Orbital speed

The orbital speed of a body, generally a planet, a natural satellite, an artificial satellite, or a multiple star, is the speed at which it orbits around the barycenter of a system, usually around a more massive body. It can be used to refer to either the mean orbital speed, the average speed as it completes an orbit, or instantaneous orbital speed, the speed at a particular point in its orbit.

The orbital speed at any position in the orbit can be computed from the distance to the central body at that position, and the specific orbital energy, which is independent of position: the kinetic energy is the total energy minus the potential energy.

Thus, under standard assumptions the orbital speed ([itex]v\,[itex]) is:

• in general: [itex]v=

\sqrt{2\left({\mu\over{r}}+{\epsilon}\right)} [itex]

\sqrt{\mu\left({2\over{r}}-{1\over{a}}\right)} [itex]

where:

Note:

In the case of radial motion:

• if the energy is non-negative: the motion is either for the whole trajectory away from the central body, or for the whole trajectory towards it. For the zero-energy case, see escape orbit and capture orbit.
• if the energy is negative: the motion can be first away from the central body, up to r=μ/|ε|, then falling back. This is the limit case of an orbit which is part of an ellipse with eccentricity tending to 1, and the other end of the ellipse tending to the center of the central body.

Transverse orbital speed

The transverse orbital speed is inversely proportional to the distance to the central body because of the law of conservation of angular momentum, or equivalently, Kepler's second law. This states that as a body moves around its orbit during a fixed amount of time, the line from the barycenter to the body sweeps a constant area of the orbital plane, regardless of which part of its orbit the body traces during that period of time. This means that the body moves faster near its periapsis than near its apoapsis, because at the smaller distance it needs to trace a greater arc to cover the same area. This law is usually stated as "equal areas in equal time."

Mean orbital speed

The mean orbital speed can be derived either from observations of the orbital period and the semimajor axis of its orbit, or from knowledge of the masses of the two bodies and the semimajor axis.

[itex]v_o = {2 \pi r \over T}[itex]
[itex]v_o = \sqrt{m G \over r}[itex]

where [itex]v_o[itex] is the orbital velocity, r is the length of the semimajor axis, T is the orbital period, m is the mass of the other body, and G is the gravitational constant. Note that this is only an approximation that holds true when the orbiting body is of considerably lesser mass than the central one.

More precisely,

[itex]v_o = \sqrt{m_2^2 G \over (m_1 + m_2) r}[itex]

where [itex]m_1[itex] is now the mass of the body under consideration, [itex]m_2[itex] is the mass of the body being orbited, and r is specifically the distance between the two bodies (which is the sum of the distances from both to the barycenter). This is still a simplified version; it doesn't allow for elliptical orbits, but it does at least allow for bodies of similar masses.

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