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, i.e., the average speed as it completes an orbit, or 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:^{[citation needed]}

• If the specific orbital energy is positive, the body's kinetic energy is greater than its potential energy: The orbit is thus open, following a hyperbola with focus at the other body. See radial hyperbolic trajectory
• For the zero-energy case, the body's kinetic energy is exactly equal to its potential energy: the orbit is thus a parabola with focus at the other body. See radial parabolic trajectory.
• If the energy is negative, the body's potential energy is greater than its kinetic energy: The orbit is thus closed. The motion is on an ellipse with one focus at the other body. See radial elliptic trajectory, 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 law is usually stated as "equal areas in equal time."^{[citation needed]}

This law implies 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.

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.^{[citation needed]}

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

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

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:

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

The mean orbital speed decreases with eccentricity.

Earth orbits

orbit	center-to-center distance	altitude above the Earth's surface	speed	Orbital period	specific orbital energy
Earth's surface (for comparison)	6,400 km	0 km	7.89 km/s (17,650 mph)	85 minutes	-62.6 MJ/kg
Low Earth orbit	6,600 to 8,400 km	200 to 2,000 km	circular orbit: 7.8 to 6.9 km/s (17,450 mph to 15,430 mph) respectively elliptic orbit: 8.2 to 6.5 km/s respectively	89 to 128 min	-29.8 MJ/kg
Molniya orbit	6,900 to 46,300 km	500 to 39,900 km	10.0 to 1.5 km/s (22,370 mph to 3,335 mph) respectively	11 h 58 min	-4.7 MJ/kg
GEO	42,000 km	35,786 km	3.1 km/s (6,935 mph)	23 h 56 min	-4.6 MJ/kg
Orbit of the Moon	363,000 to 406,000 km	357,000 to 399,000 km	1.08 to 0.97 km/s (2,416 to 2,170 mph) respectively	27.3 days	-0.5 MJ/kg

See also

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