Vis-Viva equation

The celestial mechanical Vis-Viva equation provides the local speed of bodies on Kepler orbits around a dominant celestial body, which influences the other bodies through its gravitation . Due to the dominating celestial body, the system can be described approximately as a two-body problem for each body , whereby the influences of the different bodies on one another are neglected. The Kepler orbits are conic sections , i.e. ellipses, parabolas and hyperbolas, around the common center of gravity , which are described by the two parameters of the semi-major axis and the eccentricity .

The Vis-Viva equation is based on the law of conservation of energy and the law of conservation of angular momentum , according to which the sum of the kinetic and potential energy or the angular momentum in the gravitational potential is constant. The conservation laws follow from the fact that the gravitational potential is constant over time and only depends on the distance from the center, but not on the angle; the Vis-Viva equation itself only requires as a requirement of the potential that the radial dependence is inversely proportional to the radius.

The kinetic energy depends only on the speed of the body on the path and the potential energy only on the distance. In this way, the Vis-Viva equation enables the speed and current position of the body to be linked. In addition to the gravitational parameter of the system, only the major semi-axis is included as a further parameter in the equation, but not the eccentricity of the path of the rotating object.

Etymologically, the Vis-Viva equation refers to the vis viva , in German living force , in modern terminology double the kinetic energy.

Mathematical formulation

The Vis-Viva equation for the instantaneous speed of a body that is on a path around another body is:

${\displaystyle v^{2}=G(M+m)\left({\frac {2}{r}}-{\frac {1}{a}}\right)}$

Here, the distance of the two bodies, the square of the relative velocity between the bodies, the semi-major axis of the web ( for an ellipse, a parabola and a hyperbola), the gravitational constant , and as well as the masses of the two bodies. ${\displaystyle r}$ ${\displaystyle v^{2}}$ ${\displaystyle a}$ ${\displaystyle a>0}$ ${\displaystyle a\to \infty }$ ${\displaystyle a<0}$ ${\displaystyle G}$ ${\displaystyle m}$ ${\displaystyle M}$

Derivation

The derivation of the Vis-Viva equation follows the law of conservation of energy and angular momentum. In the gravitational potential of two bodies, the total energy is through

${\displaystyle E={\frac {1}{2}}(m+M)v_{s}^{2}+{\frac {1}{2}}\mu v^{2}-G{\frac {mM}{r}}}$

given, where describes the speed of the center of gravity and is the reduced mass of the system, defined by ${\displaystyle v_{s}}$ ${\displaystyle \mu }$

${\displaystyle \mu ={\frac {mM}{m+M}}}$.

If one of the two bodies is significantly heavier than the other , then is . ${\displaystyle M\gg m}$ ${\displaystyle \mu \approx m}$

Due to the conservation of angular momentum, the total energy can increase with the amount of angular momentum and the Pythagorean theorem ${\displaystyle L}$

${\displaystyle E={\frac {1}{2}}(m+M)v_{s}^{2}+{\frac {1}{2}}\mu {\dot {r}}^{2}+{\frac {L^{2}}{2mr^{2}}}-G{\frac {mM}{r}}}$

are reshaped, where the speed in the radial direction referred to the center of gravity. ${\displaystyle {\dot {r}}}$

From the law of conservation of energy it follows for any two distances and ${\displaystyle r_{1}}$ ${\displaystyle r_{2}}$

${\displaystyle \mu {\dot {r}}_{1}^{2}+{\frac {\mu L^{2}}{2m^{2}r_{1}^{2}}}-G{\frac {mM}{r_{1}}}=\mu {\dot {r}}_{2}^{2}+{\frac {\mu L^{2}}{2m^{2}r_{2}^{2}}}-G{\frac {mM}{r_{2}}}}$,

whereby the contribution of the energy by the movement of the center of gravity cancel each other out. At the two points that are closest and most distant to the central body on a Kepler orbit, the periapsis and the apoapsis , the radial component of the velocity disappears and it is therefore valid ${\displaystyle r_{P}}$ ${\displaystyle r_{A}}$

${\displaystyle {\frac {\mu L^{2}}{2m^{2}r_{P}^{2}}}-G{\frac {mM}{r_{P}}}={\frac {\mu L^{2}}{2m^{2}r_{A}^{2}}}-G{\frac {mM}{r_{A}}}}$

This gives the square of the angular momentum to

${\displaystyle L^{2}=2G{\frac {Mm^{3}}{\mu }}\left({\frac {r_{A}r_{P}}{r_{A}+r_{P}}}\right)}$

and the energy too

${\displaystyle E=-G{\frac {Mm}{r_{A}+r_{P}}}}$

From the geometry of the conic sections follows with ${\displaystyle \textstyle a={\frac {r_{P}+r_{A}}{2}}}$

${\displaystyle E=-G{\frac {Mm}{2a}}}$.

From this equation follows with the definition of the total energy

${\displaystyle v^{2}={\frac {2E}{\mu }}+2G{\frac {Mm}{\mu r}}=G(m+M)\left({\frac {2}{r}}-{\frac {1}{a}}\right)}$

Cosmic speeds

Is for everyone , the Kepler orbit degenerates into a circular orbit; the body has the same distance from the center of gravity everywhere and accordingly the same speed everywhere ${\displaystyle a=r}$ ${\displaystyle r}$ ${\displaystyle r}$

${\displaystyle v_{1}={\sqrt {\frac {G(m+M)}{r}}}}$,

the orbital speed or first cosmic speed .

In order for a body to be able to overcome the influence of the central star, the major semi-axis must be infinitely large, so the following applies : ${\displaystyle a\to \infty }$

${\displaystyle v_{2}={\sqrt {\frac {2G(m+M)}{r}}}=v_{1}\cdot {\sqrt {2}}}$

This speed is called the escape speed or second cosmic speed . The body's orbit is then no longer closed, but open. If the orbital speed of the body is exactly the same , the orbit is a parabola, whereas at higher orbital speeds (with otherwise constant distance ) it is a hyperbola and the major semi-axis becomes (formally) negative. ${\displaystyle v_{2}}$ ${\displaystyle r}$ ${\displaystyle a}$

Example: orbital speeds in the solar system

In the solar system , the sun is the dominant central body. The mass of the earth is z. B. only 1 / 330,000 of the solar mass and can be neglected when applying the Vis-Viva equation - the error is smaller than the neglect of orbital disturbances by Jupiter. With neglected there is a constant for the respective central star, and it makes sense to extract this constant up to a unit of length from the root and to calculate it as a pre-factor. ${\displaystyle m}$ ${\displaystyle GM}$

Distances in the solar system are often in astronomical units . The prefactor then has the value

${\displaystyle {\sqrt {GM/1\,{\text{AE}}}}=2\pi \,{\text{AE pro Jahr}}\approx 0{,}01720\,{\text{AE pro Tag}}\approx 29{,}785\,{\text{km/s}}\,}$,

the mean orbital speed of the earth around the sun, which is also called the Gaussian gravitational constant .

For the velocities of the earth in perihelion and aphelion applies with a distance or and the major semi-axis ${\displaystyle r_{P}=0{,}983\,{\text{AE}}}$ ${\displaystyle r_{A}=1{,}017\,{\text{AE}}}$ ${\displaystyle a=1\,{\text{AE}}}$

${\displaystyle v_{\text{P}}=29{,}785\,{\text{km/s}}{\sqrt {{\frac {2}{0{,}983}}-{\frac {1}{1}}}}=30{,}296\,{\text{km/s}}}$

${\displaystyle v_{\text{A}}=29{,}785\,{\text{km/s}}{\sqrt {{\frac {2}{1{,}017}}-1}}=29{,}284\,{\text{km/s}}}$

and for the comet Tschurjumow-Gerassimenko in perihelion with , in aphelion with a major semi-axis of : ${\displaystyle r_{P}=1{,}289\,{\text{AE}}}$ ${\displaystyle r_{A}=5{,}717\,{\text{AE}}}$ ${\displaystyle a=3{,}503\,{\text{AE}}}$

${\displaystyle v_{\text{P}}=29{,}785\,{\text{km/s}}{\sqrt {{\frac {2}{1{,}289}}-{\frac {1}{3{,}503}}}}=33{,}51\,{\text{km/s}}}$

${\displaystyle v_{\text{A}}=29{,}785\,{\text{km/s}}{\sqrt {{\frac {2}{5{,}717}}-{\frac {1}{3{,}503}}}}=7{,}56\,{\text{km/s}}}$

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 , pp. 71-86.