Orbital speed

In gravitationally bound systems, the orbital speed of an astronomical body or object (e.g. planet, moon, artificial satellite, spacecraft, or star) is the speed at which it orbits around either the barycenter or, if the object is much less massive than the largest body in the system, its speed relative to that largest body. The speed in this latter case may be relative to the surface of the larger body or relative to its center of mass.

The term can be used to refer to either the mean orbital speed, i.e. the average speed over an entire orbit, or its instantaneous speed at a particular point in its orbit. Maximum (instantaneous) orbital speed occurs at periapsis (perigee, perhelion, etc.), while minimum speed for objects in closed orbits occurs at apoapsis (aphelion, apogee, etc.). In ideal two-body systems, objects in open orbits continue to slow down forever as their distance to the barycenter increases.

When a system approximates a two-body system, instantaneous orbital speed at a given point of the orbit can be computed from its distance to the central body and the object's specific orbital energy. (Specific orbital energy is constant and independent of position.)

Radial trajectories

In the following, it is assumed that the system is a two-body system and the orbiting object has a negligible mass compared to the larger (central) object. In real-world orbital mechanics, it is the system's barycenter, not the larger object, which is at the focus.

Specific orbital energy = K.E. + P.E. (kinetic energy + potential energy). Since kinetic energy is always non-negative (greater than or equal to zero, ≥0) and potential energy is always non-positive (less than or equal to zero, ≤0), the sign of this may be positive, zero, or negative and the sign tells us something about the type of orbit:

If the specific orbital energy is positive the orbit is open, following a hyperbola with the larger body the focus of the hyperbola. Objects in open orbits do not return; once past periapsis their distance from the focus increases without bound. See radial hyperbolic trajectory
If the specific orbital energy is zero, (K.E = −P.E.): the orbit is thus a parabola with focus at the other body. See radial parabolic trajectory. Parabolic orbits are also open.
If the energy is negative, K.E. + P.E. < 0: The orbit is closed. The motion is on an ellipse with one focus at the other body. See radial elliptic trajectory, free-fall time. Planets have closed orbits around the Sun.

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]

v\approx {2\pi a \over T}\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.

When one of the bodies is not of considerably lesser mass 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 $v_o$ as:

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]

^[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:

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. This expression is called the vis-viva equation. For the Earth at perihelion,

v={\sqrt {1.327\times 10^{20}~{\text{m}}^{3}{\text{s}}^{2}\cdot \left({2 \over 1.471\times 10^{11}~{\text{m}}}-{1 \over 1.496\times 10^{11}~{\text{m}}}\right)}}\approx 30,300~{\text{m}}/{\text{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
Earth's own rotation at surface (for comparison – not an orbit)	6,378 km	0 km	465.1 m/s (1,674 km/h or 1,040 mph)	23 h 56 min	−62.6 MJ/kg
Orbiting at Earth's surface (equator)	6,378 km	0 km	7.9 km/s (28,440 km/h or 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 (28,080–24,840 km/h or 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 (5,400–36,000 km/h or 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 (11,600 km/h or 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 (3,492–3,888 km/h or 2,170–2,416 mph) respectively	27.3 days	−0.5 MJ/kg

References

↑ 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.
↑ 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.
↑ Horst Stöcker; John W. Harris (1998). Handbook of Mathematics and Computational Science. Springer. p. 386. ISBN 0-387-94746-9.

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[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.