Computation of the transmitted and polarized scattered fluxes by the exoplanet HD 189733b in X-rays

1.5 MB·

by  F. Marin

Draft version February 2, 2017
PreprinttypesetusingLATEXstyleemulateapjv.12/16/11
COMPUTATION OF THE TRANSMITTED AND POLARIZED SCATTERED FLUXES
BY THE EXOPLANET HD 189733b IN X-RAYS
Fre´de´ric Marin1,2 and Nicolas Grosso2
1AstronomicalInstituteoftheAcademyofSciences,Boˇcn´ıII1401,CZ-14100Prague,CzechRepublic
2Universit´edeStrasbourg,CNRS,ObservatoireastronomiquedeStrasbourg,UMR7550,F-67000Strasbourg,France
Draft version February 2, 2017
ABSTRACT
Thousands of exoplanets have been detected, but only one exoplanetary transit was potentially observed in X-rays
7 from HD 189733A. What makes the detection of exoplanets so difficult in this band? To answer this question, we run
1 Monte-CarloradiativetransfersimulationstoestimatetheamountofX-rayfluxreprocessedbyHD189733bDespiteits
0 extendedevaporating-atmosphere,wefindthattheX-rayabsorptionradiusofHD189733bat0.7keV,themeanenergy
2 of the photons detected in the 0.25–2 keV energy band by XMM-Newton, is ∼1.01 times the planetary radius for an
atmosphere of atomic Hydrogen and Helium (including ions), and produces a maximum depth of ∼2.1% at ∼±46 min
b
fromthecenteroftheplanetarytransitonthegeometricallythickandopticallythincorona. Wecomputenumerically
e
in the 0.25–2 keV energy band that this maximum depth is only of ∼1.6% at ∼±47 min from the transit center, and
F
little sensitive to the metal abundance assuming that adding metals in the atmosphere would not dramatically change

1 thedensity-temperatureprofile. RegardingadirectdetectionofHD189733binX-rays,wefindthattheamountofflux
reprocessed by the exoplanetary atmosphere varies with the orbital phase, spanning between 3–5 orders of magnitude

] fainter than the flux of the primary star. Additionally, the degree of linear polarization emerging from HD 189733b is
P
<0.003%, with maximums detected near planetary greatest elongations. This implies that both the modulation of the
E
X-ray flux with the orbital phase and the scattered-induced continuum polarization cannot be observed with current
.
h X-ray facilities.
p Keywords: Planetary systems — radiative transfer — stars: individual (HD 189733) — techniques:
- polarimetric — X-rays: stars
o
r
t
s 1. INTRODUCTION Jupiters is expected to be just about detectable given
a
the current limits of current optical, broad-band, stel-
[ Hitherto,themainexoplanetdetectiontechniquerelies
on spectroscopic and photometric transits1. If a planet lar polarimeters (Schmid et al. 2005; Hough et al. 2006;
2 Wiktorowicz 2009). Berdyugina et al. (2008, 2011) re-
passesinfrontofastar,thestarwillbepartiallyeclipsed
v ported to have successfully achieved a B-band polari-
and its flux will be dimmed by the successive coverage
0 metric detection of HD 189733b and found a peak po-
and uncoverage of the stellar surface by the transiting
9 larization of 2×10−4 (i.e. 0.02 %) and a cosine-shaped
planet, resulting in detectable dips in the stellar light
9 distributionofpolarizationovertheorbitalperiod. From
curve. In the optical and infrared bands, where most of
0 thoseresults,Berdyuginaetal.(2008)inferredadditional
the transit detection occurred, those transit depths are
0
constraints on the size of the scattering atmosphere of
. of the order of 1–3%, e.g., 1.6% for HD 209458b (Char-
1 bonneau et al. 2000; Brown et al. 2001), and 2.45% for HD 189733b, and on the orientation of the system with
0 respect to the Earth-primary line of sight. Yet there is
HD 189733b (Lecavelier Des Etangs et al. 2008; Bouchy
7 noreliablemeasurementofthetruescatteredlightpolar-
et al. 2005). If the photospheric disk has a uniform
1 izationfromHD189733bintheB-band,asWiktorowicz
emission (i.e., there is no limb-darkening), these depths
: et al. (2015) and Bott et al. (2016) reported an absence
v are equal to the square of the ratio of the exoplanet
of large amplitude polarization variations. Even if their
i radius (R ) to the stellar radius (R ). From multi-
X p (cid:63) observations are consistent with a plausible polarization
wavelength light curves, it becomes possible to charac-
amplitude from the planet, the low significance of the
r terize the atmosphere and exosphere of giant exoplan-
a polarimetric observations cannot be used as a claim for
ets (e.g., Redfield et al. 2008), but additional measure-
detection. This highlights the difficulty of polarimetric
ments are needed to estimate the mass of the planet,
measurements of exoplanets in the optical band. How-
the orbital parameters, and the inclination angle. These
ever,spectroscopic detectionofexoplanetshasbeenquite
results can be achieved thanks to radial velocity mea-
successful at all wavebands, except at X-rays energies.
surements, orbital brightness modulations, gravitational
ItisunfortunatesinceX-rayphotometry,spectroscopy,
microlensing, direct imaging, astrometry, and polarime-
andpolarimetrycouldhelptoconstrainthecomposition
try. Thelaterisprobablyoneofthemostdelicatemeth-
andmorphologyoftheexoplanet’satmosphere. Poppen-
ods since the anticipated white light polarization degree
(about 10−5, Carciofi & Magalh˜aes 2005; Kostogryz et haegeretal.(2013)reportedforHD189733thedetection
inthe0.25–2keVenergybandofatransitdepthestimate
al. 2014; Kopparla et al. 2016) resulting from scatter-
ranging from 4.0±2.7% to 7.3±2.5% (see in their Ta-
ing of stellar photons onto the atmospheric layers of Hot
ble 6 the limb-brightened model), which might be larger
than the transit depth in optical of 2.391±0.001%, com-
[email protected]
puted for a uniformly emitting disk using the planetary
1 In December 2016, 2695 planets out of a total of 3547 have
beendiscoveredbytransittechniques,seehttp://exoplanet.eu/ radius of Torres, Winn & Holman (2008). These values
2 Marin & Grosso
of transit depths in X-rays were obtained by fitting the
X-ray broad-band light curve using the analytic chro-
mospheric transit of Schlawin et al. (2010) that assumes Y (R*)
a geometrically thin emission. However, the corona of
the Sun (a normal star) is by definition located above 2
HD 189733A
the chromosphere and the solar coronal plasma is ex-
1
tended above the photosphere (see for a review on the
solar corona, e.g., Aschwanden 2004), therefore, a zero 0
thickness is not appropriate to describe the geometry of
the corona of HD 189733A, a moderately active star. -1 HD 189733b
Moreover, thenormalizedphase-foldedX-raylightcurve
-2
of HD 189733A is affected for a phase binning of 0.005
(corresponding to 958 s) by large statistical noise (∼5%;
PinoiTssaobnlen5oiosef Pfroopmpeandhdaeedg-eurpertaawl.c2o0u1n3t)raantedoausttrtorpahnyssit- -2 -1 0 1 2 3 4 5 6 X (R*)
ical (white noise) scatter (∼4%; see bottom right panel
of Fig. 12 in the Appendix of Poppenhaeger et al. 2013)
thathindersanyfitting. Therefore, moresensitiveX-ray
Figure 1. Artist representation of the exoplanet HD 189733b.
observationsofthetransitofHD189733baremandatory
HD 189733b orbits in nearly circular motion around its host-star
to obtain an accurate measurement of the shape of the HD 189733A; its orbital plane is parallel within about 4◦ of the
planetary transit of the corona of HD 189733A and to Earth-primarylineofsight. Notethelimbbrighteningeffectemerg-
ingfromtheopticallythin,stellarcorona.
better constrain its maximum depth.
In any case, the first indirect detection of an extra- observations and numerical modeling.
solar planet in the 0.1 – 10 keV energy range opens new Hence, in this paper, we aim to investigate the
questions: WhatistheexpectedX-rayabsorptionradius HD 189733A planetary system in terms of photome-
and scattered flux from giant exoplanets? Is a direct de- try and continuum polarization. Focusing our study on
tection of giant exoplanets feasible in this energy band? soft X-rays, where the stellar coronal emission peaks,
Is,similarlytotheopticalband,apolarimetricdetection we build in Sect. 2 a template X-ray spectrum for the
within the reach of future instruments? To solve those corona of HD 189733A and a model for the atmosphere
issues,itisnecessarytoestimatethehost-starfluxrepro- of HD 189733b according to the latest observational
cessed by the exoplanet gaseous surface first, and then constraints. We present the Monte-Carlo code that
tocomputetheresultingscattering-inducedpolarization. we used to perform our calculations of radiative trans-
Itisaparticularlyrelevantissuesincewehaveknownfor fer and report the very first estimations of the repro-
decadesthatstellarX-raypolarimetryisaverypowerful cessed flux emerging from the outer atmospheric layers
diagnostic tool (e.g., Manzo 1993). of HD 189733b, along with its resulting X-ray polariza-
In this regards, HD 189733b is one of the best lab- tion, in Sect. 3. In Sect. 4, we discuss our results in the
oratories for giant exoplanet studies. HD 189733 is a context of present and future X-ray facilities, and con-
binary system, situated at a distance of about 19.45 pc clude our paper in Sect. 5.
from the Sun (van Leeuwen 2007). The primary compo-
2. MODELINGTHEHD189733ACORONAANDTHE
nent HD 189733A is a main-sequence star of stellar type
HD189733bATMOSPHERE
K1.5 V, mass 0.846 M (de Kok et al. 2013), radius
(cid:12)
Probingthegasesandplasmasthatformexoplanetat-
0.805 R (van Belle & von Braun 2009; Boyajian et al.
(cid:12)
mospheresisachallengingtask,asextra-solarbodiesare
2015) and rotational period 11.953 days (Henry & Winn
2008). HD189733Aemitsabout1028 erg.s−1 inthe0.25 usually very faint. Yet, constraints on the atmospheric
composition and temperature of the external layers of
–2keVband,correspondingto10−5 timesitsbolometric
hot Jupiter and hot Neptune-like planets can be derived
luminosity, which is about 10 times larger than the ratio
from primary and secondary eclipses for transiting ex-
ofX-raytobolometricluminosityobservedfromtheSun
oplanets (Seager & Sasselov 2000). Numerical models
at its maximum (Poppenhaeger et al. 2013). The sec-
arethusveryusefultoevaluatethedetectioncapabilities
ondarycomponentofspectraltypeM4Vislocatedatan
of our current and forthcoming generation of telescopes,
angularseparationof11(cid:48).(cid:48)4(∼216au)fromHD189733A
together with fitting observed spectra with atmospheric
with an orbital period is 3200 years (Bakos et al. 2006)
models. It is in this scope that we now present a plan-
anditisapproximativelytwoordersofmagnitudefainter
etary system model for HD 189733 derived from obser-
in X-rays than HD 189733A (Poppenhaeger et al. 2013).
vations. A sketch of our model is presented in Fig. 1.
TheexoplanetHD189733bwithmass1.162M (deKok
J We list in Tab. 1 the physical properties of HD 189733A
et al. 2013) and radius 1.26 R , was discovered by tran-
J andHD189733bthatweuseinthiswork. Notethatour
sitobservationsinoptical(Bouchyetal.2005). Orbiting
resultsarelikelytovaryifanewsetofinputparameters
around HD 189733A in 2.219 days (Triaud et al. 2009),
is derived from future observations.
thisHotJupiterissituatedatonly0.031aufromitshost
star. Due to its proximity from Earth and its favorable
2.1. Template X-ray spectrum of the HD 189733A
observing geometrical conditions — its orbital plane is
quiescent corona
parallel within about 4◦ of our line of sight (Triaud et
al. 2009) and its orbit is nearly circular (Berdyugina et The moderately active corona of HD 189733A emits
al. 2008) — HD 189733b stands as the perfect object for X-ray photons from an optically thin plasma in colli-
sional ionization-equilibrium; the X-ray spectrum is a
X-ray Photo-polarimetry of HD 189733b 3
Table 1
PhysicalpropertiesofHD189733AandHD189733b
Property Value Reference
R(cid:63)/R(cid:12) 0.756±0.018 Torres,Winn&Holman(2008)a
M(cid:63)/M(cid:12) 0.806±0.048 Torres,Winn&Holman(2008)b
Rp/R(cid:63) 0.15463±0.00022 Torres,Winn&Holman(2008)b
Rp/RJd 1.138±0.027 Torres,Winn&Holman(2008)b
Mp/MJe 1.144+−00..005576 Torres,Winn&Holman(2008)b
a/au 0.03099+0.00060 Torres,Winn&Holman(2008)b
−0.00063
RRoche/Rp 4.35 Salzetal.(2016)
P/days 2.21857312+0.00000036 Triaudetal.(2009)
−0.00000076
d/pc 19 Bouchyetal.(2005)b,c
i /deg 85.58±0.006 Torres,Winn&Holman(2008)
orbit
b 0.680±0.005 Torres,Winn&Holman(2008)
aBoyajianetal.(2015)obtained0.805±0.016frominterferometric
measurementusingthedistanceof19.45pc(vanLeeuwen2007).
b We use this reference to be consistent with the physical prop-
erties used by Salz et al. (2015) to compute their atmospheric
model; see the header of the file hd189733.dat available at
ftp://cdsarc.u-strasbg.fr/pub/cats/J/A%2BA/586/A75/.
c Works reporting X-ray observations used a distance of 19.3 pc
(Pillitteri et al. 2010, 2011, 2014; Poppenhaeger et al. 2013). van
Leeuwen (2007) reported a distance of 19.45±0.26 pc from the
newreductionoftheHipparcosparallax.
dRJisthenominaljovianequatorialradiusfollowingIAU’srecom-
mandation(RN =7.1492×107 m;Mamajek2015).
e The jovian meJass is computed following IAU’s recommandation
(Mamajek 2015): MJ = (GM)NJ/G with (GM)NJ = 1.2668653×
1023 cm3 s−2 the nominal jovian mass and G = 6.67384 ×
10−11 cm3 kg−1 s−2 theNewtonianconstant.
bremsstrahlung continuum emission plus line emission
from metals. The X-ray spectra of the quiescent corona
obtained with XMM-Newton and Chandra are well de-
scribed by two-isothermal plasma (Pillitteri et al. 2010,
2011, 2014; Poppenhaeger et al. 2013). Each isother-
mal plasma is characterized by an electronic tempera- Figure 2. XMM-Newton/pntemplatespectrumofHD189733A
ture and a corresponding emission measure, defined as usedinthiswork. Theenergyrangeis0.25–2keV.Toppanel: in-
EM =(cid:82) n2dV for a fully ionized plasma where the elec- strumental spectrum simulated from X-ray observations, grouped
e withaminimumof25counts. Thedashed-dottedanddottedver-
tronicdensity,n ,isequaltotheiondensity. Thephoto-
e tical lines are the energies of the median (50th-percentile), lower
electric absorption above 0.2 keV along the line of sight and upper quantiles (25th and 75th-percentiles) of the observed
is negligible since the distance of this star is low. count rates, respectively. The red, blue, and black lines are the
Weusethespectralfittingparametersofthequiescent cool, warm, and total plasma components, respectively. Bottom
panel: unfolded spectrum. The red, blue, and black dots are the
corona obtained on April 17, 2007 with XMM-Newton
cool, warm, and total plasma components used for our modeling
by Pillitteri et al. (2011) for a distance of d = 19.3 pc: oftheX-raycorona,respectively.
a cool plasma component with kT = 0.24±0.01 keV
1
(corresponding to T = 2.8 ± 0.1 MK) and EM = of 2.9×10−13 erg.cm−2.s−1 (corresponding to an X-ray
e,1 1
4.7+0.4×1050 cm−3, and warm plasma component with luminosity of 1.3×1028 erg.s−1 for d=19.3 pc) with a
−0.3
kT =0.71+0.04keV(correspondingtoT =8.2+0.5MK) count rate of 0.188 pn count s−1.
2 −0.03 e,2 −0.3
The count rate distribution versus energy, i.e., the
and EM =(2.8±0.3)×1050 cm−3.
2 simulated instrumental spectrum, is shown in the top
We model this quiescent coronal emission with XSPEC
panel of Fig. 2. Among the total counts detected in the
(version 12.9.0) using two vapec models (Smith et al.
0.25–2.0 keV energy range, 25% have energy lower than
2001) with element abundances of Anders & Grevesse
0.51 keV (see lower dotted vertical line), 50% have en-
(1989) and Fe=0.57, O=0.51, and Ne=0.3 (Pillitteri et
ergy lower than 0.69 keV (see lower dashed-dotted verti-
al. 2011). We simulate an XMM-Newton observation of
calline),75%haveenergylowerthan0.86keV(seeupper
this coronal model with the European Photon Imaging
dotted vertical line). The bottom panel of Fig. 2 is the
Camera (EPIC) pn instrument (Stru¨der et al. 2001) and
so-calledunfoldedspectrumwithanenergysampling3 of
the medium filter2 (Stru¨der et al. 2001) which deter-
∼5 eV.
mines in the 0.25–2.0 keV energy range an X-ray flux
Wewillusethistemplatespectrumtocomputenumer-
icallythedepthofthetransitobservedinthe0.25–2keV
2 We use the most recent canned response matrix for pn
(epn e2 ff20 sdY9 v14.0.rmf) and the ancillary response file for
a 40(cid:48)(cid:48)-radius extraction region centered on this on-axis source, 3 Ourenergygridisdefinedbythenominalrangesofenergyfor
obtained from the XMM-Newton Serendipitous Source Catalogue the detector channels that are listed in the EBOUNDS extension of
3XMM-DR4(Watsonetal.2009). theresponsematrixfile.
4 Marin & Grosso
energy range with XMM-Newton (Sect. 2.4).
We will perform in Sect. 2.5 Monte-Carlo radia-
tive transfer simulations at the mean energy of the
photons detected in the 0.25-2 keV energy band,
E = 0.70 keV, where the X-ray emission is mainly
mean
bremsstrahlung continuum emission. The coronal pho-
ton flux at this energy is 11.8×10−5 photons cm−2 s−1,
with F = 9.2×10−5 and F = 2.6×10−5 pho-
ph,1 ph,2
tons cm−2 s−1, the photon flux from the cool and warm
plasma, respectively (bottom panel of Fig. 2). We will
assesstheimpactonthetransmittedfluxofX-rayobser-
vation in a broad energy-band in Sect. 2.4.3. Figure 3. Coronalelectronicdensityversustheheightabovethe
corona. Theredandbluelinesareforthecoolandwarmcoronal
2.2. Model of the HD 189733 quiescent corona plasma,respectively.
The X-ray emitting plasma of the Sun corona is visi-
blebothinactiveregionswhereitisconfinedinmagnetic
loopsandinthemosthomogeneouspartsofthequietSun
coronawherethereisanydetectablefeaturesofmagnetic
loops;bothcoronalregionsextendingwellabovethepho-
tosphere. For HD 189733A only the large-scale mag-
netic topology is accessible from Zeeman-Doppler imag-
ing, Moutou et al. (2007) showed that “it is significantly
more complex than that of the Sun ; it involves in partic-
ular a significant toroidal component and contributions
from magnetic multipoles of order up to 5”. However,
it is beyond the scope of this article to use this mag-
Figure 4. Ratioofwarm-to-coolplasmaemissivity(withidentical
netictopologyasascaffoldtobuildamodeloftheX-ray
normalization)versusenergy. Thecrossisthevaluecorresponding
emitting plasma of the HD 189733A corona (see, e.g., toEmean=0.7keV(verticaldashedline).
Jardine et al. 2006). We focus on the quiescent and dif-
fuse corona, which is more representative of the average
plasmadistributiononthestellarsurface. Therefore,fol-
lowing our template spectrum, we model the quiescent
corona as two isothermal-plasma in hydrostatic equilib-
rium (see Sect. 3.3 of Aschwanden 2004).
In the isothermal approximation, the pressure scale
height is defined as:
2kT
λ (T )= e , (1)
p e µm g
H (cid:63)
where k is the Boltzmann’s constant, µ ≈ 1.27 is
the mean particle weight, mH is the mass of the hy- Figure 5. NormalizedphotonemissionatEmean=0.7keVversus
drogen atom, and g = GM /R2 is the stellar grav- the height above the corona. The red and blue are for the cool
(cid:63) (cid:63) (cid:63) and warm coronal plasma, respectively. The solid line is for the
itational field. For HD 189733, we obtain: λp(Te) = cool+warmcoronalplasmaandisnormalizedatthepeak.
3.4×109×(T /MK) cm. Therefore, λ (T ) = 0.18R
e p e,1 (cid:63)
and λ (T ) = 0.53R . Since the pressure is obtained
p e,2 (cid:63)
by p = 2n kT , the electronic density scale height is
e e
identical to the pressure scale height in the isothermal
approximation. Neglecting the variation of the gravita-
tionalfieldwiththeheightabovethephotosphere,h,the
electronic density, n , follows an exponential profile:
e
(cid:26) (cid:27)
−h
n (h)=n (0)exp . (2)
e e λ (T )
p e
We introduce, f , the fraction of the corona outer
obs
areathatisnoteclipsedbythestellardiskforanobserver
Figure 6. Coronal emission profile integrated along the line of
on Earth (Eq. 5 of Schmitt 1990):
sight versus the distance from the stellar center. The vertical
dashedlineisthesolarlimb. Alltheprofileshavebeennormalized
(cid:112)
1+ x(2+x) atthesolarlimb(verticaldashedline). Theredandbluelinesare
f (x)=0.5× , (3) themostcompactandextendedcoronalemissionprofilesoccuring
obs 1+x at 0.57 and 2.00 keV, respectively. The green line is the coronal
emissionprofileat0.7keVthatweuseinoursimulation.
with x = h/R . Obviously, only half of the coronal
(cid:63)
area is visible when h=0 (f (0)=0.5), and the coro- The observed emission measure must be computed on
obs
nal area is fully visible when h=∞ (f (∞)=1.0). thecoronalvolumethatisnoteclipsedbythestellardisk:
obs
X-ray Photo-polarimetry of HD 189733b 5
troscopyofHD189733A(forareviewofexoplanetaryat-
(cid:90) ∞ dEM(h) (cid:18) h (cid:19) mospheresseeMadhusudhanetal.2014). Assumingthat
EM = f dh, (4)
obs dh obs R the transmission spectrum of HD 189733A from opti-
0 (cid:63)
caltonear-infraredisdominatedbyRayleighscattering,
with the differential emission measure LecavelierDesEtangsetal.(2008)deriveanatmospheric
temperature of 1340±150 K at 0.1564 R . They pre-
dEM(h) (cid:63)
dh =ne(h)24π(R(cid:63)+h)2. (5) freardsiucastbteertiwnegebny∼co1n0d−e2nsaantdes∼of1E0−ns1taµtmit)er(aMthgeSriOth3,anwibthy
FromEq.1–5andthevaluesoftheobservedelectronic molecular hydrogen. Assuming solar abundances, they
temperatures and emission measures, we derive numer- derive at 0.1564 R(cid:63) for particle sizes of about 10−2 –
ically the electronic density at the base of the corona: 10−1 µm, apressureof2×10−3–2×10−6 barandapar-
n (0) = 1.9×109 cm−3 and n (0) = 6.7×108 cm−3; ticle density of 1.1×1016–1.1×1013 cm−3. The revised
e,1 e,2
Fig. 3 shows the corresponding profiles of the coronal transmission spectrum across the entire visible and in-
electronic density. frared range is well dominated by Rayleigh scattering,
We obtain the profile of the photon number emitted which is interpreted as the signature of a haze of con-
by the corona for an energy, E, and a height, h, above densed grains extending over more than two orders of
the corona: magnitude in pressure, with a possible gradient of grain
sizes in the atmosphere due to sedimentation processes
(Pont et al. 2013).
dN(E ,h) 10−14
i =∆E × ×vapec(E ,kT )× ThegasoftheHD189733batmosphereisphotoionized
dh i 4πd2 i 1 by the UV emission of HD 189733A (e.g., Sanz-Forcada
dEM (h) et al. 2011) and cools by radiation from collisionally ex-
( 1 cited atomic hydrogen, which leads similarly to Hii re-
dh
gions to a temperature of about 10,000 K. This high
vapec(E ,kT ) dEM (h)
+ i 2 × 2 ), (6) temperature produces a (slow) evaporative-wind (Yelle
vapec(E ,kT ) dh
i 1 2004; Murray-Clay et al. 2009).
where∆E isthewidthofthespectralbinoftheancil- Huitson et al. (2012) detected an upper atmospheric
i
laryresponsefileatenergyE ,andvapec(E ,kT )isthe heating of HD 189733b from HST sodium observations,
i i j
i.e., evidence of a thermosphere. Salz et al. (2015) ob-
X-ray photon flux (in unit of photons cm−2 s−1 keV−1)
tained a 1D, spherically symmetric hydrodynamic simu-
for a plasma temperature of kT and a normalization of
j lations of the escaping atmosphere of HD 189733b, pre-
1 (i.e., vapec parameter norm≡10−14EM/4πd2 =1).
dicting gas velocity of ∼9 km s−1 at the Roche’s radius,
Theratioofwarm-to-coolplasmaemissivityinEq.6is
R = 4.35 R (Fig. 7), by coupling a detailed pho-
plottedversustheenergyinFig.4,usingtheenergysam- Roche p
toionization and plasma simulation code with a general
pling of the unfolded spectrum. Peaks and dips are due
MHD code, and assuming only atomic Hydrogen an He-
toemissionlinesfromthewarmandcoolplasma,respec-
liumintheatmosphere. Sincetheresultingtemperature-
tively. In the 0.25-2 keV energy band, the minimum and
pressureprofile(seebelowFig.12)isconsistentwiththe
maximum values are 0.06 and 262 achieved at 0.57 and
rise of temperature with the altitude observed by Huit-
2.00 keV, respectively. For a spectral energy resolution
sonetal.(2012),weadopttheSalzetal.(2015)’sprofiles
of∆E =5eV,vapec(E,kT )/vapec(E,kT )isequalto
i 2 1 todescribeforr ≥R thethermosphereofHD189733b.
0.45 at E = E = 0.7 keV. Figure 5 shows the re- p
mean Line et al. (2014) showed from secondary eclipse spec-
sulting normalized profile of the photons emitted by the
troscopy that for R ≥ r ≥ r , corresponding to
corona at E obtained using Eq. 6. Figure 6 shows p in
mean an atmospheric pressure increasing from p(R ) ∼10
thecorrespondingemissionintensityintegratedalongthe p
to p(r ) ∼ 1000 dyn cm−2, the lower atmosphere of
line of sight versus R, the distance from the stellar cen- in
HD189733bisisothermal,withanequilibriumtempera-
ter. The emission intensity is maximum at the stellar
ture of T ∼ 1200 K. Therefore, we use the hydrostatic
limb and decreases from it with the distance, more or eq
equilibrium to extent the pressure and density profiles
less slowly depending of the small or large contribution,
into this lower atmospheric layer. The pressure scale
respectively, of the warm plasma compared to the cool
height of this neutral atmospheric layer is:
plasma to the coronal emission. Therefore, the size of
thecoronalemissionvarieswithenergy: thesmallestand
kTeq
largest sizes are observed at 0.57 and 2.00 keV, respec- λatm(T )= (7)
p eq µ(R )m g
tively. p H p
We define the quiescent corona has layers of constant,
where g = GM /R2 is the planetary gravitational
isotropic, and optically thin emission, above an opti- p p p
field, and T = 1190 K and µ(R ) = 1.28015 (Salz et
cally thick sphere of radius R (Fig. 5). Since ordinary eq p
(cid:63) al. 2016), which leads to λatm(T ) = 0.0043 R (i.e.,
bremsstrahlung emission from plasma shows no intrin- p eq p
sic polarization due to the random motions of electrons 350 km). The pressure and density follow the same ex-
(Dolan 1967), we consider only unpolarized X-ray pho- ponential profile:
tons from the stellar corona.
p(r)=p(R )×f(r) (8)
2.3. Model of the HD 189733b atmosphere p
n (r)=n (R )×f(r) (9)
The physical properties of the upper atmosphere of H H p
HD 189733b can be constrained with transmission spec- where
6 Marin & Grosso
Figure 8. Definitionofthegeometricalparametersoftheplane-
taryatmosphere.
andincoherent(Compton)scatteringbyboundelectrons
in atoms/molecules (Hubbell et al. 1975). For compar-
ison, we compute these cross-section components for an
atmosphere of neutral Hydrogen and Helium with so-
lar abundances (Asplund et al. 2009)4 with the XCOM
(Berger & Hubbell 1987)5. At 1 keV the total photon
cross-section is 46.5 barn/H atom, composed at 98.1%
bythephotoelectricabsorption(45.6barn/Hatom),this
proportion decreases with higher energy. The scatter-
ing cross-section at 1 keV is dominated by the coher-
ent (Rayleigh) process with 0.8 barn/H atom compared
to 0.1 barn/H atom for the incoherent (Compton) scat-
tering. Therefore, we will neglect the scattering cross-
sections in our analytic computation at 0.7 keV.
The simplified solution of the absorption radius pro-
posed byFortney (2005)is only valid for ageometrically
thin atmosphere and can not be used a priori here due
to the extended evaporative-wind. From the geometri-
cal parameters introduced in Fig. 8, the optical depth at
energy E through the planetary atmosphere along the
line-of-sight is computed versus the distance from the
planetary center, r, as follows:
τX(r,E,Z) =(cid:90) Lσ ((cid:112)r2+l2,E,Z)n ((cid:112)r2+l2)dl
2 X H
Figure 7. AtmosphericmodelofHD189733bofSalzetal.(2016) 0
(11)
usedinourX-raysimulation. Fromtoptobottompanels,theto-
(cid:112)
talHydrogennumberdensity,temperature,pressure,velocity,and with L = R2 −r2, n the total Hydrogen num-
Hydrogen(green)andHelium(red)ionicfractionsoftheplanetary Roche H
ber density, and σ (r,E,Z) the photoelectric cross-
atmospherearerepresentedversusthedistancefromtheplanetary X
center,rangingfromtheplanetaryradiustotheRoche’sradius. section per Hydrogen atom, where Z is the metallicity.
The photoelectric cross-section at the distance r from
the planetary center and energy E is computed as:
(cid:26) (cid:27)
r−R
f(r)=exp − p (10)
λatm(T )
p eq σ (r,E,Z)=f (r)σ (E)
X H X,H
when neglecting the variation of the planetary gravi- (cid:0) (cid:1)
+A f (r)σ (E)+f (r)σ (E)
tional field with the depth below the planetary radius, He He X,He He+ X,He+
(cid:88)
and p(Rp) = 12.7 dyn cm−2 and nH(Rp) = 7.7 × +Z AkσX,k(E) (12)
1013 cm−3 (Salz et al. 2016). Therefore, r = 0.981R
in p k∈/[H,He]
and n (r )=6.0×1015 cm−3.
H in
with f , f , and f the ionic fractions (see bot-
H He He+
tom panel of Fig. 7); A the abundance in number of
2.4. Analytic computation of the transmitted flux i
the neutral element i relative to the hydrogen (Asplund
2.4.1. X-ray absorption radius
The planetary absorption radius during transit is de- 4 ThelargestsolarabundancesrelativetohydrogeninAsplund
fined as the distance in the plane of the sky from the et al. (2009) are: AH =1, AHe =8.51×10−2, AO =4.9×10−4,
planetary center where the optical depth through the AC=2.69×10−4,ANe=8.51×10−5,AN=6.76×10−5,AMg=
atmosphere along the line-of-sight, τ, is equal to unity. 3.98×10−5,ASi=3.24×10−5,andAFe=3.16×10−5.
5 The XCOM: Photon Cross Section Database (version
The total photon extinction cross-section in X-rays is
1.5; Berger et al. 2010) is available at http://physics.nist.gov-
thesumofphotoelectricabsorptionandscatteringcross- /PhysRefData/Xcom/Text/intro.html for the energy range of
sections. The latter is the sum of coherent (Rayleigh) 1keV–100GeV.
X-ray Photo-polarimetry of HD 189733b 7
Figure 9. Photoncross-sectionofphotoelectricabsorptioninX-
raysversusmetallicity. Thegreenandbluesolid-linesaretheX-ray
cross-sectionsofatomicandmolecularHydrogen,respectively. The
solid and dashed red-lines are the X-ray cross-sections of neutral Figure 11. Cumulativecontributionoftheatmosphereateleva-
and one-times ionized Helium, respectively. The black solid lines tionzabovetheplanetaryradiustotheopticaldepthattheX-ray
are the X-ray cross-section for neutral atomic Hydrogen and He- absorptionradiusat0.7keV(verticaldashedline). Themaximum
lium with no metals, solar metallicity and ten times solar metal- elevationvalueonthex-axiscorrespondstotheatmosphericlayer
licity (no metallic ions). The main ionization-edges of the inner contributingfor99%totheopticaldepth.
electronic-shells are labeled with element names and shell name
exceptforK.
Figure 10. Optical depth through the planetary atmosphere
along the line-of-sight at Emean = 0.7 keV versus the distance
to the planet center. The planetary atmosphere is composed of
atomic H, He and He+ with no metals, solar metallicity and ten
timessolarmetallicity(nometallicions). Theverticaldottedlines
indicatethecorrespondingX-rayabsorptionradius.
et al. 2009) and σ the photoelectric cross-section of
X,i
the neutral or ionized element i relative to the hydrogen
(Verner & Yakovlev 1995). Since the dust grains in the
planetaryatmosphereareopticallythintoX-raysdueto
their small sizes ((cid:28) 1 µm), there is no self-blanketing Figure 12. AtmosphericstructureofHD189733b(adaptedfrom
effect, i.e., no decrease of the X-ray cross-section due to Salz et al. 2016). The top and bottom panels are the pressure-
some element in the solid phase (Bethell et al. 2011). temperatureandthedensity-temperatureprofile,respectively. The
crossesaretheHSTSodiumobservationsofHuitsonetal.(2012).
FollowingSalzetal.(2016),wewillonlyconsiderinour
The vertical labels give the distance from the planetary center in
simulation atomic Hydrogen and Helium photoelectric planetary radius. The thick grey region with percent-labeled as-
cross-sections (Fig. 9). Since σ = 2.85σ (Wilms terisks indicates the contribution of the atmospheric layers to the
etal.2000)andA =0.5,thecXo,nHt2ributionoXft,Hhemolec- absorptionattheX-rayabsorptionradiusat0.7keV(Fig.11).
H2
ular Hydrogen in the low atmosphere to an increase of
the photoelectric cross-section is small. However, metal at R (0.7 keV) is produced by the upper (z=634–4243)
X
abundanceshasastrongimpactonthephotoelectricab- atmospheric layers that intersect the line of sight (e.g.,
sorption. Since ions (e.g., He+) have lower photoelec- the dashed circle in Fig. 8). In this absorbing atmo-
tric cross-section than neutral elements (fully ionized el- sphericshellwithawidthof3,609kmthetotalHydrogen
ementshaveobviouslyanullphotoelectriccross-section), number density and temperature ranges are 1.1×1013–
the ionized escaping atmosphere is more transparent to 1.2×1011 cm−3 and1870–7004K,respectively(Fig.12).
X-rays than the lower atmosphere. The physical properties of the atmospheric layers that
Figure 10 shows the variation of the optical depth contribute to the bulk of the X-ray absorption are con-
(Eq. 12) at 0.7 keV versus the distance to the planetary strained by the optical measurements of Huitson et al.
center. We find that the planetary absorption radius (2012).
at 0.7 keV, R (0.7 keV) is 1.008 R (corresponding to Assumingthataddingmetalsintheatmospherewould
X p
0.156 R ), i.e., located at z =634 km above the plane- not change dramatically the density-temperature profile
(cid:63)
tary radius. Figure 11 shows that 99% of the absorption (see,however,thediscussioninSalzetal.2016),weesti-
8 Marin & Grosso
mate that R (0.7 keV) increases from 1.008 to 1.059 R
X p
(orequivalentlyfrom0.156to0.164R )whentheneutral
(cid:63)
metals increase from zero to ten times the solar metal-
licity (Fig. 10).
2.4.2. Transmitted flux at 0.7 keV during the planetary
transit on the stellar corona
For comparison purpose, we compute the phases of
the first, second, third and fourth contacts of the pri-
mary eclipse in the optical6: φ = −φ = −0.0169 and
1 4
φ = −φ = −0.0091 (vertical dotted lines in Fig. 13).
2 3
The corresponding photospheric transit of a uniformly
emitting disk (e.g., Mandel & Agol 2002) has a maxi-
mum transit depth of (R /R )2 =2.39% (dotted line in
p (cid:63)
Fig. 13). Figure 13. LightcurvesoftheplanetarytransitsofHD189733b.
The chromospheric transit was computed analytically Theblacksolidlineisthecoronaltransitat0.7keVcomputedfor
by Schlawin et al. (2010), assuming an optically thin RX(0.7keV)=1.008Rp(correspondingtoaplanetaryatmosphere
of H, He, and He+). For comparison, the vertical dotted lines
and geometrically thin shell of emission (zero thick- indicatedthefirst,second,third,andfourthcontactsofthephoto-
ness). ItischaracterizedbyaW-profilewithamaximum spherictransitintheoptical. Thedottedanddashedlinesarethe
depthofapproximatelyδchrom =0.5(R /R )3/2,i.e., photospherictransit(uniformlyemittingdisk,e.g.,Mandel&Agol
max,approx p (cid:63) 2002)andthechromospherictransit(Schlawinetal.2010),respec-
0.5(R /R )−1/2 times deeper than the maximum tran- tively. The cross data are the coronal transit signature obtained
p (cid:63)
sit depth of a uniformly emitting disk. These two min- byoursimulationat0.7keV(seeFig.17).
imums occur slightly before and after the second and use light curves in a broad energy band to increase the
third contacts, respectively, with a depth here of exactly signal-to-noise ratio. The extension and depth of the
δmcharxom =3.10% (dashed line in Fig. 13). This character- coronal transit is modified by the variations with energy
isticshapeisduetothestronglimbbrighteningthatoc- of the coronal emission profile, the photon extinction
curssincethezero-thicknesschromospherehasitslargest cross-section (determining the absorption radius), and
column density at the disk’s edge. the observed stellar flux. It is therefore important to as-
We compute numerically the normalized light curve sess in a broad energy band the effective transit. This
of the planetary transit on the stellar corona from the effective transit is the weighted average of the monoen-
coronalemissionprofileintegratedalongthelineofsight, ergetic transit on the broad energy-band, using the ob-
I (Fig. 6), and the optical depth through the planetary served X-ray count rate versus energy (i.e., the observed
atmosphere, τ (Fig. 10), using the formula: X-ray spectrum; top panel of Fig. 2) as weight.
For a given metallicity, we compute at each grid en-
(cid:82)+∞(cid:82)+∞I(√X2+Y2)e−τ(cid:16)r(XR,pY,t)(cid:17)dXdY ethrgeycotrhreescpoornondainlgplriogfihltecaunrvdetohfethaebscoorrpotnioanltrraadnisuits., Tanhde
I (t)= −∞ −∞ √
transit (cid:82)+∞(cid:82)+∞I( X2+Y2)dXdY toppanelofFig.14showstheabsorptionradiusdecreas-
−∞ −∞ ing with the energy, with local jump associated to the
(13)
ionization edges of metals. When there is no metals, the
where X and Y are the cartesian coordinates in the
X-ray absorption radius becomes lower than the plane-
plane of the sky expressed in stellar radius (Fig. 1),
tary radius –the lower boundary of Salz et al. (2016)’s
(cid:112)
and r(X,Y,t) = (X−Xp(t))2+(Y −Yp(t))2 is the simulation– at energies larger than ∼1.1 keV. Therefore,
distance from the position of the planetary center the optical depth used to compute the absorption factor
at time t, Xp(t) = acos(i)sin(2πt/P) and Yp(t) = inEq.13hasalsotobeestimatedinsidetheloweratmo-
−acos(i)cos(2πt/P). We obtain a similar W-profile sphere, whichrequiresanextensionofthedensitymodel
since the coronal emission is also optically thin (solid as we did in Sect. 2.3. This leads for comparison at r
in
line in Fig. 13). But, the coronal transit is less deep and to τ(Z =0,E =2.0 keV)∼12.
more extended than the chromospheric transit since the The mid and bottom panels of Fig. 14 show the max-
corona is extended above the photosphere (green line in imum depth of the coronal transit and its temporal
Fig. 6). We find δmax = 2.06% at φ = ±0.0143, cor- shift from the transit center, respectively, vs. energy.
responding to a temporal shift of ±45.8 min from the Bothvaluesaremainlycorrelatedwiththewarm-to-cool
transitcenter. Atthetransitcenterthedepthisreduced plasma emissivity ratio (Fig. 4) that controls the exten-
to only 0.77%. sion of the corona.
Then, we compute the weighted average light curve of
2.4.3. Impact of the broad energy-band on the observed the coronal transit in the 0.25–2 keV energy band for
transmitted flux XMM-Newton/pnusingthetemplateinstrumentalspec-
DuetothelimitedsensitivityofthecurrentX-rayfacil- trumofHD189733Aasweight(Fig.15). Weobtainmax-
ities, any tentative detection of planetary transits must imum transit depths of 1.62%, 1.67%, and 1.82%, for no
metals,andoneandtentimessolarmetallicity(nometal-
6 The formulae for the phases of the first and second contacts lic ions), respectively, at φ = ±0.0148 corresponding to
arerespectively: a temporal shift of ±47.4 min from the transit center; at
φ1=−arcsin((cid:112)(R(cid:63)+Rp)2−a2cosi2/asini)/2π and the transit center the depth is reduced to 0.61%, 0.64%,
φ2=−arcsin((cid:112)(R(cid:63)−Rp)2−a2cosi2/asini)/2π. and 0.72%, respectively.
X-ray Photo-polarimetry of HD 189733b 9
2.5. Monte Carlo radiative transfer
To simulate the transit of HD 189733b in front of its
host star, we used the radiative transfer code stokes.
Originally written for polarimetric investigations of ac-
tive galactic nuclei, stokes has been extended over the
last years to cover a much larger panel of astrophysi-
cal sources (a detailed list of examples can be found in
Marin & Goosmann 2014). The code and its character-
istics are extensively described in Goosmann & Gaskell
(2007),Marinetal.(2012)andMarinetal.(2015)forthe
optical/UV part, and in Marin & Dovˇciak (2015) for the
X-ray band, so we will simply summarize the principal
aspects of stokes relevant for our computations.
The code can handle a large panel of emitting, scat-
tering and/or absorbing geometries in a fully three-
dimensional environment. Multiple scattering enables
the code to radiatively couple all the different regions of
agivenmodel,andawebofvirtualdetectorsarrangedin
a spherical geometry is used to record the photon wave-
length,intensity,andpolarizationstate,accordingtothe
Stokes formalism (S(cid:126) = (I,Q,U,V)T). About 1011 pho-
tonshavebeensampledforouranalysis. Thetotalinten-
(cid:112)
sity(flux),polarizationdegreeP (= Q2+U2+V2/I),
and polarization angle ψ (= 0.5tan−1(U/Q)) are com-
putedbyaveragingallthedetectedphotons,ateachline-
of-sightinpolarandazimuthaldirections. Mie,Rayleigh,
Thomson, Compton and inverse Compton scattering are
included from the near-infrared to hard X-ray bands.
For planetary atmospheres in the X-ray domain,
RayleighandComptonscatteringofphotonsontobound
electrons will be the major mechanism to alter the net
polarization,andavisualrepresentationoftheintegrated
and differential scattering cross sections produced by
Figure 14. Characteristics of the coronal transit versus X-ray stokes is presented in Fig. 16. Note that for molec-
energy. Top panel: X-rayabsorptionradiusgiveninplanetaryra- ular hydrogen the incoherent scattering cross-section is
dius. Middlepanel: maximumdepthofthecoronaltransitinper- two-times larger than for atomic hydrogen (Sunyaev et
cent. Bottom panel: temporal shift of the maximum depth from
al. 1999).
the transit center. Black, green, and red lines are for a planetary
atmosphere composed of atomic H, He, He+ with no metals, so- Photo-ionizationandrecombinationeffects,whichalso
lar metallicity and ten times solar metallicity (no metallic ions), influence the resulting polarization by absorption and
respectively. unpolarized re-emission processes, are based on quan-
tum mechanics computations from Lee (1994) and Lee
etal.(1994). Weusefluorescentyieldsandenergyofthe
subsequently re-emitted Auger photons from the Opac-
ity Project (Cunto & Mendoza 1992), and X-ray pho-
toelectric cross-sections and inner-shell edge energies of
Verner & Yakovlev (1995). Finally, we opt for elemen-
talabundancesfromAsplundetal.(2009)andtentimes
solar metallicity to allow direct comparison with Pop-
penhaeger et al. (2013).
Since the X-ray scattering by dust grains (Mathis &
Lee 1991) is not yet implemented in stokes, we con-
sider only gaseous components. We do not consider ad-
ditional soft X-ray emission from the exoplanet itself in-
duced by the stellar wind charge exchange mechanism
Figure 15. Planetary transit in the 0.25–2.0 keV energy band. since it was estimated in the case of the close-in exo-
For comparison purpose, the vertical dashed lines are the eclipse planetHD209458btobeaboutfiveordersofmagnitude
contactintheopticalandthegreylineistheplanetarytransitat
fainter than the emission from the host star (Kislyakova
0.7keVfornometals.
et al. 2015).
Therefore, the maximum transit depth in the 0.25– We divide our two-zone model with constant-density
2 keV energy band is little sensitive to the metal abun- shellssamplingany0.1-dexvariationofthedensityfrom
dance. Moreover,thetransitlightcurveat0.7keV(grey adensityof7.697×1013cm−3correspondingtoapressure
line in Fig. 15) is a correct approximation of the transit of 12.739 bar, a temperature of 1190.2 K, and a 0.7 keV
light curve in the 0.25–2 keV energy band.
10 Marin & Grosso
sit, the 0.7 keV flux decreases by 2.00±0.15 % between
Free electron the first and second contacts, as the exoplanet starts to
Fe
10 cover the X-ray corona, then rises to 1.07±0.15 % at or-
n]
ar bital phase 0. It then decreases back to 2.02±0.15 %
Z [b O C between the third and fourth contacts before resuming
n * 1 He to the non-eclipsed flux, showing no more variation dur-
ctio H ing the remainder of the orbit of HD 189733b. A close
oss se 0.1 uclparriteyprpesuernptoasteiso.nWofethfiendtraanpseitrfiesctshaogwreneminenFtigb.e1tw3efeonr
cr
d numericalandsimulationresultswithintheMonte-Carlo
e
at statistical error of our modeling.
ntegr 0.01 Using the stokes code, we isolated photons that are
I H - Rayleigh scat. re-emitted/scattered from HD 189733b, and plotted the
resultinglightcurve(normalizedwithrespecttotheuni-
0.0010.1 1 10 100 tary flux of HD 189733A) in Fig 17 (second panel from
Energy [keV]
thetop). Toincreasethestatisticsofthephoton-starved
Forward scattering Backward scattering results, the phase resolution of the re-emitted/scattered
barn/sr] 0.1 Rayleigh rtvoaaditlihaaettisotsnroofhtnagXsp-bhraeoeytnoeednleeeccrtgrreiieacss,aebdthscoeorpmXt-piroaanryepdflroutcoxesFfsrieogsm.t1ht3ah.teDperuxee--
V [ Free electron oplanet is found to be three to five orders of magni-
e
k
1 O tude lower than the photon flux from the main star.
oss section at 0.01 C HHeFe t1Fah0treo−0m1p.87ltaehknreeegVtH.c,mDTt−h1h2e8e9sfl7fl−u3u1x3x,AdorefeflppuHerxonDcdoefi1sn2s8ge.99do7×n3o3n1tbt0ho−eis1to3harebbeioratguatmtlcmpo1hs0−pa−2hs1ees6r−oi–1cf
al cr layers of HD 189733b strongly depends on the angle
nti0.001 between the source, the exoplanet atmosphere and the
e
er observer; this effect is due to the differential scatter-
Diff ing phase function of incoherent scattering, as presented
in Fig. 16 (bottom). In the case of Rayleigh scatter-
0 45 90 135 180
Scattering angle [°] ing, forward and backward reprocessing prevails while,
for incoherent scattering onto bound electrons, forward
Figure 16. Elemental photon scattering cross-section versus en-
and perpendicular scattering are less probable. How-
ergy. Top: integrated cross-section for Compton scattering onto
a variety of elements chosen accordingly to Asplund et al. (2009) ever,backscatteringisverystrongandwhenthegaseous
for their high abundance. Solid black line: coherent (Compton) planetpassesbeyondtheplane-of-the-sky,itsreprocessed
scatteringonfreeelectrons(Klein-Nishinacross-section). Incoher- fluxrisesuptoanormalizedfluxof∼4.7×10−4. Then,
ent (Compton) scattering on bound electrons: long-dashed violet
HD189733bdisappearsbehinditsstarandfluxdropsto
lineofH,short-dashedbluelineofHe,dot-short-dashedgreenline
ofC,double-dot-dashedorangelineofO,dot-long-dashedredline zero.
ofFe. Theblackdottedlineshowscoherent(Rayleigh)scattering
ontoanhydrogenatomforcomparison. Bottom: Emean=0.7keV
differential cross section versus scattering angle for the same free 3.2. Polarization
andboundelectrons.
The middle panel of Fig 17 is a visualization of the
optical depth of ∼4.608.
linear continuum polarization degree P, ranging from 0
3. PHOTO-POLARIMETRICRESULTS (unpolarized) to 100 % (fully polarized). It is the polar-
izationresultingfromscatteredlightonly,i.e. notdiluted
We present the outcomes of our simulations in Fig 17.
by the unpolarized stellar flux. The shape of the polar-
Results are plotted with respect to the orbital phase of
ization curve is very similar to the results obtained by
HD 189733b, also converted into transit hours. Each
Stam et al. (2004), who evaluated the spectra of visible
data-point is associated with its 1σ Monte Carlo error
to near-infrared starlight reflected by Jupiter-like extra-
bars resulting from the simulation. The Monte Carlo
solarplanets. AsStametal.(2004),wefindpolarization
statistical error is estimated using the standard devia-
(cid:112) √ maximums close to orbital phases ±0.25, i.e., near plan-
tion σ that is roughly proportional to Var(X)/ N for
etary greatest elongations, with P = 64.9±2.9 % which
large N, with N the number of photons sampled in the
is slightly larger than in infrared. Since the planet’s
direction of the observer and Var(X) the variance of the orbital plane is inclined by 4◦, the polarization degree
estimate.
does not drop to zero at zero orbital phase as a small
fraction of photons scatters onto the top of the atmo-
3.1. Photometry
spheric layers towards the Earth. Hence, we are sure
The top panel of Fig 17 shows the light curve of that our model produces correct results and extends the
HD 189733A at 0.7 keV, normalized to unity. The exo- near-infrared/optical investigation of Stam et al. (2004)
planet orbits in front of HD 189733A from orbital phase to X-ray energies.
-0.25toorbitalphase+0.25,withaneclipseofthestellar The fourth panel of Fig 17 also shows the star+planet
corona(i.e.,aplanetarytransit)occuringbetweenorbital linear polarization, now taking into account polariza-
phases-0.0169and+0.0169. Duringthisplanetarytran- tion dilution by the unpolarized stellar photons, i.e.,