A neutron star of radius rotates with angular velocity and carries a surface magnetic…

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

Stars from the Inside, Ch Neutron stars and pulsars: the densest matter, the maximum mass and the spinning magnet

Problem

A neutron star of radius

R

rotates with angular velocity

\Omega

and carries a surface magnetic field

B

aligned with its rotation axis (an aligned rotator, the simplest case). Treat the star as a perfect conductor, so charges inside it rearrange until the electromagnetic force on them vanishes in the frame that corotates with the star:

\vec E + \vec v\times\vec B = 0,

where

\vec v = \vec\Omega\times\vec r

is the corotation velocity at position

\vec r

measured from the rotation axis, and near the star

\vec B

is taken as uniform and parallel to

\vec\Omega

. ((a)) Write

\vec v

and

\vec B

in cylindrical coordinates

(\varpi,\phi,z)

about the rotation axis and show that the corotation electric field is

\vec E = -\Omega\varpi B\,\hat\varpi

. ((b)) Apply Gauss's law in vacuum,

\nabla\cdot\vec E = \rho/\epsilon_0

, in cylindrical coordinates to find the charge density

\rho_{\rm GJ}

this field requires, and hence the number density

n_{\rm GJ} = |\rho_{\rm GJ}|/e

of charge carriers, the Goldreich-Julian density. ((c)) Evaluate

n_{\rm GJ}

for the Crab pulsar, for which

\Omega = 188\,\mathrm{s^{-1}}

and

B = 3.8\times10^{8}\,\mathrm{T}

at the surface. ((d)) A millisecond pulsar has

P = 5.76\,\mathrm{ms}

and surface field

B = 5.8\times10^{4}\,\mathrm{T}

. Taking

R = 10\,\mathrm{km}

, compute its light-cylinder radius

R_L = c/\Omega

and its polar-cap voltage

\Delta\Phi \approx \Omega^2BR^3/2c

, and compare each with the Crab's

1600\,\mathrm{km}

and

2\times10^{16}\,\mathrm{V}

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