# The Phases of an Extrasolar Planet

The transit of a planet across the disc of its star (see here) produces a characteristic dip in the observed brightness of the system. This can be understood simply as a light source being occulted by another object. Extending this to a slightly more extreme case, we can see that a similar event occurs when the star occults the planet, at least as far as the appearance of the light curve is concerned. Planets don’t typically emit much light on their own but they do of course reflect light from their parent stars. So in this sense, they are light sources. When a planet passes behind a star, the star blocks the light reflected off the planet from reaching the telescope on Earth.

While the shape of the effect in the light curve will be about the same, there are notable changes, one being the obvious — the effect is far more diluted. The other notable difference is that the “floor” of the light curve shape is flat instead of curved. This is of course because the total brightness of the system does not change depending on where behind the star the planet is, all else held constant, whereas in a primary transit, the apparent stellar disc is unevenly illuminated due to limb darkening. Below is the example of the light curve of the transiting planet HAT-P-7 b as obtained by the Kepler spacecraft.

The Light Curve of HAT-P-7 b

In this graph, all the data is folded to the period of the planet, and so therefore repeats each orbit. That little dip half way between the two transits corresponds to the secondary eclipse (it might help to click on the light curve to enlarge it). That is when the planet HAT-P-7 b passed behind the star HAT-P-7, which blocked its light from reaching the telescope. The extreme difference in the depths of the transit and eclipse speak to the difference in brightness of the planet and the star. Such detections require photometers of much more precision than is needed to simply detect the planet transit itself.

The secondary eclipse depth can be expressed as

$\displaystyle a = \left( \frac{R_p}{R_*} \right)^2 \left( \frac{T_p}{T_*} \right)$

where $T_p$ is the effective temperature of the planet, and $T_*$ is the effective temperature of the star.

If we vertically stretch this data to make the secondary eclipse more visible, another phenomena reveals itself.

Light Curve of HAT-P-7 b

After the transit, we see the system brightening all the way up toward the secondary eclipse, and then after the eclipse, we see the system dimming back down. This effect can be understood when considering the appearance of the system throughout this light curve and considering the phases of Venus. As Venus orbits its star as seen from Earth, it shows to us varying amounts of its illuminated hemisphere. The exact same effect explains the apparent changes in brightness of the HAT-P-7 system (for the telescope cannot resolve which light comes from the planet and which comes from the star). During the transit of HAT-P-7 b, only its unilluminated hemisphere is facing us. After the transit, we see the planet as a crescent, then half phase, then a gibbous phase. The “full” phase of the planet occurs when the planet is behind the star so we do not expect to detect light from the planet during this time. Of course the phases of Venus are a bit different because we are close enough to Venus to see it grow in apparent size toward its crescent phase. For the changing brightness of a planet due to its phases, it’s perhaps best to think of the Moon, which is always brighter near full phase than at a crescent.

A light curve that is folded over the period of the planet which reveals its phases may be called a “phase curve.” It’s best to think of a phase curve as a special type of light curves.

The phases of Venus approximate the phases of an exoplanet

Consider observations of this type in the infrared. If we assume the planet and star radiate as blackbodies (which is more reasonable for longer wavelengths), you can estimate the day side equilibrium temperature of the planet with

$\displaystyle T_{eq} = T_* \left( \frac{R_*}{2a} \right) ^{1/2} [\alpha (1 - A_B)]^{1/4}$

Where $\alpha$ is a constant that describes the heat recirculation efficiency of the atmosphere and $A_B$ is the Bond albedo of the planet. If $\alpha = 1$, then the circulation of the planet is maximally efficient, redistributing heat to the night side of the planet enough to even the day and night side temperatures (you might consider Venus a good example of a planet with a value of $\alpha$ very near to unity). The Bond Albedo quantifies the fraction of radiation that reaches the planet which is reflected back off into space. A value of $\alpha = 2$ implies that only the day side is emitting radiation. For a Bond Albedo of 1, the planet reflects all energy back into space and stays at absolute zero. While this situation is unphysical of course, high albedos are achievable. Snow and water clouds have a high albedo, while coal and asphalt has a low albedo.

Notice for the HAT-P-7 b phase curve above, the secondary eclipse depth is actually deeper than the brightness of the system just before and just after primary transit, which are the next best proxies for the brightness of the star without the planet. This subtle effect betrays the night-side brightness of the planet. The only time that the star is the only (known) contribution to the light curve is when the planet is hidden behind it. Right before transit and right after, though, only the night side of the planet is facing the observer. So we must conclude that the extra source of light is from the night side of the planet. This can be understood by the physical process of heat redistribution due to atmospheric winds (again, think Venus). For gas giant planets, the process is typically not as efficient, however.

However this observation of HAT-P-7 b with the Kepler telescope is in optical light. This phase curve therefore reveals to us that the heat redistribution to the planet is at least efficient enough to cause the night side of the planet to visibly glow red hot.

Let us turn our attention away from HAT-P-7 b for now to a hot Jupiter with less extreme irradiation, HD 189733 b. The Spitzer spacecraft observed the planet over an entire orbit to construct an infrared phase curve. An anomaly was noted in that the peak excess infrared brightness did not occur immediately before and after the secondary eclipse as would be expected if the sub-stellar point on the planet were the hottest. Instead, it was shifted over slightly.

8 µm Phase Curve of HD 189733 b

Note that not only does the transit not occur at the point of the least infrared excess, but the secondary eclipse does not occur exactly at the peak infrared excess. It turns out that you can construct a crude infrared map of an extrasolar planet by making the reasonable assumption that the planet is tidally locked to its star, such that the same longitude always faces the star. If this is the case, then it’s easy to figure out what longitude of the planet is facing the telescope, as it is a simple function of the observed orbital phase. Subtracting out the brightness of the star from the phase curve gives you just the observed brightness of the the planet. The brightness of the planet versus its longitude can therefore be represented graphically.

HD 189733 b longitudinal 8 µm brightness

This is, of course, only longitudinally resolved, and tells us nothing about where the warm spots are on the planet in latitude. Nevertheless, making various assumptions and simplifications, you can work up a crude 8 µm map of the planet.

8 µm map of HD 189733 b

We see, therefore, that the hot spot of the planet is pushed away from the substellar point at 0° longitude by 16 ± 6 degrees east. It seems reasonable to invoke upper atmospheric winds to explain this.

A more extreme case of this kind of anomaly can be seen for the innermost planet of the Upsilon Andromedae system, where the hottest spot has been pushed over Eastward a remarkable 80°(!). As of the time of this writing, it is not clear if winds alone can produce this extreme a discrepancy. This planet does not transit, as observed from Earth, however the detection methodology is similar. A phase curve can be clearly detected, only the transit and secondary eclipse are absent (here is a decent video that shows the dynamics of what goes into measuring this infrared offset).

In summary, the detection of the secondary eclipse of a planet can shed light on its reflectivity and, if measured in the infrared, temperature and heat redistribution properties of the atmosphere (and by extension a rough idea of its upper atmospheric wind behaviour).
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