Editing Orbital speed

{{Short description|Speed at which a body orbits around the barycenter of a system}}
{{Distinguish|Escape velocity}}
{{refimprove|date=September 2007}}
{{Astrodynamics}}

In [[gravitational binding energy|gravitationally bound]] systems, the '''orbital speed''' of an [[astronomical object|astronomical body]] or object (e.g. [[planet]], [[natural satellite|moon]], [[satellite|artificial satellite]], [[spacecraft]], or [[star]]) is the [[speed]] at which it [[orbit]]s around either the [[barycenter]] (the combined center of mass) 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 [[primary body|most massive body]].

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 [[apsis|periapsis]] (perigee, perihelion, etc.), while the minimum speed for objects in closed orbits occurs at apoapsis (apogee, aphelion, etc.). In ideal [[two-body problem|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.<ref name="lissauer2019" />

==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]], or total energy, is equal to ''E''<sub>k</sub>&nbsp;−&nbsp;''E''<sub>p</sub> (the difference between kinetic energy and potential energy). The sign of the result may be positive, zero, or negative and the sign tells us something about the type of orbit:<ref name="lissauer2019">{{Cite book |title=Fundamental Planetary Sciences: physics, chemistry, and habitability |last1=Lissauer |first1=Jack J. |last2=de Pater |first2=Imke |year=2019 |publisher=Cambridge University Press |isbn=9781108411981 |location=New York, NY, US |pages=29–31 }}</ref>
* If the [[specific orbital energy]] is positive the orbit is unbound, or open, and will follow a [[hyperbola]] with the larger body the [[focus (geometry)|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 total energy is zero, (''E''<sub>k</sub>&nbsp;=&nbsp;''E''<sub>p</sub>): the orbit is a [[parabola]] with [[focus (geometry)|focus]] at the other body. See [[radial parabolic trajectory]]. Parabolic orbits are also open.
* If the total energy is negative, {{nowrap|''E''<sub>k</sub> − ''E''<sub>p</sub> < 0}}: The orbit is bound, or closed. The motion will be on an [[ellipse]] with one [[focus (geometry)|focus]] at the other body. See [[radial elliptic trajectory]], [[free-fall time]]. Planets have bound orbits around the Sun.

==Transverse orbital speed==

The [[transverse direction|transverse]] orbital speed is inversely proportional to the distance to the central body because of the law of conservation of [[angular momentum]], or equivalently,  [[Johannes Kepler|Kepler]]'s [[Kepler's laws of planetary motion|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.<ref>{{cite book
 | last = Gamow
 | first = George
 | author-link = George Gamow
 | title = Gravity
 | url = https://archive.org/details/gravityclassicmo00gamo
 | url-access = registration
 | publisher = Anchor Books, Doubleday & Co.
 | orig-date = 1962 |date=2002
 | location = New York, NY, US
 | pages = [https://archive.org/details/gravityclassicmo00gamo/page/66 66]
 | isbn = 0-486-42563-0
 | quote = "...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." }}</ref>

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.<ref name="lissauer2019" />

==Mean orbital speed==

For orbits with small [[eccentricity (orbit)|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 [[mass]]es of the two bodies and the semimajor axis.<ref>{{cite book |editor-last1=Wertz |editor-first1=James R. |editor-last2=Larson |editor-first2=Wiley J. |title=Space mission analysis and design |date=2010 |publisher=Microcosm |location=Hawthorne, CA, US |isbn=978-1881883-10-4 |page=135 |edition=3rd }}</ref>

:<math>v \approx {2 \pi a \over T}  \approx \sqrt{\mu \over a}</math>

where {{math|''v''}} is the orbital velocity, {{math|''a''}} is the [[length]] of the [[semimajor axis]], {{math|''T''}} is the orbital period, and {{math|1=''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 <math>v_o</math> as:<ref name="lissauer2019" />

:<math>v_o \approx \sqrt{\frac{GM}{r}}</math>

or:

:<math>v_o \approx \frac{v_e}{\sqrt{2}}</math>

Where {{math|''M''}} is the (greater) mass around which this negligible mass or body is orbiting, and {{math|''v<sub>e</sub>''}} is the [[escape velocity]] at a distance from the center of the primary body equal to the radius of the orbit.

For an object in an eccentric orbit orbiting a much larger body, the length of the orbit decreases with [[orbital eccentricity]] {{math|''e''}}, and is an [[ellipse#Circumference|ellipse]]. This can be used to obtain a more accurate estimate of the average orbital speed:<ref>{{cite book |first=Horst |last=Stöcker |first2=John W. |last2=Harris |date=1998 |title=Handbook of Mathematics and Computational Science |pages=[https://archive.org/details/handbookofmathem00harr/page/386 386] |publisher=Springer |isbn=0-387-94746-9 |url-access=registration |url=https://archive.org/details/handbookofmathem00harr/page/386 }}</ref>
 
:<math> 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] </math>

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:

:<math> v = \sqrt {\mu \left({2 \over r} - {1 \over a}\right)}</math>

where {{math|''μ''}} is the [[standard gravitational parameter]] of the orbited body, {{math|''r''}} is the distance at which the speed is to be calculated, and {{math|''a''}} is the length of the semi-major axis of the elliptical orbit. This expression is called the [[vis-viva equation]].<ref name="lissauer2019" />

For the Earth at [[perihelion]], the value is:

:<math> \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}</math>

which is slightly faster than Earth's average orbital speed of {{convert|29,800|m/s|mph|abbr=on}}, as expected from [[Kepler's laws of planetary motion#Second law|Kepler's 2nd Law]].

{{earth orbits}}

== 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.<ref name="Mercury">{{cite web
  |title      = Horizons Batch for Mercury aphelion (2021-Jun-10) to perihelion (2021-Jul-24)
  |type       = VmagSn is velocity with respect to Sun.
  |url        = https://ssd.jpl.nasa.gov/horizons_batch.cgi?batch=1&COMMAND=%271%27&START_TIME=%272021-Jun-10%27&STOP_TIME=%272021-Jul-24%27&STEP_SIZE=%271%20day%27&QUANTITIES=%2720,22%27&CENTER=%27@Sun%27
  |work       = [[JPL Horizons On-Line Ephemeris System|JPL Horizons]]
  |publisher  = Jet Propulsion Laboratory
  |access-date= 26 August 2021}}</ref>

{|class="wikitable" style="text-align:center; font-size:0.9em;"
|+Orbital velocities of the Planets<ref>{{Cite web|url=https://public.nrao.edu/ask/which-planet-orbits-our-sun-the-fastest/|title=Which Planet Orbits our Sun the Fastest?}}</ref>
! Planet
! Orbital<br />velocity
|-
| [[Mercury (planet)|Mercury]] || {{convert |47.9|km/s|mi/s|abbr=on}}
|-
| [[Venus]] || {{convert |35.0|km/s|mi/s|abbr=on}}
|-
| [[Earth]] || {{convert |29.8|km/s|mi/s|abbr=on}}
|-
| [[Mars]] || {{convert |24.1|km/s|mi/s|abbr=on}}
|-
| [[Jupiter]] || {{convert |13.1|km/s|mi/s|abbr=on}}
|-
| [[Saturn]] || {{convert |9.7|km/s|mi/s|abbr=on}}
|-
| [[Uranus]] || {{convert |6.8|km/s|mi/s|abbr=on}}
|-
| [[Neptune]] || {{convert |5.4|km/s|mi/s|abbr=on}}
|}

[[Halley's Comet]] on an [[Orbital eccentricity|eccentric orbit]] that reaches beyond [[Neptune]] will be moving 54.6&nbsp;km/s when {{convert|0.586|AU|e3km|abbr=unit|lk=on}} from the Sun, 41.5&nbsp;km/s when 1 AU from the Sun (passing Earth's orbit), and roughly 1&nbsp;km/s at aphelion {{convert|35|AU|e9km|abbr=unit}} from the Sun.<ref>{{nowrap|1=''v'' = 42.1219 {{radic|1/''r'' − 0.5/''a''}}}}, where ''r'' is the distance from the Sun, and ''a'' is the major semi-axis.</ref> Objects passing Earth's orbit going faster than 42.1 km/s have achieved [[Escape velocity#List of escape velocities|escape velocity]] and will be ejected from the Solar System if not slowed down by a [[Perturbation (astronomy)|gravitational interaction]] with a planet.

{|class="wikitable sortable" style="text-align:center; font-size:0.9em;"
|+Velocities of better-known numbered objects that have perihelion close to the Sun
! Object
! Velocity at perihelion
! Velocity at 1 AU<br/>(passing Earth's orbit)
|-
| [[322P/SOHO]] || 181 km/s @ 0.0537 AU || 37.7 km/s
|-
| [[96P/Machholz]] || 118 km/s @ 0.124 AU || 38.5 km/s
|-
| [[3200 Phaethon]] || 109 km/s @ 0.140 AU || 32.7 km/s
|-
| [[1566 Icarus]] || 93.1 km/s @ 0.187 AU || 30.9 km/s
|-
| [[66391 Moshup]] || 86.5 km/s @ 0.200 AU || 19.8 km/s
|-
| [[1P/Halley]] || 54.6 km/s @ 0.586 AU ||  41.5 km/s
|}

==See also==
*[[Escape velocity]]
*[[Delta-v budget]]
*[[Hohmann transfer orbit]]
*[[Bi-elliptic transfer]]

==References==
{{Reflist}}

{{Orbits}}

[[Category:Orbits]]

[[hu:Kozmikus sebességek#Szökési sebességek]]