Specific relative angular momentum

In celestial mechanics the specific relative angular momentum h → {displaystyle {vec {h}}} plays a pivotal role in the analysis of the two-body problem. One can show that it is a constant vector for a given orbit under ideal conditions. This essentially proves Kepler's second law.The orbit of a planet is an ellipse with the Sun at one focus.The line joining the planet to the Sun sweeps out equal areas in equal times.The square of the period of a planet is proportional to the cube of its mean distance to the Sun. In celestial mechanics the specific relative angular momentum h → {displaystyle {vec {h}}} plays a pivotal role in the analysis of the two-body problem. One can show that it is a constant vector for a given orbit under ideal conditions. This essentially proves Kepler's second law. It's called specific angular momentum because it's not the actual angular momentum L → {displaystyle {vec {L}}} , but the angular momentum per mass. Thus, the word 'specific' in this term is short for 'mass-specific' or divided-by-mass: Thus the SI unit is: m2·s−1. m {displaystyle m} denotes the reduced mass 1 m = 1 m 1 + 1 m 2 {displaystyle {frac {1}{m}}={frac {1}{m_{1}}}+{frac {1}{m_{2}}}} . The specific relative angular momentum is defined as the cross product of the relative position vector r → {displaystyle {vec {r}}} and the relative velocity vector v → {displaystyle {vec {v}}} . The h → {displaystyle {vec {h}}} vector is always perpendicular to the instantaneous osculating orbital plane, which coincides with the instantaneous perturbed orbit. It would not necessarily be perpendicular to an average plane which accounted for many years of perturbations. As usual in physics, the magnitude of the vector quantity h → {displaystyle {vec {h}}} is denoted by h {displaystyle h} : The following is only valid under the simplifications also applied to Newton's law of universal gravitation. One looks at two point masses m 1 {displaystyle m_{1}} and m 2 {displaystyle m_{2}} , at the distance r {displaystyle r} from one another and with the gravitational force F → = G m 1 m 2 r 2 r → r {displaystyle {vec {F}}=G{frac {m_{1}m_{2}}{r^{2}}}{frac {vec {r}}{r}}} acting between them. This force acts instantly, over any distance and is the only force present. The coordinate system is inertial. The further simplification m 1 ≫ m 2 {displaystyle m_{1}gg m_{2}} is assumed in the following. Thus m 1 {displaystyle m_{1}} is the central body in the origin of the coordinate system and m 2 {displaystyle m_{2}} is the satellite orbiting around it. Now the reduced mass is also equal to m 2 {displaystyle m_{2}} and the equation of the two-body problem is