Specific angular momentum

The specific angular momentum is a physical conserved quantity in the celestial mechanics and serves as an important auxiliary variable in solving the two-body problem . The specific angular momentum is defined as the angular momentum of a body on a Kepler orbit in relation to the other body, normalized to the reduced mass of the system and therefore has the SI unit m ² · s ⁻¹ . If one of the two bodies has a significantly greater mass than the other, the specific angular momentum of the lighter body is a characteristic of the orbit and independent of its other properties. ${\displaystyle {\vec {h}}}$

The property of the conservation quantity follows from the fact that the gravitational potential as the cause of the force that the body experiences is a central potential , i.e. it only depends on the distances between the two bodies, but not on the angle between them. The second Kepler law follows from the conservation of the specific angular momentum .

Mathematical formulation

The specific angular momentum of a body is

${\displaystyle {\vec {h}}={\frac {m_{\text{ges}}}{M}}{\vec {r}}\times {\vec {v}}}$

where the total mass of the system, the mass of the other body, the distance vector between the two bodies and the relative speed denotes. Since, in addition to the conservation of angular momentum, energy conservation applies in the gravitational potential and the paths represent geometrically conic sections , the square of the amount of the specific angular momentum can be used as ${\displaystyle m_{\text{ges}}}$ ${\displaystyle M}$ ${\displaystyle {\vec {r}}}$ ${\displaystyle {\vec {v}}}$

${\displaystyle h^{2}={\frac {m_{\text{ges}}^{3}}{M^{2}}}Ga(1-\varepsilon ^{2})}$

be expressed, where is the gravitational constant , the semi-major axis and the numerical eccentricity of the orbit. If the other body is significantly heavier, this equation simplifies to: ${\displaystyle G}$ ${\displaystyle a}$ ${\displaystyle \varepsilon }$

${\displaystyle h^{2}=MGa(1-\varepsilon ^{2})}$

The size is also called the half parameter of the orbit. These equations apply to circular , elliptical and hyperbolic trajectories ; for parabolic orbits they give an indefinite expression, since their semi-major axis is infinitely large, while their eccentricity is one. Nevertheless, the half-parameter also has a definite value for parabolas. ${\displaystyle a(1-\varepsilon ^{2})}$ ${\displaystyle p}$

The direction of the specific angular momentum, like the direction of the angular momentum, is always perpendicular to the plane of the orbit .

Connection with Kepler's Second Law

The area of the relative position vector swept over in an infinitesimal time step from a heavy mass to a light mass ${\displaystyle F}$

${\displaystyle F(t,t+\mathrm {d} t)={\frac {1}{2}}|{\vec {r}}(t)\times {\vec {v}}(t)|\mathrm {d} t={\frac {1}{2}}h\mathrm {d} t}$

Since the specific angular momentum is constant, integration results for a finite period between and : ${\displaystyle t_{1}}$ ${\displaystyle t_{2}}$

${\displaystyle F(t_{1},t_{0})=\int _{t_{0}}^{t_{1}}F(t,t+\mathrm {d} t)\mathrm {d} t={\frac {1}{2}}h(t_{1}-t_{0})}$

From this follows the second Kepler law: "The driving beam sweeps over the same areas at the same time."

See also

• Specific orbital energy , 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 .