Transits of Transparent Planets - Atmospheric Lensing Effects
aa r X i v : . [ a s t r o - ph . E P ] A ug Transits of Transparent Planets - Atmospheric Lensing Effects
Omer Sidis Re’em Sari , Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel Theoretical Astrophysics, Caltech 350-17, Pasadena, CA 91125, USA
ABSTRACT
Light refracted by the planet’s atmosphere is usually ignored in analysis ofplanetary transits. Here we show that refraction can add shoulders to the transitlight curve, i.e., an increase in the observed flux, mostly just before and aftertransit. During transit, light may be refracted away from the observer. Therefore,even completely transparent planets will display a very similar signal to thatof a standard transit, i.e., that of an opaque planet. We provide analyticalexpression for the amount of additional light deflected towards the observer beforethe transit, and show that the effect may be as large as 10 − of the stellarlight and therefore measurable by current instruments. By observing this effectwe can directly measure the scale height of the planet’s atmosphere. We alsoconsider the attenuation of starlight in the planetary atmosphere due to Rayleighscattering and discuss the conditions under which the atmospheric lensing effectis most prominent. We show that, for planets on orbital periods larger thanabout 70 days, the size of the transit is determined by refraction effects, and notby absorption within the planet. Subject headings: planetary systems – techniques: photometric – planets andsatellites: general
1. Introduction
While the majority of extrasolar planets were found using the radial velocity technique(Mayor et al. 2004; Marcy et al. 2008), the method of finding planets by observing theirtransit in front of a star (Borucki & Summers 1984) is now becoming a powerful tool. Thetransit method is based on the observation of a small drop in the brightness of a star, thatoccurs when the orbit of a planet passes (’transits’) in front of it. The amount of light lostdepends on the ratio of the radius of the planet to the radius of the star. It is of order 0.01% 2 –(for earth-size planets) and 1% (for Jupiter-size planets). Transiting extrasolar planets revealmore information. Since the star’s size can be estimated from spectroscopic observations,the planet’s size can be determined. The main effect of planetary transit can be modeledgeometrically (e.g., Seager 2008). Both star and planets are projected on the plane of thesky, producing two circles of radius R ∗ and R p . Then, the fractional overlap between thetwo determines the fractional decrease in stellar light. Within transit, the size of the effectis ( R p /R ∗ ) . The high accuracy obtained, especially by space instruments (Brown et al.2001; Borucki et al. 2009) demand more advanced models, which take into account the nonuniform stellar brightness over the stellar disk. This results in a curved, rather than a flat,lightcurve around the eclipse minimum. In addition, reflection and thermal emission fromthe planet (L´opez-Morales & Seager 2007) cause an out of eclipse sinusoidal variation in theflux (Borucki et al. 2009; Knutson et al. 2009). Burrows et al. (2007) stress that care shouldbe taken in the definition of the planet radius. While the canonical definition uses radialoptical depth of τ = 2 /
3, the light in a transit is absorbed tangentially, and therefore passesa longer path through the atmosphere of the planet than a radial ray would if it were topenetrate to the same depth. This results in an increase of the apparent radius of the planetby about 5 planetary scale heights.Seager & Sasselov (2000) showed that the lengthening of the light path due to refrac-tion in the atmosphere is small and therefore does not change the absorption significantly.Hui & Seager (2002) considered refraction by planetary atmospheres, taking the effects ofoblateness and absorption into account. In this paper we consider the effects of refractionin the planetary atmosphere independent of absorption. Outside of the transit, refractionmay redirect light through the planetary atmosphere towards the observer. This results inincreased observed brightness especially right before and after a transit. In contrast, duringtransit, refraction deflects light that could otherwise arrive to the observer. This results in adiminishing of light even for a completely transparent planet, where no light is absorbed. Weshow that for planets occupying obits with large enough semimajor axis, this refraction effectduring transit results in an effective planetary radius, R p , that is larger than that dictatedby opacity. Therefore, in some cases we would detect transits of completely transparentplanets.The structure of the paper is as follows. In § § § §
2. Model assumptions
We assume a planet in hydrostatic equilibrium with a spherically symmetric exponen-tially decaying density profile. Since the deviation of the index of refraction of gas from unity δn = n − δn . Theindex of refraction n as function of the distance from the center of the planet, r , is therefore: n ( r ) = 1 + δn ( r ) = 1 + exp (cid:18) − r − R H (cid:19) , (1)where H is the scale height of the planetary atmosphere. Here we used the parameter R as anominal normalization of the index of refraction. It is the distance from the center where δn is unity. In planets like Jupiter, it is roughly halfway to the center of the planet. Therefore, R is many scale heights inward of the usual definition of a planet’s radius (e.g., where theoptical depth radially outward is τ = 2 / R and the radius of theplanet will be clarified in § H is related to theplanet’s atmospheric temperature T p and its surface gravity g by H = k B T p µg , where k B is theBoltzman constant, T p is the planet’s atmospheric temperature, and µ is its mean molecularweight. Note that we have used an exponential profile throughout the planet, but only theouter atmosphere matters.We are mainly interested in light trajectories that pass a few scale heights near theradius of the planet, meaning in its atmosphere. Inside the atmosphere, the light is diverteddue to the changing index of refraction. Our analytical results will be facilitated in the limitthat the scale height of the planet is much smaller than the planet’s radius which in turn ismuch smaller than the stellar radius R ∗ : H ≪ R p ≪ R ∗ . Finally, we assume that the latteris much smaller than the semimajor axis of the planet a . This last assumption implies thatthe angular deflection of the light, that bends over the planet’s atmosphere and reaches aviewer at infinity is ∆Θ ∼ R ∗ /a ≪
3. Light trajectories in a spherically symmetric medium3.1. Order of magnitude
For small deflections, the innermost scale height towards which the light was directeddominates the deflection. The distance traversed by an unbent ray within that scale heightis 2 p R p H . By definition of a scale height, or equation (1), δn changes significantly overa radial distance H . Therefore, in a wide light ray, the part traveling at the top of thescale height will be ahead of that at the bottom of the scale height by a distance of order 4 –( n − p HR p . We can therefore estimate the deflection angle to be ( n − p R p /H . Thiscalculation is valid for deflections which would shift the ray less than a single scale heightover a distance of p R p H . Therefore, for this result to hold, the deflection angle must beless than δ Θ ≪ p H/R p . Such small deflections can direct light towards the observer beforetransit only if R ∗ /a ≪ p H/R p . An elegant way of calculating the trajectory of light in the medium is by using Fermat’sprinciple. Following the spherical symmetry of the refraction index the trajectory takes placeon a plane. Hence, we may use polar coordinates: dt = n ( r ) c √ dr + r dθ (2)and Fermat’s principle reads δ Z dt = δ Z n ( r ) c s r (cid:18) dθdr (cid:19) dr = 0 . (3)Using the Euler-Lagrange equation, we obtain ddr n ( r ) r dθdr q r (cid:0) dθdr (cid:1) = 0 . (4)or ∂θ∂r = ∓ b p r n ( r ) − b r . (5)Here b is a constant of integration set by the initial conditions. Far away from the planet, n ( r ) r →∞ = 1, and the light travels on a straight line. Solving for n ( r ) = 1 gives a straightline trajectory with a minimal radius b and therefore b is the impact parameter. Takingthe limit of equation (5) at large r we obtain dθ/dr = b/r , and therefore b is the impactparameter of the straight line that describes the trajectory far away from the planet, i.e. theimpact parameter of the non deflected incoming light ray.For a non-homogenous index of refraction, ( n = 1), the impact parameter b and theminimum radius r min do not coincide. Since dθ/dr diverges at the minimum radius, we canfind the minimum distance by setting the denominator in equation (5) to zero. We obtain r min n ( r min ) = b (6) 5 –An alternative way to describe the light trajectory, is to denote by α (0 ≤ α ≤ π )the angle between the propagation direction of the light ray and the radial direction. Then,tan α = rdθ/dr . Therefore, equation (5) can be elegantly written as rn ( r ) = b sin α. (7)Equation (6) is then a special case of equation (7) that describes the turning points, wherethe light ray is tangential so, by definition, α = π/
2. The functional form of rn ( r ) thereforedetermines the properties of the light trajectories. Interestingly, for our assumed densityprofile, the function rn ( r ) increases linearly steeply, reaching a maximum at about ˆ r ∼ = H ,then decays roughly exponentially, reaching a minimum around ˜ r ∼ = R + H ln R /H , where˜ rn (˜ r ) ∼ = ˜ r + H . Outward of that, it increases roughly linearly with rn ( r ) ∼ = r . Note thatwe have assumed that n − rn is notat ˆ r ∼ = H , but much farther out. The minimum we found at ˜ r ∼ = R + H ln R /H is upin the atmosphere, where the exponential assumption is realistic, and therefore is correctlyestimated.We therefore conclude, that a light coming from infinity, with impact parameter b , willbe deflected from the atmosphere of the planet, if b > ∼ ˜ r + H . In contrast, for b < ∼ ˜ r + H lightrays will deeply penetrate the planet and, may only be reflected after arriving at r ≪ ˆ r ∼ = H .We note that all such rays will be absorbed in the planet, under any realistic opacity. Wenow briefly mention two other solutions to equation (5) which are fully contained within theplanet, and do not emerge out of it. At the minimum of rn or at r = ˜ r ∼ = R + H ln R /H (8)a light ray can have a circular orbit around the planet. However, such orbits are unstable,and will ultimately either escape the atmosphere or penetrate to large depth. Since rn may never increase above b (as sin α <
1) we can have trapped rays thatpropagate between two radii, one close to the atmosphere, and one close to the center of the 6 –planet. Such rays will be trapped between r and r satisfying:ˆ r < r ≤ r ≤ r < ˜ r (9)Again, these are likely to be absorbed for any realistic planet. We determine the angular deflection of light traveling through our spherically symmetricmedium by integrating equation (5). For convenience, we subtract the result of a lighttraveling within a homogenous medium, n = 1 with the same impact parameter b :∆Θ = 2 Z ∞ r min ( b/r p r n ( r ) − b − b/r √ r − b ) dr (10)Now, using a change of variables, x = rr min , and using (6) we get∆Θ = 2 Z ∞ ( 1 /x q x n ( x ) n ( r min ) − − /x √ x − dx (11)Under the assumption that r min /H ≫ δnr min /H ≪ n ( x ) n ( r min ) to thefirst order in δn and get an expression for the angular shift:∆Θ = δn ( b ) r π bH (12)This is the basic result we will use next. It is equivalent to what we found using order ofmagnitude derivation in § √ π .
4. Lensed Stellar Images
Equipped with an analytic expression for the angular deflection, we determine belowthe star’s images to a viewer at infinity. Using these images we then estimate the change inthe observed luminosity due to atmospheric lensing.A light arriving from any point on the star could pass by both sides of the planet and bedeflected towards the observer at infinity. For example in figure 1, point A has two images A ′ and A ′′ . If the center of the planet is in between the source and the image as projected onthe sky we refer to the deflection as deflection through the far side of the planet. Otherwise, 7 –if, in projection, the source and the image are both on the same side of the center of theplanet we refer to the deflection as deflection through the near side of the planet. Out oftransit, the deflection through the far side of the planet would yield a secondary image of thestar that appears within the planet’s atmosphere and has a crescent shape. The light that isdeflected though the near side of the planet and reaches an observer at infinity, is generallyway up in planet’s atmosphere and is hardly deflected given the exponential nature of theindex of refraction (for example, points A ′′ and C ′′ in figure 1). These rays produce thesimple, almost unperturbed, disk-like image of the star away from transit. The secondary,crescent like, image changes the ordinary light curve away from, but close to, transits. Inorder to estimate the change in the observed flux we find the area of the crescent image asrefraction preserves the specific intensity. We denote the star radius as R ⋆ , the semi-major axis of the planetary orbit as a andthe projected distance between the star and the planet on the major-axis as XR ⋆ .Now we must formulate a proper way of integrating over the star’s edge which will bemapped, due to refraction through the far side of the planet’s atmosphere, to the crescentshaped image. Using Green’s theorem for a closed curve integral we can calculate the crescentarea. If we denote the crescent’s area and the distance of each point on the crescent fromthe planet’s center by S ′ and b respectively, then S ′ = Z α ′ max α ′ min b dα ′ . (13)To evaluate the integral, we change integration variable from α ′ to α usingtan α ′ = sin αX − cos α . (14)For the light to arrive at the observer, the deflection angle must satisfysin ∆Θ = l + ba (15)where from Figure 1, we see that l = ( X − cos α ) R ∗ / cos α ′ . (16)We can therefore obtain b ( α ) by equating equations (12) and (15). While in § b , analytical expressions are given in § § A , B , C and D are mapped by the atmospheric lensing onto points A ′ B ′ C ′ and D ′ . On the lower panel, points A and C are also shown to arrive practically undeflected topoints A ′′ and C ′′ . An arbitrary source point in the star located at an angle α , is mappedinto α ′ in the atmosphere of the planet. We have used the parameter R to normalize our density or index of refraction profiles.During the transit itself, light is deflected away from the observer. We can now calculate theeffective radius of the planet as the radius of the dark spot in the star’s image caused by thisdeflection. In the middle of a central transit, i.e., when the center of the planet covers thecenter of the star as projected on the sky, the deflection angle that determines the effective 9 –radius is given by ∆Θ CT = ( R ⋆ + R p ) /a . This is the deflection of light coming from the edgeof the star and refracted in the far side of the planet in order to reach the distant observer.Using this deflection angle we obtain R p ≡ b (∆Θ CT ) ∼ = R + H (cid:18) πR p a HR ⋆ (cid:19) . (17)This is an implicit equation since R p appears on the right hand side as well. However, sinceit only appears in the logarithm, any value for the radius can be used there with negligibleerrors. Exact solutions can be found iteratively with ease. Note, that the above is not astandard definition of the planet radius as it depends explicitly on its distance from the star.Yet, our R p as defined above is quite similar to the planet’s radius defined in other ways.Comparison between our effective radius for occultations R p with other definitions of theradius of the planet related to atmospheric absorption or scattering are given in §
5. Thisdefinition of the planet radius, allows for a simpler equation of the deflection angle at smalldeflections: ∆Θ = R ∗ a exp (cid:18) R p − bH (cid:19) (18) We numerically integrate equation (13) and plot the images of the star as seen by anobserver at infinity. In order that the effect will be easily noticed we chose the following (notrealistic) parameters a = 100 R ⋆ , R ⋆ = 8 R and R = 40 H . For these parameters, equation(17) then implies R p = 1 . R . The image shapes are plotted in Figures (2) and (3). We cansee that as X diminishes the secondary image becomes larger.Fig. 2.— The image of the star as refracted through an exponential planetary atmosphereplotted for different values of planet star separation, X . The axis are measures in units ofstar radius. For X >
1, an additional crescent shape image is seen through the planet’satmosphere. When
X < X . In this figure the origin is fixed to the planet center, and thecoordinates are in units of the planet’s radius R p . As X decreases the crescent moves awayfrom the planet, becomes wider and covers a bigger angle.For X < R p from the center of the planet (perthe definition of R p ). Therefore, a dark spot will be observed in the center of the star, withradius R p . The change in observed flux affected by the eclipse is given by the ratio of theareas S ′ /S = R p /R ⋆ . Symmetry, of course, dictates a bright spot at the center of the darkone since a light ray that would come from the center of the star passing through the centerof the planet will arrive without deflection to the observer. However, as we have discussedbelow equation (7) such rays penetrate so deep into the planet and are always absorbed. Wetherefore ignore them.This decrease of ( R p /R ∗ ) for transparent planets is similar to the one of opaque planets.Transparent planets produce similar transits to opaque ones! The light curve that correspondto the parameters mentioned earlier is shown in Figure (4). The crescent shaped secondaryimage of the star does not yield an analytical expression for a generic of X . An analyticalexpression for the crescent’s size can be obtained under several limits. First we study thecase where the planet is far away from occultation regime, which corresponds to X ≫
1, andthen we study the crescent size just before transit, where R p /R ∗ < X − ≪
1. 11 –Fig. 4.— The light curve of a star during a completely transparent planetary transit whileconsidering atmospheric lensing, for a = 100 R ∗ , R ∗ = 8 R p and R p = 100 H . Just beforeand just after the transit, the stellar light is increased due to refraction in the atmosphereof the planet. Analytical expressions for these shoulders are given in equations (23), (24) &(30). The curved bottom of the transit appears without taking limb darkening into account.Similar curvature in transit was obtained by Hui & Seager (2002). From Figure (1), we obtain∆Θ = R ⋆ a √ X − X cos α + 1 ∼ = XR ⋆ a (cid:16) − cos αX (cid:17) (19)We also have an expression for ∆Θ from (18). Since b ≫ H and both R p and b differ by afew H ’s we solve for b ( α ) as b ∼ = R p − H ln ( X − cos α ) ∼ = R p − H ln X + HX cos α (20)Noting that for ( X ≫
1) equation (14) can be approximated as dα ′ = cos αX dα (21)We can estimate the secondary image size S ′ = I b dα ′ = πHR p X (22)Now if we assume that the light is not scattered or absorbed, and ignore limb darkening ofthe star, the brightness will be uniform and identical over both the direct and lensed stellarimages and the fractional increase in light arriving to the observer due to atmospheric lensingis S ′ S = HR p X R ⋆ . (23) 12 –or, substituting the expression for H , S ′ S = 6 . × − (cid:16) a (cid:17) − / (cid:18) R p R J (cid:19) × (cid:18) M p M J (cid:19) − (cid:18) LL ⊙ (cid:19) / (cid:18) R ∗ R ⊙ (cid:19) − X − (24) Observing the numerical plot of the planetary transit light curve shown at Figure (4),and the crescent images plotted in Figure (2) and (3), one can easily notice that the effect ismost substantial when X is near unity. Hence, we derive now asymptotic expressions in thelimit of R p R ⋆ < X − ≪
1. In this limit the crescent radial width does not change significantlyuntil its edges at α ′ of almost ± π . The area of the crescent is therefore πHR p ∆ b where ∆ b is it’s central width. To estimate ∆ b , we use equation (18): b ∼ = R p − H ln (cid:18) ∆Θ aR ⋆ (cid:19) (25)Following the notation of Figure (1), we get∆ b ∼ = b A ′ − b C ′ = H ln (cid:18) ∆Θ A ′ ∆Θ C ′ (cid:19) (26)Noting that ∆Θ A ′ = ( X + 1) R ⋆ + R p a ∆Θ C ′ = ( X − R ⋆ + R p a (27)together with R p R ⋆ < X − ≪ R ⋆ ≫ R p we finally get∆ b ∼ = H ln X − R p R ⋆ ! . (28)The crescenst’s area is then S ′ ∼ = πR p H ln X − R p R ⋆ ! , (29)so that the fractional increase in observed flux is given by S ′ S ∼ = R p HR ⋆ ln X − R p R ⋆ ! . (30)This is up to a factor of ln R ⋆ R p ≈ . X = 1 would imply. Figure (5) shows how the asymptotic estimates (23) and (30) coincidewith the numerical calculation. 13 –Fig. 5.— The shoulders in transit light curve, ignoring extinction. The numerical curveis plotted as blue line for a = 100 R ⋆ , R ⋆ = 10 R p and R p = 100 H . For comparison, ouranalytical estimates far away from transits (green line, equation (23)) and very close totransit (red line, equation (30)) are also shown for the same parameters. It can be seen thatthese provide excellent approximations within their range of validity. We have shown so far that light is added to the observer when the planet is out oftransit. There, the effect is of order HR p /R ∗ , and when averaged over all possible positionsof the planet around the star we have HR p /a . However, during transit, refraction causesa decrease of order ( R p /R ∗ ) . Averaged over all possible planet positions we have R p /a .The two effects, therefore, do not cancel. To settle this apparent discrepancy, one must takeinto account the light that penetrated deeply into the planet. The amount of that light is R p /a , but it is being spread roughly equally over 4 π steradians as viewed from the planet.Therefore, for a non realistic, truly transparent planet, the planet will shine out of transitwith brightness smaller than that of the star by ( R p /a ) .The effect of the crescent that we calculated, which result in shoulders around transits,is therefore not offsetting the light refracted away during transit. The latter is offset mostlyby deeply penetrating rays, that would allow that planet to shine anywhere in the orbit. Theshoulders provide only a small correction to the above. Together, averaged over all possibleplanet positions, the shoulders and the deeply penetrating rays which show up away fromtransit, cancel the decrease of light during transit exactly. However, in any realistic situation This average is not over an orbit, but over all orbits with all inclinations
14 –deeply penetrating rays will be absorbed, and therefore less light, on average, will arrive tothe observer.
5. Extinction by Rayleigh scattering
In order to determine the relevance of refraction one must estimate the starlight extinc-tion as it passes through the planet’s atmosphere. Rayleigh scattering is a minimal cause ofextinction. We ignore molecular absorption and clouds. We also ignore limb darkening, anddenote the uniform, unextincted, stellar surface brightness by I . The observed brightnessafter a given path through the planet’s atmosphere is I = I exp ( − τ ), where τ is the opticaldepth: τ = Z σN dl. (31)Here, σ = π λ ( n − N is the total cross section for Rayleigh scattering per molecule of thegas, n is the index of refraction, N is the number of molecules per unit volume, λ is the wavelength. The integration takes part over the path of light. Here we assume that the planethas a Hydrogen atmosphere, in this case we have d ≡ n − N = 3 .
046 cm mol (32)For an atmosphere in hydrostatic equilibrium the gas density decays exponentially over thescale height H ≪ R p , so that the principal contribution to the optical depth comes from theintegration over a few scale heights near the impact parameter b ∼ = R p . Therefore, for a lightray with negligible deflection, τ = Z ∞−∞ σN max exp − √ l + b − b H ! dl, (33)where l is the coordinate along the light trajectory measured from the point of closestapproach to the planet center. By expanding the root around small l/b , and assuming b ≫ H , we obtain an analytic expression for the optical depth: τ = σd H ∆Θ = 32 π d λ H ∆Θ (34)Now, using the expression for the cross section σ and assuming that the planet’s tem-perature is determined from thermal equilibrium with albedo A , τ = 32 π dk B T ⋆ R p √ λ µGM p (cid:18) R ⋆ a (cid:19) / (1 − A ) / ∆Θ . (35) 15 –We now derive asymptotic results far away from transit, X ≫
1, as well as just beforetransits R p /R ∗ < X − ≪ When X ≫
1, the refraction angle of the close and distant edges of the star do not varyconsiderably, ∆Θ = XR ⋆ a . Therefore, from equation (35), the optical depth is uniform. Thecrescent specific intensity is constant, attenuated simply by e − τ , where τ = 0 . (cid:18) λ red λ (cid:19) (cid:18) L ⋆ L ⊙ (cid:19) / (cid:18) R p R (cid:19) × (cid:18) M p M (cid:19) (cid:18) R ⋆ R ⊙ (cid:19) (cid:18) AUa (cid:19) / (1 − A ) / X. (36)Here we have used red light with λ red = 650 nm as our fiducial wavelength. Clearly, theoptical depth in the blue part of the spectrum is significantly larger.The fractional flux from the crescent before extinction, is given by equation (23), itdecreases with the planet’s semimajor axis a as a − / . On the other hand τ decreases as a − / . The transmitted light is therefore maximal when τ = 1 / . (37)We therefore define a / = 0 .
34 AU (cid:18) λ red λ (cid:19) / (cid:18) L ⋆ L ⊙ (cid:19) / (cid:18) R p R (cid:19) / × (cid:18) M p M (cid:19) / (cid:18) R ⋆ R ⊙ (cid:19) / (1 − A ) / (38)to be the semimajor axis where optical depth is 1 / R ∗ /a . For ourfiducial parameters, a / corresponding to an orbital period of 74 days.For a ≫ a / the refraction effect will be practically unextincted, with magnitude givenby equation (23), for any X < X max given by X max ∼ = ( a/a / ) / . (39)For X > X max the crescent flux will decay exponentially with X due to Rayleigh scattering. 16 – When the planet center as projected on the sky is less than a stellar radius away fromthe star’s edge (point A in figure 1), the refraction angles of the close and distant edges ofthe star (points A and C on figure 1) differ considerably. The light from point A is deflectedmuch less than R ∗ /a and therefore will be significantly less extincted than we estimate in § S ′ S (cid:12)(cid:12)(cid:12)(cid:12) ext = S ′ S R R out R in e − τ ( r ) dr R R out R in dr = R p R ⋆ Z R out R in e − τ ( r ) dr, (40)where R out and R in the maximal and minimal radius of the crescent as measured from thecenter of the planet, respectively. The denominator integral is simply the crescent radialwidth in the R p R ⋆ < X − ≪ r to τ we obtain S ′ S (cid:12)(cid:12)(cid:12)(cid:12) ext = R R out R in e − τ ( r ) dr R R out R in dr = R p HR ⋆ Z τ in τ out dτ e − τ τ (41)Here τ in and τ out refer to the optical depth associated with the refracted light that sets theinner and outer edges of the crescent. Explicitly τ in = σd H ∆Θ A ′ τ out = σd H ∆Θ C ′ (42)From equation (27) we see that τ in is given by equation (36) with X = 2 while τ out is obtainedif X is replaced by X − R p /R ∗ . We examine the attenuation scaling in a few differentregimes. • Both τ in , τ out ≫
1. In this regime, the crescent luminosity will undergo considerableextinction: S ′ S ext ∼ = HR p R ⋆ e − τ out τ out (43)This applies for a ≪ (2 R p / R ∗ ) / a / or, for our fiducial parameters a ≪ .
06 AU,i.e., a period shorter than 5 days. Most current transits, therefore, fall in this category,making refraction effects difficult to see. The additional light in this case grows expo-nentially in time as the planet approaches transit. Most light arrives when the planetis about R p away from transit. Therefore, the duration of the increased light would beshorter than the duration of the transit by R p / R ∗ , i.e. about a factor of 20. For hotJupiters, transits, which last about 3 hours, should be accompanied by slightly brightedges for about 10 minutes. 17 – • The crescent is only partially extincted. τ in ≫ τ out ≪
1. Here the crescentluminosity changes considerably along its radial width, so the crescent luminosity isdominated by regions where τ <
1. We obtain: S ′ S ext ∼ = HR p R ⋆ ln (cid:18) τ out (cid:19) , (44)relevant for 3 − / a / ≫ a ≫ (2 R p / R ∗ ) / a / or for our fiducial parameters 0 .
16 AU >a > .
06 AU or periods between 5 and 25 days. The fractional flux undergoes two com-peting effects. On one hand the area of the crescent decreases as a − / , on the otherhand τ out ∝ a − / decreases with a , allowing a larger fraction of the light to passthrough. The maximum is obtained where τ out = e − ∼ = 0 .
05. However, at such largesemimajor axis, given our fiducial parameter, τ in = ( R ∗ /R p ) τ out ∼ = 0 . <
1. There-fore, the maximum fractional luminosity of the crescent is obtained at the transitionbetween this and the next regime, when τ in = 1, with orbital periods of 25 days. There, S ′ S ∼ = HR p R ∗ ln( R ∗ /R p ) ∼ = 3 . × − . (45)This is the largest effect refraction can produce. The numerical value can be slightlyhigher if the planet’s radius is larger than that of Jupiter and if its mass is somewhatlower, allowing for a larger scale height at a given temperature. • Both τ in , τ out ≪
1. Here the crescent luminosity is hardly attenuated, so that itsfractional luminosity is given by (30). It decreases with semimajor axis showing again,that the maximum is obtained around τ in = 1.
6. Summary
We have derived the possible paths of light passing through a planet with exponentialdeviation of the index of refraction from unity. Though we explore exotic trapped rays andrays following circular orbits, we focus on small deflections, as given by our equation (12).We explore the size and shape of the image of the star as refracted through the planet’satmosphere. The size of the image is given by equation (23) far away from transit whilethe largest and least extinct effect should be seen just before transit, and its magnitude isgiven by equation (30). For planets with short orbital periods, refraction involves relativelylarge deflections, and will therefore be reduced by extinction. On the other hand, planetswith large orbital period have cold surfaces resulting in a small effect. The most noticeableeffect would be for planets with orbital period of about 25 days. For extrasolar planets with 18 – R p ∼ = 1 . R J , like HD 209458 where R p ∼ = 1 . R J and M ∼ = 0 . M J (Brown et al. 2001), thefractional flux increase would be as large as 10 − , if it was located on a 25 days orbit. Thisis readily observable with current instruments.Hui & Seager (2002) have also considered refraction by planetary atmospheres. How-ever, they focused on caustics, which are important for point sources. These are relevant forrefraction of background stars by atmospheres of planets in our solar system. We instead,were interested in the case where the source is very large, smearing out the effects of caustics.We have also ignored the effect of oblateness that they considered.For HD 209458 on its observed 3.5 days orbit, the refraction effect would be extinctedwith optical depth of about τ out = 1 .
7. The refractive effect, even given this extinction wouldbe about 10 − . It would last for about 10 minutes just before and just after transit. This isstill detectable in existing or future data. If the enhancement of light is detected, it couldprovide a direct measurement of the scale height of extrasolar planet.The timescale for the brightest part of the refractive effect is somewhat shorter thanthe duration of the transit, especially if extinction is involved for planets with short orbitalperiods.We also show, that during the eclipse, refraction of stellar light away from the observermay cause a signal similar to that of a regular geometric eclipse. We define the radius R p asthe effective radius of the planet as observed by a refractive eclipse. Indeed, for any planeton orbital periods longer than 25 days, R p , the radius where refraction effect control thetransit occur higher up in the atmosphere that R ray , where Rayleigh scattering is important.Using equation (38) we find that these two radii are related by R p = R ray + 32 H ln (cid:18) / aa / (cid:19) (46)Of course, other kinds of extinction may be in place, most notably clouds, which we haveignored. Those may increase the effective extinction radius above R ray and perhaps beyondour effective radius R p .Finally, we mention that refraction effects would have only a slight modification on theRossiter - McLaughlin effect, but perhaps a large effect on the absorbed spectrum duringtransits. If R p − R ray ≫ H such that refraction rather than extinction determine the eclipse,less absorption would be observed in the spectrum.We thank Oded Aharonson and Peter Goldreich for helpful discussions. RS is partiallysupported by an ERC & IRG grants and a Packard Fellowship. 19 – REFERENCES
Borucki, W. J., Koch, D., Jenkins, J., Sasselov, D., Gilliland, R., Batalha, N., Latham,D. W., Caldwell, D., Basri, G., Brown, T., Christensen-Dalsgaard, J., Cochran, W. D.,DeVore, E., Dunham, E., Dupree, A. K., Gautier, T., Geary, J., Gould, A., Howell,S., Kjeldsen, H., Lissauer, J., Marcy, G., Meibom, S., Morrison, D., & Tarter, J. 2009,Science, 325, 709Borucki, W. J., & Summers, A. L. 1984, Icarus, 58, 121Brown, T. M., Charbonneau, D., Gilliland, R. L., Noyes, R. W., & Burrows, A. 2001, ApJ,552, 699Burrows, A., Hubeny, I., Budaj, J., & Hubbard, W. B. 2007, ApJ, 661, 502Hui, L., & Seager, S. 2002, ApJ, 572, 540Knutson, H. A., Charbonneau, D., Cowan, N. B., Fortney, J. J., Showman, A. P., Agol, E.,Henry, G. W., Everett, M. E., & Allen, L. E. 2009, ApJ, 690, 822L´opez-Morales, M., & Seager, S. 2007, ApJ, 667, L191Marcy, G. W., Butler, R. P., Vogt, S. S., Fischer, D. A., Wright, J. T., Johnson, J. A.,Tinney, C. G., Jones, H. R. A., Carter, B. D., Bailey, J., O’Toole, S. J., & Upadhyay,S. 2008, Physica Scripta Volume T, 130, 014001Mayor, M., Udry, S., Naef, D., Pepe, F., Queloz, D., Santos, N. C., & Burnet, M. 2004,A&A, 415, 391Seager, S. 2008, Space Science Reviews, 135, 345Seager, S., & Sasselov, D. D. 2000, ApJ, 537, 916