Orbital speed
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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, the average speed as it completes an orbit, or instantaneous orbital speed, the speed at a particular point in its orbit.
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.
Thus, under standard assumptions the orbital speed (<math>v\,</math>) is:
- in general: <math>v=
\sqrt{2\left({\mu\over{r}}+{\epsilon}\right)} </math>
- elliptic orbit: <math>v=
\sqrt{\mu\left({2\over{r}}-{1\over{a}}\right)} </math>
- parabolic trajectory: <math>v=\sqrt{\mu\left({2\over{r}}\right)}</math>
- hyperbolic trajectory: <math>v=\sqrt{\mu\left({2\over{r}}+{1\over{a}}\right)}</math>
where:
- <math>\mu\,</math> is the standard gravitational parameter
- <math>r\,</math> is the distance between the orbiting body and the central body
- <math>\epsilon\,</math> is the specific orbital energy
- <math>a\,\!</math> is the semi-major axis
Note:
- Velocity does not explicitly depend on eccentricity but is determined by length of semi-major axis (<math>a\,\!</math>),
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[edit] Radial trajectories
In the case of radial motion:
- if the energy is non-negative: the motion is either for the whole trajectory away from the central body, or for the whole trajectory towards it. For the zero-energy case, see escape orbit and capture orbit.
- if the energy is negative: the motion can be first away from the central body, up to r=μ/|ε|, then falling back. This is the limit case of an orbit which is part of an ellipse with eccentricity tending to 1, and the other end of the ellipse tending to the center of the central body. See also free-fall time.
[edit] 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. This means that the body moves faster near its periapsis than near its apoapsis, because at the smaller distance it needs to trace a greater arc to cover the same area. This law is usually stated as "equal areas in equal time."
[edit] 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.
- <math>v_o \approx {2 \pi a \over T}</math>
- <math>v_o \approx \sqrt{\mu \over a}</math>
where <math>v_o\,\!</math> is the orbital velocity, <math>a\,\!</math> is the length of the semimajor axis, <math>T\,\!</math> is the orbital period, and <math>\mu\,\!</math> is the standard gravitational parameter. Note that this is only an approximation that holds true when the orbiting body is of considerably lesser mass than the central one, and eccentricity is close to zero.
Taking into account the mass of the orbiting body,
- <math>v_o \approx \sqrt{m_2^2 G \over (m_1 + m_2) r}</math>
where <math>m_1\,\!</math> is now the mass of the body under consideration, <math>m_2\,\!</math> is the mass of the body being orbited, <math>r\,\!</math> is specifically the distance between the two bodies (which is the sum of the distances from each to the center of mass), and <math>G\,\!</math> is the gravitational constant. This is still a simplified version; it doesn't allow for elliptical orbits, but it does at least allow for bodies of similar masses.
For an object in an eccentric orbit orbiting a much larger body, the length of the orbit decreases with eccentricity <math>e\,\!</math>, and is given at ellipse. This can be used to obtain a more accurate estimate of the average orbital speed:
- <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 - \dots \right] </math>[1]
The mean orbital speed decreases with eccentricity.
[edit] Earth orbits
| orbit | center-to-center distance | altitude above the Earth's surface | speed | period/time in space | specific orbital energy |
|---|---|---|---|---|---|
| minimum sub-orbital spaceflight (vertical) | 6500 km | 100 km | 0.0 km/s | just reaching space | 1.0 MJ/kg |
| ICBM | up to 7600 km | up to 1200 km | 6 to 7 km/s | time in space: 25 min | 27 MJ/kg |
| LEO | 6,600 to 8,400 km | 200 to 2000 km | circular orbit: 6.9 to 7.8 km/s elliptic orbit: 6.5 to 8.2 km/s | 89 to 128 min | 32.1 to 38.6 MJ/kg |
| Molniya orbit | 6,900 to 46,300 km | 500 to 39,900 km | 1.5 to 10.0 km/s | 11 h 58 min | 54.8 MJ/kg |
| GEO | 42,000 km | 35,600 km | 3.1 km/s | 23 h 56 min | 57.5 MJ/kg |
| Orbit of the Moon | 363,000 to 406,000 km | 357,000 to 399,000 km | 0.97 to 1.08 km/s | 27.3 days | 61.8 MJ/kg |
[edit] See also
[edit] References
- ^ H. St̀eocker, J. Harris (1998). Handbook of Mathematics and Computational Science. Springer, p. 386. ISBN 0387947469.
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