﻿ Orbital speed

# Orbital speed

In celestial mechanics, path velocity refers to the speed at which moves an astronomical object. For orbits is also called orbital velocity or rotational speed.

## Path velocity of the ideal Keplerian orbit

Encountered a small body in space a large, so is the trajectory due to their gravity, respectively - in the ideal case - a Keplerian orbit ( ellipse, hyperbola or parabola). Because of the conservation of energy, the web speed is not constant, but increases as the distance between the bodies is small. Johannes Kepler discovered that although distance and train speed may vary, but the driving beam ( the line connecting Gravizentrum a wrap-around body ) at the same time the same area sweeps (second Kepler law, constancy of the surface speed). His solution is only valid for the two-body problem ( Kepler problem ) itself, the restriction of a massless rotating body, and only as a non- relativistic approximation. In addition, it is always the relative velocity with respect to the Gravizentrums, never an absolute speed.

For the special case of a circular orbit, the force of attraction between heavenly bodies each brings just the necessary centripetal force for the circular path, whereby the speed is fixed ( and amount constant).

The route along the Keplerian orbit, which is used for direct displacement-time context (speed = distance per time ), has only in special cases, an analytical solution. Through observation of kinetic and potential energy, the derivation of the Vis -Viva equation succeeds. It provides a connection between the mass of the central body, the gravitational constant, the semi-major axis of the elliptical orbit, the distance of the rotating body and the velocity of said body forth:

This equation is valid for all conic sections, closed paths ( orbits ) as well as open ( parabolic and hyperbolic orbits, one-time encounters between two bodies ).

According to Vis Viva equation for the circular path and the parabolic trajectory in the crown are the special cases:

Below, between and above these two limiting cases are spiral and hyperbolic orbits ( to fall and leaving a celestial body or passages). The elliptical orbits are the most energetically stable orbits of the first cosmic velocity, ie correspond to the effect of the circular path, and are seen only geometric special cases of the same.

For one involving mass orbiting body ( mass ) applies:

For the two main vertices of the ellipse, there are also analytical solutions that can be calculated solely using the ellipse geometry and do without the gravitational parameter:

Because the radius vector differentially hardly changes in the apices, the following applies:

The Perizentrumsgeschwindigkeit is the maximum, the minimum Apozentrumsgeschwindigkeit the web speed.

## Mean orbital velocity

The mean orbital velocity results simply from the context of distance per time. Now the circumference of the ellipse is not analytically determined ( with an elliptic integral function of the eccentricity). For circular orbits or nearly circular orbits ( e small and m significantly smaller M) the first or second cosmic speed can be used with:

A first approximation taking into account the numerical eccentricity is

For an invoice with a in astronomical units and T in days, a conversion factor AU • d is given to km • s of 5439.5 ( π miteingerechnet ):

The expansion is

With increasing eccentricity e, the average path velocity decreases. Here the eccentricity is but a maximum square, so it can be neglected rapidly at small eccentricities.

It is also a simple average of Peri - and Apozentrumsgeschwindigkeit:

However, this approximation is contrary to the first, because with increasing eccentricity e increases, the average path velocity.

## Orbital velocities of artificial satellites

The web speed at satellites have nearly circular orbits is, depending on the orbit class:

• On Low Earth Orbit (LEO ) at 200 km altitude about 7 km / s (25000 km / h)
• On geostationary orbit (GEO, 42,164 km orbital radius, 35,786 km above the equator) about 3 km / s ( 11,000 km / h )

Typical launcher systems make a driving capacity of 7-11 km / s The focal length of the system is entirely dependent on the technology, ie the thrust (acceleration), and then total to achieve the necessary speed ( first cosmic velocity of the earth ) for a stable orbit. This also applies to the following drive systems.

In contrast to the kepler between Ideally satellites are subjected to friction in the atmosphere, especially at lower orbits a significant braking force, which makes them more slowly ( and consequently should not fall back to Earth ): It must orbit corrections are made periodically. Therefore, each Stellite must be equipped with propulsion systems, and its fuel supply limits its lifetime. Therefore, it is also standard for satellites (where the average angular velocity is referred to as the satellite orbit element mean motion ) at least in a seventh track element, for example, the braking effect ( as a change in the mean motion, descent rate per unit of time ), or indicate (based on the ballistic coefficient ) over the the loss of speed can be calculated. Typical path correction systems make a driving capacity of 10-600 m / s, ie a 10.000stel to 10ths of the launcher, depending on the altitude of the mission.

There are also numerous other disturbances that require further path corrections. These are summarized as position control, such systems are capable of around 20 m / s Here are - in a geostationary satellite - for the gravitational influence of the Earth and Moon 40-51 m / s per year necessary for the radiation pressure of the sun ( solar wind ) up to 30 m / s per year, the other faults remain in the single digits.

In some missions, an explicit path change is necessary, what systems with 1 to several km / s drive capacity are necessary. Engines for this task are not as path correction and position control systems to the secondary, but as the means of delivery to the primary systems expected.

## Path velocities of small bodies and space missions

In small bodies we group asteroids, comets and meteoroids together. Most asteroids follow as regular solar system objects like planets circular orbits, and there are numerous but irregular objects on highly eccentric ellipses and aperiodic objects on hyperbolic orbits. Due to the size are a lot of undiscovered, and a precise orbits of one-time observation often not possible.

A crucial factor for the origin of these bodies is the escape velocity of the sun ( or the total mass of the solar system ). This is - at the height of the Earth's orbit - at 42 km / s, which is about 150,000 km / h ( Third cosmic velocity ), until the sun's surface it grows to 620 km / s ( 2.2 million km / h ) at. All objects that are faster, leave the solar system, either by an enormous train fault, or they are indeed extrasolar origin. The escape velocity increases - after the above-mentioned formulas - with as the distance to the sun from: So goes the Voyager probes are now far behind the orbit of Saturn, a speed that is less than the orbital speed of the Earth around the sun to the solar system leave: in fact, the distance between Earth and Voyager probes decreases at times when the earth moves in the direction of their flight path. But a separate drive is necessary, or a gain in speed solar system away, as it can be reached by swing-by maneuvers ( the Voyagers were the swing-by of Saturn around 18 km / s speeds ), or even converted into a very strong impact of another body.

In Earth's orbit cruisers, including meteors and meteor showers ( shooting stars swarms ), you are in deviation from the previous, not a barycentric velocity, but the much more relevant speed relative to the earth. Depending on the angle of arrival to the Earth's orbit these objects have velocities between 11.2 ( trailer ) to 72 km / s ( frontal hits).

## Examples

• Mean path velocity of the Earth ( the Sun / barycenter of the solar system ): for comparison: speed of rotation of the earth's surface at the equator ( center of the earth ): - the speed of the observer at the equator around the sun, that is the same as the Earth ± 1.7 % diurnal (daily)
• Mean orbital velocity of the moon ( to the Earth-Moon center of gravity): For comparison: orbit velocity around the sun: the same as the Earth ± 3.4 % mensal (monthly)
• Web speed of the ISS ( the Earth ): for comparison: relative velocity ( to the observer on the Earth's surface ):
• Web speed of the " Voyager 1 " probe ( to the sun ):
• Path velocity of the comet Tempel-Tuttle at perihelion (ie the sun): for comparison: relative speed of the Leonids, the meteor shower produced by him to the earth, - that is 250 times the speed of sound
• Path velocity of the solar system ( around the galactic center ): for comparison: web speed of the earth around the galactic center: the same as sun ± 12 % annual (yearly)
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