINUE IJJ=INDEX (1,1) TOPOUT(1)=PPHOT (IJJ) + (CS (I JJ)+SLTE (I J J) ) *DE NE IF(I. EQ. 1) TOPOUT(1)=TOPOUT <1)+-BDO*X <2,5)*ABUND (5) $ +BDN*X (2,4) *ABUND (4) IF (I, EQ. 4) TOPOUT (1)=TOPOUT (1) +BUN*X (2,1)/AEUND (4) IF (I. EQ.5) TOPOUT (1)=TOPOUT ( 1) +EUO*X (2, 1) /ABUND (5) TOPIN (1) = ABEC (UJ) *BENE IF (I. EQ. 1) TOPIN (1)=TOPIN (1) +EUO*X (1 ,5) * ABUND (5) $ +BDN*X(2,4) *ABUND (4) IF (I. EQ.4) TOPIN(1)=T0PIN (1) +EDO*X (1,1) /ABUND (4) IF (I.EQ.5) TOPIN (1)=TCPIN (1) +BDN*X(1, 1) /AB UND (5) IF (AREC (UJ) ,EQ. 0.0) GO TO 24 RATIO (1)=TOPOUT (1)/TOPIN { 1) GO TO 25 24 RATIO (1) = 1. 0 25 EEL (1) =RATIO (1) IF (I. EQ, 1) GO TO 8 DO 4 JJ=2,IZ IJ=INDEX (JJ,I) TOPOUT (JJ) =PPHOT(IJ) + (CS (IJ) +SLTE (IJ) ) *DENE TOPIN(JJ) = AEEC(IJ)*EEN E IF (TOPIN (JJ) ,EQ. 0.0) GO TO 5 RATIO (JJ) =TOPOUT(JJ) /TOPIN (JJ) GC TO 4 5 RATIO (JJ) = 1. 0 4 CONTINUE C C RATIO (J) IS POPULATION LEVEL J + 1 / LEVEL J C REI(J) IS POPULATION RELATIVE TC LEVEL 1 C EEL (1) IS POP LEVEL 2 / POP LEVEL 1 C DC 6 JJ=2,II REL (JJ) =EATIO (JJ) *BEI (JJ-1) 6 CONTINUE 8 SUM-1.0 IF (AREC (INDEX (1,1) ) .EQ. 0.0) SUM=0.0 IF (I.EQ. 1) GO TO 31 C C IF RATE INTO LEVEL FROM TOP IS 0 SET POPUIATICN TO 0. C DO 7 JJ=2,II IF (AREC (INDEX (JJ, I) ) . EQ. 0.0) REL (JJ-1) =0. 0 S0M=SUM*REL (JJ- 1) 7 CONTINUE IF (AREC (INDEX (II,I)) .EQ.O.) REL(II) = 0. 31 SUM=SUM + BEL (II) FNOEM=1./SUM C C X(J,I) IS RELATIVE POP OF IONIZATION LEVEL J IN ATOM C SUM HITH I CONSTANT IS 1 C DO 9 J=1,II IF(AREC(INDEX(J,I)).GT.0.0) GO TO 16 X (J,I)=0.0 9 CONTINUE 16 NB=J NB1=NB+1 X (NE,I) = FNOEM DO 17 J=NB1,111 X (J,I) = REL (J-1) *FNOEM 17 CONTINUE 2 CONTINUE C C ELECTRON DENSITY CALCULATION c EENE=0. EO 21 1= 1,NELMNT II1=IZED <I)+1 DO 22 J=2,II1 DENE=DENE+X (J, I) * (J-1) * JIB ON C (I) 22 CONTINUE 21 CCNTINUE DENE=DENE*DEN*WTNE+(1.-WTNE)*DEOLD EE= DENE/DEN WRITE (6,1010) DENE 1010 FOEMAT (• NEW ELECTRON DENSITY IS«,E16.7) IF(TERSE) GO TO 113 DO 20 1=1,NELMNT II=IZED(I) IZ 1 = 11 + 1 IF (ULTRA. OR. ( (IT, EQ, NIT.OR. QUIT) . AND, CN VG) ) WRITE ( $ 6, 1005) II 1005 FORM AT(' RELATIVE AEUNEANCES FOR ELEMENT Z=»,I3) IF (ULTRA. OR, (<IT. EQ.NIT.OR, QUIT) . AND. CN VG) ) WRITE(6 $ ,1004) $ (J,X(J,I) ,J=1,IZ1) 20 CONTINUE 1004 FORMAT (1X,5 (15, £15.5) ) 113 IF(QUIT) GO TO 111 C C CHECKING FOR CONVERGENCE OF ELECTRON DENSITY C CONVERGENCE SEEMS TO EE SLOW WITH THIS METHOD C IF(AES(BEOLD-DENE)/DENE.LT.TOL) QUIT=.TBUE. DEOLD=DENE 1 CCNTINUE 111 CONTINUE C C NOTE THAT THERE IS NO LINE COOLING OF BARE IONS C IF (TOLD.EQ.T) GO TO 23 CALL LINCOL C C LIKE COOLING CALCULATED ONLY IF TEMPERATURE HAS CHANGED C IF (.NOT. EQUIM) TOLD=T 23 CONTINUE C C HYDROGEN LINE COOLING LOSSES C DONE AS ACCURATELY AS PGSSIELE SINCE COOLING IN 1E4 C 3E4 TEMPERATURE C RANGE IS CRUCIAL C ICOCL (1)=0. RNNOT=RNNOTH CALL HLINE (BLCOOL, 1) C THE 1 REFERS TO LOWER LEVEL FOR TRANSITIONS DO 30 N=1,9 LCOOL (1) =LCOOL (1) +HICOOL (N) HLCCOL (N) = HLCCOL(N) *X ( 1, 1) * ABUND( 1) *DENE/DEN 143 30 CONTINUE IF (WLIN. AND. CNVG) WEITE(6, 10 11) HLCOOL 1011 FORMAT(' HYDROGEN LINE LOSSES ARE:»/1X,9E14. 4) IIKLCS=0. LRRAD=0. PHEET=0., DO 10 1=1,NELMNT II=IZED (I) DO 10 J=1,II IJ=INDEX (J,I) C C ADJUSTMENT OF RECOMBINATION RATE TO ENERGY RECOMBINATION C ON RATE C USING FACTORS GIVEN BY SEATON FOR HYDROGEN C UL=IF1 (IJ)*TKI ULL2=.5*ALOG(UL) UL3=UL**(-.3333333) ABFACT= (-0. 0713+ULL2+0. 640*UL3) /(G. 4288+ULL2+.469* $ UL3) BRCCGL=RBEC (IJ)*(IP 1 (IJ) +TK)*ABFACT C C DIELECTBONIC COOLING ASSUMES LOWEST ENERGY TRANSITION C IS C DOMINANT STABILIZING TRANSITION C RDCCCL=DEEC(IJ)*(IP 1(IJ) +LOWLIN(IJ) ) LRRAD=LRRAD+ (RRCGOL + RDCOOL) *X (J+1 ,1) *ABUND(I) LCLX(IJ)=X (J,I) *LCGOL (IJ)*A FUND (I) *FE IIEICS=LINLCS+LCLX (IJ) CPHEAT(IJ) =PPHEAT(IJ) *X (J,I) * ABUND (I) PHE ET= PfiEET+CPHEAT (IJ) 10 CONTINUE C C ALL ENERGY RATES ARE IN ERG CM+3 S-1 C BEATING IS IN ERG S-1 C I-BBEMS=2.29E-27*SQBT (T) * ABUND (1) *FE LRRAD=LRRAD*FE*T.6 02192E-12 COCL= LBB EMS+LRR AD+LINLOS PHE'ETD=P BEET/DEN IF (.NOT.EQUIM) GO TO 38 IF ( .NOT. TERSE) WRITE (6,1021) PHEETD,CCOI,T 1021 FORM AT(* BEATING, COOLING RATES CM + 3 S-1«,2E15.5, $ 5X,»AT TEMPERATURE *,E15.5) DIFCLD=DIFF IF(PHEEID.GT.1.0E6*CCOL) DIFCLD=0. IF ( FfiEETD. LT. 1.0 E-6*COOL) D3FOLD=0. DIFF=COOL-PBEETD IF(CNVG) GO TO 20002 C C RADIATIVE EQUILIBRIUM TEMPERATURE CALCULATION C IF (.NOT. FIRST) GO TO 20000 TOID=T T=T*1.01 FIRST-. FALSE. GC TC 20001 20000 DERIV=(DIFF-DIFOLD)/(T-TOLD) IF(DERIV,EQ.O.) GO TO 20002 DELT=-DIFF/DERIV IF (. NOT. TERSE) MBIT I {6, 1022) T,DELT 1022 FORMAI(* T, DELTA T (DELT) * ,2E15. 5) IF (ABS(DELT/T)•GT.I'M AX) GO TO 20003 IF (ABS (DELT/T) .LT. TOL) CNVG=.TRUE. TOL D= T T=T+DELT GO TO 20001 20003 IOID=T T—I+DMAX*DELT/ABS(DELT)*T GO TO 20001 20002 CONTINUE 38 IF (.NOT. WLIN) GO TO 11 C C FEINTED OUTPUT OF DETAILS OF BEATING AND COOLING BATES C DG 12 1=1,NELMNT IZ=IZED (I) IJB=INDEX (1 ,1) IJE=IJB+IZ-1 WRITE (6,1009)IZ 1009 FORMAT (• LINE COOLING LOSSES FOB ATOM GF Z»,I3) WBITE(6,1003) (LCIX(U) , IJ= IJE, IJE) 1003 FORMAT(1X,8E15.3) 12 CONTINUE IF(WFJ.LE. 0.0. OB.. NOT. WLIN) GO TO 11 DO 13 1=1,NELMNT IZ=IZED(I) IJB=INDEX (1 ,1) IJE=IJB+IZ-1 WRITE (6,1 019) IZ 1019 FORMAT(» PHOTOIONIZATION SEATING BATES FOB ATOM Z=« $ ,13) WBITE(6,1003) (CPHEAT(IJ),IJ=IJB,IJE) 13 CONTINUE C C INTERNAL ENERGY AND ENTHALPY IN UNITS OF EEG PER C CUBIC CM C 11 EINT=0. DO 14 1=1,NFLMNT IZ=IZED(I) DO 14 J=1,IZ IJ=INDEX (J,I) EINI=EINT + IE1 (IJ) *X (J + 1,I) *ABUND(I) 14 CONTINUE EINT=(EINT*DEN+1.5*TK*(DEN+DENE)) * 1.602192E-12/DEN ENTHAL=EINT+(DEN+DENE)*TK*1.602192E-12/DEN CEINT=£INT/(TK*1.602192F-12) CENTHP=ENTHAL/ (TK* 1. 602192E- 12) C C OUTPUT CALCULATED QUANTITIES c IF (OUTPUT) WBITE(7) DIN ,T, HF, FJ, PHEET, COOL, L BE EM S , $ LRRAD,LINLOS, $ EI NT , ENTHAL,DENE, AEUND, X ,LCLX,CP HEAT, PFTOT IF{EQ DIM) WRITE (6,1051) ICLCOP 1051 FOBMAT (» NUMBER OF LOOPS TO CONV EBGENCE1,II) WRITE (6,1006) DEN,T,»FJ,PHEETD,COOL,LBREHS,LRBAD, $ IINLOS, $ EINT,ENTHAL,CEINT,CENTHE,DENE 1006 FOBMAT (»- PABAMETEBS WIRE: DENSITY, TEMPEBATOBE, • $,'DILUTION FACTOB•,3E15.5/ $ » TOTAL HEATING/DENSITY AND COOLING BATE1,2E15.5/ $ » THE COOLING BAT IS FOB EE EMSSTRAHLUNG, », $ 'R ECO MBIN ATION RADIATION, AND LIKE LOSSES *,3E15.5/ $ 'INTERNAL EN ERG Y , EN TH ALP Y» ,2E1 5. 5 ,5X ,' AND COEFFI $ ,» CIENTS* ,2115.7/ $ • ELECTRON DENSITY*,E15.5) IF (. NOT.TERSE) WRITE (6, PABAM) NOWAST=.TRDE. GO TO 10000 3OCO0 WBITE (6,1049) NLOOP 1049 FOBMAT (« SOBBY EUT MAX NUMBIB OF TEMPEBATUBE LOOPS $ ^EXCEEDED',13) GG TO 10000 1000 1 STOP END PROGRAM SjgECHJAT SUBROUTINE CHGEX(BBC,BDO,BUN,BDN,T) C C CHARGE EXCHANGE TAKEN FEOM: C 0 FIELD AND STEIGMAN C N STEIGMAN, HEFNER, AND GELDON C C EUO IS EETA FOR OI TO Oil THAT IS UP FI(X)=ERF(SQRT(X) ) -1. 1 2 83 8*EXP (-X) *SQBT {X) XAC=6.034/T XAD=732.8/T XC=0.812336/T XD=S8.64/T BDO=1. 97E-0 9* {, 3864 15**1 (XAC) ••0. 5* (PI (XAD) -Fl (XAC)) $ + 0.529412* (1.-$ EI (XAD)) ) +2. 11E-09*( (0.115385*EXP (- XC) * $ FI(XAC-XC) + I 0.0294118*EXE (-XD) *FI (XAD-XD) ) ) £UO=EXP (-227. 45/T) * (i. 97E-0 9-BDO) BUN=1. S7E~0 9*(EXP (-11031. 5/T) *.333333+EXP(- 11102. 3/ $ T)*.333333+ $ EXP (-11220. 7/1) *. 151515) EDN=1.97E-09-EXP(11031.5/T)*BUN BETUBN END ELOCK DATA COMMON /A/ INDEX,IZED,DEN,DENE,T,IK,TKI,T4,TSQRT, $ VEBECS,LAST INTEGER IZED (9) DATA IZED /1,2,6,7,8,10, 12,14,16/ DIMENSION INDEX (16,9) DATA INDEX /1,15*0,2,3,14*0,4,5,6,7,8,9,10*0, $ 10,11,12,13,14,15,16,9*0, $ 17,18,19,20,21,22,23,24,8*0, $ 25,26,27, 28, 29, 30, 3 1,3 2, 33,34,6*0, $ 35,36,37,38,39,40,41,42,43,44,45,46,4*0, $ 47,48,49,50,51,52,53,54,55,56,57,58,59,60,2*0, $ 61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76/ END SUBROUTINE COLIGN(J,1) C C BOUTINE FOR CALCULATION OF COLLISIONAL C IONIZATION BATES FOB ALL ELEMENTS BUT C HXEEOGEN C DIMENSION INDEX (16,9) ,IZED (9) EEAI IP1 (76) ,IP2 (76) DIMENSION NUM1 (76),NUM2 (76) ,CS (76),SLTE(76) BEAL CHIT (76) LOGICAL VERBOS ,SEMICO,ULTRA COMMON /A/ INDEX,IZED,DEN,DENE,T,TK,TKI,T4,TSQRT, $ VERBOS,LAST COMMON /CION/ IP2,NUM1,NUM2,CS,SLTE COMMON /COLBEC/ IP 1,CHIT,BNNOT COMMON /CONTRO/ SEMICO,ULTRA IJ=INDEX (J,I) IF(J.NE.IZED(I) ) GO TO 199 3 IE (.NOT. SEMICO) RNC=1.0EQ6 101 CS(IJ) =COLH (RNNOT,IE 1 (IJ) ,1) SLTE(IJ)=0. RETURN C C CORRECTION FACTOR FROM P 205 MCWHIRTER ,R. W. P. , IN ATOM C IC AND C MOLECULAR PROCESSES IN ASTROPHXSICS, ED EY MCE HUBBARD C AND H C NDSSBAUMER, GENEVA OBSERVATORY, SAUVERNY, SWITZERLAND, C 1975. C 199 CS(IJ) = NUM1 (IJ) * EXP(-IF1 (IJ) *TKI) / (IP 1 (IJ)*IP1 (IJ) ) $ /(4.88+TK/IP1 (IJ) ) IF ( NUM2 (IJ) . LE. 0) GO TO 200 CS (IJ)=CS (IJ) +NUM2 (IJ) *EXP (-IP2 (IJ) *TKI) /(IP2 (IJ) * $ IP2(IJ)) / (4. 88+TK/IP2 (IJ)) 200 CS (IJ) =8.35E-08*TSQET*CS (IJ) C CHIT(IJ)=(2.8E-28*IP1(IJ)*DENE*DE NE*TKI)**. 142857143 CHIT (IJ) =IP1 (IJ) / (RNNOT*RNNOT) SITE (IJ)=4. 8E-06*CHIT (I J) / (IP 1 {IJ) *IP1 (IJ) *TSQRT) * $ EXP(-IP 1 (IJ) $ *TKI) IF(.NOT.SEMICO) SLTE(IJ)=0. BETUEN C C CHIT IS ESTIMATE OF IONIZATION- POTENTIAL OF LOWEST LEVEL C IN EQ'M WITH CONT * M C SITE FROM WILSON C END FUNCTION COLH(RN0,XIP,N) C C COLLISICNAL IONIZATION RATE FOR HYDROGEN C DIMENSION INDEX (16, 9) DIMENSION IZED (9) LOGICAL VERBOS COMMON /A/ INDEX,IZID,DEN,DENE,T,TK,TRI,T4,TSQRT, $ VERBOS,LAST X0=1.-1./(RN0*RN0) X02=1./(X0*X0) X03=X02/X0 RN=FLOAT (N) EN2=SN*RN EPKT-XIP*TKI/RN2 Y=X0*EPKT C FOR OPTICALLY THICK CALCULATIONS CAN USE N OTHER THAN 1 IF (N. EQ. 1) GO TO 2 A=1.9602805*RN*X03*(.3595-0.05798/X0+5.894E-03*X02) B=. 6666667*BN2/X0* (3. + 2./X0 •». 11 69*X02 ) Z=X0* (0.653 + EPKT) GO TO 4 2 A=1.9602805*X03*(.37767-0.1C 15/X0 + 0, 01<4028*X02) B=. 6666667/X0* (3.+2./XO-0.603*X02) Z=XO*(0.45+EPKT) 4 IF(Z.GE. 170. ) GO TO 3 COLH= 1.093055E-10*R N2/X0*TSQRT*Y* v* $ (A* (EGNE (Y,IY) /Y-EONE (Z,INZ) /Z) • $ (B-A*ALOG (2.*RN2/X0) ) * (ZETA (Y) -ZETA (2) ) ) CCLH=CCLH*(13.598/XIP) **2 IF (IY.EQ.O.OR.INZ.EQ.O) GOTO 1 EETURN 1 WRITE(6, 1000) Y,Z,IY,INZ 1000 FORMAT (' ****** ERROR IN EQNE*,4E15.4) RETURN 3 COLH=0. BETURN C C JOHNSON'S COILISICNAL IONIZATION FORMULA C CURRENTLY ONLY FOB IONIZATION FfiCM LEVELS 1 AND 2 C END FUNCTION ZETA (T) C C LITTLE FUNCTION REQUIRED BY COLE C EE-EXP (-T) E0=EF/T E1 = EONE (T, IN D) IF(IND.EQ.O) WRITE (6,100) T 100 FORMAT(• ***** ZETA EGNE ERBOR*,E15.4) I2=EF-T*E1 ZETA=E0-2. *E1*E2 BETUBN END SUBROUTINE REC(J,I) C C CALCULATION OF RECOMBINATION RA1ES FOB C ALL ELEMENTS BUT HYDROGEN C USES ALDROVANDI AND PEQUIGNOT TABLE C DIMENSION INDEX (16,9) DIMENSION IZED(9) REAL ARAD (76) , ETA (76) ,TMAX(76) ,TCRIT (76) ,ADI(76) , $ TO (76) , EDI (76) , $ Tl (76) ,RREC(76) ,DBEC (76) REAL CHIT(76),UREC(76),IP1(76) LOGICAL VEREOS ,ULTBA,SEMICO,FUDGE COMMON /A/ INDEX,IZED,DEN,DENE,T,TK,TKI,T4,TSQRT, $ VERBOS,LAST COMMON /RECO/ RREC,DREC,U8EC,ARAD,ETA,TMAX,TCRIT,AD $ I,T0,BDI,T1 COMMON /COLREC/ IP 1,CHIT,RNNOT COMMON /CONTEO/ SEMICO, ULTRA COMMON /CFUDJ/ F UD J,R N OT,F UDG E IJ=INDEX (J,I) FUDJ=1. IF (J. NE. IZED (I) ) GO TO 199 IF (IP1 (1*3) *TKI. GT. 170.) GO TO 200 101 DP.EC (IJ)=0. RREC(IJ)=0.0 Z=FLGAT (J) NNOT=BNOT NTGP=MIN0 (9,NNGT) DO 111 N=1,NTOP C CAN CHANGE DO LOOP BANGE TO 2,9 FOB OPTICALLY THICK TO C LYMAN ALPHA BBEC (IJ) = BBEC (IJ) •Z*BHII (IP 1 (IJ) , N) 111 CONTINUE IF (NTOP.EQ.9) GO TO 112 BBEC (IJ) =RREC (IJ) + (BNOT-FLOAT (NTOP) ) *Z*BHII (IP1 (IJ) $ , NTCF+1) 112 CONTINUE UBEC (IJ)=0. 0 BETUEN C C FACTOR OF 3 IS TO MAKE UP FOB TENDENCY OF TMAX QUOTED TO C EE MUCH TO LOW C 199 IF (T. GT. 3. *TMAX (IJ) ) GO TO 200 IF(T.LT.TMAX(IJ)/2000.) GG TO 200 EBEC(IJ) =ABAD(IJ) *I4** (-ETA (IJ) ) UREC (IJ) = 1. 8E-14*IP1 (IJ) *TK** (- 1. 5) *CHIT(IJ) IF (.NOT. SEMICO) UREC(IJ)=0. GO TO 299 s 200 BBEC (I J) =0.0 DBEC(IJ)=0.0 UREC(IJ)=0.0 BETORN 299 IF(T.LT.TCBIT{IJ)/10.) GO TO 300 C C FACTOR OF 10 AN ATTEMPT TO MAKE TRANSITION SMOOTHES C DREC (IJ) =ADI (IJ) *T** (-1.5) *EXP (-TO (IJ)/T) * (1. tBDI (IJ) $ *EXP (-T1 (IJ)/ $ T)) IF (» NOT. FUDGE) RETURN AEG=12. 55-7.*ALOG10 (BNOT) IF(ARG.LE.0.) ARG= 0. DELA=.01458333*ARG*ARG+0.09166667*ARG FUDJ=10.** (-DELA) BETORN 300 DREC (IJ) =0.0 C C ALL BUT HYDROGEN FROM FORMULAE OF ALDROVANI AND PEQUINO C T IN AA C H LIKE FROM JOHNSON C RETURN END BEAL FUNCTION RHII(XIP,N) C C RECOMBINATION TO HYDROGEN C DIMENSION INDEX(16,9) DIMENSION IZED (9) LOGICAL VEBBOS BEAL IP1 (76) ,CHIT (76) COMMON /A/ INDEX,IZED,DEN ,DENE,T,TK,TKI ,T4 , TS QBT , $ VEBBOS,LAST COMMON /COLBEC/ IP 1,CHIT,BNEOT COMMON /CFUDJ/ FUDJ,RNOT,FUDGE NNOT-BNNOT X0=1,-1./ (BNOT*BNOT) X02=1./(X0*X0) EIN=1./N FIN2=FIN*FIN XIN=XIP*FIN2*TKI XTIN=XO*XIN X2=XTIN*XTIN C C ABECMOWITZ AND ST EG UN EXPRESSION FOB EXP (X) EONE (X) C NOTE THAT X=X0*IPN/KT, AND NEED TO MAKE COERECTION TO C EXTEBIOB EXP(IPN/KT) C IF (XTIN. IE. 10.0) GO TO 4 EXE1=(X2+4.03640*XTIN+1.15198)/(X2+5.03637*XTIN+ $ 4.19160)/XTIN GO TO 5 4 EXE1=EXP (XTIN) *ECNE (XTIN,INX) IF(INX.EQ. 0) WBITE (6, 1000) XTIN 1000 FOBMAT(* *******EONE EBBOB IN BHII******,E15.6) 5 EXE2=1. -XTIN*EXE1 EXE3=0.5*(1.-XTIN*EXE2) IF(N.GT.2) GO TO 3 IF (N. EQ, 2) GO TO 2 G0=1. 133 G1=-0.4059 G2=.07O14 GO TC 1 2 G0=1.0785 G1=-0.2319 G2=0.02947 GO TC 1 3 G0=0.9935+0.2328*FIN-G.2196*FIN2 G1=-EIN*(0.6282-0.5598*FIN+C.5299*FIN2) G2=FIN2* (0. 3887-1. 181*FIN+1. 470*FIN2) 1 BHII=5.197E-14*XIN**1,5*EXP(XIN/(BNOT*BNOT))* $ (GO* EXE1+G1*EXE2/X0 +G2*EX E 3*X 02) C MULTIPLY ANSWER BY Z OF ION BETUBN END BEAL FUNCTION GEN(IE,IJ,IZ,Y) C C GAUNT FACTOB CALCULATION C USES MEWE AND KATO DATA C ANE MEWE APPBOX FOB EXP(Y)EGNE(Y) C DIMEKSICN A(49) ,B(49) ,C(49) ,D{49) DIMENSION INDEX (16,9) DIMENSION IZED (9) LOGICAL VERBOS REAL IP1 (76) ,CHIT (76) COMMON /A/ INDEX,IZED,DEN,DENE,T,IK,T Kl ,T4 , TS QRT , $ VERBOS,LAST COMMON /GFACI/ A,B,C,D COMMON /COLREC/ IE 1,CHIT,RNNOT X=1./(IZ-3.0001) IE (ID. GT. 28) GO TO 1 IF (ID.EQ.20) GO TO 100 IF (ID. EQ. 22) GO TO 10 3 IF (ID.EQ.21) GO TO 104 IF (ID. EQ. 23) GO TO 101 IF (ID.EQ.28) GO TO 102 1 GFN= A (ID)+ (B(ID) *Y-C(ID) *Y*Y+B(ID) ) * ( ALOG ( (Y+1.) /Y) $ -0.4/((Y+1.)*(Y+1.)))+C(ID) *Y RETORN C C ALL A LA MEWE C WITH ADDITIONS DUE TO KATO C 100 A(ID)=0.7*(1.-. 5*X) B(ID) = 1.-0.8*X C <ID)=-0.5* (1.-X) GO TO 1 101 A (ID)=0. 11* (1.+3. *X) GO TO 1 102 A(IE)=0.35*(1.+2. 7*X) B (ID) =-0. 1 1* (1. *5. 4*X) GO TO 1 103 A (ID) = -0. 16* (1.+2. *X) B (ID) =0.8* (1.0-0. 7*X) GO TO 1 104 A(ID) =-0.32* (1.-0. 9*X) B(IB)=0. 88* (1. -1. 7*X) C (ID) = 0. 27* (1.-2. 1*X) GO TO 1 END SUBROUTINE LINCOL C C LINE COOLING C RITH MODIFICATIONS FOR FINITE OPACITY C USES LINE LIST FROM MORTON C AND MORTON AND HAYDEN SMITH C REAL LCCOL (76) , ELINE (407) ,F (407) INTEGER IIND (407) ,JIND (407) ,IDENT (407) DIMENSION INDEX (16,9) DIMENSION IZED (9) INTEGER IB (16) /1, 2, 3*0, 3, 4, 5,0, 6, 0, 7, 0, 8, 0, 9/ REAL X(17,9) ,ABUND (9) COMMON /THICK/ X,ABUND,DV,TAUMAX COMMON /IINE/LCOOL,ELINE,F,I DENT,IIND,JIND,NLINE COMMON /A/ INDEX,IZED,DEN,DENE,T,TK,TKI,T4,TSQBT, $ VEREOS,LAST TAU=1. 152 IF(DV) 11,11,12 12 COLUMN=DEN/DV 11 CONTINUE DO 10 IJ=1,LAST 10 ICGOT(IJ) = 0. DO 1 L=1,NLINE II=IIND(L) IF(IR{IL) .GT.LAST) GO TO 1 IF (IL. EQ, 1) GO TO 1 C NOTE THAT THIS IS THE 2 OF THE ION JL=JIND(L) IV=IR(IL) IJ^INDEX <JL,IV) Y=ELINE(L)*TKI IF(TAU MAX) 5,5,4 4 TAU=3.2905 E-6*F(L)*ABUND(I?)*X(JL,1V) *CGLUMN/ELINE(L) IF(TAU-.OI) 6,7,7 6 TAU=1. GO TO 5 7 TAU= (1.-EXP(-TAU))/TAU 5 G=GFN (IDENT(L) ,IJ,IL,Y) 3 ICCCL (IJ) = LCCOL(IJ) + F (L) *G* EXP {-Y) *TAU 1 CONTINUE DC 2 IJ=1,LAST LCOOL (IJ)=2.71E-15/1SQRT*LCCCL(IJ) 2 CONTINUE BETUBN END SUBROUTINE HLINE(HLCCCI,NBOT) C C LINE COOLING FOR HYDBOGEN C BEAL IP1 (76) ,CHIT (76) ,HLCCCI (9) INTEGEB INDEX (16,9) ,IZED(9) LOGICAL VERBOS COMMON /COLEEC/ IP 1,CHIT,RNNOT COMMON /A/ INDEX,IZED,DEN,DENE,T,TK,TKI,T4,TSQRT, $ VEEEOS,LAST NNOT=RNNOT FB=FLOAT(N BOT) FB2=FB*FB EE4=FB2*FB2 FIN=1./FB FIN2=FIN*FIN IF(NBOT.GT.2) GO TO 3 IF (NEOT. EQ. 2) GO TO 2 G0=1. 133 Gl=-0.4059 G2=.07014 GO TO 11 2 G0=1.0785 G1=-0.2319 G2=0.02947 GO TO 11 3 G0=0.9935*0.232 8*FIN-0.2196*FIN2 G1=-FIN*(0.6282-0.5598*FIN+C.5299*FIN2) 153 G2=FIN2* (0. 3887-1. 1 81*FIN*-1 .470*FIN2) 11 SE=-0.603 IF (NBOT.EQ.2) SB=.1169 RN=0. 45 IF (NBOT.GE.2) RN-1.94*FB** (-1.57) DG 9 N=1,9 HLCOOL(N)=0.0 9 CONTINUE NTOP=MIN0(9,NNOT) KE1=NBOT+1 DO 1 N=NB1,NTOP FN= FLCflT (N) FN 2=FN*FN FN3=FN*FN2 X=1.-(FB/FN)**2 ANN=3.920561 *(FB/FN)**3/X**4*(G0+G1/X+G2/(X*X)) BNN=4.*FB4/(FN3*X*X) *(1. + 1. 233333/X + SE/(X*X) ) Y=13.59 8*TKI/FB2*X ENN=BN*X Z=RNN+Y E1Y=EGNE(Y,INY) E1Z=E0NE(Z,INZ) IF (INY. EQ.0.OB.INZ.EQ.0) WR ITE (6, 1000) Y,Z 1000 FOBMAT {• *******ECNE EEBOB IN HLINE',2E15.5) HLCOOL(N)=2.3814724E-21*TSQRT*FB2*Y*Y* (ANN* ( (1./Y+. $ 5)*E1Y $ - (1. /2 + . 5) *E1Z) • (BNN- ANN*ALCG (2.*FB2/X) )* $ ((EXP(-Y)-Y*E1Y)/Y-(EXP(-Z) -Z*E1Z)/Z) ) *X C C TEE FINAL MOST X IS FOB TBE ENERGY OF THE TRANSITION C TEE CONSTANT HAS A BOUT IN 13.598EV AND AN EV TO EBG C CONVERSION C 1 CONTINUE BETDBN END SUBBOUTINE LEVEL (J,I) C C LOWEST LEVEL IN EQUILIBRIUM WITH CONTINUUM C DIMENSION IP1 (76) ,CHIT (76) DIMENSION INDEX (16,9) , IZED (9) LOGICAL VERBOS,SEMICO,ULTRA COMMON /CONTBO/ SEMICO,ULTBA COMMON /A/ INDEX,IZED,DEN,DENE,T,TK,TKI,T4,TSQRT, $ VEBBOS,LAST COMMON /COLBEC/ IP1,CHIT,RNKCT C C SEATGN'S ESTIMATE OF LOWEST LEVEL IN EQUIIIEBIUM WITH C C CNTINUUM C FOLLOWS WILSON, BUT WILSON'S FUMBERS USED. C RJ=J BNCE= (1. 4E15*RJ**6.*TSQRT/DENE) **. 1428571 RNNOT=RNCE IF (J. NE. IZED (I) ) RETURN 154 Y=6.337053E-06*T/J**2 CCNS= <¥/(DENE*DENE))**.Q58823f3*RJ**. 823594 DO 2 IT=1,5 CQNS2=. 23529i»/(RNCE**3*Y) BNC=126.*CONS*EXP(CGNS2) DIF=BNCE-RNC IF (TABS (DIE) . LT. 0.5) GO TO 3 fiNCE=BNCE-DIF/(1.+BNC*3.*CGNS2/BNCE) 2 CONTINUE 3 IF(.NOT.SEMICO) RNC=1.0E06 IF (VEBBOS) WBITE (6, 1000) BNC, IT 1000 FOBMAT(* CONTINUUM IEVEL*,F10.1,» ITEEATION #',I3) BNNCT=BNC BETUBN END 155 FBOGBAM COEF C A C A PBOGBAM TO GFNEE ATE PHYSICAL PARAMETERS WHICH GO C INTO C THE COEFFICIENTS CALCULATED FOE THE DISPERSION RELATION C N C C INPUT IS FROM UNIT 1 AND CONSISTS OF THE DERIVATIVE C OUTPUT FEOM C THE BEATING AND COOLING PROGRAM IN THE FORM (N,T),{N C ,T+/-DELT), C (N +/-DELN,T) . C OUTPUT IS TG UNIT 7 AND CONSISTS OF THE 2EE0 ORDER C QUANTITIES AND C THEIR DENSITY AND TEMPERATURE DERIVATIVES. C C NAMELIST PARAMETERS C WFUDGE A FUDGE FACTOR FOB ALT IRING FLUX C GRAV THE GRAVITATION IN CM S-2 C VO THE GAS VELOCITY C DV VELOCITY DEBTVATIVE WHICH MAY BE CALCULATED C INTERNALLY, C BUT A STARTING VALUE IS NEEDED., C NCNEQ CALCULATION OF ENERGY FQUILIBBIUM BY MAKING OP C DIFFERENCE C WITH CON DUCT ION, C DNON AN DENSITY DIFFERENT THAN USED IN HEATING AND C CCCLING C TNGN A TEMPERATURE DIFFERENT THAN USED FOR HEATING C AND COOLING C DRHO DNON CONVERTED TO DENSITY (GM CM-3) C DT TEMPERATURE DERIVATIVE C D2T SECOND DERIVATIVE GF TEMPERATURE WITH RESPECT C TO DISTANCE, C CONKAP CONDUCTION CONSTANT. C NIINE NUMBER OF LINES IN FORCE CALCULATION C CEF THE INVERSE OF THE FACTOR TG CONVERT FROM ZERO TO C FIRST MOMENT C OF THE RADIATION FIELD C VERBOS CONTROLS PRINTING IF ON SEE DETAILS OF X DERIV C ATIVES,ETC. C SKIP IF CN NG NEW HEATING COOLING DATA READ, JUST C C ALCULATES FORCE C SLAB FOB A STATIC ATMOSPHEBE THIS IS THE EFFECTIVE C COLUMN DEPTH C CCNDUC IF OFF THE CONDUCTION QUANTITIES ABE SET TO ZE C BO C TEAD IS THE BRIGHT NESS TEMPERATURE OF THE RADIATION C FTELD/WF C USED IN THE CALCULATION OF THE WAVE DAMPI C NG DUE TO RADIATION C DTCL IS THE LEVEL GF TEE LOGARITHMIC DERIVATIVE BEL C OW WHICH IT IS C TO ZERO. 156 C DYNEQ IF ON THE MOMENTUM EQUATION IS EALANCED BY AEJ C USTING EV C DYNIT NUMBER OF ITERATIONS IK DYNEQ C DYNTCL ACCURACY BEQUIRED OF DV IN MOMENTUM BALANCE C WLINE IF ON THE FORCE CALCULATION FOB EACH LINE IS 0 C UTPUT C SLIM A CONVERGENCE AID IN DXNEQ, WHICH SHOULD BE SE C T TC ABCUT 1.5 C SPHERE IF ON THE DERIVATIVE OF THE DENSITY IS CAICULA C TED FOR A SPHERICAL C COORDINATE SYSTEM , ASSUMING SYMMETBY. C STABMU IS THE COSINE OF THE ANGLE TO THE STAR. C RSTAR IS THE DISTANCE FROM THE CENTRE CF THE STAR. 0 C NLY USED BY THE C DENSITY AND VELOCITY. DERIVATIVE. THE GRAVI C TY MUST BE ADJUSTED. C C C TEE OUTPUT QUANTITIES ABE MOSTLY SELF EXPLANATORY C EN ENDINGS ARE DENSITY DERIVATIVES, DT ABE TEMPERATU C RE C C REAL LEREMS , LRR AD, LINLCS,X (17,9) , LCLX{76) ,CPHEAT<76) $ ,FTOT INTEGER INDEX(16,9) /1, 15*0,2,3,14*0,4,5,6,7,8,9,1 $ 0*0, $ 10,11,12,13,14,15,16,9*0, $ 17,18,19,20,21,22,23,24,6*0, $ 2 5,26,27,28,29,30,31,32,33,34,6*0, $ 35,36,37,38,39,40, 4 1, 42, 43,44, 45, 46, 4*0 , $ 47,48,4 9,50,51,52,53,54,55,56,57,5 8,59,60,2*0, $ 61,62,63,64,65,66,67,68,6 9,70,71,72,73,74,75,76/ REAL AD (5) ,AT(5) ,AWFJ (5) , AGAIN (5) ,ACOOL (5) , AE (5) , AH S (5), $ ADE (5) ,AX(17,9,5) REAL KBCLTZ /1.38062E-16/ REAL RGAS /8.314E7/ REAL ABUND (9) REAL AMASS(9) /1.008,4.0026,12.0111,14.0067,15.9994 $ 20.179 $ ' 24.305,28.G86,32.06/ DIMENSION ELINE { 1000) ,F (1000) ,11 ( 1000) , JJ(1000) ,FLU $ X(100) , $ FPT(100) INTEGER IZED(9)/1,2,6,7,8,10, 12,14,16/ LOGICAL FREEZE,NONEQ,VEREOS,SKIP,CONDOC,DYNEQ,WLINE LOGICAL SPHERE BEAL*8 SLIM BEAL VTHEHM(9) EEAI*8 DLOG10,DVOLD,EGBDEV,ESQBT,DSIGN,DABS BEAL*8 N0,T0,V0,DN,DT,DV,RHOCV,SCUND, $ NEO,HO,EO,DNEDT,DNEEN,DHDT,DHDN,DEDT,DEDN, $ D2T,CONKAP,DKDT,DKDN,GRAV,GRAD,GRADE,DGRDT, $ DGRDN, $ GO,L0,DGDT,DGDN,DLDT,DLDN 157 REAL PHOT (76) , PHEAT (76) , SIGH A ( 100, 76) ,DELE (100) SEAL*8 GRIJ (16,9) ,DXDT(16,9) ,DXDN(16,9) EEAL*8 DIFF,DELV,DIE C L D , D F D E V , DP BE AL*8 V,T,DEN REAL*8 CK,FRAP,DGL,GL,TAU,DEXF INTEGEB NUO (76) ,NUF(76) INTEGEB DYNIT COMMON /?/ ELINE,F,II,JJ,FLUX,FPT,NLINE,NFLUX COMMON /A/ ABUND,X,VTHERM,SLAB COMMON /PHOBCE/ GRAD,DT,DGRDN,DXDT,DXDN,GRIJ,DGR $ DGRDDV COMMON /CONTBO/ WLINE,SPBERE,STARMU,RSTAR COMMON/DER/ DNU,DNL,DTU,DTL ,DEN,T NAM ELIST /PABAM/ F B FEZ E,W FUIG£,GR AV,V 0,DV,NONEQ,D NO $ N,TNON,DRHO, $ DT,D2T,NLINE,CHE,VERBOS,SKIP,SLAB,CONDUC^T $ RAD $ ,DTOL,DY NEQ,DYNIT,DYNTOL,HLINE,SLIM,SPHEBE,STARM $ U,RSTAR NAMELIST /PHYSIC/ NO,TO,VO,EN,DT,DV,RHOCV,SOUND, $ NE0,H0,E0,DNEDT,DNEDN,DHDT,DHDN,DEBT,DEDN, $ D2T,CONKAP,DK DT,DKDN,GRAV,GRAD,GRADE,DGRDT, $ DGRDN, $ GO,LO,DGDT,DGDN,DLDT,DLDN NFLUX=97 FREEZE=. FALSE. HFDDGE=1. GBAV=1. E4 C C V>0 AWAY FROM STAR, SIMILAB1Y Z INCREASES UPWARDS C V0=0. DV-0. NCNEQ=. TRUE. DTOL=.05 EYNEQ=. FALSE. , DYNIT-10 DYNTOL=1.E-3 SLIM=.5D0 TRAD=50000. CONDUC=.TRUE. DNON=1. E11 TNON=0. T= 1. E5 DRHO=0. DT=0. D2T=0. NIINE=874 CHF=1. SLAE=0. VERBOS=.TBUE. WLINE=. FALSE. SKIP=.FALSE. SPHEBE=. FALSE. STARMU=1. EST AR= 1. E12 158 BEHIND 1 BEHIND 2 C C BEAD IN LINE AND FLUX IAT A (USUALLY FILE L FOB C ED AT) C BEAD (2) ELINE,F,II,JJ,FLUX,FPT C C INCOMING FLUX HAS BEEN MULTIPLIED BY 4*PI/C C DG 49 18=1,96 DEL E (IN) =FPT (IN + 1) -FPT (IN) 49 CONTINUE BEHIND 3 BEAD(3) PHOT,PHEAT,SIGMA,FTOT1 BEHIND 4 BEAD (4,9774) (NDO (IJ) , NUF< IJ) , IJ = 1, 76) 9774 FOBMAT (214) C C BEADS FILE CONTAINING INTEGBATTON FBEQUENCIES WHICH HAS C USED BY PHOTION C 998 BEAD (5,PABAM, END=999) IF(.NOT.VEBBOS) WLINE=.FALSE. IF {. NOT. NONEQ. AND. VO. EQ.O.) V0=1. D7 WBITE(6,1014) C C BEAD IN SET OF 5 OUTPUT QUANTITIES FROM HEAT COOL PBGGBAM C 1014 FOBMAT(*1») WBITE(6,PABAM) IF (SKIP) GO TO 111 DC 14 1=1,5 BEAD(1) D,T A,W FJ,FJ,GAIN,COOL,LBB EMS,LBRAD,LINLOS, $ EINT,ENTH,DE,ABUND,X,LCLX,CPHEAT,FTOT AD(I) = D AT (I) =TA AHFJ (I) = WFJ AGAIN(I)=GAIN AC0CL(I)=CO0L AE(I) =EINT AH (I) = ENTH ADE (I) =DE DO 15 IZ=1,9 IQ1=IZED (IZ) + 1 DO 15 J=1,IQ1 AX(J,IZ,I)=X(J,IZ) 15 CONTINUE 14 CONTINUE FA=ABUND(3)/3.0 35918E-04 WBITE (6,1000) FJ,FA,FTOT 1000 FOBMAT('OFJ,IA,FTOT*,2F11.6,E15.7,/* DEN,TEMP,WFJ,H* $ ,*EAT,COOL,«, $ * EI NT, ENTHALPY, DEN E* ) DO 1 1=1,5 AGAIN (I) = AGAIN (I)/AD(I) 1 CONTINUE WBITE (6,1001) (SD (ID) , AT (ID) , AW FJ (ID) ,AGAIN (ID) , $ ACOOL (ID) ,AE(ID) , $ AH (I D) , ADE (IE) , ID = 1 , 5 ) 1001 FOBMAT(1X,8E15.1) C C SET UP CENTRAL VALOES C DEN=AD (1) DENE=ADE(1) T= AT (1) N0=DEN N10=DENE TG=T DEINT=AE(1) DENTH=AH (1) INTH= AH (1) ST4=SQRT(SNGL(T)*1.E-04) C C THERMAL VELOCITY ALREADY DIVIDED BY C C DO 11 1=1,9 VTHERM (I) = 4. 28 33E-5/SQRT (AM ASS (I) ) *ST4 11 CONTINUE DEN 1= ADE (1) C C CALCULATE UPPER AND LOWER DERIVATIVES DIFFERENCES C DNU=AD (4)-AE (1) DNL=AD(1)-AD(5) DTU= AT (2) - AT (1) DTL= AT (1) - AT (3) C C DERIVATIVES OF THE IONIZATION FRACTIONS C DO 16 IZ=1,9 IQ=IZED(IZ) DO 17 J=1,IQ GRIJ (J,IZ)=0. DXDT(J,IZ)=0. EXDN (J,IZ)=0. TF(AX(J,IZ, 1) .LE. 1.E-10) GO TO 17 DXDT ( J , IZ) = • 5* ( (AX (J , IZ , 2) - AX (J , IZ, 1) )/DTU + $ (AX (J,IZ,1)-AX(«3,IZ,3) )/DTL) IF ( E AES ( EX DT (J, IZ) *T/AX (J ,IZ,1) ). LE. DTOL) DXDT (J,IZ) $ =0.D0 DXDN(J,IZ) = . 5*( (AX (0,IZ,4) - AX (0 ,IZ, 1) )/DNU + $ (AX (J, IZ, 1) - AX (J, IZ , 5) )/DNL) IF (DABS(DXDN (J,IZ)*DEN/AX (J ,IZ,1)).LE.DTOL) DXDN (J, $ IZ)=0. DO 17 CONTINUE 16 CONTINUE IF ( .NOT.VERBOS) GO TO 20 DC 19 IZ=1,9 IQ=IZED(IZ) WRITE (6,1029) IQ 1029 FORMAT(* DXDT,DXDN FOB ATGM Z=*,I2) WBITE (6,1030) (DXDT(J,IZ) ,J=1,IQ) 1030 FOBMAT(1X,10E13.5) WBITE (6,1030) (0XDN(J,IZ) ,J=1,IQ) 19 CONTINUE 20 DO 18 IZ=1,9 IQ=IZED(IZ) DO 18 J=1,IQ X(J,IZ)=AX(J,IZ,1) 18 CONTINUE C C PHYSICAL DEBIV ATIV ES C DU=0. DL=0. CALL DERIV(ADE,DNEIN,DNEDT,DTOL) WBITE (6,1003) DNEDN,DNEDT 1003 FOBMAT {* DNEDN, DNEDT* , 2E1 5. 7) DN2=AD <1)*AD (1) CALX DEBIV(ACCOL,DLDN,DLDI,DTOL) DLDN= AD (1) *2.*ACOOL (1) +DL DN *DN2 DLDT=DLDT*DN2 BL=ACCCL (1) *DN2 WRITE(6,1005) DLDN,DLDT , EL 1005 FOBMAT ( * DLDN,DLDT,LOSS ES *,3E15.7) CALL DEBIV (AGAIN,DGDN,DGDT,DTOL) GAIN= AGAIN (1)*DN2 DGAINC=GAIN/2.9979E10 DGDN=AD (1) *2.*AGAIN (1) +DGDN*DN2 DGDT= DGDT* DN2 WRITE (6,1007) DGDN,DGDT,GAIN 1007 FORM AT ( * DGDN, DGDT , GAINS * , 4E15. 7) CALL DERIV(AE,DEDN,DEDT,DTGL) C= D EDT/KBOLTZ WRITE (6,1009) DEDN,DEDT,C 1009 FOB MAT (* DEDN, DEDT , CV » , 4E15. 7) CALL DERIV (AH,DHDN,DHDT ,DTOI) C=DHDT/KBOLTZ C C CCNDUCTION CALCULATION FBGM SPITZEB C WRITE (6,1011) DHDN, DHDT, C 1011 FORMAT(* DHDN,DHDT,CP»,4E15.7) IE (CCNDUC) GO TO 110 DKDT=0. DKDN=0. CONKAP=0. GO TO 113 110 CONTINUE COULOG = 9.00 + 3. 45*ALOG10(SNGI (T) )-1. 15*ALOG 10 ( DENE) CGNKAP=1.8E-5*T**2.5/COULOG DKDT=2.5*CONKAP/T+CCNKAP/ (COHLOG*T) *3.45 DKD N=CCNKAP/(COULOG*DEN£) * (-1 . 15) C C THE MEAN MASS OF AN ATOM COEFFICIENT C 113 WRITE (6, 1013) CCNKAP, DKDT, DKDN 161 1013 FORMAT(* CONKAP,DKDT,DKDN»,3E15.7) EHCCCN=0. BO 10 1=1,9 BHGCGN=RHOCGN*AMASS (I) *AEUND(I) 10 CONTINUE C C TEE FORCE DUE TO CONTINUUM RADIATION C RHOCV=RBOCON*1.660531E-2U 111 CONTINUE GRADC=0. DGCDN=0. DGCDT=0. DO 51 1=1,9 IZ=IZED(I) DO 52 J=1,IZ IJ=INDEX(J,I) XAC=ABUND(I) *X (J,I) IF(XAC.LT. 1.E-10) GO TO 52 KUB=NU0(IJ) NUE=NUF (IJ) IF(NUE. EQ« NUB) GO TO 52 GCL=0. DO 50 IN=NUB,NUE GCL=GCL+.5* (FLUX(IN + 1) *SI GM A (IN • 1, IJ) +FLUX (IN)* $ SIGMA (IN,IJ))* $ DELE (IN) 50 CONTINUE GR ADC=GR ADC+GCL*X AC DGCDN=DGCDN*GCL*DXDN<J,I) *ABUND(I) DGCDT=DGCDT + GCL*DXDT (J, I) *AEUND (I) 52 CONTINUE 51 CONTINUE DGCDN=DGCDN*CHF/RHOCV DGCDT= DGCDT*CHF/EHOCV GRADC=GRADC/RHOCV*CHF TRKAPC=0. RHO=DEN *RHOCV DDEN=0. IFIT=0 DIFF=0. C C IF DYNEQ EALANCE MOMENTUM EQUATION C IE (DYNEQ) WRITE (6, 1062) 1062 FORMAT(6X,,DIFFJ,11X,,«DDEN«,11X,,DV«,13X,«BELV»,11X I * DGRDDV',9X,•GRAD*,11X,*DP«,13X,»IFIT*) 201 CONTINUE GRAD=0. ,, GRADE=0. DGBDT=0. DGRDN=0. IF (FTOT. EQ.O. ) GO TO 112 DGRDD V=0. C C CALCULATION OF LINE FORCE 162 C CALL FORCE(V0,DV,T,DEN) EGRBDV= DGRDDV/RHOCV *CHF GRADL=GRAD/RHOCV*CHF GRADE=FTCT*2. 2 19E-35*DENE/ (DEN*RHOCV) *CHF DGEDT=GRADE*DNE DT/DENE DGR ET= DGRDT *CH F/RHO CV•DG E DT • DGCDT DGRDN=DGRDN*CHF/RHOCV+DGCDN C NOTE THAN GBIJ IS NOT MULTIPLIED BY CHF C NUMBER IS THOMSON CROSS SECTIONCVEB THE SPEED OF LIGHT GBAD=GBADL*GBADE+GRADC IF(.NOT.DYNEQ) GO TO 200 BN=~DV*N0/V0 IF(SPHERE) DN=DN~2.DG*N0/RSTAB DDEK=DN DIFOLD=DIFF DP=KBOLTZ/BHO* {DT* (NO+ NEO+DNEDT*T0/NO) + DN* (1.D0 + $ DNEDN) *I0) DIFF= DP $ *GBAV-GRAD+VO*DV IF (IFIT. EQ. 0) GO TO 202 C C ADJUSTMENT OF DV/DZ FOR MOMENTUM EALANCE C DFDEV= (DIFF-DIFOLD) / (DV-BVOLD) DELV=-DIFF/DFDDV IF(DABS(DELV).GT.DAES(EV).AND.SLIM.NE.0.DO) DELV= $ DSIGN(1.D0,DELV) *DABS (DV) *SLIM EVOLD=DV DV=DV+DELV WRITE (6, 1061) DIFF,DDEN, DV, DELV, DGR DDV, GRAD, DP, I FIT 1061 FORMAT(1X,7E15.5,15) IFTT=IFIT + 1 IF(DABS(DELV/DV).LT.DYNTOL) GC TO 200 IF (IFIT.LE. EYNIT) GO TO 201 GO TO 200 202 DVOLD=DV DV=DV*1.1D0 IFIT=1 GO TO 201 200 CONTINUE TBKAPC=GEAD/FTOT GB ATIO= GRAD/GBADE GAMMA=(GRAV-GRAD)/GRAV WRITE (6,1012) GRAD,GRADE,GBADL,GBADC,GBATIO,GAMMA, $ DGRDT,DGRDN 1012 FORMAT (* GRAD,GRADE,GRAEL,GRAEC,GRATIO,GAMMA,DGRDT, $ DGRDN */1X, $ 8E16.6) 112 CONTINUE 10= (DEN + DENE) *T*KBOLTZ DP=RHO* (-V0*DV-GRA V+GRAD) IF ( NONEQ) GC TO 46 IF (VO.EQ.O.) GO TO 44 DRHO=-RHO*DV/V0 DDEN= DRHC/EHOCV 44 IF(NONEQ.AND.(VO.EQ.O.)) GO TO 45 C C ENERGY EQUILIBRIUM FORCED C EI USING CONDUCTIVE ENERGY TRANSPORT C RARELY USED C VC1=1. C IK FRAME OF STAR VC1=1 DT=(-KBOLTZ/BHOCV*(DDEN*(1.+DNEDN) ) *T/DEN $ -V0*DV+VC1*GBAD-GRAV)/ $ (KBOLTZ/RHOCV*(T/DEN*DNEDT+1. + DENE/DEN) ) GO TO 46 45 DDEN= ( (-GRAV+GR AD) *HHO- (DEN + DENE) *KBOLTZ*DT) / $ (KBOLTZ*T*(1.+DNEDN)) DBHG= DDEN*RBOCV 46 DH=DHDT*DI+DHDN*DRHO/RHOCV DKAF=DKDT*DT*DK EN*E EFN TN= (DEN+DENE)*T SOUND=SQRT (. 5*( ( (AD{4) +ADE(4) )*AT (4)-TN)/DNU + $ (TN- (AD (5) +ADE (5) ) *AT (5) ) /DNL) *RGAS/RHOCON) IF (NONEQ) GO TO 33 D2T=(-VC1*GAIN + RL + DDEN*V0*( 1.5*V0**2+ENTH)*RHOCV $ +DEN*RHOC¥*DV* (1. 5*V0**2+ENTH) $ •RHOCV*DEN*(DHBT*DT+DHEN*DEN))/{-CCNKAE) 33 IF (NGNEQ. AND. (TNON . GT. 0 .) ) T=TNON WRITE (6,1022) PO,RHO,DRHO,DP,SOUND,NONEQ,DT, $ EKAP,D2T 1022 FORMAT (» PHYSICAL PARAMETERS CALCULATED,P0,BHO,DRHO, $ »DP,SOUND' , 5E15.7,/' NONEQ, DT, DK AP , * , $ »D2T»,/1X,L8,3E15.7) CFLUX=CGNKAP*D2T*DKAP*DT WRITE (6,1024) CFLUX 1024 FORMAT(1 CONDUCTION FLUX=«, E15.7) IF (SKIP) GO TO 122 DN=DDEN H0= ENTH/RHOCV E0=EINT/RHOCV DHDT=DHDT/RHOC¥ DHDN=DHDN/RHOCV DEDT=DEDT/RHOCV DEDN=DEDN/RHOCV G0=GAIN L0=BL 122 CONTINUE C C OUTPUT OF PHYSICAL QUANTITIES C WBITE (7) N0,T0,V0,DN,DT,DV,BHOCV,SOUND, $ NE0,H0,E0,D NEDT,D NED N,DHDT,DH DN,DE DT,DEDN, $ D2T,CONKAP,DKDT,DKDN,GRAV,GRAD,GRADE,DGRDT$ DGRDN, $ GO , LO, DGDT, DGDN, DLDT, DLDN WRITE(6,PHYSIC) C C CALCULATION OF PHYSICAL LIMITING FREQUENCIES c WRAB=5.6S97E-03*TRAD**3*TRKAPC WCCCL=6.28319*10/(£0*RHO) 8REC=1.88E-10/DSQRT (T) *DEN EQC=C0ULOG*BENE*T** (-1.5) WEQPE-2.5E-02*EQC BEQEE=4.57E01*EQC WCGND=6.28*ABS (SN GL ( DK DT * DT + CO N K A P* D 2 T) ) /(DEDT*RHO) 9SITE (6,1025) WRAD,WCOQI,WREC ,WEQPE,WEQEE,WCOND 1025 FOR MAT(• WRAD,HCOOL,WREC,WEQPE,WEQEE,WCOND*,6E15.3) IF (DDEN. EQ.O. ) GO TO 519 HSCALE=DABS(DEN/DDEN) GAM=DHDT/DEDT WACS=GAM*GRAV/(2.*SCUND) WACH=SQBT (SNGL (GAM*GRAV/ (4. *HSCALE) ) ) WBVS=SQRT (GAM-1.)*GBAV/SOUNO WBVH=SQHT(SNGL( (GAM-1.) *GBAV/(GAM*HSCALE) ) ) WBITE (6,1033) HSCALE,GAM,WACS,WACH,WBVS,WBVH 1033 FORMAT(* HSCALE,GAM,MACS,WACH,WBVS,WBVH,»,6E14.3) 519 CONTINUE GO TO 99 8 999 STOP END SUBROUTINE FOBCE(V,EV,T, DEN) C C BOOTINE TO CALCULATE LINE FORCE C WITH SIMPLE LUCY RADIATIVE TRANSFER C FOE ONE SCATTERING LINE C ALWAYS MOST BE SUPERSONIC FLOW C MO OVERLAPPING LINES ALLOWED FOB C LOGICAL VEBEOS INTEGEB INV(16) /1,2,0,0,0,3,4,5,0,6,0,7,0,8,0,9/ DIMENSION ELINE(1000),F(100C),II(1000),JJ(1000) DIMENSION ABUND (9) ,2(17,9) , VTHEBM (9) ,FLUX(100) , $ FPT(100) SEA L*8 DXDT (16,9) ,DXDN(16,9) , GRAD,DGRDT,DGRDN , $ GRIJ(16,9) BEAL*8 V,DV,T,DEN,EVI REAL*8 CK,GL,DGL,TAO,TAUC,DGBDDV,FKAP,DGLDN B E A L* 8 DABS LOGICAL SPHEBE COMMON /PHORCE/ GBAD,DGBDT,DGBDN,DXDT,DXDN,GRIJ, $ DGRDDV COMMON /A/ ABUND,X,VTHEBM,SLAB COMMON /?/ ELINE,F,II,JJ,FLUX,FPT,NLIRE,NFLUX COMMON /CONTEO/ VEREOS,SPHERE,STABMU,BSTAB IVI=DABS(DV) IF (SPHERE) EVI=£ABS (. 5*{1. •STABMU*STABM U) * $ (DV-V/RSIAR) + V/RSTAR) IF=1 FDF=1.-V/3.£10 COLUMN=DEN*SLAB IF (DVI. NE. 0.) COLUMN = 2. 9979 110/DVI*DEN IF(COLUMN.EQ.0.) GO TO 100 DO 10 L= 1,NLINE J=JJ (L) 1=11 (L) IV=INV(I) 3 IF (ELINE (L) . GT. FDF*FPT (II) ) GO TO 2 FNU= (FLUX (IF-1) +S* (ELINE (L) -FDF*FPT(IF) ) ) *FCF CK=1.0 976E-16*F (L) *ABUND (IV) C CONSTANT IS•PI*E**2/(HE*C) / (EV TO HZ CONVEBSION) TAU=CK/ELINE(L)*COLUMN*X(J,1V) IF<X (J,IV) .IE. 1.E-10) GOTO 10 DGL=CK*FNU GI=DGL*X (J,IV) C C ELINE IS IN EV BUT NOTE THAN END IS EBG CM-2 S-1 EV-1 C TAUC=TAU*DVI IF(IAU.LE. 1.E-3) GG TO 4 IF (TAU.GT.170.) GO TG 6 DGRDDV=DGBDDV+GL/TAUC*(1.-DEXP(-TAU)*(1.DO+TAU)) GL=GL*(1.-DEXP(-TAU) )/TAU DGLDN=-GL/DEN*DGL*DEXP (-TAU) * (DXDN (J, IV) +X (J,IV) / $ DEN) 4 CGNTINUE DGBDDV=DGRDDV*GL/TAUC*TAU*TAU DGL= DGL * DEX P ( -T AU) GO TO 5 2 IF=IF*1 IF (IF. GT.NFLUX) BETUEN S= (FLUX (IF) -FLUX (IF-1) ) / (FPT (IF) - FPT (IF-1) + 1. E-50) GO TC 3 6 CONTINUE DGB DDV=DGRBDV +GL/TAUC GL=GL/TAU DGL DN=-GL/DEN DGL=0. 5 GRAB=GRAD+GL GEIJ (J,IV)=GRIJ (J,IV) +GL DGB DT= DGB DT • DGL * DX DT (J , IV) DGRDN=DGRDN +DGLDN 1 IF<VEBBGS) 8BITE (6, 1000) L, ELINE(L) ,J,I,IV,FNU,TAU, $ GL,CK,FRAP,DGL 1000 FORMAT (1X#I3,F10. 5,3X5, 6E1S. 5) 10 CONTINUE C C ALSO, WHAT ABOUT THE CONTINUUM OPACITY C BETUEN C C OPTICALLY THIN CALC C 100 DO 101 L=1,NLINE J=JJ (I) 1=11 (L) IV=INV (I) 103 IF(ELINE(L) .GT.FDF*FPT (IF)) GO TO 102 FN0= (FLUX (IF-1) +S* ( ELINE (L) -FDF*FPT (IF) ) )*FDF DGL=1.0976E-16*F (L) * ABUND (IV) *FNU GL=LGL*1 (J,IV) GRAD=GRAD+GL GRIJ (J,IV) = GRIJ (J,IV) +GL D G R D1~ D G R DI + D G L * D X DT (J , IV) DGFDN=DGRDN*DGL*DXDN (J, IV) IF (VERBOS) WRITE (6, 1000) I-,EIINE(L) ,J,I,IV,ENU,GL,DGL GO 10 101 102 IE=IF+1 IF(IF.GT.NFLUX) RETURN S= (FLUX (IF) -FLUX(IF-I) ) / (FPT (IF) - FPT (IF-1) +1.E-50) GO TO 103 101 CONTINUE RETURN ENE SUBROUTINE DERIV (Q,DQDN,DQDT,DTOL) C C A ROUTINE TO CALCULATE DERIVATIVES C OF PHYSICAL QUANTITIES AND CHECK C THAT THEIR LOG DERIVATIVES EXCEED C SOKE MINIMUM, IF NOT THE ARE SET TO ZERO. REAL*4 Q(5) REAI*8 DQDN,DQDT REAL*8 DEN ,T COMMON /DER/ DNU,DNL,DTU,DTL,DEN,T IF(Q(1) .EQ.O.) GO TO 1 DU* <Q(4)-Q(1))/DNU DL=(Q(1)-Q (5))/DNL DQDN=.5*(DU+DL) DLQ=DQDN*DEN/Q(1) IF (AES (DLQ).LT.DTOL) GO TO 2 « DU=(Q(2)-Q (1))/DTU DL= <Q(1)-Q(3))/DTL DQDT=.5*(DU+DL) DLQ=DQDT*T/Q (1) IF(ABS(DLQ).LT.DTOL) GG TO 3 BETURN 1 DQDN=0.D0 DQDT=0. DO RETURN 2 WRITE (6,1001) DU,DL,DQDN,DLQ,DTOL 1001 FORMAT(* DU,DL,DQDN,DLQ,DTOL * ,5E15.7) EQDN=0. DO GO TO 4 3 WRITE (6,1002) DU,DL,DQDT,DLQ,DTOL 1002 FORMAT('DU,DL,DQDT,DLQ,DTOL * ,5E15.7) DQDT=0.DO RETURN END PROGRAM CGCALG C C C THIS PROGRAM CALCULATES THE COEFFICIENTS OF W AND K C FOB THE DISPERSION RELATION POLYNOMIAL C THE PHYSICAL QUANTITIES PBOEUCFD BY THE PROGRAM COEF C ABI USED AS INPUT C THE OUTPUT IS USFB BY TEE PBOGBAM DISPEB C THIS IS A SUBROUTINE CALLED IN THE DISPEB* C C SUEBGUTINE CCCALC(*) BEAL*8 KBOLTZ,RMU,VC1,C,VG LOGICAL FREEZE LOGICAL MANY,RESTOR BEAI*8 CMASS,CMTM, CENE BEAL*8 C,VC1,MTMKN,MTMC N,MTHKT,MTMCT,DVG, $ EW N , EKN, ECN,EHT, FKT, iCT, EH V, EKV, ECV BEAL*8 N0#TO,VQ,DN,DT,DV,BHOCV,SOUND, $ NE0,H0,EO,DNE DT,DNECN,DHDT,DHDN,DEDT,DEDN, $ D2T ,CONKAP ,DKDT,DKD K ,GEAV ,GRAD0 ,GBADE,DGBDT $ ,DGBDN, $ GO,LO,DGDT,DGDN,DLDT,DLDN BEAL* 8 CBD (5,4) , CI £(5,4) COMMON /COEFS/ CBD,CID COMMON /CNTB02/ MANY,BESTOB NAMELIST /PHYSIC/ NO,T0,VO,DN,DT,DV,BHOCV,S00ND, $ NEO,HO,EO,DNEDT,DNE DN,DHDT,DHDN,DEDT,DEDN, $ D2T,CONKAP,DKDT,DKDN,GBAV,GBADO ,GBADE, DGBDT $ ,DGBDN, $ GO,LO,DGDT,DGDN,DLDT,OLDN,FBEEZE NAMELIST /DISCO/ BMU,VC1,HTMKN,MTMCN,MTMKT,MTMCT,DVG, $ EWN,EKN,ECN,EWT,EKT,ECT,EHV,EKV,ECV IF (. NOT. BESTOB) GO TO 4 BACKSPACE 1 GG TO 3 4 CONTINUE IF (MANY) GO TO 2 3 BEAD(1,END=999) NO,TO,VO,DN,DT,DV,BHOCV,SOU ND, $ NEO,HO,EO,ENEDT,DNE DN,DHDT,DHDN,DE DT,DEDN, $ D2T,CONKAP,DKDT,DKDN,GBAV,GBAEO,GBADE,DGBDT $ ,DGBDN, $ GO,LO,DGDT,DGDN,DLDT,DLDN 2 FBEEZ£=.FALSE. C=2.9979D10 KBGLTZ=1.380626D-16 BEAD(5,PHYSIC,END=999) BMU=KEGLTZ/BHOCV VC1=1.D0 C C THE NAMES OF THESE VARIABLES COMES FBGM C C THE LINEABIZATION OF THE EQDATICNS OF MOTION C 168 C EXCEPT NOTE THAT P THERE IS REPIACED EY HTM, C CMASS=DN*V0+DV*N0 CMTM=BMU*T0/N0* (DN + (INEEN*EN+ENEDT*DT) ) +DV* VO $ +RMU*DT*(1.D0+NE0/NO)-VC1*GRAD0+GRAV CENE=-VC1*GG+L0*CGNKAP*D2T+ (D KD N* D N* DKDT*DT)* DT + DN* $ V0*BHOCV*{1.5D0* $ V0**2+H0)+N0*RHOCV*DV*(1.5D0*V0**2+H0)+ $ RHOCV*N0*(BHDT*DT*DHDN*DN) WRITE (6,1001) CMASS,CMTM, CENE 1001 FORMAT(* CONSERVATION EQUATIONS CMASS,CMTM,CENE', $ 3D25. 12) VC1=1.-V0/C C CALCULATION DONE IN FRAME MOVING WITH GAS VG=G.D0 WRITE (6,PHYSIC) MTMKN=KBOLTZ/(N0*RHOCV)*(DNEDN*T0+T0) MTMCN=KBCLTZ/(N0**2*EHOCV)*{-BN*T0+N0*DNEDN*DT $ -NE0*DT- (DN£DN*DN*DNEDT*DT) *T0) -VC1*DGRDN MTMKT=KEOLTZ/(N0*RHOCV)*(NE0+DNEDT*T0+N0) MTMCT=KBCLTZ/(NO*BHGCV)*(DN+DNEDT*DT+DNEDT*DT+DNEDN $ *DN)-VC1*DGRDT DVG=DV+GRAD0/C EWN=(DEDN*N 0+(VG**2/2.+E0))*RHOCV EKN=VG*(.5D0*VG**2 + H0+DEBN*NC)*RHOCV + |- DKDN) *DT ECN=RHOCV* (DN*VG*DHDN+1.5*DV *VG**2+D¥*H0+ $ DV*NO*DHDN+VG*(DHDN*DN+DHDT*DT)) + D2T* (-DKDN $ )-VC1*DGDN+DLDN $ +RHOCV* { VO * (DEDN*DN+DEDI*DT) *• DEDN*VO*DN) EWT=DEDT*N0*RHOCV EKT=RHOCV*DHDT*VG*N0+(-DKDT)*DT+{ (-DKDT)*DT +(-DKDN) $ *DN) ECT-DLDT-VC 1*DGDT+ (DHDT* (DN*VG+DV*N0) ) *EHOCV+ (-DKDT $ )*D2T $ +RHOCV*DEDT*VO*DN E8V=VG*N0*RHOCV EKV=(1.5*VG**2+H0)*N0*RHOCV ECV= (DN*(1.5*VG**2 + H0)*3.*DV *VG*N0+ NO *{DHDT*DT+DHDN $ *DN) ) $ *RHOCV+G0/C $ +RHOCV*(N0*V0*DV) C C C CUTFUT LINEARIZED EQUATION QUANTITIES C THIS IS USEFUL FOR EXAMINING THE MAGNITUDES C OF THE PHYSICAL QUANTITIES WHICH C ARE DOMINATING THE SITUATION C WRITE (6,DISCO) CRD (1,1) = $ -ECN*MTMCT*DN-DVG*ECT*DV+ HTMCT*ECV*DV*MTMCN*DN*EC T CID(1,1)=0. DO CRD(2,1)=0.DO CID(2,1) = 169 $ VG* (-DVG*£CT+MTMCT*ECV-ECT*DV)-ECN*MTMCT*NO-ECN* $ MTMKT*DN-EKN* XMTMCT*DN-DVG*EKT*DV +MTMCT *EKV*DV+MTMKT*ECV*DV+MTMCN $ *DN*EKT>MTMCN* XNO*ECT+MTMKN*DN*ECT CRD (3,1) = $ VG**2*ECT+VG* (DVG*EKT-MTMCT*EKV-MTMKT*ECV+EKT*DV) $ +ECN*MTMKT* XN0 + EKN*H1MCT*N0+EKN*MBKT*DK + DVG* (-CONKAP) *DV-MTMKT $ *EKV*DV $ -BTMCN*DN* X(-CCNKAP) -MTMCH*N0*EKT-MTMKN*DN*EKT-HTMKN*NO*ECT CID (3, 1)=0. DO CRD (4, 1) =O.DO CID (4,1) = I VG**2*EKT+VG* (DVG* (-CONKAP) -MTMKT*EKV* (-CGNKAP) *DV $ ) + $ EKN*HTMKT*NO-XHTMCN*N0*(-CONKAP) -MTMKN*DN* (-CGNKAP) -MTMKH*N0* EKT CED (5,1) = $ -VG**2*(-CONKAP)+ MTMO*N0* (-CCNKAP) CID (5,1)=0.DO CRD (1,2)=0.D0 CID(1,2) = $ EWN *MTMCT*DN+D VG*ECT*D VG*E KT*DV-MTMCT*ECV-MTMCT* $ EWV*DV-«IMCN* XDN*E8T*ECT*DV CRD (2,2) = $ VG* (-DVG*E8T+MTMCT*EWV-2*ECT-EHT*DV)-E8N*MTMCT*NO $ -EWN*MTMKT* XDN-DVG* EKT+MTMCT*EKV•MTMKT*£CV+MTMKT* EH V*D V+MTMCN* -$ N0*E8T + MTMO*DN X*EHT-EKT*DV CID (2,2)=0,D0 CRD (3,2) =0. DO CID (3,2) = $ -VG**2*EHT+VG* (MTMKT'*EWV-2*EKT) -EWN*MTMKT*NO-DVG* $ (-CGNKAP)+MTHKT X*EKV+MTMKN *NO*EBT-(-CONKAP)*DV CRD (4,2) = $ 2*VG*(-C0NKAP) CID (4,2) =0. DO CRD (5,2)=0.D0 CID <5,2)=0. DO CRD (1,3) = $ DVG*EwT-MTMCT*EwV4ECT+EHT*DV CID(1,3)=0.D0 CRD (2,3)=0. DO CID (2,3) = $ 2* VG* EHT- MT H KT * EH V • EKT CRD (3,3) = $ - (-COMKAP) CID (3,3)=0.D0 CRD(4,3) =0.D0 CID(4,3)=0.DO CRD(5,3)=0.D0 170 CID (5,3) = 0. DO CBD(1,4)=0.D0 CID <1,4) = $ -EST CBD (2,4) =0. DO CID(2,4)=0.D0 CBD (3,4)=0. DO CID (3,4) =0.D0 CBD <4,4)=0. DO CID(4,4)=0.D0 CBD (5,4)=0. DO CID(5,4)=0.D0 WBITE (6,1000) ((CBD (J.I) ,J=1,5) , (CID (J,I) , J= 1,5) ,1= $ 1,4) 1000 FOB H AT (' 0 * , 5D25.10 , /, 1 X, 5 D2 5.10) EETUBN 999 EETOBN1 END PROGRAM DISPER C C A EEOGBAM TO FIND THE BOOTS OF THE CUBIC DISPERSION C RELATION C FCUKD FOE THE CASE OF ONE DIMENSIONAL PLANE HAVES C SUBROUTINE COCALC USES THE OUTPUT FBOM COEF TO C ACTUAL FIND THE POLYNOMIAL C C C THE PARAMETERS CONTROLLING THE EEOGBAM ARE: C KLGG IF TBUE A LOGARITHMIC SERIES OF K ARE GENERATED C KMIN MINIMUM K VALUE (IF KLOG THEN THIS IS A LOG) C KM AX MAXIMUM K VALUE C PLREAL PLOT REAL FREQUENCIES C ELIMAG PLOT IMAGINARY IBEQUFNCIIS C KINC INCREMENT BETHEEN K VALUES C TGI IF TWO BOOTS LIE CLOSES THAN THIS THEY ABE C ASSUMED TO BE IDENTICAL C EBB ACCUBACY OF NDINVT SOL'N TO B=0, DD=0 C MAXIT # OF ITERATIONS IN NDINVT C EQTCL ACCURACY OF NEWTON ITERATION ^BOOT IMPROVER C NFILE FILE LINENUMBEB WHERE COEFFICIENTS START C USUAL 1+A MULTIPLE OF 5 C LABEL TRUE TO LABEL PLOTS C SUEDIV # OF SUBDIV USED EY NEXT BOOT ESTIMATOB C PEMIN FOB DIFFEREN BEAL PLOT Y AXIS MINIMUM C PBINC SAME EUT INCREMENT FROM MIN FOR 10" PLOT C PIMIN SAME AS PRMIN BUT FOR IMAGINARY PART C PIINC SAME AS PBINC EUT IMAG C NPRINT EVERY NPRINT'TH K VALUE AND ROOT OUTPUT TO PRINTER C ER C SEMIV IF TRUE OUTPUT Will ALLOW NPRINT TO TAKE EFFECT C REAL*8 DKR (210) REAL* 8 BD( 11), ID (11), BOOTS (4) ,ROOTI(4) ,PKR(5) ,PKI(5 $ ) EEAL*8 EDIS(12) ,IDIS (12) SEAL*8 DELTAK REAL*8 DIST,DISTOL ,RSAVE,DERR INTEGER*4 KEEG (3) , KEND ( 3) REAL*8 X(4) ,F(4),ACCEST (4) ,EEB EEAL*8 BBC (4) ,IDC(4) SEAL*8 DONE /1.D0/ EEAL*8 WIM (3,210) ,WB(3,210) BEAL*8 KMIN,KMAX,KINC,TOL BEAI*8 DSIGN, DEE AL, DIM AG, DLOG10, DABS, DMIN1 ,DMAX1 COMPLEX*16 DCMPLX,CECOT,CDLCG REAL*8 CDABS,BLOG LOGICAL FREEZE,NONEQ,KLOG,PLREAL,PLIMAG,SOLVEQ,KNEG $ ,MANY,RESTOR,FIRST,DUPLIC,LABEL LOGICAL*1 INSTAB(3,210),ALABEL(80) I NT EG EE* 4 SYM(3) /12,2,5/ REAL*8 GEEL,EQTOL,COMPAE,LDEI,DPS,DPI REAI*4 VPHASE(3, 210) ,VG(3,210) , AX (210) ,AY(210) C0MPLEX*16 CK,DC (4),DDC (4) ,8C,»C2 ,WC3,NDC(3) ,DDIS, $ DDDK C0MPLEX*16 DwDK,PRED (3) ,SLOPE (3) LOGICAL V EE EOS, ZEE IMG, FANCY, SEMIV,LPRINT INTEGER*4 SUBDIV INTEGER*4 INSTBI (3, 40) , NMAXL (3) REAL*8 BMAXL (3) ,DAWIM CGHHCN /NEWT/ DDIS,MAXIT COMMON /CCCALC/ CDC,EC,NEC COMMON /CPOL/ RDC,IDC COMMON /CONTEO/ SOLVEQ COMMON /CNTB02/ MANY,BESTOR COMMON /DIS/ RDIS,IDIS NAMELIST /PARAM/ KLGG,KMIN,KMAX,PIREAL,PLIMAG,KlNC, $ TOL $ ,ERR,MAXIT,VERBOS,FANCY,PINC,PKMIN,MANY $ , RES TOR, EQTOL, NFILE,LABEL,S OBDIV $ ,PRMIN,PBINC,PIMIN,PIINC,NPRINT,SEMIV EXTERNAL FCN C C SET UP DEFAULT VALUES C SUEBIV=3 NFILE=1 TCL=1. D-6 EQTOL=1.D-14 FLIMAG=. TBUE. PLBEAL=.TBUE. , KLGG=. TRUE. , KMIN=-12.D0 KMAX=-2. DO KINC=.1D0 VEBBCS=. FALSE. , NPBINT=5 SEMIV=.TEUE. FANCY=.FALSE. ZEBIMG= .TBUE. , MANY=.FALSE. RESTOB=. FALSE. MAXIT-100 EEB=1,D-15 FIBST-.TBUE. LABFL=. FALSE. 99S9 IF(MANY.AND..NOT.FIBST) GO TO 9990 NFIIE=1 READ (5,PARAM,END=9S98) C C READ IN PARAMETER LIST OF WHAT TO DO C NFILE 1=NFILE- 1 IF(NFILEI.LE.O) GO TO 5 DO 6 ISKIP=1,NFILE1 READ (1) 6 CONTINUE 5 CCNTINUE IF(MANY.AND..NOT.FIRST) GO TO 9990 NK= (KMAX-KMIN)/KINC+1. 5 C C CALCULATE ABBAY OF K VALUES C IF (KLOG) GO TO 2 DO 1 I=1,NK DKB (I) = KHIN*(I-1) *KINC 1 CONTINUE GO TO 3 2 NKB= 1 SKE=NK DO U I=NKB,NKE DKB (I) = 10. DO** (KMIN • (I-NK B) *KINC) <4 CONTINUE 3 CONTINUE 99S0 WRITE (6,1000) 1000 FORMAT(* 1') IF (.NOT.LABEL) GO TO 9 BEAD(5,1066,END=9998) ALABEL WBITE (6,1067) ALABEL 1067 FORMAT(1X,80A1) 1066 FOBMAT(80 A1) 9 CONTINUE C C BLOT SCALING QUANTITIES C PINC=0. FKMIN=0, PRMIN=0. PBINC=0. PIMIN=0. EIINC=0. WRITE (6,PARAM) EIRSI=. FALSE. RRMN=9.E70 C C SEE IF ROOTS ARE PROPERLY SEQUENCED C RRMX=-9.E70 EIMN=9.E70 RIMX=-9.E70 DO 222 IN=1,3 VPHASE (IN, 1)=0. WMAXL (IN) = -9. D70 NHAXL(IN)=0 VG (IN, NK) =0. 222 CONTINUE CALL COCALC (69998) C C CALCULATE POLYNOMIAL COEFFICIENTS C GREL=0. DO 1=1 SOLVEQ=. FALSE. DDPLIC=.FALSE. C 17a C FIND FIRST BOOT OF POLYNOMIAL C CALL DISPCO(DKB(I)) CALL CPOLY1 (BBC, IDC,3,BOOTB,BOOTI,6999) 185 SOL VEQ=. TRUE. 99 DO 97 IN=1,3 CALL NEWTON (ROOTR (IN) ,ROOTI (IN) ,EQTOL) HR (IN,I) = BGGTR(IN) WIM(IN,I)=ROOTI (IN) 97 CONTINUE 181 DO 183 IN=1,2 IR=IN+1 184 IF(IR.GT. 3) GO TO 183 WC=DCMPLX (WR (IN, I) ,HIM (IN, I) ) HC2=DCMPLX(HR (IR,I) ,HIM (Ifi,I) ) C0MEAB=DMAX1 (CDABS (HC) , CDABS (HC2) ) IF(CDABS(HC-HC2)/COMPAR.LT.TCI) GO TO 170 IR=Ifi+1 GO TO 184 183 CONTINUE BUPLIC=.FALSE. 189 1=1+1 C C ESTIMATE THE VALUES OF THE NEXT SET OF ROOTS C IF (I.GT.NK) GO TO 98 DELTAK=DKR(I)-DKR(1-1) DO 96 IN=1,3 CALL ADVANC (ROOTR (IN) ,ROOTT (IN) , DELTAK , DHDK ,SUBDIV) VG (IN,I-1)=SNGL (DBEAL (DHDK) ) PEED (IN)=DCMPLX (ROOTR (IN) ,ROOTI (IN) ) 96 CONTINUE CALL DISPCO (DKR (I) ) C C CAICULATE COEFFICIENTS FOB NEXT K C GO TO 99 170 IF(DUPLIC) GO TO 188 IF (I. EQ. 1) GO TO 188 SOLVEQ=.FALSE. C A LI DISPCO (DKB (I) ) C C IF DUPLICATE ROOTS ABE FOUND C GO BACK TO POLYNOMIAL ROOT FINDER C TO SEE IF ANOTHER ROOT CAN BE FOUND C IF SO TRY TO PROPERLY ORDER ROOTS C HBITE(6,1099) I,IN,IB 1099 FOBMAT(» SS6&S6SfiS8fiS6DUPLICATE ROOTS I,IN,IR«,3I5 $ ) CALL CPOLY1 (RDC,IDC,3, ROOTR,ROOTI,£999) NFIL=1 198 DISTOL=9.D70 DEBB=1.D70 DO 174 IN=1,3 IF (BOOTB (IN) . EQ.O. DO) GOTO 173 175 DEBB=DMIN1 (DABS (ROOTS (IN) ) , DEER) 173 IF (EOOTI (IN) . EQ.O. DO) GOTO 174 DERH=DMIN1 (DABS(BOOT! (IN)) ,DERR) 174 CONTINUE BEEE=DERR * 1.D-3 EPR=DLOG (CAES (DREAL (PRED(NFIL) ) ) +DERR) DPI=DLOG(DABS(DIMAG(PEED(NEIL))) + DERR) 194 DO 195 IN=NFIL,3 DIST-DABS ( (DREAL (PEED (NFIL) ) - BOOTB (IN) ) * $ (DPH-DLOG (DABS (ROOTB (IN) ) +DERR) ) ) $ +DABS ( (DIHAG (PEED (NFIL) )-ROOTI (IN) ) * $ (DPI-DLOG (DABS (ROOTI (IN ) ) +DER B) ) ) IF (DIST.GT,DISTCL) GO TO 1S5 DISTOL=DIST INFIL=IN 195 CONTINUE 196 WR (NFIL,I) = BOOTR (INFIL) WIM(NFIL,I)=ROOTI(INFIL) IF (INFI1.EQ.NFIL) GO TO 192 ROOTB(INFIL)=ROOTR (NFIL) BOOTR(NFIL)=WR(NFIL,I) ROOTI(INFIL)=ROOTI(NFIL) ROOTI(NFIL)=WIM(NFIL,I) 192 NFIL=NFIL+1 IF (NFIL. LT. 3) GO TO 198 GO TO 185 188 DUPLIC=. FALSE. GO TO 189 98 SOIVEQ=. FALSE. DO 100 1=1,NK LPRINT=. FALSE. IF (VERBOS) LP RINT=,TR U E. IF ( (• NOT. VERBOS. AN E. SEMI?) .AND. $ (MOD(I-1,NPRINT).EQ.0)) LPEINT=.TRUE. 166 DG 100 IN=1,3 C C CHECK FOB INSTABILITIES C IF(WR (IN,I) . EQ. 0. DO) GOTO 102 GR EL= WIM(IN,I)/WR (IB , I) IF (WIM (IN,I) . LT.O. DO) GOTO 102 INSTAB(IN,I)=.TBUE. IF (I. EQ. 1) GO TO 105 DAWIM=WIM (IN,I) IF (DAWIM.LT.O. DO) WMAXL (IN) = 0. DO IF(WMAXL(IN) .GT.DAWIM) GO TO 101 IF (INSTBI (IN, NMAXL (IN) ) .EQ.I-1) GO TO 104 105 NMAXL (IN) =NMAXL (IN) +1 104 INSTBI (IN, NMAXL (IN) )=I WM AXL (IN) =DAWIM GO TC 101 102 INSTAB(IN,I)=.FALSE. 101 CCNTINUE IF(DKB(I).EQ.O.DO) GO TO 288 C C FIND PHASE AND GBOUP VELOCITIES 176 C CHECK FOB ICCAL MAXIMA IF ONSIAELE C VPHSSE (IN,I)=SNGL (WE (IN , I)/IKE (I) ) 288 IF (LP.BINT) WBITE (6,1033) I , DKR (I) , WR (I N,I) ,WIM (IN, $ I) $ ,VPHASE (IN,I) ,GREI,VG(IN,I) 1033 FORMAT(1X,13,3D26.14,3D 15.5) C C SAVE PLOT SCALING MAX AND MIN C IF(KLOG) GO TO 131 BIMX=AMAX1 (BIMX,SNGL(WIM(IN,I)) ) EIMN=AMIN1(RIMN,SNGI(WIM(IN,1) ) ) EBMX=AMAX1 (RRMX ,SNGI (WB (IN, I) ) ) RBMN-=AMIN1 (RRMN,SNGL (WR (IN,I) ) ) GO TO 100 131 R=SNGL(DABS (WR(IN,I))) IF (B.EQ.O. ) GO TO 132 RRMX=AMAX1(RRMX,R) BRMN= AMIN1 (FRMN, R) 132 R= SNGL (DABS (HIM (IN ,1) ) ) IF (B.EQ.O. ) GO TO 100 RIMX=AMAX1(RIMX,R) RIMN= AMIN1 (BIMN, B) 100 CONTINUE SOLVEQ=. FALSE. IF(.NOT.PLREAL) GO TO 300 WBITE(6,1011) RRMX,EFMN,RIMX,RIMN 1011 FORMAT(' RBMX,RRMN,RIMX,RIMN',4E15.7) IF (. NOT. KLOG) GO TO 201 RIMX=ALOG10 (RIMX) RIMN=ALOG10 (RIMN) £BMX=ALOG10 (BRMX) BRMN=ALOG10 (BBM 8) 201 IE (. NOT. FANCY) GO TO 260 RMM=AINT (ALOG10 (ABS (BRMN) ) ) IM=RRMN/10.**RMM HMIN=IM*10.**RMM BMM= AlNT(ALGG10 (BRMX-RBMN)) IM= (RRMX-RRMN) /10. **RMM WDX=IM*10.3?*RMM C C PLOTTING C DO SCALING C PLOT AXES C IAE EL C PLOT ROOT LINES C APEIY SPECIAL SYMBOLS IE C REAL(8)<0, OF IMAG(W)>0. C GO 10 261 260 WMIN= ERMN WDX=(RRMX-REMN)/10. 261 IF (PINC. EQ.O.) PINC= (KMAX-KMIN)/1 0. IF(PKMIN.EQ.O.) PKMIN=K MIN INC=(NK-1)/20 177 IF(PRMIN.NE.O.) WMIN=Ffi MIN IF(PBINC.NE. 0. ) WDX=PBINC CALL AXIS(0.,0.,'HAVE NUMBEE* ,-11,10.,0.,PKMIN,PINC $ ) CALL AXIS{0.,0.,«BEAL ANG FBEQ•,13,10.,90.,WMIN,WDX $ ) IF (.NOT. KLOG) GO TO 262 CALL SYMBOL (—0.3,9.0,.14,* LOG-LOG * ,90.,7) 262 IF (.MOT. LABEL) GO TO 263 CALL SYMBOL(i.,9.75,.14,ALABEI,0. ,80) 263 CONTINUE S=10./(NK-1.) DO 206 1=1,NK AX(I) =(1-1.) *S AY (I)=0. 206 CONTINUE IF (KLOG) GO TO 250 DO 202 IN=1,3 DO 203 1=1,KK AY (I) = (SNGL (WB (IN,I) ) -WMI N) /WDX 203 CONTINUE CALL LINE(AX,AY,NK,1) 202 CONTINUE CALL PLOT(12.,0. ,-3) GO TO 300 250 DO 212 IN=1,3 NIN=0 DO 213 1=1,NK IF (SB (IN,I) . EQ. 0. DO) GOTO 214 AY (I) =(SNGL(DLOG10 (DABS (WE (IN,I) ) ) ) -WMIN) /WDX IF(HB(IN,I).GT.0.DO) GO TO 213 IF (I. EQ. 1) GO TO 240 IF(WR (IN,1-1) .LT.O.DO) GO TG 241 NIN=NIN+1 KBEG(NIN)=1 KEND (NIN) = I GO TO 213 241 KEND (NIN) —I GO TO 213 240 NIN=1 KBEG (NIN)=1 KEND (NIN) = I GO TO 213 214 AY(I)=0. 213 CONTINUE CAIL LINE (AX, AY, NK, 1) IF(NIN.EQ.O) GO TO 212 DG 248 K=1,NIN KB=KBEG(K) KE= KE ND (K) DO 249 I=KB,KE,INC CALL SYMBOL (AX (I) , AY (I) ,. 14,SYM (IN) ,0.0,-1) 249 CONTINUE 248 CONTINUE 212 CONTINUE CALL PLOT (12. ,0. ,-3) 178 300 IF(.NOT.PLIMAG) GO TC 399 DO 369 1=1,NK AY (I) =0. 369 CONTINUE IF{ .NOT. FANCY) GO TO 360 E M M= AINT(AL CG10 (ABS (BIMN) ) ) IM=EIMN/1Q.**8MM WMIN=IM*10. **.BJ1M BMM=AINT (ALOG10 (BIMX-EIMN) ) IM=(BIHX-BIMN)/10.**BMM WDX=IM*10.**BMM GO TO 361 360 WMIN= BIMN WDX=(BIMX-RIMN)/10. IF (1DX. EO. 0.) GO TO 399 361 CONTINUE IF (PIMIN. NE. 0. ) WMIN=PIMIN IF(PIINC.NE.O.) WDX=PIINC CALL AXIS(0.,0.,»WAVE NUMEES•,-11,10.,0.,PKMIN,PINC $ ) CALL AXIS(0.,0.,•IMAG ANG FEEQ»,13,10.,90.,WMIN,SDX $ ) IF(KICG) GO TO 350 DO 252 IN=1,3 DO 251 1=1,NK AY (I)= (SNGL<aiM (IN, I) ) -HMIN) /WDX 251 CONTINUE CALL LINE (AX, AY,NK, 1) 252 CONTINUE GO TO 390 350 CALL SYMBOL(-0.3,9.0,.14,* LOG-LOG',90.,7) DO 351 111=1,3 NIN=0 DO 352 1=1,NK IF(WIM(IN,I).EQ.O.) GO TO 353 AY (I)= (SNGL (DLOG10 (DABS (WIM (IN, I) ) ) ) -WMIN) /WDX IF (WIM (IN,I) .LT.O.DO) GO TO 352 IF(I.IQ. 1) GO TO 340 IF (WIM (IN, 1-1) . GT. 0. DO) GO TO 341 NIN=NIN+1 KBEG(NIN) = I KEND(NIN)=I GO TO 352 341 KEND(NIN)=1 GO TC 352 340 NIN=1 KBEG (NIN) = I KEND(NIN)=1 GO TC 352 353 AY(I)=0. 352 CCNTINUE CALL LINE(AX,AY,NK,1) IF (NIN. EQ. 0) GO TO 351 DO 348 K=1,NIN KB=KBEG (K) KE=KEND (K) DO 349 I=KB,KE,INC CALI S¥MBOL(AX(I) ,A¥(I) ,.14,SYM(IN) ,0.0,-1) 349 CONTINUE 348 CONTINUE 351 CONTINUE 390 CALL PLOT(12. ,0. ,-3) 399 DO 501 IN=1,3 WRITE (6,1013) (INSTAB (IN ,1) ,1=1, NK) 1013 FOBMAT (1X, 10 (3X , 10L1) ) NM=NMAXL (IN) IF (KH.EQ.0) GO TO 501 WfiITE(6, 1012) IN,NM, (INSTBI (IN,I) ,1=1 ,NM) 1012 FOBMAT(« INSTABILITY MAXIMA FOB GBOUP',12,110,/lX, $ 6(2X,5I4)) 501 CONTINUE C C USE THE LOCAL MAXIMA AS STARTING POINTS C FOB SOIUTICN TO D=0, D£/DK= 0 C DO 401 IN=1,3 NNX=NMAXL(IN) IF(NNX.LT.1) GO TO 401 DO 400 1=1,NNX II=INSTBI (IN,I) X(1)=WR(IN,II) X (2) = WIM (IN, II) X(3)=DKR (II) X(4)=0.DO F ( 1) = 0.D0 F (2) = 0. DO C C DS E NEWTON PROCEDURE (FEOM BBC COMPUTER CENTBE) C TO SOLVE EQUATIONS C C MAXIT 200 USUALLY USED C C C NOTE ABOUT COEFFICIENT STRUCTURE: C IF K GOES TO -K*, THEN W GOES TO -W* C WHICH MEANS THAT THE SAME PHYSICAL ROOT RETURNS C CALL DISPCO (X (3) ) WC=ECMPLX(X(1) ,X(2) ) WC2=HC*WC WC3=WC*WC2 DDDK=DDC (1) *WC3 + DDC (2) *WC2+CDC (3) *WC+DDC (4) F(3)=DREAL (DDDK) F (4) = DIM AG (DDDK) WRITE (6, 1054) I,IN,X,F 1054 FORMAT(* OSTART",2I3,8D15.7) CALL NDINVT (4,X,F,ACCEST,MAXIT,ERR,FCN,S996) WRITE (6, 1015) (X(IC) ,ACCEST (IC) ,IC=1,4) 1015 FOBMAT (* X ACCEST* , 4 (D18. 7, D10. 2) ) IF (X (2) .LT. 0.D0) GO TO 402 CALL DISPCO (X(3)) CALL CPOLY1 (BDC,IDC,3,BOOTB,ROOTI,&999) DO 505 IIN=1,3 CALL NEWTON (BOOTH (UN) , BOOTI (UN) , EQTOL) WC=DCMPLX(ROOTR (IIN) ,BOOTI (UN) ) WC2=WC*WC WC3=8C2*WC DDDK= DEC (1) *WC3+DDC(2) *WC2+IDC{3) *WC + DDC(4) DDDK=-DDDK/ (NDC (1) *WC2+NDC(2) *8C+NDC(3) ) WBITE (6, 10 55) DDDK, BOOTB (UN) , BOOTI (UN) 1055 FOBMAT (» ######## ABSOLUTE INSTABILITY, GBOUP VELCC* $ ,'ITY',2D16.8, $ 5X,»W=»,2D15.5) 505 CONTINUE GO TO 402 996 WBITE(6,1056)ERfi,X,ACCEST 1056 FOBMAT (» *******NDINVT FAILED**** EBR,X,ACCEST*/,1X, $ 9D13.5) 402 CONTINUE 400 CONTINUE 401 CONTINUE GO TO 9999 9998 CALL PLGTND STOP 997 NP=997 GO TO 990 998 KP=998 GO TO 990 999 NP=999 990 WRITE(6,1020) NP 1020 FOB MAT ( * **************CPOLY1 TROUBLES****,14) GO TO 9999 END SUBROUTINE DISPCO(K) C C CALCULATES COEFFICIENTS OF DISPEESICN RELATION FOR REAL C K C LOGICAL SOLVEQ REAL*8 DREAL,DIMAG,K,K2,K3,K4 COMELEX*16 EDC(4) , DC (4) ,NDC(3) , DCMPLX REAL*8 CRD (5,4) ,CID (5,4) ,RDC(4) ,IDC(4) COMMON /CCCALC/ DDC,£C,NDC COMMON /CPOL/ BDC,IDC COMMON /COEES/ CRD, CID COMMON /CONTRO/ SOLVEQ K2=K*K K3=K2*K K4=K3*K IF(SOLVEQ) GO TO 1 DG 100 1=1,4 IB=5-I BDC (I) = CBD (1,IB) +CBD(2, IB) *K+CRD ( 3, IB) *K2 $ +CRD(4,IB) *K3+CRD(5,IB) *K4 IDC (I) =CID(1, IB) +CID <2,IB)*K+CID(3,IB)*K2 $ +CID(4,IE) *K3*CID (5, IB) *K4 100 CONTINUE 1 CONTINUE 181 DO 1C2 1=1,4 IB=5~I DC (I) =DCMPLX (CRD (1 ,IB) ,CID (1 ,IB) ) $ +DCMPLX (CBD (2, IB) ,CID(2, IB) ) *K $ +DCMPLX(CBD (3,IB) ,CID (3 ,IB) ) *K2 $ +DCMPLX(CRD(4,IE),CID(4,IB) )*K3 $ +DCMPLX (CRD (5,IB) ,CID (5,IB) ) *K4 DDC (I) =DCMPLX(CRD (2 ,IB) ,CID (2 ,IB) ) $ +DCMFLX (CRD(3,IB) , CID(3, IB)) *2.D0*K $ +DCMPIX(CBD (4,IB) ,CID (4 ,IB)) *3 . D0*K2 $ +DCMFLX (CBD (5, IB) ,CIE (5, IB) )*4.D0*K3 IF (IB.EQ. 1) GO TO 102 NDC(I)=DC(I) *DFLOAI(IB-1) 102 CONTINUE RETURN END SUBROUTINE NEWTON(RE,SI,TCI) C C DOES NE8TON METHOD IMPBOVEMENT OF ROOTS C VAIUES FROM ESTIMATE OB BOOT FIN DEB C ABE SUBSTITUTED BACK INTO THE FULL EQUATION C BEAL*8 BB,BI,TOL,BDC (4) ,IDC (4) ,BATIO BEAI*8 DREAL,DIMAG COMPLEX*16 DDC(4) ,DC (4) ,NDC(3),DIS,DDIS,DELW,WC,WC2 $ ,WC3 COMPLEX*16 DCMPLX REAI*8 CDABS,WAES COMMON /NEWT/ DDIS,MAXIT COMMON /CCCALC/ DDC,EC,NDC ILOGP=0 HC=DCMPLX(RR,RI) 2 wc2=wc*wc WC3=WC2*WC DIS=DC (1) *WC3+DC (2) +WC2+DC (3)*WC+DC (4) DDIS=NDC(1) *HC2+NDC (2) *WC *NEC (3) IF (DREAL (DDIS) . EQ. 0. DO. AND. DIMAG (DDIS) . EQ. 0.D0) GO $ TO 3 DEIW=-DIS/DEIS WABS=CDABS(WC) IF (WABS. EQ.O. DO) GOTO 1 BATIO=CDABS(DELW)/SABS WC=WC+DELW IF(RATIO.LE.TOL) GO TO 1 ILOCP=ILOOP+1 IF (ILOOP.LT.MAXIT) GO TO 2 WRITE (6, 1000) WC, RATIO 1000 FOBMAT(* MAXIMUM NUMBEB OF ITERATIONS EXCEEDED.,W R«, f 'OCT IS NOW*, $ 2D25.15,« EBBOB= *,D 15.5) GO TC 1 3 WBITE (6,100 1) WC,DIS,BATIO 1001 FOEM AT (' DEBIVATIVE GOES TO ZEBO. WC, EIS,EEROB. • , $ 4D15.5) 1 CONTINUE BB=DBEAL (WC) 182 RI=DIMAG(WC) FETURN END SUBROUTINE ABVANC (WB, HI, DELTAK, DWBK , SUBDIV) C C ESTIMATES NEXT BOOT IN K SEQUENCE FROM PRESENT C ROOT AND DERIVATIVE C REAL*8 WR,HI,DREAL,DIMAG,DEITAK,DK,DFLOAT INTEGER*4 SUBDIV COMPLEX*16 DDC (4),DC (4) ,NDC(3),WC,DDIS,DHDK COMELEX*16 DCMPLX,WC2,WC3,WC4 COMMON /NEWT/ DDIS,MAXIT COMMON /CCCALC/ DDC,DC,NDC WC=DCMPLX(WR,WI) DK=DELTAK/DFLOA T(S U B DIV) DO 1 1=1,SUBDIV WC2=WC*WC WC3=WC2*WC DDIS=NDC(1)*WC2+NDC(2)*HC+NDC(3) £HDK=-(DDC (1) *WC3*ECC (2) *HC2+DDC (3)*HC+DDC (4) J/DDIS WC=WC+DWDK*DK 1 CONTINUE WR=DREAL(HC) WI=DIMAG(WC) RETURN END SUBROUTINE FCN(X,F) C C SUBROUTINE CALLED BY NDINVT C EVALUATES D AND DD/DK FCE COMPLEX H AND K C BEAL*8 X (4) ,F(4) COMPLEX*16 CK,C«,CW2,CW3,DD,DDDK COMPLEX* 16 DDC (4) , EC (4) ,NDC(3) COMPLEX*16 DCMPLX EEAL*8 DBEAL,DIMAG COMMON /CCCALC/ DDC,DC,NDC CK= DCMPLX (X (3) ,X (4) ) CALL DISCO (CK) CW= DCMPLX (X (1) ,X (2) ) CW2=CW*CW CW3=CW*CW2 DD=DC( 1) *CW3+DC (2) *CW2 + DC (3) *CH«-DC (4) F(1) = DBEAL (CD) F (2) = DIMAG (DD) BDDK=DDC (1) *CW3 + DDC (2) *CW2+DDC (3) *CW+DDC (4 ) F(3)=DBEAL(DDDK) F(4)=DIMAG(DDDK) BETUBN END SUEBOUTINE DISCO(CK) C C CALCULATES DISPEBSION POLYNOMIAL FOB COMPLEX K C BEAL*8 DBEAL,DIMAG C0MPLEX*16 DDC (4) ,DC(4) ,NDC(3) CCKBLEX*16 CK,CK2,CK3,CK4 C0MPLEX*16 DCMPLX BEAL*8 CBD (5,4) ,CID<5, 4) COMMON /CCCALC/ DDC,DC,NDC COMMON /COEIS/ CBD, CID CK2=CK*CK CK3=CK2*CK CK4=CK3*CK DO 100 1=1,4 IB=5-I DC (I) = DCMPLX (CBD (1, IE) , CID ( 1, IB) ) $ +DCMPLX(CBD(2,IB) ,CID (2,IB) ) *CK $ + DCMPLX (CBD(3, IB) ,CID (3, IB) )*CK2 $ +DCMPLX(CBD(4,IB) ,CID(4,IB) ) *CK3 $ +DCMPLX(CBD(5,IB),CID(5,IB))*CK4 DDC(I) = DCMPIX (CBD (2, IE) , CID (2, IS) ) $ •DCMPLX (CBD (3,IB) ,CID (3 , IB)) *2 . DO *CK $ +DCMPLX (CBD (4, IE) , CI D (4, IB)) *3.D0*CK2 $ + DCMPLX (CBD (5,IB) ,CID (5,IB) ) *4.D0*CK3 CCNTINUE BETUEN END 

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