A protostar still contracting towards the main sequence has mass and radius

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Stellar structure, energy generation and evolution · Stellar structure, energy generation and evolution

Stars from the Inside, Ch Pressure against gravity: the equation of state, polytropes and the Lane-Emden equation

Problem

A protostar still contracting towards the main sequence has mass M=1.5 M⊙M = 1.5\,M_\odot and radius R=3 R⊙R = 3\,R_\odot, and is modelled as a fully convective n=3/2n=3/2 polytrope of fully ionised gas with adiabatic index γ=5/3\gamma = 5/3. ((a)) Starting from hydrostatic equilibrium dPdr=−Gmρr2\dfrac{dP}{dr} = -\dfrac{Gm\rho}{r^2}, multiply through by 4πr34\pi r^3, integrate over the star using P(R)=0P(R)=0, and integrate the left-hand side by parts, to show that the gravitational energy can be written Ω=−3∫0RP 4πr2 dr\Omega = -3\displaystyle\int_0^R P\,4\pi r^2\,dr. ((b)) Using the standard polytrope result Ω=−35−nGM2R\Omega = -\dfrac{3}{5-n}\dfrac{GM^2}{R}, evaluate Ω\Omega for this protostar, in joules. ((c)) The internal (thermal) energy of the ideal gas is U=∫0RPγ−1 4πr2 drU = \displaystyle\int_0^R \dfrac{P}{\gamma-1}\,4\pi r^2\,dr. Combine this with part (a) to show that the total energy is E=U+Ω=3γ−43(γ−1)ΩE = U+\Omega = \dfrac{3\gamma-4}{3(\gamma-1)}\Omega, and evaluate EE, in joules. ((d)) State, in terms of Γ1\Gamma_1, why this configuration is dynamically stable, and say what the formula for EE implies would happen if Γ1\Gamma_1 were to fall to exactly 4/34/3.

Browse the problem library