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Star counts in M31 Hodder, Philip Jeremy Crichton 1994

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STAR COUNTS IN M31ByPhilip J. C. HodderB. Sc. (Astronomy & Astrophysics) University of St. AndrewsM. Sc. (Astronomy) University of British ColumbiaA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESGEOPHYSICS AND ASTRONOMYWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIA1995© Philip J. C. Hodder, 1995In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Geophysics and AstronomyThe University of British Columbia129-2219 Main MallVancouver, CanadaV6T 1Z4Date: ;Z% /f9i’AbstractThe structural properties of external galaxies may be investigated using a model togenerate the predicted number of stars as a function of apparent magnitude in any fieldin the galaxy. An implementation of such a model is explained in detail and is thentested against observations of M31. These data consist of several CCD images of severalfields in M31 along the minor axis, with one field along a “diagonal”. Additional datafor 5 fields along the major axis was also made available.Modelling of two galaxy components — the spheroid and the disk — is undertaken. Aspheroid density normalization of 3.1 x 10_6 stars pc3 at 10 kpc is derived for two of thespheroid fields. The data for the field along the diagonal gives a density approximately1.5 greater than this, possibly implying that the spheroid of M31 is inhomogenous. Thespheroid axial ratio can be constrained to between 0.4 and 0.7, values similar to previousworks. The effective radius cannot be constrained as well because it has a much smallereffect on the observed number counts. It is also noted that changes in these parameterscan compensate for changes in the density normalization.Modeffing the disk counts is more problematical — the reasons for this are discussedin some detail. The disk density normalization is found to be approximately 1.5 x iOstars pc3 at 10 kpc giving a disk to spheroid density ratio of about 48:1. Using modelsrun over grids of scale height and scale length it is found that the scale height is limitedto between 50 and 400 pc, the scale length constrained to between 5 and 7 kpc.11Table of ContentsAbstract iiList of Tables viList of Figures viiAcknowledgements ix1 Star Counts and Galactic Structure 11.1 Introduction 11.2 Early Studies of Galactic Structure 11.3 Modern Star Count Analysis 41.4 The Thick Disk 51.5 Stellar Populations and Galaxy Evolution 61.6 External Galaxies 71.7 Star Counts in External Galaxies 82 Observations and Data Reduction 92.1 Introduction 92.2 Observations 102.3 Data Reduction 132.3.1 Standard Star Frames 142.4 Calibration 142.4.1 DAOGROW 15in2. Standard stars2.4.3 Secondary StandardsIncompleteness TestsObserved Luminosity Functions &The Brewer Major Axis Fields15202022273 The External Galaxy Model3.1 IntroductionGeometrical ConsiderationsProgram operationComponent Description3.5.1 The Disk3.5.2 The Spheroid3.5.3 Absorption3.5.4 Density Normalization and Luminosity3.5.5 Colour—Magnitude DiagramsParameter ListCode DescriptionSource Code Availability4 Modelling the Spheroid4.1 Introduction4.2 The Input Luminosity Function4.3 A Default Model4.4 A Problem with the G352 field4.5 Comparing the Model and Observations .ivColour-Magnitude Diagrams . .3.2 The Bahcail—Soneira Model3. Model Sensitivity 664.6.1 Tests with Spheroid Axial Ratio 674.6.2 Tests with Effective Radius 694.7 Initial Density Estimates 724.8 Parameter Grids 754.9 Discussion 815 Modelling the Disk 855.1 Introduction 855.2 Spheroid Contributions in Disk Fields 865.3 Disk Counts along the Major Axis 905.4 Input Disk Luminosity Function 925.5 Model Sensitivity 925.5.1 Tests with Scale Length 945.5.2 Tests with Scale Height . . . 945.5.3 Effects of Inclination and Absorption 965.6 Initial Density Estimates 985.7 Parameter Grids 1005.8 Discussion 1056 Conclusions 106References 111VList of Tables2.1 List of observed fields in M31 . 112.2 List of standard star frames . . 112.3 List of standard star magnitudes 162.4 Final calibration zero points . . 222.5 Contaminant levels for M31 252.6 Observed “G” field V luminosity functions 292.7 Observed “B” field luminosity functions 343.1 The basic model parameters 485.1 Spheroid contributions to the disk fields 90viList of FiguresMap of M31 identifying observed fieldsNight 1 standard star calibrationsNight 2 standard star calibrationsNight 3 standard star calibrationsComparison of field star data and modelsEstimates of background galaxy countsObserved V LFs for G213, G263, 0302, 0312, 0352 and 0355Colour magnitude diagrams for 0263, 0302, 0312 andColour magnitude diagrams for Bi, B2, B3, B4 and B5Luminosity functions for Bi, B2, B3, B4 and B53.1 393.2 403.3 403.4 503.5 53Spheroid counts along the line of sightSpheroid input luminosity functions . .Spheroid number count variations with a64.4 Spheroid number count variations with r. .4.5 Density normalizations for 0302, 0312, 0352 and B54.6 Fits of selected models to the 0312 data2.• 121819• 212426283032• 33G352Coordinate transformationsExternal galaxy geometry — elevation .External galaxy geometry — projectionEGM flowchartSample source code for the disk component calculation4.14.24.3606268717476vii4.7 Contour plots of Q for the 0302 field 784.8 Contour plots of Q for the 0312 field 794.9 Contour plots of Q for the G352 field 804.10 Contour plots of Q for the 0302 and 0312 fields combined 824.11 Contour plots of Q for the 0302, 0312 and 0352 fields combined . . . . 835.1 Disk and spheroid count along the major and minor axes of M31 885.2 Disk and spheroid counts for an “external” Milky Way 895.3 The exponential nature of the disk 915.4 Input disk luminosity functions 935.5 Disk count variations with id 955.6 Disk count variations with hd 975.7 Contour plots of x2 for the B3 field 1015.8 Contour plots of x2 for the B4 field 1025.9 Contour plots of x2 for the B5 field 1035.10 Contour plots of x2 for the B3, B4 and B5 fields combined . . . 1046.1 Comparison between the EGM model and PvdB94 data 110vu’AcknowledgementsI would like to thank my parents and family, for their continued love and support. Thanksto all the friends I’ve made in Vancouver: Brad and Kristine, Ted, Scott and Sally, James(especially for the use of his data), Dave (all of them! - especially Dave Woods for hisgalaxy counts), Andrew, Yiman, Steve, Remi, Greg, Sandra, and all the others. Andfinally thanks to Harvey Richer and Greg Fahlman for supervising this thesis and helpingme along.‘I look at it like this,’ he said. ‘Before I heard him talk, I was like everyone else. Youknow what I mean? I was confused and uncertain about all the little details of life. Butnow,’ he brightened up, ‘while I’m still confused and uncertain it’s on a much higherplane, d’you see, and at least I know I’m bewildered about the really fundamental andimportant facts of the universe.’Terry Pratchett, Equal RitesixChapter 1Star Counts and Galactic Struêture1.1 IntroductionOne of the most fundamental problems in the field of Galactic astronomy is that of stellarpopulations: that is, what is the number and type of kinematic and chemical componentsthat make up our Galaxy and others. This problem has generated much debate andcontroversy amongst researchers, but has also led to some important discoveries aboutthe nature and history of galaxies. This chapter will briefly review the main points inthe history of this subject and how they relate to this thesis. The emphasis will be onthe subject of star counts rather than on kinematical studies.1.2 Early Studies of Galactic StructureThe first attempt to derive quantitative measurements of the Galaxy by counting starsin different areas of the sky was by William Herschel in the l8” century. His work on“star-gauging” — counting the numbers of stars down to increasingly faint magnitudes —led him to conclude that the Galaxy was roughly effiptical (with an axial ratio of about5:1) with the Sun near the center. At that time interstellar absorption was an unknownquantity and this, coupled with Herschel’s assumption that all the stars were of the samebrightness, led to this erroneous result. 11See Mihalas & Binney 1981 for a more complete description and further references.1Chapter 1. Star Counts and Galactic Structure 2With the development of large telescopes and astronomical photography (which permitted much more accurate measurements of a larger number of stars) Kapteyn 1922undertook a similar task, leading to a similar model of the galaxy. Kapteyn deducedfrom star counts and proper motion analyses that the Galaxy was a flattened spheroid, with the density of stars dropping to half the central value at about 800 pc in theplane and 150 pc perpendicular to the plane. This is now known to be rather too smallbecause the model also did not include the effects of interstellar absorption. Kapteyn,whilst aware that absorption could be present, could find no direct evidence for it at thattime.In Kapteyn’s model the Sun was close (650 pc) to the centre of the Galaxy. Thiswas disputed by Shapley (see Shapley 1919, for example) on the basis of his analysis ofthe distribution of galactic globular clusters. Shapley proposed that the Sun was in theouter parts of the Galactic plane, about 15 kpc from the centre. Although this value isnearly twice as large as the currently accepted value, the idea was basically correct.Support for Shapley’s model of a large Galaxy with the Sun near the edge was provided by the dynamical studies of Lindblad 1927, who also proposed that the Galaxymay be composed of a number of components all rotating about an axis through thegalactic centre and exhibiting various degrees of flattening depending on the speed of rotation. This dynamical reference was one of the first suggestions that the Galaxy may becomposed of two or more physical components with distinct properties. Oort 1927, 1928built on Lindblad’s ideas to produce a more complete kinematical model of the galaxy,compatible with Shapley’s.Baade’s 1944 study of M31 revealed the presence of two classes or populations of stars.Baade designated these as Population I (young objects in the spiral arms of galaxies, withcolour-magnitude diagrams similar to those of open clusters) and Population II (olderred stars found in the halo, and with globular cluster-like colour-magnitude diagrams). AChapter 1. Star Counts and Galactic Structure 3system of five populations (Blaauw 1965) was later considered by many to be necessaryto provide a complete description of the Galaxy. However this figure is now viewed asunnecessarily high — usually only two populations, a thin disk-like configuration and aspheroidal halo, are used to model Galactic structure (Bahcall 1986). However there issome evidence for a third “thick disk” component (see §1.4).Until quite recently population studies of the Galaxy concentrated mainly on analyzing samples containing detailed information on the spatial, kinematic and chemicalabundances of a number of stars (see, for example, the references in §1.4). These types ofstudies usually share a common feature: the samples consist of stars within our Galaxywith well known properties (kinematics, distances, metafficities etc.). This means thatsample size is often limited, and that samples can often be subjected to observationalbias. For example, the data set of Carney, Latham & baird 1989 is chosen from theLowell Proper Motion Catalogue. Compared to other samples (e.g. Norris 1986) thehalo stars in this set have an unusually large velocity dispersion in the radial direction,implying a bias against stars moving on more circular orbits. The stars in the sampleare therefore kinematically biased but are unbiased in metallicity.Because the sample sizes are generally small, they are susceptible to bias effectsintroduced by the presence of a few stars with extreme properties — for which there maybe no a priori reason to remove from the sample. Further complications arise due to thefact that the metafficity and kinematics of a population can only be described by someform of distribution function: therefore there will always be an overlap between the twoor more components.Chapter 1. Star Counts and Galactic Structure 41.3 Modern Star Count AnalysisRecently (as discussed in Bahcall 1986) the technique of using star counts has made acomeback — principally due to the advent of computer controlled machines to analyzephotographic plates and CCD cameras capable of fast and accurate data acquisition.When making observations of a particular field the final number of stars one sees(down to some limiting magnitude) comes from three sources: the density distributionof stars throughout the Galaxy, the distribution of stars with absolute magnitude (theluminosity function) and the interstellar absorption. Although the latter does not changethe number of stars, it does change their apparent magnitude and therefore the observedluminosity function. One may formalize this and write that the number of stars, A,having magnitudes m (mi m m2) in direction (, b) in a projected area d is givenbypm2 pooA(mi,m2,t,b)df J dm’J R2p(t,b,R,M)4(M)d2dR (1.1)mj 0where M m’ — 5log R + 5 — 0(R). In this “fundamental equation of stellar statistics”,p, 4 and 0 refer to the density and luminosity functions and the interstellar absorptionin the field. £ and b are the Galactic longitude and latitude respectively. The densityfunction, p, may also be a function of spectral type or absolute magnitude. For example,the scale height of stars in the disk varies from approximately 200 pc for the main-sequence dwarfs up to about 1 kpc for the red giants (Bahcall 1980). In addition theluminosity function is assumed to be independent of the density function and position.The classical approach to solve this equation for p, 4 and 0 — given an appropriateset of data — is to invert it mathematically. The most common methods are those of(m, log ii-) (see, for example, Mihalas & Binney 1981), and that of Malmquist 1924, 1936.However these can be unstable mathematical procedures, especially when the sampleis small (as is often the case), its photometry poor and the obscuration is patchy orChapter 1. Star Counts and Galactic Structure 5unknown (or both). It is often desirable to obtain the luminosity functions of stars ofdifferent spectral types, (M, S). These are quite well defined and can be extractedfrom the data separately. However it is often necessary to solve for a general luminosityfunction (M) to avoid reducing the sample size still further.The alternative adopted by Bahcafl & Soneira 1980, 1984 is to assume that the densitydistributions and luminosity functions are known, predict what the star counts (and colour distribution) should be, and compare this to the observations. This can be repeatedwith changes in the model parameters until a suitable match is found. This method ismuch simpler than the usual inversion techniques. However, the resulting model willprobably not be unique, but it will determine the basic structural properties of theGalaxy. Bahcall & Soneira 1984 and Bahcall 1986 have used this model, with only twostructural components (a disk and a spheroid), with considerable success in fitting theobservations from a variety of observers in a multitude of fields in our Galaxy.1.4 The Thick DiskIn recent years the existence of a third discrete component, usually referred to as a“thick disk”, has been both proposed by some and been found unnecessary by others.The presence of a thick disk was first proposed by Gilmore & Reid 1983 who found anexcess in the density distribution perpendicular to the galactic plane. Further evidenceof a thick disk has also been claimed by, amongst others, Gilmore & Wyse 1985, Norris,Bessell & Pickles 1985, Rose 1985 and Carney, Latham & Laird 1989. However theoriginal work by Gilmore & Reid has been criticised by Bahcall & Soneira 1984 whopoint out that the sample could be contaminated with giants, leading to the observationof an artificial thick disk. Bahcall 1986 reviews the case for the two component model,and finds the existence of a thick disk to be unnecessary.Chapter 1. Star Counts and Galactic Structure 6Furthermore, Norris 1987 demonstrates that a four component model can fit the data— which implies that one can construct a model composed of almost any number ofcomponents, of which few (or none) may have any physical basis.However a recent study by Majewski 1992 has shown that there appears to be athick disk, and furthermore, that it is a spatially distinct entity. He uses photometryand proper motion studies of SA 57, caJibrated with COD photometry and claims thestudy is unbiased out to a distance of up to 25 kpc from the Galactic plane. He finds asharp change in the stellar populations at a height of approximately 5.5 kpc, which heassociates with the edge of the thick disk. The chemical and kinematical properties ofthis component are found to be quite distinct from the spheroid.1.5 Stellar Populations and Galaxy EvolutionThe presence, or lack, of a discrete thick disk will have significant effects on theoriesof galaxy formation. With no thick disk, a model of monolithic galactic collapse akinto that proposed by Eggen, Lynden-Bell & Sandage 1962 is plausible. In this model,the galaxy formed out of a cloud of gas in a uniform manner, the disk rotating fasteras it collapsed and increased its angular momentum. If a thick disk is present then aslower accretion of “sub-units” (such as that favoured by Searle & Zinn 1978) would bepreferred. After the formation of the disk the collapse of spheroid components’ would bemuch more independent and inhomogenous, possibly leading to a flat metallicity gradientin the outer parts of the galaxy.If a thick disk is present, and if it has properties distinct from those of the spheroidalhalo (as the results by Majewski 1992 seem to indicate), then its formation cannot haveoccurred as a smooth transition between the formations of the (thin) disk and the halo.Each of the three Galactic components may have formed separately from the others —Chapter 1. Star Counts and Galactic Structure 7possibly the disk formed via a process similar to that proposed by Eggen, Lynden-Bell& Sandage 1962, whilst the halo formed “chaoticaJly”.1.6 External GalaxiesA number of the problems mentioned in §1.3 can be overcome by examining externalgalaxies. In practice this usually means looking at the Andromeda Galaxy, M31, whereindividual stars can be resolved. This is the most luminous galaxy in our Local Group,and the third brightest in the sky (after the LMC and SMC). van den Bergh 1960 classifiesits Rubble type as Sb I—IT, making it very similar to our Galaxy (type Sb/c). The distanceis well known — van den Bergh 1991 gives a mean distance (determined from a varietyof methods) of 725 ± 35 Kpc. Walterbos & Kennicutt 1988 (hereafter WK88) haveconducted an extensive study of the surface brightness profiles of M31, particularly thedisk and spiral arm regions. They derive several parameters for the disk and spheroidthat will be referred to in this work, usually as starting points for analysis.To investigate stellar populations in M31 it was decided to adopt an approach similarto Bahcall & Soneira — that is, assume a density. distribution and luminosity functionand predict the star counts for various fields in M31 by integrating from the outside intothe galaxy. The model is fairly straightforward — once the projection effects due to theinclination are accounted for, the distance of the field from the centre of the galaxy canbe found and a “back-bearing” can be obtained to find an effective “t” and “b”. Theintegration then proceeds in a manner similar to the Balicall — Soneira model except nowthe apparent magnitude is given by M = m — Slog(D ± R) — 0 + 5 where D is thedistance to M31 and R the distance from the plane of M31 (R is added or subtractedfrom D depending on whether the integration is proceeding away from or towards theobserver).Chapter 1. Star Counts and Galactic Structure 8The terminology used by different works to describe various galactic components canbe confusing at times. In this work the spheroid will be taken to mean the extendeddistribution of stars in a galaxy — it will not be referred to as the bulge or halo (the latteroften referring to an extended (R—2) distribution of dark matter).1.7 Star Counts in External GalaxiesStar counts in M31 have been performed by Reddish 1962, van den Bergh 1966 andBerkhuijsen & Humphreys 1989. However these are not “counts” in the same senseas the method discussed here — they derived luminosity functions from counts of OBassociations and individual OB stars.Pritchet & van den Bergh 1994 (hereafter PvdB94) have also performed a study ofthe stellar populations of M31 using star counts. However after performing the starcounts and conducting the incompleteness tests they convert these numbers to a surfacebrightness to determine the halo parameters. They achieve a good fit to the r lawand also note that the maximum disk contribution is approximately 40% at a minor axisdistance of 15’. They conclude that a single de Vaucouleurs law with a minor axis effectiveradius of 1.3 kpc can fit the spheroid from r = 200 pc all the way out to r = 2 kpc. Fromanalysis of fields along the “diagonal” they derive an axial ratio of 0.55 ± 0.05. This isfairly consistent with the values of 2.0 kpc and 0.63 derived by WK88.Chapter 2Observations and Data Reduction2.1 IntroductionAny external galaxy model must be tested against observations of real galaxies. In thiswork, observations of several fields in M31 were used as a test of the model. Theseobservations came from a program originally proposed by Fahiman & Richer (privatecommunication) and combined with a project by Christian & Heasley (private communication).The data set consists of three nights of observation of globular clusters in M31. Theseobjects take up a very small area of the frame (the typical diameter is less than 10”) andcan easily be masked out of the analysis procedure. The total amount of data availablein this set is quite large but of variable quality: only six fields were picked for study andare discussed in this chapter. These fields lie mainly along the south east minor axis,with one lying due east of the galaxy along a “diagonal”.Additionally the results of data from five fields along the south west major axis werekindly made available by Brewer (private communication). Although these data wereobtained with a very different purpose in mind they do lend themselves quite well for usein star count analysis.This chapter will describe the observations and data reduction procedures used to -analyze the first six fields. Interpretation of the results and comparison with the externalgalaxy model will be discussed in chapters 4 and 5.9Chapter. 2. Observations and Data Reduction 102.2 ObservationsThe data used in this work were obtained by Fahiman and Christian at CFHT on thenights of 1990 August 16/17, 1990 August 17/18 and 1990 August 18/19 (designated asnights 1 to 3 in the following discussion) using HRCam and the SAIC2 CCD. This devicehas a size of 1024 x 1024 pixels with a scale of O’f 13 per pixel when used with the HighResolution Camera.Table 2.1 lists the fields used in this work, identified by the globular cluster in the field(column 1, Sargent et al. 1977). Coordinates (epoch 2000.0) are given in columns 2 and 3.The projected distance from the centre of M31 (a2000 =h42m45s, 62000 = +41° 16’29”)— essentially the impact parameter, b — is given in column 4 in kpc. The number of Vand I frames are given in columns 5 and 6 along with the exposure times (in seconds) asthe second number. The size of each field was 1014 x 1023 pixels, or 132” x 133” , afterpreprocessing. At the distance of M31 each field is 420 pc on a side.In addition, the five major axis fields made available by Brewer (private communication) are listed (Bi, B2, B3, B4 and B5). These were obtained with FOCAM at CFHT.These fields are substantially larger than the HRCam fields at 7’ x 7’. They are morefully discussed later in this chapter.Table 2.2 lists the standard star observations for each night. Positions (epoch 2000.0)are given in columns 2 and 3, exposure times for each standard frame (in seconds) aregiven in column 4 (V) and 5 (I). The M92 standards consist of 10 to 12 stars in theouter regions of M92. Magnitudes in .V and I were taken from Landolt 1992, except forthe M92 stars whose data came from the work of the “Kitt Peak consortium standardsin M92” (Christian et al. 1985, Heasley and Christian 1986).‘The Canada-France-Hawai’i Telescope (CFHT) is operated by the National Research Council ofCanada, le Centre National de la Recherche Scientifique de France, and the University of Hawai’i.Chapter 2. Observations and Data Reduction 11Table 2.1: List of observed fields in M31ID a2000 62000 b V frames I frames Night0213 0h43m14•6s 41° 07’ 23” 2.2 2 x 300 5 x 120 30263 44 03.3 41 04 57 3.9 1 x 450 1 x 450 30302 45 14.9 41 06 23 6.2 3 x 1000 4 x 400 1G312 45 58.8 40 42 32 10.5 6 x 600 5 x 300 20352 50 10.0 41 41 01 18.2 3 x 600 3 x 300 3G355 51 33.9 39 57 34 26.8 1 x 450 — 3Bi 41 40.4 40 59 00 4.5 9 x 600 7 x 300B2 41 12.7 40 50 41 6.6 6 x 600 4 x 300B3 40 21.9 40 36 24 10.3 6 x 600 4 x 300B4 39 23.8 40 14 32 15.4 6 x 600 3 x 300B5 35 41.9 39 29 03 28.4 4 x 600 5 x 300Table 2.2: List of standard star framesField a2000 82000 V exp. I exp. NightSA93—317 0154m38s +00° 43’ 00” 5 2 1SA93—424 01 55 26 +00 56 43 5 2 1Feige 16 01 54 08 —06 42 54 5 5 1M92 17 17 07 +43 08 11 60 60 1M92 17 17 07 +43 08 11 60 60 2SA113-492 21 42 28 +00 38 21 2 2 2P01633+099 16 35 24 +09 47 50 20 10 2SA93—317 01 54 38 +00 43 00 5 2 3SA93—424 01 55 26 +00 56 43 5 2 3Feige 16 01 54 08 —06 42 54 5 5 3Chapter 2. Observations and Data Reduction 12Figure 2.1: A map of M31 showing the locations of the fields listed in Tables 1 and 2(the black squares). Also shown are the companions M32 and NGC 205. The shading inthe figure is only representative and does not indicate a true optical size. The centre ofM31 — shown by the small cross — is at a2000 = 0h42m and 82000 = 41° 16’ in this figure.c)I I I I1 U)0— C4J aC.)z10c)U.,c)0I I I IIC%1Cc)0aC’)0ECoc)E000. 0. c’1ECoECJLf)ECoL()cJ00 0 0 0 0C’) C1 0 0)C’)(oooo) oaChapter 2. Observations and Data Reduction 132.3 Data ReductionThe frames were preprocessed (bias subtracted and flat fielding) using IRAF 2 If threeor more frames were available for a particular field they were combined using the sigmaclip average option of the IRAF task imcombine. In this procedure each pixel of thefinal frame is the average of the same pixel in the input frames, but with deviant pixelsexcluded. This has the effect of increasing the S/N ratio and successfully removing mostcosmic rays. If only two frames were available they were averaged and the cosmic raysremoved with the cosmicray task in IRAF. This was also quite successful, and was alsothe method used to remove cosmic rays if only one frame was available. The images werethen trimmed to remove several columns of bad pixels along one edge.In a few cases the same field had been observed on two or more nights. In these casesthe frames from the best night (most frames, better seeing) were combined and used.Frames taken on different nights were not combined because of small rotations betweenthe frames. Note that in the case of the 0213 field both V and I data were available butonly the V data is presented here. The individual and combined I images suffer fromextreme crowding and large background variations due to their proximity to the centreof M31.After preprocessing and combining (where appropriate) each image was reduced usingthe DAOPHOT software package (Stetson 1987). DAOPHOT automates the process offinding objects on the image, identifying them as stars and measuring their magnitudes.A certain amount of user interaction is required to set the point spread function (PSF)for the image. In practice the ALLSTAR program in the latest version of the DAOPHOTsoftware was used in this analysis.The output from the reduction of each image is a list of star positions and instrumental2lmage Reduction and Analysis Facility (IRAF), a software system distributed by the National OpticalAstronomy Observatories (NOAO).Chapter 2. Observations and Data Reduction 14magnitudes. The final steps are to remove stars that are (a) too near (within 100 pixelsof) the globular cluster on the image; (b) stars on and around the HRCam guide staron the edge of the image (typically bright and saturated); and (c) remove stars that areobviously not stars at all — on a few of the final images one or two bright galaxies arepresent which the DAOPHOT find detection routine did not classify correctly.2.3.1 Standard Star FramesStandard star frames (see Table 2.2) were preprocessed in the same way. For all thestandards (except for the M92 field) it was necessary only to measure the aperture magnitude (see the §2.4.2, below). The M92 standard field was sufficiently crowded to requirea full reduction procedure. This enabled the “non” standard stars to be removed (subtracted) from the frame. Although this leads to noisier images (in the sense that somestars will not subtract out perfectly and will leave residual effects behind) it is necessaryto do this to get uncontaminated magnitudes for large apertures.2.4 CalibrationFrom Table 2.2 one can see that the available standards were quite limited— the lownumber of standards for each night precluded the calculation of extinction coefficients.Standard CFHT coefficients of av = 0.13 and cq = 0.05 were used instead. Checks weremade, however, to ensure that these were consistent with the data. This turned out to bethe case. The data are also consistent with a zero colour coefficient, though a non-zerocoefficient cannot be ruled out entirely.Chapter 2. Observations and Data Reduction 152.4.1 DAOGROWThe DAOGROW program was used to determine aperture magnitudes for both the standardstars and all the secondary standards on the individual images. The usual calibrationprocedure is to use an aperture magnitude and a profile fitting magnitude (PSF magnitude) for the same secondary standards to derive an offset, and to correct the PSFmagnitudes of all the stars on the image to the aperture photometry scale. If the frameis crowded then there may be few (or no) stars that are sufficiently isolated for this towork properly. In this case two aperture magnitudes can be used (at radii of, say, 3 and10 pixels). The magnitude in the smaller aperture can be found for many stars, and themagnitude in the larger one measured for selected uncrowded stars to get a magnitudethat includes all the stellar light in the wings of the profile. These two corrections canthen be combined to provide a single aperture correction. DAOGROW (Stetson 1990)takes this procedure one step further; It derives a “total magnitude” for a star usinga combination of the empirical curve of growth and an analytical function fitted to thecurves of growth of all the specified stars on the image. In this way it uses all the availableinformation about the stellar proffle — not just samples of the data at one or two radii.This total magnitude is then used as the real instrumental magnitude of the star and iscompared to the PSF magnitude to derive an offset.2.4.2 Standard starsTable 2.3 lists standard star magnitudes and V — I colours. Reference 1 is Lan-dolt 1992; reference 2 is from the Kitt Peak consortium standards in M92 (Christianet al. 1985, Heasley and Christian 1986). Instrumental aperture magnitudes were obtained using DAOGROW and corrected for exposure time and extinctions (using CFHTstandard values). Extinction coefficients were hard to measure due to the paucity ofChapter 2. Observations and Data Reduction 16Table 2.3: List of standard star magnitudesStandard V 0v V — I oTr_ ReferenceFeige 16 12.406 0.0013 —0.001 0.0021 1SA93—317 11.546 0.0007 0.592 0.0008 1SA93—424 11.620 0.0009 1.058 0.0008 1SA113—492 12.174 0.0033 0.684 0.0053 1SA113—493 11.767 0.0039 0.824 0.0039 1SA113—495 12.437 0.0024 1.010 0.0057 1PG1633+099 14.397 0.0025 —0.212 0.0111 1PG1633+099A 15.256 0.0036 1.015 0.0111 1M92—4 14.618 0.0140 0.950 0.0380 2M92—5 16.052 0.0090 0.590 0.0090 2M92—6 16.331 0.0210 —0.123 0.0110 2M92—7 16.437 0.0200 —0.127 0.0220 2M92—8 15.932 0.0210 0.653 0.0240 2M92—9 16.986 0.0100 1.311 0.0300 2M92—10 14.036 0.0100 0.933 0.0200 2M92—11 15.146 0.0230 1.152 0.0280 2M92—12 15.982 0.0270 0.844 0.0180 2M92—13 15.078 0.0180 0.681 0.0240 2M92—21 17.925 0.0090 0.636 0.0170 2M92—22 17.542 0.0150 0.680 0.0260 2M92—23 16.791 0.0230 0.771 0.0430 2Chapter 2. Observations and Data Reduction 17standards. Plotting the magnitude offset as a function of airmass showed that thestandard CFHT values of the extinction coefficients were at least consistent with thedata. There is no convincing evidence for a colour term on any of the three nights.The number of M92 standards that were usable depended on the precise field and seeingconditions and therefore varied from night to night.Night 1Standard stars available on this night included SA93—317, SA93—424, Feige 16, and sixof the M92 standards (6, 7, 8, 9, 12 and 23). The plot of standard magnitude (V) minusaperture magnitude (measured using DAOGROW, Vap) against standard V — I is shownin figure 2.2. M92 standards are plotted as filled circles, the SA93 stars as triangles andFeige 16 as a square. Panel (c) of the figure shows that there is no significant colour effect.The zero points are (V— Vap) — —0.9471 ± 0.0281 and (I lap) = 1.0712 ± 0.0936.Night 2Available standards on night 2 included SA113—492, SA113—493, SA113—495, PG1633+099,PG1633+099A and 11 M92 standards (4, 5, 7, 8, 9, 10, 11, 13, 21, 22 and 23). Figure 2.3shows the plot of standard minus instrumental magnitude as a function of standardV — I. M92 standards are plotted as filled circles, the SA113 stars as squares and thePG1633+099 stars as triangles. M92 standard 21 was not measured in V; standard 10was not measured in I (that is, the star was either too crowded or the neighbours didnot subtract cleanly enough for a good curve of growth to be measured). Again thereis no significant colour term. The zero points are: (V— Vap) = 1.0044 ± 0.0734 and(I— lap) = 1.0873 ± 0.0386.Chapter 2. Observations and Data Reduction 18I I I I I I I I I(a)—0.90 - --0.951.00 -I 1.1 I I II I I I I I I—0.9- (b):--I I I I I I I I I I I I0.0- (c)_0.1L*-—0.2 -—0.3I I I I I I I I—0.5 0.0 0.5 1.0 1.5v—IFigure 2.2: This figure shows the zero point shifts (the difference between standardmagnitude (V) and instrumental aperture magnitude (Vap)) for the standard stars fornight 1, plotted against standard (V — I) for the V and I measurements (panel (a) and(b) respectively). Panel (c) plots the colour difference (V — I) = (V— I)ap — (V — I).The M92 standards are plotted as filled circles; SA93 stars as triangles and Feige 16 asa square. The broken horizontal line shows the mean value in each case.Chapter 2. Observations and Data Reduction 19— I I I I I I I I I I I I—0.9- (a) *-1.0 =•—1.1: 1 --1.2-—1.3•••I I I I I I I I I I ——0.9- (b)I I I I I I I I I I I I I0.1 (c) -0. H -—0.1- --0.2- 1 I -• I I I I I I I I I i—0.5 0.0 0.5 1.0 1.5v—IFigure 2.3: This figure shows the zero point shifts for the standard stars for night 2,plotted against standard (V — I). The M92 standards are plotted as circles; SA113 starsas squares and P01633+099 stars as triangles. The broken horizontal line shows themean value in each case.Chapter 2. Observations and Data Reduction 20Night 3Only 3 standards were available this night: Feige 16, SA93—317 and SA93—424. No colourterm is evident in figure 2.4, which shows the zero point shift as a function of V — I. Thezero points are: (V— Vap) = —0.9600 + 0.0164 and (I — lap) = —1.1243 ± 0.06522.4.3 Secondary StandardsFor each frame secondary standards were selected from the final photometry lists usingthe criteria that they were relatively uncrowded and bright but not saturated (a selectionmethod similar to choosing PSF stars). All the stars in the photometry list except theseones were then subtracted from the frame using DAOPHOT. The instrumental aperturemagnitudes of these stars on the subtracted frames were measured using the DAOGROWprogram for several radii. The sample was further “pruned” by selecting stars withcurves of growth that flattened out reasonably well at large radii. Typically 7 or 8 starssurvived this process to provide well measured aperture magnitudes. These aperturemagnitudes were then compared with the profile fitting magnitudes (from ALLSTAR) toprovide corrections from the one to the other. Once again, no colour term was found to besignificant. By summing these corrections (PSF magnitude to aperture magnitude) withthe shifts from equations 2.1, 2.2 and 2.3 (aperture magnitude to standard magnitude)and adding terms for the extinction with air mass and for the exposure time, a set offinal zero point shifts was derived. These are shown in Table 2.4, for both the V and theI frames, along with the formal errors.2.5 Incompleteness TestsThe final observed luminosity function for each field will suffer from incompleteness,i. e., not all stars that are actually there may have been found and photometered. ThisChapter 2. Observations and Data Reduction 21I I I I I I I I I I I—0.94 - (a)—0.96 -—0.981.00 -I I I I I I I I I I I I I I(b)—1.10 - -1.15 - —1.20 - -I I I I I I I I I I I(c)—0.10 - -—0.15 - -—0.20 - -II I I I—0.5 0.0 0.5 1.0 1.5v—IFigure 2.4: This figure shows the zero point shifts for the standard stars for night 3,plotted against standard (V — I). SA93 stars are plotted as filled circles and Feige 16 asa triangle. The broken horizontal line shows the mean value in each case.Chapter 2. Observations and Data Reduction 22Table 2.4: Final calibration zero pointsField ZV 0zV ZIG213 4.1972 0.0536 — —G263 4.8804 0.0523 4.5339 0.0840G302 5.6647 0.0919 4.6389 0.0989G312 5.2903 0.1021 4.4936 0.0599G352 5.0670 0.0792 4.0952 0.1227G355 4.7995 0.0623 —problem gets worse with fainter magnitudes but can be partially alleviated by addingartificial stars to the image and re-reducing it in exactly the same way as the original.By comparing the number of artificial stars added in to those recovered it is possible toestimate how many stars are being missed as a function of magnitude. A small numberof stars are added to the original image to keep the crowding problems at the samelevel. This procedure is then done several times to build up a statistically significant testsample. Typically in this work 20 sets of 100 stars were added independently to eachframe so that errors in the incompleteness ratios could be found. These incompletenessratios can be used to correct the observed counts. A very thorough description of theprocess is found in Drukier et al. 1988 and also in Bergbusch 1993.2.6 Observed Luminosity Functions & Colour-Magnitude DiagramsThe observed luminosity function will be contaminated by two effects — foreground starsbelonging to our own Galaxy and background galaxies in the field. The latter is not alarge problem in disk fields due to the increased absorption and much larger numbersof stars, but visual inspection of such images often revealed large, obvious backgroundgalaxies. These were masked out of the data reduction in the same way as the globularclusters.Chapter 2. Observations and Data Reduction 23The number of foreground stars in various fields has been estimated by Ratnatunga &Bahcall 1985 using the standard Balicail and Soneira Galaxy model (Bahcall & Soneira1980). Figure 2.5(a) shows this model calculation compared to a version of the code(called BSM) written during the development of the External Galaxy Model described inChapter 3. The new BSM code reproduces the Ratnatunga and Bahcall results very well.An attempt was made to run this model using exactly the same parameters but becauseof the large number of parameters this was not possible. This may account for the slightdiscrepancies between the two models.Figure 2.5(b) compares the same BSM model used above with background field datafrom the study of the halo of M31 by PvdB94. The BSM model fits the data verywell for magnitudes V < 22.25 and is used to generate all the estimates of foregroundcontamination in this work.At fainter magnitudes the chief source of contamination is from background galaxiesthat have been mis-identified as stars. An attempt was made to remove galaxies explicitlyusing a variety of image classification methods. This was found to be unsatisfactory dueto the low signal to noise ratio at these faint magnitudes.It was decided to remove background galaxies from the luminosity functions (LFs)in a statistical sense, rather than from the images themselves. There have been severalstudies of galaxy counts but the majority of them utilize the I and R bands — for thiswork V data is required. Fortunately deep galaxy counts in V have recently been madeby Woods, Fahlman and Richer 1995 who give the following relation for the number ofgalaxies per 0.5 magnitude bin per square arc minute:log N = 0.41V — 9.19 (2.1)This relation is plotted in figure 2.6, together with the BSM model prediction for theforeground stars, and with the PvdB94 background field data. The fit of this relation toChapter 2. Observations and Data Reduction 2432z1064z2020 22 24VFigure 2.5: (a) A comparison of the Ratnatunga and Balicall 1985 model (filled circles)and the BSM model described in the text (solid line). N is the number of stars per 2magnitude bin. (b) A comparison of the PvdB94 background data with the BSM model.The fit to the brighter magnitudes (V < 22.75) is very good. N is the number of starsper 0.5 magnitude bin.15 20 25VI I I I I I(b)• Pritchet & van den Bergh 1994Bahcall & Soneira Model•IIIChapter 2. Observations and Data Reduction 25Table 2.5: Contaminant levels for M31V Stars Galaxies V Stars Galaxies17.0 0.53 0.03 22.5 2.32 5.0717.5 0.61 0.05 23.0 2.62 8.1318.0 0.69 0.07 23.5 2.92 13.0318.5 0.76 0.12 24.0 3.19 20.8919.0 0.84 0.19 24.5 3.41 33.5019.5 0.95 0.30 25.0 3.56 53.7020.0 1.10 0.48 25.5 3.63 86.2020.5 1.27 0.77 26.0 3.62 138.0421.0 1.49 1.23 26.5 3.52 221.3121.5 1.74 1.97 27.0 3.33 354.8122.0 2.02 3.16 27.5 3.05 568.85the faint end of the PvdB94 data is quite good.Table 2.5 lists these galaxy counts (scaled to the area of the “G” fields of 4.68 squarearc minutes) along with the foreground star counts. In the analysis of the spheroidcomponent (Chapter 4) the galaxy corrections are only applied for bins V> 20.5 (wherethere is more than one galaxy to account for). In the disk analysis (Chapter 5) it isassumed that no faint galaxies can be seen through the disk, and this correction is notapplied. It should also be noted that in Chapter 4, when the External Galaxy Model iscompared against these corrected LFs, the “goodness of fit” criterion includes a correctionfor these contaminants. Since the bins that have been corrected for galaxy contaminationare also those with a large errors in the number counts (because they are faint) thesecorrections do not play a large role in the analysis.Figure 2.7 shows the observed V luminosity functions for the six fields. The thick vertical line in each panel indicates the magnitude at which the completeness of the sample is50%, as obtained from incompleteness tests. These luminosity functions are tabulated inTable 2.6, which does not incorporate the foreground and background subtractions. Thez0Chapter 2. Observations and Data Reduction 26I I I I I I I/.• Pritchet & van den Bergh 1994 /Bahcall & Soneira Model— — —— Woods et al 1995 ///.0,50—0.5/../20 22 24VFigure 2.6: Estimate of background galaxy counts (per square arc minute) (dashed line)from Woods, Falilman and Richer 1995, together with the PvdB94 background data(filled circles) and the BSM model prediction (solid line). At V 21.25 there are equalnumbers of foreground stars and background galaxies.Chapter 2. Observations and Data Reduction 27counts in this table are the completeness corrected counts in that field (0.0013 squaredegrees) in a 0.5 magnitude bin centered at the magnitude given in column (1). Thecorresponding errors (a) are a combination of (a) the standard deviation of the mean ofthe counts in each bin, as measured over the 20 incompleteness tests, (b) the Poissoncounting errors in each bin for each test, and (c) the error in the incompleteness fraction.Figure 2.8 shows the colour-magnitude diagrams for the four fields with the best Vand I data: 0263, 0302, 0312 and 0352. These diagrams were constructed by matchingtogether the V and I photometry lists of each field. Similar diagrams for the 0213 andG355 fields are not shown. In the former case there is too much crowding and backgroundvariation across the frame; in the latter no I frame was obtained.2.7 The Brewer Major Axis FieldsTo facilitate the analysis of the disk parameters Brewer (private communication) kindlymade available V and I data from five fields along the major axis, taken at CFHT usingFOCAM. These “B” fields (as they will be referred to) are approximately ten times largerthan the “0” fields at 7’ x 7’ in size. Additional information is given in Table 2.2. Datais also available in two narrow band filters centered around CN and TiO absorptionfeatures respectively. Only stars with a measured V and I magnitude are included inthis discussion — the additional selection procedure of including stars only measured in allof the four filters was not applied as it was felt this would place unnecessary restrictionson the data sample.Figure 2.9 shows (I,V — I) CMDs for each of the five Brewer fields. The Bi andB4 CMDs show a fairly “clean” disk population — the tip of the giant branch is clearlydelineated, as is the form of the giant branch down to V 22. The situation is morecomplex for B2 and B3 — there appears to be a substantial blue component at V—I < 0.6.Chapter 2. Observations and Data Reduction 2843z01043to201043b02010VFigure 2.7: The observed V band luminosity functions for the fields G213, 0263, 0302,0312, 0352 and 0355 (filled circles). The bins are 0.5 mag wide and the ordinate is thelogarithm of the number of stars in the fields per 0.5 magnitude bin at that magnitude.The open circles are the raw counts, not corrected for completeness. The dashed line isthe contribution from foreground stars and background galaxies as described in the text.The thick vertical bar indicates the 50% completeness level.20 22 24 26 20 22 24 26VChapter 2. Observations and Data Reduction 29Table 2.6: Observed “G” field V luminosity functions.V G213 @263 @30219.0 3.40 1.94 0.00 0.00 1.95 1.4519.5 2.15 1.51 1.00 1.00 0.10 0.4420.0 5.65 2.50 3.00 1.73 1.65 1.4120.5 12.40 3.69 1.00 1.10 3.80 2.1021.0 59.00 43.20 10.00 3.20 3.25 1.8621.5 66.67 29.15 8.15 2.90 3.20 1.8422.0 225.40 42.75 42.73 14.55 7.65 3.0822.5 790.23 114.01 148.93 23.52 40.75 6.7323.0 1579.01 196.32 682.94 125.90 132.75 11.7723.5 4580.08 791.44 1302.23 261.00 330.97 44.4724.0 6246.87 1280.88 3765.73 1491.29 728.43 1066024.5 28.50 5.89 5709.30 2850.74 1520.52 277.7525.0 — — 12032.26 12202.14 2123.33 445.0325.5 — — 7356.38 2993.9426.0 — 3440.79 2122.1226.5 — — —V G312 @352 @355______19.0 2.00 1.41 2.00 1.41 0.00 0.0019.5 1.00 1.00 1.00 1.00 1.00 1.0020.0 1.00 1.00 1.00 1.00 1.00 1.0020.5 1.00 1.00 2.00 1.52 3.00 1.7321.0 1.00 1.00 3.55 2.16 1.00 1.0021.5 3.00 1.73 10.00 3.60 3.00 1.7322.0 3.20 1.84 10.40 3.46 5.00 2.3122.5 16.75 4.14 15.05 4.08 13.10 3.7123.0 35.95 6.11 33.29 9.97 10.15 3.8323.5 103.20 10.28 84.27 10.57 35.14 9.5124.0 210.65 25.36 140.61 16.00 172.57 59.5324.5 288.62 32.57 292.46 35.11 1389.11 578.1525.0 657.88 94.63 771.09 97.26 2465.12 1141.9025.5 1420.03 205.56 2208.30 349.19 — —26.0 2096.40 414.36 2330.77 924.3326.5 1647.30 608.11 —1 2 3v—IFigure 2.8: Colour magnitude diagrams in (I,V — I) for the G263, G302, G312 and G352fields.Chapter 2. Observations and Data Reduction 30G263•I.G3128..• •,.18202224 -18202224I I I- G302I.I I I I I I•I I I I I IG352•. ..%•1•• ..‘ .SI I1 32v—II I I I I i i i i___I__i I I I I i i IChapter 2. Observations and Data Reduction 31This is especially noticeable in B3 where there is also a comparative excess of stars slightlyabove the giant branch at I 20, v—I 1.8. This may be attributed to the presence of ayounger spiral arm component which may be hard to remove in the modelling describedin Chapter 5. WK88 avoid this problem by excluding these regions from their fit tosurface brightness, an option that is not available here. The B5 field has many fewerstars, appearing to be very much like a spheroid field. However appearances can bedeceptive as the analysis in Chapters 4 and 5 will show.Table 2.7 lists the observed LFs for the B fields, corrected for the difference in areasbetween the FOCAM and HRCAM fields by a factor 0.10022, but not corrected forforeground or background contamination. The units are stars per 0.5 magnitude bin per0.0013 square degrees at a magnitude centered on the magnitude given in column (1).Figure 2.10 plots these LFs along with the foreground correction from Table 2.5. Thebackground galaxy counts are not included in this plot — it is assumed that faint galaxiescannot be seen through the disk, a fact borne out by examination of the images. In theB2 and B3 fields the removal of the component blueward of V — I = 0.6 did not leadto a substantial reduction in the number of stars. These modified LFs are also listed inTable 2.7 (columns 4 and 6) as “B2(R)” and “B3(R)”. They are plotted in figure 2.10 asopen circles. Incompleteness fractions were not available for this data set.Chapter 2. Observations and Data Reduction 32Figure 2.9: Colour magnitude diagrams in (I,V—I) for the Bi, B2, B3, B4 and B5 fields.The blue (V — I < 0.6) components can be clearly seen in the B2 and B3 diagrams.18202224 B1 B2IiiiiIII.iIlIIII.IiiiiiiIiI..I.IIIII.I[I• .i.. I I I.’B318202224—i—————i———: :1: :1— : :18 .20 .. •.22 .-.24I I I I I1I I I I I I I IIB42v—I32v—I3Chapter 2. Observations and Data Reduction1043z0120 22 24V33Figure 2.10: V band luminosity functions for the Bi, B2, B3, B4 and B5 fields. The opencircles in the B2 and B3 panels indicate the counts with the blue (V—I > 0.6) componentremoved. The dashed line show the level of field star contamination predicted by theBSM program. The units are the logarithm of the number of stars per 0.5 magnitudesper 0.0013 square degrees.432I I I I I I I I I . I I I I I I- Bi -- B2&B2(R). .•.- ... ... ...-•. :8 •o 0 •.. • •O.-——ii I I I I I I I I ii I I I I I- B3 &B3(R) -- B4:. .:0.i••.0.00 — . ——0- ———— — . . ————— — — . — — —I I I I I I I I I I043-I’.. 20 22 24VB5.. . . ——- — — —-oI ¶ I I I I I210 .Chapter 2. Observations and Data Reduction 34Table 2.7: Observed “B” field luminosity functions.V Bi B2 B2(R) B3 B3(R) B4 B519.0 1.30 2.51 1.50 4.51 2.00 1.40 1.7019.5 1.20 2.81 1.50 8.72 4.31 1.20 0.5020.0 2.81 5.71 3.01 15.43 6.51 2.31 0.9020.5 5.11 9.12 4.51 34.48 18.44 2.91 1.8021.0 8.62 14.73 6.72 51.81 28.26 4.01 1.6021.5 12.23 22.55 12.93 89.80 55.82 5.91 2.9122.0 47.81 49.51 37.48 150.03 97.01 20.45 4.0122.5 369.31 187.81 177.79 286.33 252.15 100.02 13.9323.0 1299.65 329.22 326.12 451.99 442.17 301.76 28.1623.5 1315.29 147.22 146.82 256.06 254.76 187.31 13.1324.0 478.55 29.16 28.96 56.02 55.92 16.14 1.1024.5 121.57 5.41 5.41 9.32 9.32 0.80 0.10Chapter 3The External Galaxy Model3.1 IntroductionTo predict star counts in M31 a mathematical model, similar to that of Bahcall & Soneira1980, has been developed. In this chapter this external galaxy model and its implementation will be discussed.Although a version of the Bahcall & Soniera code (written in FORTRAN 77) wasavailable in source form, it was quite basic in operation, and was not very flexible interms of changing parameters and running extensive sequences and grids of models in“batch” mode. In any event a number of modifications would have been necessary toapply it to an external galaxy.It was decided to write the External Galaxy Model program (referred to hereafter asEGM) in the ANSI C programming language (Kernighan & Ritchie 1988), rather than inFORTRAN. It was felt that the improved I/O and memory management facilities of Coffset the slight advantages in coding formul in FORTRAN. The computations requiredfor this modelling are not excessive or complex — however, there is a substantial amountof array management, “book-keeping” tasks and parsing of input parameters, for whichC is eminently suitable.The EGM code was written from scratch using the Bahcall & Soniera Export Code(Bahcall 1986) only as a reference. The final code, approximately 3000 lines of ANSI C,bears little resemblance to the latter except in algorithms, especially in terms of how it35Chapter 3. The External Galaxy Model 36is actually used.This chapter will first describe the Bahcail & Soneira model used to predict starcounts in our own Galaxy, and will then go on to detail how EGM uses the same kindof techniques for an external galaxy. The modifications that must be made and theassumptions implicit in the model will be described. Finally a detailed description of theoperation of the core modules of the code will be given.3.2 The Bahcall—Soneira ModelBahcaJl & Soneira 1985 developed a model to predict star counts for a run of magnitudesalong a given line of sight in our Galaxy. For a given galactic longitude and latitude,£ and b, the model computed the number of stars per unit magnitude per unit area fora range of apparent magnitudes. There are, of course, several important factors thatneed to be considered. Chief among these are the density distribution of the Galaxy(i.e., number of stars per cubic parsec as a function of r) and the luminosity function(LF, the number of stars as a function of absolute magnitude, M). The BahcaJl & Soneiramodel performs numerical integration on the fundamental equation of stellar statistics(equation 1.1, which is repeated below).p00A(mi,m2,t,b)df= J dm’J R2p(R)cI(M)dfZdR (3.1)0where M m’ — 5 log R + 5 — 0(R) and the volume element is of size R2dRctl. Anydependence of p on spectral type or luminosity class will be ignored. Except for verysimplistic (and unrealistic) forms of 4’ and p this equation has no analytic solution,requiring the use of numerical integration for its solution. The Bahcall & Soneira model,and its variants, has been used with considerable success by Bahcall & Soneira 1985 andRatnatunga & Bahcall 1985 as described in Bahcall 1986 and by Pritchet 1983. Thereare several practical points that may be raised here.Chapter 3. The External Galaxy Model 371. The integration does not have to be carried out to r = oo. In practice (and forsensible models for p) the integration is stopped at some distance rmax where thecontribution to the counts becomes negligible.2. It is possible to obtain a predicted colour distribution for the model if one hasa colour — absolute magnitude diagram. As the integration proceeds (see §3.7)through each magnitude bin in the LF , the colour can be found using a look—up—table of colours and magnitudes. The number of stars with that colour is given byequation 3.1. By convolving the final colour distribution with a Gaussian having awidth of the typical observed colour error, an expected colour distribution can befound.3. 4’ does not have to be an analytic function because discrete steps in M are beingtaken. LFs taken from real data can be used.4. One can integrate in any order (R then M, or vice versa). However it can be usefulto integrate 4’ “inside” the integration of p so that the distribution along the lineof sight can be obtained.5. Absorption effects are important and must be calculated for each step in R.6. There are usually a sufficient number of parameters that a change in one can becompensated for by changing one or more other parameters, resulting in the samepredicted number counts. However it should be noted that this approach is, inessence, predictive — it does not try to recover p and 4’ from an inversion of thedata set.7. There is some evidence for a triaxial bulge — the position angles of the bulge and thedisk are offset by some 100. The EGM model currently does not model a spheroidChapter 3. The External Galaxy Model 38with this shape. Variations of the axial ratio with radial distance are also notmodelled.8. The presence of spiral arms in the disk is not accounted for because there is noconvenient functional form to represent them. Neither is any warping of the diskmodelled, though it would be possible to include one by modifying the form of thedisk density distribution.9. There is no nuclear bulge component to model fields very close to the centre. However such a component can be easily added to the code if required.3.3 Geometrical ConsiderationsAs shown in the previous section, the Bahcall & Soneira code integrates the densitydistribution and luminosity function along a line of sight from the Sun defined by agalactic latitude b and galactic longitude £. This same procedure can be applied toexternal galaxies if the assumption is made that the observer is in a field in the planeof the (external) galaxy and is looking back along the line of sight. By calculating theeffective b and £ for that field almost exactly the same procedure can be used — the chiefdifference being that the volume element is now a function of the distance along the lineof sight to the galaxy, rather than the distance from the plane of the galaxy.The first step is to convert the right ascension and declination of a field to rectangularcoordinates, as shown in figure 3.1. The x and y axes are (somewhat arbitrarily) definedto lie along the apparent major and minor axes of the galaxy. The x—axis is inclined tothe axis of right ascension by the position angle 7. The transformation of the coordinatesof P from (a, 8) to (x, y) is given by equation 3.2.Chapter 3. The External Galaxy Model 39yaFigure 3.1: Converting the Right Ascension and Declination of the field at P to x andy coordinates. These are defined to lie along the major and minor axes of the galaxyrespectively. The shaded ellipse is a schematic representation of an inclined spiral galaxy.= acos7—Ssin7 (3.2)y = asin7+Scos7To calculate the- effective galactic latitude and longitude of the observer as seen fromthe field (P, in figure 3.1) the procedure is as follows. Figure 3.2 is a side view of thegalaxy, looking along the major (x) axis. An observer at 0 sees a field P in the galaxywhose plane (G) is inclined at an angle i to the plane of the sky (S). The angle b isthe “galactic latitude” of 0 as seen from P. Provided D >> z then d D and e 0.Therefore, to a good approximation, b = 900 —Figure 3.3 shows a projection of the three dimensional geometry. The z—axis is definedto lie along the line of nodes of the intersection of the galaxy with the plane of the sky.p. )Chapter 3. The External Galaxy Model 40Figure 3.2: A simplified view of the geometry of the situation. The plane of the galaxy(C) is inclined at an angle i to the plane of the sky (S) (as seen from 0). AssumingD >>zwehaveb=90°—i.Figure 3.3: As seen by the observer the field Q on the sky projects to P in the galaxy. rand 1 can be found by geometrical analysis.dV4;,Go“4PChapter 3. The External Galaxy Model 41The plane of the galaxy is defined by the orientation of the disk component in space. Forthe spheroid component such a plane has no physical meaning, but does provide a usefulreference for the coordinate system — this is the plane about which the spheroid may beflattened by changing the axial ratio. The field Q on the plane of the sky projects to P(in the galaxy). The angle 4 and the distance of Q from the galactic centre, p, are foundsimply from x and y. r (the distance of P from the galactic centre, in the plane of thegalaxy) and £ (the effective galactic longitude) are then found from:r =p(cos2+ sec2isin2)8 = arctan(tañ (3.3)seci J£ = 900_8Of course this value of £ is vaJid only for this particular quadrant (x > 0, y > 0 and Pbeing “behind” the plane of the sky, as seen in figure 3.3). Values of r, 8 and £ for otherquadrants and orientations follow easily, and in a similar fashion, once 8 has been found.The EGM program described later in this chapter also calculates the distance PQ infigure 3.3 — the true distance to the plane of the galaxy which is added to the distancederived from the distance modulus (which is taken to be to the centre of the galaxy.)Although small compared to the true distance to M31, the extra calculation is trivial andadds to the completeness of the model.3.4 Program operationThe major design goals for EGM were to make it easy to use, suitable for batch fileprocessing and to allow for a large number of adjustable parameters. These were accomplished by having EGM being driven only by command line options — no direct interactionby the user is necessary.Chapter 3. The External Galaxy Model 42The main method of data entry to the program is via “parameter files” — files containing lists of keyword and value pairs. Virtually every parameter in the model is adjustable,though in most cases the program supplied default values are more than adequate. Forsome of the “key” parameters (inclination, x and y position of the field etc.) thesedefaults, and the values in the parameter file, can be overridden using command lineoptions. This design makes it easy to change key parameters (e.g. the inclination of theexternal galaxy) when running several similar models (e.g. with the same input LFs).The output of the EGM program is a file containing the number counts as a functionof apparent magnitude — both differential (number of stars in that magnitude bin) andintegral (total counts down to that magnitude). Also given is the colour distribution ofthe stars for that field. Preceding these data is a header section, listing the values of allthe parameters in the model. This feature allows the model parameters to be recoveredfrom a data file.3.5 Component DescriptionThere are currently only two types of components in EGM — an exponential disk andthe spheroidal halo. A third component, the thick disk, is functionally equivalent to thethin disk model and may be optionally included in the model with separate parametersthan thern thin disk. The functional forms discussed below have been hard-coded intothe program and cannot be changed without rewriting the appropriate subroutines. Theparameters in these formu1 can, of course, be set at run-time.Chapter 3. The External Galaxy Model 433.5.1 The DiskThe density distribution of the disk stars in the model is represented by the followingdouble exponential function:pd(r) = pexp [—i-- — _ro] (34)where lid is the scale height of the disk and 1d is the scale length. TO is the distance of thefield from the galactic centre (in pc) and z is the height perpendicular to the plane. Thelid, id and r0 parameters can be adjusted in the galaxy model described below. Similarparameters for the thick disk component (h and i) can also be defined.3.5.2 The Spheroidde Vaucouleurs 1959 found that the projected brightness distribution of elliptical galaxieswas well represented by:log = —3.3307 — ij (3.5)where r is the spatial distance that projects to an angle containing half the total luminosity. Other functional forms (such as power laws) may be used, however in this work thede-projected form of this will be used. Young 1976 gives an asymptotic approximationfor the spatial density of stars which leads to this form of de—projection:ir 2P8(T) = 2r (;:;) (3.6)where r1 = and the constant b = 7.6692. This formulation is accurate for7’ O.2Te. At the distance of M31 this equation can therefore be applied to fields thatare more than approximately 1’ from the centre. Pritchet and van den Bergh 1994, intheir study of surface brightness profiles, showed that a de Vaucouleurs profile can fitChapter 3. The External Galaxy Model 44the spheroid over a wide range of distances — from 200 pc to more than 20 kpc. Bahcall1986 presents a similar function with more terms for greater accuracy. The spheroid canalso be made oblate by introducing an axial ratio parameter a,. In this case the neweffective radius is re/../. Note that this is the true, not the projected, axial ratio. Boththe effective radius, 7’e, and the axial ratio of the spheroid, a,, can be adjusted in theprogram.3.5.3 AbsorptionThere are three cases where absorption may be included in the model. Absorption fromour Galaxy is included as a fixed value to be added in to the apparent magnitude calculations. The model can integrate both sides of the galaxy (i.e., the nearer and the farthersides) — it is possible to include a fixed value of absorption to add to the calculationsof the far side integration. This can represent a very thin dust layer throughout thewhole disk, or can be applied to a particular field if it is known that there is an excessiveamount of absorption in that area.Three different models for internal galactic absorption are currently supported — noabsorption, the “cosecant” law and the Sandage absorption law, the “cosecant” formbeing the default model. Each case (except for no absorption) requires various additionalparameters. The cosecant law calculates absorption based on the formulaA(b)=ai(90°)cscb (3.7)where ai(90°) 0.15 magnitudes in V (the default) and Ày = 0.75AB.The Sandage absorption model (Sandage 1972) in the V band isA(b) a2(a3 — tanS) csc S b 50° (3.8)A(b)= 0 Ib>50°Chapter 3. The External Galaxy Model 45where for the V filter a2 = 0.165 and a3 1.192. In the B filter AB = 1.33Av. Thisformulation is only defined for the V and B ifiters.In both the “cosecant” and Sandage absorption laws, the absorption in magnitudesat distance R from the centre of the galaxy is calculated fromA(R) = A(b) [i — (39)where a0 is the scale height of the absorbing material. The default is 100 pc. It isassumed there is no variation with distance from the galactic centre, only an exponentialvariation perpendicular to the plane.WK88 perform a detailed analysis of optical extinction and reddening in M31 usingmulti-colour surface photometry. A detailed analysis is quite difficult because of theunknown nature of the properties of the dust in the disk, coupled with poor knowledgeof the geometrical configuration and sources of illumination. Despite these problems theanalysis can be done and results in a reddening law very similar to that of the Milky Way.They derive a reddening law of AB/Av = 1.35 + 0.03, Rv = 2.8 ± 0.3 for a dust lane inthe inner region of M31. One may assume that, at least to a first-order approximation,the reddening laws are the same for both the Milky Way and M31.3.5.4 Density Normalization and Luminosity FunctionsOne item of program input that needs careful consideration is the density normalizationto choose for the disk, thick disk and spheroid components. The initial source of thisinformation is the luminosity function used by the program (and specified by the inputparameters) for each component. The LF used for input into EGM is simply a file containing magnitude and number information. Each LF is normalized to some number ofstars per cubic parsec in some magnitude range. Also specified is the distance from thegalactic centre at which that normalization is valid. For example, Bahcall & SoneiraChapter 3. The External Galaxy Model 461980 use a Wielen LF that is normalized to the local density of the disk: 0.13 stars pc3between M = —6 and M = 16.5 at a distance of 8 kpc.The LF is then scaled (using equation 3.3) to r0, which is the reference point for theintegration (recall that this is the distance, in the plane of the galaxy, from the centre tothe projection of the field onto the plane).Different LFs can be specified for the disk, thick disk and spheroid components. TheseLFs are stored in files and are read by EGM at runtime. Thus the user can specify anyLF desired. It is usually necessary to have these LFs normalized to a value appropriateto the density normalization being used.3.5.5 Colour—Magnitude DiagramsEGM uses CMDs to calculate the expected colour distribution for the stars in the fieldin question. These CMDs are also stored in files and may be specified in a parameterfile. EGM uses them to calculate, at each absolute magnitude in the LF, the colour of thestars in the volume element. At the end of the computation the colours are convolvedwith a Gaussian of some width °ce (the typical error in colour measurement) to simulatethe effect of magnitude errors in measuring colour. Further complications arise if a mainsequence component is included (in the majority of cases it is not, as external galaxiesare in general too far away, however the capability is included in EGM). In this case a filegiving the fraction of stars on the main-sequence as a function of absolute magnitude mustbe given. EGM then uses this file to distribute stars in the volume element amongst thegiant branch and main sequence components. This only affects the colour distribution.Chapter 3. The External Galaxy Model 473.6 Parameter ListThere are over 50 parameters in the model that may be set by the user — however this alsoincludes some of the more mundane “management” parameters, such as setting limits onthe output apparent magnitude range and so on. Table 3.1 lists some of the more usefuland important parameters (i.e., ones that affect the model in a meaningful sense, suchas the inclination of the disk, rather than parameters used to define output, such as thefaintest magnitude to carry the integration down to), together with a brief description.Note that all distances are in parsecs. In the following text “parameter” is taken to meanone of these parameters that has some impact on the final result.3.7 Code DescriptionThe operation of the program is fairly straightforward. The basic procedures followedby the code are shown in the flowchart in figure 3.4. Firstly the model parameters areinitialized to default values. Then the parameter file and command line are checked forchanges to these parameters. The luminosity functions and colour magnitude diagramsare read in, spline fits are made to them and the results stored in arrays, the size of whichis determined by the ranges of absolute magnitudes the user desires. The projectionparameters (as described earlier in this chapter) are then calculated, as are the scalefactors for the density normalization. This is so that the density normalization radiuscan be kept constant when running several different models with the same LF in thesame way that the Bahcall & Soneira model defines the solar circle to be 8 kpc in radius.The program then proceeds to calculate the differential number counts for the diskand spheroid, and for the thick disk, if appropriate. If the far side of the galaxy is to beincluded in the model (the default case in the following chapters) then these calculationsare repeated, with the required change of projection parameters. The raw data for theChapter 3. The External Galaxy Model 48Table 3.1: The basic model parameters.Parameter Descriptionthe position of the field in arc minutes along the major axisthe position of the field in arc minutes along the minor axisfield size in square degreesD distance to the galaxyi inclination of the galaxy in degreesdistance of projected field to galactic centredistance of peak of distribution from galactic planedm apparent magnitude intervaldM absolute magnitude intervalp the density normalization of the disk at rthe normalization radius for the disk LFscale length in the plane for the disk componentgd scale height of the disk giantscalculated disk normalization at r0the density normalization of the thick disk at rthe normalization radius for the thick disk LFit scale length in the plane for the thick disk componentgt scale height of the thick disk giantscalculated thick disk normalization at rop the density normalization of the spheroid at rthe normalization radius for the spheroid LFre the de Vaucouleurs effective radius of the spheroida3 the spheroid axial ratiocalculated spheroid normalization at r0Chapter 3. The External Galaxy Model 49colour distribution are collected during the calculations of the differential number counts— the next step for EGM is to collate this information, and to convolve the raw colourdistribution with a Gaussian function to represent the “colour error” of the data to becompared. Finally the integral counts are found by accumulating the differential counts,the results are written out and the program ends.Program useThe first thing EGM does is initialize all the parameters to default values. Then itscans the command line for either parameter files or options. If it finds and can read aparameter file it does so, one line at a time. Each line of this ifie is parsed for a keywordwhich is checked against an internal list of valid keywords. If it comes across an invalidkeyword it will stop with an error message, otherwise it will overwrite the default valueof that parameter with the value in the parameter file. If it detects a valid option on thecommand line it updates the appropriate parameter directly. A typical invocation of theprogram may look like this:$ egm m31.pm -x 23.0-y 45.0 -i 77.0This tells the program to read the keyword — value pairs in the file “m31.pm”, set theXo and yo parameters to 23f 0 and 45f 0 respectively, and to set the inclination of themodel to 77°. If the parameter file “m31.pm” sets any of these three parameters itselftheir values will be overwritten by the subsequent command line options. Full programdocumentation and the source code can be obtained from the author upon request.InitializationEGM obtains values for LFs and CMDs by reading in the appropriate file (specified inthe parameter file or on the command line) and using spline interpolation to fill in aChapter 3. The External Galaxy Model 50Figure 3.4: A flowchart for the EGIvt code showing the principle logical steps in constructing a model.Chapter 3. The External Galaxy Model 51“look up table” (LUT). This is done so that the input LF need not be regularly sampled,or sampled at the same intervals as any other LF . It then uses this LUT to obtaininformation needed when calculating the star counts. Initialization of the various LUTsoccurs after the parameters have been read because the LTJT sizes and limits depend oninput parameters, such as the brightest and faintest apparent magnitude to calculate themodel for. For example, if the brightest and faintest absolute magnitudes for the modelto consider are set to —3 and 4 respectively then a LUT of 15 entries is needed if theinterval is set to 0.5 magnitudes. The input LF specified in the parameter file is splinefit to this array between these limits.Several LUTs (which are dynamically allocated at run—time) are needed — one eachfor the disk and spheroid LFs and the disk giant and spheroid giant color-magnitudediagrams and possibly two more for the main sequence CMD and for the fraction of starson the main-sequence (FMS) data. These latter two are only needed if the main-sequencecomponent is being modelled. Corresponding arrays will also be needed for the thick diskif it is to be included. Other arrays used to hold the results of the integration (numbercounts and colour distribution) are also set up at this time but are merely initialized tozero.EGM now calculates the “projection” parameters — the effective galactic latitude andlongitude based on the input parameters (as described in section 3.3) as well as r0. Thedensity normalization is calculated at this point based on the normalization radius (givenon the command line or parameter file). As a convenience it is possible to indicate thatthe normalization radius is equal to the distance in the plane from the galactic centre tothe field. EGM will then calculate and use the appropriate value.Chapter 3. The External Galaxy Model 52Integration ProcedureThe best way to understand how the program works is to examine the source code butsince there are approximately 3000 lines of it a shortcut will be taken! In this sectionconsiderable reference will be made to figure 3.5, which is a listing of the module forcalculating the disk number counts. This is taken directly from the source code, withminor modifications made to the layout and variable names for clarity. In particular,variable declarations have been left out. The bracketed numbers down the left hand sideof the code listing denote key points and will be referred to in the following discussionin brackets. Variable names will be typeset in teletype font. Functions are indicated bya set of parentheses containing a number of arguments, such as obscure (amode, b, r,a), or more conveniently, obscure 0.The procedure starts ([1]) with the calculation of the area of the field in steradiansfarea. SQD2ST is a macro defined to be 0.0003046174 (in square degrees per steradian).riuax_d is the maximum distance to carry the integration out to and dr_d is the size ofthe distance step (default is 25 pc) — both may be set by the parameter file. The numberof steps to take along the line of sight is derived from these two parameters.The main integration starts at point [2]. tot holds the total number of stars computed,at any point in the integration. (Its main use is at point [17]). The integration proceedsfrom r = 0 to r = rmax in steps of dr_d ([3]). If r is less than some minimum radius(r..iuin, also specified in the parameter file) then that distance step is skipped. Thedefault value for rmin (and the value used in subsequent chapters) is 0 pc, so this neverusually occurs.The statement at line [4] computes the distance of the volume element from theobserver along the line of sight, d. The value of DIR is either +1 (integrating from the“plane” away from the observer — the far side of the galaxy) or —1 (towards the observerChapter 3. The External Galaxy Model 53void calc_dsk() {[1] farea = omega * SQD2ST;[2] for (tot = 0.0, j = 0; 3 < (int) r_iuax_d / dr_d ; j = 3 + 1) {[3] r= 3 * dr_d; if Cr < r_rnin) continue;[4] ddist+(DIR*r);[5] absmag = obscureCamode, b, r, a) + abs;[6] z = r * sin(b);x = sqrtCro * rO + r * r * cos(b) * cos(b) — 2 * r * rO * cos(b) * cos(l));vol = d * d * farea * dr_d * drn_abs;trnp = vol * exp(—(x — rO)/ psi_d) * exp(—z / gsh_d) * dn[DSK];[7] for (znrn = 0.0, k = 0; k < ni; k = k + 1) {[81 rn_abs = m_bri + (k * dm_abs);rn_app = rn_abs + 5.0 * loglO(d / 10.0) + absrnag;[9] if (DIR == FAR) rn_app + dabs;[10] if ((rn_app > ma_dim) II (rn_app < rna_brt)) continue;[11] if (dorns == TRUE) {frns = frns[kl; fg = 1.0 — fms;} else {fms = 0.0; fg = 1.0;}[12] dmus = tmp * ifd[k] * frns; dug = tmp * lfdtkl * fg;chit = dxuus + dug; znm = ziun + dnt;[13] md = (int) ((rn_abs — rn...bri) / chu_abs) + 1;[14] if (doms == TRUE)coi_dist(DSK, m_abs, rn_app, chuus, rns[ind], dug, dg[indl, pm, nm);elsecoi_dist(DSK, rn_abs, rn_app, 0.0, 0.0, dug, dgEindl, pm, mu);[15] md = Cint) fioor((m_app — (ma_brt — thu_app / 2.0)) / diu_app);if (md >= 0 && md < nu) nuin[DSK] [mdl = num[DSKI [mdl + dnrns + dug;1[16] tot = tot + zluu; if (zmu < c_f ac * tot) break;return;}Figure 3.5: The source code listing for the calculation of the disk component. Functionprototypes and variable declarations have been removed for clarity.Chapter 3. The External Galaxy Model 54— the near side). In this way the same routine can be used to determine counts on boththe near and far side of the galaxy. dist is the distance to the external galaxy in parsecs.The default is 725000.0 (which, by no coincidence, is appropriate for M31).At this point the amount of absorption affecting the volume element is calculated([5]) in the obscure() function. This uses the galactic latitude, b, and r to calculatethe absorption at that point. ainode tells it to use no absorption, the cosecant law or theSandage model, as discussed in §3.6.3. a is an array containing the various absorptionparameters. Then any fixed value of absorption (abs) is added to the value returned byobscure C).At [6] a number of variables whose values remain fixed for that distance step arecalculated: z is the distance above the “plane”; x is the “horizontal” distance in the planefrom the galactic centre; vol is the size of the volume element; and trnp is a temporaryvariable used to hold the product of the volume element and the scale factors arisingfrom the exponential nature of the disk (the exp(-x/psLd) and exp(-z/gsh_d)). Alsoincluded is the scaling due to the density normalization, d.n EDSK], which is computedas exp (-Cr0 - rn) / psl_d) where rn is the normalization radius. Strictly speakingthe rO term in both these expressions is redundant, but it is calculated this way so thatrelative density values may be investigated directly if desired. By including all thesefactors into one variable the function executes faster.The program then loops through the disk luminosity function ([7]), which has beeninitialized in a previous function. The array ifd E] contains a look up table of ni elements,where ni is calculated from the LF limits (m_bri and rn_dim) and the desired absolutemagnitude interval din_abs. These are set by the user to cover the desired range overwhich the LF is used. To ensure that the correct magnitude bins are used this array is aspline fit to the input disk luminosity function. Steps in absolute magnitude are taken,starting at m_bri ([8]). At each step the apparent magnitude rn_app is calculated fromChapter 3. The External Galaxy Model 55m_abs, d and absmag. If the integration is taking place on the far side then the valueof dabs is added ([9]). This is to account for any absorption taking place in the disk ofthe external galaxy (a very thin dust component, for example, and is in addition to theabsorption calculated by ob s cure 0).ma_dim and ma_brt define the apparent magnitude desired for output. If the calculatedapparent magnitude is outside this range then that step through the LF is skipped ([10]).One point to note is that the integral counts at a particular magnitude are the sum ofthe differential counts (per magnitude bin) up to that magnitude. Thus if the magnituderange is too restrictive (for example, if ma_brt is set too faint) the integral counts willnot include all the stars that they should. For models run in this work the. limits onapparent magnitude were set at a level to avoid this effect.[11] For most models of an external galaxy such as M31 one may expect to see onlythe giant branch, the main sequence turn-off being at V 27. However the model doesallow the user to include a main-sequence component in the CMD if desired. At a givenabsolute magnitude bin (k), the proportion of stars on the main sequence is given byfms Ek] (a number between 0 and 1), where ±ms LI is an array used as a bUT and filled atprogram initialization using spline fitting to a user defined table specified as a parameter.The fraction of stars on the main-sequence and giant branch, ±ms and fg, are then usedin step [12].diuus and dng are the counts contributed by the main-sequence and giant branchesfor the volume element at that absolute magnitude (k). dut is the sum of these twovariables and zxua is the total for that volume element.The next step ([13]) is to fill in arrays holding the colour distribution in the model.This is done in the col_dist C) function ([14]). md is an index into the main-sequenceand giant branch CMD arrays (ms E] or dg E]) which hold the colours for a particularabsolute magnitude bin.Chapter 3. The External Galaxy Model 56The final step in the LF loop is to store the number counts ([15]). nuanE] t] is a twodimensional array containing the differential counts (i.e., counts per magnitude bin, k)for the disk, thick disk and spheroid components. The macro DSK is defined to point tothe correct column for storage of the disk counts. md is now set to be an index into thenuinE] E] array. The somewhat baroque looking formula for md ensures proper alignmentof bin boundaries.Before proceeding to the next volume element a check is made ([16]). tot is the totalnumber of counts so far for the integration of this component. If the counts from thecurrent volume element, znm, are less than some fraction of the total (tot), the procedurefinishes. The fraction c...±ac has a default value of 1 x 1O, which is sufficient for mostpurposes.Post-IntegrationThe integration procedure for the thick disk and spheroid is very similar, differing onlyin the choice of parameters (in the former case) and density distribution (in the latter).The far side of the external galaxy is integrated separately after the near side — thisis also the default mode of operation. The calculations for the far side of the galaxyproceed in an identical manner, except for the possible inclusion of extra absorption andthe fact that now as distances from the galaxy increases, so do the distances from theobserver.In the final steps the computed colour distribution is convolved with a Gaussian errordistribution to obtain the predicted colour distribution. Various other statistics of thecolour distribution (mean colours of each component and of the total distribution, etc.,)are also worked out. The cumulative number counts in each magnitude bin for eachcomponent is calculated from the differential counts. The results are then written outon the standard output stream or to a file.Chapter 3. The External Galaxy Model 573.8 Source Code AvailabilityA copy of the source code and full documentation may be obtained from the author, orvia anonymous FTP from ±tp.astro.ubc.ca:/pub/hodder/. ZIPped and compressedTAR ifies are available for the External Galaxy Model (EgmSrcl7 . zip, EgmSrcl7 .tar. z),a version of the Bahcafl & Soneira Model developed during the first stage of EGM(BszuSrc32 . zip, BsmSrc32 . tar. 1), and a set of utility programs useful in creating andnormalizing LFs and CMDs (GmuSrcl2.zip, GmuSrcl2.tar.Z).Chapter 4Modelling the Spheroid4.1 IntroductionThis chapter will detail the results of using the EGM model to obtain various parametersof the M31 spheroid. The sensitivity of the model to these parameters (mainly the axialratio of the spheroid and its effective radius) will also be discussed. The observed Vband luminosity functions (hereafter abbreviated to “LFs”) of the fields G302, G312,G352 and B5 will be used and the limitations of this data set will be explored. It wasthought that the B5 field might be a pure spheroid field — this turns out not to be thecase, but part of the B5 analysis will be discussed here as well as in Chapter 5.The intent is to determine spheroid parameters first using fields uncontaminated bythe disk. These model parameters can then be used to “subtract” spheroid counts fromthe disk field observations. Deconvolution of the disk and thick disk counts (if modelled)would be more problematical because the two systems are physically coincident.4.2 The Input Luminosity FunctionOne of the most important input parameters to the EGM model is the luminosity function(LF) — i.e., the number of stars per unit volume (cubic parsecs for example) per magnitude bin. However, all that is known initially is the observed LF of the field (i.e., thenumber of stars per unit area (square parsecs) per magnitude bin). To get the actualdensity of stars one must either deconvolve the observed LF and density distribution,58Chapter 4. Modelling the Spheroid 59or select an assumed LF and density normalization, compare the resulting model prediction to the observations and iterate to a solution. This density normalization mustbe specified at some normalization radius — the distance from the galactic centre wherethat density applies. However, because the form of the density distribution follows (oris assumed to follow) the r1 law they are in fact coupled and can be taken as a singleparameter.Figure 4.1(a) shows the spheroid contribution from points along the line of sightfor a typical (the default) model. As might be expected most of the counts originatefrom a comparatively small region where the spheroid peaks in density but because thedistribution still has some finite width it is not possible to easily convert from the observedLF to a density. If the LF has simple linear form (such as a power law) then adding thecontributions from different densities along the line of sight will not distort the shape ofthe final LF. However if the LF is not a simple function, or is a combination of linearpieces of different slopes, then the final LF will not be the same shape as the input LF. Inpractice, because the main contributions to the final observed LF all come from a narrowregion around the peak, any such distortion is minimal. In addition, if the normalizationis restricted to a narrow magnitude range then this effect is greatly reduced. This isshown in figure 4.1(b), which compares the DaCosta 47 Tuc LF (discussed below) to thepredicted number counts resulting from running it through the EGM model. As can beseen there is a small amount of overall distortion introduced by the integration process,and the effect at brighter magnitudes (of interest in this work) is even smaller.The problem is then deciding upon the density normalization of the LF. One possibility is to assume the observed LF has the correct shape of the actual LF (with thecaveats mentioned above) and choose some arbitrary normalization for it. When runthrough the model this will produce a set of predicted counts that can be scaled to theoriginal normalization. This scaling factor can give the density normalization that willChapter 4. Modelling the Spheroid 601.00.8E 0.6z0.40.20.0600 9000 I1z0—2—3—2 0 2MFigure 4.1: (a) The spheroid counts for a model 50’ along the minor axis is plotted againstdistance from the observer. The counts have been normalized to the maximum value.(b) The input DaCosta 47 Tuc LF (filled circles) plotted with the final number counts(open circles) resulting from “convolving” the input LF with the density distribution infigure 4.1(a).700 800D (kpc)I I I I(b)• 0o•.•0•0• original0 convolved•0 I I I i i IChapter 4. Modelling the Spheroid 61return the correct number counts, and can be used to investigate the effect on the numbercounts of a particular field for various model parameters. This works because the numbercounts reported by EGM scale directly with the density normalization of the input LF.That is, if the number of observed counts in a particular magnitude range is N0 and theobserved LF is scaled to n stars pc3 , then if the model results in a predicted countof N the actual density normalization is nN0/ . One interesting result of this is thatforeground absorption “drops out” of the calculation in the sense that if the observedLF is de-reddened by, say 0.1 magnitude, this must be added into the model in orderthat the same LF is recovered. This is not to say that it is not important in convertingapparent to absolute magnitudes.Alternatively, a known LF can be used — provided that the properties of the objectfrom which it was derived are similar to those assumed for the spheroid in the model.The difficulty then is finding a LF corresponding to that metallicity from, say, a suitableglobular cluster. Pritchet and van den Bergh 1988 derive a metallicity of the halo of M31of [Fe/H] —1. A reasonable approximation to this is to use the 47 Tuc LF presentedby DaCosta 1982. 47 Tuc has an [Fe/H] of —0.71 (Lang 1991) and its LF has beensuccessfully used in the original Bahcall & Soneira model, although that was for theMilky Way. Figure 4.2 plots the DaCosta 47 Tuc LF together with the observed numbercounts of the G302, G312 and G352 fields. Also shown is the mean observed LF forthe three “C” fields. This was derived by normalizing each LF to some arbitrary valueand taking the mean. The resulting LF represents the mean shape of the LFs, not thenumber counts. In the figure some of the “mean G” points lie outside all three C fieldLFs — this is because of the arbitrary normalization used to compare the shapes. Theobserved LFs are somewhat steeper than the 47 Tuc LF at brighter magnitudes — thisimplies that even the best fitting model may not fit very well. Figure 4.1(b) shows thatthere is little distortion introduced by assuming the observed LF has the same shapeChapter 4. Modelling the Spheroid0z0—262Figure 4.2: The observed LFs for the 0302, 0312 and 0352 fields are shown togetherwith the 47 Tuc LF from DaCosta 1982. The observed LFs have been shifted by the M31distance modulus and reddening and all the LFs have been arbitrarily normalized to thesame value between —2 M +1 so the shapes can be compared.M—2 —1 0 1Chapter 4. Modelling the Spheroid 63as the underlying luminosity function: it is therefore possible to use the “mean G” LFas one of the input parameters. It should be noted that this method is guaranteed toachieve a better fit between the model and observations than using the 47 Tuc LF, andthis should be borne in mind during the following discussions. Note also that the “meanC” LF has been extended to faint magnitudes by using a standard Wielen 1974 LF formagnitudes Mv > 1. This has been done for completeness and does not affect the resultssignificantly — the contributions of these faint stars (below the main-sequence turn off)is negligible given the type of observations available. For the majority of the analysisperformed in this chapter the “mean G” luminosity function will be used.All density normalizations have been performed at a distance of 10 kpc and betweenthe magnitudes —1 Mv +1. These choices are arbitrary but normalizing at differentradii and magnitude ranges merely serves to scale the input LF — the actual values arenot important as long as they are consistent.4.3 A Default ModelTo facilitate the discussion of the response of the model to different parameters it is usefulto define an initial model using a default set of parameters. Some of these parametersare collected from a variety of previous works which use studies of the galaxy’s integratedlight rather than star counts. This default model can be used as a starting point for eachfield.The two most important spheroid parameters are the axial ratio (cr9) and the effectiveradius (re). Pritchet & van den Bergh 1994 use measurements of integrated light ofstar counts to derive values of these parameters. (see Chapter 1 for a more completedescription). They derive a3 = 0.55 ± 0.05 and an effective radius of the minor axis of= 1.3 kpc. It is interesting to compare these values to the results by WK88 derivedChapter 4. Modelling the Spheroid 64using surface photometry: a, = 0.63 and Te = 2 kpc. It should be noted that in themodel the parameter a, refers to the true axial ratio, not the projected axial ratio asmeasured by surface brightness studies. At the low inclination of M31, however, therewill be little difference between the two.A third parameter important in the model is the inclination, i, of the galaxy. Althoughthis strictly speaking refers to the inclination of the disk relative to the plane of the skyany non-spherical halo is assumed to be oblate — that is, flattened about the plane ofthe disk. The inclination of the model is set to 775 (van den Bergh 1991). The otherparameters for the default model were chosen to be a, = 0.6 and Te 2 kpc.van den Bergh 1991 reviews foreground reddening estimates of M31. He adopts avalue of EB_v = 0.08, consistent with Burstein & Heiles 1984 (EB_v = 0.08), Walterbos& Schwering 1987 (0.06 EB_v 0.09) and Massey, Armandroff & Conti 1986 (fromthe minimum reddening for OB associations, EB_v 0.08). A value of EB_v = 0.08,leading to an absorption in V of 0.248 magnitudes will be assumed in the following work.It is also assumed that there is no reddening in the M31 halo itself.As a final point the final field size is set to 0.0013 square degrees (or 4.68 square arcminutes). This is slightly smaller than the CCD area discussed in Chapter 2 because itaccounts for the masking out of regions of the image containing the guide star, globularcluster and bright galaxies from the incompleteness tests.4.4 A Problem with the G352 field.The data from the G352 field presented something of a problem at the very outset:even using an approximate r1/4 law calculation there are too many stars by a factorapproximately 2 to 3!One possible reason for this was an error in the data reduction procedure. TheChapter 4. Modelling the Spheroid 65final image used in the analysis for the 0352 field was an average of 3 separate frames.As a check each of the three V frames was re-reduced, calibrated and incompletenesscorrected separately. All three frames gave the same results as each other and with thecombined frame (to within the calculated errors), reducing the possibility of an error inthe reduction process.During this procedure it was noted that there were a large number of fairly bright andobvious galaxies on the frame, indicating possible contamination by a background cluster(in excess of the normal background field galaxies). An attempt was therefore made toremove these galaxies from the observed LF using image shape statistics. This workswell for the brighter objects but becomes increasingly difficult at fainter magnitudes.The numbers of galaxies indicated by this method could not account for the excess.A search for galaxies and cluster of galaxies in the region of 0352 was made usingthe NASA Extragalactic Database. 1 There are several individual galaxies and galaxyclusters in the region but none is near enough to account for the excess objects.Another possibility is that the area is being contaminated by some other unknownobject external to M31 (e.g. a dwarf galaxy captured by M31, a stellar stream or otherexotica). However a contour plot (not shown) of the number of stars per unit area showedno indication of any other “physical” component — but given the small size of the fieldthis is hardly surprising. The CMD also does not reveal any additional components.4.5 Comparing the Model and ObservationsWhen comparing a particular model to a set of observed number counts it is necessaryto define some sort of “goodness-of-fit” criterion. A simple x2 test is quite suitable.1The NASA/IPAC EXTRAGALACTIC DATABASE (NED) is operated by the Jet PropulsionLaboratory, California Institute Of Technology, under contract with the National Aeronautics and SpaceAdministrationChapter 4. Modelling the Spheroid 66The model predictions m can be compared to the observed number, n., noting that ineach magnitude bin there are some foreground and background contaminants, c, to besubtracted. The x2 statistic used in the remainder of this analysis is:(41o•n + ciwhere the sum is taken over i (all the magnitude bins where n2 > 0). Here o-?I has beentaken from Table 2.6 and is the quadratic sum of the Poisson counting errors for eachincompleteness test, the standard deviation in the mean of each bin for each test and theerror in the incompleteness fraction. It is then possible to find the probability, Q(x2Iv),that the observed cu-squared exceed the value x2 even if the model is correct. Note thatthis definition of x2 assumes that all the errors are Gaussian. This is not quite the casehere, which has the effect of increasing the minimum x2, but does not change its positionis parameter space.Using this x2 statistic has the effect of giving the bins at the faint end of the LF onlya small influence over the quality of the fit but fortunately this is also the region in whichthe background galaxy contamination becomes large and possibly uncertain. The finalestimate of how well the model matches the observed counts is therefore only weaklyinfluenced by these corrections.4.6 Model SensitivityThis chapter is principally involved with trying to determine the effective radius, axialratio and density of the spheroid. The EGM model was tested to see if it was possibleto determine these parameters to sufficient accuracy. For example, one could take anassumed spheroid density and vary Te or a3 and see what effect this has on the resultantnumber counts. If the variation is much less than the errors in the observed LFs it maybe difficult to obtain a unique result.Chapter 4. Modelling the Spheroid 67These sensitivity tests will be presented for sets of test fields along the major andminor axes. All the fields discussed here are close to one of these axes and it is useful to seethe general trends of the model. In all these tests the “mean 0” luminosity function wasused with a normalization of 3.1 x 10_6 between the magnitude range —1 M +1.This normalization value provides the best fit to the 0302 and 0312 data in the V = 22.5to V = 24.5 range using the standard model. See §4.7 below for a detailed discussion ofthe density normalization estimates.4.6.1 Tests with Spheroid Axial RatioIn this test the sensitivity of the minor axis counts to the spheroid axial ratio wereinvestigated. The default model was run for 20 minor axis fields at distances of 10’, 20’,190’, 200’ on the same side of M31 as the observed minor axis fields (which have apositive value of yo as defined by the coordinate system given in Chapter 3). The G312field (at x0 = -4f473, yo = 49f648) is sufficiently close to the minor axis that comparisoncan be made directly.In the first test the, axial ratio was varied from 0.1 to 1.0 in steps of 0.1. The a3 = 0case is not physical and was not considered. Figure 4.3(a) shows a large variation in thecounts in the 0.5 magnitude wide bin centered at V = 22.5 with a, over the whole rangeof the minor axis. 2 Also shown are the counts in that bin for the 0302, 0312 and 0355fields (corrected for foreground and background contamination) — there are 33.36 + 6.73,19.36 ± 4.14 and 5.72 ± 3.71 stars respectively. The 0302 and 0312 counts both fit themodel quite well, though not exactly because these fields are not aligned precisely alongthe minor axis. The 0355 field shows a poor fit — this may be attributed to the poorerquality of the data (see Table 2.1) and the fact that the counts at this distance from2A11 figures of this nature have been plotted with the same axis scales for each of the major andminor axes, to facilitate comparison.Figure 4.3: This figure shows the variation of spheroid counts with axial ratio, a9, forthe minor axis (panel (a)) and the major axis (b). The remaining parameters used thedefault values. The heavy line shows the default models. (a) There is a large variationin number counts with a3. The points represent the number counts in the V = 22.5 binfrom the G302, G312 and G352 fields. (b) There is much less of a variation along themajor axis, even over the whole range of a9. The counts from the B5 field are also shownhere.Chapter 4. Modelling the Spheroid 68321z0_0—1—232z0100 50 100 150 200d (‘)Chapter 4. Modelling the Spheroid 69the centre of M31 ( 127’) are very close to the contaminant levels. At the 50’ fieldthe counts are 0.3 for a, = 0.1 rising to 33.1 for a, = 1.0 — a factor of approximately110. Of course it seems unlikely that the spheroid of M31 has an axial ratio as flat as0.1 but even taking a, = 0.4 (where the number count equals 3.0) as a reasonable lowerbound there is still a possible variation by a factor of 10. The error in the counts is suchthat any change in axial ratio by more than 0.1 could be detected (although this maybe masked by changed in re and the normalization). Furthermore, the variation of thenumber counts is approximately equal over the whole minor axis range, implying thatthis parameter may be tested at fields closer to the centre, where the number counts arehigher and are less susceptible to foreground and background contamination.Figure 4.3(b) shows the variation of major axis number counts with spheroid axialratio. The models were run at the same galacto-centric distances as above but alongthe major axis. There does not appear to be a great deal of change from one modelto the next. This is because the flattening of the spheroid with decreasing a, is not“foreshortened” along the major axis as it is along the minor axis (and as a disk wouldbe). The counts (corrected for foreground stars and background galaxies) for the B5 fieldare also shown on this diagram — the agreement is poor because of the significant diskcontribution at this position in the galaxy. At xo = 130’, which corresponds closely tothe B5 field (see Chapter 2) the spheroid counts change from 2.0 (at a, = 0.3) to 1.4 ata, = 1. Note the slightly peculiar behaviour of the a, = 0.1 and a, = 0.2 models whichintersect the rest of the curves.4.6.2 Tests with Effective RadiusThe effect of the effective radius, re, on the minor axis counts was tested next. Valuesranged from 1 kpc to 4 kpc in steps of 250 pc — otherwise the same set of fields anddefault parameters were used. The results are presented in figure 4.4(a). The crossingChapter 4. Modelling the Spheroid 70point of all the models is at Yo 24f 3, which corresponds to a distance of 4639.1 pc onthe sky and 21433.7 pc in the plane of the galaxy. On close examination it was seen. thatthe curves did not all intersect at the same point — in this region they are just closesttogether.The larger the effective radius of the spheroid (which, recall, is the half light radius),the higher the counts at a particular field. Changing re has much less of an effect atsmall distances but quite a large one further out in the halo. At Yo = 50’ the countsonly change from 11.6 (for re = 4 kpc) to 5.7 (for Ve = 1 kpc). However at distances ofmore than approximately 150’ the number counts become small enough that they can beaffected greatly by foreground contamination regardless of re. (Note that increasing thefield size to try and compensate for this will also scale both the number counts and thecontaminants.)The 0302, 0312 and 0355 data for the V = 22.5 bin are also shown on figure 4.4(a).It can be seen that, for the default model, the ability to detect a deviation in re is poor.This implies that it will be difficult to derive an exact value for the effective radius ofthe spheroid.Similar tests were performed for the major axis — although there is no available datafor halo fields along the major axis it may be useful to know how the halo behaves in thatregion. These results may also be used in the analysis of the disk fields where subtractionof the spheroid counts may be important. Models were generated for fields at 10, 20,190 and 200’ along the major axis. Note that the EGM model is symmetrical about theminor axis (i.e., models run at x0 and —x0 will be identical.)The changes in counts due to changes in effective radius (figure 4.4(b)) can, on theother hand, be more substantial along the major axis than the effects of axial ratio. AtXo = 130’ the observed spheroid counts change from 0.9 when r = 1 kpc to 3.1 when= 4 kpc — a factor of 3.4. However the effect of changing re when the effective radius isChapter 4. Modelling the Spheroid 71321z—1—232Ff2z010Figure 4.4: This figure shows the variation of the spheroid counts with effective radius,re, for the minor axis (panel (a)) and the major axis (b). The remaining parametersused the default values. The heavy line shows the default models. The variation isapproximately equal for both axes, and is not as large as the variations caused by axialratio (figure 4.3). In (a) the counts in the V = 22.5 bin for the G302, G312 and G355fields are shown; in (b) the B5 counts are plotted.0 50 100 150 200d (‘)Chapter 4. Modelling the Spheroid 72already large is minimal. Again note the apparent crossing point of the models at aO38f4 corresponding to a distance of 7330.9 pc.4.7 Initial Density EstimatesThere are three parameters that have to be determined for the spheroid model: theaxial ratio of the spheroid (a3), its effective radius (re) and the density normalizationp at some radius r. Accordingly data from at least three different fields is needed.Unfortunately, as noted above, the G352 field shows a large excess in the number counts,and the B5 has a significant contribution from the disk. However an attempt will bemade to at least set out the problem nd describe a method for solving for a uniqueparameter set.In order to make some initial estimates of the density normalization, one can conductthe following experiment using the scaling method discussed above: normalize the inputLF to some arbitrary value (1.0, say) between some magnitude range and use it in theEGM model. The ratio of observed to predicted counts gives the scaling factor for thenormalization. This scaling factor value can be used to compute the normalization thatwould predict the same number of stars as is actually observed, for that field and set ofparameters a3 and re.Using this procedure for the four spheroid fields at 10 kpc and normalizing to countsusing the bins from V 22.5 to V = 24.5 (except for B5 which was to be normalized overthe range 22.5 to 23.0 to avoid using data from faint magnitudes) the following densities(—1 M < +1) are obtained:G302: 3.11 x 106 stars pc3 G352: 5.00 x 106 stars pc3G312: 3.07 x 10_6 stars pc3 B5: 1.81 x iO stars pc3Note that. the densities for the G302 and G312 fields are very similar, the value for theChapter 4. Modelling the Spheroid 73G352 field is too high by a factor of approximately 1.6, and that density returned by theB5 data is a factor of approximately 6 too large because disk stars are being included inthe calculation. A spheroid density normalization of 3.1 x 10_6 will be used as a defaultin the tests that follow.This procedure was also carried out for the four fields G302, G312, 0352 and B5 overa grid of c (varying from 0.1 to 1.0 in steps of 0.1) and ?e (varying from 1 kpc to 4 kpcin steps of 250 pc). The results are shown in figure 4.5. It was assumed that the B5 fieldis a pure spheroid field — this is not the case (see Chapter 5) but it is still instructive torun the test. In these tests the “mean 0” LF was used and was normalized to a valueof 1.0 between —1 < M < +1. For each “0” field the number counts between theV = 22.5 and V = 24.5 bins were used to derive the required density normalization (at10 kpc). For B5 the counts in the V = 22.5 and V = 23.0 bins were used. These valuesare plotted as contours in figure 4.5. The contour values range from 1 x 106 stars pc3to 1 x iO. Heavy lines indicate contour values of 106, i0 and 1O and are labelled.Figure 4.5 can also be viewed as a plot of density surfaces — at each value of a8and re on the grid the height of the surface gives the density normalization. The set ofparameters (a., r, r) that results in the correct densities for all the fields concerned canbe found where two of these surfaces intersect. This line of intersection when projectedonto the (a9, r) plane will determine the set of values that gives the same densitynormalization for each field. Therefore at least three fields are needed: two pairs of fieldswill give two lines which will themselves intersect at the solution (indicating that this pairof (a3, re) values satisfies all three models). However this is not a very rigorous procedureand provides no information on how well this solution fits the data. In figure 4.5 theslopes of these “surfaces” are nearly the same for the 0302, 0312 and G352 fields whichfurther compounds the problem.Chapter 4. Modelling the Spheroid 74Figure 4.5: A contour plot of the log10 of the density normalizations that have to beapplied to the data for fields G302, G312, G352 and B5. The contours range from1 x 10—6 to 1 x iO in steps of 0.2 in the log — the contours at 106, iO and i0 areplotted in heavy lines and are labelled.432143V21Chapter 4. Modelling the Spheroid 754.8 Parameter GridsAn alternative to the above approach is to run the EGM model over a grid of densitynormalizations as well as a grid of a3 and re. The “mean C” LFs was normalized to asequence of values running from 2.1 x 10_6 to 6.1 x 10_6 in steps of 5 x io stars pc3,over the magnitude bins centered at —1 M +1. A grid over a3 and re (as describedabove) was run for each of these values. For each model the x2 statistic described in§4.5 was calculated over the range 22.5 V <24.5 and from this, the probability Q. Inessence a three dimensional “data-cube” is built up over parameter pace.Figure 4.6 shows the observed G312 LF and four different models, one of which —using default parameters and a spheroid normalization of 3.1 x 106 stars pc3 — fitsquite well indeed. The x2 value is shown for fitting the model and observations using themagnitude bins from V = 22.5 to V = 24.5. Note that a x2 of less than 5 (the numberof degrees of freedom for this fit) does not necessarily imply a “more than perfect” fitbut is indicative of an overestimation of the errors. The chi-square probability, Q, is 0.64for the first model (the default G312 model) and ranges from 5 x iO to 1.5 xnumber for the other models shown here. The principle effect of changing parameters isto increase and decrease the predicted counts — a situation that can be compensated forby changing the normalization. It can be concluded, then, that it is possible to fit almostany parameter set.Figures 4.7 to 4.9 show contour plots of the probability, Q, as calculated from thex2 statistic, for each of the G302, G312, G352 fields. The fit can also be computed forcombinations of these fields: for each grid point in (a3, re, p) the EGM model is run foreach field and the x2 statistics added together (i.e., the sum in equation 4.1 extendsacross magnitude bins in each model). These are shown in figure 4.10 (for G302 andG312) and figure 4.11 (for G302, G312 and G352). In each of the figures the contoursChapter 4. Modelling the Spheroidz0321076Figure 4.6: This figure shows the fit of four models to the G312 data. x2 refers to thechi squared statistic taken over the V = 22.5 to V = 24.5 bins.G3 12——///V//////—64. 1E—62. 1E—62. 1E—65. 1E—6x23.4022.1823.0278.9052.8122.5 23 23.5 24 24.5VChapter 4. Modelling the Spheroid 77vary from 0.1 in steps of 0.1 and the maximum value of Q for that grid is shown in theupper right-hand corner of each plot.The models for the 0302 field (figure 4.7) show that at all normalizations there isvery little sensitivity to r,. Over the density range shown the axial ratio is constrainedto between 0.7 and 0.4 — values close to the default model. r is not strongly constrainedand there is evidence for more than one best fit for the densities 5.1 x 10—6 and 5.6 x 106where there is more than one peak. It is possible that this reflects the difference betweenthe observed LF and the “mean 0” LF. The “best fit” — denoted by the maximum valueof Q — is approximately the same for all grids. The large value of Q for these grids is dueto an overestimation of the errors.The 0312 models (figure 4.8) typically show two strong peaks — the position of thesepeaks can be seen to “drift” as the density normalization changes. Again, a, seems tobe constrained to between 0.7 and 0.4. Te is constrained quite well for any particularmodel, but overall varies from 1250 to 3500 pc. This reaffirms the conclusions drawnearlier from figure 4.6, and from the sensitivity tests. The best fit is, as for the 0302field, reasonably constant across all the grids. The maximum is lower because the “meanG” LF does not fit the 0312 observations as well as the 0302 data. This can be clearlyseen in figure 4.2.Figure 4.9 shows the 0352 field models. In contrast to the previous plots there is a“ridge” of high Q values containing 2 or 3 individual peaks that “drifts” across the (a,,?‘e) grid as the normalization increases. This makes it difficult to constrain any of theparameters.The fits to a combination of the 0302 and 0312 fields, shown in figure 4.10 are moreinteresting. Comparing the broad features of this plot to previous figures it can be seenthat the G302 dominates slightly because it fits the “mean 0” LF better. However theaddition of the 0312 data “sharpens” the peaks (especially in along the re axis) andChapter 4. Modelling the SpheroidV43214321432178Figure 4.7: Contour plots of Q for the G302 field. Contours start at 0.1 and increase insteps of 0.1. Values of r are given in kpc. The “mean G” LF was used — the normalizationand maximum value of Q are shown in the top right-hand corner of each plot.1 0.2 0.4 0.6 0.8Chapter 4. Modelling the Spheroid 79143Figure 4.8: Contour plots of Q for the 0312 field. Contours start at 0.1 and increase insteps of 0.1. The “mean 0” LF was used — the normalization and maximum value of Qare shown in the top right-hand corner of each plot.4321432210.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1Chapter 4. Modelling the Spheroid4)43214321432180Figure 4.9: Contour plots of Q for the 0352 field. Contours start at 0.1 and increase insteps of 0.1. The “mean 0” LF was used the normalization and maximum value of Qare shown in the top right-hand corner of each plot.0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1Chapter 4. Modelling the Spheroid 81typically produces only one maximum. It appears to constrain re to between 2000 and3750 Pc.The fits to a combination of the three fields, G302, G312 and G352, are shown infigure 4.11. This combination favours higher densities because, as has been noted before,there are too many stars in the G352 field, leading to a correspondingly higher density.It also constrains re to be larger than 2750 pc, though this figure should be treated withcaution due to this excess in the G352 counts.4.9 DiscussionThe results described in the previous sections show that models can be used to predictstar counts in external galaxies but that, unfortunately, the available data is not ofsufficient quality, i.e., the exposures are not long enough, and the variance between thedata in different fields is too great to allow reasonable parameters to be recovered.In addition it has been seen that changes in parameters can compensate for each other.This is illustrated in figure 4.6 which shows that changes in the density normalization,axial ratio and effective radius the all conspire to change only the magnitude of the finalcounts, not the shape of the luminosity function. In figures 4.7 to 4.9 the minimumx2 value is similar for a particular data set over a wide range of densities — though thec and re which give this minimum can vary considerably. A similar study using theDaCosta 47 Tuc LF shows a similar effect, with the mean x2 value being higher due tothe different shapes.Despite this, because it is possible to place limits on the axial ratio and effectiveradius (a3 cannot be greater than 1, for example), it is possible to place limits on howmuch the density normalization can be affected by these other parameters. For examplewhen considering the G312 field, figure 4.3 shows that the density can vary by no moreChapter 4. Modelling the SpheroidV43214321432182Figure 4.10: Contour plots of Q for the 0302 and 0312 fields combined. Contours startat 0.1 and increase in steps of 0.1. The “mean 0” LF was used — the normalization andmaximum value of Q are shown in the top right-hand corner of each plot.0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1Chapter 4. Modelling the Spheroid0.2 0.4 0.6 0.883Figure 4.11: Contour plots of Q for the 0302, 0312 and 0352 fields combined. Contoursstart at 0.1 and increase in steps of 0.1. The “mean 0” LF was used — the normalizationand maximum value of Q are shown in the top right-hand corner of each plot.4321432143212.1E—6L ... I I2.6E—6L — I 13.1E—60.03 0.55 0.17I I I I I I I I I i I I I I ‘I I I I I I I I I I I I I I I I I I3 6E6 4 1E6 4 6E6IIIIIIIIIIIIIIIII[IIIIIIIIIIIIIIIIEIIIIIIIIIIIIIIIIIIr5.1E—65.6E6 6.1E6® -I I I .i,li,i l_ I I I I1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1Chapter 4. Modelling the Spheroid 84than a factor of 10, if the axial ratio of the spheroid is limited to the range 0.4 < a, < 1.0.Similarly the effective radius of the spheroid can change the density by no more than afactor of 2 over the range 1 kpc to 4 kpc if a, is assumed to be known (and is 0.6). Whencoupled together (see figure 4.5) the density can vary between approximately 1 x 10_6and 1 x io stars pc3.The excess of counts in the 0352 field is also puzzling. If contamination by backgroundgalaxies or galaxy clusters can be ruled out then it appears that the spheroid of M31 maybe inhomogenous. Interestingly enough recent work by Majewski 1992 seems to indicatethat the spheroid of our own galaxy may contain similar structures — in fact it may be thecase that most halo stars are part of stellar “streams”, left over from structures formedor captured after the collapse of the disk. It is also possible that the excess may be dueto a fluctuation in the background galaxy counts due to large scale structure. Triaxiaiityin the spheroid is another possibility but the spheroid would have to have an extremelynon-spherical shape to account for the 0352 discrepancy.If indeed this is the case for M31 then a possible means of detection would be toobtain data for a series of halo fields covering as many different areas as possible. If thespheroid distribution is smooth (and the G352 field counts are indeed anomalous) thenthe de Vaucouleurs model should fit the data. Star counts from some fields may lead to aconsistent solution, with other fields showing up as excesses or deficiencies in the overallscheme. It may even be the case that no consistent model can be found to model thespheroid of M31.Chapter 5Modelling the Disk5.1 IntroductionThe analysis of the disk parameters will be discussed in this chapter. From the outsetthe disk is harder to model than the spheroid for the following reasons:1. The disk of M31 is known to be warped. Baade 1963 notes an asymmetry in thedisk and Walterbos & Kennicutt 1987 show that the isophotes of the southernmajor axis exhibit a change in position angle at distances beyond 90’.2. Spiral arms are present in the disk. WK88 compensate for the surface brightnessof spiral arms by excluding those regions from the fit of an exponential profile —that luxury is not available in this analysis.3. The spheroid was assumed to have negligible internal reddening. This is obviouslynot the case for any reasonable model of the disk. Unfortunately absorption in M31is poorly known, although WK88 do present some estimates for internal reddening.4. In some parts of the disk the stars on the far side of the plane of the galaxy fromthe observer will be shining through the disk. The effect of this, in addition to theinternal absorption in the disk, is not well understood.5. In some fields the spheroid counts will be sufficiently high that they will contribute asignificant proportion of the total counts and must be subtracted from the observed85Chapter 5. Modelling the Disk 86LF. However this will have to be done by assuming a density normalization froma spheroid field, assuming a set of spheroid parameters and finding the spheroidcounts in that field. This will lead to a dependency of the disk results on thespheroid.The entire issue is further complicated because the star formation history and relativenumber of disk and spheroid stars is bound to vary throughout the disk because ofdifferent star formation histories in and out of spiral arms.5.2 Spheroid Contributions in Disk FieldsAs noted above it is necessary to subtract the spheroid contribution from the disk countsin order to derive the disk parameters. At this stage it is useful to examine a model ofthe disk constructed from what are assumed to be plausible default parameters. Thismay be used as a stepping stone to more complex and realistic models.The default spheroid model (c = 0.6, re = 2 kpc and using the “mean C” LFnormalized to 3.1 x 10—6 between —1 My +1) was used to generate the number ofspheroid counts in the B4 field. From an examination of the CMD, this field appears tobe clear of spiral arm contamination. The spheroid counts in that magnitude bin weresubtracted from the B4 observed LF and the resulting number (presumed to be all diskstars) was used to normalize the Wielen LF so as to reproduce the B4 observed countsat V 22.5. Only the counts on the near side of the galaxy were used. Default values forthe scale length and height of the exponential disk were taken to be 5500 pc (from §5.3)and 250 pc (a value comparable with that for giants in the Milky Way) respectively. The“cosecant” absorption model using the default (i.e., MW parameters) was also used.1See §5.4 for a more complete description of the choices for the disk LF — the Wielen LF , while nota perfect fit, is at least a good first order approximation.Chapter 5. Modelling the Disk 87Figure 5.1 shows the relative contribution along the major axis from the disk and thespheroid models using these parameters and normalizations. In panel (a) the total countsin the magnitude range 18 Mv 28 are shown; in (b) only the counts at V = 22.5are plotted. Note how, as might be expected, the disk dominates over the spheroid alongmost of the major axis (for 10 < xo < 150’).An interesting experiment can be done by modelling how the Milky Way would appearif it “replaced” M31. The EGM model was run using reasonably standard parameters forthe Milky Way (see Bahcall & Soneira 1980): a3 = 0.8, re = 2670 pc, 1d = 3500 pcand hd = 1200 pc. The Wielen LF was used for the disk, normalized to 0.13 starspc3 between —6 M 16.5 at 8 kpc and the DaCosta 47 Tuc LF was used for thespheroid, normalized to th of that at the same distance. (Typical values of the ratioof disk to spheroid densities in the solar neighborhood vary between 500 and 1000 to 1).The results are shown in figure 5.2. Again it can be seen that the disk stars significantlydominate the spheroid stars at distances of x0 < 200’. In practice one would expect thedisk counts to drop abruptly at a distance corresponding to the physical edge of the disk.M31 has measured disk isophotes out to 100 ‘along the major axis. If the radius ofthe Milky Way disk component is, say, 30 kpc, this corresponds to approximately 150 ‘.Table 5.1 compares the number of stars seen in each “B” field and in the G213 andG263 fields (and corrected for foreground stars) in the magnitude bin at V = 22.5 tothe number of spheroid counts predicted using the default spheroid model (a3 = 0.6,= 2 kpc and the “mean G” LF normalized to 3.1 x 106 between —1 M +1).Also shown are the remaining disk counts and, in the last column, the number of diskstars predicted using the default parameters (id = 5500 pc, hd = 250 pc) and a WielenLF normalized to 1.5 x iO (see §5.6). As noted in §3.5.5 values of the absorptionparameters appropriate for the Milky Way were used, in the absenceof better data.Chapter 5. Modelling the Disk4- 2zto0042zto0088Figure 5.1: The disk (dotted line), spheroid (dashed line) and total (solid line) countsin the V = 22.5 magnitude bin for (a) the major axis and, (b) the minor axis. The diskLF was normalized using the B4 counts; the spheroid using the G312 observations asdescribed in the text.0 50 100 150 200or 370 C)Chapter 5. Modelling the Disk4__2z004zbE000or y0 (‘)89Figure 5.2: The disk (dotted linc), spheroid (dashed line) and total (solid line) counts inthe V = 22.5 magnitude bin for (a) the major axis and, (b) the minor axis for a modelof the Milky Way galaxy placed at the distance of M31.50 100 150 200Chapter 5. Modelling the Disk 90Table 5.1: Spheroid contributions to the disk fieldsField Obsd. Sphd. Disk Pred.Bi 367.0 191.1 175.9 763.2B2 185.5 82.2 103.3 518.0B3 284.0 26.0 258.0 260.9B4 97.7 8.9 88.8 82.0B5 11.6 1.0 10.6 10.3G213 787.9 460.5 327.4 346.2G263 146.6 121.1 25.5 73.65.3 Disk Counts along the Major AxisTable 5.1 shows that there is a reasonably good match between the calculated and observed disk counts, at least for the B3, B4 and B5 fields — it should be remembered thatthe spheroid contribution is model dependent and that these figures are for the defaultmodel parameters. The disk counts are, by and large, of an exponential nature: figure 5.3(a) shows a plot of the raw counts in the V = 22.5 bin (corrected for foregroundcontamination) against distance from the centre of M31. A scale length of 6600 pc derived when including the B2 data; a value of 5500 kpc is obtained using the B3, B4and B5 data only. Figure 5.3(b) plots the same counts also corrected for the spheroidcontribution given in Table 5.1. This has only a small effect on the scale length if B2is omitted from the fit (5510 pc) due to the relatively small contribution in those fields.The Bi and B2 data clearly do not fit an exponential, possibly because of the patchynature of absorption in the disk, inter-arm gaps, and so on. WK88 obtain a value of thescale length of 5300 pc, in very good agreement with this data.Chapter 5. Modelling the Disk 91I — I I I I I I I6 (a) Uncorrected5 •4 .-6.60kpc—— ——5.50 kpc .2 I I I I I I I I6 (b) Corrected5.—.-.--..z -.--.—4 ..7.73kpc:5.51 II 210 Ir (kpc)Figure 5.3: A least squares fit to (a) the uncorrected counts and; (b) the B field datacorrected by the predicted spheroid counts. In each case dotted lines indicate a fit madeignoring the B2. Dashed lines are fits to only the B3, B4 and B5 data.Chapter 5. Modelling the Disk 925.4 Input Disk Luminosity FunctionIn the previous sections all the models were calculated using a Wielen LF. The actualdata for this LF comes from the original Bahcall & Soneira model code— it amounts to aLuyten 1968 LF based on studies of the solar neighbourhood, modified to incorporate the“Wielen dip” (Wielen 1974), and then extended to fainter magnitudes. In figure 4.5 thisfunction is shown along with the observed LFs from the “B” fields (shifted in magnitudeand normalized to the same value as the Wielen LF). The “B” data has been correctedfor foreground and spheroid contamination. Bahcall & Soneira 1984 make a good casefor using the Wielen LF in preference to the simpler Luyten 1968 LF, finding that itgives a much better fit to Galactic star count data.It is obvious that the shapes of the observed disk LFs differ much more than theG302, G312 and G352 LFs differ from the “mean G” spheroid LF (see figure 4.2). Thesedifferences in shape may represent differences in the star formation history of these regions. However the Wielen LF does provide a moderately good representation of the diskLFs and will be used throughout the analysis in this chapter. This has the implicationthat any calculated x2 values may be unusually large. However the main interest is inthe position in parameter space of the minimum x2, not the actual value.5.5 Model SensitivityThe discussion will now turn to the question of the sensitivity of the disk model to itsinput parameters (scale length, 1d, scale height, hd, and inclination, i, as well the asdensity normalization). In a similar fashion to the spheroid tests a default model waschosen. This was the B4 field corrected for spheroid counts (as derived from the defaultspheroid model) using id = 5500 pc, hd = 250 pc and i = 775 and a density normalizationof 1.5 x iO (used to generate the disk counts for figure 5.1(a)). A “cosecant” absorptionChapter 5. Modelling the Disk 932 I I I I I IBiHZ.1-—— B3B5Wielen -0- .*— -—__7___:_ 7- /—1 - I -//_-.-•,._--v--Z/f . -/. /-2 -‘-.-3 I I I I I I I—6 —4 —2 0MFigure 5.4: The Wielen LF is shown here (filled circles) together with the observed (andspheroid corrected) data from the “B” fields, as indicated in the legend. The observeddata have been shifted in magnitude and re-normalized to the same counts in the range—5Mv—2.Chapter 5. Modelling the Disk 94model was used in the default model and absorption parameters for the Milky Way wereused. The effect of increased absorption will be to reduce the number counts at a givenmagnitude (as the LF shifts to fainter magnitudes) — as with the spheroid tests theimportant consideration here is the relative change in the counts as the parameters vary.5.5.1 Tests with Scale LengthThese first set of tests were run over 20 fields along the both axis, ranging from distancesof 10’ to 200’ in steps of 10’. The scale length of the disk was varied from 2500 pc to7500 pc in steps of 500 pc, all other variables being the same.Figure 5.5(a) shows the variation along the minor axis with scale length. The largerthe scale length, the higher the disk counts, and the further the disk extends, as mightbe expected. At this steep inclination, however, the even a large disk does not extendvery far along the minor axis.For the major axis test (seen in figure 5.5(b)) varying the scale length by has a largeeffect on the the model at all distances, changing by, for example, a factor of 19.5 at 100’.However changing the scale length a small amount (by a few hundred parsecs, say) mayhave a minimal effect. The intersection of the models occurs at x0 47’, or 9870 pc.5.5.2 Tests with Scale HeightThe next test (shown in figure 5.6) shows the variation with scale height, hd, for theminor axis (a) and major axis (b). The scale height was varied from 50 to 1050 pc insteps of 100 pc.The minor axis tests, shown in figure 5.6(a), reveal that changes in hd will have largeeffect on the observed number counts, but only in a limited range of distances. Thevariation between hd 50 and hd 1050 pc at Xo = 20’ is by a factor of approximatelyz0Chapter 5. Modelling the Disk 95210—1—2—332z010Figure 5.5: The variation of number counts (for the V = 22.5 bin) with scale length, id,for (a) the minor and; (b) the major axes. The first, last, and default models are labelledwith the scale length in kpc. In (a) the counts in the V = 22.5 bin for the G213 fieldhave been plotted; in (b) the B3, B4 and B5 counts are shown.0 50 100d (‘)150 200Chapter 5. Modelling the Disk 9634 but beyond 100’ along the minor axis the disk has effectively vanished because M31has a small (12<5) inclination to the line of sight.The counts along the major axis (at the default inclination) are comparatively insensitive to changes in disk scale height, changing only by a factor of 8 to 10 over theentire major axis. The effect of changing the scale height diminishes as the scale heightincreases, reflecting the decrease in density of the disk component.5.5.3 Effects of Inclination and AbsorptionThe tests for changes in inclination are not shown — the inclination in all the remainingmodels is set to 77’5. However the model does behave as expected to changes in i. Alongthe minor axis the changes in number counts will depend on the scale height as well asthe inclination. For the default scale height and at a minor axis distance of 20’ the countschange by a factor of 11.6 over a range of inclinations from 0° (face on to the observer)to 80°. At 40’ this ratio has increased to 460 as the inclination becomes lower (more“face on”) the counts increase more further out along the axis as the line of sight travelsthrough more of the disk. Along the major axis the effect is constant (the ratio being0.4). because there is no scale length in the absorption modelThe effects of absorption and internal reddening in the disk of M31 will obviouslyhave an effect on the observed LF . However the determination of the exact absorptionparameters (as described in §3.5.3) is not well understood, Noting that WK88 find thatM31 has a similar reddening law to the Milky Way, this test is conducted using thesame absorption parameters for our own Galaxy. The results will be a reasonable firstapproximation. Applying this absorption model reduces the counts at a given distanceby a factor of 1.97 for “cosecant” absorption and 2.04 for the Sandage model. This is alarge effect considering that the scale height is set to 250 pc but it must be rememberedthat the disk is highly inclined and the line of sight will pass through a significant fractionlog(N0)log(ND)0—t\)C)CDCD c.•CDp,c_.b CDCDo o_•I—Ci)c.,_•Ci)c4-I-0i1CDIs)C.YtoCDI-4.•° I-•+-Ci)CD.Is)Ci)CD)—c,C.3‘-CDCD.-CDc’.CD CD‘—CDe4-0CD I-’.‘CD.0 0 C 0 Z\)0 0Co —1Chapter 5. Modelling the Disk 98of the dust, even if it is confined to regions near the plane. Of course, this does allow fortremendous leeway in fitting the model — in the extreme each disk field could be modelledwith a different set of absorption parameters.Changing the “cosecant” absorption model parameters causes the model to behave asone might expect. For example, halving the absorption scale height, a0, to 50 Pc reducesthe counts to 88% of the default model along the major axis. Increasing it to 250 pcincreases the counts by a factor of 1.23. Halving the coefficient, a1, reduces the countsby 57%.5.6 Initial Density EstimatesAs has been mentioned before, the normalization of the input LF is very important tothe success of the model fitting the data. An initial estimate of the density normalizationfor the default model (id = 5500 pc, hd = 250 pc) was made for each field, using theWielen LF discussed above. As with the spheroid, the normalization was performed overthe range —1 M ç +1 at a distance of 10 kpc. In each case the LF was scaled toa value of 1.0 over this range and the resulting number counts used to determine thenormalization that would return the correct counts in the V = 22.5 magnitude bin. Thevalues obtained were:Bi: 3.46 x iO stars pc3 B5: 1.55 x iO stars pc3B2: 2.99 x 10 stars pc3 0213: 1.42 x iO stars pc3B3: 1.48 x i0 stars pc3 0263: 5.20 x iO stars pc3B4: 1.62 x iO stars pc3The values returned by the B3, B4 and B5 fields are quite consistent, the 0213 density(along the minor axis) is also in good agreement. In the following discussion a value of1.5 x iO (between —1 M +1 at a radius of 10 kpc) will be used as a defaultChapter 5. Modelling the Disk 99model density.It is of interest to compare these densities to the values derived for the spheroid,and to compare the ratio of these two to the density normalizations obtained for thesolar neighbourhood. The ratio of disk to spheroid density over the magnitude range—1 M +1 is 48.4 at 10 kpc. The LFs used for each component are, however,of different shapes. Using these density values to normalize the LFs it is possible tocalculate the density over the range —6 M +16.5, i.e., down to the H—burninglimit. The spheroid density is then 1.4 x iO and the disk density is 7.5 x 10_2 pc3leading to a ratio of 53.6.Wielen 1974 gives a normalization for the Wielen LF of 0.13 stars pc3 down tothis limit. Bahcail, Schmidt & Soneira 1983 derive a density of the spheroid of from(1 — 9) x iO stars pc3 depending on the model being used. This leads to a disk tospheroid density ratio, at 8 kpc, of between approximately 1300:1 and 140:1, althoughthe lower limit usually used is about 500:1.It appears then that the spheroid of M31 is more massive than the Milky Way’swhilst the disk is less massive, leading to a relative normalization about an order ofmagnitude lower than that of the Galaxy. However it should be noted that a directcomparison between the Milky Way densities observed in the solar neighbourhood (at8 kpc) and the M31 densities predicted at 10 kpc may not be completely meaningful.Possible reasons for this apparent discrepancy may include: the different sizes of thegalaxies; the differences in disk and spheroid parameters; possible differences between theMW and M31 luminosity functions (recall that only the giant branch is being observedhere); and the difficulty in observing a “clean” disk sample in M31.Chapter 5. Modelling the Disk 1005.7 Parameter GridsTo more fully investigate the properties of the B4 field disk counts models were run overa grid of density normalizations, scale lengths and scale heights. The scale length, id, wasvaried from 2500 Pc to 7500 Pc in steps of 500 pc; the scale height, hd, was varied from50 pc to 1050 pc in steps of 100 pc. Nine density normalizations were used for the WielenLF (in the range —1 M +1) ranging from 5.0 x iO stars pc3 to 2.5 x iO insteps of 2.5 x iO stars pc3.The resulting grids for the B3, B4 and B5 fields are shown in figures 5.7 to 5.9respectively. Figure 5.10 is a fit to a combination of those three fields. Unlike thespheroid models, these grids plot x2 not Q. This is because even the minimum x2 valuesare quite large: the Wielen LF is not asgood a fit to the B field data as the “mean C”LF is to the spheroid fields. The contours for each field have been chosen to highlightthe “vaJley” of minimum x2 values. The fits were made over the range 21.5 V 23.0using equation 4.1. Poisson (/) errors were applied to the raw counts, propagated bythe area scaling factor (0.10022) and used as an estimate of 0N. Foreground star countsand spheroid counts from the default model were used as the contaminants.The x2 fits to the B3 data are shown in figure 5.7. At higher density normalizationsthe scale height is quite well constrained between 50 pc and 1000 pc. However the entirerange of tested scale lengths fits this data. Figure 5.8 shows the results for the B4 field.Unfortunately neither the scale length or the scale height is constrained very well atall, the minimum x2 values lying diagonally across the (id, hd) grid. The results for B5(figure 5.9) are similar, though the fit is generally better with a minimum x2 5.Figure 5.10 shows the results of fitting all three of the B3, B4 and B5 data simultaneously. Although even the minimum x2 is questionably large, it appears to constrainhd between 50 and 400 Pc and the scale length, id, between approximately 5 and 7 kpc.-I.—.Cl)-C5I—.©Cl)Cl)0CDI—• CDTh“Cl)CD CD••0 0I-’.p Cl)CMCM CD0 I-..0CD0CDa-’-0C..3CD0CDCDi-.CDI0CM 00CDI.-.0‘ CDe CDI-.C.,hd01’3‘0)0)000)0)00‘)0)0)0CJ 0)IA _4I-’.oqIAcOq CDC3Chapter 5. Modelling the Disk420108102Figure 5.8: This figure shows contours of x2 for the B4 field over a grid of scale lengthin kpc (id) and scale height (hd) in 100 Pc units. Contour levels range from 300 to 500in steps of 20. The Wielen LF was used — the density normalization is shown in theupper right hand corner of each panel. The fit was made to observed counts in the range21.5 V 23.0.1086420108664203 4 5 6 7 3 4 5 6 7 3 4 5 6 7Chapter 5. Modelling the Disk0108103Figure 5.9: This figure shows contours of x2 for the B5 field over a grid of scale lengthin kpc (id) and scale height (hd) in 100 pc units. Contour levels range from 4 to 104in steps of 10. The Wielen LF was used - the density normalization is shown in theupper right hand corner of each panel. The fit was made to observed counts in the range21.5 V 23.0.108642010864264203 4 5 6 7 3 4 5 6 7 3 4 5 6 7rCDq1Oi.+.0o-+-I-.o-cn,cn‘-.0c,cCDa’0CDCD‘—‘)gEc.,i--+-CDIA‘- 0C.YP0a’CDCD 1 o‘-•CDCDp.,-0 Z.“‘.j)(C7EI••‘•‘•-‘).+-cIp.,BCDI-’ p.,.0a..-CDCD‘•hd0t’3.O00.O00Z’30C,’CD01 01I-.Chapter 5. Modelling the Disk 1055.8 DiscussionThe analysis of the disk of M31 can be summed up in one word: difficult! The choice ofappropriate spheroid parameters (in order to correct the counts to true disk counts) iscritical. In addition, the inhomogenous character of the disk itself conspires to complicatethe problem. In some respects studies of the surface brightness distribution rather thanstar counts may be better suited to the determination of the properties of the disk becausea smooth average over large portions of the disk can be taken (see, for example, WK88).The spheroid counts appear to be so high because the assumption that all, or most,of the giant stars seen in these fields belong to the disk population is not necessarily true.The actual proportion of disk giants to spheroid giants will depend heavily on the starformation history in that region of the galaxy and deconvolving the two populations willrequire this information. The studies of the disk of M31 that use integrated light andsurface brightness profiles do not suffer from this problem because most of the disk lightcomes from dwarf stars, which considerably outnumber the giants.In order to effectively use this kind of star count analysis on the disk fields one musteither obtain data that reaches extraordinarily faint magnitudes, or one must use a bright“tracer” of the (old) dwarf population whose relative proportion amongst the disk is wellknown. Rose 1985 has used RHB stars to find evidence for a thick disk in the Milky Way,but this has been criticised by Norris & Green 1989 because it is difficult to distinguishbetween them and core He burning “clump” stars. These types of studies in M31 arelikely to remain technically infeasible for quite some time. It has also been assumed thatthere is no contribution to the disk counts from stars on the far side of the galaxy shiningthrough the disk. Inclusion of this effect may also improve matters but it will be difficultto derive the parameters for the absorption.Chapter 6ConclusionsThe EGM code as described in Chapter 3 has been successfully developed and implemented. Tests have shown that it can be used to reproduce and predict counts in variousfields in M31. However, the use of the code to derive parameters for the disk and thespheroid depends heavily on the data available. The data on the spheroid counts canbe used directly (under the assumption that the spheroid LF , axial ratio and effectiveradius does not change with radius) because there are few complications involved whenlooking at this component. Modelling the disk is much more challenging — the disentanglement of the disk and spheroid giants, the knowledge of the proportion of giants todwarfs in each component, and the effects of different star formation on different fields isa very complex problem and cannot be attempted in this work with the data currentlyavailable.A more precise determination of the spheroid parameters using the EGM code is quitefeasible but requires better data. In particular the data set should be as homogeneous aspossible — exposure times should be the same in each field to ensure that incompletenessestimates are only affected by crowding. In addition to data along the minor axis astudy along a “diagonal” (between the major and minor axes) is required to make amore accurate determination of the spheroid axial ratio. Finally, background fields ofthe same exposure time should be taken to remove foreground star and backgroundgalaxy contamination directly without resorting to modelling their contributions.Majewski 1992 has studied the dynamics of stars in the Milky Way near the North106Chapter 6. Conclusions 107Galactic Pole using proper motion measurements. As well as evidence of a thick diskcomponent dominating at heights of 1 kpc to 5.5 kpc above the plane he notes thepresence of a “moving group” of spheroid stars. This group of stars occupies a verydistinct region in velocity space at a height of -- 4550 Pc above the galactic plane. Thisseems to indicate that the galactic spheroid is not homogeneous — possibly the result ofa formation scenario similar to the Searle & Zinn 1978 model.To determine if this is indeed the case for the Andromeda Galaxy it would be necessaryto conduct a systematic survey of the spheroidal component of M31. Very deep exposureswould not be necessary as long as a sufficient number of stars was recorded. The fieldswould be distributed over the region around M31 in a grid-like pattern — the number ofthese “grid points” and their spacing would select the scale of structure that could bedetected. Because the nature of these inhomogeneities would be essentially stochasticit might be the case that even an extensive survey would miss any deviations from asmooth distribution of stars. However if the “background spheroid” distribution is itselfpatchy then it may have a reasonable chance of success. Using CFHT, for example, tocompletely map out the halo in one quadrant of M31 (as seen from the Earth) out to adistance of 100’ would require approximately 150 FOCAM frames if a 2048 x 2048 CCDwas used. Clearly this is a massive undertaking and reducing the coverage of the gridthen sets limits on the sizes of deviations that can be detected.It has been demonstrated that modelling the disk counts is problematic. The contamination of the disk giant sample by the spheroid giants seems to preclude any simpleanalysis because the relative numbers of giants to dwarfs is likely to change with positionin the disk. Varying star formation histories and the possibility of an inhomogenousspheroid compound the problem. It seems as if the only way to successfully determine disk parameters is to sample the dwarf population directly or to discover and usea luminous tracer of the old disk population. In the latter case the relative numbersChapter 6. Conclusions 108of this “tracer” component and the disk component would have to be well known, andvariations with star formation history would still have an effect. It should be noted thatthese problems will be particularly acute if the galaxy is face-on, such as M33, which isanother possible candidate for this sort of study. In this case there will be no “pure”spheroid field whose fit can be subtracted from the disk fields. This presupposes thatM33 has a significant spheroid.It was hoped initially to detect and determine the parameters of a thick disk component in M31. With the data currently available this is not possible. The problem ofseparating two components with a similar scale height is still present. This is compounded with the problem noted for the disk analysis — if a thick disk is present then threesets of LFs and density distributions must be de-convolved. This would only be possibleif both the spheroid and the disk components were well known in that field so that anydeviations from the two component model could be detected. This would require that thespheroid be homogeneous (so that models derived from several areas produce a consistentsolution) and that any non-uniformities in the disk (different star formation histories forexample) had been accounted for. It is unlikely that the anomalous counts in the 0352field are due to a thick disk — the field is too far away from the disk of M31 for a thickdisk with any plausible scale height and density normalization to have such a large effect.Studies of a possible Milky Way thick disk benefit from being able to look straight upthrough the thin disk and into the regions of interest. Studies of edge on spiral galaxiesare also more suited to this task, presenting a profile of both the thin and thick disks.Investigation of other functional forms for the disk (isothermal, with a sech2 dependenceon height) and spheroid (a power law) can be done using the EGM model but are unlikelyto solve this problem.It would be desirable to compare the results of the EGM model to some independentdata set. Fortunately such data is available in the form of the Pritchct & van den BerghChapter 6. Conclusions 1091994 study. Figure 6.1 plots this data, corrected for contamination using the PvdB94background field data, as connected points (logarithm of the number per square arcminute per 0.5 magnitudes) for four of the PvdB94 fields (MO, Ml, M2 and El). Thedashed line is the EGM model results using the “mean G” LF, c = 0.6, re = 2 kpcand a density normalization for the spheroid LF required to reproduce the G312 counts(3.1 x 106 stars pc3).As can be seen the fit to the MO and Ml fields is very good indeed, especially considering that the spheroid LF density was derived completely independently. The fit to thediagonal field El is worse, probably because the axial ratio of the model is not exactlyright. The M2 field is approximately l5 from the centre of M31 — at this point the numbers of contaminating objects is becoming relatively large and the fit correspondinglypoorer.It may be concluded that, broadly speaking, this type of analysis — using star countsrather than surface brightness — does work! Given reasonable model parameters andnormalizations the EGM code will predict the counts one would expect to see. Solving forthese parameters, on the other hand, is much more difficult and requires high quality data(and plenty of it). It is hoped that this model can be used in future studies to properlyinvestigate the underlying physical structure of external galaxies as a complement to themethod of examining the observed properties using surface brightness proffles.Chapter 6. Conclusionsz0zbL0210210V110Figure 6.1: A comparison between the EGM model (dashed line) and the Pritchet & vanden Bergh 1994 data (connected dots). 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