= Cumulative periastron-time shift from orbital decay
{title2=$\delta T\simeq\frac12(\dot P_b/P_{b0})t^2$}
A tiny <orbital period> change accumulates into a larger timing shift. Choose the original model $P_b(t)=P_{b0}+\dot P_b t$, with $P_{b0}=28{,}800\,\mathrm{s}$ and $\dot P_b=-2\times10^{-12}\,\mathrm{s/s}$. Expanding the accumulated cycle count gives
$$
N(t)=\int_0^t\frac{dt'}{P_b(t')}\simeq\frac{t}{P_{b0}}-\frac{\dot P_b t^2}{2P_{b0}^2},\qquad
\delta T\simeq\frac{\dot P_b t^2}{2P_{b0}}.
$$
Negative $\delta T$ means an event of fixed orbital cycle count arrives earlier than a constant-<period> prediction. After twenty Julian years, the chosen <period> has decreased by $1.262\,\mathrm{ms}$, while the accumulated event shift is about $-13.83\,\mathrm{s}$. This is a time shift of <periapsis> passage; it is different from a rotation of the direction of <periapsis>.
\Image[/binary-pulsar-kepler-delay-and-period-drift.png]
{title=A chosen eccentric light-travel delay and an accumulated orbital-period change}
{id=binary-pulsar-kepler-delay-and-period-drift}
{description=Original calculations, not observations or a historical fit. The upper panel solves Kepler's equation for x=2 seconds, an 8 hour orbital period, eccentricity 0.6 and argument of periapsis 40 degrees; the dashed curve sets eccentricity to zero. The lower panel independently chooses a constant period derivative of minus 2 times 10 to the minus 12 seconds per second. It plots the leading event-time shift against a constant-period ephemeris; no masses or radiation mechanism are fitted.}
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