## Mapping the Galaxy

The Milky Way Galaxy (Source)

Measuring the distance to a star makes use of astrometry – the careful monitoring of a position of a star over time. As Earth orbits the sun, it has a maximum displacement from any given position along its orbit of about 2 AU (i.e., being on the other side of the orbit). By observing the angular change in the apparent position of a star 2 AU apart, simple trigonometry can allow you to calculate the distance to the star.

In the middle of the last century (not terribly long ago from a historical perspective), we knew the distances to very few stars and knew their positions with much poorer accuracy. The FK4 catalogue catalogued the position of stars in the year 1950 with a precision of position of about 0.04 arcsec in the northern hemisphere, and a dismal 0.08 arcsec precision in the southern hemisphere. It was suggested that using a network of astrolabes over ten years could reduce the errors to about 0.03 arcsec, only marginally better. Major obstacles to the advance of stellar cartography was the typical issues that plague amateur astronomers now — atmospheric distortion of stellar images, instrumental instability, and inability for a ground-based observatory to view the entire sky.

In 1966, Pierre Lacroute came up with an idea (that he himself called “weird”) of performing the necessary measurements from a spacecraft, orbiting Earth outside the atmosphere. The idea was presented in 1967 to the IAU where it received a great deal of interest, but the technological capacity at the time (and available rocketry in France) was not accommodating to the idea. The satellite, a 140 kg spacecraft designed to observe 700 stars all over the sky with a precision of 0.01 arcsec, had stability requirements that could not be met by the Diamant rocket used by France at the time.

The idea of a spacecraft to catalogue the distances and positions of a large number of stars evolved over time and was revised and improved for the next decade, while the rest of astrophysics advanced and continued running into the problem of distance scales being poorly known.

“The determination of the extragalactic distance scale, like so many problems that occupy astronomers attention, is essentially an impossible task. The methods, the data, and the understanding are all too fragmentary at this time to allow a reliable result to be obtained. It would probably be a wise thing to stop trying for the time being and to concentrate on better establishing such things as the distance scale in our Galaxy.” — Hodge (1981)

Support for a space-based astrometry mission continued to grow and recognising that France alone did not have the resources necessary to complete the task, the European Space Agency planned and devised a new spacecraft, Hipparcos, to catalogue the positions of 100,000 stars and to determine their positions with an accuracy of 0.001 arcsec (1 milliarcsec).

Hipparcos

Hipparcos was launched on August 8, 1989 on a 3.5 year mission. It determined the positions of stars, monitored the position over the course of a half year to determine the parallax and thus distance to the star, monitored the position over the course of the entire mission to determine the proper motion of the star in space, measured the spectrum of stars to determine their composition, and performed radial velocity measurements on these stars to determine their motion toward or away from Earth. In total, 118,200 stars were observed with high precision observations (published in 1997), with another 2.5 million stars observed with lower precision (published in 2000).

Hipparcos data has practically revolutionised astronomy. With the knowledge of the positions and motions of over a hundred thousand stars in hand, we’ve been able to understand the structure and dynamics of nearby clusters, understand the local structure of the Galaxy, understand the orbits and true orientations of binary star systems, and more. Even an extrasolar planet transit was observed (though it was not known until the planet was discovered later).

This brings us to today. This Hipparcos catalogue remains as the best available source of uniform parallaxes and positions. It is time, however, to take another step forward, with greater precision, a larger sample, and newer science. The successor to Hipparcos is called GAIA – Global Astrometric Interferometer for Astrophysics – however it will not use interferometry due to a design change.

Gaia will essentially do exactly what Hipparcos did, but better. Whereas Hipparcos only measured a hundred thousand stars down to brightnesses of V = 9, Gaia will observe over a billion stars with brightnesses down to V = 20. Gaia will measure the angular position of all stars of magnitude 5.7 – 20. For stars brighter than V = 10, it will determine the position with a precision of 7 µas (microarcseconds), a precision of 12 – 25 µas down to V = 15, and 100 – 300 µas down to V = 20. It will acquire their spectrum (from 320 – 1000 nm) to determine their temperature, age, mass, and composition. It will also measure the radial velocity of stars with a precision of 1 km s-1 for V = 11.5, and 30 km s-1 for V = 17.5. Tangential velocities for 40 million stars will be measured with a precision better than 0.5 km s-1.

Gaia (Source: ESA)

While the stellar astrophysics enabled by Gaia will be revolutionary in its own right, the unprecedented astrometric precision also makes the mission interesting from an extrasolar planet perspective. Hipparcos was not able to discover any planets on its own, but it was marginally helpful for extrasolar planet science. Planets detected with radial velocity have unknown true masses. The greater the true mass of the planet, the greater the astrometric amplitude of the barycentric motion of the star is (see this post where astrometry is discussed in the context of planet detection). Planets of especially high true masses would therefore have a chance of having their star’s barycentric motion detectable to Hipparcos. Otherwise, Hipparcos data could be used to set upper limits to the true mass of the planet, by knowing that it’s astrometric effect must be sufficiently low so as to not have been detected by Hipparcos (an upper limit to the astrometric amplitude and thus the planetary mass).

The astrometric precision and vast number of targets available to Gaia will allow for the detection of a large number of planets. Astrometry is, of course, less biased toward high values of the planetary orbital inclination, and will permit us to know the true mass of the planet and orientation of the orbit in 3D space. Still, several complications are expected to arise based on nearly two decades of radial velocity experience.

Just like with radial velocity (and, actually, science in general), models will need to be fitted to data points to yield high-quality fits, however as Doppler spectroscopy has shown us, planetary systems can often feature several components all contributing to the barycentric velocity profile of the star, complicating radial velocity fitting in the same way it can be expected to complicate astrometric fitting. Radial velocity surveys can often produce more than one model that fit the data nicely, where both models may disagree on certain aspects of the orbit, or even number of planets. Astrometry is likely to be prone to the same problems. In the case of astrometry, it may even be harder because of the greater number of free parameters – ascending node, inclination, etc, issues that need to be modelled for an astrometric fit that could usually be ignored for a radial velocity fit.

These challenges can be addressed and handled, and the Gaia data will be wonderfully productive to extrasolar planet science. It is hard to know how many planets we can expect Gaia to discover, because statistics for planets in intermediate-period orbits are still unconstrained, but with the accuracy and large number of stars Gaia will observe, it is likely that Gaia will discover thousands of giant planets. It will be sensitive to Jupiter analogues out to 200 parsecs.

Gaia Results (Source: Sozzetti (2010)

What about transiting planets? A transit of HD 209458 b was squeezed out of Hipparcos data, which was not at all optimised for transiting planet science. Can Gaia be expected to detect transiting planets? As far as photometric precision, Gaia is expected to achieve 1 mmag precision for most objects Gaia will observe, down to V ~ 15, and 10 mmag precision at the worst case of V ~ 20. For most hot Jupiter systems, mmag precision is indeed sufficient for transit detections. The next major issue is cadence.

Focused transit searches tend to be high-cadence, narrow field observations, whereas Gaia is an all-sky, low cadence observatory. On average, each star will be observed by Gaia 70 times, giving us 70 measurements for a light curve of any given star with a baseline of five years. While 70 measurements spread out over five years seems dismal (and let’s not sugar-coat the issue — for a transit search, it is dismal, but Gaia is not designed to be a transit search mission), but for a planet in a short period orbit, perhaps three or four measurements may occur while the planet is transiting. Obviously, the longer the orbital period, the less a fraction of the planet’s orbital period is spent in-transit, and the fewer transits will be observed by Gaia. Since only 70 measurements will be taken, Gaia is severely biased toward short-period transiting planets.

Early studies suggested wildly fantastic transiting planet yields. Høg (2002) estimated over a half million hot Jupiters and thousands of planets in longer periods would be found, based on the (unrealistic) assumption that a transit could be identified based on a single data point and other oversimplifications. Robichon (2002) suggested that Gaia will detect 4,000 – 40,000 transiting hot Jupiters under the assumption that each star would receive an average of 130 measurements, however the currently planned Gaia mission has instead 70 measurements per star.

Dzigan & Zucker suggest that Gaia could potentially detect sub-Jupiter-sized planets around smaller stars, and that a ground-based follow-up campaign can easily observe hints of transiting planets that show up in Gaia data. They also suggest that a few hundred to a few thousand hot Jupiters could be found in Gaia photometry.

While Gaia will perform km s-1 radial velocity measurements on millions of stars, this precision level is simply not sufficient to detect even hot Jupiters. It will, however, be able to tell if a transiting planet candidate is a brown dwarf instead, or an eclipsing binary star, allowing for one method of ruling out false positives. Interestingly, the astrometric fit to the orbit of a planet will have the inclination of the planetary orbit sufficiently well-characterised that a list of planets that are likely to transit can be compiled and followed-up with ground-based radial velocity and photometry. These long-period transiting planets will certainly prove valuable – they will be likely to host detectable rings and moons.

ESA will launch Gaia on a Soyuz ST-B rocket in November of this year. It will take five years after a commissioning phase for the total extrasolar planets science results to become known. It will be very exciting to see what giant planets exist in the solar neighbourhood. They will attract interest in follow-up observations to discover smaller, inner worlds that may exist. Gaia has the potential for flagging the first solar system analogues in the solar neighbourhood for dedicated study.

## Transits Across g-Darkened Stars

Titan transiting its oblate fluid planet (Credit: HST)

We have previously looked at modelling the transit of an extrasolar planet incorporating limb darkening. How can we build upon this? For starters, it would be nice to actually take into consideration more physical phenomena – nature is not always simple. Take for example the case of a non-spherical star. Consider the case of Saturn, whose rapid rotation distorts the planet away from a spherical shape and more into an oblate shape. Stars, being fluid bodies like gas giant planets, are subject to similar physics, and have measurable effects on the transit light curve.

How does this affect a system? A rotating fluid body like a gas giant planet or a star is subject to centrifugal force which causes the equatorial radius to be larger than the polar radius. This effect, called oblateness, can be measured with

$\displaystyle \text{oblateness} = \frac{a-b}{a}$

Where a is the equatorial radius and b is the polar radius. In the case of Saturn, the oblateness is about 0.1. Despite a higher rotational velocity, Jupiter’s higher density and surface gravity are able to keep it’s oblateness lower, but still noticeable, at about 0.06. All of the other planets have very different, non-gas-dominated compositions, and have much a lower oblateness (in all cases, less than 0.025). The sun, however, with it’s high surface gravity (28 times that of Earth) and long rotation period (about a month), is almost perfectly spherical, with an oblateness of 0.000009.

In the case of a self-luminous body like a star, oblateness will cause the surface gravity and therefore surface temperature of the star to be a function of latitude. Parts of the surface closer to the poles will be hotter than those near the equator which are pushed further from the stellar centre due to the centrifugal force imposed by the stellar rotation. The temperature of the surface at a given latitude, θ, is given by

$\displaystyle T_{eff}(\theta)=T_p\left(\frac{g(\theta)}{g_p}\right)^\beta$

Where Tg and Tp are the surface gravity and temperature at the pole, respectively, and β is the gravity darkening coefficient, which much like limb-darkening coefficients, are dependent on the properties of the star and will have to be tinkered with to get a good fit.

An example of this comes from the nearby and well-known rapidly rotating, A-type star Altair, which has been spatially resolved using interferometry, allowing for the construction of a low-resolution temperature map.

Altair (Source)

Notice the oblateness of Altair is greater than even that of Saturn. This has a strong effect on the surface temperature of the star, with the equator being over 1,000 K cooler than the poles!

Much like in the case of limb-darkening, gravity-darkening will cause the luminosity profile of the stellar disc to be concentrated in one particular area. Unlike the case of limb-darkening, however, this area need not be the centre of the disc, but could instead be anywhere on the disc. This allows for transit light curves to actually be asymmetric in the case of a misaligned planetary orbit. A transiting planet in a polar orbit of a gravity-darkened star with, for example, a 45 degree rotation axis inclination, will have its planet occulting the bright polar region before moving over the more equatorial regions, causing the minima of the transit light curve to be displaced from the transit centre over toward the ingress.

Consider for example the case of KOI-13, a rapdily rotating A-type star, much like Altair, which was discovered to have a transiting hot Jupiter by the Kepler mission.

 KOI-13 Light Curve (Source) KOI-13 Light Curve Residuals

You may need to click on the light curve to expand it to full size to see the slight difference between the gravity-darkened and non-gravity-darkened model. If nothing else, the residuals to the non-gravity-darkened model are clear that the star is not uniformly luminous.

The stars where gravity darkening will be most pronounced will of course be the stars which have high absolute rotation rates. While a high v sin Istar implies a rapid rotation, the converse is not necessarily true since the stellar inclination angle is not known a priori. However, because a gravity-darkened star appears rotationally symmetric when viewed pole-on, the effects it has on the transit light curve are greatly diminished for low values for v sin Istar. This conspires to mean that the stars for which gravity-darkening effects on the light curves of transiting planets will be those stars where Doppler spectroscopy is less effective. Thus, while the Rossiter-McLaughlin effect may not be measurable due to the stellar spectral line broadening and corresponding lack of radial velocity precision, one can still in some cases use gravity-darkening to determine the projected stellar obliquity (or misalignment between the planetary orbit axis and stellar spin axis, however one chooses to interpret the system). Your typical system for which this is applicable is a bloated hot Jupiter transiting a rapidly rotating A-type star, and at present, only a couple such such systems are known.

From the perspective of modelling, gravity darkening is essentially asymmetric limb-darkening. This post was designed to give you the idea behind gravity darkening of stars and how it can affect the light curve for a transiting planet. In a future post, we will look at modelling transiting planets orbiting gravity-darkened stars.

## Probability of Transit

Transiting Planets. Credit: NASA

Transiting planets are valuable items to explore the properties of planetary atmospheres. Planet searches like Kepler that focous on fields of sky tend to reap rewards amongst dimmer stars simply because there are many more dim stars in a given patch of the sky than bright ones. Transiting planets around bright stars are of particular value, though, as the increased brightness makes the system easier to study.

Radial velocity surveys tend to monitor brighter stars since spectroscopy is even more severely limited by stellar brightness than photometry, but it is not limited to observing patches of sky – telescopes performing Doppler spectroscopy tend to observe a single object at a time due to technical and physical limitations. Radial velocity surveys are also much less sensitive to the inclination angle of a planet orbit with respect to the plane of the sky. The planet doesn’t have to transit to be spectroscopically detectable. As such, radial velocity surveys tend to generate discoveries of planet candidates with unknown inclinations and true masses, but around much brighter stars than those planets discovered by the transit method.

As such, planet candidates discovered by radial velocity, especially planet candidates in short orbital periods are excellent targets for follow-up observations to attempt to detect transits. Transiting planets that have been discovered first through radial velocity have been of great scientific interest due to their host stellar brightness and thus ease of study. If more such systems are found, it would be of great benefit to understanding extrasolar planet atmosphere. While only a hand-full of transiting planets have been discovered first through radial velocity, they all orbit bright stars and are some of the best-characterised planets outside our solar system.

The probability that a planet will transit is, as has been discussed previously, given by
$\displaystyle P_{tr} = \frac{R_*}{a}$
where a is the semi-major axis of the planet orbit. This is the distance between the centre of the star and the centre of the planet. However, due to the inclination degeneracy – the reoccurring evil villain constantly plaguing radial velocity science – the star-planet separation is unknown. Remember that the period of the RV curve gives only the orbital period of the planet. If the orbital period is held constant, increasing the mass of the planet increases the star-planet separation. An increase in the total system mass requires greater separation between the two bodies to preserve the same orbital period.

For example, if radial velocity observations of a star reveal the presence of a mp sin i = 1 ME planet candidate, but the inclination is actually extremely low such that the true mass of the companion is in the stellar regime, then because the mutual gravitational attraction between the two stars will be much greater than the mutual gravitational attraction between the star and an Earth-mass planet at the same period, the two stars must have a wider separation, otherwise their orbital period would be smaller.

Mathematically, the true semi-major axis is given by
$\displaystyle a = \left(\frac{G[M_*+M_{\text{pl}}(i)]}{4\pi^2}\right)^{1/3}T^{2/3}$
Where G is the gravitational constant, and Mpl(i) is the mass of the planet at a given inclination i, and T is the period of the system. It is worth noting that the true semi-major axis is not significantly different from the minimum semi-major axis as long as the mass of the star is much greater than the mass of the planet – which is typically the case.

The fact that the true semi-major axis is a function of the unknown inclination makes for an interesting clarification: The probability that a planet of unknown inclination will transit is not simply given by Rstar/a, but is only approximated by it. If we assume that the distribution of planet masses is uniform (and extending through into the brown dwarf mass regime), then you would expect a planet with a minimum mass equal to Earth to have a much greater chance of being a bona-fide planet than a planet with a minimum-mass of 10 MJ, simply because there is a greater range of inclinations the former planet can be while still remaining in the planetary mass regime. Taking this a step further, even if both the Earth-mass planet candidate and the 10 Jupiter-mass planet candidate have the same orbital period, the probability that the latter planet transits ends up being less than the Earth-mass planet simply because of its high mass. Since its inclination is unknown, the probability that its mass is so high that the true semi-major axis is noticeably larger than the minimum semi-major axis is much higher, resulting in a likely lower transit probability.

Except it turns out that the mass distribution of planets and brown dwarfs isn’t constant. Earth-sized planets are significantly more common than Jupiter-sized planets, and super-Jupiters appear rare. It isn’t clear yet what the mass distribution planets actually is, with significant uncertainty in the sub-Neptune regime, but it is clear that for a highly accurate estimate of the transit probability, the inclination distribution cannot be thought of as completely random as it is fundamentally tied to the planet mass distribution.

Planet Mass Distribution given by Ida & Lin (Left) and Mordasini (Right)

Consider the case of a super-Jovian planet candidate, perhaps with a minimum mass of 7 or 8 Jupiter-masses. Because a significant fraction of physically allowable inclinations would place the true mass planet into a mass regime that is in reality sparsely populated, it is less likely that the planet candidate’s orbit is in those inclinations. It is thus more likely that the planet candidate’s orbit is edge-on than would be expected from the probability function of randomly oriented orbits. As such, the transit probability of a super-Jovian planet is actually boosted by ~20 – 50% over what you would expect from Ptr = Rstar/a. If this is the case, then we would expect to find an excess in the fraction of transiting planets in this mass regime then would be expected purely from the standard transit probability function. Indeed this is what we see.

Candidate planets with masses in the terrestrial planet regime are similarly affected, with broadened transit probabilies owing to the fact that terrestrial planets are more common than higher mass planets, arguing in favour of a higher inclination than the random inclination distribution function.

On the other hand, planet or brown dwarf candidates of minimum masses in the most sparsely populated region of the mass distribution are unlikely to truly have that mass. They are quite likely in orbits with low inclinations and with much higher true masses. The transit probability for companion candidates with minimum masses in this mass regime are actually reduced from the standard transit probability function.

Geometric and a posteriori transit probabilities

In the table above, taken from this preprint, we see that the geometric transit probability, Ptr,0, can be much less than the a posteriori transit probability, Ptr. The transit probability for 55 Cnc e, for example, jumps up from 28% to 36%. With these higher a posteriori transit probabilities, these short-period low-mass planets should be followed-up for transits. If transits are found, it would be of significant benefit to the extrasolar planet field.

In summary, there are various additional effects that can cause the a posteriori transit probability to be significantly different from the geometric transit probability. Planets with only minimum masses known can be more accurately assigned a transit probability when taking into account the uneven planetary mass distribution. Low-mass planets and super-Jupiters are more likely to transit than their geometric transit probability because a significant range of the inclination space is consumed by planets of masses that are simply rare. These planet candidates are more promising targets for transit follow-up than, for example, Jupiter-mass planets or intermediate-mass brown dwarfs.

As anyone with any contact to the outside world knows, the planet Venus transited its own star as seen from our perspective this week. The only thing separating this from being nearly identical to an exoplanet (aside from the obvious difference in distance) is that as Earth orbits the same star, our perspective keeps changing, denying us the ability to observe a transit every Venusian revolution (there are also some issues with the differences in the inclinations between the two orbits that play a vital role in making this an uncommon phenomenon for us).

The above image, taken by the JAXA/NASA Hinode spacecraft shows for Venus what happens essentially each time an exoplanet transits its star. The Kepler spacecraft watches thousands of planets do this, but not quite as dramatically, for Kepler cannot resolve the discs of its 150,000 target stars, and instead has to rely on the characteristic dimming of the star as the planet blocks some of the star’s light.

Notice the entire limb of the planet is visible, even the part that is not yet over the solar disc. Sunlight is passing through the Venusian atmosphere and is scattered every which way, some of it reaching us for us to observe. The colour of the sunlight will be changed as it filters through the atmosphere, just like on Earth where our atmosphere turns sunsets and sunrises red. This is exactly what is observed from Earth when we determine the compositions of extrasolar planet atmospheres through transmission spectroscopy.

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 $R_b$ = 2$R_c$, 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.