When exoplanets pass in front of their host star (as seen from Earth), a portion of the start light is blocked out and a decrease in the photon flux is measured. Measuring the change in flux over time, allows for the creation of a light curve. Fitting models to the light curve, various characteristics such as orbital motions and atmospheric composition can be extracted. Both the size of the host star and the planet will determine the decrease in flux during the transit. The orbital distance between the exoplanet and its host star does not affect the transit depth due to the enormous distance from Earth.

The transit method is particularly useful for calculating the radius of an exoplanet. To first order (assuming the stellar disc is of uniform brightness, and neglecting any flux from the planet) the ratio of the observed change in flux, \Delta F, to that of the stellar flux F can be expressed as:

  \frac{\Delta F}{F} = \frac{R_p^2}{R_*^2}

where R_p and R_* are the planetary and stellar radii respectively. As described below, limb-darkening will have an affect on the transit light curve, but to first order, the equation above holds.

Impact Parameter:

The total transit duration is heavily dependent on the impact parameter b, which is defined as the sky-projected distance between the centre of the stellar disc and the centre of the planetary disc at conjunction* and is shown in Fig. 1**. Assuming a circular orbit the impact parameter is expressed as:

  b = \frac{a\cos i}{R_*}

impact-parameter
Fig. 1: The impact parameter b varies from b=0 centre of stellar disk with b=1 being on the cusp of the disc.

Transit Duration:

The total transit duration, T_{\mathrm{dur}}, defined as the time during which any part of the planet obscures the disc of the star, depends on how the planet transits the host star. If the exoplanet crosses the centre of the stellar disc (b = 0), the transit duration is the longest with b \ne 0 signifying a shorter transit duration. With the help of Fig. 2 and using Pythagoras’s theorem, the length the planet has to travel across the disk of the star can be expressed as,

  2l = 2\sqrt{(R_*+R_p)^2-b^2}

star-next-to-planet
Fig. 2: Star-planet geometry showing the distance traversed by the planet, 2l, impact parameter of the system, b and the stellar and planetary radii, R_* and R_{\rm{p}} respectively.

With the aid of Fig. 3, the exoplanet moves from A to B around its orbit, creating an angle \alpha (measured in radians) with respect to the centre of the host star.

complicado
Fig. 3: The orbital geometry of a transiting exoplanet system showing the projected distance travelled across the surface of the star, 2l, between the points A to B and the angle \alpha this geometry forms, with the inclination, i, and semi major axis, a, shown.

With the assumption of a circular orbit, the distance around an entire orbit is 2\pi a, where a is the radius of the orbit. The arclength between points A and B is \alpha a and the distance along a straight line between A and B is 2l.

From the triangle formed by A, B and the centre of the star,

  \sin \left( \frac{\alpha}{2} \right)= \frac{l}{a}

thus

  T_{dur} = P\frac{\alpha}{2\pi} = \frac{P}{\pi}\sin^{-1} \left( \frac{l}{a} \right) = \frac{P}{\pi}\sin^{-1} \left( \frac{\sqrt{(R_* + R_P)^2 - b^2}}{a} \right)  \label{eq:transit-duration}

an expression of the full transit duration.

The Effects of Limb Darkening:

The effect caused by the stellar disk being brighter in the centre compared to the limb of the disk is called limb darkening. Photons emitted from the limb of the stellar disc at a certain atmospheric depth L, follow a more oblique path through the stellar atmosphere compared to the photons emitted from the centre of the stellar disc as seen in Fig. 4. For the photons escaping the edge of the stellar disc, an optical depth of unity is reached at a higher altitude where the temperature is cooler (T_{\rm{LO}}) and the radiation is less intense causing the apparent darkening.

limb-darkening
Fig. 4: Limb darkening of a star showing how the intensity and temperature diminishes as an observer looks towards the limb of the star.

The limb darkening effect is largest at short wavelengths where a highly rounded light curve is observed. For longer wavelengths the effect is less severe and the centre of the transit takes on a flatter shape (see Fig. 5).

lc_limb
Fig. 5: Two model light curves of the super-Earth GJ~1214b for observations at 5000 Å (blue) and 1~\mu m (orange) using a tunable filter with a width of 12 Å. Observations at shorter wavelengths result in a deeper and narrower transit.

Orbital Inclination:

Radial velocity observations provide information about the minimum mass, of M_p \sin i, assuming the stellar mass is known. To constrain the actual mass of an exoplanet, the orbital inclination, i, has to be measured. This is done by fitting a analytical transit light curve to the data using the transit equation of \cite{mandel02}. A transiting exoplanet which has an impact parameter b \ne 0 or i < 90, will have a shorter transit duration, a shallower transit depth and longer ingress and egress times. This is seen in Fig. 6 where the effects of varying i from 90^{\circ} to 80^{\circ} is shown.

lc_impact
Fig. 6: Inclination values ranging from 90^{\circ} to 80^{\circ} at 1^{\circ} intervals, with the shallowest light curve corresponding to i = 80^{\circ}.

* Conjunction: The point in the orbit where two objects are most closely aligned, as viewed from Earth
** The figures and derivations are adapted from “Transiting Exoplanets”, by Carole A. Haswell.