Ir r Tf

where Tp and T* are the effective temperatures of the planet and star, respectively. For Tp = 1100 K and T* = 5800 K, and Rp the same as above, this gives Ip ^ 0.003Is.

Figure 7. IRAC data from Charbonneau et al. (2005), showing (top panel) the 8 ¡m flux variation during secondary transit as a function of time, and (bottom panel) the x2 surface for fitting the data, as a function of transit center time and transit depth. Contours in the lower panel correspond to confidence levels of 1, 2, and 3 standard deviations.

Figure 7. IRAC data from Charbonneau et al. (2005), showing (top panel) the 8 ¡m flux variation during secondary transit as a function of time, and (bottom panel) the x2 surface for fitting the data, as a function of transit center time and transit depth. Contours in the lower panel correspond to confidence levels of 1, 2, and 3 standard deviations.

Though not a huge signal, this ratio is much more promising than the value for reflected light.

Recently such observations of the planets TrES-1 and HD 209458b have been undertaken using the Spitzer Space Telescope at wavelengths of 4.5, 8, and 24 ¡m (Charbonneau et al. 2005; Deming et al. 2005). The 8 ¡m observations of TrES-1 are shown in Figure 7; the top panel shows the observed light curve, while the bottom panel shows the x2 surface resulting from a fit to the data, with the transit depth and center time taken as free parameters. Although the fluxes involved are much larger, the 4.5 ¡m observations of TrES-1 and the 24 ¡m observations of HD 209458b both have similar statistical quality, because the relative transit depths grow larger at wavelengths longer than 8 ¡m, and smaller at shorter wavelengths.

The depths of secondary transits are, in effect, measurements of the IR fluxes from the respective planets, averaged over the wavelength bands in question. It is thus possible to compare these fluxes with those predicted by models of the planetary atmospheres (augmented with information about the planets' sizes derived from primary transit data). Such comparisons have been done by Burrows et al. (2005); Fortney et al. (2005), and Seager et al. (2005), with interesting results, as seen in Figure 8, which has been adapted from Burrows et al. (2005).

Figure 8. Model thermal emission spectra of HD 209458b (upper curve) and TrES-1 (lower curve) as computed by Burrows et al. (2005), overlain with observed fluxes from Charbonneau et al. (2005) and Deming et al. (2005). The observations are shown as points with both vertical and horizontal error bars, the latter representing the wavelength range sampled by the observations. Points with only horizontal bars represent averages of the model fluxes over the indicated wavelength range.

Figure 8. Model thermal emission spectra of HD 209458b (upper curve) and TrES-1 (lower curve) as computed by Burrows et al. (2005), overlain with observed fluxes from Charbonneau et al. (2005) and Deming et al. (2005). The observations are shown as points with both vertical and horizontal error bars, the latter representing the wavelength range sampled by the observations. Points with only horizontal bars represent averages of the model fluxes over the indicated wavelength range.

Model planetary emission spectra show a marked dip in the flux near 4.6 ¡m, due to strong absorption by the CO vibration-rotation band at that wavelength. The relatively low flux observed at 4.5 ¡m may, therefore, be interpreted as evidence for the presence of CO in the planetary atmosphere (e.g., Burrows et al. 2005). This result is not inconsistent with the failure of ground-based observations to find evidence for CO, because the 4.6 ¡m band is the fundamental vibration-rotation band, which is much stronger than the 2.3 ¡m first overtone band. Similarly, the broad dip in the model spectra between 4.5 ¡m and 10 ¡m results mainly from water vapor. The slightly lower flux at this wavelength compared to 10 ¡m is evidence (albeit weak) for water absorption.

The observed fluxes at all wavelengths are uniformly greater than those computed in the models by Burrows et al. (2005). These models were computed assuming that the incoming stellar heat flux distributes itself uniformly over the entire surface of the planet, i.e., that heat transport from the planet's day side to its night side is very efficient. Comparison of the observed fluxes with the models by Burrows et al. (2005) therefore suggests that the heat transport to the planet's night side may be quite inefficient, thereby raising the temperature of the planet on the day side. This conclusion appears model dependent, however. Thus, Fortney et al. (2005) find that agreement with their atmospheric models demands efficient heat transport (else the observed fluxes would be larger than they are), while Seager et al. (2005) find an intermediate case, in which the heat transport must be moderately efficient. This disparity among models is a bit discouraging, but it likely is an inevtiable state of affairs, given the untested nature of both the models and the observations.

Finally, the timing of the secondary transits can be used to set limits on the eccentricities of the planetary orbits. If an orbit is elliptical and the Earth does not lie near the projection of its major axis, then the secondary eclipses do not occur midway in time between primary eclipses. This timing difference turns out to be quite large, even for seemingly small eccentricities—as much as 37 minutes for eccentricity of 0.01 and a four-day orbital period. Since the primary transit center times can be estimated with errors of only a few tens of seconds, and the secondary ones to better than 10 minutes, it is possible to set stringent bounds on the possible combination of orbital eccentricity and major axis orientation. For both HD 209458b and TrES-1b, Charbonneau et al. (2005) and Deming et al. (2005) find that the timings are consistent with circular orbits, within the observational errors. If the Earth lay along the minor axes of the orbits, these observations would imply orbital eccentricities of less than about 0.003 for both planets. Since the orientations of the hypothetically elliptical orbits are not known, substantial eccentricities are not formally excluded; they are, however, very unlikely. The small limits on the probable eccentricities strongly suggest that damping of orbital eccentricity does not provide an internal energy source that might cause the inflated radius of HD 209458b.

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