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, i.e. the average speed as it completes an orbit, or the speed at a particular point in its orbit such as perihelia.
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.
Contents
 Radial trajectories 1
 Transverse orbital speed 2
 Mean orbital speed 3
 Precise orbital speed 4
 Tangential velocities at altitude 5
 See also 6
 References 7
Radial trajectories
In the case of radial motion:
 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 zeroenergy 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, freefall 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.^{[1]}
This law implies that the body moves slower near its apoapsis than near its periapsis, because at the smaller distance along the arc it needs to trace 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.
 v_o \approx {2 \pi a \over T}
 v_o \approx \sqrt{\mu \over a}
where v is the orbital velocity, a is the length of the semimajor axis, T is the orbital period, and μ=GM 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{G (m_1 + m_2) \over r}
where m_{1} is the mass of the orbiting body, 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 assuming r equal to the body's radius
 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 orbital eccentricity e, and is an 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] ^{[2]}
The mean orbital speed decreases with eccentricity.
Precise orbital speed
For the precise orbital speed of a body at any given point in its trajectory, both the mean distance and the precise distance are taken into account:
 v = \sqrt {\mu \left({2 \over r}  {1 \over a}\right)}
where μ is the standard gravitational parameter, r is the distance at which the speed is to be calculated, and a is the length of the semimajor axis of the elliptical orbit. For the Earth at perihelion,
 v = \sqrt {1.327 \times 10^{20} ~m^3 s^{2} \cdot \left({2 \over 1.471 \times 10^{11} ~m}  {1 \over 1.496 \times 10^{11} ~m}\right)} \approx 30,300 ~m/s
which is slightly faster than Earth's average orbital speed of 29,800 m/s, as expected from Kepler's 2nd Law.
Tangential velocities at altitude
orbit 
Centertocenter distance 
Altitude above the Earth's surface 
Speed  Orbital period  Specific orbital energy 

Standing on Earth's surface at the equator (for comparison  not an orbit)  6,378 km  0 km  465.1 m/s (1,040 mph)  1 day (24h)  −62.6 MJ/kg 
Orbiting at Earth's surface (equator)  6,378 km  0 km  7.9 km/s (17,672 mph)  1 h 24 min 18 sec  −31.2 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 (15,430 mph to 17,450 mph) respectively elliptic orbit: 6.5 to 8.2 km/s respectively 
1 h 29 min to 2 h 8 min  −29.8 MJ/kg 
Molniya orbit  6,900 to 46,300 km  500 to 39,900 km  1.5 to 10.0 km/s (3,335 mph to 22,370 mph) respectively  11 h 58 min  −4.7 MJ/kg 
Geostationary  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  0.97 to 1.08 km/s (2,170 to 2,416 mph) respectively  27.3 days  −0.5 MJ/kg 
See also
References
 ^
 ^ Horst Stöcker; John W. Harris (1998). Handbook of Mathematics and Computational Science. Springer. p. 386.
