Orbital speed

Not to be confused with Escape Velocity.

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.

Radial trajectories

In the case of radial motion:

• If the energy is non-negative: The orbit is open. The motion is either directly towards or away from the other body, the motion never stops or reverses direction. See radial hyperbolic trajectory
• For the zero-energy case, see radial parabolic trajectory.
• If the energy is negative: The orbit is closed. The motion can be first away from the central body, up to r=μ/|ε| (apoapsis), 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. See radial elliptic trajectory, radial trajectories, free-fall time.

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

For orbits with small eccentricity, the length of the orbit is close to that of a circular one, and the mean orbital speed can be approximated 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.

$v_o \approx {2 \pi a \over T}$

$v_o \approx \sqrt{\mu \over a}$

where $v_o\,\!$ is the orbital velocity, $a\,\!$ is the length of the semimajor axis, $T\,\!$ is the orbital period, and $\mu\,\!$ is the standard gravitational parameter. Note that this is only an approximation that holds true when the orbiting body is of considerably lesser mass than the central one, and eccentricity is close to zero.

Taking into account the mass of the orbiting body,

$v_o \approx \sqrt{m_2^2 G \over (m_1 + m_2) r}$

where $m_1\,\!$ is now the mass of the body under consideration, $m_2\,\!$ is the mass of the body being orbited, $r\,\!$ is specifically the distance between the two bodies (which is the sum of the distances from each to the center of mass), and $G\,\!$ is the gravitational constant. This is still a simplified version; it doesn't allow for elliptical orbits, but it does at least allow for bodies of similar masses.

When one of the masses is almost negligible compared to the other mass as the case for Earth and Sun, one can approximate the previous formula to get:

$v_o \approx \sqrt{\frac{GM}{r}}$

or

$v_o \approx \frac{v_e}{\sqrt{2}}$

Where M is the (greater) mass around which this negligible mass or body is orbiting, and v_e is the escape velocity.

For an object in an eccentric orbit orbiting a much larger body, the length of the orbit decreases with eccentricity $e\,\!$, and is given at ellipse. This can be used to obtain a more accurate estimate of the average orbital speed:

$v_o = \frac{2\pi a}{T}\left[1-\frac{1}{4}e^2-\frac{3}{64}e^4 -\frac{5}{256}e^6 -\frac{175}{16384}e^8 - \dots \right]$^[1]

The mean orbital speed decreases with eccentricity.

Earth orbits

orbit	center-to-center distance	altitude above the Earth's surface	speed	period/time in space	specific orbital energy
minimum sub-orbital spaceflight (vertical)	6,500 km	100 km	0.0 km/s	just reaching space	-61.3 MJ/kg
ICBM	up to 7,600 km	up to 1,200 km	6 to 7 km/s	time in space: 25 min	-27.9 MJ/kg
Low Earth orbit	6,600 to 8,400 km	200 to 2,000 km	circular orbit: 6.9 to 7.8 km/s elliptic orbit: 6.5 to 8.2 km/s	89 to 128 min	-17.0 MJ/kg
Molniya orbit	6,900 to 46,300 km	500 to 39,900 km	1.5 to 10.0 km/s	11 h 58 min	-4.7 MJ/kg
GEO	42,000 km	35,786 km	3.1 km/s	23 h 56 min	-4.6 MJ/kg
Orbit of the Moon	363,000 to 406,000 km	357,000 to 399,000 km	0.97 to 1.08 km/s	27.3 days	-0.5 MJ/kg

See also

• Escape velocity
• Delta-v budget
• Hohmann transfer orbit
• Bi-elliptic transfer

References

1. ^ Horst Stöcker, John W. Harris (1998). Handbook of Mathematics and Computational Science. Springer. pp. 386. ISBN 0387947469.