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.


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5 thoughts on “Of Light and Shadow

  1. […] a planet transits its star (see here), and we’re able to determine it’s mass through some other means, typically Doppler […]

  2. […] a planet transits its star (see here), and we’re able to determine it’s mass through some other means, typically Doppler […]

  3. […] 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 […]

  4. […] sequence stars continuously, using the transit method to discover extrasolar planet candidates (see here for a description of how planets are found this way). It has uncovered thousands of planet […]

  5. […] looked at a simplistic understanding of planetary transits and transit light curves (here). In the interests of accuracy, it would be worth investigating a more formal understanding of the […]

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