# Cosmology - Transit Method

The Transit Method **(Kepler Space Telescope)** is used to find out the size. The dip in brightness of a star by a planet is usually very less unlike a binary system.

**F**is flux of the star before the planet occults it._{0}**F**is the flux after the entire planet is in front of star._{1}

The following image will be used for all the calculations.

$$\frac{F_0 - F_1}{F_0} = \frac{\pi r_p^{2}}{\pi R^2_\ast}$$

$$\frac{\Delta F}{F} \cong \frac{r^2_p}{R^2_\ast}$$

$$\left ( \frac{\Delta F}{F} \right )_{earth} \cong 0.001\%$$

$$\left ( \frac{\Delta F}{F} \right )_{jupiter} \cong 1\%$$

This is not easy to achieve by ground based telescope. It is achieved by the Hubble telescope.

Here, $t_T$ is time between position A and D and $t_F$ is time between position B and C.

The geometry of a transit related to the inclination **i** of the system. Transit latitude and inclination are interchangeable.

From the above images, we can write −

$$\frac{h}{a} = cos(i)$$

$$\frac{h}{R_\ast} = sin(\delta)$$

$$cos(i) = \frac{R_\ast sin(\delta)}{a}$$

$$y^2 = (R_\ast + R_p)^2 - h^2$$

$$y = [(R_\ast + R_p)^2 - h^2]^{\frac{1}{2}}$$

$$sin(\theta) = \frac{y}{a}$$

$$\theta = sin^{-1}\left [ \frac{(R_\ast + R_p)^2 - a^2cos^2(i)}{a^2} \right ]^{\frac{1}{2}}$$

$$t_T = \frac{P}{2\pi} \times 2\theta$$

Here, $t_T$ is the fraction of a time-period for which transit happens and (2θ/2π) is fraction of angle for which the transit happens.

$$sin(\frac{t_T\pi}{P}) = \frac{R_\ast}{a}\left [ \left ( 1+ \frac{R_p}{R_\ast}\right )^2 - \left ( \frac{a}{R_\ast}cos(i)\right )^2 \right ]^{\frac{1}{2}}$$

Usually, a >> R∗ >> Rp. So, we can write −

$$sin(\frac{t_T\pi}{P}) = \frac{R_\ast}{a}\left [ 1- \left ( \frac{a}{R_\ast}cos(i) \right )^2\right ]^{\frac{1}{2}}$$

Here, **P** is the duration between two successive transits. The transit time is very less compared to the orbital time-period. Hence,

$$t_T = \frac{P}{\pi}\left [ \left ( \frac{R_\ast}{a}\right )^2 - cos^2(i)\right ]^{\frac{1}{2}}$$

Here, **t _{T}, P, R∗** are the observables,

**a**and

**i**should be found out.

Now,

$$sin(\frac{t_F\pi}{P}) = \frac{R_\ast}{a}\left [\left (1 - \frac{R_p}{R_\ast} \right )^2 - \left ( \frac{a}{R_\ast}cos\:i \right )^2\right ]^{\frac{1}{2}}$$

where, $y^2 = (R_\ast − R_p)^2 − h^2$.

Let,

$$\frac{\Delta F}{F} = D = \left ( \frac{R_p}{R_\ast} \right )^2$$

Now, we can express,

$$\frac{a}{R_\ast} = \frac{2P}{\pi} D^{\frac{1}{4}}(t^2_T - t^2_F)^{-\frac{1}{2}}$$

For the main sequence stars,

$$R_\ast \propto M^\alpha_\ast$$

$$\frac{R_\ast}{R_0} \propto \left ( \frac{M_\ast}{M_0}\right )^\alpha$$

This gives **R∗**.

Hence, we get value of ‘a’ also.

So, we get ‘R_{p}’, ’ap’ and even ‘i’.

For all this,

$$h \leq R_\ast + R_p$$

$$a\: cos\: i \leq R_\ast + R_p$$

For even 𝑖~89 degrees, the transit duration is very small. The planet must be very close to get a sufficient transit time. This gives a tight constraint on ‘i’. Once we get ‘i’, we can derive ‘m_{p}′from the radial velocity measurement.

This detection by the transit method is called as chance detection, i.e., probability of observing a transit. Transit probability (probability of observing) calculations are shown below.

The transit probability is related to the solid angle traced out by the two extreme transit configurations, which is −

$$Solid \: angle \:of \:planet \: = 2\pi \left ( \frac{2R_\ast}{a} \right )$$

As well as the total solid angle at a semi-major axis a, or −

$$Solid \:angle \:of \:sphere \: =\: 4\pi$$

The probability is the ratio of these two areas −

$$= \: \frac{area \:of\: sky \: covered \:by\: favourable \: orientation}{area\: of\: sky \:covered\: by\: all\: possible\: orientation\: of\: orbit}$$

$= \frac{4\pi a_pR_\ast}{4\pi a^2_p} = \frac{R_\ast}{a_p}$ $\frac{area\: of\: hollow \: cyclinder}{area\: of\: sphere}$

This probability is independent of the observer.

## Points to Remember

- The Transit Method (Kepler Space Telescope) is used to find out the size.
- Detection by Transit Method is a chance detection.
- The planet must be very close to get sufficient transit time.
- Transit probability is related to the solid angle of planet.
- This probability is independent of observer frame of reference.