Work each short estimate using the isolation-mass

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

Work each short estimate using the isolation-mass, reflex-velocity, scale-height, transit-timing and occurrence-rate formulas above. Idealise all orbits as circular unless an eccentricity is given, and treat every star as spherical. ((a)) A protoplanetary disc at

r = 2\,\mathrm{AU}

around a star of mass

M_* = 1\,M_\odot

has a solid surface density

\Sigma_s = 40\,\mathrm{kg\,m^{-2}}

. Using a feeding zone

\Delta r = 10\,r_H

wide, estimate the isolation mass

M_{\rm iso}

in Earth masses. ((b)) A planet with

m_p\sin i = 1.5\,M_J

orbits a star of mass

M_* = 0.9\,M_\odot

with period

P = 2.0\,\mathrm{yr}

and eccentricity

e = 0.3

. Find the star's reflex velocity semi-amplitude

K

. ((c)) Estimate the semi-amplitude

K

that Venus (

m_p\sin i = 0.815\,M_\oplus

P = 0.615\,\mathrm{yr}

, circular orbit) induces on the Sun (

M_* = 1\,M_\odot

). ((d)) A warm Neptune has

R_p = 0.35\,R_J

, atmospheric temperature

T = 900\,\mathrm{K}

, surface gravity

g_p = 15\,\mathrm{m\,s^{-2}}

and mean molecular weight

\mu = 2.3

. Its host star has

R_* = 0.9\,R_\odot

. Estimate the extra transit depth

\Delta\delta

contributed by one atmospheric scale height (

N_H = 1

). ((e)) Two transiting planets around the same star have periods

P_1 = 12.0\,\mathrm{d}

and

P_2 = 25.0\,\mathrm{d}

, close to a 2:1 resonance (

j = 2

). Find the super-period of their transit-timing variations. ((f)) A transit survey monitors

N_* = 8000

Sun-like stars and finds 5 planets, each with transit probability

p_{\rm tr} = 0.006

and pipeline completeness

\eta = 0.4

. Estimate the occurrence rate

f

of such planets per star.

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