= Binary-pulsar periastron advance
{title2=$\dot\omega$}
<Binary-pulsar periastron advance> is the gradual <rotation> of the orbit's <argument of periapsis> $\omega$. A closed <Kepler orbit> has a fixed direction of closest approach. In <general relativity>, the leading conservative correction makes the relative orbit precess even before <gravitational radiation> causes appreciable <binary-pulsar orbital decay>.
For nonspinning point <masses> $m_1,m_2$, total <mass> $M=m_1+m_2$, semimajor axis $a$ of the relative orbit and <orbital eccentricity> $0<e<1$, the first post-Newtonian advance per orbit is
$$
\Delta\omega=\frac{6\pi GM}{ac^2(1-e^2)}.
$$
Divide by the <orbital period> $P_b$ and use <Kepler's third law> $n^2a^3=GM$, where $n=2\pi/P_b$, to obtain
$$
\dot\omega=\frac{3n^{5/3}}{1-e^2}\left(\frac{GM}{c^3}\right)^{2/3}.
$$
Here $\dot\omega$ is in radians per unit <time>. The expansion assumes $GM/[ac^2(1-e^2)]\ll1$.
A <pulsar> acts as an orbital clock: the timing delay changes as the eccentric orbit turns relative to the <line of sight>. If this measured advance is dominated by the relativistic point-mass term, then $P_b,e,\dot\omega$ determine the total <mass> $M$, independently of the unknown <orbital inclination>. They do not separately determine both component <masses>. A second effect, such as the <binary-pulsar Einstein delay>, supplies another <mass> relation; additional timing parameters can then test the theory instead of merely fitting it.<Binary Pulsars and Relativistic Gravity by Joseph Hooton Taylor Jr.>[Section V, equations (5)–(6), pp. 84–85, periastron advance and the two-mass timing test]{ref}
Spin-induced quadrupoles, tidal deformation and higher relativistic orders can contribute in other regimes and must be modeled before interpreting all observed precession as this term. At $e=0$, a separate periastron direction is undefined. Advance of this direction is conceptually distinct from decreasing $P_b$: conservative precession by itself is not evidence of radiated orbital <energy>.
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