Analytic Lightcurves for Planetary Transit Searches

TL;DR: In this paper, exact analytic formulae for the eclipse of a star described by quadratic or nonlinear limb darkening are presented for the HST observations of HD 209458, showing that the ratio of the planetary to stellar radii is 0.1207+-0.0003.

Abstract: We present exact analytic formulae for the eclipse of a star described by quadratic or nonlinear limb darkening. In the limit that the planet radius is less than a tenth of the stellar radius, we show that the exact lightcurve can be well approximated by assuming the region of the star blocked by the planet has constant surface brightness. We apply these results to the HST observations of HD 209458, showing that the ratio of the planetary to stellar radii is 0.1207+-0.0003. These formulae give a fast and accurate means of computing lightcurves using limb-darkening coefficients from model atmospheres which should aid in the detection, simulation, and parameter fitting of planetary transits.

Summary

1. INTRODUCTION

• The eclipse of the star HD 209458 by an orbiting planet was recently used to measure the size and mass of the planet, which had been found with velocity measurements (Charbonneau et al. 2000; Henry et al. 2000).
• Several large surveys that aim to find planets using the transit signature are now being carried out or planned and will soon yield large numbers of light curves requiring fast computation of eclipse models to find the transit needles within the haystack of variability (Borucki et al.
• The limb darkening of main-sequence stars is typically represented by functions of , where v is the angle betweenm p cos v the normal to the stellar surface and the line of sight to the observer (Fig. 1a).
• In § 3, the authors derive the light curve for eclipses of nonlinear limb-darkened stars.

2. UNIFORM SOURCE

• In what follows, d is the center-to-center distance between the star and the planet, is the radius of the planet, is the stellar radius, is the normalized separationr r z p d/rp ∗ ∗ of the centers, and is the size ratio (Fig. 1b).
• The flux relative to the unobscured flux is F.p p r /rp ∗.
• The authors next consider the effects of limb darkening.

3. NONLINEAR LIMB DARKENING

• Limb darkening causes a star to be more centrally peaked in brightness compared to a uniform source.
• This leads to more significant dimming during eclipse and creates curvature in the trough.
• The authors partition the parameter space in z and p into the regions and cases listed in Table 1.
• In case 7, the edge of the planet’s disk touches the stellar center, but the planet is not entirely contained inside the area of the stellar disk.
• This and the previous case apply when the planet is larger than half the size of the star.

4. QUADRATIC LIMB DARKENING

• For linear limb darkening, , Merrill (1950) presents an equivalent analytic expression ing p 02 terms of an “eclipse function,” a.
• The expressions here require fewer evaluations of the elliptic integrals, which decreases computation time, and include quadratic limb darkening.
• The authors expression for eclipse with quadratic limb darkening decreases computation time by more than an order of magnitude compared to evaluating the expressions in § 3 or numerical integration of the unocculted flux.

5. SMALL PLANETS

• For a small planet, , the interior of the light curve, , can be approximated by assuming the surface brightnessp 0.1 z !.
• Taking the ratio of the depth of the eclipse at these points yields .
• This is a very fast means of computing transit light curves with reasonable accuracy and may be used for any limb-darkening function, generalizing the approach of Deeg, Garrido, & Claret (2001).

6. DISCUSSION

• Note that the higher order functions{m ( n} p p 0.1 have flux that is concentrated more strongly toward the center of the star and thus have a more gradual ingress and a deeper minimum as more flux is blocked at the center than the edge.
• Contribution of flux from the planetary companion or other companions may be added to the light curve, reducing the transit depth.
• To illustrate the utility of their formulae, the authors have fitted the nonlinear limb-darkened light curve to the HST Space Telescope Imaging Spectrograph (STIS) data of HD 209458 (Brown et al. 2001).
• The ’s are poorly constrained given the small differences in the basis functions relative to the observed errors and2x p 1.046 cn the large impact parameter for this system.
• The value for p in the quadratic case is consistent with the nonlinear case, indicating that the fit is independent of the assumed limb-darkening law.

7. CONCLUSIONS

• The authors have derived analytic expressions for an eclipse including quadratic limb darkening and nonlinear limb darkening.
• The authors have written a code that takes the properties of a host star, finds the limbdarkening coefficients in the tables of Claret (2000), and computes light curves for the parameters of a given planetary transit.
• The authors thank Sara Seager and Leon Koopmans for useful discussions.
• Support for E. A. was provided by the National Aeronautics and Space Administration through Chandra Postdoctoral Fellowship award PF0-10013 issued by the Chandra X-Ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-39073.
• K. M. was supported by a Caltech Summer Undergraduate Research Fellowship.

Figures

TABLE 1 Limb-darkened Occultation

Fig. 2.—Transit light curves for and (solid line), and all coefficients equal zero but (dotted line), (dashed line),p p 0.1 c p c p c p c p 0 c p 1 c p 11 2 3 4 1 2 (dash-dotted line), or (dash–triple-dotted line). The thinner lines (nearly indistinguishable) show the approximation of § 5.c p 1 c p 13 4

Fig. 1.—(a) Geometry of limb darkening. The star is seen edge-on, with the observer off the top of the page. The star has radius , andv is defined as ther∗ angle between the observer and the normal to the stellar surface, while . (b) Transit geometry from the perspective of the observer.m p cosv

Frequently asked questions

Q1. What are the contributions in "Analytic light curves for planetary transit searches" ?

The authors present exact analytic formulae for the eclipse of a star described by quadratic or nonlinear limb darkening. In the limit that the planet radius is less than a tenth of the stellar radius, the authors show that the exact light curve can be well approximated by assuming the region of the star blocked by the planet has constant surface brightness. The authors apply these results to the Hubble Space Telescope observations of HD 209458, showing that the ratio of the planetary to stellar radii is. 

Q2. What are the future works in "Analytic light curves for planetary transit searches" ?

The authors have written a code that takes the properties of a host star, finds the limbdarkening coefficients in the tables of Claret ( 2000 ), and computes light curves for the parameters of a given planetary transit. Using the appropriate limb-darkening coefficients for each star ’ s spectral type will help to distinguish these contaminants from true planetary transits, which can be accomplished using the formulae presented here. 

Q3. What are the main problems of a planetary search?

Planetary searches suffer from two important backgrounds: grazing eclipsing binaries and triple systems in which two stars eclipse while the flux from the third reduces the depth of the eclipse. 

Q4. What is the egress/ingress of the light curve?

All of the curves cross near , which means that accuratez ∼ 0.7 observations are required near minimum and egress/ingress to constrain the coefficients of the various basis functions. 

Q5. What was the recent use of the transit light curve?

The eclipse of the star HD 209458 by an orbiting planet was recently used to measure the size and mass of the planet, which had been found with velocity measurements (Charbonneau et al. 2000; Henry et al. 2000). 

Q6. What is the purpose of this Letter?

In this Letter, the authors compute analytic functions for transit light curves for the quadratic and nonlinear limb-darkening laws and make available their codes to the community (§ 7). 

Q7. What is the effect of adding flux from the planetary companion to the light curve?

Contribution of flux from the planetary companion or other companions may be added to the light curve, reducing the transit depth. 

Q8. What is the fit for the quadratic light curve?

The value for p in the quadratic case is consistent with the nonlinear case, indicating that the fit is independent of the assumed limb-darkening law. 

Q9. What is the method for finding the light curves?

The nonlinear law (§ 3) provides an accurate fit to realistic stellar limb darkening, while the quadratic fit (§ 4) provides a fast means of obtaining a relatively accurate light curve. 

Q10. What is the way to determine the limb darkening coefficients for a given?

Using the appropriate limb-darkening coefficients for each star’s spectral type will help to distinguish these contaminants from true planetary transits, which can be accomplished using the formulae presented here. 

Q11. What is the subject of this Letter?

The recent activity in this new field of astronomy motivates a return to the equations describing the transit light curve, the subject of this Letter. 

Q12. What is the way to find the light curves?

For an extremely fast and fairly accurate approximation for any limb-darkening law,the equations in § 5 may be used to derive light curves. 

Q13. What is the limb darkening of main-sequence stars?

The limb darkening of main-sequence stars is typically represented by functions of , where v is the angle betweenm p cos v the normal to the stellar surface and the line of sight to the observer (Fig. 1a).