![[diagram of orbital parameters]](orbital-parameters-clip7.jpg)
lower case omega (ω), sometimes "w"
(Nobody knows why it's called "Argument". The earliest known use of the term "argument" to describe an angle was by Chaucer ~1391 in section 44 of his unfinished "A Treatise on the Astrolabe")
![[dragon's head]](images/dragons_head.png)
- A circle has an eccentricity of 0.
- An ellipse has eccentricity between 0 and 1.
- A parabola has an eccentricity of 1.
- A hyperbola has an eccentricity greater than 1.
- sqrt (1-(b^2/a^2))
where a is the semimajor axis and b is the semiminor axis - (apocenter - pericenter) / (apocenter + pericenter)
- 1-(pericenter / semimajor axis)
![[\bar omega ]](images/bar_omega.png)
This sometimes is called a "broken angle" since it includes angular measurements in two different planes.
L = M + = M + Ω + ω,
where
-
M is the orbit's Mean Anomaly,
is the Longitude of Pericenter,
Ω is the Longitude of the Ascending Node and
ω is the Argument of Pericenter.
-
For a closed orbit: e < 1.0
a = q/(1-e)
n = sqrt (G[M+m]/a^3) = elliptical mean motion
n = 1/P -
For a hyperbola: e > 1.0
which is not a closed orbit, one can use
a = q/(1-e)
n = sqrt (G[M+m]/[-a]^3) = hyperbolic mean motion.
(note that a is negative for hyperbolic orbits; hence the use of the minus sign to make the value positive.)
where
n is the Mean Motion
a is the length of the semimajor axis
q is the pericenter distance
e is the eccentricity of the ellipse
G is the "gravitational constant"
M is the mass of the central object
m is the mass of the orbiting object
P is the period of the orbting object
See also: http://www.tamuk.edu/math/scott/stars/docs/classical.pdf
Note: n is measured in Radians, not degrees.
- q = a * (1 - e)
where q is the pericenter distance
a is the length of the semimajor axis and
e is the eccentricity of the ellipse.
- For a closed, elliptical solar orbit,
P = sqrt (( q / ( 1 - e)) ^3)
or = a^1.5
P = 1/n = 1/ Mean Motion - For an open, hyperbolic orbit, one can use
P = sqrt (( q / ( e - 1)) ^3)where
P is the period measured in years
q is the pericenter measured in AU
e is the eccentricity
and a is the semimajor axis of the orbit, also measured in AU (which has a negative value for hyperbolic orbits). - More generally,
P = sqrt( K a^3 )
where a is the semimajor axis of the orbit,
and K is inversely proportional to the sum of the masses of the central body and the orbiting body.When the mass of an orbiting body is small enough to ignore, the same value of K can be used for all small objects orbiting around the same central body. In other words, given the orbital period of one satellite, it's easy to calculate the periods of others. This is also known as "Kepler's Third Law."