# 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).