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 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.

In the case of radial motion:[citation needed]

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 move faster 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.[2]

${\displaystyle v\approx {2\pi a \over T}}$
${\displaystyle v\approx {\sqrt {\mu \over a}}}$

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. This is an approximation that only holds true when the orbiting body is of considerably lesser mass than the central one, and eccentricity is close to zero.

In case one of the bodies is not of considerably lesser mass then see: Gravitational two-body problem

So, when one of the masses is almost negligible compared to the other mass, as the case for Earth and Sun, one can approximate the orbit velocity ${\displaystyle v_{o}}$ as:

${\displaystyle v_{o}\approx {\sqrt {\frac {GM}{r}}}}$

or assuming r equal to the body's radius

${\displaystyle v_{o}\approx {\frac {v_{e}}{\sqrt {2}}}}$

Where M is the (greater) mass around which this negligible mass or body is orbiting, and ve 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:

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

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:

${\displaystyle v={\sqrt {\mu \left({2 \over r}-{1 \over a}\right)}}}$

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 semi-major axis of the elliptical orbit. For the Earth at perihelion,

${\displaystyle v={\sqrt {1.327\times 10^{20}~m^{3}s^{-2}\cdot \left({2 \over 1.471\times 10^{11}~m}-{1 \over 1.496\times 10^{11}~m}\right)}}\approx 30,300~m/s}$

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 Center-to-center
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 6,600–8,400 km 200–2,000 km circular orbit: 7.8–6.9 km/s (17,450–14,430 mph) respectively
elliptic orbit: 6.5–8.2 km/s respectively
1 h 29 min – 2 h 8 min −29.8 MJ/kg
Molniya orbit 6,900–46,300 km 500–39,900 km 1.5–10.0 km/s (3,335–22,370 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 363,000–406,000 km 357,000–399,000 km 0.97–1.08 km/s (2,170–2,416 mph) respectively 27.3 days −0.5 MJ/kg

References

1. ^ Gamow, George (1962). Gravity. New York: Anchor Books, Doubleday & Co. p. 66. ISBN 0-486-42563-0. ...the motion of planets along their elliptical orbits proceeds in such a way that an imaginary line connecting the Sun with the planet sweeps over equal areas of the planetary orbit in equal intervals of time.
2. ^ Wertz, edited by James R. Wertz; Larson, Wiley J. (2010). Space mission analysis and design (3rd ed.). Hawthorne, Calif.: Microcosm. p. 135. ISBN 978-1881883-10-4.
3. ^ Horst Stöcker; John W. Harris (1998). Handbook of Mathematics and Computational Science. Springer. p. 386. ISBN 0-387-94746-9.