# Orbital speed

Last updated

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 one body is much more massive than the other bodies of the system combined, its speed relative to the center of mass of the most massive body.

## Contents

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. The maximum (instantaneous) orbital speed occurs at periapsis (perigee, perihelion, etc.), while the minimum speed for objects in closed orbits occurs at apoapsis (apogee, aphelion, 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, sometimes called "total energy". Specific orbital energy is constant and independent of position. [1]

In the following, it is thought 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, or total energy, is equal to Ek  Ep. (kinetic energy  potential energy). The sign of the result may be positive, zero, or negative and the sign tells us something about the type of orbit: [1]

## 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. [2]

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. [1]

## 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. [3]

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

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 ${\displaystyle v_{o}}$ as: [1]

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

or assuming r equal to the radius of the orbit[ citation needed ]

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

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: [4]

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

The mean orbital speed decreases with eccentricity.

## Instantaneous orbital speed

For the instantaneous orbital speed of a body at any given point in its trajectory, both the mean distance and the instantaneous distance are taken into account:

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

where μ is the standard gravitational parameter of the orbited body, 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. [1]

For the Earth at perihelion, the value is:

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

which is slightly faster than Earth's average orbital speed of 29,800 m/s (67,000 mph), 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 km0 km 465.1 m/s (1,674 km/h or 1,040 mph)23 h 56 min 4.09 sec−62.6 MJ/kg
Orbiting at Earth's surface (equator) theoretical6,378 km0 km7.9 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 km200–2,000 km
• Circular orbit: 6.9–7.8 km/s (24,840–28,080 km/h or 14,430–17,450 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 km500–39,900 km1.5–10.0 km/s (5,400–36,000 km/h or 3,335–22,370 mph) respectively11 h 58 min−4.7 MJ/kg
Geostationary 42,000 km35,786 km3.1 km/s (11,600 km/h or 6,935 mph)23 h 56 min 4.09 sec−4.6 MJ/kg
Orbit of the Moon 363,000–406,000 km357,000–399,000 km0.97–1.08 km/s (3,492–3,888 km/h or 2,170–2,416 mph) respectively27.27 days−0.5 MJ/kg

## Planets

The closer an object is to the Sun the faster it needs to move to maintain the orbit. Objects move fastest at perihelion (closest approach to the Sun) and slowest at aphelion (furthest distance from the Sun). Since planets in the Solar System are in nearly circular orbits their individual orbital velocities do not vary much. Being closest to the Sun and having the most eccentric orbit, Mercury's orbital speed varies from about 59 km/s at perihelion to 39 km/s at aphelion. [5]

Orbital velocities of the Planets [6]
PlanetOrbital
velocity
Mercury 47.9 km/s (29.8 mi/s)
Venus 35.0 km/s (21.7 mi/s)
Earth 29.8 km/s (18.5 mi/s)
Mars 24.1 km/s (15.0 mi/s)
Jupiter 13.1 km/s (8.1 mi/s)
Saturn 9.7 km/s (6.0 mi/s)
Uranus 6.8 km/s (4.2 mi/s)
Neptune 5.4 km/s (3.4 mi/s)

Halley's Comet on an eccentric orbit that reaches beyond Neptune will be moving 54.6 km/s when 0.586  AU (87,700 thousand  km ) from the Sun, 41.5 km/s when 1 AU from the Sun (passing Earth's orbit), and roughly 1 km/s at aphelion 35 AU (5.2 billion km) from the Sun. [7] Objects passing Earth's orbit going faster than 42.1 km/s have achieved escape velocity and will be ejected from the Solar System if not slowed down by a gravitational interaction with a planet.

Velocities of better-known numbered objects that have perihelion close to the Sun
ObjectVelocity at perihelionVelocity at 1 AU
(passing Earth's orbit)
322P/SOHO 181 km/s @ 0.0537 AU37.7 km/s
96P/Machholz 118 km/s @ 0.124 AU38.5 km/s
3200 Phaethon 109 km/s @ 0.140 AU32.7 km/s
1566 Icarus 93.1 km/s @ 0.187 AU30.9 km/s
66391 Moshup 86.5 km/s @ 0.200 AU19.8 km/s
1P/Halley 54.6 km/s @ 0.586 AU41.5 km/s

## Related Research Articles

In celestial mechanics, the Lagrange points are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of the restricted three-body problem.

In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.

In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically stated as an ideal speed, ignoring atmospheric friction. Although the term "escape velocity" is common, it is more accurately described as a speed than a velocity because it is independent of direction. The escape speed is independent of the mass of the escaping object, but increases with the mass of the primary body; it decreases with the distance from the primary body, thus taking into account how far the object has already traveled. Its calculation at a given distance means that no acceleration is further needed for the object to escape: it will slow down as it travels—due to the massive body's gravity—but it will never quite slow to a stop. On the other hand, an object already at escape speed needs slowing for it to be captured by the gravitational influence of the body.

An apsis is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, for orbits about the Sun the apsides are called aphelion (farthest) and perihelion (nearest).

The orbital period is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting planets, exoplanets orbiting other stars, or binary stars. It may also refer to the time it takes a satellite orbiting a planet or moon to complete one orbit.

In astronautics, the Hohmann transfer orbit is an orbital maneuver used to transfer a spacecraft between two orbits of different altitudes around a central body. Examples would be used for travel between low Earth orbit and the Moon, or another solar planet or asteroid. In the idealized case, the initial and target orbits are both circular and coplanar. The maneuver is accomplished by placing the craft into an elliptical transfer orbit that is tangential to both the initial and target orbits. The maneuver uses two impulsive engine burns: the first establishes the transfer orbit, and the second adjusts the orbit to match the target.

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control.

The Hill sphere of an astronomical body is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hill sphere of its own. Any object within that distance would tend to become a satellite of the moon, rather than of the planet itself. One simple view of the extent of the Solar System is the Hill sphere of the Sun with respect to local stars and the galactic nucleus.

Earth's orbit is an ellipse with the Earth-Sun barycenter as one focus and a current eccentricity of 0.0167. Since this value is close to zero, the center of the orbit is relatively close to the center of the Sun. Earth orbits the Sun at an average distance of 149.60 million km in a counterclockwise direction as viewed from above the Northern Hemisphere. One complete orbit takes 365.249 days, during which time Earth has traveled 940 million km.

In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1 and is an unbound orbit that is exactly on the border between elliptical and hyperbolic. When moving away from the source it is called an escape orbit, otherwise a capture orbit. It is also sometimes referred to as a C3 = 0 orbit (see Characteristic energy).

In astrodynamics or celestial mechanics, a hyperbolic trajectory or hyperbolic orbit is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the fact that according to Newtonian theory such an orbit has the shape of a hyperbola. In more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one.

In astrodynamics, the characteristic energy is a measure of the excess specific energy over that required to just barely escape from a massive body. The units are length2 time−2, i.e. velocity squared, or energy per mass.

In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a wider sense, it is a Kepler orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1.

A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.

In astrodynamics, an orbit equation defines the path of orbiting body around central body relative to , without specifying position as a function of time. Under standard assumptions, a body moving under the influence of a force, directed to a central body, with a magnitude inversely proportional to the square of the distance, has an orbit that is a conic section with the central body located at one of the two foci, or the focus.

In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values between 0 and 1 form an elliptic orbit, 1 is a parabolic escape orbit, and greater than 1 is a hyperbola. The term derives its name from the parameters of conic sections, as every Kepler orbit is a conic section. It is normally used for the isolated two-body problem, but extensions exist for objects following a rosette orbit through the Galaxy.

Spacecraft flight dynamics is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag.

The two-body problem in general relativity is the determination of the motion and gravitational field of two bodies as described by the field equations of general relativity. Solving the Kepler problem is essential to calculate the bending of light by gravity and the motion of a planet orbiting its sun. Solutions are also used to describe the motion of binary stars around each other, and estimate their gradual loss of energy through gravitational radiation.

In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semi-minor axis of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle.

C/1980 E1 is a non-periodic comet discovered by Edward L. G. Bowell on 11 February 1980 and which came closest to the Sun (perihelion) in March 1982. It is leaving the Solar System on a hyperbolic trajectory due to a close approach to Jupiter. In the 42 years since its discovery only two objects with higher eccentricities have been identified, 1I/ʻOumuamua (1.2) and 2I/Borisov (3.35).

## References

1. Lissauer, Jack J.; de Pater, Imke (2019). Fundamental Planetary Sciences: physics, chemistry, and habitability. New York, NY, USA: Cambridge University Press. pp. 29–31. ISBN   9781108411981.
2. Gamow, George (1962). . New York, NY, USA: Anchor Books, Doubleday & Co. pp.  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.
3. Wertz, James R.; Larson, Wiley J., eds. (2010). Space mission analysis and design (3rd ed.). Hawthorne, CA, USA: Microcosm. p. 135. ISBN   978-1881883-10-4.
4. Stöcker, Horst; Harris, John W. (1998). . Springer. pp.  386. ISBN   0-387-94746-9.
5. "Horizons Batch for Mercury aphelion (2021-Jun-10) to perihelion (2021-Jul-24)". JPL Horizons (VmagSn is velocity with respect to Sun.). Jet Propulsion Laboratory. Retrieved 26 August 2021.
6. v = 42.1219 1/r − 0.5/a, where r is the distance from the Sun, and a is the major semi-axis.