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Mapping Extrasolar planets II. Astrophysical Effects on Eclipse Scanning

Credit: NASA/JPL-Caltech

We looked at how carefully monitoring the secondary eclipse of a transiting planet can reveal deviations from a uniformly bright disc here. We considered the system to be a “perfect” transiting planet system, with a perfectly spherical planet on a perfectly circular orbit with a perfectly tidally locked rotation, but nature need not be so conveniently arranged and there is room for many different complex scenarios. A paper submitted to Astronomy & Astrophysics takes a look at some of the astrophysical phenomena that can affect the interpretations of a planet’s brightness distribution in the context of deriving a map of the planet.

Lest we forget, the eclipse scanning method for deriving a two-dimensional map of an extrasolar planet involves careful, high-precision monitoring of the secondary eclipse ingress and egress of the planet. Differences in the brightness distribution of the day-side observable surface of the planet will produce an asymmetric ingress/egress light curve.

8µm Ingress and Egress LC of HD 189733b

Above is the section of the light curve showing the ingress (left) and egress (right) of HD 189733 b’s secondary eclipse behind it’s host star. The red line is what would be expected if the planet’s day-side surface were of uniform brightness. The fact that there are significant residuals to this fit indicate that the planet is not adequately described as a uniformly bright disc.

One contribution to an anomalous ingress/egress light curve could of coruse be the shape of the planet. An oblate planet will, for all orientations that are not pole-on, have ingress/egress light curve shape that differs from that of a spherical planet in a way that is different from a localised bright spot on the planet.

The eccentricity of the planet will also have an effect. Circular orbits have an equal time between transits and eclipses, with the eclipse occurring at half-phase. Eccentric orbits for most orientations (longitudes of periapsis of \omega \neq \pm 90^{\circ}) will have the secondary eclipse away from half-phase. Obvious eccentricities may reveal themselves through their radial velocity -derived orbit fits, but very tiny ones may not, yet may still add complications to secondary eclipse scanning. If, due to the eccentricity of the planet, the planet’s position is slightly offset from its expected position by an amount that is rather small, on the order of the size of surface features on the planet, then this can cause some ambiguity in the surface brightness distribution map. This is especially a problem if the brightest feature on the planet is shifted away from the substellar point, as indeed is the case for HD 189733 b, however phase curve observations of reflected light from the planet can be used to constrain the true position of the brightest spot.

Because of this so-called “brightness distribution-eccentricity degeneracy” effect, it can be difficult to find a unique solution to the surface brightness distribution of the planet. Assumptions of the underlying brightness distribution can permit estimates of the eccentricity of the orbit (see this paper by de Wit et al).

Using the simplest model (below) to explain both the observed ingress/egress curves and the phase curve, shown below and to the left, is well supported by the amplitude as derived from the secondary eclipse depth (effectively the total brightness of the planet), and in longitudal resolution as derived from the phase curve. The standard deviation from the model (right) is easily seen to be small, much more so than other, more complex models of the underlying brightness distribution.

One of several HD 189733b 8 µm brightness distribution models

A more complex underlying brightness distribution model (below), and a much poorer fit to the observed data, allows for the resolution of structures that are less constrained by the secondary eclipse depth and in longitudinal resolution as derived from the phase curve. However, this model is well-constrained by the secondary eclipse scanning.

One of several HD 189733b 8 µm brightness distribution models

Increasingly complex models for the underlying brightness distribution produce worse global fits to the data, however a consistent theme of a longitudinally displaced hot spot remains. To illustrate how the day-side mapping of the planet can constrain the parameters of the system, if we assume this brightness model (below) to be a true representation of the underlying brightness distribution of the planet’s dayside surface, then it requires a larger planetary orbital eccentricity. Because of an increased eccentricity, the orbital velocity will be different despite the constant (measured) eclipse duration. Thus, the radius of the star will need to be slightly adjusted to fit this model (in this case made smaller), and accordingly the impact parameter of the planet’s secondary eclipse will be affected (in this case increased), while changing the density of the planet (recall that the planet’s radius is known only as a ratio of the star’s).

One of several HD 189733b 8 µm brightness distribution models

Another source of complexity in the analysis of eclipse scanning can come from limb-darkening of the planet, in much the same way a star is limb-darkened. As with stars, the severity of this limb-darkening will be wavelength dependent, and in the 8 µm wavelength that these Spitzer results are derived from, the limb-darkening of a hot Jupiter is expected to be negligible.

The detailed mapping of extrasolar planets, even now, cannot be said to even be in its infancy yet. It is still being born. Small astrophysical effects beyond our current ability to measure can cause profound changes to the derived map of a planet, requiring extreme caution. As of the time of this writing, only two planets – HD 189733 b and υ And Ab – have had surface brightness distributions modelled with Spitzer phase curve photometry, and for both of them, unique models have been put forward to explain the observations. Direct imaging will, in the future, provide more definitive means of mapping extrasolar planets, but until then, we are forced to use tricks requiring very high quality data to tease out such information from planets we can’t even see.


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).