Orbit equation

In astrodynamics an orbit equation defines the path of orbiting body m 2 {displaystyle m_{2},!} around central body m 1 {displaystyle m_{1},!} relative to m 1 {displaystyle m_{1},!} , without specifying position as a function of time. Under standard assumptions, a body moving under the influence of a force, directed to a central body, with a magnitude inversely proportional to the square of the distance (such as gravity), has an orbit that is a conic section (i.e. circular orbit, elliptic orbit, parabolic trajectory, hyperbolic trajectory, or radial trajectory) with the central body located at one of the two foci, or the focus (Kepler's first law).Consider a two-body system consisting of a central body of mass M and a much smaller, orbiting body of mass m, and suppose the two bodies interact via a central, inverse-square law force (such as gravitation). In polar coordinates, the orbit equation can be written asIf the central body is the Earth, and the energy is only slightly larger than the potential energy at the surface of the Earth, then the orbit is elliptic with eccentricity close to 1 and one end of the ellipse just beyond the center of the Earth, and the other end just above the surface. Only a small part of the ellipse is applicable.Consider orbits which are at one point horizontal, near the surface of the Earth. For increasing speeds at this point the orbits are subsequently: