, 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).
If the conic section intersects the central body, then the actual trajectory can only be the part above the surface, but for that part the orbit equation and many related formulas still apply, as long as it is a freefall (situation of weightlessness).
Consider a two-body system consisting of a central body of mass M and a much smaller, orbiting body of mass
, 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 as[1]
describes a conic section.
controls what kind of conic section the orbit is: The minimum value of
If the maximum is less than the radius of the central body, then the conic section is an ellipse which is fully inside the central body and no part of it is a possible trajectory.
If the maximum is more, but the minimum is less than the radius, part of the trajectory is possible: If
becomes such that the orbiting body enters an atmosphere, then the standard assumptions no longer apply, as in atmospheric reentry.
If 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.
The energy at the surface of the Earth corresponds to that of an elliptic orbit with
the radius of the Earth), which can not actually exist because it is an ellipse fully below the surface.
The maximum height above the surface of the orbit is the length of the ellipse, minus
, minus the part "below" the center of the Earth, hence twice the increase of
minus the periapsis distance.
times this height, and the kinetic energy is
This adds up to the energy increase just mentioned.
The width of the ellipse is 19 minutes[why?]
The part of the ellipse above the surface can be approximated by a part of a parabola, which is obtained in a model where gravity is assumed constant.
This should be distinguished from the parabolic orbit in the sense of astrodynamics, where the velocity is the escape velocity.
Consider orbits which are at one point horizontal, near the surface of the Earth.
For increasing speeds at this point the orbits are subsequently: Note that in the sequence above[where?
first decreases from 1 to 0, then increases from 0 to infinity.
The reversal is when the center of the Earth changes from being the far focus to being the near focus (the other focus starts near the surface and passes the center of the Earth).