visvivaBeziehung

VisvivaBeziehung, commonly called the vis-viva equation, is a core relation in celestial mechanics. It links the orbital speed v of a body orbiting a central mass to its distance r from the focus and to the orbit’s semi-major axis a. The standard form is v^2 = μ (2/r − 1/a). In this equation μ is the standard gravitational parameter, equal to GM, where G is the gravitational constant and M is the mass of the central body. Thus μ depends on the central body, with typical values such as μ_sun ≈ 1.327×10^20 m^3/s^2 and μ_earth ≈ 3.986×10^14 m^3/s^2.

The Vis-vivaBeziehung applies to all conic sections: elliptical orbits (a > 0), parabolic (a → ∞), and hyperbolic (a < 0). It follows from the conservation of mechanical energy and angular momentum; specifically, the specific orbital energy ε = v^2/2 − μ/r equals −μ/(2a), which, when rearranged, yields the vis-viva equation.

Uses of the relation include determining the speed of a spacecraft at a given distance from a planet, planning transfer trajectories, and analyzing orbital properties. It provides a direct link between the orbit’s size (a) and its instantaneous speed at any radius r, making it a fundamental tool in orbital design and astrodynamics.