As I understand it from Astronomical Algorithms, by Jean Meeus, the Longitude of Perihelion is a very common numeric value associated with planets, even used as one of the planetary orbital elements. As I understand it, the Long.ofPeri is the sum of the Argument of Perihelion and the Longitude of the Ascending Node.
Here is my dilemma. The Long.ofAsc.Node is an angle measured from within the Ecliptic. The Arg.ofPeri is an angle measured in the associated planets orbital plane. These planes are separated by the Inclination between them, as they intersect only at the line of nodes.
Im a mathematician, primarily. I dont understand the basis of this measure, the Long.ofPeri. This seems invalid to me. It may be used, but perhaps this is where some inaccuracy comes from in your predictions? Why use it at all if its invalid?
From mathematics/geometry, we learn that you CANNOT add two angles in the sort of way that we are apparently doing here. These two angles are in different planes. They are not parallel, and at their point of adjacency they split off in entirely different directions.
I prefer to use the Arg.ofPeri and the Long.ofAscNode separately. Conceptually it makes sense and the mathematics Im more certain about.
Best Answer
The sum of two angles in different planes is known as a dogleg angle. A dogleg angle is not the same as an angle. The operation of adding two angles in different planes to get a dogleg angle is well-defined mathematically.
The reasons astronomers use the the longitude of the perihelion instead of the argument of the perihelion are circular orbits and equatorial orbits.
The inclination of an equatorial orbit is 0 by definition. Equatorial orbits do not have a line of nodes. Since the argument of the perihelion is the angle between the (non-existing) ascending node and the perihelion point, it is undefined.
Circular orbits do not have apses (points of perihelion and aphelion). Since the argument of perihelion is the angle between the ascending node and the (non-existing) point of perihelion, it is undefined. Without a perihelion, the longitude of perihelion, the true anomaly and the mean anomaly are all undefined.
To handle all possible orbits, astronomers use another dogleg angle, the mean longitude instead of the conceptually simpler mean anomaly. The mean longitude is the sum of the longitude of periapsis and the mean anomaly.
When applying Hamiltonian mechanics to the Kepler problem for the first time, a set of canonical angle-action variables was required. The first set, known as Delaunay elements, were defined in the 1860s. They are based on the argument of perihelion, the mean anomaly, and the longitude of the ascending node. Delaunay variables are singular for equatorial or circular orbits.
To remedy this, Poincaré elements were defined in the 1890s. They are based on the longitude of the perihelion, the mean longitude, and the longitude of the ascending node. They exist for all orbits (except when $ i\!=\!\pi $).