A star of mass drives a steady

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

Stars from the Inside, Ch The Sun as a test: calibration, sound, neutrinos and a magnetised atmosphere

Problem

A star of mass MM drives a steady, isothermal, spherically symmetric, fully ionised stellar wind (no rotation, no magnetic field). At radius rr the wind has density ρ(r)\rho(r) and outward speed v(r)v(r), and obeys the isothermal relation P=ρcs2P=\rho c_s^2 with cs2=kBT/(μmH)c_s^2=k_BT/(\mu m_H) constant throughout the flow (TT the coronal temperature, μ\mu the mean molecular weight in units of mHm_H, kBk_B the Boltzmann constant, mHm_H the mass of the hydrogen atom). ((a)) Starting from mass conservation, 4πr2ρv=M˙=const4\pi r^2\rho v=\dot M=\text{const}, and the steady radial Euler equation ρv dv/dr=−dP/dr−GMρ/r2\rho v\,\mathrm dv/\mathrm dr=-\mathrm dP/\mathrm dr-GM\rho/r^2, derive the wind equation (v−cs2v)dvdr=2cs2r−GMr2.\left(v-\frac{c_s^2}{v}\right)\frac{\mathrm dv}{\mathrm dr}=\frac{2c_s^2}{r}-\frac{GM}{r^2}. ((b)) Argue that a solution with dv/dr\mathrm dv/\mathrm dr finite everywhere must pass through a critical point (rc,vc)(r_c,v_c) at which both sides of the wind equation vanish together, and find rcr_c and vcv_c in terms of GG, MM and csc_s. ((c)) Evaluate csc_s, rcr_c and the ratio rc/Rr_c/R for a star with M=1.2 M⊙M=1.2\,M_\odot, R=1.5 R⊙R=1.5\,R_\odot, a corona at T=2.0×106 KT=2.0\times10^6\,\mathrm K and μ=0.6\mu=0.6.

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