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Orbital elements
The previous analysis suffices when considering a single planet orbiting around the Sun. However, it becomes inadequate when dealing with multiple planets whose
orbital planes and perihelion directions do not necessarily coincide. Incidentally, for the time being, we are neglecting interplanetary gravitational interactions, which allows us to assume
that each planet executes an independent Keplerian elliptical orbit about the Sun.
Let us characterize all planetary orbits using a common Cartesian coordinate
system
,
,
, centered on the Sun. (See Figure 4.6.) The

plane defines a reference plane,
which is chosen to be the ecliptic plane (i.e., the plane of the Earth's orbit), with the
axis pointing towards the ecliptic north pole (i.e., the direction normal to the ecliptic plane in a northward sense).
Likewise, the
axis defines a reference direction, which is chosen
to point in the direction of the vernal equinox (i.e., the point in the Earth's sky at which the apparent orbit of the
Sun passes through the extension of the Earth's equatorial plane from south to north). Suppose that the plane of a given
planetary orbit is inclined at an angle
to the reference plane. The point
at which this orbit crosses the reference plane in the direction of
increasing
is termed its ascending node. The angle
subtended between the reference direction and the direction of the ascending node
is termed the longitude of the
ascending node. Finally, the angle,
, subtended between the
direction of the ascending node and the direction of the orbit's perihelion is termed
the argument of the perihelion.
Figure 4.6:
A general planetary orbit.

Let us define a second Cartesian coordinate system
,
,
, also
centered on the Sun. Let the

plane coincide with the plane of a particular planetary orbit,
and let the
axis point towards the orbit's perihelion point. Clearly,
we can transform from the
,
,
system to the
,
,
system via a series of three rotations of the coordinate system: first,
a rotation through an angle
about the
axis (looking down the
axis); second, a rotation through an angle
about the new
axis; and
finally, a rotation through an angle
about the new
axis.
It thus follows from standard coordinate transformation theory (see Section A.6) that





(4.71) 
However,
,
, and
. Hence,
Thus, a general planetary orbit is determined by Equations (4.67)(4.70) and (4.72)(4.74), and is therefore parameterized
by six orbital elements: the major radius,
; the time of perihelion passage,
; the eccentricity,
;
the inclination (to the ecliptic plane),
; the argument of the perihelion,
; and the longitude of the ascending node,
. [The mean orbital angular velocity, in radians per year, is
, where
is measured in astronomical units. Here, an astronomical unit is the mean EarthSun distance, and
corresponds to
(Yoder 1995).]
In lowinclination orbits, the argument of the perihelion is usually replaced by

(4.75) 
which is termed the longitude of the perihelion. Likewise, the
time of perihelion passage,
, is often replaced by the mean
longitude at
otherwise known as the mean longitude at epochwhere the mean longitude is defined

(4.76) 
Thus, if
denotes the mean longitude at epoch (
) then

(4.77) 
where
. Another common alternative to
is the mean anomaly at epoch,

(4.78) 
The orbital elements of the major planets at the
epoch J2000 (i.e., at 00:00 UT on January 1, 2000) are given in Table 4.1.
1pt
Table:
Planetary data for J2000.
 major radius;
 mean longitude
at epoch;
 eccentricity;
 inclination to ecliptic;
 longitude of perihelion;
 longitude
of ascending node;
 orbital period;
 planetary mass / solar mass. Source: Standish and Williams (1992).









Planet 






T( yr) 










Mercury 
0.3871 
252.25 
0.20564 
7.006 
77.46 
48.34 
0.241 

Venus 
0.7233 
181.98 
0.00676 
3.398 
131.77 
76.67 
0.615 

Earth 
1.0000 
100.47 
0.01673 
0.000 
102.93 
 
1.000 

Mars 
1.5237 
355.43 
0.09337 
1.852 
336.08 
49.71 
1.881 

Jupiter 
5.2025 
34.33 
0.04854 
1.299 
14.27 
100.29 
11.87 

Saturn 
9.5415 
50.08 
0.05551 
2.494 
92.86 
113.64 
29.47 

Uranus 
19.188 
314.20 
0.04686 
0.773 
172.43 
73.96 
84.05 

Neptune 
30.070 
304.22 
0.00895 
1.770 
46.68 
131.79 
164.9 


The heliocentric (i.e., as seen from the Sun) position of a planet is most conveniently expressed in terms of
its
ecliptic longitude,
, and ecliptic latitude,
. This type of longitude and latitude is referred to the
ecliptic plane, with the Sun as the origin. Moreover, the vernal equinox is defined to be the zero of longitude. It follows that
where (
,
,
) are the heliocentric Cartesian coordinates of the planet.
Next: Planetary orbits
Up: Keplerian orbits
Previous: Elliptic orbits
Richard Fitzpatrick
20160331