Specific rail energy

The specific orbital energy is a physical conservation quantity in celestial mechanics . It is defined as the energy that a body has in an orbit around another body, normalized to the reduced mass of the system and therefore has the SI unit m ² · s ⁻² . In the context of the two-body problem , which serves as a mathematically solvable model of celestial mechanics, the specific orbital energy is a characteristic of the orbit that the body travels and is independent of its other properties. In particular, its mass is only included in the specific orbital energy in the form of the total mass of the system. The property as a conservation quantity follows from the law of conservation of energy , which states that the sum of kinetic energy and potential energy in the gravitational potential is constant. ${\displaystyle \epsilon }$

Mathematical formulation

The specific orbital energy of a body is by definition

${\displaystyle \epsilon ={\frac {1}{2}}v^{2}-{\frac {Gm_{\text{ges}}}{r}}}$

with the distance between the two bodies , the amount of the relative speed between the bodies , the total mass of the system and the gravitational constant . Since the solutions of the orbits on which a body can move in celestial mechanics are the Kepler orbits and thus geometrically conic sections and since the conservation of angular momentum applies in the gravitational potential , the specific orbital energy can be expressed by the major semi-axis of these conic sections. The following applies: ${\displaystyle r}$ ${\displaystyle v}$ ${\displaystyle m_{\text{ges}}}$ ${\displaystyle G}$ ${\displaystyle a}$

${\displaystyle \epsilon =-{\frac {Gm_{\text{ges}}}{2a}}}$

In this form, the property is manifest as a conserved quantity, since the specific orbital energy no longer contains time-dependent variables. For bound orbits , i.e. ellipses and circles , the semi-major axis is positive and the specific orbital energy is therefore negative. The further away a body orbits the central star, the greater the specific orbital energy becomes. If it is equal to zero, then the orbit is a parabola with an infinitely large semi-axis and the two bodies can be as far away from each other as desired. For hyperbolic orbits , the semi-major axis is negative and the specific orbital energy is positive; these railways are also unbound.

By equating the two formulations for the specific orbital energy, the Vis-Viva equation results

${\displaystyle v^{2}=Gm_{\text{ges}}\left({\frac {2}{r}}-{\frac {1}{a}}\right)}$

Examples

The orbit height, tangential speed, orbital time and specific orbital energy of some orbits around the earth

Orbit	Distance from center to center	Height above the surface of the earth	Track speed	Orbital time	Specific rail energy
Standing on the earth's surface at the equator (comparative value, no orbit)	6,378 km	0 km	465.1 m / s	1 day (24h)	−62.6 MJ / kg
Orbit at the level of the earth's surface (equator)	6,378 km	0 km	7.9 km / s	1 h 24 min 18 sec	−31.2 MJ / kg
Low earth orbit	6 600 to 8 400 km	200 to 2000 km	Circle: 6.9 to 7.8 km / s Ellipse: 6.5 to 8.2 km / s	1 h 29 min to 2 h 8 min	−29.8 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	−4.7 MJ / kg
Geostationary orbit	42,000 km	35,786 km	3.1 km / s	23 h 56 min	−4.6 MJ / kg
Lunar orbit	363,000 to 406,000 km	357,000 to 399,000 km	0.97 to 1.08 km / s	27.3 days	−0.5 MJ / kg

Orbital energy in general relativity

For a small mass in orbit around a large non-rotating mass, the Schwarzschild metric applies ; the conservation magnitude of the total energy is made up of

${\displaystyle -E=mc^{2}+E_{\rm {kin}}+E_{\rm {pot}}}$

So the rest, the kinetic and the potential energy together, whereby

${\displaystyle E_{\rm {kin}}={\frac {mc^{2}}{\sqrt {1-v^{2}/c^{2}}}}-mc^{2}\ }$and . ${\displaystyle \ E_{\rm {pot}}={\frac {mc^{2}\ {\sqrt {1-r_{\rm {s}}/r}}-mc^{2}}{\sqrt {1-v^{2}/c^{2}}}}}$

This gives the specific orbital energy

${\displaystyle \epsilon =-c^{2}\ {\sqrt {\frac {1-r_{\rm {s}}/r}{1-v^{2}/c^{2}}}}+c^{2}}$

with . For the orbit around a strongly rotating dominant mass, the Kerr metric must be used. ${\displaystyle r_{\rm {s}}=2GM/c^{2}}$

See also

• Specific angular momentum , different conservation quantity in the two-body problem.

literature

• Ernst Messerschmid , Stefanos Fasoulas: Space systems. An introduction with exercises and solutions. 2nd updated edition. Springer, Berlin et al. 2005, ISBN 3-540-21037-7 .