Let $a_{\rm ns}$ be the neutron star's distance from the <centre of mass>. In a circular orbit,
$$
a_{\rm ns}=a\frac{M_{\rm wd}}{M_{\rm ns}+M_{\rm wd}},
\qquad
K=\frac{2\pi a_{\rm ns}}P\sin i.
$$
Combining this with <Kepler third law>,
$$
\frac{4\pi^2a^3}{P^2}=G(M_{\rm ns}+M_{\rm wd}),
$$
gives the <binary mass function>
$$
\boxed{F\equiv\frac{PK^3}{2\pi G}
=\frac{M_{\rm wd}^3\sin^3i}
{(M_{\rm ns}+M_{\rm wd})^2}}.
$$
Since $\sin i\leq1$ and $(M_{\rm ns}+M_{\rm wd})^2>M_{\rm wd}^2$,
$$
\boxed{F<M_{\rm wd}}.
$$

When a low-mass red giant undergoes stable <Roche-lobe overflow>, the neutron star receives matter with substantial <specific angular momentum>, usually through an <accretion disk>. The accretion torque spins it up and weakens its external magnetic field, producing a <recycled pulsar> with a millisecond period. Removal of the donor's envelope exposes its helium core as a low-mass <white dwarf>.

At detachment, the donor mass and core mass both become $M_{\rm wd}$. Its luminosity is fixed by the red-giant core-mass--luminosity relation, and the supplied radius law therefore makes its final radius $R_L$ a function only of $M_{\rm wd}$. Cubing the Roche-lobe relation gives
$$
\frac{R_L^3}{a^3}
=\frac{M_{\rm wd}}{M_{\rm wd}+M_{\rm ns}}.
$$
Eliminating $a$ with <Kepler third law> yields
$$
\boxed{P^2=\frac{4\pi^2R_L(M_{\rm wd})^3}{GM_{\rm wd}}}.
$$
This is the <white-dwarf mass--orbital-period relation>: dependence on the neutron-star mass cancels.

For randomly oriented binaries, the <isotropic binary inclination distribution> is
$$
\boxed{p(i)=\sin i,\qquad0\leq i\leq\frac\pi2}.
$$
The observed period gives $M_{\rm wd}$ from the preceding relation. If $M_{\rm ns}=1.35M_\odot$ is adopted, the measured mass function then gives
$$
\boxed{\sin i=
\left[\frac{F(M_{\rm ns}+M_{\rm wd})^2}
{M_{\rm wd}^3}\right]^{1/3}},
$$
so the inclination is inferred without astrometry.

Two plausible causes of an apparent shortage of edge-on systems are:

* High-inclination radio systems are preferentially obscured or eclipsed by ionized gas near the companion, producing an observational selection effect.
* Accretion makes recycled neutron stars systematically heavier than $1.35M_\odot$. Using too small an assumed $M_{\rm ns}$ makes the inferred $\sin i$ too small and shifts truly high-inclination systems to lower inferred inclinations.

Scatter or bias in the core-mass--period relation can reinforce the second effect.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 322 / 1 / Solution

{title2=$F$}
{wiki=Binary_mass_function}

For a single-lined circular spectroscopic binary whose visible component has radial-velocity amplitude $K_1$ and orbital period $P$, the binary mass function is
$$
F=\frac{PK_1^3}{2\pi G}
=\frac{M_2^3\sin^3i}{(M_1+M_2)^2}.
$$


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