Medium movement

The mean movement or mean daily movement n is the average angular velocity of an object in an elliptical orbit and is often given as one of the orbital elements or satellite orbital elements .

Objects in the solar system and artificial satellites in earth orbit move in an elliptical orbit around the center of gravity z. B. the earth. The speed varies during a revolution because of the conservation of the angular momentum .

The mean movement is the angular velocity of a theoretical object on the auxiliary circle of the satellite ellipse, but does not correspond to the mean anomaly according to Kepler, which is related to the pericenter passage, but relates to an observer on the equator, i.e. the time. It is given in d ⁻¹ or rad / d.

Derivation

To get the mean movement, one measures the time it takes the satellite to orbit the earth.

The following applies:

${\displaystyle n={\frac {2\pi }{T}}}$

with T = cycle time

calculation

${\displaystyle n={\sqrt {\frac {G\cdot M}{a^{3}}}}\,\!}$

The unit is rad / s .

G	=	Gravitational constant
M.	=	Mass of the circled object
a	=	major semi-axis

Conversions

If n is given, we know T and the semi-major axis a . If the numerical eccentricity  ε is also known, it follows:

Orbital time	${\displaystyle T={\frac {2\pi }{n}}}$	(1)
major semi-axis	${\displaystyle a={\sqrt[{3}]{\mu \cdot {\frac {T^{2}}{4\pi ^{2}}}}}}$	(2)
small semi-axis	${\displaystyle b={\sqrt {a^{2}-(\varepsilon \cdot a)^{2}}}}$	(3)
Distance of the perigee	${\displaystyle r_{\mathrm {Peri} }=a\cdot (1-\varepsilon )}$	(4)
Distance of apogee	${\displaystyle r_{\mathrm {Apo} }=a\cdot (1+\varepsilon )}$	(5)

example

${\displaystyle n={\frac {2\pi }{T}}}$

The International Space Station ISS has an orbital time of around 91 minutes.

${\displaystyle n={\frac {2\pi }{91\,\mathrm {min} }}\approx 1{,}15\cdot 10^{-3}\,{\frac {\mathrm {rad} }{\mathrm {s} }}}$

or

${\displaystyle \!\ n={\sqrt {\frac {G\cdot M}{a^{3}}}}}$

The major semi-axis of the ISS-Bahn has a length of around 6,720 km (earth radius + orbit height).

${\displaystyle \!\ G\cdot M=398200\,\mathrm {km} ^{3}/\mathrm {s} ^{2}}$ (Gravitational constant × mass of the earth)

${\displaystyle n={\sqrt {\frac {398200\,{\frac {\mathrm {km} ^{3}}{\mathrm {s} ^{2}}}}{(6720\,\mathrm {km} )^{3}}}}\approx 1{,}15\cdot 10^{-3}\,{\frac {\mathrm {rad} }{\mathrm {s} }}}$

See also

• Medium anomaly

literature

• Andreas Guthmann: Introduction to celestial mechanics and ephemeris calculus . 2nd Edition. Spectrum Academic Publishing House, Heidelberg / Berlin 2000, ISBN 3-8274-0574-2 , p. 143, 183 .