Photometric Phase Variations of Long-Period Eccentric Planets
aa r X i v : . [ a s t r o - ph . E P ] S e p Submitted for publication in the Astrophysical Journal
Preprint typeset using L A TEX 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
ABSTRACTThe field of exoplanetary science has diversified rapidly over recent years as the field has progressedfrom 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 planetarystructure 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 targetingof 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 opticalwavelengths for long-period planets, particularly those in the high-eccentricity regime and multiplesystems in resonant and non-coplanar orbits. We apply this to the known exoplanets and discussdetection prospects and how observations of these signatures may be optimized by refining the orbitalparameters.
Subject headings: planetary systems – techniques: photometric – techniques: radial velocities INTRODUCTION
The currently known diversity of exoplanets is greatlyattributable to the revolution of the transit detectionmethod over the past 10 years. The measurement ofradius and hence density were the first steps from theresults of this technique, but soon to follow were atmo-spheric studies from both primary transit and secondaryeclipse. However, the unknown inclination of the plan-etary orbits makes this technique only applicable to arelatively small fraction of the known exoplanets. Forthe non-transiting planets, reflected light and phase vari-ations present an additional avenue through which toinvestigate planetary atmospheres (Charbonneau et al.1999; Leigh et al. 2003). The net result of this new in-formation has lead to an unprecedented ability to char-acterize exoplanets.Phase functions in the Infra-Red (IR) primarily mea-sure the thermal properties of the planet whereas op-tical measurements probe the planetary albedo. Therelation between giant planet atmospheres and phasecurves have been described in detail by Sudarsky et al.(2005). Iro & Deming (2010) further investigate thetime-variation of the atmospheres in the IR for eccentricplanets using radiative transfer models. The phase vari-ation of our own solar system has been investigated byDyudina et al. (2005), and it has been shown that phasefunctions can be used to produce longitudinal thermalmaps of exoplanets (Cowan & Agol 2008). In addition,phase curves of terrestrial planets have been consideredby Mallama (2009).Phase variations of exoplanets in the IR and op-tical regimes have had success due to increased ac-cess to improved instrumentation and space-based ob-servatories. This has primarily been investigated fortransiting planets since the edge-on orbital plane pro-duces the highest phase amplitude. Examples of ob-served phase variations in the IR (using Spitzer) includeHD 189733b (Knutson et al. 2009a) and HD 149026b
Electronic address: [email protected] (Knutson et al. 2009b). Examples in the optical includeKepler observations of HAT-P-7b (Welsh et al. 2010) andphase variations detected in the light curve of CoRoT-1b(Snellen et al. 2009).Kane & von Braun (2008) and Kane & von Braun(2009) showed that planets in eccentric orbits haveinflated transit probabilities, as demonstrated byHD 17156b (Barbieri et al. 2007) and HD 80606b(Laughlin et al. 2009). These types of planets will pro-duce relatively high phase amplitudes during a brief pe-riod (periaston passage) of the orbit. However, phasevariations of non-transiting planets have been restrictedto hot Jupiters, including υ And b (Harrington et al.2006) and HD 179949b (Cowan et al. 2007). Therehave been searches for phase variations of, for exam-ple, HD 75289Ab (Rodler et al. 2008) and τ Boo b(Charbonneau et al. 1999; Rodler et al. 2010) but no sig-natures were detected in either case. Exploring the at-mospheric properties of longer-period planets requirestaking advantage of highly-eccentric non-transiting sys-tems. Thus we also constrain the inclination and hencethe mass of such planets.Here we investigate the expected photometric phaseamplitude of long-period eccentric planets and show howplanetary orbits in resonance can result in ambiguousphase variation detections. We further apply this anal-ysis by calculating maximum flux ratios for the knownexoplanets and considering several interesting case stud-ies. We also determine the effective orbital phase regimeover which detectability is maximised and show how re-finement of orbital parameters can allow efficient tar-geted observations at these times. This study is intendedfrom an observers point of view in so far as these observ-able variations can be reconnected back to the theoreticalmodels of exoplanetary atmospheres. EXOPLANET PHASE VARIATIONS
In this section, we establish the theoretical frameworkwhich will be applied in the remainder of the paper,similar to the formalism used by Collier Cameron et al. Stephen R. Kane & Dawn M. Gelino to observerstar ω planet α = 0°α = 90° α = 180° α = 270° Fig. 1.—
Orbit of eccentric planet, showing orbit phase anglescorresponding to full ( α = 0 ◦ ), first quarter ( α = 90 ◦ ), new ( α =180 ◦ ), and third quarter ( α = 270 ◦ ) phases. (2002) and more recently by Rodler et al. (2010). Figure1 shows a top-down view of an elliptical planetary orbit.The phase angle α is described bycos α = sin( ω + f ) (1)where ω is the argument of periastron and f is the trueanomaly. The phase angle is defined to be α = 0 ◦ whenthe planet is at superior conjunction (“full” phase). Interms of orbital parameters, this location in the orbitoccurs when ω + f = 270 ◦ .The flux at wavelength λ incident upon the planet isdescribed by F i ( λ ) = L ⋆ ( λ )4 πr (2)where L ⋆ is the luminosity of the star and r is the star–planet separation. This separation is given by r = a (1 − e )1 + e cos f (3)where a is the semi-major axis and e is the orbital ec-centricity. The geometric albedo of a planet is defined at α = 0 ◦ as follows A g ( λ ) = F r (0 , λ ) F i ( λ ) (4)where F r is the reflected light from the planet. The plan-etary flux received at Earth is then f p ( α, λ ) = A g ( λ ) g ( α, λ ) F i ( λ ) R p d (5)where R p is the planetary radius, d is the distance to thestar, and g ( α, λ ) is the phase function. Since the stellarflux received at Earth is f ⋆ ( λ ) = L ⋆ ( λ )4 πd (6)then the flux ratio of the planet to the host star is definedas ǫ ( α, λ ) ≡ f p ( α, λ ) f ⋆ ( λ ) = A g ( λ ) g ( α, λ ) R p r (7)and thus contains three major components; the geomet-ric albedo, the phase function, and the inverse-squarerelation to the star–planet separation. Note that for acircular orbit, only the phase function is time dependent. Wavelength Dependence
As noted in the previous section, the observed fluxratio from an exoplanet is wavelength dependent. Inparticular, the atmospheric composition drives the scat-tering properties and thus the forms of the geometricalbedo and phase function. This dependence has beenconsidered in great detail by Sudarsky et al. (2005) inwhich they construct empirical models for exoplanet at-mospheres and intergrate over the surface with assumedopacities depending upon atmospheric composition. Thisthorough analysis is not reproduced here, but we douse the results of their analysis for a restricted wave-length, particularly with regards to the albedo functiondiscuessed in the following sections. Here we confine ourstudy to optical wavelengths centered on 550 nm. Thisbroadly encompasses the results obtained by such stud-ies at Collier Cameron et al. (2002), Leigh et al. (2003),and Rodler et al. (2010). This also places the study nearthe peak response of the Kepler CCD, the relevance ofwhich will be discussed in later sections.
Geometric Albedo
It has been shown through atmospheric models thatthere is a dependence of the geometric albedo ofgiant planets on the semi-major axis of the orbit(Sudarsky et al. 2000, 2005; Cahoy et al. 2010). We di-rect the reader to Figure 9 of Sudarsky et al. (2005)which details the wavelength and star–planet separationdependence of the geometric albedo. Jupiter is knownto have a visual geometric albedo of ∼ .
52. However,the strong irradition of the atmospheres of giant planetsin short-period orbits results in the removal of reflectivecondensates from the upper atmospheres and thus a sig-nificant lowering of the geometric albedo. Observationsof HD 209458b using the Microvariability and Oscilla-tions of STars (MOST) satellite by Rowe et al. (2008)failed to detect phase variations and thus they were ableto place an upper limit of A g < .
08, subsequently in-vestigated using model atmospheres by Burrows et al.(2008). More recent observations of HAT-P-7b using Ke-pler by Welsh et al. (2010) revealed phase variations inthe light curve from which they were able to deduce ageomtric albedo of 0.18.Through the examples mentioned above, and the con-sideration of the theoretical models of Sudarsky et al.(2005), we have constructed a model which approximatesthe albedo of giant planets as a function of star–planetseparation (see Equation 3). This is a hyperbolic tan-gential function of the form A g = ( e r − − e − ( r − )5( e r − + e − ( r − ) + 310 (8)Equation 8 is plotted in Figure 2, showing the semi-majoraxes of HAT-P-7b and Jupiter for reference. This func-tion well represents the rapid rise in optical albedo be-tween 0.2 and 1 AU described by Sudarsky et al. (2005),as well as the continued rise beyond 2 AU whereby waterclouds begin to be present. Though broadly robust toencompass both the theoretical calculations and the lim-ited number of measured examples mentioned above, thisempirical function does not account for planetary grav-ity whose variation may affect the atmospheric propertiesand thus the albedo properties.hase Variations of Eccentric Planets 3 Fig. 2.—
Approximation of the geometric albedo distribution forgiant planets, where λ ∼
550 nm.
Phase Function
A planetary phase function can be considered to bedefined by the continuous presence of an imaginary lineconecting the center of the star and planet which is nor-mal to the day–night terminator of the planet. The phasefunction of a Lambert sphere assumes the atmosphereisotropically scatters over 2 π steradians and is describedby g ( α, λ ) = sin α + ( π − α ) cos απ (9)and is thus normalized to lie between 0 and 1. For acircular orbit, the phase function applied this to the fluxratio relation (Equation 7) results in both a phase func-tion and flux ratio which are maximum at a phase angleof zero. Generalizing the phase angle (see Equation 1)and thus the phase function for an eccentric orbit requiresfirst solving Kepler’s equation M = E − e sin E (10)where M is the mean anomaly and E is the eccentricanomaly. The true anomaly is then related to the eccen-tric anomaly by cos f = cos E − e − e cos E (11)where the true anomaly establishes the time-dependentvariation of the phase function.For the analysis performed here, we adopt the ap-proach of Collier Cameron et al. (2002) and Rodler et al.(2010) which utilizes the empirically derived phase func-tion of Hilton (1992). This is based upon observations ofJupiter and Venus and incorporates substantially moreback-scattering due to cloud-covering. This approachcontains a correction to the planetary visual magnitudeof the form∆ m ( α ) = 0 . α/ ◦ ) + 2 . α/ ◦ ) − . α/ ◦ ) (12)which leads to a phase function given by g ( α ) = 10 − . m ( α ) (13)where the wavelength dependence has been removed (seeSection 2.1). This Hilton phase function is used through-out the remainder of this paper. Shown in Figure 3 are phase functions and normalizedflux ratios for various eccentricities and orbital orienta-tions. Note that the maximum flux ratio does not neces-sarily occur at zero phase angle for a non-circular orbit.This is because the orbital distance is changing and in-deed we shall show in later sections that the star–planetseparation component of Equation 7 becomes dominantfor highly eccentric orbits. This time-lag between max-imum flux ratio and maximum phase was also noted bySudarsky et al. (2005). Orbital Inclination
For interacting systems, many of the system param-eters depend on the orbital inclination angle, i , of thesystem (for example, see Gelino et al. (2006)). For ex-oplanetary systems, given an assumed albedo, the trueamplitude of the phase variation can be used to estimatethe inclination angle, and therefore constrain the massof the planet derived from radial velocity data. To addthe effect of inclination angle to the phase function, thephase angle (Equation 1) is modified as follows:cos α = sin( ω + f ) sin i (14)At 1st and 3rd quarter ( α = 90 ◦ and α = 270 ◦ ), theflux ratio is completely independent of inclination angle(see Figure 1). The effect of inclination on the shape ofthe phase function is quite small and has negligible effecton the location of the minimum and maximum values ofthe flux ratio, as shown by Figure 20 of Sudarsky et al.(2005). One complicating factor in this simple inclina-tion consideration is that any additional light sources (i.e.planets) in a given system will dilute the signature fromthe dominant light-reflecting planet. The dilution of thesignature will remain constant if the additional planetsare in face-on ( i = 0 ◦ ) orbits. This will be discussedfurther in Section 3.1. MULTI-PLANET SYSTEMS
Detection of multi-planet systems is becoming morefrequent as we are increasingly able to probe into smallermass and longer period regimes of parameter space. Ifindeed single planet systems are rare, then it is highlylikely that the phase curve from a particular planet willbe “contaminated” by the reflected light of other planetsin the system. For planets which are similar in size, theeffect of an outer planet to the combined phase curve willbe small since (from Equation 7) ǫ ( α, λ ) ∝ r − . Herewe discuss the specific cases of orbital coplanarity andresonant orbits. Coplanarity
Additional planets in a system serve to dilute the sig-nature from the dominant light-reflecting planet. De-pending upon planet formation scenarios, it cannot betaken for granted that planets within a system will lie incoplanar orbits. As shown in Section 2.4, the phase func-tion may be reduced in amplitude significantly for orbitsinclined relative to the line-of-sight, to the extreme ofeliminating a time-variable photometric signature of theorbit if it is face-on ( i = 0 ◦ ).Consider the case of the planetary system orbiting thestar υ And, which was first discovered by Butler et al.(1999). A search for reflected light from the inner-most Stephen R. Kane & Dawn M. Gelino
Fig. 3.—
The phase functions (dashed line) and normalized flux ratios (solid line) for various eccentricities and periastron arguments; e = 0 . ω = 0 ◦ (top-left), e = 0 . ω = 90 ◦ (top-right), e = 0 . ω = 0 ◦ (bottom-left), e = 0 . ω = 90 ◦ (bottom-right). planet was carried out by Collier Cameron et al. (2002),but only a marginal detection was produced leaving anambiguity of the result concerning the degeneracy be-tween the planetary radii and assumed albedos. The υ And system is one of the few systems which has beenmonitored astrometrically as well as spectroscopically inorder to provide constraints on the orbital inclinations ofthe planets (McArthur et al. 2010). The inclinations ofthe outer two planets (c and d), with semi-major axes of0.83 AU and 2.53 AU, were measured to be ∼ ◦ and 24 ◦ respectively. The factor of 3 increase in orbital distanceof the outer planet results in a factor of 9 less contribu-tion to the total planetary reflected light from the system.However, the relative inclinations of the planets causesthe phase function of the outer planet to be almost 3times stronger than that of the inner planet. Resonance
A number of systems have now been found to con-tain planets in eccentric orbits with some kind of reso-nant behaviour. An interesting example is the HD 82943planetary system (see Section 7.3), the 2:1 resonance ofwhich has been studied in detail by Lee et al. (2006).The periodic simulataneous periastron passage of twoplanets produces a distinct signature from a phase ampli-tude perspective, although those moments will not nec-essarily be those of maximum flux ratio. In the con-text of radial velocity measurements, Giuppone et al. (2009) describe how resonant orbits can affect the de-tectability of exoplanets. Furthermore, it was shown byAnglada-Escud´e et al. (2010) that 2:1 resonant systemscan be mis-interpreted as single-planet eccentric orbitswhen performing a fit to the radial velocity data. Thesame is true for phase curves of multi-planet systemswhere resonant orbits can effectively hide the presenceof the outer planet in the resulting phase curve since thecombined phase variation will be periodic with time. Incontrast, non-resonant planets will in general producea non-periodic combined signal that will resolve as twoseparate phase functions with time.Shown in Figure 4 are two example systems, each withtwo Jupiter radii planets. The orbits of the first systemare in 2:1 resonance with e = 0 . e = 0 .
5. Givensufficient photometric precision and observing cadence,it may be possible to distinguish the deviant secondarypeak of the first system and deduce the presence of theouter planet. However, this will be a difficult endeavoursince the two peaks are relatively similar in amplitude.The second system presents an even more difficult prob-lem, with the combination of high eccentricity and largerrelative semi-major axis of the outer planet leading toa limited observation window, higher required cadence,and modest phase signature from the outer planet (seenclose to an orbital phase of 0.95 in Figure 4). If one ishase Variations of Eccentric Planets 5
Fig. 4.—
Normalized flux ratio (solid line) for a multi-planet system in which both planets are in 2:1 resonance with e = 0 . e = 0 . unable to monitor this highest peak and also discern thedifference in amplitude with the other three peaks thenthe presence of the outer planet will remain hidden tothe observer. Thus the derived system architecture basedpurely upon the phase variations will be incorrect. Res-onant systems such as these currently comprise a smallfraction of the total number of exoplanet systems. Therelevance of this issue will increase as radial velocity sur-veys sample to longer periods and as Kepler discoversmulti-planet systems, of which candidates have alreadybeen announced (Steffen et al. 2010). APPLICATION TO KNOWN EXOPLANETS
Here we apply the results of the previous sections to theknown exoplanets. The orbital parameters of 370 plan-ets were extracted using the Exoplanets Data Explorer .The data are current as of 22nd May 2010. Since theflux ratio is ∝ R p , the unknown planetary radii for thenon-transiting planets injects a degree of uncertainty intothese calculations. The models of Bodenheimer et al.(2003) and Fortney et al. (2007) show that there is aclear planetary radius dependence upon stellar age aswell as incident flux and planetary composition. How-ever, Fortney et al. (2007) also showed that, for a givenplanetary composition, planetary radii should not varysubstantially between orbital radii of 0.1–2.0 AU. Sincemost of the planets we are considering here lie beyond0.1 AU from their parent stars (by virtue of their eccen-tricity) and the mass distribution peaks at one Jupitermass in this region, we fix the radius for each of the plan-ets in this sample at one Jupiter radius, with the caveatsmentioned above in mind.As shown earlier, the orbital phase at which the max-imum flux ratio, ǫ max , occurs for an eccentric orbit de-pends upon the orbital orientation. Since Kepler’s equa-tion is a transcendental function, the integral of Equation7 must be solved numerically in order to determine themaximum flux ratio for each planet. These calculationsassume an orbital inclination of i = 90 ◦ and thus themaximum flux ratios are upper limits in most cases, eventhough the radial velocity technique is biased towards de-tection of edge-on orbits since these produce larger radial http://exoplanets.org/ velocity semi-amplitude signatures. We have also cal-culated the minimum time difference in units of orbitalphase between where maximum flux occurs and wherethe flux drops to less than 5% of the difference betweenmaximum and minimum flux. This quantity is desig-nated ∆ t and represents the minimum time over whichobservations of maximum effectiveness can be made.These calculated values are plotted in Figure 5, both asa function of period and eccentricity. The value of ∆ t is ∼ .
37 for all planets in circular orbits since this is wherethe phase difference for a simple cosine variation crossesthe <
5% threshold described above. For eccentric or-bits ∆ t can be larger than expected, particularly where ω ∼ ◦ . Even so, the distribution shown in the top-leftpanel of Figure 5 mirrors the distribution of orbital ec-centricities. The bottom-left panel shows that there is aminimum value of ∆ t that may occur for a given eccen-tricity, but once again we see that this can float upwardsdepending upon the value of ω . The evident linear re-lation in log-space of the flux ratio on period shown inthe top-right panel demonstrates that the flux ratio isindeed dominated by the star-planet separation as onewould expect. However, note the significant outliers be-yond a period, P , of 200 days which are caused by thehighly eccentric planets which pass through periastronclose to a phase angle of 0 ◦ . Several of these systems arediscussed in detail in Section 7.The calculated values of ǫ max and ∆ t for ∼
70 of themost eccentric known exoplanets are tabulated in Table1. Of the planets represented in this table, the planetwith the highest eccentricity, HD 80606b, is also theplanet with the highest predicted flux ratio. This is notsurprising considering that this planet’s periastron pas-sage is behind the star, leading to the high secondaryeclipse probability and the subsequent observation ofthat eclipse by Laughlin et al. (2009). As described ear-lier and demonstrated by Figure 5, the flux ratios of theplanets in Table 1 are dominated by the period and there-fore the semi-major axis of the orbits. ORBITAL PARAMETER REFINEMENT
As described by Kane et al. (2009), the refinement oforbital parameters is not only an essential componentfor succussful detection of features which only appear Stephen R. Kane & Dawn M. Gelino
Fig. 5.—
Calculated values of maximum phase amplitude ǫ max and approximate time (in phase) between minimum and maximumamplitude ∆ t for the 370 known exoplanets included in this sample. The top two panels show these calculated quantities plotted as afunction of orbital period, and the bottom two show their variation as a function of orbital eccentricity. for a small fraction of the orbit, it is also achievable withrelatively few additional radial velocity measurements.This is particularly true of long-period planets whose or-bits tend to have higher associated uncertainties and forwhich opportunities to observe at a particular place inthe orbit are far less frequent.For the goal of attempting to detect a planetary tran-sit, it is the time of predicted transit mid-point whichneeds to be constrained. For optimal observations ofphase variations, it is the time span during which themaximum change in flux ratio occurs which needs to beaccurately determined, previously defined by the quan-tity ∆ t . The reason for this is because, even though thephase variation occurs over the entire orbit, it is assumedthat the high cadence and precision needed will necessi-tate limited observing time using highly-subscribed in-struments (discussed further in Section 7).Using the analogy of the transit window described byKane et al. (2009), we introduce the concept of the phaseprediction window, which is the time period during whicha particular phase of the orbit could occur according tothe uncertainties associated with the orbital period andthe time of periastron passage. In Figure 6 we plot or-bital eccentricity as a function of the phase window forthe 370 exoplanets for which the necessary uncertaintieswere available (358). It is clear that the planets with the Fig. 6.—
Orbital eccentricity as a function of phase predictionwindow for the known exoplanets. The dashed vertical line indi-cates a reasonable boundary beyond which observational attemptsduring a predicted phase location become difficult. highest eccentricities are the most difficult cases to pre-dict orbital phase locations. This is important becausethe most eccentric orbits tend to have much smaller ∆ t values and so it is essential that orbital refinement beused to reduce the uncertainty in the phase predictionand hence the size of the associated window.hase Variations of Eccentric Planets 7 TABLE 1 ǫ max and ∆ t for eccentric exoplanets. Planet P (d) e ω ( ◦ ) ∆ t ǫ max (10 − )HD 80606 b 111.43 0.93 300.60 0.005 3.8029HD 20782 b 585.86 0.93 147.00 0.006 0.0946HD 4113 b 526.62 0.90 317.70 0.008 0.1939HD 156846 b 359.51 0.85 52.23 0.017 0.0249HD 45350 b 963.60 0.78 343.40 0.030 0.0160HD 30562 b 1157.00 0.76 81.00 0.050 0.0022HD 20868 b 380.85 0.75 356.20 0.035 0.0388HD 41004 A b 963.00 0.74 97.00 0.060 0.0030HD 37605 b 54.23 0.74 211.60 0.038 0.5608HD 222582 b 572.38 0.73 319.01 0.043 0.0263HD 2039 b 1120.00 0.71 344.10 0.045 0.0091iota Dra b 511.10 0.71 91.58 0.073 0.0036HD 96167 b 498.90 0.71 285.00 0.055 0.0281HD 86264 b 1475.00 0.70 306.00 0.055 0.0078HAT-P-13 c 428.50 0.69 176.70 0.050 0.0167HD 159868 b 986.00 0.69 97.00 0.081 0.0021HD 43848 b 2371.00 0.69 229.00 0.056 0.0058HD 17156 b 21.22 0.68 121.90 0.053 0.254916 Cyg B b 798.50 0.68 85.80 0.092 0.0025HD 89744 b 256.78 0.67 195.10 0.053 0.0298HD 39091 b 2151.00 0.64 330.24 0.067 0.0043HD 131664 b 1951.00 0.64 149.70 0.070 0.0024HD 74156 b 51.65 0.63 176.50 0.065 0.1567HD 171028 b 538.00 0.61 305.00 0.080 0.0172HD 154672 b 163.94 0.61 265.00 0.090 0.0715HD 16175 b 990.00 0.60 222.00 0.082 0.0074HD 3651 b 62.22 0.60 245.50 0.083 0.2431HD 190984 b 4885.00 0.57 318.00 0.085 0.0014HIP 2247 b 655.60 0.54 112.20 0.159 0.0035HD 175167 b 1290.00 0.54 325.00 0.101 0.0051HD 190228 b 1136.10 0.53 101.20 0.142 0.0010HD 87883 b 2754.00 0.53 291.00 0.113 0.0033HD 142022 b 1928.00 0.53 170.00 0.102 0.0025HD 108147 b 10.90 0.53 308.00 0.100 1.3664HD 168443 b 58.11 0.53 172.95 0.097 0.0988HD 81040 b 1001.70 0.53 81.30 0.171 0.0016HIP 5158 b 345.72 0.52 252.00 0.119 0.0241HD 4203 b 431.88 0.52 329.10 0.104 0.0127HD 217107 c 4270.00 0.52 198.60 0.100 0.0012HAT-P-2 b 5.63 0.52 185.22 0.100 2.5595HD 1237 b 133.71 0.51 290.70 0.117 0.0650HD 142415 b 386.30 0.50 255.00 0.132 0.0170HD 34445 b 1000.00 0.49 131.00 0.145 0.0025HD 215497 c 567.94 0.49 45.00 0.150 0.0048HD 106252 b 1531.00 0.48 292.80 0.135 0.0045HD 33636 b 2127.70 0.48 339.50 0.117 0.0025HD 33283 b 18.18 0.48 155.80 0.120 0.2591HD 196885 b 1333.00 0.48 78.00 0.179 0.0010HD 181433 d 2172.00 0.48 330.00 0.119 0.0030HD 210277 b 442.19 0.48 119.10 0.190 0.0046HD 154857 b 409.00 0.47 59.00 0.187 0.0039HD 187085 b 986.00 0.47 94.00 0.238 0.0014HD 147018 b 44.24 0.47 335.97 0.118 0.1681HD 66428 b 1973.00 0.47 152.90 0.130 0.0015HD 50554 b 1224.00 0.44 7.40 0.137 0.0031HD 23127 b 1214.00 0.44 190.00 0.135 0.0035HD 202206 b 255.87 0.44 161.18 0.146 0.0114HD 74156 c 2473.00 0.43 258.60 0.156 0.00224 UMa b 269.30 0.43 23.81 0.153 0.0096HD 213240 b 882.70 0.42 201.00 0.144 0.0048HD 117618 b 25.83 0.42 254.00 0.156 0.3417HD 141937 b 653.22 0.41 187.72 0.150 0.0058HD 65216 b 613.10 0.41 198.00 0.148 0.0071HD 126614 A b 1244.00 0.41 243.00 0.162 0.004270 Vir b 116.69 0.40 358.71 0.149 0.0307HD 171238 b 1523.00 0.40 47.00 0.182 0.0015HD 5388 b 777.00 0.40 324.00 0.155 0.0055HD 11977 b 711.00 0.40 351.50 0.154 0.0041HD 181433 b 9.37 0.40 202.00 0.147 1.105614 Her b 1754.00 0.39 19.60 0.163 0.0016HD 125612 b 510.00 0.38 21.00 0.180 0.005442 Dra b 479.10 0.38 218.70 0.163 0.0091 Note . — ∆ t in units of orbital phase. Fig. 7.—
The percentage change in the values of ∆ t and ǫ max as afunction the percentage variation in the planetary visual magnitudecorrection (see Equation 12) for the planets in Table 1. ROBUSTNESS OF PHASE MODELS
Although the phase function and albedo formulationadopted in this study to compute the expected phasevariations are physically motivated, the sample of exo-planets with accurate measurements for these functions isrelatively small. For example, the Hilton phase function(Equation 13) is based upon the cloud maps of Venus andJupiter which, although they produce similar phase func-tions, have their own unique cloud configurations withresulting slight variations in their respective phase func-tions. Likewise, real exoplanets could potentially exhibita range of phase functions and albedo distributions be-yond those considered here. As is clear from Equation7, the flux ratio is linearly dependent on the geomet-ric albedo. In other words, a 1% change in the valueof A g translates into a 1% change in the value of ǫ max .The geometric albedo has no effect on the time betweenminimum and maximum flux, ∆ t , since the shape of thephase variation is not altered.A change in the phase function however, will alter boththe calculated values of ∆ t and ǫ max . In order to quantifythis effect, we varied the value of the planetary visualmagnitude correction, ∆ m , as described in Equation 12.This was performed for all of the planets shown in Table1, from which the the mean of the percentage changein ∆ t and ǫ max was calculated for those planets. Theresults of this simulation are shown in Figure 7. Evena substantial change in the phase function of 10% leadsto a relatively minor change in the expected value of ∆ t and an even smaller impact on the expected flux ratio.Additionally, the magnitude of these changes are largestfor eccenctric planets which are more sentitive to thelocation of the peaks in the phase curve. CASE STUDIES AND DETECTABILITY
Here we present specific examples of predicted phaseamplitudes and detectability for several of the known ex-oplanets. In evaluating whether these signatures are de-tectable or if instead calculating these signatures will re-main a theoretical exercise for the immediate future, con-sider the precision of the Hubble Space Telescope (HST)and the MOST satellite. Both of these telescopes haveobserved the V = 7 .
65 star HD 209458. The HST ob-servations by Brown et al. (2001) achieved a precision of Stephen R. Kane & Dawn M. Gelino1 . × − and the MOST observations by Croll et al.(2007) achieved a precision of 3 . × − . In these cases,the necessarily high cadence resulting from the brightnessof the host stars could be used to advantage by binningthe data to produce higher precision. In addition, the el-lipsoidal variations detected by Welsh et al. (2010) usingKepler data is of amplitude 3 . × − . The HD 37605 System
Our current knowledge of the HD 37605 system con-sists of a single planet in a ∼
54 day, highly eccentricorbit (Cochran et al. 2004). This places this planet closeto the top of the list shown in Table 1. The peak fluxratio from this planet is expected to be 0 . × − , morethan a factor of 10 smaller than that for HAT-P-7b. Evenso, the peak flux ratio is helped substantially by the peri-astron argument of ω = 211 ◦ which places the periastronpassage close to the observer–star line-of-sight on the farside of the star.The minimum time between minimum and maximumflux ratio is 0.04 of the orbital phase, or ∼ . ∼ . ∼ ∼
40 orbits which have occurred sincethen. Thus, the precision of the orbital parameters re-quire further improvement before a robust attempt toonly observe the ∆ t section of the orbital phase is made. The HAT-P-13 System
The HAT-P-13 system (Bakos et al. 2009) is particu-larly interesting because it consists of an inner transitingplanet ( P = 2 .
92 days) in a circular orbit with an outercompanion in a long-period ( P = 428 . . × − , compared to 0 . × − for the outerplanet. Detecting the signature of the outer planet wouldbe a challenging detection task even if the signature ofthe inner planet were not present. Secondly, since the in-clination of the outer planet is not known, the predictedflux ratio becomes an upper limit, potentially makingthe detection criteria even more dire. The substantiativedominance of the inner planet over the phase signaturemakes this far too challenging a task for any current in-struments. The HD 82943 System
The HD 82943 system contains two planets in eccentricorbits in a well-studied 2:1 resonance (Lee et al. 2006).The inner and outer planets have periods of 219 and 441days respectively but have near identical maximum fluxratios of ∼ . × − ; well beyond the detection limitsof current instruments. The reason for the similarity inthe peak flux ratios is due to the orbital elements in thesystem. The eccentricity of the orbits combined with the Fig. 8.—
The predicted normalized flux ratio (solid line) for theHD 82943 system, the planets of which are in 2:1 resonance. Thedashed line represents the phase function for the outer planet. resonance has led to the axes of the orbits being almost π out of phase with each other. This leads to the complexstructure for the total flux ratio variation of the system,shown in Figure 8. Note that the periastron passage ofthe outer planet occurs behind the star, whereas it occursin front of the star for the inner planet, yielding a relativeincrease in the maximum flux ratio for the outer planet,partially compensating for the larger semi-major axis. CONCLUSIONS
Current generation space-missions are already detect-ing exoplanet phase variations in the optical (eg., Kepler)and the IR (eg., Spitzer). The steps these produce to-wards characterizing the atmospheres of these exoplanetsare significant since they provide direct measurementsof the atmospheric albedo and thermal properties. Thishas currently been primarily undertaken for short-periodplanets since many of these transit and produce phase va-rations on easily observable timescales. However, currentradial velocity surveys are biased towards planets whoseorbits are closer to edge-on, since larger semi-amplitudesignatures are produced, and therefore biased towardsplanets with larger predicted phase amplitudes.We have shown here how time and position depen-dent functions for the geometric albedo and phase canbe used to describe expected phase variations for long-period eccentric giant planets. There is a clear degener-acy with orbital inclination and resonace when consid-ering multi-planet systems and care must be taken toaccount for these possibilities. Applying these resultsto the known exoplanets shows that many long-periodeccentric planets can have significant peak flux ratios,comparable to those of short-period planets. Addition-ally, the phase prediction windows of eccentric planets,during which observations will be optimally placed, willtend to be poorly contrained. The refinement of orbitalparameters for the known exoplanets is clearly a key com-ponent for optimal observations of the eccentric planetsduring maximum phase amplitude. Improving the mea-sured orbits of long-period planets is already being un-dertaken by such projects as the Transit Ephemeris Re-finement and Monitoring Survey (TERMS) (Kane et al.2009). However, most of the predicted flux ratios forthe known planets push heavily against the boundariesof what is achievable with current ground and space-hase Variations of Eccentric Planets 9based instruments. A thorough search of all these planetswill therefore likely need to await future generation tele-scopes, such as the European Extremely Large Telescope(E-ELT), the Thirty Meter Telescope (TMT), the GiantMagellan Telescope (GMT), and the James Webb SpaceTelescope (JWST).As more science results are released by the Kepler mis-sion, the study of photometric phase variations of long-period planets will become increasingly relevant. Notonly is it an existing mission which has already detectedphase variations in the light curve of HAT-P-7b, it isspecifically looking for transiting planets where there isan a priori knowledge that the orbital inclination is fa-vorable towards maximum phase amplitude. Addition-ally, Kepler will eventually detect transiting long-period (
P >
100 days) planets where the bias will certainly betowards eccentric orbits since those have a higher prob-ability of transiting (Kane & von Braun 2008). Thesefuture discoveries will be prime candidates to detect thephase variations described here.
ACKNOWLEDGEMENTS