Polarimetric study of transiting extrasolar planets

Mon. Not. R. Astron. Soc. 415, 695–700 (2011) doi:10.1111/j.1365-2966.2011.18746.x

Polarimetric study of transiting extrasolar planets

N. M. Kostogryz,� T. M. Yakobchuk, O. V. Morozhenko and A. P. Vid’machenkoMain Astronomical Observatory of the National Academy of Sciences of Ukraine, 27 Zabolotny Str., 03680, Kyiv, Ukraine

Accepted 2011 March 17. Received 2011 March 17; in original form 2010 November 26

ABSTRACTWe present the results of modelling the polarization produced during planetary transits inthe systems HD 189733, TrES-3, Wasp-4 and Wasp-25, using the Monte Carlo method.Polarization maxima at the limb are calculated to be ∼0.022 per cent for HD 189733 withstellar polarization according to Chandrasekhar. The polarization for the system HD 189733 of∼0.022 per cent is close to that previously published, although this was attributed to scatteringof starlight, rather than produced in transit. Using three-dimensional modelling data for thelinear polarization of the Sun’s continuous spectrum, the limb polarization of the solar-typestars Wasp-25 was calculated to be ∼0.00018 per cent, ∼0.00024 per cent for TrES-3 and∼0.00016 per cent for Wasp-4 in the B band. It is noted that observations of the Sun-like starsin the Ti I 4536 Å spectral line are particularly suitable for distinguishing between differentcontributions to the polarization. Also, the shape of the polarization curves, at the near limbtransits, can be used for obtaining the inclination of the planet orbit, as a good alternative tostandard transit methods.

Key words: polarization – methods: numerical – planetary systems.

1 IN T RO D U C T I O N

More than 500 extrasolar planets have been discovered to date bydifferent indirect (radial velocity, photometric transit, microlens-ing, pulsar timing) and direct imaging methods. Polarimetry isa promising technique (Hough et al. 2006; Keller 2006; Schmidet al. 2006) for studying the planetary atmospheres of known ex-trasolar planets, by means of scattered central starlight. The basicprinciple is that light reflected from the planetary atmosphere be-comes linearly polarized (Seager, Whintney & Sasselov 2000; Stam,Hovenier & Waters 2004; Stam et al. 2006; Stam 2008), whereasthe direct starlight itself has negligible linear polarization.

Recently, Berdyugina et al. (2008) reported the possible first de-tection of polarized scattered light from the planetary atmosphereHD 189733b. They made polarimetric measurements in the B bandover the orbital period and derived two polarization maxima nearplanetary elongations with a peak amplitude of ∼2 × 10−4. How-ever, Lucas et al. (2009) considered this value to be too large.They presented very high sensitivity polarimetry of the unresolvedstar–planet systems 55 Cnc and τ Boo, with measured standarddeviations of the nightly averaged Stokes Q/I and U/I parametersof 2.2 × 10−6 and 5.1 × 10−6, respectively. These results contrastmarkedly with that obtained by Berdyugina et al. (2008), being twoorders of magnitude lower. Lucas et al. (2009) note that the largeamplitude periodic polarization signal from HD 189733 cannot beexplained in terms of reflected light from the planet HD 189733b

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and ascribed it to the possible contribution of starspots to the polar-ization of the system. Using the Polarimeter for Indination Studiesof High Mass X-ray Binaries/Hot Jupiters (POLISH) instruments,Wiktorowicz (2009) did not reproduce the large amplitude polari-metric observations of Berdyugina et al. (2008) and showed thatpolarimetric modulation of HD 189733 could not be due to exo-planets. Later, Berdyugina et al. (2011) confirmed the detection ofpolarized light from the planet in the blue B and U bands close to10−4 ± 10−5. They assumed typical parameters of starspots on a Kdwarf and evaluated a maximum polarization increase of 3 × 10−6.Berdyugina et al. (2011) also considered the symmetry breaking ef-fect of the tidal interaction of the planet with the star. However, theauthors did not take into account the effect of the eclipsing systemon the Chandrasekhar polarization. This effect was first detected inthe eclipsing binary Algol (Kemp et al. 1983).

Carciofi & Magalhaes (2005) first estimated this effect for tran-siting exoplanetary systems. Evidently, this effect cannot repro-duce the whole orbital curve, but it makes a significant contribu-tion to the polarization during the transit. Carciofi & Magalhaes(2005) presented the results of the numerical simulations of thepolarization produced in the planetary systems of G-K-M-T dwarfstars, with the planet sizes ranging from that of the Earth to thetwo times of the Jupiter size. Such occultation breaks sphericalsymmetry over the projected stellar disc and thus results in linearpolarization.

In this study, we model the linear polarization that results fromthe transit of extrasolar planets using the method of Carciofi &Magalhaes (2005). We consider four planetary systems, HD 189733,TrES-3, Wasp-4 and Wasp-25, selected according to the star andplanet sizes, orbit configurations and stellar spectral classes.

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696 N. M. Kostogryz et al.

2 PO L A R I Z AT I O N

The normalized flux absorbed by the planet can be formulated as

F (p0, ϕ0) = 1 − 1

π

∫ 2π

0dϕ′

∫ Rp/R∗

0p′f (p′, ϕ′) dp′ (1)

where Rp, R∗ are the planetary and star radii, respectively, and p′

and ϕ′ are polar coordinates of a star–planet system and p0, ϕ0 arethe polar coordinates of the centre of the planet (see fig. 1 in Carciofi& Magalhaes 2005).

Assuming that the limb polarization is defined by μ alone, i.e. P =P(μ), and the polarization direction is centrosymmetric with respectto the direction between the centre of the star and the observer, weuse the following expressions for the Stokes parameters Q and U:

Q(p, ϕ) = I0f (μ)P (μ) cos(2ϕ),

U (p, ϕ) = −I0f (μ)P (μ) sin(2ϕ), (2)

where μ = cos θ is the angle between the normal to the surfaceand the direction to the observer, f (μ) is a limb-darkening law. Itfollows from equations (1) and (2) that the observed normalizedStokes parameters are

q(p0, ϕ0) = 1

π

∫ 2π

0dϕ′

∫ Rp/R∗

0p′f (p′, ϕ′)P (p′, ϕ′) cos(2ϕ) dp′

u(p0, ϕ0) = − 1

π

∫ 2π

0dϕ′

∫ Rp/R∗

0p′f (p′, ϕ′)P (p′, ϕ′) sin(2ϕ) dp′.

(3)

3 L I M B - DA R K E N I N G LAW A N DCENTER-TO-LIMB POLARIZATION

Limb darkening is taken into account using the non-linear limb-darkening law (Claret 2000)

f (μ) = 1 − a1(1 − μ1/2) − a2(1 − μ)

− a3(1 − μ3/2) − a4(1 − μ2) (4)

where ai are constants that depend on wavelength, spectral type,surface gravity and metallicity.

For the stars considered in this paper the limb-darkening curvesin B band are shown in Fig. 1. The upper curve is for the solar-typestar Wasp-25, while the lower two nearly coinciding curves are forWasp-4 (spectral type G8) and TrES-3(G). The bottommost curveis for HD 189733(K2).

Figure 1. Limb-darkening curves in B band for Wasp-25, Wasp-4, TrES-3and HD 189733 from top to bottom, respectively. The curves for Wasp-4and TrES-3 are nearly coinciding.

Fig. 1 shows that the difference between the Wasp-25 curve andthose for Wasp-4 and TrES-3 is considerably smaller than it isfor Wasp-25 and HD 189733. It should be noted that the bright-ness is inversely dependent on the polarization degree (Dyck 1968;Harrington 1969).

Naturally, observational and theoretical studies of the limb po-larization have largely concentrated on the Sun (e.g. Trujillo Bueno& Shchukina 2009; Shchukina & Trujillo Bueno 2009; Gandorfer2002), while moderate number of theoretical studies have focusedon other spectral types (Harrington 1969; Magalhaes, Coyne &Benedetti 1986; Collins 1988).

For the solar-type stars Wasp-25, Wasp-4 and TrES-3, we adoptthe centre-limb continuum polarization calculated in Trujillo Bueno& Shchukina (2009), who solved the 3D radiative transfer prob-lem for the linear polarization of the solar continuous radiation,which is produced by Rayleigh and Thomson scattering. For the starWasp-25 we use the observed Q/I values for the line Ti I (4536 Å)at μ = 0.1 (Gandorfer 2002).

Based on the difference between limb-darkening curves in Fig. 1,we do not apply the solar limb polarization for the later spectral typestar HD 189733. Unfortunately, there is no limb polarization calcu-lated for this spectral type; therefore, we take the Chandrasekhar’sdata as the first approximation (Chandrasekhar 1950). Accordingto these data, the outgoing intensities of two polarization states areequal in the centre of stellar disc (μ = 1) and vary up to 25 percent in the disc edge (μ = 0). Thus, the polarization degree variesfrom zero in the centre of the stellar disc to 11.7 per cent at the edge(Table 1).

4 MO N T E C A R L O SI M U L AT I O N

We used the Monte Carlo method for solving equations (3) as pro-posed by Carciofi & Magalhaes (2005). For a known set of pa-rameters (i.e. Rp, R∗, a and ψ) that describe the configuration ofthe star–planet system, we adopted the following procedure. Everyphoton packet (PP), emitted from a randomly chosen point on thestellar surface in the direction towards the observer, was added theweight

εi = f (μ)L∗N

, (5)

where L∗ is the stellar luminosity. Basically, the given weight relatesto the probability of the PP being emitted towards the observer. Inaddition, the PP is characterized by linear polarization.

Qi = εiP (μ) cos(2ϕ),

Ui = −εiP (μ) sin(2ϕ). (6)

For the PP, emitted in the opposite hemisphere with respect tothe observer, the weight is set to zero. If, on the other hand, the PPis emitted in the visible hemisphere, a check is made to see if itstrajectory crosses the planet. If it does, the PP, being absorbed bythe planet, is also given zero weight. The projected coordinates ofthe planet centre, in the plane of the sky, are

p0 = a

R∗

√sin2 ψ + cos2 ψ cos2 i

ϕ0 = arctancos i

tgψ

,

(7)

where a is the semimajor axis of the planet’s orbit.The PPs which were emitted from the part of stellar disc covered

by the planet were given zero weight according to the following

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Polarimetry of transiting exoplanets 697

Table 1. Polarization degree of outgoing radiation for an electron scattering stellar atmosphere (Chandrasekhar 1950).

μ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0P (per cent) 11.71 7.45 5.41 4.04 3.03 2.25 1.63 1.11 0.68 0.32 0

expression:

p2 + p20 − 2pp0 cos(ϕ − ϕ0) <

(Rp

R∗

)2

. (8)

We adopted a circular planetary orbit (eccentricity = 0), andaccording to this assumption we calculate the ingress position onthe planetary orbit. Then, we divide the transit duration into equalintervals (the zero-point is the centre of the stellar disc) and calculatethe time of the planetary transit over the stellar disc.

As a result of the N iterations of the procedure, the flux andStokes parameters can be found from the following formulae:

Fψ =∑

N

εi,

qψ =∑

N Qi

Fψ

,

uψ =∑

N Ui

Fψ

(9)

where the subscript ψ denotes the planet position. In order to simu-late the temporal progression of the transit, F, q and u are calculated,using given time-steps, for the different planet positions.

While being simple to implement, the described procedure hasthe advantage of avoiding the complications arising from the ingressand egress phases of the transit, for which only part of the planetintersects with the stellar disc. Moreover, this model has the possi-bility of easily accounting for other physical effects, such as stellarspots.

5 R E S U LT S O F C O M P U TAT I O N SAND DISCUSSION

In this section we present the results of Monte Carlo simulations forfour planetary transiting systems. They were selected according totheir component sizes, configurations and stellar spectral classes.

HD 189733 is currently the brightest (mV = 7.67 mag) known starto harbour a transiting exoplanets (Bouchy et al. 2005). This, alongwith the short period (2.2 d), makes it very suitable for differenttype of observations including polarimetry. Since we are interestedin occultation effects, a very important parameter is the ratio ofplanetary to star radii. In Table 2, the main physical parametersof the selected extrasolar planetary systems are listed. The stellarradius of HD 189733 is equal to 0.788 Rsun (Baines et al. 2009) andthe planetary radius is 1.151 RJ (Southworth et al. 2010), giving aratio Rp/R∗ of 0.148.

The panel (a) of Fig. 2 illustrates the size of the planet comparedto the star as well as the path of the transit across the stellar disc

for the given inclination angle. Panel (b) shows the time depen-dence of the stellar flux reduction with the planet transit. The nextthree panels (c, d and e) demonstrate changes of Stokes parametersand polarization degree during occultation. The maximum polar-ization observed on the stellar limb is 2.2 × 10−4. This value isvery close to that obtained by Berdyugina et al. (2008) from di-rect polarimetric observations, but the authors ascribe this result tothe polarization arising from the starlight scattered in the planetaryatmosphere. However, as Lucas et al. (2009) concluded this effectcannot be explained by reflected light from the planet HD 189733b.They suggested a more likely explanation is a contribution fromstellar spots, given that HD 189733 is an active star. Based on ourmodelling, we argue that polarization of the system during a planettransit is significant and it should be taken into account. Knowingprecise moments of ingress and egress, the effect can be verifiedthrough polarimetry.

Fig. 3 shows the results for planetary system TrES-3. It is re-markable for having a planetary companion with one of the shortestorbital periods and one of the largest radii ratio (Rp/R∗ = 0.167)among known transiting exoplanets, i.e. stellar radius is 0.813 Rsun

(Sozzetti et al. 2008) and planetary radius is 1.305 RJ (Southworth,2010; Table 2). With these parameters it is not surprising that TrES-3provides one of the highest polarizations. Moreover, in this system,the planet passes over the very limb of the star, which leads tothe maximum polarization degree observed during the whole oc-cultation period, in contrast to the two-peak dependence for othersystems presented here. However, TrES-3 is not a very suitable ob-ject for detailed polarimetric studies, because of its faintness (mV =12.4 mag).

In Fig. 4, the same results are presented for Wasp-4. It was chosenfor the inclination angle of the planet’s orbit, which is close to 90◦

(89.◦35), and small Rp/R∗ ratio of 0.126 (Table 2), which results inthe minimum polarization degree in our sample. Since the planetarypath is nearly diametrical, the amplitude of flux change in this caseis the largest and its time dependence is more subjected to thelimb-darkening effect (Fig. 4b). It is generally known that Stokesparameter u is equal to zero for configurations with i = 90◦, whichis useful for testing models. As evident from Fig. 4(d), Wasp-4 givesthe smallest scatter of this parameter among the selected systems.

The host star of the Wasp-25 system is very similar to the Sun interms of mass (M/Msun = 1), radius (R/Rsun = 0.95) (Table 2) andtemperature (T = 5750 K). Because of the similarity to the Sun,we used the results of Trujillo Bueno & Shchukina (2009), whoproduced three-dimensional modelling of the linear polarizationof the Sun’s continuous spectrum, which is principally producedby the Rayleigh scattering from neutral hydrogen atoms and theThompson scattering from free electrons. In our study we used

Table 2. Physical parameters of the extrasolar planetary systems (Schneider 2011).

Star Rs(Rsun) Rp(RJ) T, K a (au) i (◦)

HD 189733 0.788 (±0.051) 1.151 (±0.038) 4980 0.03142 (±0.00052) 85.76TrES-3 0.813 (−0.027 + 0.012) 1.305 (−0.09 + 0.028) 5720 0.0226 (±0.0013) 82.15WASP-4 1.15 (± 0.28) 1.416 (−0.068 + 0.043) 5500 0.023 (±0.001) 89.35Wasp-25 0.95 (±0.04) 1.26 (±0.06) 5750 0.0474 (±0.0004) 87.7


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Figure 2. Modelling of the flux and polarization (Stokes q = Q/I, u = U/Iand polarization degree) for the planetary transit for HD 189733.

their centre-to-limb variations of the linear continuum polarization(Stokes Q/I) calculated for the wavelength range between 3000and 7000 Å and at the solar positions μ between 0.1 and 1. Fig. 5shows the results of the modelling in the B band. Linear polarizationof the Sun is significantly smaller than the polarization calculatedby Chandrasekhar for the Thompson scattering, thus resulting ina much smaller occultation polarization for the Wasp-25 system.Although the polarization degree for Wasp-25 is calculated to bevery small, it could be detected with a high sensitivity polarimeter(Lucas et al. 2009).

Since Trujillo Bueno & Shchukina (2009) made their simulationsfor several spectral wavelengths, we used these data to find thepolarization degree at different wavelengths. Fig. 6 shows the timedependence of polarization degree for standard Johnson–Cousins U,B, V , R filters. Despite the larger decrease in limb surface brightness,the resulting polarization in the U band is higher by one order ofmagnitude than in the R band and, overall, the effect weakens fromU to R.

Figure 3. The same as in Fig. 2 for TrES-3.

It is generally known that linear polarization is much more promi-nent in individual spectral lines than in the continuous spectrum.One of the most polarized lines is Sr I 4607 Å (Trujillo Bueno,Shchukina & Asensio Ramos 2004; Trujillo Bueno & Shchukina2007). Using linear polarization for the Sun at μ = 0.1 from obser-vations of Gandorfer (2002), we calculated the polarization degreefor Wasp-25 close to the limb, as 0.013 per cent. It is much largerthan those in Figs 5 and 6. On the other hand, Sr I 4607 Å is shownto be very sensitive to the magnetic field and, therefore, to the Hanleeffect, suppressing the linear polarization. More suitable for obser-vations is the spectral line Ti I 4536 Å, which is insensitive to themagnetic field. Our modelling shows that the maximum polariza-tion degree in this line is 0.0013 per cent, an order of magnitudesmaller than for Sr I 4607 Å. None the less, in the case of Ti I 4536the polarization of the system primarily occurs from the planet tran-sit rather than from any starspots and other manifestations of stellaractivity.


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Polarimetry of transiting exoplanets 699

Figure 4. The same as in Fig. 2 for Wasp-4.

Until recently, the only determination of inclination angle i hasbeen made by using the shape of the flux drop caused by the planettransit. One of the drawbacks of this method is that at smaller i (i.e.high latitudes transits) flux signatures are weak and observationallyhard to detect. In contrast, the linear polarization during the occul-tation is more sensitive to the inclination angle. In particular, themaximum polarization does not depend on i, but is defined by theratio of planet-to-star radius. This allows the planet orbit inclina-tion to be determined even for the near limb transits. Furthermore,the shape of the Stokes parameters time curves can be useful indetermining the planet orbit orientation in space, similarly to theRossiter–McLaughlin effect method.

6 C O N C L U S I O N S

We have presented the results of Monte Carlo simulations for thefour planetary transiting systems: HD 189733, TrES-3, Wasp-4 andWasp-25, calculating the polarization that results as the planet transitover the stellar disc. There are two polarization maxima at the limb:

Figure 5. The same as in Fig. 2 for Wasp-25.

∼0.022 per cent for HD 189733. For this system, we adopted thepolarization of the star as calculated by Chandrasekhar (1950) forThompson scattering only. The maximum polarization calculatedfor HD 189733 is very close to that observed by Berdyugina et al.(2008). Although they attributed this polarization to scattering ofstarlight by the planetary atmosphere, this is thought to be mostunlikely (Lucas et al. 2009). We suggest the transit polarizationis large enough and should be accounted for. This can be readilyverified knowing the moments of ingress and egress.

For the solar-like stars Wasp-25, TrES-3 and Wasp-4, we usedthe results of three-dimensional modelling of the linear polariza-tion of the Sun’s continuous spectrum made by Trujillo Bueno &Shchukina (2009). The linear polarization of Wasp-25 is very weak,∼0.00018 per cent, ∼0.00024 per cent for TrES-3 and ∼0.00016 percent for Wasp-4 at the limb in the B band.

We modelled the occultation polarization at the limb ofWasp-25 in two spectral lines Sr I 4607 Å and Ti I 4536 Å, in whichthe linear polarization is much more prominent than in the continu-ous spectrum. Observational data of secondary spectra were taken


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Figure 6. Dependence of polarization degree during transit for Wasp-25 infilters U, B, V , R.

from Gandorfer’s atlas (Gandorfer 2002). Despite the limb polar-ization in Sr I 4607 Å (0.013 per cent) being an order of magnitudelarger than in Ti I 4536 Å (0.0013 per cent), the latter was notedto be particularly suitable for distinguishing between polarizationproduced by a transiting planet or through starspots. We also con-sidered the polarization in different filters and found that linearpolarization is higher in the U band than in the R band, implyingthat the former is more convenient for detection.

We find that the maximum polarization does not depend on theinclination angle, being determined by the planet-to-star radii ratioRp/R∗ only, increasing towards the higher values. However, theshape of the polarization curves, at the near limb transits, can beused to determine the inclination of the planet orbit, more reliablythan observing the change in brightness. Furthermore, the timevariations of the Stokes parameters can be used to determine theorientation of the planet orbit in space, similarly to the Rossiter–McLaughlin effect method.

Finally, we emphasize that this technique can be additionallyused for confirmation of exoplanets.

AC K N OW L E D G M E N T S

The authors gratefully acknowledge Dr. Sci. Nataliya Shchukinafor providing the data on solar polarization and Professor James H.Hough for checking the paper text and having constructive discus-sions. The authors thank the anonymous referee for helpful com-ments. This work has been partially funded by National Academyof Sciences of Ukraine through project 1.4.6/5-261B.

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Polarimetric study of transiting extrasolar planets