A&A 481, L83-L86 (2008)
DOI: 10.1051/0004-6361:200809388
LETTER TO THE EDITOR
A. Lecavelier des Etangs^{1,2} - F. Pont^{3} - A. Vidal-Madjar^{1,2} - D. Sing^{1,2}
1 -
CNRS, UMR 7095,
Institut d'Astrophysique de Paris,
98bis boulevard Arago, 75014 Paris, France
2 -
UPMC Univ. Paris 6, UMR 7095,
Institut d'Astrophysique de Paris,
98bis boulevard Arago, 75014 Paris, France
3 -
Physikalisches Institut, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
Received 11 January 2008 / Accepted 18 February 2008
Abstract
The transit spectrum of the exoplanet HD 189733b has recently been obtained between 0.55 and 1.05 m.
Here we present an analysis of this spectrum.
We develop first-order equations to interpret absorption spectra.
In the case of HD 189733b, we show that the observed slope of the absorption as a function
of wavelength is characteristic of extinction proportional to the inverse
of the fourth power of the wavelength (
). Assuming an
extinction dominated by Rayleigh scattering, we derive an atmospheric
temperature of 1340 150 K.
If molecular hydrogen is responsible for the Rayleigh scattering, the atmospheric
pressure at the planetary characteristic radius of 0.1564 stellar radius
must be 410 30 mbar. However the preferred scenario is
scattering by condensate particles. Using the Mie approximation,
we find that the particles must have a low value
for the imaginary part of the refraction index. We identify MgSiO_{3} as a
possible abundant condensate whose particle size must be between 10^{-2}
and 10^{-1} m.
For this condensate, assuming solar abundance, the pressure at 0.1564 stellar radius
is found to be between a few microbars and few millibars, and the temperature is found
to be in the range 1340-1540 K, and both depend on the particle size.
Key words: stars: planetary systems - scattering - techniques: spectroscopic - stars: individual: HD 189733b
Transiting extrasolar planets, like HD 189733b offer unique opportunities to scrutinize their atmospheric content (e.g., Charbonneau et al. 2002; Vidal-Madjar et al. 2003). In particular, primary transits can reveal tenuous quantities of gas or dust by spectral absorption leading to variations in the apparent planet size as a function of the wavelength. HD 189733 b is presently the best target for such studies because of its nearby and bright host star, its large absorption depth in its transit light curve (Bouchy et al. 2005), and a short period (Hébrard & Lecavelier des Etangs 2006).
Using transit observations with the ACS camera of the Hubble Space Telescope, the apparent radius of HD 189733b has been measured from 0.55 to 1.05 microns, thus providing the ``transmission spectrum'' of the atmosphere (Pont et al. 2008). In this wavelength range, strong features of abundant atomic species and molecules were predicted but not detected, leading Pont et al. (2008) to the conclusion that a haze of sub-micron particles is present in the upper atmosphere of the planet.
Here we use these measurements of the planet radius as a function of the wavelength from 550 to 1050 nm. In Sect. 2 we describe how a transmission spectrum can be interpreted to derive basic quantities. We then estimate the temperature of HD 189733b from fit to the ACS spectrum and show that the observed absorption can be caused by Rayleigh scattering (Sect. 3). Several species that are possibly responsible for the Rayleigh scattering are discussed, and the corresponding pressure at the absorption level are estimated in Sect. 4.
To interpret transit spectra, one can use detailed atmosphere models that include temperature, pressure, and composition as a function of the altitude and numerically integrate the radiative transfer equations to obtain a theoretical spectrum to compare to the observations (e.g., Ehrenreich et al. 2006). Here we propose another approach by deriving first-order equations to obtain a better feeling for the basic quantities that can be obtained from these measurements.
Following Fortney's (2005) derivation of the path length,
the optical depth, ,
in a line of sight grazing the planetary limb at an altitude z
is given by
,
where
is the assumed planet radius,
H the atmosphere scale height, and
the
volume density at the altitude z of the main absorbent
with a cross section
.
For a temperature T, H is given by
,
where
is the mean mass of atmospheric particles taken to
be 2.3 times the mass of the proton, and g the gravity.
Using planetary transit measurements, a fit to the light curve
provides the ratio of the effective
planetary radius as a function of wavelength to
the stellar radius:
.
Here we define
by the optical depth at altitude
such that a sharp occulting disk of radius
produces
the same absorption depth as the planet with its translucent atmosphere;
in other words,
is defined by
.
Using a model atmosphere, and numerically integrating over
the whole translucent atmosphere, we can calculate the
effective planet radius, and then obtain the corresponding
optical thickness at the effective radius,
.
For various atmospheric scale heights,
we calculate
by numerical integration.
For a wide range of atmospheric
scale height, provided that
is between 30 and 3000,
the resulting
is roughly constant at a value
(Fig. 1).
In the case of HD 189733b,
varies between 280 and 560
when the temperature varies from 1000 to 2000 K; therefore,
the approximation of a constant
at 0.56 fully applies.
This demonstrates that, for a given atmospheric structure
and composition, estimating the altitude
at which
is all that is needed to calculate
the effective radius of the planet at a given wavelength.
Figure 1: Plot of as a function of the ratio of the planet radius to the atmosphere scale height. | |
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For a given atmospheric structure and composition,
the effective altitude of the atmosphere
at a wavelength
is calculated by finding
,
which solves the equation
.
Using the quantities defined above, the effective altitude z is given by
If the variation of the cross section as a function of wavelength is known, the
observation of the altitude as a function of wavelength allows the derivation of
H and, therefore, of the temperature T, given by:
In summary, the effective planetary radius is characteristic
of the pressure and abundance, and the variation in this
radius as a function of wavelength is characteristic
of the temperature.
Figure 2: Plot of the altitude in the atmosphere of HD 189733 b corresponding to the measured planetary radius as a function of wavelength. Here the zero altitude is defined for a planetary radius of 0.1564 times the stellar radius. The measured values with error bars are those of Pont et al. (2008). The fit to the data (thick line) is obtained assuming an extinction proportional to and for an atmospheric temperature of 1340 K. Fits using H_{2} Rayleigh scattering or MgSiO_{3} grains with sizes between 0.01 and 0.1 m give the same plots. Larger grain diameters are not consistent with the observations (e.g., 0.4 m, shown with dotted line). | |
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In HD 189733b, the plot of altitude as a function
of the wavelength shows
an increase in absorption toward shorter wavelengths
(Fig. 2).
Using Eq. (2)
and assuming a scaling law for the cross section in the form
,
the slope of the planet radius as a function of the wavelength
is given by
Therefore, we have
Assuming
as expected for Rayleigh scattering
and using Eq. (2), we derive a temperature
For condensates, the scale height is often found to be significantly smaller than the gaseous scale height ) (Ackerman & Marley 2001; Fortney 2005). If this were the case, because of the observed slope in the spectrum, the temperature (or ) should be about three times higher than found above, which seems unlikely. We therefore conclude that, if produced by dust condensates, the observed transit spectrum of HD 189733 b shows that these condensates must be unusually well-mixed vertically with the atmospheric gas with a similar scale height.
One possible carrier of the Rayleigh scattering is the most abundant molecule, molecular hydrogen. The effective altitude at which the Rayleigh scattering of molecular hydrogen dominates only depends on the mean density. Therefore, measurements of the altitude in the regime where Rayleigh scattering by H_{2} dominates over other absorbents allows the determination of not only the temperature, but because the abundance of H_{2} is close to 1 ( ) also the total density and consequently the total pressure at the reference zero altitude.
Following Eq. (3), the pressure P_{0} at the altitude corresponding
to the radius at wavelength
is
A fit with two free parameters (temperature and pressure) and assuming that Rayleigh scattering by molecular hydrogen dominates leads to a very good fit to the data published by Pont et al. (2008) with a equals 9.8 for 8 degrees of freedom. We thus obtain a temperature of T=1340 150 K and a pressure at z=0 (defined by ) of P=410 30 mbar.
Another possible carrier of the Rayleigh scattering is haze condensate typically smaller than the wavelength.
Assuming that the absorption measured between 550 and 1050 nm is due to particle condensate, we calculate the transmission spectrum using the Mie approximation. In transmission spectroscopy, given the small solid angle defined by the star seen from the planet, we can use the single scattering approximation. The glow produced by off-axis scattering is also negligible (Hubbard et al. 2001). In this configuration, the Mie extinction efficiency ( ) is used and calculated as the summation of the absorption efficiency ( ) and scattering efficiency ( ).
Rayleigh scattering in the form
is obtained only
for particle sizes (a) much smaller than the wavelength
(
).
In this last case, the absorption and scattering efficiencies
are given by the approximation:
,
and
,
where
is the complex refraction index of the condensate.
Figure 3: Plot of the temperature and pressure at zero altitude as a function of the size of MgSiO_{3} grains needed to obtain a satisfactory fit to the data as in Fig. 2. Here we assume solar abundance for magnesium atoms to derive the grain abundance. For particles of 0.03 m, temperature is found to be the same as with H_{2} Rayleigh scattering : T=1340 150 K. At 0.01 and 0.1 m temperature increases slightly to T=1540 170 K and T=1390 150 K, respectively. For particles smaller than 0.01 m and larger than 0.1 m, Rayleigh scattering does not dominate the extinction ( ), and the temperature must be higher to fit the slope of the measured radius as a function of wavelength. At 0.3 the fit is bad with for 8 degrees of freedom; increases steeply for larger sizes. | |
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We see from the two last equations that Rayleigh scattering
dominates the transit spectrum only if
.
Assuming that the imaginary part of the refraction index is much smaller
than the real part, we find that this condition occurs for particle
size, a, larger than a minimum size:
A broader range of particle size is allowed if the imaginary part of the refraction index is significantly smaller than 0.1. In other words, the condensate must be transparent enough for the absorption to be negligible relative to scattering. Among the possible abundant condensates, MgSiO_{3} presents such a property (Dorschner et al. 1995; Fortney 2005). If MgSiO_{3} contains no Fe atoms, the imaginary part of the refraction index is found to be in the considered wavelength range. Using this value, we find . Therefore, we conclude that MgSiO_{3} is the candidate species possibly responsible for the observed Rayleigh scattering in the HD 189733b spectrum, and its size must be in the range 10^{-2}-10^{-1} microns (see upper panel of Fig. 3).
Assuming an abundance for the MgSiO_{3} grains and using Eq. (3), we can evaluate the pressure at the effective planetary radius if the Rayleigh scattering by these grains dominates the transit spectrum. For MgSiO_{3}, the refraction index is taken to be in the range 550-1050 nm (Dorschner et al. 1995). This gives a cross section . The grain abundance is obtained using solar abundance and assuming that it is limited by the available number of magnesium atoms. Thus the grains have an abundance , where is the magnesium abundance, is about 100.4 times the mass of the proton (), g cm^{-3}is the grain density, a the particle size, and N Avogadro's number. Therefore we have . Finally, at z=0 defined by the effective radius at nm, with T=1340 K we obtain a pressure bar if the size of MgSiO_{3}grains is about 0.01 m, and bar if the size is about 0.1 m. These values obtained with Eq. (3) are similar to those obtained with a general fit to the data (Fig. 3). It is noteworthy that these values are consistent with the condensation pressure of MgSiO_{3} which is 10^{-4} bar at 1400 K (Lodders 1999).
Finally, Rayleigh scattering by MgSiO_{3} is preferred to scattering by H_{2}to explain the observed spectrum. Indeed, molecular hydrogen requires higher pressure, at which the signature of abundant species like Na I, K I, H_{2}O, and possibly TiO should overcome the Rayleigh-scattering signature, except when they have anomalously low abundances.
Because the cross section varies as a^{6} in the Rayleigh regime, the scattering is largely dominated by the largest particles in the size distribution. Therefore, the typical sizes quoted above must be considered as the size of the largest particles in the distribution.
This work assumes that the whole transit spectrum from 550 to 1050 nm can be interpreted by the absorption by a unique species. Alternatively, there could be a combination of absorptions by various species (e.g., Na I, K I, and H_{2}O) in different parts of the spectrum to mimic a spectrum of Rayleigh scattering with the expected atmospheric temperature. This would be coincidental, but it cannot be excluded from the present data. This question needs higher resolution data to be solved.
The temperature and pressure found above are mean values integrated along the planetary limb, where temperature is expected to vary from the pole to the equator and from one side to the other. A numerical integration of the absorption through the planetary atmosphere assuming a variation in temperature as a function of altitude of -60 K per scale height (Burrows et al. 2007) or as a function of longitude of 350 K over 120 degrees (Knutson et al. 2007) shows that temperatures found by solving Eq. (1) are slightly overestimated by no more than about 3% and 2%, respectively.
If the measurements of planetary radius in the 550-1050 nm wavelength band are extrapolated to the infrared, where Spitzer measurements are available, we can obtain a minimum effective radius due to Rayleigh scattering. From at 750 nm, we extrapolate that must be larger than 0.1539, 0.1531, and 0.1526 at 3.6 m, 5.8 m, and 8.0 m, respectively (with 0.0003 (stat.) 0.0006 (syst.) error bars). These values are consistent with the Spitzer/IRAC measurements given by Knutson et al. (2007) and Ehrenreich et al. (2007), and withmeasurements by Beaulieu et al. (2007) at the limit of the upper error bars.
Rayleigh scattering has also been proposed to explain the detection of polarized scattered light (Berdyugina et al. 2007). However, this detection concerns the high upper atmosphere and cannot be directly compared to the present work on scattering deeper in the atmosphere.
Ground-based, high-resolution spectra allowed the detection of sodium in the atmosphere of HD 18973b (Redfield et al. 2008). Assuming a pressure at a given altitude, the measured absorption allows the determination of the sodium abundance. For that purpose, a fit of the sodium line shape is needed, and still has to be developed. Alternatively, by assuming the sodium abundance, measuring sodium absorption should constrain the pressure and allow for differentiating between the several carriers of the Rayleigh scattering proposed in the present paper. Presently, extinction high in the atmosphere by MgSiO_{3} with its low condensation pressure of 10^{-4} bar appears a natural explanation for the absence of broad spectral signature and is the preferred scenario.
Acknowledgements
We thank the anonymous referee for the constructive remarks. We warmly thank J.-M. Désert and G. Hébrard for fruitful discussions.