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More precisely, what are the factors that influence that size (mass of the star, spin velocity, accretion disk around it, ... do any of those things matter?)

I tried reading this. I cannot: [paper about relative size of neutron stars' magnetospheres]

Is there an equation for the size of neutron-stars magnetospheres? If so, can someone with just barely passable knowledge of differential calculus solve it?

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The size of a neutron star's magnetosphere (in-so-far as it can be approximated as something spherical) is, to within a small numerical factor, given by the Alfven radius. This is where the magnetic energy density equals the kinetic energy density of the surrounding gas/plasma.

Using Gaussian units, the energy density is $B^2/8\pi$, where $B$ is the magnetic field. If we assume that the magnetic field is dipolar and the magnetic dipole moment is $m$, then $B \simeq m/r^3$, where $r$ is the radial distance from the neutron star centre.

If the neutron star is accreting then the kinetic energy density is $\rho v^2/2$, where $\rho$ is the mass density and $v$ the inflow velocity. If we further assume spherically symmetric inflow and that the velocity would just be given by gravitational freefall, then $v = \sqrt{2GM/r}$ and the density is given by the conservation of mass equation: $$ \dot{M} = 4 \pi r^2 \rho v\ ,$$ where $\dot{M}$ is the total mass accretion rate.

If you now equate the magnetic and kinetic energy densities at $r=r_A$ $$ \frac{m^2}{8\pi r_{A}^{6}} = \rho \frac{GM}{r_A} = \frac{\dot{M}}{4\pi r_A^2}\left(\frac{GM}{2r_A}\right)^{1/2}\ ,$$ then the Alfven/magnetosphere radius is given by $$r_A = \left( \frac{m^4}{2G M \dot{M}^2}\right)^{1/7}\ .$$

The paper you refer to contains this formula but also notes that it is only good to a factor of two or so. That is because of various factors - the field may not be dipolar (and a dipolar field is of course not spherically symmetric!), the accretion may not be spherically symmetric and so on. These in turn can depend on how fast the neutron star is spinning.

So the magnetospheric radius is quite strongly dependent on the magnetic dipole moment of the neutron star and quite weakly, and inversely, dependent on the mass of, and the mass accretion rate onto, the neutron star. This is all that matters to first-order.

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  • $\begingroup$ I'm very grateful for your answer. I've been struggling to understand it without success, and I hope your answer helps people with the same doubt that are more educated than me. I'm not completely sure what the letters mean. Is B the magnetic field vector? If so, what is r? I think I'll try to find some books on magnetism first. $\endgroup$
    – Davi
    Jul 4 '21 at 11:46
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    $\begingroup$ @Davi see edits. $\endgroup$
    – ProfRob
    Jul 4 '21 at 11:51

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