|
In the gravitational two-body problem, the specific orbital energy (or vis-viva energy) of two orbiting bodies is the constant sum of their mutual potential energy () and their total kinetic energy (), divided by the reduced mass. According to the orbital energy conservation equation (also referred to as vis-viva equation), it does not vary with time: : : where * is the relative orbital speed; * is the orbital distance between the bodies; * is the sum of the standard gravitational parameters of the bodies; * is the specific relative angular momentum in the sense of relative angular momentum divided by the reduced mass; * is the orbital eccentricity; * is the semi-major axis. It is expressed in J/kg = m2·s−2 or MJ/kg = km2·s−2. For an elliptical orbit the specific orbital energy is the negative of the additional energy required to accelerate a mass of one kilogram to escape velocity (parabolic orbit). For a hyperbolic orbit, it is equal to the excess energy compared to that of a parabolic orbit. In this case the specific orbital energy is also referred to as characteristic energy. ==Equation forms for different orbits== For an elliptical orbit, the specific orbital energy equation, when combined with conservation of specific angular momentum at one of the orbit's apsides, simplifies to: : is the standard gravitational parameter; * is semi-major axis of the orbit. Proof: ::For an elliptical orbit with specific angular momentum ''h'' given by :: ::we use the general form of the specific orbital energy equation, :: ::with the relation that the relative velocity at periapsis is :: ::Thus our specific orbital energy equation becomes :: ::and finally with the last simplification we obtain: :: or the same as for an ellipse, depending on the convention for the sign of ''a''. In this case the specific orbital energy is also referred to as characteristic energy (or ) and is equal to the excess specific energy compared to that for a parabolic orbit. It is related to the hyperbolic excess velocity (the orbital velocity at infinity) by : It is relevant for interplanetary missions. Thus, if orbital position vector () and orbital velocity vector () are known at one position, and is known, then the energy can be computed and from that, for any other position, the orbital speed. 抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)』 ■ウィキペディアで「Specific orbital energy」の詳細全文を読む スポンサード リンク
|