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 unpke a binary system.

F₀ is flux of the star before the planet occults it.

F₁ is the flux after the entire planet is in front of star.

The following image will be used for all the calculations.

$$frac{F_0 - F_1}{F_0} = frac{pi r_p^{2}}{pi R^2_ast}$$

$$frac{Delta F}{F} cong frac{r^2_p}{R^2_ast}$$

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

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

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 incpnation i of the system. Transit latitude and incpnation are interchangeable.

From the above images, we can write −

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

$$frac{h}{R_ast} = sin(delta)$$

$$cos(i) = frac{R_ast sin(delta)}{a}$$

$$y^2 = (R_ast + R_p)^2 - h^2$$

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

$$sin( heta) = frac{y}{a}$$

$$ heta = sin^{-1}left [ frac{(R_ast + R_p)^2 - a^2cos^2(i)}{a^2} ight ]^{frac{1}{2}}$$

$$t_T = frac{P}{2pi} imes 2 heta$$

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.

$$sin(frac{t_Tpi}{P}) = frac{R_ast}{a}left [ left ( 1+ frac{R_p}{R_ast} ight )^2 - left ( frac{a}{R_ast}cos(i) ight )^2 ight ]^{frac{1}{2}}$$

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

$$sin(frac{t_Tpi}{P}) = frac{R_ast}{a}left [ 1- left ( frac{a}{R_ast}cos(i) ight )^2 ight ]^{frac{1}{2}}$$

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

$$t_T = frac{P}{pi}left [ left ( frac{R_ast}{a} ight )^2 - cos^2(i) ight ]^{frac{1}{2}}$$

Here, t_T, P, R∗ are the observables, a and i should be found out.

Now,

$$sin(frac{t_Fpi}{P}) = frac{R_ast}{a}left [left (1 - frac{R_p}{R_ast} ight )^2 - left ( frac{a}{R_ast}cos:i ight )^2 ight ]^{frac{1}{2}}$$

where, $y^2 = (R_ast − R_p)^2 − h^2$.

Let,

$$frac{Delta F}{F} = D = left ( frac{R_p}{R_ast} ight )^2$$

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,

$$R_ast propto M^alpha_ast$$

$$frac{R_ast}{R_0} propto left ( frac{M_ast}{M_0} ight )^alpha$$

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 ?~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., probabipty of observing a transit. Transit probabipty (probabipty of observing) calculations are shown below.

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

$$Sopd : angle :of :planet : = 2pi left ( frac{2R_ast}{a} ight )$$

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

$$Sopd :angle :of :sphere : =: 4pi$$

The probabipty is the ratio of these two areas −

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

$= frac{4pi a_pR_ast}{4pi a^2_p} = frac{R_ast}{a_p}$ $frac{area: of: hollow : cycpnder}{area: of: sphere}$

This probabipty 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 probabipty is related to the sopd angle of planet.

This probabipty is independent of observer frame of reference.