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:

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

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,

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

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.

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

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

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.

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).

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.

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.

on The exoplanet transit method.
1. […] planet’s semi-major axis divided by the star’s radius (a/R_*), and b = (a/R_*) * Cos i . It’s important to keep in mind though that while I have freedom to choose i, a is […]

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3. Max Nighswander
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Does the variable P refer to the orbital period here?

4. Angel
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Is there any method to calculate the orbital inclination from it?

5. Roque Caballero
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Hi Paul,
Firstly many thanks for your blog. It’s quite interesting!

I’ve a request regarding the topic “Orbital inclination”. I’ve noticed that the reference to the transit equation is missing. Please, could you provide it? I’d like to know more about it.

Many thanks,

Roque

6. |

whats the first equation called?

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Not sure it has a name. It is simply a ratio of fluxes where the flux is expressed as: F = L / 4*pi*R^2 (the inverse square law, where L is luminosity).

7. |

Is it possible to replace flux with lux ? Or are they just the same.

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In astronomy the flux term is commonly defined as the amount of energy transferred in the form of photons at a certain distance from the source per unit area per second. Check out Radiative flux. Lux is similar in that it also measures a flux per unit area, however, lux is used as a measure of the intensity, as perceived by the human eye. Since we are interested in the total loss of flux and not just the loss of flux perceived by the human eye, we choose to not use lux but flux.

8. Boris
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Hello! Sorry for my off-the-wall amateur questions.

First off, I couldn’t find a definition of “transit depth”. Is it basically an amount of the “dip” in the star’s brightness?
Is the “transit depth” roughly proportional to the ratio of the planet’s disk area to the star’s disk area? TD ≈ ((Rp)^2) / ((R*)^2) ?

The reason I’m asking – I was looking at one of the many discussions about the famous Tabby’s star’s 20% dips in brightness, and according to Wiki, the leading theory is a “swarm of comets”. One of the papers Wiki ( https://en.wikipedia.org/wiki/KIC_8462852 ) refers to ( https://arxiv.org/abs/1511.08821 ), says “To reach a transit depth of ∼0.2, the comets need to be in a close group of ∼30, if they are ∼100 km in radius”. I can’t see how this could be: Tabby’s star radius is 1.58 of our Sun, or about 1000000km, so if my understanding of the “transit depth” is correct, TD of such a swarm should be 0.0000002, not 0.2. Am I missing something basic?

Thank you so much!

Boris

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Hi Boris.

When we are talking about transit depth we are referring to how much flux has been blocked out compared to some reference. Usually the reference is the continuum (basically when nothing is transiting in front of the star). Say then that the star is transited by an exoplanet, comet etc.. then this will cause a decrease in the measured flux and we will see a dip in the light curve. The difference in flux from the continuum to the lowest point of the light curve is what we call the transit depth. Typically hot Jupiters block out 1-2 % percent of the flux. This is equivalent to a transit depth of 0.01 – 0.02.

Tabby’s star is an extraordinary case where a lot more light is being blocked out at times. A transit depth of 0.2 means that 20% of the flux is being blocked out. Have a look at Fig 1. from the paper you are referring to. They have normalised the flux so that 1 is their reference. Then the transit event occurs and they measure 0.8. This means a 1 – 0.8 = 0.2 transit depth = 20% decrease.

9. Boris
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(An update to my previous comment)
Found what I was missing: I read the paper itself rather than just the abstract, and it’s not the comets themselves that are supposed to cause the dip in brightness, but rather, the clouds of dust they produce when grazing the star. They assume a very high dust-production rate and large groups of comets, all grazing the star simultaneously.

Thank you so much!

Boris

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That is correct Boris. The comets are tiny in comparison with the star and block out a negligible amount of light. It is what they leave behind such as dust which blocks the light. Good on you for continuing to read the paper!

10. Emil
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Hey Paul.
Im a student and i wondered if you could help me calculate the radius of an exoplanet.
I know you touched it in the first part of the text, but i wondered if you could go into a little bit more detail, or atleast point me in the right direction.

11. |

No problem Emil. Just contact me directly with your problem by selecting ‘Contact’ from the top menu.