Keplerian Orbital Velocity
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Circular Orbital Velocity under Gravitational Forces
In the case of a two-body problem and simple circular motion due to only gravitational forces, the Keplerian orbital velocity can be found by simply equating centrapetal force to gravitational force. In general the two masses, say $m$ and $M$, orbit around the center of mass of the system, with an effective mass equal to the reduced mass. In the case that $m \ll M$ the center of mass is near the center of gravity of $M$, and the Keplerian orbital velocity is
\[
v = \sqrt{\left(\frac{GM}{r}\right)}
\]
or
\[
v \sim 30 \left(\frac{M}{M_{\odot}}\right)^{\frac{1}{2}}
\left(\frac{1 {\rm AU}}{r}\right)^{\frac{1}{2}} \ \ {\rm km \ s^{-1}}
\]
or
\[
v \sim 66,600 \left(\frac{M}{M_{\odot}}\right)^{\frac{1}{2}}
\left(\frac{1 {\rm AU}}{r}\right)^{\frac{1}{2}} \ \ {\rm miles \ per \ hour}
\]
where 1 AU is 1 astronomical unit, and $M_{\odot}$ is a solar mass. In all of the above, $r$ is the radius of the orbit.
Another convenient form (again for $m \ll M$) that expresses the
Keplerian velocity as a fraction of the speed of light, $c$, is
\[
\frac{v}{c} = \sqrt{ \frac{r_{g}}{r} }
\]
where $r_{g}$ is the gravitational radius, for which convenient
forms are
\[
\begin{array}
rr_{g} & = 1.4822 \times 10^{13}M_{8} \ {\rm cm} \\
\\
r_{g} & = 1.4822 (M/M_{\odot}) \ {\rm km} \\
\\
r_{g} & \sim M_{8} \ {\rm AU}, \\
\end{array}
\]
where $M_{8}$ is the central is in units of $10^{8}$ solar masses.
Orbital Velocity when One Mass is Not Negligible
In the case that one of the masses is not negligible compared to the other, simply use the total mass ($M+m$) in place of $M$ in the above formulas for orbital velocity.
Orbital Velocity for Elliptical Orbits
In the general case of elliptical orbits, simply multiply any of the above equations for orbital velocity by the following factor, $f_{\rm elliptical}$:
\[
f_{\rm elliptical} \ \equiv \frac{v_{\rm elliptical}}{v_{\rm circular}}
\ = \ \left(2 – \frac{r}{a}\right)^{\frac{1}{2}}
\]
where $a$ is the semimajor axis of the ellipse, and $r$ is the length of the line joining the two masses (obviously, for elliptical orbits the speed varies around the ellipse).
