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The albedo of an exoplanet: spacebased photometry of the transiting system HD 209458 Rowe, Jason 2008

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The Albedo of an Exoplanet Spacebased photometry of the transiting system HD 209458 by Jason Frank Rowe B.Sc., The University of Toronto, 2000 M.Sc., The University of British Columbia, 2003 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Astronomy)  The University Of British Columbia January, 2008 c Jason Frank Rowe 2008  Abstract Very precise photometry of transiting extrasolar planets can be used to learn about the physical structure of Jupiter-like planets in an exoplanetary systems. The fraction of light reflected from the planet (albedo) provides crucial insight into the chemical structure of atmospheres and global thermal properties of a planet, including heat dissipation and global weather patterns. Measuring the albedo of an exoplanet requires very precise photometry with high time sampling and nearly continuous time coverage spanning may orbits, which can be achieved at present only from space. We present spacebased photometry of the transiting exoplanetary system HD 209458 obtained with the MOST (Microvariablity and Oscillations of STars) satellite during 2004 and 2005. The data span 14 and 44 days respectively, and have nearly complete time coverage for both spans. The HD 209458 photometry was obtained in MOST’s Direct Imaging mode, not part of the original mission but implemented to make possible measurements of stars in the brightness range 6.5 ≥ V ≥ 13. The photometric reduction techniques developed for this thesis have become the standard pipeline for processing MOST Direct  Imaging data, in particular the corrections for stray Earthshine reaching the MOST instrument focal plane. Our analysis of MOST photometry of HD 209458 sets a 1σ upper limit on the depth of the optical eclipse of the planet of 17 parts per million (ppm) = 17 micromagnitudes. This corresponds to a 1σ upper limit on the flux ratio of planet to star of 1.57 × 10−5 and an upper limit on the geometric  albedo in the MOST bandpass (400 to 700 nm) of 8%. This is the most sensitive measurement of an exoplanetary albdeo ever obtained. The limit on the albedo of the gas giant HD 209458b means it much less reflective as Jupiter. This result rules out the presence of reflective clouds in the ii  Abstract atmosphere of HD 209458b and has already enabled theoretical modeling of far-infrared measurements to contrain the planet’s equilibrium atmospheric temperature to be 1550 ± 150K.  The MOST albedo analysis demonstrates the potential of spacebased  photometry missions like CoRoT (launched in December 2006) and Kepler (due for launch in early 2009) and has already provided important lessons for both missions.  iii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  Abstract  List of Tables  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . .  x  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xi  Glossary  Dedication  Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . xii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  . . . . . . . . . . . . . .  14  2.1  Data Format . . . . . . . . . . . . . . . . . . . . . . . . . . .  15  2.2  Photometric Reduction  . . . . . . . . . . . . . . . . . . . . .  17  2.2.1  Dark Correction . . . . . . . . . . . . . . . . . . . . .  17  2.2.2  Flatfield Corrections . . . . . . . . . . . . . . . . . . .  18  2.2.3  Star Detection and Centroiding  19  2.2.4  Stray Light and Background Determination  . . . . .  22  2.2.5  PSF Fitting and Adding Up the Starlight . . . . . . .  22  2.2.6  Removal of Stray Light Effects . . . . . . . . . . . . .  25  2.2.7  Crosstalk Corrections . . . . . . . . . . . . . . . . . .  26  2 MOST Direct Imaging Photometry  . . . . . . . . . . . .  iv  Table of Contents 2.2.8  Image Stacking: Aperture Photometry and Partial Pixels . . . . . . . . . . . . . . . . . . . . . . . . . . .  31  Photometric Errors  . . . . . . . . . . . . . . . . . . .  32  3 MOST Direct Imaging in Action . . . . . . . . . . . . . . . .  36  2.2.9  3.1  3.2  BD +18 4914 . . . . . . . . . . . . . . . . . . . . . . . . . . .  37  3.1.1  Photometry  38  3.1.2  Frequency Analysis  . . . . . . . . . . . . . . . . . . .  38  3.1.3  A hybrid pulsator . . . . . . . . . . . . . . . . . . . .  42  3.1.4  Summary . . . . . . . . . . . . . . . . . . . . . . . . .  45  AQ Leo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  45  3.2.1  Nonlinear least-squares fitting and bootstrapping  47  3.2.2  The largest intrinsic variations observed by MOST -  . . . . . . . . . . . . . . . . . . . . . . .  an unexpected challenge 3.2.3  . .  . . . . . . . . . . . . . . . . . . .  47  HD 189733 . . . . . . . . . . . . . . . . . . . . . . . . . . . .  48  3.3.1 3.3.2  MOST photometry of HD 189733 . . . . . . . . . . . System Parameters . . . . . . . . . . . . . . . . . . .  49 50  3.3.3  Discussion  . . . . . . . . . . . . . . . . . . . . . . . .  53  4 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  57  3.3  Intrinsic frequencies and new modes in AQ Leo  47  4.1  The Phased Light Curve  . . . . . . . . . . . . . . . . . . . .  58  4.2  Modeling the Light Curve . . . . . . . . . . . . . . . . . . . .  60  4.2.1  Bootstrap Error Analysis . . . . . . . . . . . . . . . .  68  5 Results and significance . . . . . . . . . . . . . . . . . . . . . .  72  5.1  Transit Timing Predictions . . . . . . . . . . . . . . . . . . .  80  5.2  Summary and Future  . . . . . . . . . . . . . . . . . . . . . .  81  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  85  v  List of Tables 3.1  Detected frequencies from BD+18 4914 observations . . . . .  41  3.2  Fitting Parameters for HD 189733 Lightcurve. . . . . . . . . .  53  4.1  Best fit parameters and errors for the HD 209458 system . . .  71  5.1  Albedos of Giant Planets . . . . . . . . . . . . . . . . . . . .  73  vi  List of Figures 1.1  Comparison of the orbit of HD 209458b compared to the Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3  1.2  Libration and Tides . . . . . . . . . . . . . . . . . . . . . . .  8  1.3  Transit Timing Model . . . . . . . . . . . . . . . . . . . . . .  10  2.1  FITS file layout . . . . . . . . . . . . . . . . . . . . . . . . . .  16  2.2  Dark current correction map. . . . . . . . . . . . . . . . . . .  19  2.3  MOST point performance . . . . . . . . . . . . . . . . . . . .  21  2.4  Data correction steps . . . . . . . . . . . . . . . . . . . . . . .  27  2.5  Background level correlation . . . . . . . . . . . . . . . . . . .  28  2.6  Identifying crosstalk . . . . . . . . . . . . . . . . . . . . . . .  30  2.7  Handling partial pixels in photometry . . . . . . . . . . . . .  33  2.8  Photometric Error corrections . . . . . . . . . . . . . . . . . .  35  3.1  Photometry and frequency analysis for BD+18 4914 . . . . .  39  3.2  No detection of amplitude changes in BD+18 4914 . . . . . .  44  3.3  AQ Leo lightcurve . . . . . . . . . . . . . . . . . . . . . . . .  46  3.4  Light curve for HD 189733 . . . . . . . . . . . . . . . . . . . .  51  3.5  HD 189733 filtered data phased to the planetary orbital period 52  3.6  Fitted parameters and errors for HD 189733 system parameters 54  3.7  Constraints on the inclination angle of rotation for HD 189733a 56  4.1  Light curve model. . . . . . . . . . . . . . . . . . . . . . . . .  4.2  HD 209458 light curve phased at the planetary orbital period. 60  4.3  Observations of the transit . . . . . . . . . . . . . . . . . . . .  62  4.4  Modeling the eclipse . . . . . . . . . . . . . . . . . . . . . . .  63  59  vii  List of Figures 4.5  MOST system throughput . . . . . . . . . . . . . . . . . . . .  64  4.6  Comparing HST and MOST . . . . . . . . . . . . . . . . . . .  67  4.7  Parameter fits and uncertainties  . . . . . . . . . . . . . . . .  70  5.1  Estimating the temperature of HD 209458b . . . . . . . . . .  77  5.2  Relationship between the Bond albedo and geometric albedo  78  5.3  N-body results for maximum transit timing deviation vs. orbital period of the perturbing planet. The colour scale as defined on the right side of the plot indicates the mass of the perturbing planet in Earth masses. . . . . . . . . . . . . . . .  82  viii  Glossary MOST EGP AB Ag AgMOST M⊙ MJ Teq Fp F∗ pc AU FWHM CCD PSF ADU HST O-C τ  Microvariability and Oscillations of STars Extrasolar Giant Planet Bond Albedo Geometric Albedo Geometric Albedo measured over the MOST band pass Solar Mass Mass of Jupiter Equilibrium Temperature Planetary Flux Stellar Flux Parsec Astronomical Unit Full Width Half Maximum Charged Coupled Device Point Spread Function Analog Digital Unit Hubble Space Telescope Observed minus Calculated Optical Depth  ix  Acknowledgements Thank you to my Brother, Mother and Father. Your continued and unequalled love and support have made me the person I am today. Thank you for encouraging my studies and allowing me to follow my dreams. I hope I make you proud. I am very happy to have an RR Lyrae type star included in this thesis as related to my work with Christine Clement during my undergraduate days in Toronto. Her early direction and guidance have proved invaluable. In Vancouver I’ve made many great friends. I would like to thank all my fellow graduate students. Thanks to Jason Kalirai, Tyron Tsui, and Robert Ferdman for originally sharing an office with me. Those were great times. Thanks to Anna Sajina, Marie-Andree Dumais, Mark Huber and Thomas Pfrommer for all the coffee, it helped keep up the addiction. A big thank you to Kelsey Hoffman for all the support, you mean the world to me. I would like to thank the entire MOST Science team - Werner Weiss, Gordon Walker, Dimitar Sasselov, Anthony Moffat, Slavek Rucinski and David Guenther. Also a thank you to Sara Seager and Eliza Miller-Ricci for the rewarding collaborations. To Rainer Kuschnig for all his help with my work. He is the Don Cherry of Austria. I also want to thank Chris Cameron for his friendship and extremely helpful scientific discussions we have had over the years. I know that we will collaborate again in the future. The MOST lab would of been a lonely place without his daily appearance. Finally, there is the Grand Poobah, Jaymie Matthews. I could not ask for a better Ph.D. supervisor. That is why he gets this entire paragraph. He is a great scientist and will be a friend for life. I’ve had too many memorable experiences and some experiences that I’ll never be able to remember. His noble spirit embiggens the smallest man. x  Dedication To my Brother, Mother and Father.  xi  Statement of Co-Authorship In keeping with the guidelines described in the Faculty of Graduate Studies Publication “Instructions for the Preparation of Graduate Theses”, this preface is intended to notify the reader of publications in which work from this thesis appears. The majority of the work pertaining to the photometry and planetary albedo limits from the 2004 observations (Chapter 2 and portions of Chapters 4 and 5) was published in Rowe, Jason F.; Matthews, Jaymie M.; Seager, Sara; Kuschnig, Rainer; Guenther, David B.; Moffat, Anthony F. J.; Rucinski, Slavek M.; Sasselov, Dimitar; Walker, Gordon A. H.; Weiss, Werner W. 2006, ApJ, 646. 1241. The paper was written primarly by myself with helpful comments from my supervisor Jaymie Matthews. The table of planetary albedos for the MOST bandpass was calculated by Dr. Sara Seager, one of the co-authors. The work on the 2005 observations is not yet published, but is in the process of being submitted to the Astrophysical Journal and can be found in Chapter 5. The interuptation and analysis was done myself. The production of HST photometry to match the MOST bandpass and used to compare the photometric results was computed by Eliza Miller-Ricci a co-author on the upcoming paper. In addition to these papers, I have presented work in the form of poster papers and oral presentations at various conferences. The photometry data reduction pipeline was presented as an oral at the Stellar Pulsation and Evolution Meeting (Rome, Italy June 2005) and as a poster at the 36th CASCA Meeting (Montreal, Q.C. June 2005). The planetary albedo results have been presented as an oral at the 37th CASCA Meeting (Calgary, A.B. June 2006) and as a poster at the 2007 AAS/AAPT Joint Meeting (Seattle, U.S.A. January 2007).  xii  Chapter 1  Introduction In 1995, on the same day, the discoveries of the first bonefide brown dwarf, Gliese 229B [58], and a planet orbiting the solar type star 51 Peg [54] were announced. The evolution of planets and brown dwarfs are not dominated by thermonuclear processes, which differentiates them from stellar objects such as the Sun. In terms of mass, brown dwarfs and stars separate at approximately 0.08 M⊙ which is the minimum mass to generate temperatures and pressures high enough to support the fusion of hydrogen in their cores [42]. At 13 MJ (1 M⊙ = 1047 MJ ) the minimum mass for deuterium-burning is reached which allows one to distinguish between a planet and a brown dwarf [7]. The formation process of planets may also distinguish them from brown dwarfs and stellar objects, as much as their non-thermonuclearness. With over 250 planets discovered around nearby stars1 there are many new and exciting puzzles to be solved. These include understanding the origin of gas giant planets in short-period orbits closer to their host stars than where they could of formed and the structure, spectrum, interior fluid dynamics and thermal evolution of extrasolar planets. To answer these questions, observations can be made to determine the mass, radius and bulk spectra features of the planet. For example, the mass function can be determined from radial velocity Doppler measurements that give the motion of the host star around the centre-of-mass of the system. Doppler measurements, pioneered by the work of Campbell and Walker [11], look for reflex motion of the star as it moves about the centre-of-mass of the system projected towards our line-of-sight. The first strong evidence for a planet outside our Solar System was from radial velocity measurements of the yellow giant γ Cephei [12, 83] which combined with recent spectroscopic followup 1  The Extrasolar Planets Encyclopedia: http://www.exoplanet.eu.  1  Chapter 1. Introduction observations point to a substellar companion with a minimum mass of 1.7 times the mass of Jupiter in 2.14 year orbit [34]. As the inclination of the planet’s orbit, relative to our viewing perspective, cannot be obtained from spectroscopy, only a minimum mass (mass times the sine of the rotational inclination axis) can be inferred. In some cases the planet will also transit the star from our perspective. The first transiting extrasolar planet was discovered orbiting the solarlike star HD 209458 [13] and gave the first definitive measurement of the mass of an extrasolar planet. The effects of limb darkening allow multicolour photometry of the transits to precisely measure the orbital inclination, and hence the mass and radius of the planet with respect to the parent star [43]. Far-infrared measurements of the eclipses in thermal emission [23] give brightness temperatures at specific wavelengths to constrain the atmospheric properties. Measurement of the optical eclipse of such a system provides an albedo value which, when combined with the thermal emission data, sensitively probes the exoplanetary atmosphere to search for high-altitude clouds (such as silicate condensates) and possibly even weather patterns [66]. HD 209458 is a nearby (47 pc), bright (V = 7.65) Sun-like star (G0V) hosting a transiting hot extrasolar giant plant (EGP) [18]. Figure 1.1 shows the orbit of the planet in the HD 209458 system compared the orbits of Mercury, Venus and the Earth in our Solar System. The proximity of the planet to the star means the planet has a high surface temperature and hence is a relatively strong thermal emitter, which can be measured at infrared wavelengths. Spitzer observations at 24 µm detected an eclipse at a depth of 0.26% ± 0.046% which gives a brightness temperature (1130 ± 150 K)  for the observed band [23]. The brightness temperature is the temperature at which a blackbody would have to be in order to duplicate the observed intensity of an object at the observed frequency. Estimating the blackbody equivalent temperature is model dependent because of strong, continuous H2 O absorption [23]. Atmospheric models predict an effective temperature of 1442 K, assuming 4π reradiation and solar metallicity [27]. Physical models of the interiors of gas giants can be constructed based on equations of gas equilibrium (like the hydrostatic equations of stellar struc2  Chapter 1. Introduction  Figure 1.1: The orbit of HD 209458b at 0.047 AU is compared to the orbit of Mercury, Venus and Earth which are at distances of 0.39, 0.72 and 1 AU respectively.  3  Chapter 1. Introduction ture), and in the outer envelopes, the equations of hydrodynamics. Current studies commonly adopt physics and chemistry based on our understanding of the gas giants in our own Solar System, such as Jupiter, and adapting a consistent equation of state and opacity for the matter. By comparing observations to such a model one can begin to classify and gather statistics on physical parameters of exoplanets to test unified formation and structural models and places our own Solar System in the context of distant worlds. A numerical model is based on finding a solution to a set of differential equations which require appropriate boundary conditions. Photometric observations place constraints on the boundary conditions, specifically, estimates of the planetary radius and temperature. At distances from a star of 5 or more AU, the photosphere of a planet will cool to ∼ 100K in a few  gigayears [7]. For irradiated planets, such as HD 209458b2 which orbits its host star at a mere 0.047 AU every 3.5 days, the thermal equilibrium temperature at its photosphere can be estimated as Teq = T∗ (R∗ /2a)1/2 [f (1 − AB )]1/4 ,  (1.1)  where the star and planet are considered blackbodies and T∗ and R∗ are the effective temperature and radius of the host star and the planet has a Bond albedo AB and is at a distance a = (GM∗ P 2 /4π 2 )1/3 with an orbital period, P where G is the gravitational constant. The Bond albedo is the ratio of the total amount of light reflected from a sphere to the whole amount incident on it [68]. Here f is the proxy for atmospheric circulation, f = 1 if the absorbed stellar radiation is redistributed evenly throughout the planet’s atmosphere (e.g., due to strong winds rapidly redistributing the heat) and f = 2 if only the heated sunlit side reradiates the energy. The equilibrium temperature (Teq ) gives an appropriate boundary condition and quickly demonstrates how the distance and albedo dictate the interior structure of the planet. For solar type stars the peak of emitted radiation is in the optical range, 2 extrasolar planets are named based on the star they are found to orbit around. In the case of the HD 209458 system, the star is designated with an ”a” and the planet with a ”b”. Additional planets found orbiting the star will be named ”c”, ”d” as they are discovered  4  Chapter 1. Introduction thus the optical albedo gives a good approximation to the Bond albedo. The Bond albedo is a sensitive measurement of the total energy input from irradiation, but is difficult to measure. The geometric albedo Ag is the ratio of brightness of the planet (Fp ) at zero phase angle compared to an idealized flat, fully reflecting and diffusively scattering disk. From the definition of Ag , one can derive the ratio of the stellar and planetary fluxes, Fp = Ag F∗  Rp a  2  ,  (1.2)  where Rp is the planetary radius. In the case of the HD 209458 system the flux contrast is no more than about 1 part in 10 000 in the optical range, as shown in this thesis; see also [66]. Planets at increasingly larger radii with similar albedos become increasingly difficult to measure. Our studies conducted with the MOST satellite [82] (including this one) have shown that the optical albedo for irradiated EGPs such as HD 209458b are much lower than previously expected [52]. We will show that HD 209458b has an optical geometrical albedo of 3.8% ± 4.5%. This can be compared to  Jupiter which has an albedo of approximately 50% in the MOST bandpass [66]. If the composition of EGPs is similar to Jupiter, than the temperature of the planet is the dominent variable driving the atmospheric chemistry and expected albedo. This means that as exoplanets are discovered at different radii around stars of different luminosities we can expect the albedo to vary. The temperatures at which various albedo values are measured or constrained are a good tracer of chemical processes (such as the formation of molecules) in planetary atmospheres [79]. A current curiosity for a handful of irradiated extrasolar planets (for  example HD 209458b, HAT-P-1b [3] and Wasp-1b [75]) is that the radius of the planet sometimes appears bloated compared for its mass, age and temperature. The radius and mass give a mean density of 0.33 g cm−3 which is 27% of the Jovian value. Explanations include transportation of energy to deep layers of the planet or tuning physical parameters of models [74], other orbiting planets [5], or if the planet is in a Cassini State with a large obliquity [86]. 5  Chapter 1. Introduction Recent models of the interior structure of irradiated EGP that invoke enhanced atmospheric opacities and a range of planet core sizes suggest that there is no discrepancy between current models and observations [8]. The radius of an EGP depends on the incident stellar flux, atmospheric composition, presence of a core, its age and atmosphere circulation and possible external heat sources (such as tidal heating, which is described below). The atmospheres of Jupiter and Saturn have supersolar abundances of dominant elements such as carbon, nitrogen and sulfur [2]. The higher metallicity of planets may be caused by the collapse of primordial nebular gas into a gravitational well that was initiated by a dense core of refractory elements [36]. Thus it is not far-fetched to infer that the atmospheres of EGPs are rich in heavy elements. This in turn raises the atmospheric opacities which enables the planet to retain heat over longer time scales which in turn gives a larger radius. The radius of a gaseous object, such as a star or planet, is determined by the optical depth (τ ). The optical depth is a measure of transparency and is defined by the amount of radiation that scattered or absorbed along the line of sight. Typically the boundary condition adopted for an interior model does not correspond to the radius from observations as τ is wavelength dependent. At 0.6 µm a radius difference of ∼ 0.05 RJ  (RJ = radius of Jupiter) can be expected for HD 209458b [4]. When using 10× solar metallicity in model calculations and accounting for the variation of τ with wavelength radii consistent with observations of HD 209458 can be produced [8]. The caveat is that the metallicity enhancements are currently a free parameter of the models for which we currently do not have observations to serve as constraints. The relatively small distance between the planet and the star means that tidal interaction time scales will be shorter than or roughly equal to the expected ages of the star and planet. The time scale for synchronization of the planetary rotation period and orbital period is approximately 106 years and approximately 108 to 109 years for orbital circularization [48, 61]. The synchronization of the planetary rotation period and orbital period mean that the planet is tidally locked to the star, so the same side of the planet will always face towards the centre of mass, which is close to the 6  Chapter 1. Introduction centre of the star. If the planet is in a circular orbit then the tidal bulge and the face of the star will both point in the same direction. However, if the orbit is non-circular, then libration would be observed from the centre-ofmass. Libration is a very slow oscillation, real or apparent, of a satellite as viewed from the larger celestial body around which it revolves. The Moon is observed to show libration. If one were to observe the Moon over its 28 day period, the surface would appear to rotate back and forth. Approximately 59% of the lunar surface can be observed from the Earth as the Moon’s rotation leads or lags its orbital position. Libration is observed if the body is in an ellipical orbit. The rotation period of the body is constant, but the speed of the body along its orbit changes from perihelion (fastest) to aphelion (slowest). The tidal bulge will point towards the centre-of-mass (close to the more massive object) and the surface of the object will point towards the empty focus of the orbit. In Figure 1.2, the orbit of an object is deplicted. Two lines, normal to the surface, are drawn, with one pointing towards the centre of mass and the other towards the empty focus. As the object completes its orbit the angle between the two lines changes. This demonstrates how a tidal budge can oscillate across a planetary surface. The resulting frictional loss deposits thermal energy into the interior of the planet. Assuming energy disipation rates based on the Io-Jupiter interaction the eccentricity of the orbit of HD 209458b must be somewhere between 0.03 and 0.1, depending on whether the model assumes a planetary core [5]. For this mechanism to work, an additional orbiting body (such as another planet) is required to continuously force the eccentricity. Timing of the infrared detection of the secondary eclipse places its occurence to within 7 minutes (1 σ) of the expected midpoint from the transit. This suggests that the orbital eccentricity is less than 0.03, unless the viewing angle is parallel to the major axis of the orbit. However, dynamical simulations show that the eccentricities of two planets can oscillate with time scales ∼ 105 yr as  they gravitationally interact with each other [6], thus we may be observing the system during a minimum in inclination. In the case of HD 209458b, long term Doppler and photometric mea-  surements have ruled out, as an explanation, additional planets down to a 7  Chapter 1. Introduction  Figure 1.2: Libration is a very slow oscillation of a satellite as viewed from the larger celestial body around which it revolves. If the less massive body is in an elliptical orbit, then the tidal bulge will point towards the centre-ofmass (close to the more massive object) and the surface of the object will point towards the empty focus of the orbit. In this figure, the orbit of an object is depicted. Two lines, normal to the surface, are drawn, with one pointing towards the centre of mass and the other towards the empty focus. As the object completes its orbit the angle between the two lines changes. This demonstrates how a tidal budge can move across the surface an object. The resulting frictional loss deposits thermal energy into the interior of the planet.  8  Chapter 1. Introduction few Earth masses that have orbital periods ranging from 1 to 9 days [1, 55]. These studies timed the occurrence of planetary transits to search for gravitational perturbations of additional planets in the system. Figure 1.3 shows the results from a 3-body simulation for the HD 209458 system where an additional Earth mass planet has been placed in a circular 2 day orbit. The timing residuals are expressed as Observed minus Calculated (O-C) which is the difference in the occurrence of the transit event compared to the expected time of the transit event from the mean orbital period. Long term photometric observations (on year timescales) are essential to find perturbations from Earth sized masses. The expected transiting timing variations are on the order of a few seconds, but are periodic. With enough observed transits, such signals can be detected. If a planet is caught in a Cassini state, which is a resonance between spin precession and orbital precession the orbit of the planet can have a large obliquity. A planet’s spin axis and orbit will precess in the presence of a torque. The torque could be caused by additional planets or satellites. The outcome of tidal evolution is that the orbit normal and the spin axis will precess at the same rate about the same axis. Two stable configurations are that the orbit normal and spin axis are on the same side (State 1) or opposite sides (State 2) of the common axis [86]. In a Cassini State 2 configuration the pole of the planet rotation axis will be aligned in the orbital plane with the pole pointing in the same direction. From the planet perspective, at the north and south poles, the parent star rises and sets once per orbital period, and at mid-day passes directly overhead in the sky. As the same part of the planet is no longer always facing the star, tidal friction will quite strong and add an additional energy source [86]. Recently studies by Rabrycky et al. [24] suggest that the rate of dissipation is much less than hypothesized as large dissipation rates can destroy the Cassini equilibrium. Determining whether a planet is in a Cassini State may be possible only with infrared observations, which depend on how absorbed stellar radiation is redistributed throughout the planet’s atmosphere (day/night temperature gradient). Any atmospheric flow model of HD 209458b must include the effects of irradiation. Current attempts to model the temperature distribution 9  Chapter 1. Introduction  Figure 1.3: The expected O-C (Observed - Calculated) transit times for HD 209458b if an additional 1 Earth mass planet is placed in an orbit with a period of 2 days. The x-axis scale in the orbit number, which for HD 209458b corresponds to 3.5d per orbit. The amplitude of the variation is only a few seconds, but is periodic, hence long term measurements are required to detect such a signal.  10  Chapter 1. Introduction of the planet [15, 25, 45] assume a temperature distribution of T = Tef f +  ∆T cos α 2  (1.3)  where Tef f is the temperature given by Equation 1.1, α is the zenith angle of the star as observed from the planetary surface and ∆T is the maximum day/night temperature difference and is expected to be between 300 and 1000K [25]. The thermal forcing is then be given by FT = −  Tm − T τrad  (1.4)  where Tm is the model temperature at a specific model point on the surface and τrad is the radiative timescale. Thus, there are three free parameters in these simple models: the Bond albedo which determines the thermal input from the star; the day/night temperature contrast which is dependent on the interior structure of the planet (i.e. the temperature of the planet in the absence of stellar irradiation); and the radiative timescale which depends on the thermal characteristics of the atmospheric composition. The MOST (Microvariability and Oscillations of STars) satellite [53, 82] houses a 15-cm optical telescope feeding a CCD or Charged Coupled Device photometer through a single broadband filter (350 - 750 nm), which is capable of sampling target stars up to 10 times per minute. From the vantage point of its 820-km-high circular Sun-synchronous polar orbit with a period of 101.413 minutes, MOST can monitor stars nearly continuously for up to 8 weeks in a Continuous Viewing Zone (CVZ) with a declination range of +36◦ ≥ δ ≥ −18◦ . The satellite was designed to achieve photometric precision of a few parts per million (ppm ∼ µmag) at frequencies > 1 mHz.  The necessary photometric stability is achieved by projecting an extended image of the instrument pupil illuminated by the Primary Science Target via one of an array of Fabry microlenses above the MOST Science CCD (a  1K × 1K backside illuminated E2V 47-20 detector). For exposure times up to 1 minute long, and observing runs of at least 1 month, this Fabry Imaging mode can reach the desired precision of about 1 ppm at frequencies greater 11  Chapter 1. Introduction than 1 mHz for stars brighter than V ∼ 6.5.  Fainter stars can be monitored (simultaneously, or independently of the  Fabry Imaging mode) in an open area of the Science CCD not covered by the 6 × 6 Fabry microlens array and its chromium field stop mask. In the Direct Imaging field, defocused star images are projected, with a FullWidth Half-Maximum (FWHM) of about 2.5 pixels. (The focal plane scale of MOST is about 3 arcsec/pixel.) The photometric precision possible in the Direct Imaging Mode is not as good as for the Fabry photometry as Direct Imaging targets are fainter and more sensitive to CCD calibrations, but much better than expected because of high pointing precision. Combined with the unprecedented duty cycle of the MOST observations and the thermal/mechanical stability of the instrument the results have been impressive. The point-to-point precision of the photometry reported here is about 3 millimag for a 1.5s exposure during low stray light conditions, which is approximately the photon shot-noise limit. Sampling rates are as high as 10 exposures per minute and a duty cycle of 81% for the 2004 observations and 89% for the 2005 observations, where a duty cycle of 100% represents no loss of data acquisition. Photometry of this quality and coverage represents a unique opportunity to explore the HD 209458 transiting system. The star is not too bright for the MOST Direct Imaging mode (V = 7.65) and it is well placed in the MOST CVZ, observable for up to about a month and a half. MOST data have many applications to this system: (1) accurate transit timing to refine the near-zero orbital eccentricity and check for effects of orbital precession; (2) searching for transits of other smaller planets in the systems with different orbital periods, as predicted by some models to explain the dynamical stability of HD 209458b [37]; (3) revealing subtle intrinsic variability in the star HD 209458a and possible interactions with the close-in planet; and (4) detection of the eclipse of HD 209458b in optical light to directly measure the geometric albedo of the planet. We report here an attempt to detect the eclipse of HD 209458b in reflected light at optical wavelengths, with photometry from the MOST satellite. The reflected light signal from an exoplanet is sensitive to the com12  Chapter 1. Introduction position of its atmosphere, as well as the nature and filling factor of cloud particles suspended in that atmosphere [9, 29, 73]. The proportion of scattered to absorbed radiation is critical to the planet’s energy balance and hence an albedo measurement is key to understanding its atmosphere.  13  Chapter 2  MOST Direct Imaging Photometry MOST is a microsatellite (mass = 54 kg; peak solar power = 39 W) with limited onboard processing capability, memory, and downlink. Hence it is not possible to transfer the entire set of 1024 × 1024 pixels of the Science  CCD to Earth at a rapid sampling rate and with an ADC (analogue-to-  digital Conversion) of 14 bits (necessary to preserve variability information at the ppm level). Small segments of the CCD (”subrasters”) are stored, which contain key portions of the target field. This usually includes the Primary Science Target Fabry Image, 7 adjacent Sky Background Fabry Images, and subrasters for dark and bias readings. CCDs are usually constructed out of silicon3 that operate via the photoelectric effect. The fundamental element of a CCD is the metal insulator semiconductor (MIS) capacitor which starts with a neutral silicon crystal upon which a p-type epitaxial silicon is grown. This layer is usually borondoped to create positive carriers which dramatically increases the conductivity of the semiconductor. For a buried channel device an n-type silicon layer is formed on top where photoelectrons can collect. An insulator layer such as silicon-dioxide is then added with a conductive gate layer such as doped poly-silicon to form the surface4 . When a positive voltage is applied to the gate a potential well for electrons is created within the n-type silicon layer[38]. Constructing an array of MIS capacitors creates a CCD. Through manipulation of the voltage on the gates the collected charge can 3  Germanium is sometimes used as well For a surface channel device the n-type silicon layer is not present, causing charges to gather at the Si − SiO2 interface. 4  14  Chapter 2. MOST Direct Imaging Photometry be transferred down columns of the array towards a perpendicular array that transfers the charge to an onboard amplifier and then digitally encoded. The huge advantage of the CCD is its low signal detection capacity because of low read out noise, which means better photometry.  2.1  Data Format  The MOST satellite collects photometry in three ways: (1) Fabry Imaging, projecting an extended image of the telescope pupil illuminated by a bright target [53, 62]; (2) Guide Star photometry, based on onboard processing of faint stars used for telescope pointing [84]; and (3) Direct Imaging, where defocused star images (FWHM ∼ 2.5 pixels) are projected onto an open area of the Science CCD.  The MOST satellite can sample several nearby Science Targets in the Direct Imaging field (less than about 0.5◦ away from a Fabry target), by placing subrasters of typical dimensions 20 × 20 pixels over those stars.  These targets are automatically monitored with the same sampling as the Primary Fabry Imaging target. The Fabry Image illuminates about 1500 pixels, while each Direct Image illuminates a PSF out to a radius of several pixels. When simultaneously observing Fabry and Direct Imaging targets to avoid saturation, the Direct Imaging Targets must be at least 5.5 magnitudes fainter than the Fabry Target. For example, for MOST’s first Primary Science Target, Procyon, the V = 8 star HD 61199 was chosen for the Direct  Imaging field and was discovered to be a multiperiodic δ Scuti pulsator [53] with a peak amplitude of about 1 millimag. It is also possible to select a star as bright as V = 6.5 as the principal science target in the Direct Imaging field, without a brighter Fabry target. Then the exposure time and sampling rate can be optimized for the Direct Imaging target. This was the case for HD 209458 (V=7.7). When the binary data stream is transferred from the satellite it is converted to a FITS format image. The layout is shown in Figure 2.1. The locations of each subraster in the MOST Science CCD (in x-y pixel coordinates) are specified in the FITS file header. Typically, the FITS file contains 15  Pixels  Chapter 2. MOST Direct Imaging Photometry  Pixels  Figure 2.1: The FITS file layout is shown. There are 5 subrasters visible in this example, 3 containing stellar direct images, the Fabry projection and region shielded from light. The axis units are in pixels. The Fabry image has been binned 2x2.  16  Chapter 2. MOST Direct Imaging Photometry resolved subrasters for the Fabry image, 2-4 Direct images and 1 dark. The header also contains the pixel sums for various bias and dark regions, as well as satellite and instrument telemetry (e.g., spacecraft attitude control data; CCD focal plane temperatures) to allow additional photometric calibrations on the ground. The data format is unique to MOST, which necessitated the development of custom software to handle and process MOST data. Examples of MOST raw data and a document describing the FITS file and header format in detail are available in the MOST Public Data Archive at the MOST Mission website: www.astro.ubc.ca/MOST.  2.2  Photometric Reduction  In general, the approach to reducing MOST Direct Imaging photometry is similar to groundbased CCD photometry, applying traditional aperture and PSF (Point-Spread Function) approaches to the subrasters. However, there are several aspects and challenges specific to MOST data. For example, the MOST instrument has no on-board calibration lamp for correction of pixelto-pixel sensitivity variations (“flatfielding”). In its orbit, MOST passes through a region of enhanced cosmic ray flux known as the South Atlantic Anomaly (SAA). There are also phases of increased stray light from scattered Earthshine which repeat regularly during each satellite orbit (P = 101 min).  2.2.1  Dark Correction  To lower cost and increase reliability, the MOST instrument has no moving parts, so there is no mechanical shutter which can cut off light to the entire CCD for dark measurements; exposures are ended by rapid charge transfer into a frame buffer on the CCD. Dark measurements are obtained from portions of the CCD shielded from light by a chromium mask above the focal plane. One-dimensional dark current correction is done by using averages from these dark subrasters; the averages are computed on the satellite as the full raster information is not available on the ground (to meet downlink  17  Chapter 2. MOST Direct Imaging Photometry limitations while maximizing the amount of stellar data available). The values for each dark region are weighted in the average by the number of pixels in each subraster. There are usually 4 dark measurements available. If any of these individual readings deviates strongly (by more than 3σ) from the mean (likely due to a particularly energetic cosmic ray hit), then that value is discarded.  2.2.2  Flatfield Corrections  In data from the satellite commissioning and its early scientific operations, pointing errors of up to about 10 arcsec led to wander of the Direct Images of up to 3 pixels on the CCD. This made precision photometry vulnerable to uncalibrated sensitivity variations among adjacent pixels. The pointing performance of the satellite has been improved dramatically since its early operation, now consistently giving positional errors of about ±1 arcsec ∼  0.3 pixel rms (see §2.2.3).  Despite the lack of an on-board calibration lamp, it is possible to re-  cover some flatfielding information for each subraster by exploiting intervals of high stray light during certain orbital phases. Frames containing no detectable stars are chosen from the data set. Such frames can be obtained when the satellite is commanded to point to an “empty” field (with only stars much fainter than V = 13), or occasionally, when the satellite loses fine pointing and the stars wander outside their respective subrasters. During two weeks of observing, over which 50,000 - 100,000 individual exposures are typically obtained, about 2000 - 4000 flatfielding frames are usually available from when the satellite loses fine pointing. The stray light can produce a strong spatial gradient across the CCD, so a 2-dimensional polynomial is fitted to all the available flatfielding pixels and removed. For each subraster, the mean pixel value is measured and compared to each individual pixel value; a linear fit to the correlation is made. The slope of this relationship is the relative gain and the zero point is the dark current. The relative pixel gains are found to vary by less than 2% across any individual subraster, with the standard deviation for any  18  Chapter 2. MOST Direct Imaging Photometry  5 0 -5 -10 -15 -20 -25 -30 -35  ADU  164 162 160 158 156 154 Pixels 152 150 148 146 144 90  5 0 -5 -10 -15 -20 -25 -30 -35  92  94  96  98  108 110 104 106 100 102 Pixels  Figure 2.2: A map of the dark current correction for the 20x20 subraster containing HD 209458. Hot pixels show up as negative corrections. These corrections are applied before photometry is extracted for the star, minimizing pointing errors. individual pixel less than 0.5%. In Figure 2.2 the dark current correction is shown for the 20x20 subraster that was centred on HD 209458. These values represent the zero point from the linear fits. The maximum correction is ∼ 35 ADU which is quite small compared to the peak value of the stellar PSF (∼ 103 ADU).  2.2.3  Star Detection and Centroiding  We employ two methods for star identification. The first method involves deconvolving the individual subrasters with a model PSF. The model PSF is created by registering subrasters containing stars and mapping the pixel  19  Chapter 2. MOST Direct Imaging Photometry values onto an oversampled grid to create a model profile with twice the resolution of a real image. After deconvolution, the strongest source is chosen as the target star for photometry. The second method starts by prompting the user to identify by eye stars on each subraster via a graphical interface, which is particularly useful for fields containing several stars. Once the stars have been identified, centroids are computed by an intensityweighted average on a 5x5 grid around each object of interest. This process is iterated until the centroid converges. On the first frame, the centroids for all the stars are saved as a master grid and used to check the positioning for all subsequent frames. After the first frame the strongest source is always selected using the deconvolution technique and its position is calculated with intensity-weighted means. This is done for each subraster and the offsets are compared to the master grid of centroids. If the offset of a star varies by more than 2 pixels compared to the average offset for all stars, then that centroid is corrected by using the average offset from the remaining objects. This correction is particularly important when cosmic rays interact with the detector and can mimic a stellar PSF, leading to the occasional spurious star identification. The pointing performance of MOST has been dramatically improved since its original on-orbit commissioning, thanks to upgraded software and a better understanding of the mechanical performance of the reaction wheels in its attitude control system. In Figure 2.3, the distribution of pointing centroids for three fields observed from December 2003 to August 2004 are shown. The first is for the star HD 263551 in the field of ξ Geminorum (observed during 20 - 28 December 2003); the second, for HD 61199 in the field of the primary Target Procyon (8 January - 9 February 2004); and the third, for HD 209458 itself (14 - 30 August 2004). The standard deviation for each target in x and y pixel directions are (0.985,1.945), (0.947,1.837), and (0.309,0.403), respectively. The substantial improvement in tracking stability has resulted in improved photometry precision as flat fielding errors have become less relevant.  20  Chapter 2. MOST Direct Imaging Photometry  Figure 2.3: The pointing performance for three different targets. HD 263551 is shown in the top panel, HD 61199 in the middle panel and HD 209458 in the bottom panel. The top panel represents the typical point performance in early satellite operations. There is a substantial drift in the Y-direction. In the middle panel, the Y-axis drift has been largely eliminated. The bottom panel shows the current and much improved pointing performance.  21  Chapter 2. MOST Direct Imaging Photometry  2.2.4  Stray Light and Background Determination  In order to remove the strong background gradients associated with stray light, a 2-dimensional 2nd order polynomial is fitted to the subrasters based on the pixel co-ordinates on the CCD and subtracted. A sky radius is defined (=8 pixels for HD 209458), centred on selected stars. Only pixels outside this radius are included in the polynomial fit, to minimize influence from stellar sources. Once the gradient has been suppressed, the background level for each subraster is determined by rejecting pixels with levels more than 2.5 standard deviations from the median to eliminate cosmic ray hits. The rejection of pixels is iterated until the median converges, and the background is defined as the mean of the remaining pixel set. The median is chosen for the calculation of the standard deviation since the small subraster sizes limit the total number of pixels available for background determination, making the mean sensitive to errant values, mostly due to cosmic ray events.  2.2.5  PSF Fitting and Adding Up the Starlight  After determination of the star positions and the background, the PSF for each star is fitted by either a Moffat profile [56] or a Gaussian function. The Gaussian function is given by, (x − xcoo )2 (y − ycoo )2 − σx2 σy2 (x − xcoo )(y − ycoo ) +2σxy ], σx σy  G(x, y) = Ib +Ic Exp[−  (2.1)  where Ib is the sky intensity, Ic is the scaled intensity of the star, xcoo and ycoo are co-ordinate of the star and σx , σy and σxy describe the width and rotation of the PSF. The Moffat profile is given by M (x, y) = Ib +Ic /(1 +  (x − xcoo )2 (y − ycoo )2 + σx2 σy2  +(x − xcoo )(y − ycoo )σxy )β  (2.2)  22  Chapter 2. MOST Direct Imaging Photometry where β controls the sharpness of the profile. For β = 1 the profile is Lorentzian in shape and for larger values of β the profile becomes similar to a Gaussian profile. When fitting the Moffat profile β is held fixed at 3.5. In general it is found that the Moffat profile provides a better approximation of the PSF shape5 . The difference between a Gaussian and Moffat profile is that the wings of the latter fall off much more slowly accounting for the scattering profile at large off-axis distances, whether in the Earth’s atmosphere or in space, including telescope optics. The PSF is computed independently for each subraster using the Levenberg-Marquardt approach [59] to find the best fit parameters. This procedure requires the derivatives of Equations 2.1 and 2.2 which for the Gaussian fit are  dG = γIc dxcoo  dG = 1, dIb  (2.3)  dG = γ, dIc  (2.4)  2(x − xcoo ) 2σxy (y − ycoo ) − σx2 σx σy  2σxy (x − xcoo ) 2(y − ycoo ) dG = γIc − + dycoo σx σy σy2 dG = γIc dσx  ,  (2.5) ,  2(x − xcoo )2 2σxy (x − xcoo )(y − ycoo ) − σx3 σx2 σy  2σxy (x − xcoo )(y − ycoo ) 2(y − ycoo )2 dG = γIc − + dσy σx σy2 σy3 dG 2γIc (x − xcoo )(y − ycoo ) = , dσxy σx σy  (2.6) ,  (2.7) ,  (2.8) (2.9)  where γ = Exp −  (x − xcoo )2 2σxy (x − xcoo )(y − ycoo ) (y − ycoo )2 + . (2.10) − σx2 σx σy σy2  5 Even though the Moffat profile was designed to reproduce stellar images smeared by atmospheric seeing. We are essentially comparing a 15 cm space telescope with small pointing gitter to a large groundbased telescope at an average site.  23  Chapter 2. MOST Direct Imaging Photometry Likewise, for the Moffat function the necessary derivatives are, dM = 1, dIb  (2.11)  dM = ζ −β , dIc  (2.12)  2(x − xcoo ) dM = −ζ −1−β Ic β − − σxy (y − ycoo ) , dxcoo σx2  (2.13)  dM 2(y − ycoo ) = −ζ −1−β Ic β −σxy (x − xcoo ) − dycoo σy2  (2.14)  ,  dM 2ζ −1−β Ic β(x − xcoo )2 = , dσx σx3  (2.15)  2ζ −1−β Ic β(y − ycoo )2 dM = , dσy σy3  (2.16)  dM = −2ζ −1−β Ic β(x − xcoo )(y − ycoo ), dσxy  (2.17)  dM = −ζ −β Ic Log [ζ] , dβ  (2.18)  with, ζ =1+  (x − xcoo )2 (y − ycoo )2 + σxy (x − xcoo )(y − ycoo ) + . 2 σx σy2  (2.19)  The background level can be allowed to vary with the fit minimization procedure, but we find that better photometry is obtained by fixing the level as determined by the procedure outlined in §2.2.4. When fitting the PSF the shape parameters are sensitive to the FWHM of the PSF. So the fitting radius (for determining pixels included in the fit) is typically 1 to 2 times the FWHM (2-5 pixels). As the true background level is not reached until beyond a radius of 8 pixels, the non-linear fitting procedure cannot properly converge to a proper solution for the background level (Ib ). For data sets obtained early in the mission lifetime, when the tracking performance was not ideal, such as with ξ Gemini, multiple images of the 24  Chapter 2. MOST Direct Imaging Photometry same source can appear on the subraster. In these cases, once the initial PSF fit has been made, the fit is removed and the residual image is deconvolved using a predetermined instrumental PSF. The strongest peak is then identified and the original subraster data are fitted with both centroids. This process is iterated until the deconvolved image has no significant peaks. Once the stellar source has been modeled, the total flux is estimated by using aperture photometry for the centre of the model fit and using the model fit for the faint extended wings. For cases such as ξ Gemini, this step is important as the aperture accounts for the smearing effects of pointing jitter not included in the model and the model allows an estimation of stellar flux located outside the subraster. The FWHM of the PSF under good tracking is found to be about 2 pixels, but the stellar source can be easily traced out to a radius of 8 pixels, meaning for large pointing deviations a significant portion of flux can lie outside the subraster. With HD 209458 where the pointing accuracy is greatly improved, only the PSF model is necessary to determine the brightness of the source. The final magnitude is defined as mag = 25.0 − 2.5 ∗ log  Fp + Fa g Et  (2.20)  as the standard conversion between magnitude and flux, where Fp is the flux in ADU (Analogue-to-Digital Units) measured from the PSF fit, Fa is the flux residual inside a small aperture in ADU, Et is the exposure time in seconds and g is the gain in e− /ADU. The zero point of 25 has been arbitrarily chosen to give instrumental magnitudes that resemble the true magnitude.  2.2.6  Removal of Stray Light Effects  Once the instrumental photometry has been extracted, variable stray light effects must be removed. It was discovered that the background as determined in §2.2.4 needs to be scaled to properly remove the contribution from  stray light. The cause of this effect may because the 2D fits (see §2.2.4)  are not sufficient. With the small number of subraster pixels a higher order  25  Chapter 2. MOST Direct Imaging Photometry polynomial fit is not stable enough, hence the estimation of the background level is not optimum. The problem may also be related to the incident angle of photons reaching the detector. Star light will reach the detector face-on, whereas stray light enters the system from light-leaks in the satellite housing sweeping across the detector as the position of the stray light source, such as the Earth, moves relative to the satellite. The angle at which the light strikes the detector can be close to grazing. The projected area and thickness of the CCD will appear different to grazing photons and may play a role in understanding the correlation. Regardless, it can be corrected. The top panel of Figure 2.4 shows the data without any corrections (raw photometry). There are a large amount of data seen at brighter magnitudes. The inflections occur at the 101 minute period of the MOST satellite orbit. In Figure 2.5, the relationship between instrumental magnitude and the background level is shown. A fit is made between the stellar flux and the background level either with a polynomial or a cubic spline depending on the complexity of the relationship. If the relationship is modeled with a cubic spline, then the original data are binned before the fit, with 500 data points per bin and a minimum bin width of 10 ADU per pixel. The light curve for HD 209458 after the correction for the correlation is shown in the middle panel of Figure 2.4. In most cases the peak-to-peak amplitude of stray light variations is reduced by this approach to a few × 0.1 millimag.  It can be further reduced by techniques such as subtracting a running, averaged background phased to the orbital period [67]. The advantage of the simple approach taken here is that no prior knowledge of the orbital period is required and the amplitude of the stray light component is allowed to vary from orbit to orbit as caused by changes in the Earth’s albedo (such as clouds or snow cover).  2.2.7  Crosstalk Corrections  The effects of crosstalk noise on the photometry can be seen in the upper and middle panels of Figure 2.4 as small dips (∼ 2-4 mmag) in the light curve at times of 16, 18, 20 and 22.5 d. Crosstalk (or “video noise”) oc-  26  Chapter 2. MOST Direct Imaging Photometry  magnitude  A  Time (days)  magnitude  B  Time (days)  magnitude  C  Time (days)  Figure 2.4: Plots of the 2005 HD 209458 light curve after various stages of data reduction. All the light curves have been binned with 40 minute intervals for clarity. The top panel shows the raw photometry. The middle panel shows the photometry after corrections for the stray light correlations. The bottom panel shows the data after crosstalk corrections. The x-axis is marked in days and the y-axis is marked in magnitudes.  27  Chapter 2. MOST Direct Imaging Photometry  Figure 2.5: The relationship between the instrumental magnitude and the background level as measured on the CCD frame.  28  Chapter 2. MOST Direct Imaging Photometry curs when the CCD controller electronics interacts with another electrical component due to incomplete grounding [28]. In the MOST instrument, the Science and Startracker CCDs are located close beside each other in the camera focal plane, and are electrically isolated from one another. The aluminum structure of the spacecraft bus provides excellent grounding for the camera electronics, and crosstalk levels tested prior to launch were well below the original mission science requirements. However, the transit and eclipse analysis of HD 209458 were not part of the original plan for MOST and these results are more sensitive to the intermittent crosstalk. Crosstalk happens when both Science and Startracker CCDs read out at the same time. The duration of frame transfer and readout for the Science CCD is about 1 s, while the Startracker has a readout time of ∼0.1 s.  A noise band approximately 10 pixels wide appears on the Science CCD (corresponding to the ratio of readout times) which can overlap the subraster containing the target star. This noise source does not obey Poisson statistics. Without crosstalk, the noise for pixels used to estimate the sky background level in the subraster can be predicted from Poisson noise expected from the incident stray light plus read noise inherent to the detector electronics. This can be compared directly to the standard deviation of the pixel intensity values. Crosstalk introduces additional noise which changes the ratio of measured to predicted noise, which would be constant outside of crosstalk events. The correlation of background light and instrumental magnitude (see §2.2.6) above does introduce changes in the noise ratio, but the time  scale of this variation (the satellite orbital period of about 101.4 min) and the durations of crosstalk events (about 0.5 d) are so different that they are easily distinguished from each other. The positive signal added by crosstalk noise will traverse through the  downloaded Science CCD subraster as the times of the independent CCD readouts move out of sync. The background level will be overestimated, but the PSF (Point Spread Function) fit to the stellar image is largely unaffected. Hence, the stellar flux is underestimated and the width of the crosstalkaffected pixel region moving across the CCD gives the characteristic sawtooth shape of the crosstalk artifacts seen in the upper two panels of Figure 2.4. In 29  Chapter 2. MOST Direct Imaging Photometry  20  Sky Standard Deviation  15  10  5  0 0  5  10  15  20 Days  25  30  35  40  Figure 2.6: The standard deviation of the sky pixels is plotted as a function of time, the crosstalk events are easily identifiable as rapid increases that last for half a day. For example, the deflections at day 16, 18, 20 and 22.5 are crosstalk events. The general scatter is caused by the stray light correlation outlined in §2.2.6. Figure 2.6 the standard deviation of the sky pixels is plotted as a function of time, the crosstalk events are easily identifiable as rapid increases that last for half a day. The general scatter is caused by the stray light correlation outlined in §2.2.6. By correlating the background pixel scatter with the  stellar photometry, the background level is rectified and the correct stellar  flux value is recovered. The bottom of Figure 2.4 shows the 2005 photometry after this correction. In particular, notice the adjustment to the depth of the fifth transit during day 16, which coincided with a crosstalk event.  30  Chapter 2. MOST Direct Imaging Photometry  2.2.8  Image Stacking: Aperture Photometry and Partial Pixels  For the first half of the current mission lifetime, individual images where downloaded. Bandwidth limitations meant that only 1 image every 10 seconds could be transferred. A new observing mode was added where images can be stacked onboard before transmission, thus allowing the full interval for integration. Depending on the single exposure time, there can be a dramatic increase in the number of photons captured. The photometry for stacked images is the same as described above, except that measurements of the stellar signal are obtained within an single aperture (typically with a radius of 4 pixels) rather than through PSF fitting. This change is due to the onboard image stacking procedure. Small spacecraft pointing jitter of less than one pixel means the PSF shape of a stacked image is not well fitted by a Gaussian or Moffat profile. Comparisons of PSF and aperture photometry showed the latter technique to give superior results. When summing photometry with a circular aperture, the problem of partial pixels arises. This is when the aperture only partially covers a pixel. In the popular photometry computer code DAOPhot [77] an approximation is made. If the intensity of a pixel labeled i, j is Iij , and rij is the distance of that pixel from the centre of the star and R is the radius of the photometric aperture, then if rij < R - 0.5 then all of Iij is included in the photometry as the pixel is complete inside the aperture. If rij > R + 0.5 then Iij is excluded from the photometric sum as the pixel is complete outside the radius. If R−0.5 < rij < R +0.5 then we include the fraction [(R+0.5)−ri,j ] of Iij in the sum. From §2.2.6 it was shown how the intensity of the star is  correlated to background level. It is likely that correlation applies to each  individual pixel. As the star centroid moves around, the effective number of pixels included in the aperture will vary by about a full pixel as a circle is being approximated by an irregular polygon. This approximation can be seen in the final photometry. To reduce the problem of partial pixels an exact procedure is implemented to calculate the fraction of a pixel that is included in the intensity sum.  31  Chapter 2. MOST Direct Imaging Photometry First the intersection points of the aperture and pixel need to be found. If the centre of a pixel is defined by (rx, ry) and the centre of the photometry aperture is at xc, yc, then we check the four sides of the pixel for intersection with the edge of the aperture. Figure 2.7 shows an example where two intersection points are found. If less than two intersection points are found then the pixel is either completely inside or outside the aperture, which can be determined by the criteria from the DAOPhot algorithm. If more than two intersection points are found then the pixel is subdivided into 4 pixels and the search intersection search algorithm is repeated for the 4 new pixels. If two intersection points (c1 and c2 ) are found, then the area (A) of the pixel covered by the aperture is given by the integral of the arc bounded by c1 and c2 ,  c2  A= c1  ± R2 − (x − xc )2 + yc dx,  (2.21)  where the solution is, A=  ± 21  R2 − (x − xc )2 (x − xc ) + xyc √ 2 c2 R −(x−xc )2 (x−xc ) ± 12 R2 ArcTan −R2 +(x−xc )2 c1  (2.22)  where x is evaluated at c1 and c2 . We then remove the area from the bottom of the pixel to the centre of the integral (rx-0.5). In the common case that the intersection points lie on non-adjacent parts of the pixel, then the area will have an additional rectangular component added to the integral. The algorithm was tested by randomly selecting 105 positions for the aperture on a 100x100 pixel grid and summing up the pixel areas. In all cases, where the aperture did not extend beyond the grid, the returned area was as expected. Thus, our aperture photometry algorithm always has the same number of effective pixels eliminating problems with the stellar intensity correlations.  2.2.9  Photometric Errors  It is still unclear what causes the correlation of sky background to instrumental stellar magnitude readings in the data. To investigate this, we tested  32  Chapter 2. MOST Direct Imaging Photometry  Figure 2.7: Handling partial pixels with aperture photometry. A partial pixel, as shown, is the overlap of an aperture centred at (xc ,yc ) with a radius of R and a square pixel of area unity centred at (rx ,ry ). The intersection of points of the aperture and pixel are labeled c1 and c2.  33  Chapter 2. MOST Direct Imaging Photometry the validity of the predicted Poisson errors to the observed scatter, outside of times of crosstalk and times surrounding exoplanetary transits (about 5 hr every 3.5 d). We binned the data by sky background value, with an equal number of data points for each bin. We then calculated the expected scatter in each bin based on Poisson statistics and the measured standard deviation. Figure 2.8 shows the ratio of predicted to measured scatter as a function of the background level. At background levels of about 3000 ADU, the peak of the stellar PSF becomes becomes saturated, since the 14-bit ADC of the MOST CCD electronics has a saturation limit of 16 384 ADU. This causes the large discrepancy between theory and measurement seen at this threshold in the figure. At low background levels there is a linear increase up to about 1000 ADU, followed by a slower rising plateau until saturation. This change could be interpreted as a variation in CCD gain with signal. We have corrected the error estimates with a linear interpolation over the binned sky background values for each photometric measurement. This provides the correct weight for each measurement and the proper statistics for subsequent fitting to models of the light curve.  34  Chapter 2. MOST Direct Imaging Photometry  3.5  Error Correction (Emeas/Ecal)  3  2.5  2  1.5  1  0.5 500  1000  1500  2000  2500  3000  3500  4000  4500  Background Level (ADU/s)  Figure 2.8: Here we show the error corrections. The y-axis shows the ratio of the expected error versus the Poisson estimate of the photometric error. This ratio is used as the correction factor applied to original estimate of the photometric error. The x-axis show the background level. The sharp increase after 3000 ADUs is non-linearity of the detector.  35  Chapter 3  MOST Direct Imaging in Action As a proof of concept I present light curves and results for observations of various stars that have resulted in publications or upcoming publications. All the light curves have been reduced using the methods described in §2.2.  Observations of BD +18 4914 where obtained simultaneously with HD  209458. These observations where processed using the same techniques. The discovery of the p-modes and g-modes in this star make it a member of a small population of asteroseimologically interesting stars. The quality of the observations for a star that is approximately 3 magnitude fainter than HD 209458 demonstrate the robustness of the photometric reductions. The lack of amplitude changes in the detected pulsation frequencies also demonstrate the photometric stability of the MOST satellite on year timescales. These results have been published in Rowe et al. [64]. During observations of the primary science target ι Leo the only MOST observations of a RR Lyrae type variable where obtained of the double mode pulsator AQ Leo which happened to be within the field of view. It exhibits large amplitude variations (∼ 0.8 mag) typical of this class of variable. The analysis of this light curve shows the weakness of the data reduction scheme presented in §2. In particular, the decorrelation of the background level  with the instrumental magnitude can produce an artifact signal sidelobe mirroring the MOST orbital frequency, but at an amplitude reduced by a factor of 100. For this reason, the decorrelation fit for HD 209458 was performed with data excluding occurrences of the transit and eclipse. These results have been published in Gruberbauer et al. [30].  36  Chapter 3. MOST Direct Imaging in Action The discovery of a EGP in a 2.2 day orbit presented an exciting opportunity for MOST observations. The close orbit and larger planet-to-star radius ratio meant that the eclipse would be potentially easier to detect. However, the stellar component is a chromospherically active K star that shows evidence for star starspots. The large variability of the star overwhelms the albedo signal and demonstrates the venerability of the albedo measurement to stellar activity.  3.1  BD +18 4914  γ Doradus stars pulsate with typical periods of about 0.8 days [40], consistent with high-overtone nonradial g-modes. They represent one of the newest classes of pulsating variable stars, and about half of the currently known γ Doradus stars lie within the δ Scuti instability strip [31]. The δ Scuti variables exhibit p-modes of low radial order, seen only in low degree photometrically, with typical periods of a few hours. It has been shown that the pulsation characteristics of the two classes can be clearly separated by their values of the pulsation constant Q [33]. The overlap in physical properties of γ Doradus and δ Scuti stars suggested the possibility that hybrid pulsators may exist. The astroseismic implications are exciting since the g-modes would probe the deep interior of the star and the p-modes, its envelope. This has lead to photometric monitoring of γ Doradus and δ Scuti stars to search for hybrid behaviour. The first such hybrid to be discovered was in the binary system HD 209295 [32], from careful monitoring of 26 γ Doradus variables [33] but the γ Doradus pulsations in the primary component are likely caused by tidal interactions with the secondary. The first convincing case of a single hybrid star, the Am star HD 8801, was discovered from monitoring 39 stars from a volumelimited sample of 114 γ Doradus candidates [35]. HD 8801 shows frequencies clustered around 3, 8 and 20 cycles/day (c/d). There were only two frequencies in the γ Doradus range and 4 frequencies classified as δ Scuti in nature. The low number of frequencies makes this star a challenging subject for asteroseismic modeling, but just the existence of a hybrid single star, and the 37  Chapter 3. MOST Direct Imaging in Action chemical peculiarity of HD 8801, point to new and interesting astrophysics.  3.1.1  Photometry  Photometry of the star BD+18 4914 (α = 22h 02m 38s , δ = +18o 54′ 03′′ [J2000], V=10.6, B=11.1)6 by the MOST satellite is shown in Figure 3.1 in which we detect frequencies consistent with hybrid pulsations. The observations of BD+18 4914 were carried out during 14-30 August 2004 and 1 Aug - 15 Sept 2005, for a total of 57 days, during Direct Imaging photometry of the transiting exoplanet system HD 209458. Panel D of Figure 3.1 shows the 2005 observations of BD+18 4914 using 40-min bins to clearly demonstrate the low-frequency pulsations with periods around 1 day.  3.1.2  Frequency Analysis  A preliminary analysis of the 2004 observations of BD+18 4914 showed the dual nature of its pulsations [65]. Here we describe a more methodological approach to the frequency analysis. We begin by computing the discrete Fourier transform (DFT) of the 2005 time series. The amplitude spectrum is shown in panel A of Figure 3.1. The data are then fitted using an equation of the form Aj cos(2πfj t + φj ),  mag = A0 +  (3.1)  j=1,n  Where A0 is a linear offset and fj , Aj and φj are the frequency, amplitude and phase for each successive peak found in the amplitude spectrum using the Leveberg-Marquardt approach [59]. After each fit, the DFT of the residuals is recalculated and the next largest amplitude is chosen, but only if the Signal-to-Noise (S/N) is greater than 3.6. The S/N is defined as the amplitude of the peak in the spectrum divided by the mean of nearby frequencies. We use a window about 3 c/d wide in frequency space, centred on the highest peak, to calculate the mean which we use as an estimate of 6  This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.  38  Chapter 3. MOST Direct Imaging in Action  Frequency (mHz)  Amplitude (mmag)  A  Frequency (c/d) Frequency (mHz)  Amplitude (mmag)  B  Frequency (c/d) Frequency (mHz)  Amplitude (mmag)  C  Frequency (c/d)  mmag  D  Time (HJD−2453586.2)  Figure 3.1: Panel A shows the amplitude spectrum of the 2005 data set. Panel B shows the corresponding window function for the highest peak in Panel A. Panel C shows the amplitude spectrum after prewhitening of the 8 strongest frequencies. The detection limit of 3.6 times the noise is also shown. (The MOST orbital frequency is marked as a dashed vertical line.) Panel D presents the photometric data in 40-min bins to highlight the lowfrequency (2 c/d) oscillations.  39  Chapter 3. MOST Direct Imaging in Action the local noise floor. We detect 16 frequencies in the 2005 photometry with a S/N greater than 3.6. If the same procedure is repeated for the 2004 photometry, then the relatively short duration (14 days) of the data set causes degeneracies in the nonlinear solution because of poor frequency resolution. Specifically, the solutions for frequencies j = {2, 3} in Table 3.1.2 converge to identical values  with phases offset by π radians. The amplitudes in turn grow unreasonably large. This is due to the degeneracy of the solution at these low frequencies in the shorter time series. The 2005 data set has a higher resolution due to its longer duration. To avoid the degeneracy problem, the frequency solution from the 2005 dataset is used as the initial solution for the non-linear routine applied to the 2004 data, with the frequencies held as fixed parameters to derive the amplitudes to be determined. If one examines the DFT of the residuals from the best fit there are no significant peaks remaining, thus the 2005 frequencies solution is valid for the 2004 data set. The best fit parameters are presented in column 5 of Table 3.1.2. This does not imply that the frequencies are constant from 2004 to 2005, only that the 2004 data set is too short to give meaningful results for a change in frequency in the low frequency regime. To estimate the errors in our fitted parameters, we perform a bootstrap analysis. This method involves redetermining the fitted parameters with randomly generated data sets. The new datasets are created by randomly selecting data from the original time series with replacement. In other words, any individual data point can be chosen more than once but the total number of selected points is always the same as the original data set. The bootstrap method is effective since it preserves the same noise profile in each random set as exists in the original data and given enough iterations will produce error distributions for each fitted variable. We generated 22083 and 18595 bootstrap iterations for the 2004 and 2005 data sets, respectively. The 1-σ error distributions using a Gaussian model are given in Table 3.1.2 for both data sets.  40  Chapter 3. MOST Direct Imaging in Action  j  fj (c/d) σf j  Aj (mmag) σA j  φj (rad) σ φj  A2004 (mmag) j σA2004  S/N  Q (days)  6  0.9496 0.0004 1.1586 0.0004 1.6150 0.0002 1.6924 0.0002 1.7829 0.0002 2.9286 0.0007 7.2530 0.0012 7.4354 0.0015 8.9122 0.0010 9.0348 0.0003 9.3847 0.0010 9.7156 0.0005 10.0043 0.0013 14.6977 0.0010 14.8967 0.0010 15.4106 0.0018  1.97 0.06 2.07 0.06 4.74 0.06 4.60 0.06 5.25 0.06 1.22 0.06 0.65 0.06 0.53 0.06 0.84 0.06 3.09 0.06 0.84 0.06 1.56 0.06 0.60 0.06 0.91 0.06 0.87 0.06 0.46 0.06  6.08 0.07 3.60 0.07 0.55 0.03 2.05 0.03 5.99 0.03 3.08 0.12 0.22 0.21 0.03 0.27 4.66 0.17 0.43 0.05 0.90 0.17 3.95 0.09 3.09 0.23 3.07 0.16 3.09 0.16 3.76 0.31  1.65 0.10 1.75 0.10 4.57 0.10 4.48 0.10 4.98 0.10 0.87 0.10 1.04 0.10 0.42 0.10 0.94 0.10 3.08 0.10 1.10 0.10 1.85 0.10 0.50 0.10 0.64 0.10 0.85 0.10 0.24 0.09  10.0  0.34  9.9  0.28  19.0  0.20  20.0  0.19  19.4  0.18  7.0  0.11  3.9  0.04  3.8  0.04  4.9  0.04  16.3  0.04  5.1  0.03  9.4  0.03  3.9  0.03  7.0  0.02  6.9  0.02  3.8  0.02  5 2 3 1 8 14 15 11 4 12 7 13 10 9 16  j  Table 3.1: Observed frequencies and parameters for BD+18 4914. The epoch is HJD=2453586.20349121.  41  Chapter 3. MOST Direct Imaging in Action  3.1.3  A hybrid pulsator  The frequencies found in BD+18 4914 cluster in the two ranges typical of γ Dor and δ Sct oscillation modes, making this a clear candidate for a hybrid pulsator. We can quantify this assessment using the criterion established from observations of δ Scuti and γ Doradus populations that the pulsation constant Q distinguishes the g− and p− modes in this type of star [33]. It was shown that although the pulsation periods of δ Scuti and γ Doradus overlap there is a clear separation when Q is considered. To calculate Q, we require basic properties of the star: log g, Mbol and Tef f . We obtained a 10˚ A/mm spectrum with the 1.8m Plasket telescope at the Dominion Astrophysical Observatory7 covering a range of 6473-6716˚ A. Our initial analysis gives values of Tef f = 7250 K, log g = 3.7 cgs and Mbol = 2.5. We compute values of Q for each frequency, and they are presented in Table 1. From analogy with the Am star hybrid pulsator HD 8801 [35], we assume that all frequencies less than 3.0 c/d are of γ Doradus type and the frequencies higher than 6 c/d are of δ Scuti type. Although we do not have enough information (multibandpass photometry or spectral line profile variation data) to make pulsation mode identifications, we can apply the Frequency Ratio Method (FRM) [57] to the 6 lowest frequencies. This assumes that the observed γ Dor pulsations can be described by the asymptotic approximation under the assumption of adiabaticity and spherical symmetry [80]. If the modes all share the same degree ℓ, then the ratio of the frequencies can be approximated by σα1 n2 + 1/2 , ≈ σα2 n1 + 1/2  (3.2)  Under these assumptions, we have searched for 6 overtone n values which satisfy Equation 3.2, taking an error of ±1.3 × 10−2 for calculation of the  sets of possible overtones [78]. Restricting our search to overtones up to and including n = 60, we find only one viable solution: n = {12, 20, 21, 22, 31, 7 Based in part on observations obtained at the Dominion Astrophysical Observatory, Herzberg Institute of Astrophysics, National Research Council of Canada  42  Chapter 3. MOST Direct Imaging in Action and 38} for the frequencies labeled j = {8, 1, 3, 2, 5, and 6} in Table 3.1.2. If we search up to n = 100, then the density of natural number ratios compared  to our error bounds allows for 78 more solutions. Regardless, other mode identification methods need to be applied to restrict the possible values of degree ℓ. Observations of other γ Doradus pulsators have shown that the amplitudes can be variable, such as seems to be the case with 9 Aurigae [41]. Using our results for the best-fit parameters of the 2004 and 2005 photometry, we can examine the possibility of amplitude changes over a 1-year interval. In Figure 3.2 we plot the measured amplitudes from 2004 versus those from 2005 (see Table 3.1.2) with 1σ error bars. No significant amplitude changes have occurred. We define detection of an amplitude change as ∆Aj = σ  Aj − A2004 j  2 + σ2 σA j A2004  (3.3)  j  where the definitions are same as presented in Table 3.1.2. The distribution of ∆Aj /σ appears to be non-Gaussian. We can test this with a student T-test and an F-test. To do so, we generated 10000 sets of 16 random Gaussian deviates and calculated the student T-test probability and F-test probability for the distributions to have similar means to our ∆Aj /σ sample. Our adapted criteria (for the samples to differ) is a probability less than 0.0026 (3 σ). For the T-test, none of the 10000 sets were rejected; thus the means are statistically similar. For the F-test, 26.7% of cases were rejected (i.e., only 27.6% produced a probability less than 0.0026). At our chosen 3σ threshold, our sample has a Gaussian distribution thus an amplitude change is not statistically significant. If we apply the tests to our 2004 and 2005 amplitude distributions and ask if the two distributions differ, then the student T-test gives a probability of 90.9% and the F-test, a probability of 88.3% that the two have similar means and variances, respectively.  43  Chapter 3. MOST Direct Imaging in Action  Figure 3.2: The measured amplitudes for the 2004 and 2005 observation campaigns plotted against each other, with 1σ error bars.  44  Chapter 3. MOST Direct Imaging in Action  3.1.4  Summary  We have presented 14 and 43 days of nearly continuous photometry obtained by the MOST satellite. From this photometry, we detect 16 frequencies whose amplitudes have S/N > 3.6, clustered in two ranges consistent with γ Doradus-type and δ Scuti-type pulsations. With 6 frequencies in the γ Doradus range, application of the FRM method assuming a common degree ℓ yields a unique set of radial orders of the pulsations. Comparison of the 2004 and 2005 data sets show no statistically significant changes in the pulsation amplitudes. However, this star is scheduled to be observed for a third time by MOST in the fall of 2006 to gain insight into the stability of the observed frequencies and remove any degeneracies from our fits. Groundbased spectroscopy and multicolour photometry will be necessary to obtain independent mode identifications to confirm whether the FRM assumptions we have made are valid and to take advantage of the potential of BD+18 4914 for asteroseismology.  3.2  AQ Leo  RR Lyrae variables are horizontal branch (HB) stars found within the instability strip of the Hertzsprung-Russell diagram. With masses of ∼ 0.5  M⊙ RR Lyrae variables are found to pulsate in the first overtone (RRc) or fundamental radial modes (RRab). There are also RRd stars that pulsate in a combination of first overtone and fundamental radial modes. The photometric amplitudes can be as large as full magnitude. AQ Leo (V ∼ 12.6)  is a RRd type variable that happened to fall in the field around the MOST space mission Primary Science Target, ι Leo. This alignment presented an  opportunity to obtain the first new light curve of this star in three decades, with very complete time coverage and unequalled photometric precision. AQ Leo was observed for 34 days during 2005 from February 14 to March 22 with exposure times of 25s obtained every 30s. The MOST Direct Imaging photometry reduction pipeline was used. To minimise the effects of pointing errors on the photometry, dark and flatfield corrections were performed by  45  Chapter 3. MOST Direct Imaging in Action  -7.4  mag  -7.2 -7.0 -6.8 -6.6 1875  1880  1885  1890 JD-JD2000  1895  1900  1905  -7.2  mag  -7.0 -6.8 -6.6 -6.4 1886  1887  1888  1889  JD-JD2000  Figure 3.3: The light curve of AQ Leo. The top panel shows the entire 34 days of photometry. The bottom panel shows a small 3.5d expanded view of the light curve. monitoring individual pixel responses during test exposures on fields empty of stars brighter than the background. The correlation in the raw photometry between the instrumental magnitude light curve and the estimated sky background was removed as described in §2. The reduced lightcurve is  presented in Figure 3.3. The top panel shows the entire lightcurve. The large amplitude (∼ 0.8 mag) is apparent. The large amplitude of variations presented new challenges for the photometric reductions.  46  Chapter 3. MOST Direct Imaging in Action  3.2.1  Nonlinear least-squares fitting and bootstrapping  The nonlinear least-squares approach we used for determining the sinusoid parameters from the AQ Leo time series is the same as presented for the star BD+18 4914 as presented in §3.1.2. Discrete Fourier Transforms (DFTs) are calculated and an updated fit is subtracted from the data successively, until  there is no meaningful change in the fit residuals. The bootstrap method was used to assess the uncertainties in these fitted parameters as described in §4.2.1.  3.2.2  The largest intrinsic variations observed by MOST an unexpected challenge  MOST was designed to detect and characterise very low-amplitude stellar oscillations, down to a few µmag. AQ Leo is an exception to the MOST target list, varying over a range of a few × 0.1 mag. In this case the stel-  lar variability is large enough that this introduces a modulation because the relative contribution of the stray light variation changes with the mean brightness of the star. By fitting sky values versus instrumental magnitude, the effects of straylight can be largely removed, but the intrinsic stellar variation serves to modulate the relative contribution of the stray light to the light curve. This introduces a small artifact signal sidelobe mirroring the MOST orbital frequency but reduced in amplitude compared to the intrinsic stellar signal by a factor of about 100. The MOST team is now alert to this effect for any possible future observations of large-amplitude variable stars.  3.2.3  Intrinsic frequencies and new modes in AQ Leo  From the time series 42 frequencies where detected. The two dominant frequencies are the first radial overtone f1 and the fundamental radial mode f0 . Of the remaining frequencies, 32 are linear combinations of these two up to the 6th order. The 42 frequencies include 11 which are not linear combinations of f0 and f1 . These include 1.96 d−1 , and its first harmonic at 3.92 d−1 (and which has a higher amplitude), and their linear combinations with f0 and f1 , we hereafter refer to them as fi and fii . 47  Chapter 3. MOST Direct Imaging in Action The MOST observations of AQ Leo are the first in which additional frequencies besides the fundamental radial mode, the first overtone radial mode and their linear combinations are detected. The two additional frequencies are fi ≃ 1.96d−1 and fii ≃ 3.92d−1 . The amplitude ratio of fi to fii is 0.64  ± 0.083. There is increasing evidence that some RR Lyrae stars pulsate ”beyond radial modes” [17]. Our observations represent the first detection of additional modes in an RRd star.  3.3  HD 189733  When a planet transits its host star one can immediately infer the ratio of the radii of the two bodies. With estimates of the stellar radius, the distance of the planet from the star and the mass of the planet, sufficient boundary conditions are set to construct preliminary models for a giant planet, assuming its interior structure is similar to the gas giants in our own Solar System. The observation of EGPs around bright Sun-like stars has made possible the first direct tests of models of the physical structures and chemical processes in giant planets. When a planet transits a chromospherically active star such as HD 189733 [88] with photospheric spots, the optical light curve of the system will show periodic variations as the spots are carried across the stellar disk by rotation. This direct measurement of the rotation period of the star, combined with accurate measurements of vsinI via the Rossiter-McLaughlin Effect and an estimate of the stellar radius from other data, provides a value of the inclination I of the stellar rotation axis with respect to the line of sight. By combining this determination of (or constraint on) I with the amplitude of the Rossiter-McLaughlin Effect, it is possible to infer the three-dimensional alignment of the stellar rotation axis. The transit observations offer additional opportunities to learn about the properties of the host star. If the planet occults a star spot during transit, the light curve represents a one-dimensional profile of the spot’s flux contrast against the rest of the photosphere. The depth of the transit relative to others measures the temperature difference in and out of spot 48  Chapter 3. MOST Direct Imaging in Action transit. Timing the ingress and egress of subsequent transits across the same spot monitors the evolution of the spot over several stellar rotations. Photometry with the MOST satellite offer a unique vantage point for monitoring active stars with planets that lie within the Continuous-ViewingZone for the satellite allowing consequent transits of the planet to be recorded and no ambiguity about maximum and minimum in the spot modulation light curve for an estimation of the rotation period of the host star.  3.3.1  MOST photometry of HD 189733  HD 189733 was observed by MOST for 21 days from 31 July − 21 August  2006. Individual exposures lasted 1.5 s, with 14 consecutive images stacked  on board the satellite to produce one co-added exposure sampled every 21 seconds. The first 14 days of observation have a duty cycle of 94% and the final 7 days have a reduced duty cycle of 46%. The last week of observations was shared with another MOST Primary Science Target field. When the intrinsic variability of the star was recognised in the early photometry, the originally planned 14-day run was extended by a week, by switching with another target during every MOST satellite orbit (P = 101.4 min). This meant that these observations of HD 189733 could only be carried out during phases of the highest scattered Earthshine, resulting in increased scatter in the light curve which can be treated in the reduction. The photometry is similar to the Direct Imaging mode described in §2,  except that measurements of the stellar signal were obtained within an aperture of radius 4 pixels rather than through PSF (Point-Spread-Function) fitting. This change is due to the onboard image stacking procedure, since small spacecraft pointing jitter of less than one pixel means the PSF shape of a stacked image is not well fitted by a Gaussian or Moffat profile. Comparisons of PSF and aperture photometry showed the latter technique to give superior results. The photometry also shows a correlation between the centroid position of the PSF and instrumental magnitude. The positional correlation in the photometry was removed by computing the average magnitude at a scale of 0.25 pixel. This sub-pixel map was then used to  49  Chapter 3. MOST Direct Imaging in Action interpolate the position of the PSF and calculate the corrected instrumental magnitude. The data reduction techniques are the same as described in §2, except  that bias/dark current measurements are not available from stacked images.  However, from extensive previous single-image photometry with MOST, we know that the bias level is approximately 500 ADU per exposure. We use this value in our reduction of the HD 189733 data. The only consequence of this bias correction estimate is that a small systematic error is introduced into the estimated photometric errors due to an offset in the background (sky) level from the true value. To estimate the true photometric errors, we binned the data into 25 groups based on the background level. We excluded data points that included a transit and smoothed the light curve with 0.5-d bins to remove the 3% intrinsic stellar variations caused by spot modulation. The standard deviation of each bin was then compared to the value predicted from our estimated photometric errors. The ratio of these two values gives a correction factor to our photometric uncertainties as a function of sky background. By interpolating over these values, the photometric uncertainties were corrected to better represent the true scatter in the data. The reduced light curve is presented in Figure 3.4. The transits appear as sharp drops in the magnitude of the star every 2.2d. The long term modulation (timescale  10d) is interpreted as spots rotating across the  visible disk of the star.  3.3.2  System Parameters  Modeling the transit requires removal of the star spot influence of the light curve. The spot modulation was filtered out by filtering all significant frequencies not related to the planetary orbit or its harmonics. The DFT of a lightcurve with a periodic transit will show the primary period of the transit and its harmonics as the shape of the transit is non-sinusodal. The filtered light curve is presented in Figure 3.5. We modeled the transit of the lightcurve using Equation 4.4. Limb Dark-  50  Chapter 3. MOST Direct Imaging in Action  -0.04  magnitude  -0.02  0  0.02  0.04  2405  2410  2415 time (HJD-2451545):  2420  2425  Figure 3.4: The 21 day HD 189733 light curve. The transit is seen at sharp drops in the magnitude of the star every 2.2d (the orbital period of the star). The long term modulation is interpreted as starspots. At the star rotated the star spots move around the visible disk of the star.  51  Chapter 3. MOST Direct Imaging in Action  Figure 3.5: The phased light curve for the filtered dataset. The top panel shows all data using 40 minute bins, the line shows the best fit model. The bottom panel shows the data heavily binned. 52  Chapter 3. MOST Direct Imaging in Action  M∗ MP R∗ RP P i AG φ  Prior 0.82 ± 0.03 1.15 ± 0.04 0.758 ± 0.016 1.154 ± 0.032 2.218573 ± 2 × 10−5 85.79 ± 0.25  Bestfit 0.80 ± 0.02 1.15 ± 0.02 0.75 ± 0.01 1.19 ± 0.02 2.21859 ± 2 × 10−5 85.65 ± 0.1 0.34 ± 0.1 −1.5520 ± 2 × 10−4  Units M⊙ Mj R⊙ Rj days deg rad  Table 3.2: Fitting Parameters for HD 189733 Light curve.  ening parameters for a non-linear model [16] were found by using a Kurucz model representative of HD 189733 with and effective temperature of 5000 K, log g = 4.5 cgs and solar abundances combined with the bandpass for the MOST optical system. Errors for our fitted parameters are estimated using the bootstrap technique described in §4.2.1. Our best fit parameters and uncertainties are presented in Table 3.3.2. The correlations and constraints on various parameters is also shown in Figure 3.6  3.3.3  Discussion  The radius for active stars can be 10-20% larger than expected. This is seen from observations of binary stars (such as V1061 Cygni [81]) where spot activity and magnetic fields are detected. Our prior for the radius of the star listed in Table 3.3.2 is based on a stellar model that is tuned to the physical parameters of the Sun. This includes the efficiency of convection based on the mixing length theory. When a strong magnetic field is present, the ability of the convection to transport material (and energy) is dampened, which in turn affects the radius of the star. If the radius of the star is systematically underestimated, then the planetary radius is as well. The middle panel of Figure 3.6 shows the strong correlation in determining the radius of the planet and star. A 20% increase in the radius of the planet would make it comparible in density to HD 209458b and making it an additional example of another planet with an inflated radius. 53  Chapter 3. MOST Direct Imaging in Action  Figure 3.6: Bootstrap results for HD 189733 parameter fits to MOST photometry. Contours outline the 95% confidence region. Top: Planet radius vs. geometric albedo through the MOST passband. Middle: Constraints on the radii of the star and planet. Bottom: Constraints on the masses of the star and planet.  54  Chapter 3. MOST Direct Imaging in Action If we clean the lightcurve of the transit signal using the DFT method above, we can fold the spot modulated light curve to estimate the period of rotation for the star. Using a Phase-Dispersion-Minimization (PDM) algorithm we obtain a rotation period of ∼ 11.8d. This is shorter than  the period from groundbased observation which estimate a period of 13.2d.  Spectroscopic measurements provide vsinI=2.97±0.22 km/s [87], where I is the inclination angle of the stellar rotation angle. Using the stellar radius from Table 3.3.2 we can estimate the inclination angle the star. Figure 3.7 shows the 1, 2, 3 σ confidence regions for the period and inclination based on the radial velocity (vsinI) and the stellar radius. The two vertical lines indicate a period of 11.8 and 13.2d. If the period of rotation is 11.8 days then the inclination angle of the star is greater that ∼ 55◦ at the 1 σ limit (where an angle 0◦ means we are observing the star pole on). The longer  rotation period points to an inclination angle greater than 65◦ . In addition the arrow show the shift in rotation period if the radius is increased by 10%. The top panel of Figure 3.6 indicates the presence of large albedo signature at high significance. The bottom panel of Figure 3.5 shows phase light curve heavily binned which shows a systematic variation present. At first glance looks like phase variations of a planet. Infrared detection of the planetary eclipse shows that the eclipse should occur at phase of 0.25 in Figure 3.6. Thus we should see a drop in the magnitude of the star equal to the amplitude of the phase variations (or even larger if the atmospheric material produces strong backscatter). No such variation is detected. The likely explanation is that the stellar activity cannot be fully removed. Using a DFT to remove significant periodicies in the data, requires that the stellar activity is periodic. Large spots produce this behaviour from stellar rotation, but flare activity and short lived small spots will not produce this kind of behaviour and will remain in our filtered light curve. Thus active stars seriously inhibits our ability to measure reflected light from the planet. With continued observations of HD 189733, one can hope that the stellar activity phased to the orbital period of the planet will average out leaving behind a signature of a reflected light signature, an essential parameter for modeling the atmosphere of EGPs. 55  Chapter 3. MOST Direct Imaging in Action  Figure 3.7: Constraints on the inclination angle as a function of the rotation period for star HD 189733. The shaded regions give the 1,2 and 3 σ confidence intervals from constraints on the stellar radius and vsini. The vertical lines mark rotation periods of 11.8 and 13.2d and the arrow shows the shift in Period if the radius is increased by 10%.  56  Chapter 4  Observations Photometry of HD 209458 was obtained in two runs, during 14 - 30 August 2004 and 1 August - 15 September 2005, for a total of 58 days of highcadence photometry. The data were collected in MOST’s Direct Imaging mode (see §2), where a defocused image of the star is recorded in a subraster of the Science CCD, with 1.5 s exposures sampled every 10 s. Two  fainter stars were simultaneously observed, HD 209346 (A2, V = 8.33) and BD +18 4914 (A0, V = 10.6) in the same way. The latter was found to be a hybrid (δ Scuti + γ Dor) pulsator (see §3.1). We rejected exposures with high cosmic ray fluxes which occur when MOST passes through the South Atlantic Anomaly (SAA), as well as data with background illumination values greater than 3000 ADU due to scattered Earthshine modulated with the satellite orbital period (see §2.2.9). For the 2004 dataset there were data losses in the first third of the run due to crashes of the satellite’s  control system (ACS) when subtle bugs in new software (which had operated smoothly during the previous month of observations on another target) manifested themselves. Once the problem was traced, the previous version of the software was uploaded to the satellite and observations continued with only one brief interruption due to a cosmic-ray-induced crash. This resulted in data sets with 106,752 and 334,245 points and duty cycles of 81% and 89% respectively for the 2004 and 2005 runs. For a transiting system the simplest method for determining the geometric albedo is to compare the brightness of the planetary system before and after eclipse, where an eclipse is defined as the planet passing behind the star from our perspective. Another method is to use the entire light curve to follow the phase changes of the planet. Figure 4.1 shows the expected light curve for a transiting extrasolar planet. The transit is shown at point 57  Chapter 4. Observations E and has a depth of 2% for a planet with a radius equal to Jupiter. This is approximately what is observed for HD 209458. The eclipse is shown at point A. The sinusodal shape seen between transit and eclipse is due to the changing phase of the planet as it moves along its orbit from points A to H as demonstrated in the lower portion of the figure. The benefit of utilizing the entire lightcurve is that random error is minimized. Measuring the phase changes requires the data to have a stability to better than 1 part in 10 000 over the planetary orbit. The star’s intrinsic brightness may vary, because of, but not limited to: stellar pulsations, star spot modulation or star planet interactions due to magnetic fields. Period analysis of the phenomena can differentiate pulsations and spots from planets, but for some EGPs MOST observations have shown that interactions between the planet and the stellar surface can produce a photometric signal which will have the same period as the planet’s orbit and can mimic the phase modulated shape caused by reflected light from the planet [85]. Multi-epoch observations from MOST have shown a strong evolution in the lightcurve shape and amplitude for interacting stars and planets and thus multi-epoch observations will be necessary to differentiate between planetary phase changes and planet-star interactions or intrinsic variability of the star itself.  4.1  The Phased Light Curve  Figure 4.2 presents the 2004 and 2005 photometry plotted in phase with the orbital period of the exoplanet (see Table 4.2.1). The top panel shows the unbinned data. The repetitive pattern seen in these data is due to the increased photometric scatter during intervals of highest stray light modulated with the MOST orbital period of about 101.4 min. This appears in the phased lightcurve because, by coincidence, the orbital frequencies of the MOST satellite and HD 209458b are in a near-harmonic ratio of 50:1. The middle panel shows the data binned in 40-min intervals. The effects of stray light are less obvious here, since we use weighted averages based on the photometric errors described above in §2.2.9. The planetary transit is at phase 0.75; the eclipse corresponds to phase 0.25. The bottom panel 58  Chapter 4. Observations  1.0001 Eclipse  1.00005  Transit  H  1  B C  A  G D  F  Relative Flux  0.99995 0.9999 0.99985 0.9998 0.99975 0.9997 E  0.99965 x1  x3  x2  x4  0.9996 0  0.2  0.4  0.6  0.8  1  Phase  x1  A  x2  H  B 11 00 00 11 00 11 00 11 000 111 000 111 000 111  111 000 000 111 000 111  G F x4  E  11 00 00 11 C 00 11 00 11  D  x3  Figure 4.1: A model of the flux changes over the orbital period of HD 209458b. At the phase point labeled A is the total flux of the planet and star when the planet is eclipsed by the star. Point E shows the total flux when then planet transits in front of the star. Note the flux drop at E drops to approximately 0.98, far off the plotted scale. The sinusoidal curve for points B, C, D, F and H are due to the changing phase of the planet as illustrated in the lower portion of the figure. The points x1 though x4 mark the ingress and egress of the eclipse and transit.  59  magnitude  Chapter 4. Observations  magnitude  Phase  Phase  magnitude  111111 000000 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111  Phase  Figure 4.2: The data phased to the period of the planet. Top panel shows the raw data. The second panel shows the data binned with 40 minute bins and the bottom panel shows the data heavily binned. shows the data averaged in bins of width 0.04 cycle in phase (∼3.4 hr) with 1σ error bars of about 25 ppm (µmag).  4.2  Modeling the Light Curve  The light curve of a system with one transiting planet contains the following variations in phase with the exoplanet orbital period: (1) the transits themselves; (2) the eclipses; and (3) the changing flux from the planet as it goes through illumination phases during its orbit. There are also possible intrinsic variations in the star due to rotational modulation which may be in sync with the planet [85] but are not necessarily so. 60  Chapter 4. Observations If we momentarily ignore the effects of limb darkening and assume the star and planet are observed as flat uniform disks, then the shape of the transit is dictated by the area of the star covered by the planet. Assuming no radiation from the darkside of the planet the maximum depth of the transit is simply the difference between the projected areas of the star and planet, Rp R∗  fmax = 1 −  2  (4.1)  where Rp and R∗ are the radius of the planet and star and f has been normalized to unity when the star is not being transited. Figure 4.3 shows a closeup of the observed transit for the HD 209458 system. During the ingress and egress of the transit, only a portion of the planetary disk is covering the star. This gives the characteristic large slope to the shape of the light curve. If we have two circular disks of radius R1 and R2 that represent the star and planet located at (x1 , y1 ) and (x2 , y2 ) respectively as projected towards the observer (see Figure 4.4) then the x-intersection of the disks during ingress or egress is given by, γ = −R12 α + R22 α + (x1 + x2 )(α2 + β 2 ), ζ=  β 2 ((R1 − R2 )2 − α2 − β 2 )(−(R1 + R2 )2 + α2 + β 2 ), Rt =  γ±ζ , 2(α2 + β 2 )  (4.2)  where α = x1 − x2 and β = y1 − y2 and Rt are the intersection points. A partial transit occurs only if ζ is real. The angle θ defines our integration  range. We want to find the areas shown at A1 and A2 in Figure 4.4, which are given by, An =  1 2  Rn2 − (x − xn )2 (x − xn ) √ 2 Rn −(x−xn )2 (x−xn ) − 21 Rn2 ArcTan −R2 +(x−xn )2 n  x=xi1 ,xi2  (4.3)  for n = 1, 2 where xi1 and xi2 are defined by the intersection points. For the eclipse light curve of the planet disappearing behind the star 61  Chapter 4. Observations  Figure 4.3: Closeup view of the transit using 40 minute bins. The line shows the model based on the parameters in Table 4.2.1  62  Chapter 4. Observations  Figure 4.4: Drawing to understand modeling the planetary eclipse. the light curve shape defined above is adopted in our light curve model. The shape of the egress and ingress can be seen in Figure 4.1 at point A. For the star, it is a poor approximation to assume that the stellar surface appears as a uniform disk. When looking at a star, we see to an average vertical optical depth of τ ∼ 2/3 over the disk of the star. To estimate  the brightness change over the disk profile Kurucz atmospheric models [70] were generated for various atmospheric depths for a star with Tef f = 6100 K and log g=4.38 cgs, representative of HD 209458a [43]. The atmosphere models are multiplied by the MOST transmission function (the transmission function and spectra are shown in Figure 4.5) and integrated and fitted to a non-linear limb darkening law [16]. Our adopted parameters are given in Table 4.2.1. The disk of the star will appear fainter towards the edge and will give the transit shape a smoothed look, opposed to a boxy shape from an uniformly illuminated disk. To model the shape of the transit we adopt the analytic derivation by Mandel & Agol [50]. The derivations are similar to our work above, but include the effects of a non-linear limb darkening model. For a planet in a circular orbit like HD 209458b, we have approximated the reflected light variation as a sinusoid (see §5). Our function to describe  63  Chapter 4. Observations  Figure 4.5: The total system throughput for the MOST satellite optics and CCD detector is shown as the dashed line. A Kurucz model spectrum with Tef f =6100 K and log g = 4.38 cgs, representative of HD 209458a, is also plotted.  64  Chapter 4. Observations the light curve, including transits, is given by f=  M A (R∗ , Rp , a, i, cn , t) + 1+  Fp 2F∗ 1 + Fp F∗ sin(i)  cos  2πt P  +φ  sin(i)  ,  (4.4)  where MA is the normalized flux computed using the adopted analytic expressions for a transiting planet including the effects of limb darkening [50]. F∗ , Fp , R∗ and Rp are the fluxes and radii of the star and planet, a and P are the semi-major axis and period of the planet’s orbit, i is the inclination of the orbit relative to our line of sight, cn are the nonlinear limb darkening parameters [16] and φ is an arbitrary starting phase to coincide with the transit in the phase diagram. During the eclipse of the planet, we use Equation 4.3 to model the light curve. A schematic of this light curve model is shown in Figure 4.1. The transit is marked at point E and for the case of HD 209458, the transit depth extends to a relative flux of 0.98, far below the plotted scale. The amplitude of the flux variations due to phase changes of the planet is equal to the depth of the eclipse of the planet by the star8 . We adapt a Bayesian approach to our best-model-fit minimization so that we can incorporate priors in our fits. The probability function that we wish to maximize is p(y1 , ..., yn ) =  p(y1 |I)...p(yn |I)p(D|y1 , ..., yn , I) , p(D, I)  (4.5)  where y1 to yn are the 13 model parameters listed in Table 4.2.1, p(y1 |I)  to p(yn |I) are the corresponding priors, p(D|y1 , ..., yn , I) is the likelihood function and p(D, I) is a normalization factor. Having priors is important as single-band photometry gives few constraints on the orbital inclination of the exoplanet due to degeneracy of this parameter with the radii of the planet and star. Our adapted priors for the orbital inclination, orbital period, mass and radius of the planet are from observations based on multi-band Hubble Space Telescope (HST) photometry with limb-darkening information to estimate the inclination angle [43]. We take our prior for the mass of the 8  This model assumes a Lambertian phase function. This approximation is discussed further in §5  65  Chapter 4. Observations planet from spectroscopy obtained at the Keck telescope [46]. Priors are given by p(yi ) = √  1 Exp 2πei  −(yi − µi )2 2e2i  ,  (4.6)  where µi is the prior value of yi , with a 1σ uncertainty of ei , as listed in Table 4.2.1. We do not fit explicitly for the limb darkening parameters. In the case of fixed parameters, p(yi ) = δ(yi − µi ) and p(yi ) = 1 when no prior  information is available. The likelihood function is given by n  p(D|y1 , ..., yn , I) = Exp[−χ2 ] i=1 n  χ2 = i=1  √  1 , 2πσi  (di − f (xi ; y1 , ..., yn ))2 , σi2  (4.7) (4.8)  where σi is the photometric uncertainty as determined in §2.2.9.  We find the maximum of equation 4.5 using a downhill simplex model based on the Amoeba routine in Numerical Recipes [59]. We did not adopt any prior for the radius of the planet, as a direct comparison of MOST and Hubble Space Telescope (HST) photometry of the HD 209458 transits, scaled to the same bandpass, produce different depths of transit. Using the HST spectroscopic observations [43] and the MOST bandpass, HST photometry was generated to match MOST photometry. Figure 4.6 compares the averaged transit for the two sets of photometry, where the open diamonds are MOST measurements and the smaller filled diamonds are HST measurements. The MOST data indicate a deeper transit than the HST data. Keeping the planetary radius as a free parameter, our best-fitted value for the model is given in column 3 of Table 4.2.1. The comparison of the HST and MOST photometry may indicate a systematic error in the measured amplitude of the transit. It may also be related to the non-differential photometry from HST. Unlike MOST, HST cannot continuously observe HD 209458. Instead the complete transit must be pieced together from many different orbits as the HST field-of-view will typically be occulted by the Earth during some portion of the orbit.  66  Chapter 4. Observations  Figure 4.6: Comparison of HST and MOST observations for the transit of HD 209458b. The open diamonds are MOST photometry and the smaller, filled diamonds are HST photometry.  67  Chapter 4. Observations This causes a thermal change in the optical telescope assembly which in turns changes the optical focus, producing the well documented breathing effect which is reflected in the photometry [49]. There are no additional stars bright enough to be observed simultaneously with HD 209458, thus the observations are non-differential and the flux changes created by the thermal instability can only be estimated. When reconstructing the transit light curve, it was assumed that the egress and ingress are symmetric (no change in shape) and zero-point offsets in the photometry from each orbit where corrected until a symmetric transit was produced. There is no reason why the true amplitude of the transit should be preserved. Thus the ∼ 1  mmag difference in the transit depths observed by MOST and HST may be an instrumental artifact of the HST orbital environment.  4.2.1  Bootstrap Error Analysis  To estimate uncertainties in our fitted parameters, we use a bootstrap technique. This involves randomly selecting data from the original time series and generating a new time series, with replacement. Replacement means that any data point can be chosen more than once, but the total number of points is always the same, so some points will not be included in the new datasets. Each generated data set has a noise profile similar to the original time series and repeating the fitting procedure on a series of these randomized data sets produces a robust error distribution for our best-fitted parameter values [10]. We performed ∼20000 bootstrap iterations. Figure 4.7 shows the boot-  strap results for some key parameters in our model fit. The contours outline  the 68% confidence region. The top panel shows the planet radius versus the geometric albedo. The middle planet shows the constrains on the radii of the star and planet, in particular the strong correlation between these two parameters is evident. The bottom panel shows the constraints on the mass of the star and planet, which is highly dependent on the adopted priors. Table 4.2.1 gives 1σ errors for all fitted parameters. These uncertainties were estimated by assuming a normal distribution and calculating the standard  68  Chapter 4. Observations deviation of the bootstrap sample. Our best fit to the flux ratio of the planet and star (Fp /F∗ ) is (7 ±  9) × 10−6 . This gives a 1σ upper limit of 1.6 × 10−5 . Applying the best-  fitted value to equation 1.2 gives the geometric albedo measured through the MOST filter: AgMOST = 0.038 ± 0.045. The top panel of Figure 4.7  shows the bootstrap error analysis for AgMOST as a function of the planet radius Rp . The use of priors heavily constrains the system parameters for the planet (such as radius and mass) that are allowed. For instance, the limbdarken parameters, cn should be allowed to vary with the stellar parameters. Computing limb-darkening parameters for the MOST bandpass over a large range of parameters constitutes one area of future work in the interpretation of transit light curves. A glance at the errors stated in Table 4.2.1 reveals that these errors are unrealistically small. This probably indicates that our priors are incompatible with each other. Thus, we also tested our fits by removing the prior information. The derived errors are more in tune with other studies such as those based on HST data [43]. Photometry alone gives no information about the mass of the planet or star as the orbital period (measurable from transit photometry) only gives the combined mass of the star and planet. This is show by our unconstrained fits which give uncertainties on the mass of the star which is larger than the mass of the planet from spectroscopy. The main subject in this paper is the flux ratio of reflected light from the planet compared to the host star and most of the parameters are for fitting the shape of the transit, which, as labeled in Figure 4.1, applies for only a small fraction of the data. Thus flux ratio of the planet and star is largely independent of the transit fitting parameters.  69  Chapter 4. Observations  A  B  C  Figure 4.7: Bootstrap results for HD 209458 parameter fits to MOST photometry. Contours outline the 68% confidence region. Top: Planet radius vs. geometric albedo through the MOST bandpass. Middle: Constraints on the radii of the star and planet. Bottom: Constraints on the masses of the star and planet. 70  Chapter 4. Observations  yi M∗ MP R∗ RP P i Ag φ zpt c1 c2 c3 c4  Prior (ui ± ei ) 1.101 ± 0.064 0.69 ± 0.05 1.125 ± 0.02 – 3.52474859 ±3.8 × 10−7 86.929 ± 0.01 – – – fixed fixed fixed fixed  Bestfit with Priors 1.083 ± 0.005 0.69 ± 0.01 1.118 ± 0.002 1.339 ± 0.002 3.5247489 ±2 × 10−7 86.937 ± 0.003 0.038 ± 0.045 −1.57206 ± 0.0001 −0.00001 ± 1 × 10−5 0.410769 -0.108929 0.904020 -0.437364  Errors without Priors 0.1 1.0 0.03 0.04 1 × 10−6  Units  0.2 0.050 0.0002 1 × 10−5  deg  M⊙ Mj R⊙ Rj days  rad mag  Table 4.1: Table of parameters to describe the MOST HD 209458 light curve. Column (1) lists the 13 parameters used to describe the light curve. The priors in column (2) refer to constraints used to derive the best fit parameters listed in column (3). Column (4) gives the errors on the fitted parameters if no priors are used. Column (5) gives the units for each row in the table.  71  Chapter 5  Results and significance Photon scattering and molecular absorption are the dominant mechanisms that determine the reflected and emitted spectra of an extrasolar Giant Planet (EGP) [51]. In the blue portion of the optical (wavelengths shorter than 600nm) Rayleigh scattering in a clear atmosphere will reflect a large fraction of the stellar flux outwards. In the red portion (wavelengths greater than 600 nm) photons will be absorbed deep in the atmosphere that will make the reflected spectrum appear relatively dark. The strong incident UV flux can produce a rich mixture of compounds from molecules producing a haze that can absorb incident UV photons and will darken the appearance of the planet in the blue part of the spectrum [52]. Jupiter, which receives much less UV flux, contains unidentified species at the part in 1010 level that decrease the blue and green geometric albedos by a factor of 2. The chemical make up of an EGP’s atmosphere under such a strong UV source, such as experienced by HD 209458b, is highly speculative[8]. The process of photochemistry in the atmosphere is currently not well understood thus its importance or significance can not be underestimated. The geometric albedo, Ag is the quantity relevant to the MOST measurement. Ag is defined as the ratio of the planet’s luminosity at full phase to the luminosity from a Lambert disk9 with the same cross-sectional area as the planet. Ag is equivalent to the fraction of incident stellar radiation scattered in the direction of the observer for planetary full phase (if the stellar intensity is spatially uniform). Ag is usually specified at a particular wavelength, so we use AgMOST for the geometric albedo integrated over the MOST bandpass (400 to 700 nm, see Figure 4.5) and AgTOT for the geometric albedo integrated over all wavelengths. 9  A Lambertian surface is an ideal, isotropic reflector at all wavelengths.  72  Chapter 5. Results and significance Planet HD 209458b Jupiter Saturn Uranus Neptune  Geometric Albedo MOST Bandpass ≤ 0.08 0.50 0.47 0.43 0.38  Geometric Albedo All Wavelengths – 0.274 ± 0.013 0.242 ± 0.012 0.208 ± 0.048 0.25 ± 0.02  Bond Albedo – 0.343 ± 0.032 0.342 ± 0.030 0.290 ± 0.051 0.31 ± 0.04  Table 5.1: Albedo of HD 209458b compared to albedos of the Solar System giant planets. Solar system planet geometric albedo in the MOST bandpass computed from Karkoschka[39] and other Solar System planet albedos from Voyager, Pioneer, and ground-based measurements described in Conrath et al.[20].  HD 209458b’s geometric albedo of ≤ 0.08 (1 σ) is a relatively low value.  The Solar System giant planets have albedos in the MOST bandpass with Jupiter’s value 0.5 (see Table  4.2.1).10  0.4,  HD 209458b is therefore less  than one fifth as bright as Jupiter in the MOST bandpass. The Solar System giant planets all have bright cloud decks (water ice or ammonia ice) which cause them to be bright in the visible band of the spectrum, which is essentially the same as the MOST filter bandpass. HD 209458b is an order of magnitude hotter than Jupiter, far too hot for water or ammonia clouds to be present. HD 209458b may have clouds in its atmosphere, but composed of high-temperature condensates such as silicates or solid iron, instead of ices. Clouds at high altitude are consistent with previous observations of the HD209458b atmosphere including: a primary transit low sodium absorption [14]; a primary transit CO non-detection [22]; and a secondary eclipse non-detection of H2 O at 2.2 µm [63]. If clouds are present in the HD 209458b atmosphere, the low AgMOST rules out any bright clouds at high altitudes. Unlike ice clouds, high temperature-condensate clouds may be dark if they consist of small particles or are predominantly Fe [73]. 10  The Karkoschka albedos are measured at 6.8, 5.7, 0.7, and 0.3 degrees away from full phase for Jupiter, Saturn, Uranus, and Neptune respectively. The albedos have an uncertainty of 4%. Jupiter’s and Saturn’s albedo are probably about 5% higher at full phase where the definition of geometric albedo formally applies [39].  73  Chapter 5. Results and significance If HD 209458b does not have clouds, strong sodium and potassium atomic absorption could be present on the day side and cause a low albedo in the MOST bandpass. While AgMOST is not definitive, it is a key constraint on atmosphere models of HD 209458b with specific regard to the thickness, altitude, composition, and particle size distribution of clouds. Our measurements are consistent with other albedo studies [19, 47] that also find low albedo upper limits for the short period planetary companions of υ And and HD 75289 which have orbital periods of 4.6d and 3.5d, similar to HD 209458b. With an upper limit determined for the measured geometric albedo a natural question is: Can we estimate the Bond albedo? The Bond albedo, AB , is the total radiation reflected from the planet compared to the total incident radiation; i.e. the total amount of radiation reflected in all directions integrated over all wavelengths. The Bond albedo can be separated in two quantities, AB = Ag q  (5.1)  where q is the phase integral which is defined as π  φ(α)sinα dα  q=  (5.2)  0  where φ(α) is the phase function, or the brightness variation of the planet at different phases. The phase angle, α, is the angle between the star and Earth as seen from the planet (where α = 0 corresponds to opposition). From the Earth one can measure q for Mercury, Venus, Mars and Moon as we can observe these bodies over a large range of the phase angle. For the outer planets, which can only be seen a few degrees from full phase illumination, the measurement of q requires satellite missions. The Lambert sphere is a reflecting surface with a reflection coefficient that is constant for all angles of incidence. The reflection coefficient is the ratio of the amount of light diffusively reflected in all directions by an element of the surface to the incident amount of light that strikes that element. The phase variations of a Lambert sphere are due only to phase  74  Chapter 5. Results and significance effects, so the phase function is, φ(α) =  π 1 (sin α + (π − α) cos α), 0 < α < , π 2  (5.3)  which we approximate by eye as, φ(α) =  1 + cos(α) . 2  (5.4)  This is an adequate approximation to the phase variations from a Lambertian sphere for the current set of observations. Our geometric albedo estimate based on the light curve model given by Equation 4.4 assumes the simplistic case of a Lambert Sphere. The actual shape of the reflected light signature will depend on the absorptivity and directional scattering probability of the atmospheric material [73]. For small particles (∼ 0.01 µm) the size of the particle is smaller than the wavelength of radiation considered and will scatter as Rayleigh scatting which produces more backscattering of photons (reflected towards observer) that is slightly different when compared to isotropic scattering. For larger particles (∼ 1 µm) the reflected light curve can show a narrowly peaked backscatter (ie. strong reflection at opposition). The amplitude of the phase variation can vary from 0.2 ppm to 80 ppm depending on the particle size and composition [73]. To measure the secondary eclipse for the planet disappearing behind the star, we average the photometry over phase width of 0.044 centered 0.5 phase away from the transit (at phase 0.25 in Figure 4.2). We then find the average of the photometry from two adjacent bins with the same width. The planetary transit lasts for ∼ 3.7 hours which is the approximate width of the bins. Our error is estimated by bootstrapping the means for each  bin. Our observations find a drop in flux of 22 ± 29 ppm as the planet is  eclipsed by the star. The non-detection rules out models that produce large backscattering. In principle AB could be measured for a transiting extrasolar  planet if its brightness at all phases could be measured in a wavelength range that encompasses all the planet’s scattered light. However, HD 209458 is too faint for such a measurement by MOST, and MOST’s bandpass has a  75  Chapter 5. Results and significance cutoff at 0.7 microns (Figure 4.5). From the bottom panel Figure 4.2 we can see that the scatter of any individual binned data point is less 100 ppm. Thus our observations are approaching the realm of determining q and in turn an accurate determination of the Bond Albedo AB . Nevertheless, we estimate an AB upper limit for HD209458b, based on the Solar System planet albedos, the AgTOT /AB relation, and model atmosphere considerations. AB is an important physical parameter because it specifies the amount of stellar radiation absorbed by the planet and hence the equilibrium temperature of the planet as given by Equation 1.1. Figure 5.1 shows Teq for the HD209458 parameters listed in Table 4.2.1. The upper left corner represents our parameter range space based on 1 σ limits. The Solar System giant planets all have AB > AgTOT , as illustrated in Figure 5.2. This can be understood by considering a Lambertian planet, with AB = 1. AgTOT would have to be less than one, since Ag includes only the radiation scattered back toward the observer. More precisely, for a Lambert sphere AB = 1.5 AgTOT (Lambert’s law is the dotted line in Figure 5.2). Under the reasonable assumption that HD 209458b is a gas giant planet with a thick atmosphere and no reflecting surface, we can confine our attention to the very general theoretical case described by a semi-infinite atmosphere model. In this case the physically relevant albedo parameter space in Figure 5.2 is bound by the dashed and dotted lines, 0.67 < AgTOT /AB < 1 [76]. Indeed, the solar system giant planet albedos comply [20]. Therefore, under the simplest case assumptions about the atmosphere of HD 209458b, we can use the isotropic scattering limit (dotted line) in order to derive an upper limit on its AB . Multi-layered atmosphere models and other complications can produce geometric albedos below that limit by no more than ∼10% [76]. One further assumption is required in order to estimate HD 209458b’s  AB from the AgMOST upper limit: AgTOT = AgMOST . We first note that the AgTOT < AgMOST for Solar System planets because of strong CH4 absorption redward of the MOST bandpass, but blueward to the wavelength where their thermal emission dominates over scattered radiation. HD 209458b is an order of magnitude hotter than the Solar System giant planets and should differ. If HD209458b were a blackbody emitter, its thermal emission 76  Chapter 5. Results and significance  Figure 5.1: The dayside Teq for HD209458b as a function of AB for different values of f (see equation 1.1). The approximate estimate of AB is shown as a vertical dashed line. The 24 µm brightness temperature of 1130 K is shown [23] and can be considered a lower limit to Teq [71]. The upper left quadrant is the parameter space range for HD209458b based on our 1 σ limits.  77  Chapter 5. Results and significance  Figure 5.2: Relationship between the Bond albedo (AB ) and total geometric albedo (AgTOT ) for a Lambertian sphere and Solar System giant planets. The points are for Uranus, Neptune, Saturn, and Jupiter (in order of increasing AB ). The dotted line (AgTOT /AB = 0.67) is for Lambertian isotropic reflectance (i.e., constant for all angles of incidence). The dashed line is the line of equivalence where AgTOT = AB (all gas giant planets with deep atmospheres must lie to its right). The wedge between the dotted and dashed lines defines a useful limiting region: it bounds the photometric properties of most spherical bodies with deep atmospheres (with, e.g., Rayleigh scattering, clouds, dust, etc.). Hence for HD 209458b, with the assumption of AgTOT = Ag MOST and AgMOST ≤ 0.08, we estimate that AB ≤ 0.12.  78  Chapter 5. Results and significance would peak around 2–5 microns (depending on its actual equilibrium temperature). HD209348b, however, is expected to deviate significantly from a blackbody. Near-IR molecular absorption could induce thermal emission as short a wavelength as 0.8 or 0.9 microns [71]. In low-geometric-albedo models, the thermal emission could dominate over scattered radiation at such short wavelength and AgTOT ≃ AgMOST is not too unreasonable. With AgTOT and the 0.67 < AgTOT /AB < 1 argument (Figure 5.2), we estimate  for HD 209458b that AB  0.12.  From Equation (1.1) this value of AB gives Teq > 1400 K. In Figure 5.1 we show how this estimated AB value together with the Spitzer/MIPS 24 µm brightness temperature measurement of 1130 K [22] constrain the overall Teq of HD209458b. As HD 209458b is expected to be tidally locked. Infrared measurements at 8 µm place 2 σ limits on the phase variations at 0.0015 [21]. The low Bond albedo estimate requires that planet distribute at least 35% at the 1 σ level of the absorbed stellar energy on the night side of the planet to reduce the thermal day/night constrast. This means that f from Equation 1.1 must be less than 2 which agrees with models which give a poor fit to the 24 µm measurement of the eclipse when no circulation is assumed [23, 27]. Recent models predict Bond albedos of 0.05 to 0.1 for EGP at distances less than 0.1 AU primarily from sodium and potassium which absorb strongly in the optical [26]. However, these models ignore the formation of clouds in the atmosphere which also strongly modify the reflected spectra. The presence of silicates and iron in the atmosphere can significantly alter the planetary spectrum [72] and the same self consistent irradiated atmospheric models allow the condensation of enstalite (MgSiO3 ) from comparisons of the temperature-pressure profile compared to expected condensation conditions [27]. However, the physics of cloud formation is a process that is not well understood or constrained both for particle size and densities which are governed by the competing effects of condensation and coagulation versus sedimentation [52]. The low reflected fraction of incident radiation readily rules out highly reflective clouds. Our observations are consistent with atmospheric model predictions for the Bond albedo [26], but our observations 79  Chapter 5. Results and significance cannot distinguish between a clear or dark and cloudy atmosphere (i.e. at what atmospheric depth are the photons absorbed?).  5.1  Transit Timing Predictions  The idea of using transit timing variations to detect additional planets is that two planets in close orbit will gravitationally interact to perturb the orbits such that the change in orbital parameters can be detected from timing the occurrence of planetary transits. For a system with only one planet, that planet will transit at a constant rate. This is not true if a 2nd planet is present. Using consecutive transits, as observed by MOST, one can search for these transit timing deviations, which would be evidence for the presence of additional planets. For HD 209458, the limits of transit deviations are approximately 30 seconds [55]. With this constraint, one would like to know what parameter range of additional planets is excluded. To answer this question the classical N-body problem was solved, Gmj (xi − xj ) d2 xi = −Σj=1;j=iN , 2 dt |xi − xj |3  (5.5)  where for 3 bodies, N=3 and x describes the initial positions of the particles. For the HD 209458 system, we assumed a stellar mass of 1.101 M⊙ and planetary mass of 0.69 Mjup in a circular orbit with a period of 3.52474859 days. A third body was inserted with an initial circular orbit with periods ranging from 1 to 9 days in increments of 0.01 days and masses from 1 - 100 M⊕ in 1 M⊕ increments. The solution was advanced at 1.0 second intervals for 100 orbits of HD 209458b (∼ 2 × 107 s) using the LSODA routine from  ODEPACK [60]. The O-C values where calculated by a linear interpolation to estimate the integration time when the true anomaly of HD 209458b  returns to its initial value (zero). We then computed a Fourier Transform of the O-C series for each value of period and mass of the 3rd body and extracted the largest amplitude. Figure 1.3 plots period vs amplitude for various masses. At orbital resonances, where the period of the two planets is an integer 80  Chapter 5. Results and significance ratio (eg 2:1, 3:2, ...) we can rule out the presence of Earth-mass planets and also rule out Earth-mass planets in orbits close to the EGP where the gravitational field of the EGP dominates over the star11 . The mass limits we place are also related to the eccentricity of the EGP over large time scales and addressing whether tidal heating is important source of energy to explain the inflated radii of some extrasolar planets, such as HD 209458b. Long term monitoring of transiting EGP will eventually drive down detection limits to an Earth mass over a large range of orbital parameters space, and hopefully one day discover one.  5.2  Summary and Future  We have presented 58 days of high-cadence photometry for the MOST satellite of the nearby transiting EGP hosting star HD 209458. We then fit this light curve to determine physical characteristics of the transiting planet. The depth of the transit observed with MOST is approximately 1 mmag deeper than HST observations suggesting a small systematic error in one of the analyses but one cannot rule out the possibility of atmospheric changes in HD 209458b. In terms of the radius of the planet, the uncertainty in the radius of the star is the predominant error. The MOST satellite reobserved the HD 209458 system for approximately 4 weeks in August - September 2007. A new observing mode has been added that allows images to be stacked onboard the satellite before downloading (see §2.2.8 and §3.3). This will dramatically increase the number of photons  detected. Instead of obtaining a 1.5 second exposure every 10 seconds one can obtain one exposure every 15 seconds composed of 10 stacked 1.5 second exposures. This should decrease scatter in the data by a factor ∼ 3. MOST  will either detect the secondary eclipse or place a significant limit of a few percent. More importantly, we will begin to sample the phase function, q,  which in turn can tell us about the reflective properties of the atmospheric composition, such as grainsize. 11  We have yet not studied the long term stability of possible additional planet configurations  81  Chapter 5. Results and significance  Figure 5.3: O-C predictions for HD 209458  82  Chapter 5. Results and significance HD 209458b is less reflective than Solar System giant planets, such as Jupiter. Our measurements place a 1 σ upper limit on Fp /F∗ of 1.6 × 10−5 . The inferred low Bond albedo < 0.12 rules out the presence of highly reflec-  tive clouds in the atmosphere of HD 209458b and is consistent with noncloudy atmospheric models. However, the effects of photochemistry and cloud formation are poorly understood. The largest unknown in determining Teq is the efficiency for the planet to distribute heat from the dayside to nightside of the planet, which we model by the parameter f in Equation 1.1 determines whether the planet reradiates of 2π or 4π steradians. Further infrared measurements at different phases of the planet orbit will place stronger constraints on this parameter. An interesting observation will be monitoring the amplitude of the albedo signature with time. Comparison of the eclipse depth and amplitude of flux changes due to the changing phase of the planet is a consistency check to determine if indeed the measured albedo is actually changing. Starplanet interactions will not produce an eclipse, thus the eclipse will give a robust albedo measurement but with a lower signal-to-noise ratio. A positive detection of a changing albedo can be interpreted as atmospheric changes on the planet, such as cloud formation (or dissipation) or weather changes. Such measurements may be possible with new spacebased photometric missions. Larger optical space telescopes dedicated to the study of exoplanets have just recently launched or are on the horizon. The CoRoT satellite, which launched in December 2006, consists of 27cm aperture optics to monitor a field of view of 3.9◦ for 6 months at a time. The Kepler project is a onemetre aperture Schmidt telescope with a 12◦ field of view. The detector consists of 21 2k x 2k CCD detectors and will enable 160 000 of visual magnitude ≤ 14 stars to be simultaneously measured for 4 years [44]. Of  the few hundred transiting EGPs that Kepler and CoRoT may find, a 10%  sample will provide reliable albedo measurements to sample the enviroments in which the planets are found and address the question of how the albedo varies with temperature. This result will be of fundemental importance to understand the physics of hot Jupiters. An expected observation is that albedo will vary with stellar type. For late type stars the peak of the stellar 83  Chapter 5. Results and significance flux will be absorbed by molecular transitions, whereas for early type stars the peak of the stellar flux will be scattered, suggesting that the albedo will rise with the stellar temperature. Long term observations are essential to monitor the stellar activity of an exoplanet-hosting star. In the case of HD 189733 (see §3.3), which is known to be chromospherically active [88], observations show strong spot activity  which can be used to determine the spin-orbit alignment of the star and planet, complementary to results based on the Rossiter-McLaughlin effect [87]. Error limits on the geometric albedo for HD 209458 have now reached a few percent. A basic model for the interior of a planet currently uses an equation of state based on a mixture of hydrogen and helium including effects of ionization and partial disassociation [69] and opacities based on the Solar composition. Such a model cannot account for the inflated radii already detected in some known extrasolar planets without invoking supersolar metallicities or additional heating effects. Future observations not only help constrain/verify the albedos of an EGP, but will also monitor the long term photometry stability of planet hosting stars. 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