Submitted for publication in the Astrophysical Journal A Preprint typeset using LTEX style emulateapj v. 03/07/07 PHOTOMETRIC PHASE VARIATIONS OF LONG-PERIOD ECCENTRIC PLANETS Stephen R. Kane, Dawn M. Gelino NASA Exoplanet Science Institute, Caltech, MS 100-22, 770 South Wilson Avenue, Pasadena, CA 91125, USA Submitted for publication in the Astrophysical Journal ABSTRACT The field of exoplanetary science has diversified rapidly over recent years as the field has progressed from exoplanet detection to exoplanet characterization. For those planets known to transit, the pri- mary transit and secondary eclipse observations have a high yield of information regarding planetary structure and atmospheres. The current restriction of these information sources to short-period plan- ets may be abated in part through refinement of orbital parameters. This allows precision targeting of transit windows and phase variations which constrain the dynamics of the orbit and the geomet- ric albedo of the atmosphere. Here we describe the expected phase function variations at optical wavelengths for long-period planets, particularly those in the high-eccentricity regime and multiple systems in resonant and non-coplanar orbits. We apply this to the known exoplanets and discuss detection prospects and how observations of these signatures may be optimized by refining the orbital parameters. Subject headings: planetary systems – techniques: photometric – techniques: radial velocities 1. INTRODUCTION (Knutson et al. 2009b). Examples in the optical include The currently known diversity of exoplanets is greatly Kepler observations of HAT-P-7b (Welsh et al. 2010) and attributable to the revolution of the transit detection phase variations detected in the light curve of CoRoT-1b method over the past 10 years. The measurement of (Snellen et al. 2009). radius and hence density were the first steps from the Kane & von Braun (2008) and Kane & von Braun results of this technique, but soon to follow were atmo- (2009) showed that planets in eccentric orbits have spheric studies from both primary transit and secondary inflated transit probabilities, as demonstrated by eclipse. However, the unknown inclination of the plan- HD 17156b (Barbieri et al. 2007) and HD 80606b etary orbits makes this technique only applicable to a (Laughlin et al. 2009). These types of planets will pro- relatively small fraction of the known exoplanets. For duce relatively high phase amplitudes during a brief pe- the non-transiting planets, reflected light and phase vari- riod (periaston passage) of the orbit. However, phase ations present an additional avenue through which to variations of non-transiting planets have been restricted investigate planetary atmospheres (Charbonneau et al. to hot Jupiters, including υ And b (Harrington et al. 1999; Leigh et al. 2003). The net result of this new in- 2006) and HD 179949b (Cowan et al. 2007). There formation has lead to an unprecedented ability to char- have been searches for phase variations of, for exam- acterize exoplanets. ple, HD 75289Ab (Rodler et al. 2008) and τ Boo b Phase functions in the Infra-Red (IR) primarily mea- (Charbonneau et al. 1999; Rodler et al. 2010) but no sig- sure the thermal properties of the planet whereas op- natures were detected in either case. Exploring the at- tical measurements probe the planetary albedo. The mospheric properties of longer-period planets requires relation between giant planet atmospheres and phase taking advantage of highly-eccentric non-transiting sys- curves have been described in detail by Sudarsky et al. tems. Thus we also constrain the inclination and hence (2005). Iro & Deming (2010) further investigate the the mass of such planets. arXiv:1009.4931v1 [astro-ph.EP] 24 Sep 2010 time-variation of the atmospheres in the IR for eccentric Here we investigate the expected photometric phase planets using radiative transfer models. The phase vari- amplitude of long-period eccentric planets and show how ation of our own solar system has been investigated by planetary orbits in resonance can result in ambiguous Dyudina et al. (2005), and it has been shown that phase phase variation detections. We further apply this anal- functions can be used to produce longitudinal thermal ysis by calculating maximum flux ratios for the known maps of exoplanets (Cowan & Agol 2008). In addition, exoplanets and considering several interesting case stud- phase curves of terrestrial planets have been considered ies. We also determine the effective orbital phase regime by Mallama (2009). over which detectability is maximised and show how re- Phase variations of exoplanets in the IR and op- finement of orbital parameters can allow efficient tar- tical regimes have had success due to increased ac- geted observations at these times. This study is intended cess to improved instrumentation and space-based ob- from an observers point of view in so far as these observ- servatories. This has primarily been investigated for able variations can be reconnected back to the theoretical transiting planets since the edge-on orbital plane pro- models of exoplanetary atmospheres. duces the highest phase amplitude. Examples of ob- served phase variations in the IR (using Spitzer) include 2. EXOPLANET PHASE VARIATIONS HD 189733b (Knutson et al. 2009a) and HD 149026b In this section, we establish the theoretical framework which will be applied in the remainder of the paper, Electronic address: [email protected] similar to the formalism used by Collier Cameron et al. 2 Stephen R. Kane & Dawn M. Gelino 2.1. Wavelength Dependence As noted in the previous section, the observed flux ratio from an exoplanet is wavelength dependent. In particular, the atmospheric composition drives the scat- tering properties and thus the forms of the geometric albedo and phase function. This dependence has been considered in great detail by Sudarsky et al. (2005) in planet α = 0° which they construct empirical models for exoplanet at- mospheres and intergrate over the surface with assumed opacities depending upon atmospheric composition. This thorough analysis is not reproduced here, but we do use the results of their analysis for a restricted wave- α = 90° star length, particularly with regards to the albedo function discuessed in the following sections. Here we confine our α = 270° study to optical wavelengths centered on 550 nm. This ω broadly encompasses the results obtained by such stud- α = 180° ies at Collier Cameron et al. (2002), Leigh et al. (2003), and Rodler et al. (2010). This also places the study near to observer the peak response of the Kepler CCD, the relevance of which will be discussed in later sections. Fig. 1.— Orbit of eccentric planet, showing orbit phase angles corresponding to full (α = 0◦), first quarter (α = 90◦), new (α = 180(2002)◦), and and third more quarter recently (α = by 270 Rodler◦) phases. et al. (2010). Figure 2.2. Geometric Albedo 1 shows a top-down view of an elliptical planetary orbit. It has been shown through atmospheric models that The phase angle α is described by there is a dependence of the geometric albedo of cos α = sin(ω + f) (1) giant planets on the semi-major axis of the orbit where ω is the argument of periastron and f is the true (Sudarsky et al. 2000, 2005; Cahoy et al. 2010). We di- anomaly. The phase angle is defined to be α =0◦ when rect the reader to Figure 9 of Sudarsky et al. (2005) the planet is at superior conjunction (“full” phase). In which details the wavelength and star–planet separation terms of orbital parameters, this location in the orbit dependence of the geometric albedo. Jupiter is known occurs when ω + f = 270◦. to have a visual geometric albedo of ∼ 0.52. However, The flux at wavelength λ incident upon the planet is the strong irradition of the atmospheres of giant planets described by in short-period orbits results in the removal of reflective L (λ) condensates from the upper atmospheres and thus a sig- F (λ)= ⋆ (2) i 4πr2 nificant lowering of the geometric albedo. Observations of HD 209458b using the Microvariability and Oscilla- where L is the luminosity of the star and r is the star– ⋆ tions of STars (MOST) satellite by Rowe et al. (2008) planet separation. This separation is given by failed to detect phase variations and thus they were able a(1 − e2) r = (3) to place an upper limit of Ag < 0.08, subsequently in- 1+ e cos f vestigated using model atmospheres by Burrows et al. where a is the semi-major axis and e is the orbital ec- (2008). More recent observations of HAT-P-7b using Ke- centricity. The geometric albedo of a planet is defined at pler by Welsh et al. (2010) revealed phase variations in α =0◦ as follows the light curve from which they were able to deduce a geomtric albedo of 0.18. Fr(0, λ) Ag(λ)= (4) Through the examples mentioned above, and the con- Fi(λ) sideration of the theoretical models of Sudarsky et al. where Fr is the reflected light from the planet. The plan- (2005), we have constructed a model which approximates etary flux received at Earth is then the albedo of giant planets as a function of star–planet 2 Rp separation (see Equation 3). This is a hyperbolic tan- fp(α, λ)= Ag(λ)g(α, λ)Fi(λ) (5) gential function of the form d2 r−1 −(r−1) where Rp is the planetary radius, d is the distance to the (e − e ) 3 Ag = + (8) star, and g(α, λ) is the phase function. Since the stellar 5(er−1 + e−(r−1)) 10 flux received at Earth is L (λ) Equation 8 is plotted in Figure 2, showing the semi-major f (λ)= ⋆ (6) ⋆ 4πd2 axes of HAT-P-7b and Jupiter for reference. This func- then the flux ratio of the planet to the host star is defined tion well represents the rapid rise in optical albedo be- as tween 0.2 and 1 AU described by Sudarsky et al.