A protoplanetary disc has a solids surface density at a distance from a star of mass

PHYSICS · Subjective · 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 protoplanetary disc has a solids surface density Σs=40 kg m−2\Sigma_s = 40\,\mathrm{kg\,m^{-2}} at a distance r=3 AUr = 3\,\mathrm{AU} from a star of mass M∗=1.0 M⊙M_* = 1.0\,M_\odot. Taking a feeding zone of width Δr=10 rH\Delta r = 10\,r_H, the isolation mass of an embryo growing there is Miso=(20πr2Σs)3/2(3M∗)1/2.M_{\rm iso} = \frac{(20\pi r^2 \Sigma_s)^{3/2}}{(3M_*)^{1/2}}. Compute MisoM_{\rm iso} in units of Earth masses, and state whether this embryo alone could trigger runaway gas accretion (which needs a core of roughly ten Earth masses).

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