A transiting planet around a star gives a light curve with transit depth

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Celestial mechanics and planetary systems · Celestial mechanics and planetary systems

Stars from the Inside, Ch Planetary systems: formation estimates, transits and the atmospheres of other worlds

Problem

\delta = 0.0100

, period

P = 1.20\,\mathrm{d}

, total transit duration

T_{14} = 1.30\,\mathrm{h}

and flat-bottom duration

T_{23} = 1.05\,\mathrm{h}

. Assume a circular, near edge-on orbit so that

T_{14}\approx (P/\pi)(R_*/a)\sqrt{(1+k)^2-b^2}

and

T_{23}\approx (P/\pi)(R_*/a)\sqrt{(1-k)^2-b^2}

, where

k = R_p/R_*

. ((a)) Find

k

b

and

a/R_*

. ((b)) Find the stellar mean density

\rho_*

. ((c)) The Rossiter-McLaughlin anomaly for this system is antisymmetric about mid-transit, positive (redshifted) in the first half of the transit and equal and opposite in the second, with a peak amplitude of

15\,\mathrm{m\,s^{-1}}

, and the star has

v\sin i_* = 3.0\,\mathrm{km\,s^{-1}}

. Using

\Delta v \approx k^2v\sin i_*\sqrt{1-b^2}

, comment on whether the data are consistent with

\lambda\approx0

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