Orbit equation

In astrodynamics an orbit equation defines the path of orbiting body $m_2\,\!$ around central body $m_1\,\!$ relative to $m_1\,\!$, 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).

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).

In polar coordinates the orbit equation is:

$r={{h^2}\over{\mu}}{{1}\over{1+e\cos\theta}}$ or $u={{1+e\cos\theta}\over{a(1-e^2)}}$

where we have the polar coordinates:

• $r\,$ is the distance between the orbiting body and the central body
• $u={{1}\over{r}}\,$ is the reciprocal distance (see the Binet's Equation [1])
• $\theta\,$ is the direction of the orbiting body, called the true anomaly

and the parameters:

• $h\,\!$ is the specific relative angular momentum of the orbiting body
• $e\,\!$ is the eccentricity of the orbit, also called the orbital eccentricity
• $a\,\!$ is the semi-major axis of the orbit (taken negative for a hyperbola)
• $\mu\,\!$ is the constant which, divided by the distance squared, gives the magnitude of the acceleration; in the case of gravity this is the standard gravitational parameter

Note that ${{h^2}\over{\mu}}$ is the semi-latus rectum of the conic section. This ratio, together with $e\,\!$, fully determines the geometry of the orbit. For a given orbit, the larger $\mu\,\!$, the faster the orbiting body moves in it: twice as fast if the attraction is four times as strong.

The minimum value of r in the equation is

$r={{h^2}\over{\mu}}{{1}\over{1+e}}$

while, if e < 1, the maximum value is

$r={{h^2}\over{\mu}}{{1}\over{1-e}}$

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 the energy is non-negative (parabolic or hyperbolic orbit): the motion is either away from the central body, or towards it.
• if the energy is negative: the motion can be first away from the central body, up to

$r={{h^2}\over{\mu}}{{1}\over{1-e}}$

after which the object falls back.

If r becomes such that the orbiting body enters an atmosphere, then the standard assumptions no longer apply, as in atmospheric reentry.

Low-energy trajectories

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.

If the horizontal speed is $v\,\!$, then the periapsis distance is $\frac{v^2}{2g}$. The energy at the surface of the Earth corresponds to that of an elliptic orbit with $a=R/2\,\!$ (with $R\,\!$ the radius of the Earth), which can not actually exist because it is an ellipse fully below the surface. The energy increase with increase of a is at a rate $2g\,\!$. The maximum height above the surface of the orbit is the length of the ellipse, minus $R\,\!$, minus the part "below" the center of the Earth, hence twice the increase of $a\,\!$ minus the periapsis distance. At the top the potential energy is g times this height, and the kinetic energy is $\frac{v^2}{2}$. This adds up to the energy increase just mentioned. The width of the ellipse is 19 minutes times $v\,\!$.

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. See also trajectory.

Categorization of orbits

Consider orbits which are at one point horizontal, near the surface of the Earth. For increasing speeds at this point the orbits are subsequently:

• part of an ellipse with vertical major axis, with the center of the Earth as the far focus (throwing a stone, sub-orbital spaceflight, ballistic missile)
• a circle just above the surface of the Earth (Low Earth orbit)
• an ellipse with vertical major axis, with the center of the Earth as the near focus
• a parabola
• a hyperbola

Note that in the sequence above, h, $\epsilon$ and a increase monotonically, but e first decreases from 1 to 0, then increases from 0 to infinity. The reversal is when the center of the Earth changes from apoapsis to periapsis (the other focus starts near the surface and passes the center of the Earth). We have

$e=\left | \frac{R}{a}-1\right |$

Extending this to orbits which are horizontal at another height, and orbits of which the extrapolation is horizontal below the surface of the Earth, we get a categorization of all orbits, except the radial trajectories, for which, by the way, the orbit equation can not be used. In this categorization ellipses are considered twice, so for ellipses with both sides above the surface one can restrict oneself to taking the side which is lower as the reference side, while for ellipses of which only one side is above the surface, taking that side.

See also

• Kepler's first law
• Circular orbit
• Elliptic orbit
• Parabolic trajectory
• Hyperbolic trajectory
• Rocket equation