Tag Archives: HD 189733

Mapping Extrasolar Planets III. Visible Light Surface Features from Kepler



In our previous looks at mapping extrasolar planets, we have focused on spatial variations in infrared brightness, and thus temperature, on the visible surface of extrasolar gas giant planets. The reason for this bias toward longer wavelengths is two-fold: 1) At the time, Spitzer phase curve photometry has been the dominant means of deriving crude longitudinal maps of extrasolar planets. 2) The ratio between the flux from the star and the planet is an order of magnitude less in infrared than it is in visible light – also the reason that the successes from direct imaging of extrasolar planets have been almost exclusively in infrared (the nature of the object imaged at Fomalhaut by HST in visible is unknown). Short-period extrasolar planets are thus prime candidates for secondary eclipse observations in the infrared. The following table lists several prominent planets and the eclipse depths as measured in four infrared channels by Spitzer, known as the IRAC channels (Infrared Array Camera) which have been a cornerstone of the building up of the foundation of our understanding about extrasolar planets in the past decade.

Spitzer IRAC Eclipse Depths
Planet 3.6µm 4.5µm 5.8µm 8.0µm
HAT-P-7 Ab 0.098% 0.159% 0.245% 0.225%
Kepler-5 b 0.103% 0.107%
Kepler-6 b 0.069% 0.151%
HD 189733 b 0.256% 0.214% 0.310% 0.391%
HD 209458 b 0.094% 0.213% 0.301% 0.240%

Of course the eclipse depth of a planet depends on its intrinsic brightness at the observed wavelength, the brightness of the star itself, and the radius ratio between the two, and given the comparitive brightness and size of a star, it is not hard to see why the drop in flux from the system during the planet’s eclipse behind the star would be so miniscule, < 1%.

The difficulty involved is exaggerated in visible wavelengths, where the flux is dominated less by thermal emission of the two bodies and more by intrinsic processes within the star and how reflective the planet is of the star’s light. Here are some secondary eclipse depth measurements from Kepler, which observes in a filter that is approximately visible light (0.400 – 0.865µm).

Kepler Eclipse Depths
Planet ΔF
HAT-P-7 Ab 0.0069%
Kepler-5 b 0.0021%
Kepler-6 b 0.0022%
Kepler-7 b 0.0042%
Kepler-12 b 0.0031%

Notice that the eclipse depths in visible light are much lower than the eclipse measurements in infrared. For planets with both Spitzer (infrared) and Kepler (visible light) eclipse depth measurements, the contrast is clear: the star-planet brightness ratio is less in infrared than it is in visible light and therefore hot Jupiters are essentially relatively brighter in infrared than in visible.

This does not exclude extrasolar planets from visible light mapping using the phase curve analysis and eclipse scanning techniques described before, it only makes it harder. Optical phase curves are weaker than infrared phase curves, so either a bigger telescope is needed to observe with enough photometric precision to resolve the phase curve in one orbit, or several orbits must be observed with existing instrumentation and allow the data to build up until a phase curve can be resolved.

Fortunately, Kepler has offered quasi-continuous coverage of the transits of many hot Jupiters over the course of four years, and in some cases, it is enough to confidently detect a phase curve. One such planet is Kepler-7 b.

Kepler-7b Phase_Curve

Kepler-7b Phase_Curve

In this image, the green curve corresponds to the expected phase curve if the planet reflected light in a geometrically symmetric way. The red and blue curves are fitted to the data and incorporate a longitudinal offset (see Demory et al for details). The primary transit is on the left side, and the planet passes behind the star on the left side (the secondary eclipse). The depth of the primary transit is so deep on this scale that it is off the image. Notice that the phase curve is not perfectly phased to the orbit of the planet – the secondary eclipse does not occur at the peak apparent brightness. This would be comparable to the Moon being brightest not when it is full, but rather at a gibbous phase. Keep in mind that this is an optical phase curve, the brightness variations on the planet’s dayside hemisphere suggested by the phase curve corresponds to actual features you would be able to see with the human eye (and maybe a welding helmet).

The Kepler-7 b phase curve shows us that the brightest part of the planet’s dayside atmosphere is on the westward side of the dayside hemisphere. Because this phase curve is an accumulation of 14 quarters of Kepler data, it is best thought of as an “average” phase curve over the course of 3.5 years, and therefore the longitudinally resolved visible light map of the planet is an average of the planet’s surface brightness over 3.5 years. Since the planet does not contribute to the phase curve of the system during secondary eclipse, the scatter of the data during that secondary eclipse is a good representative of the overall data scatter. It may come from instrumental noise or stellar noise, but whatever its origin, the fact that it is not obviously different from the scatter in the phase curve when the planet’s brightness is contributing implies that the surface features resolved here are both stable and long-lived.

Kepler-7b Visible Map

Kepler-7b longitudinal brightness distribution

What could be the cause of this bright area on the planet? Demory et al explain:

Kepler-7b may be relatively more likely to show the effects of cloud opacity than other hot Jupiters. The planet’s incident flux level is such that model profiles cross silicate condensation curves in the upper, observable atmosphere, making these clouds a possible explanation. The same would not be true for warmer planets (where temperatures would be too hot for dayside clouds) or for cooler planets (where silicates would only be present in the deep, unobservable atmosphere). Furthermore, the planet’s very low surface gravity may play an important role in hampering sedimentation of particles out of the atmosphere.

Now how do we know that this bright spot is not simply due to thermal emission? Some hot Jupiters are sufficiently hot that the glow from their heat in visible light can affect, or even dominate their eclipse depths. The obvious answer would be to check Kepler-7 b’s eclipse depths in the infrared, and this is a job for Spitzer. Spitzer observed the secondary eclipse of Kepler-7 b in both 3.6 µm and 4.5 µm, and in both wavelengths, the eclipse of the planet behind the star was not confidently detected. This means that the planet isn’t just not hot enough to produce the brightness asymmetry in the Kepler phase curve, but it’s eclipse couldn’t even be confidently detected in infrared. This firmly rules out thermal emission as the source of the optical phase curve asymmetry.

This represents the first time that visible light “surface” features have been identified on an extrasolar planet. This is but a baby-step forward in our ability to map extrasolar planets, but it is a milestone nonetheless. Kepler data may be able to detect phase curve asymmetries in other hot Jupiter systems (or if we are lucky, smaller worlds!), and this can significantly contribute to our understanding of the atmospheres of these planets.


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.

Mapping Extrasolar Planets I. A 2D 8µm Map of HD 189733b

Abraham Ortelius's 1564 map of the world "Typus Orbis Terrarum"

Last December, we looked at how monitoring the phases of an extrasolar planet can permit a crude longitudinally-resolved map of an extrasolar planet in a tight synchronous orbit of its star, (here). Specifically, we focoused on the well-studied transiting hot Jupiter HD 189733 b. This map was what you would call a one-dimensional map of the planet because it was only longitudinally resolved. Assumptions about atmospheric circulation were put into it to make it appear realistic, latitudinally, because it looks more representative of what is probably the case. In the case of the 8µm map for HD 189733 b, the polar regions of a planet are not likely to be as warm as the equator for example, so these modifications are probably not too far off. But for the sake of illustration, let’s take a comparative look at a one-dimensional and two-dimensional map of a planet that we’re a bit more familiar with.

1D and 2D Map of Earth

The production of a longitudinally resolved, one-dimensional map for an extrasolar planet does not require that planet to be tidally locked. Observing the brightness of a planet over an entire rotation period will also permit such a map to be made. Above is an example of what such a map would produce for Earth if it were observed like an extrasolar planet might be. In the one-dimensional, we are permitted only longitudinal information about surface features. We can tell there are two major land masses, which are of course the Americas and the Eurasian+African continents. Other than that, we can’t tell much else. Adding latitudinal information lets us distinguish between North and South America, for example.

It is possible to extract latitudinal information about the distribution of surface features on an extrasolar planet if the planet undergoes secondary eclipses and has a non-zero impact parameter through very careful monitoring of the ingress and egress of the eclipse, as described by Rauscher et al (2006). It’s really rather clever. The limb of the eclipsing star is in an opposite orientation for the egress and ingress. Surface features on the planet that contribute to the light curve may therefore by eclipsed asymmetrically.

Ingress and Egress of the eclipse of an exoplanet

In the above diagram, a planet with a surface feature, say a cloud system for example, passes behind its eclipsing host star. During the ingress, the cloud system is eclipsed about half way through the ingress due to the geometry of the stellar limb and the surface of the planet. During the egress, the cloud system is one of the first features to emerge. An “ingress map” can be made charting where on the surface the features contributing to the light curve may have been during ingress. Clearly this would not be generally known, except for the constraint that a surface feature must have been somewhere on the planet along a line curved to the stellar limb. The same can be done for an “egress map.” Combing the two can permit one to form a two-dimensional map of the eclipsed planet’s sun-facing hemisphere.

Looking through the daily postings on arXiv the other day, I was delighted to find that Majeau et al have done just that, taking Spitzer 8µm data of HD 189733 b to produce a new map of the planet, this time, one in two dimensions. The title of their paper says it all, this is the first two dimensional map of an extrasolar planet, and a huge congratulations to them for the achievement.

The hottest spot on the planet is again found to be displaced from the sub-stellar point by a few degrees in longitude, in agreement with previous work. They also find that the hottest point is on or very near the equator, which isn’t exactly surprising, but it does let us confidently rule out any significant axial tilt for the planet’s rotation.

A 2D 8µm map of HD 189733 b

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).

Of Light and Shadow

Detecting planets around other stars is hard. One of the easiest ways is to detect the apparent dimming of a star as a planet crosses between the observer and the star. The planet will block some of the photons from reaching the telescope, resulting in an apparent dimming of the star. This event is known as a “transit,” and the planets that do this are called “transiting planets.” The amount of light that is blocked is easy to calculate when one considers the problem from the perspective of simple 2-dimensional geometry. The amount of light blocked will be directly proportional to the amount of surface area of the star covered by the planet. For a star of radius R_{star}, and a planet of radius R_{pl}, the amount of the change in flux, \Delta F, is simply ratio of the area of the two bodies:

\displaystyle \Delta F = \left( \frac{R_*}{R_pl} \right) ^2

Clearly, then, larger planets will produce larger ΔF than smaller planets, which directly translates to a more detectable planet. But the difficulty does not scale linearly with the radius of the planet, as one can tell from the exponent in the above equation. Consider two planets, b and c, with radii R_b and 4_c, such that R_b = 2R_c, essentially planet b is twice the radius of planet c. With the above equation it is seen that \Delta F_b = (2^2)\Delta F_c = 4\Delta F_c – One-half the planet produces one-fourth the dip in brightness. Detecting Earth-sized planets can be understood to be far more difficult than detecting Jupiter-sized planets. The typical Jupiter-sized planet will be about 10 times the radius of the typical Earth-sized planet. This translates to 100 times the \Delta F. Can you detect a transiting Jupiter-sized planet? You need a hundred-fold increase in sensitivity to detect Earths.

Not all planets will transit. Assuming a random planetary inclination, the geometric probability that a planet will transit its star may be expressed (somewhat over-simplistically*) as

\displaystyle P_{transit} = \frac{a}{2R_*}

Where a is the semi-major axis of the planet’s orbit. We can see, therefore, that planets which orbit closer to their stars are more likely to transit than those further away. For Mercury, this works out to a ~1.19% transit probability. For Earth, this is an even more disappointing ~0.5%. This also means that if we assume that other stars are randomly distributed across the sky (which is not completely unreasonable out to distances where the structure of the galaxy does not become apparent), then we can say that ~0.5% of stars will have the right perspective to view Earth as a transiting planet. Similarly, we might also say that if all solar-radius stars have Earth-radius planets, then 0.5% of them are detectable through this method.

An important feature in determining how long transits last, aside from their orbital period, is the “impact parameter,” b, which is a measure of how far the transit chord is from the centre of the stellar disc, as measured in units of the stellar radius. A transit with b = 0 will be a perfectly edge-on orbit with the transiting planet passing straight through the centre of the stellar disk, with higher values of b being less dead-on transits.

Values of b > 1 imply a non-transiting planet, with the maximum attainable value for b being achieved for a face-on orbit (i = 0° or 180°) at a/R_*. There is a very small range of values of b > 1 for which transits still occur, depending on the radius of the planet. While the centre of the planetary disc may not intrude upon the stellar disc for these values of b, planets are not points, and so have a radius of their own. Transits with values of b close to 1 are called “grazing” transits, because the planet just grazes the stellar disk.

Mathematically, the impact parameter may be calculated by b \equiv a \cos{i}.

These transits may be plotted as brightness as a function of time, leading to a “transit light curve.” A transit event will have four events called “contacts.” The “first contact” is when the planetary disc first reaches the stellar disc. The “second contact” occurs when the entire planetary disc has moved onto the stellar disc. “Third contact” occurs when the planetary disc has reached the other edge of the stellar disc on its way out of the transit. Finally, “fourth contact” occurs when the planetary disc has moved completely off the stellar disc. The time from first to second contact is characterised by a significant drop flux, followed by a comparatively constant flux from second to third contacts. From third contact to fourth contact, the flux jumps up to the pre-transit value. The time between first and fourth contacts is the total transit duration, tt, while the time between second and third contacts is the “full transit duration,” tf, denoting the amount of time the planet is fully transiting the star.

For the image below, two transits are shown, one with a high impact parameter and one with an impact parameter of b = 0. Notice how differing the value of b changes both the duration and shape of the transit light curve. For both, all four contacts are labelled as vertical lines.

How does this look for the case of a real planet transiting a real star? Below is the transit light curve of the planet HD 189733 b.

The transit of HD 189733 b

Notice that the light curve between second and third contact is curved. This is because of stellar limb darkening, where the light coming from the limb of the star is darkened compared to the light from the centre of the stellar disc.

* Taking the radius of the planet, the eccentricity of its orbit, and the longitude of periastron into account, the geometric probability that a planet with a randomly oriented orbit will transit is expressed as

\displaystyle P_{transit} = \left( \frac{a}{2(R_*+R_{pl})} \right) \left( \frac{1-e \cos{\pi/2 - \omega}}{1-e^2} \right)

This dependence on the longitude of perihelion can be understood from the consideration of eccentric orbits. In reality, what is the dominant driver as to the probability that a planet will transit is more of the planet’s distance from the star during the transit window, as opposed to the planet’s distance from the star as measured by the semi-major axis. Below shows two planets with identical orbits, except for the latter having a higher value of ω. Note that because of this, the latter planet’s distance from the star during the transit window, shown in light-grey, is much further away than the first planet, and so its transit probability is considerably lower.