A planet of radius ratio transits its host star with impact parameter (the projected…

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Radiation, radiative transfer and stellar atmospheres · Radiation, radiative transfer and stellar atmospheres

Stars from the Inside, Ch Radiative transfer solved: the grey atmosphere, the diffusion limit and the curve of growth

Problem

A planet of radius ratio

k \equiv R_p/R_* = 0.10

transits its host star with impact parameter

b = 0.50

(the projected distance of the planet's centre from the star's centre at mid-transit, in units of

R_*

). The star's disc obeys the grey Eddington limb-darkening law

I(\mu)/I(1) = (2+3\mu)/5

, where

\mu = \cos\theta

is the direction cosine at the point on the disc being blocked. Treat the planet as small enough that at mid-transit it blocks a patch of the stellar disc of local intensity

I(\mu_p)

, where

\mu_p

is the value of

\mu

at the sub-planet point. ((a)) Using the transit geometry, find

\mu_p = \sqrt{1-b^2}

and evaluate the local intensity ratio

I(\mu_p)/I(1)

from the grey limb-darkening law. ((b)) The star's disc-averaged intensity for a linear law

I(\mu)/I(1) = 1-u(1-\mu)

\langle I\rangle/I(1) = 1-u/3

. Evaluate this for

u = 3/5

. ((c)) The mid-transit relative flux decrement is

\delta = k^2\,I(\mu_p)/\langle I\rangle

. Compute

\delta

and compare it with the flat-disc value

k^2

that would be used if limb darkening were ignored.

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