# Orbital speed (astronomy)

In celestial mechanics , orbit speed describes the speed at which an astronomical object moves. In orbits we also speak of orbital speed or velocity of circulation.

The movement is specified in a suitable coordinate or reference system , usually in the center of gravity system of the celestial bodies involved:

## Orbit speed of the ideal Keplerbahn

If a small body encounters a large one in space, its trajectory is due to gravity - in the idealized case of the two-body problem  - a Kepler orbit (ellipse, hyperbola or parabola) around the large celestial body or around the common center of gravity. Due to the conservation of energy , the path speed is not constant, but increases when the distance between the bodies becomes smaller. Johannes Kepler discovered that the distance and the orbital speed vary, but the driving beam (the line connecting the center of gravity and the revolving body) sweeps over the same area at the same time ( second Kepler law , constancy of surface speed ). Its solution only applies to the two-body problem (Kepler's problem) itself, the restriction to spherically symmetrical bodies and only as a non-relativistic approximation. In addition, it always gives the relative speed with respect to the center of gravity, never an absolute speed.

For the special case of a circular orbit, the force of attraction between the celestial bodies applies the centripetal force necessary for the circular orbit , whereby the speed is fixed (and constant in terms of amount).

The route along the Keplerbahn, which is needed for the direct distance-time relationship (speed = distance per time ), only has an analytical solution in special cases . The Vis-Viva equation can be derived by considering kinetic and potential energy . It establishes a connection between the mass of the central body, the gravitational constant , the major semiaxis of the orbiting ellipse, the distance of the rotating specimen and the speed of this specimen: ${\ displaystyle v = s / t}$${\ displaystyle M}$ ${\ displaystyle G}$ ${\ displaystyle a}$${\ displaystyle r}$${\ displaystyle v}$

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

Taking into account the mass of the surrounding body, the following applies: ${\ displaystyle m}$

${\ displaystyle v = {\ sqrt {G (M + m) \ left ({\ frac {2} {r}} - {\ frac {1} {a}} \ right)}}}$

For the circular path and the parabolic path, the total mass results : ${\ displaystyle M}$

${\ displaystyle v _ {\ mathrm {K}} = {\ sqrt {\ frac {GM} {r}}}}$ …  Circular orbit, 1st cosmic speed
${\ displaystyle v _ {\ mathrm {P}} = {\ sqrt {\ frac {2GM} {r}}}}$ …  Speed ​​of escape, 2nd cosmic speed

Below ( ) and above ( ) these two borderline cases are spiral and hyperbolic orbits (falling onto and leaving a celestial body or passages). There are elliptical trajectories between the two values ​​( ). ${\ displaystyle v ${\ displaystyle v> v _ {\ mathrm {P}}}$${\ displaystyle v _ {\ mathrm {K}}

There are also analytical solutions for the two main vertices of the ellipse :

${\ displaystyle \ omega _ {\ mathrm {pz}} = \ omega _ {\ mathrm {m}} \ cdot p ^ {2} / (ae) ^ {2}}$ ... angular velocity in the pericenter (point closest to the gravic center)
${\ displaystyle \ omega _ {\ mathrm {az}} = \ omega _ {\ mathrm {m}} \ cdot p ^ {2} / (a ​​+ e) ​​^ {2}}$ ... angular velocity in the apocenter (point furthest from the center of gravity)
${\ displaystyle \ omega _ {\ mathrm {m}}}$ ... mean angular velocity, angular velocity of a body on a circular path with the same period of rotation = mean anomaly (according to Kepler)${\ displaystyle \ omega _ {\ mathrm {m}} = 2 \ pi / T}$
${\ displaystyle T}$ ...  period of circulation
${\ displaystyle a}$ ...  major semi-axis of the orbit ellipse
${\ displaystyle e}$ ...  linear eccentricity ${\ displaystyle e = {\ sqrt {a ^ {2} -b ^ {2}}}}$
${\ displaystyle p}$ ...  half parameters ${\ displaystyle p = b ^ {2} / a}$
${\ displaystyle b}$ ...  small semiaxis of the orbit ellipse

The Vis-Viva equation gives:

${\ displaystyle v _ {\ mathrm {pz}} = {\ sqrt {GM (2 / r _ {\ mathrm {pz}} -1 / a)}} = {\ sqrt {GMp}} / r _ {\ mathrm {pz }}}$ ...  pericenter speed
${\ displaystyle v _ {\ mathrm {az}} = {\ sqrt {GM (2 / r _ {\ mathrm {az}} -1 / a)}} = {\ sqrt {GMp}} / r _ {\ mathrm {az }}}$ ...  apocenter velocity

The pericenter speed is the maximum and the apocenter speed is the minimum orbit speed. Since the movement in the main vertices is tangential, the specific angular momentum can be easily read in both cases , which is constant over the entire path:

${\ displaystyle \ rho = L / m = v _ {\ perp} \ cdot r = {\ sqrt {GMp}} = {\ frac {2 \ pi} {T}} p ^ {2}}$

Thus, the speed of an equivalent circular orbit (mean anomaly, but with the same specific angular momentum ) can also be determined: ${\ displaystyle v _ {\ mathrm {o}} = 2r _ {\ mathrm {o}} \ pi / T}$${\ displaystyle \ rho}$${\ displaystyle GM = \ rho ^ {2} / r_ {o} = \ rho v_ {o} = v_ {o} ^ {2} r_ {o}}$

${\ displaystyle v _ {\ mathrm {o}} = {\ frac {\ sqrt {v _ {\ mathrm {o}} ^ {2} r _ {\ mathrm {o}} p}} {r _ {\ mathrm {o} }}} \ to r _ {\ mathrm {o}} = p \, \, {\ text {and thus}} \, \, v _ {\ mathrm {o}} = {\ frac {\ rho} {p} } = {\ frac {2 \ pi} {T}} {\ frac {b ^ {2}} {a}}}$

Inserting the results in the respective path speed with the distance to the second focal point: ${\ displaystyle GM / p = v _ {\ mathrm {o}} ^ {2}}$${\ displaystyle r '= 2a-r}$

${\ displaystyle v = {\ sqrt {GM \ left ({\ frac {2} {r}} - {\ frac {1} {a}} \ right)}} = {\ sqrt {v _ {\ mathrm {o }} ^ {2} p \ cdot {\ frac {2a-r} {a \ cdot r}}}} = v _ {\ mathrm {o}} {\ frac {b} {a}} {\ sqrt {{ \ frac {2a} {r}} - 1}} = v _ {\ mathrm {o}} {\ frac {b} {a}} {\ sqrt {\ frac {r '} {r}}}}$

The speed results in the side vertices:

${\ displaystyle v _ {\ mathrm {N}} = v _ {\ mathrm {o}} {\ frac {b} {a}} = {\ frac {\ rho} {b}}}$

## Mean orbital velocity

The mean orbital velocity results from the relationship between distance and time. The circumference of the ellipse cannot be determined in a closed manner; with the elliptic integral of the 2nd kind it applies : ${\ displaystyle E (k)}$

${\ displaystyle {\ bar {v}} = {\ frac {U (\ varepsilon)} {T}} = {\ frac {4a} {T}} E (\ varepsilon) = {\ frac {4a} {T }} {\ int _ {0} ^ {\ pi / 2}} {\ sqrt {1- \ varepsilon ^ {2} \ sin ^ {2} (t)}} \, \ mathrm {d} t = { \ frac {2 \ pi} {T}} a \ left [1 - {\ frac {1} {4}} \ varepsilon ^ {2} - {\ frac {3} {64}} \ varepsilon ^ {4} - {\ frac {5} {256}} \ varepsilon ^ {6} - {\ frac {175} {16384}} \ varepsilon ^ {8} + {\ mathcal {O}} (\ varepsilon ^ {10}) \ right]}$

With increasing eccentricity , the mean path speed decreases with the same specific angular momentum . ${\ displaystyle \ varepsilon}$${\ displaystyle \ rho}$

In addition, there is a simple approximation for the rotational speed

${\ displaystyle {\ bar {v}} \ approx {\ frac {\ pi} {T}} (a + b) = {\ frac {\ pi} {T}} \ cdot a \ left (1 + {\ sqrt {1- \ varepsilon ^ {2}}} \ right) = {\ frac {U (\ varepsilon)} {T}} - {\ frac {2 \ pi a} {T}} {\ frac {\ varepsilon ^ {4}} {64}} + {\ mathcal {O}} (\ varepsilon ^ {6})}$,

which is therefore more precise for small eccentricities than the termination according to the in quadratic term. ${\ displaystyle \ varepsilon}$

## Orbital speeds of artificial earth satellites

The orbital speeds for satellites that have almost circular orbits are, depending on the class of the satellite orbit :

Typical launch vehicles have a propulsion capacity of 7-11 km / s. The burning time of the system depends entirely on the technology, i.e. the thrust (acceleration), in order to then achieve the necessary speed (1st cosmic speed of the earth) for a stable orbit. This also applies to the drive systems mentioned below. ${\ displaystyle \ Delta v}$

In contrast to Kepler's ideal case, satellites are subject to a significant braking force, especially at low orbits, due to friction in the high atmosphere, which means that the orbit height continuously decreases and the mean angular velocity increases. Therefore, at least a seventh orbit element is specified as standard for the satellite orbit element mean movement , for example ${\ displaystyle n}$

• the braking effect (as a change in the mean movement, rate of descent per unit of time)${\ displaystyle {\ dot {n}} / 2}$
• or a ballistic coefficient that can be used to calculate the loss of speed.${\ displaystyle B ^ {*}}$

However, in order to prevent re-entry (burning up in the atmosphere), path corrections must be made regularly . That is why many satellites are equipped with propulsion systems, but their fuel supply limits their service life. They do 10–600 m / s, i.e. a 10,000th to 10th of the launcher, depending on the height of the mission.

There are also numerous other disturbance variables that require further path corrections and position control with powers of around 20 m / s. In the case of a geostationary satellite, 40–51 m / s per year are required for the gravitational influence of the earth and moon, up to 30 m / s per year for the radiation pressure of the sun ( solar wind ), the other disturbances remain in the single-digit range.

In some missions an explicit change of path is necessary, for which systems with 1 to a few km / s drive capacity are necessary. Engines for this task are not counted as secondary systems like orbit correction and attitude control systems, but as primary systems like the engines of the launcher.

## Orbital velocities of small bodies and space missions

Among small bodies one summarizes asteroids (minor planets), comets and meteoroids together. Most asteroids run - as regular objects of the solar system - on circular ellipses like the planets, albeit with greater orbital inclinations . In addition, there are numerous irregular objects on strongly eccentric ellipses and aperiodic objects on hyperbolic orbits . Because of their small size, most of them are still undiscovered, and a precise orbit determination is often not possible with a single observation.

A decisive factor for the origin of these bodies is the escape speed to the sun (or the total mass of the solar system). At the height of the earth's orbit, this is 42 km / s, i.e. around 150,000 km / h ( third cosmic speed ), up to the surface of the sun it increases to 620 km / s (2.2 million km / h). All objects that are faster leave the solar system, either due to severe orbital disturbances or they are actually of extrasolar origin. According to the formulas mentioned at the beginning, the escape speed decreases with the distance to the sun: For example, the Voyager probes , which are now far beyond Saturn's orbit, need a speed that is less than the orbital speed of the earth to leave the solar system. For this, however, a separate drive is necessary, or a gain in speed to the outside, as can be achieved through swing-by maneuvers (the Voyagers were accelerated by around 18 km / s through swing-by on Saturn). Some small bodies can also leave the solar system through violent collisions. ${\ displaystyle {\ sqrt {r}}}$

In the case of earth orbit cruisers , including meteors and meteor streams (swarms of falling stars), in contrast to the above, one does not specify a barycentric speed, but the more relevant relative speed to the earth. Depending on the angle of incidence to the earth's orbit, these objects have speeds between 11.2 (trailing) to 72 km / s (frontal hit).

## Orbital speeds of comets

The velocities of long cometary orbits are extremely different. One example is Comet Halley , whose 76-year orbit ellipse extends from within Venus' orbit to beyond Neptune. In perihelion (0.59  AU ) it moves at 55 km / s, in aphelion (35 AU) only at 0.9 km / s, which is why it lingers for decades beyond the orbit of Saturn and is unobservable. Even more extreme are “ comets of the century ” from the Oort cloud , which can drift from there at a few m / s towards the sun and finally (like McNaught in early 2007) orbit it at over 100 km / s.

## Examples

• Mean orbital speed of the earth (around the sun / barycenter of the solar system ):${\ displaystyle v \ approx 29 {,} 780 \ \ mathrm {km} / \ mathrm {s} \ approx 107 \, 000 \, \ mathrm {km} / \ mathrm {h} \ \ pm 1 {,} 7 \, \% \ {\ text {annual}}}$
For comparison: Rotation speed on the earth's surface at the equator (to the center of the earth):  - Speed ​​of the observer at the equator around the sun, i.e. the same as the earth ± 1.7% diurnal (daily)${\ displaystyle v _ {\ text {observer}} \ approx 460 \ \ mathrm {m} / \ mathrm {s} \ approx 1 \, 670 \ \ mathrm {km} / \ mathrm {h}}$
• Mean orbital speed of the moon (around the earth-moon center of gravity ):${\ displaystyle v \ approx 1020 \ \ mathrm {m} / \ mathrm {s} \ approx 3670 \ \ mathrm {km} / \ mathrm {h} \ \ pm 5 {,} 5 \, \% \ {\ text {mensal}}}$
For comparison: Orbital speed around the sun: the same as the earth ± 3.4% mensal (monthly)
• Orbital speed of the ISS (around the earth ):${\ displaystyle v \ approx 7 \, 770 \ mathrm {m} / \ mathrm {s} \ approx 28 \, 000 \ \ mathrm {km} / \ mathrm {h}}$
For comparison: relative speed (to the observer on the earth's surface):${\ displaystyle v _ {\ text {Boden}} \ approx 7 \, 500 \ mathrm {m} / \ mathrm {s} \ approx 27 \, 000 \ \ mathrm {km} / \ mathrm {h}}$
• Orbital velocity of the Voyager-1 probe (to the sun):${\ displaystyle v \ approx 17 \, 000 \ \ mathrm {m} / \ mathrm {s} \ approx 61 \, 400 \ \ mathrm {km} / \ mathrm {h}}$
• Orbital speed of the comet Tempel-Tuttle in perihelion (i.e. around the sun):${\ displaystyle v \ approx 41 \, 600 \ \ mathrm {m} / \ mathrm {s} \ approx 150 \, 000 \ \ mathrm {km} / \ mathrm {h}}$
For comparison: the relative speed of the Leonids , the meteor shower generated by them, to earth:  - that is, 250 times the speed of sound${\ displaystyle v \ approx 71 \, 000 \ \ mathrm {m} / \ mathrm {s} \ approx 255 \, 000 \ \ mathrm {km} / \ mathrm {h}}$
• Orbital speed of the solar system (around the galactic center):${\ displaystyle v \ approx 250 \, 000 \ \ mathrm {m} / \ mathrm {s} \ approx 900 \, 000 \ \ mathrm {km} / \ mathrm {h}}$
For comparison: the orbital speed of the earth around the galactic center: same as the sun ± 12% annual (annually)

## literature

• Hans Rolf Henkel: Astronomy - A Basic Course. Harry Deutsch publishing house, Frankfurt / Main 1991.

## Individual evidence

1. An absolute speed does not exist: The Earth orbits the sun , this the galactic center, the Milky Way is moving in the multi-body problem of the local group, that the gravitational field of large structures, and the universe expands total: In astronomy, there is no excellent zero, to which one could measure movements "absolutely". The zero point is always problem-related: in the solar system its barycentre, with satellites and the moon the earth, with Jupiter moons the Jupiter, with binary stars their center of gravity. Statements about speeds other than relative speeds to the barycentre are rather irrelevant, see speed and reference system . Exceptions are e.g. B. Relative speeds to the observer (mostly to earth), or collision speeds in general.
2. Norbert Treiz: How fast does the sun see a planet moving? In: Spectrum of Science . tape 04/09 . spectrum Academic publishing house, April 2009, Physical entertainments. Solar system (III): No sundial for Mercury. , S. 36–38 (box p. 37 - with derivation of the formulas about energy conservation).
3. ^ Horst Stöcker, John W. Harris: Handbook of Mathematics and Computational Science . Springer, 1998, ISBN 0-387-94746-9 , p. 386.
4. a b c d Ernst Messerschmid, Stefanos Fasoulas: Space systems: An introduction with exercises and solutions. 4th edition, Verlag Springer DE, 2010, ISBN 978-3-642-12816-5 , section 7 Drive systems for path and position control, in particular p. 266 ( limited preview in the Google book search).
5. a b Detailed tabular overview in:
Messerschmid, Fasoulas: Raumfahrtsysteme. Table 7.3 Requirements for orbit and attitude control of a three-axis stabilized geostationary satellite. P. 290 ( limited preview in Google Book search).
6. Where are the Voyagers? At: voyager.jpl.nasa.gov. With the live data.
7. Isaac Asimov: The Return of Halley's Comet. Publishing house Kiepenheuer, Cologne 1985.
8. A mean circumference of the earth of approx. 40,000 km in approx. 24 hours; the speed is latitude dependent ,  = geographical latitude ; at the pole it is 0.${\ displaystyle v_ {B} = \ cos B \ cdot v _ {\ mathrm {{\ ddot {A}} q}}}$${\ displaystyle B}$
9. Calculation of the ISS orbit time. In: physikerboard.de. Dec. 15, 2008, 19:58 ff.
The calculation is based on the fact that the ISS is following a heading (to the equator) of 38.4 °.