A planet around a star of spectroscopically determined mass is detected both by…

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

A planet around a star of spectroscopically determined mass

M_* = 1.00\,M_\odot

is detected both by radial velocity and by transit photometry, on a circular orbit of period

P = 5.00\,\mathrm{d}

. The radial-velocity curve has semi-amplitude

K = 40\,\mathrm{m\,s^{-1}}

. The transit light curve has depth

\delta = 1.00\%

, total duration

T_{14} = 4.04\,\mathrm{h}

(first to fourth contact) and flat-bottom duration

T_{23} = 3.24\,\mathrm{h}

(second to third contact). ((a)) The spectroscopic-binary mass function for a single-lined system is

f(m) = \dfrac{PK^3}{2\pi G} = \dfrac{(m_p\sin i)^3}{(M_* + m_p)^2}

for a circular orbit. Take the limit

m_p \ll M_*

to find the planet's minimum mass

m_p\sin i

, in Jupiter masses. ((b)) From

\delta

T_{14}

and

T_{23}

find the radius ratio

k = R_p/R_*

, the impact parameter

b

, and

a/R_*

. ((c)) Use

a/R_*

and Kepler's third law to find the star's mean density

\rho_*

, and compare it with the Sun's mean density,

1.41\times10^{3}\,\mathrm{kg\,m^{-3}}

. State whether the comparison is consistent with an unevolved Sun-like star of this mass.

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