A newly formed proto-neutron star has mass and radius

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Stellar death and compact objects · Astrophysical fluid dynamics and plasmas

Stars from the Inside, Ch Stellar explosions: core collapse, the neutrino-driven shock, radioactive light and remnants

Problem

A newly formed proto-neutron star has mass

M = 1.4\,M_\odot

and radius

R = 25\,\mathrm{km}

. Treat it as uniform at its mean density for an order-of-magnitude estimate. Neutrinos deep in the interior have a typical energy

E_\nu = 50\,\mathrm{MeV}

and random-walk out by scattering from, and absorption on, individual nucleons. Take a representative cross-section per nucleon

\sigma_\nu \approx 1\times10^{-47}\left(\frac{E_\nu}{1\,\mathrm{MeV}}\right)^2\,\mathrm{m^2}.

((a)) Find the mean density

\bar\rho

and the number density of nucleons

n = \bar\rho/m_H

. ((b)) Find the neutrino mean free path

\lambda = 1/(n\sigma_\nu)

. ((c)) Estimate the diffusion time

t_{\rm diff} \approx R^2/(\lambda c)

. ((d)) Compare

t_{\rm diff}

with the roughly

13\,\mathrm{s}

span over which Kamiokande II, IMB and Baksan recorded neutrinos from SN 1987A, and comment on whether the trapped-diffusion picture is consistent with that duration.

Browse the problem library