= Projected pulsar semimajor axis
{title2=$x=a_1\sin i/c$}
The <semimajor axis> $a_1$ of the <pulsar>'s orbit about the binary <barycenter>, projected along the line of sight and divided by the <speed of light>. The <orbital inclination> $i$ is zero for a face-on <orbit>. Thus $x$ is a time, often quoted as the numerical number of <light-seconds> in $a_1\sin i$, not the true binary separation. The relative-orbit <semimajor axis> is instead $a=a_1+a_2$; <conservation of momentum> gives $a_1=a m_2/(m_1+m_2)$.
Combining <Kepler's third law> with that <barycenter> relation gives the timing form of the <binary mass function>, also valid for eccentric <Kepler orbits>:
$$
f=\frac{4\pi^2(cx)^3}{GP_b^2}=\frac{(m_2\sin i)^3}{(m_1+m_2)^2}.
$$
Choose $x=2\,\mathrm{s}$ and $P_b=8\,\mathrm{h}$ for an original example: $a_1\sin i=599{,}585\,\mathrm{km}$ and $f\simeq0.0773M_\odot$, where $M_\odot$ is a <solar mass>. For a specified pulsar <mass> $m_1=1.4M_\odot$, the smallest allowed companion <mass> occurs at $\sin i=1$ and is approximately $0.698M_\odot$. A different assumed $m_1$ changes this bound. One measured $f$ does not determine both <masses> and <orbital inclination>.
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