In a globular cluster

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

Stars from the Inside, Ch Accretion power: mass transfer, discs and the weighing of black holes

Problem

In a globular cluster, a metal-poor

0.8\,M_\odot

giant fills its Roche lobe and feeds a neutron star of mass

M = 1.4\,M_\odot

, radius

R_* = 10\,\mathrm{km}

, moment of inertia

I = 1.0\times10^{38}\,\mathrm{kg\,m^2}

and surface dipole field

B_* = 3.0\times10^{4}\,\mathrm{T}

. The giant's nuclear evolution drives transfer at

\dot M = 1.0\times10^{-9}\,M_\odot\,\mathrm{yr^{-1}}

, and the neutron star starts out spinning slowly. ((a)) Compute the accretion efficiency

\eta = GM/R_*c^2

and the accretion luminosity

L = \eta\dot Mc^2

, and compare

L

with the Eddington luminosity

L_{\rm Edd} = 4\pi GMm_pc/\sigma_T

. Can the neutron star take all the gas it is offered? ((b)) Compute the magnetospheric radius

r_m = \left[2\pi B_*^2R_*^6/(\mu_0\dot M\sqrt{2GM})\right]^{2/7}

and the equilibrium spin period

P_{\rm eq} = 2\pi\sqrt{r_m^3/GM}

. ((c)) Gas that joins the star from the disc's inner edge brings with it the specific angular momentum of a Keplerian orbit at

r_m

. Show that the spin-up torque is

N = \dot M\sqrt{GMr_m}

, and estimate the time needed to spin the star up from a slow spin to

P_{\rm eq}

, holding

\dot M

r_m

and

I

fixed. ((d)) How much mass does the neutron star accrete in that time? Compare it with the

0.6\,M_\odot

envelope the giant can give before only its

0.2\,M_\odot

helium core is left, and say whether this binary can recycle the neutron star into a millisecond pulsar.

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