If an opaque spherical planet were to pass across a uniformly illuminated stellar disk, the fractional decrease in light arriving from the star 51/1 would be simply
where Rp is the radius of the planet and R* is that of the star. The lowest-order information from a stellar transit is thus the ratio between the planetary and stellar cross-sectional areas.
Two useful complications arise right away. The first is that stellar disks are not uniformly bright; rather, they are limb-darkened (at visible and near-IR wavelengths, anyway), with the amount and functional form of the limb darkening depending on the wavelength of observation. Thus, even a simple-seeming measurement of the ratio of radii is unreliable (at the 20% level) unless we know the transit's impact parameter, i.e., the minimum separation between the stellar and planetary centers, projected on the plane of the sky. For circular orbits, this is the same as knowing the orbit's semi-major axis a and inclination to our line of sight i. Fortunately, the orbital inclination has other effects on the observed light curve (decreasing i from 90° shortens the total duration of the transit and lengthens the times required for ingress and egress when the planet's disk lies only partly on that of the star). The upshot is that one can characterize a transit light curve with four parameters as illustrated in Figure 1, two of which (l and d) are measureable in practice with relative ease and accuracy, and two of which (w and c) are more difficult and error prone. Given the orbital period, an estimate of the stellar mass, and these four observables, it is possible to make separate estimates of R*, Rp, i, and a limb-darkening parameter cl . If one is willing to stipulate the value of any of these parameters (cl , say, based upon stellar atmospheres models), then the others can be determined more accurately, or with poorer data. These considerations (combined with mass estimates from radial velocities) have led to our present knowledge about the mean densities of extrasolar planets, which we shall discuss briefly below.
The second complication is that, for purposes of a transit light curve, Rp is the radius at which the planet becomes opaque to tangential light rays at the wavelength of observation. This is generally not the same as the radius of the photosphere (which relates to light rays propagating more or less vertically, and wavelength averaged to boot), and it is certainly not the same as some arbitrary (e.g., one bar) pressure level in the atmosphere (Burrows et al. 2003). The dependence on pressure level is important for comparisons with self-consistent models of the planetary structure. The wavelength dependence, however, provides diagnostics for the composition, dynamics, and thermodynamic state of the planetary atmospheres.
Consider first a "continuum" wavelength Xc, at which the atmosphere is relatively transparent, and let the zero point for measuring height in the atmosphere be that point at which the optical depth tc along a tangential ray is unity. If we now choose a nearby wavelength where the opacity Kl is larger (say, the center of a strong molecular absorption line), then unit tangential optical depth at this wavelength will occur at a greater height, contacts:
Figure 1. The light curve of a planet transiting the disk of a star, showing four parameters (transit depth, total duration, ingress/egress time, and bottom curvature) that can be measured with photometry of suitable quality.
such that the decrease in mass density compensates for the increase in opacity per gram. If the atmosphere is locally exponentially stratified with a density scale height H, and the absorbing species is well mixed, then one can express the necessary height change 5z as
Thus, to compensate for an increase in opacity by a factor of e, one must move upward in the atmosphere by one scale height. This means that the apparant radius of the planet depends upon wavelength, and so too does the fraction of stellar light blocked by the planet during a transit.
How large is this effect? In principle, it can be substantial. The opacity ratio between strong lines and continuum for relevant molecular vibration-rotation lines is pretty often 104, corresponding to 10 atmospheric scale heights. Moreover, the atmospheres of close-in giant planets likely have low molecular weight, high temperature, and modest surface gravity. As a result, H can exceed 500 km, which may be more than 0.5% of the planetary radius. Thus, according to this simple calculation, small changes in wavelength might change the total light obstructed by the planet by as much as 10%. For further details, see Seager & Sasselov (2000) and Brown (2001). The wavelength-dependent variations seen so far are not as large as this, but they are, nevertheless, big enough to detect in at least one case.
Another interesting situation is that of a cloud deck composed of large (larger than A) particles, distributed in a layer with a well-defined top at height zp. Such a cloud deck will indiscriminantly scatter or absorb tangential rays of all wavelengths. The planet will then appear to have the same radius at all wavelengths, except those for which the
cloud-free continuous opacity is so large that the tangential optical depth is greater than unity at zp. High clouds can, therefore, suppress the atomic and molecular absorption signatures that one might otherwise see.
Finally, consider thermal emission from the planet. At visible wavelengths, most of the light emitted from the planet's day side is expected to be reflected starlight. The maximum possible reflected signal relative to the stellar flux is simply (Ip/I*) = nR /(4na2), which for close-in giant planets typically gives (Ip/I*) ^ 3 x 10~5. If the planetary albedo is less than unity, this number is, of course, even smaller. In the thermal infrared, however, things are quite different. For the expected planetary equilibrium temperatures of roughly 1000 K, the peak of the Planck function falls at about 3.5 ¡m wavelength. Wavelengths in the 4-25 i range, observable by the Spitzer Space Telescope, thus begin to approach the Rayleigh-Jeans long-wavelength limit, in which Ip/T = (Rp/R*)(Tp/T*), where Tp and T are the effective temperatures of the planet and star, respectively. For most known transiting extrasolar planets, this intensity ratio is a few times 10~3, a much more accessible signal than in visible light. This, then is the motivation behind recent successful attempts to measure the diminution in IR flux during the secondary transits of HD 209458b and TrES-1b.
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