The validity of the Longitude of Perihelion

Question

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.

Answer 1

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 $).

Answer 2

Orbital Mechanic's formulation is correct, but the usage is not limited to textbooks. For example, the third edition of the Explanatory Supplement to the Astronomical Almanac, in its "low accuracy" algorithm for locating planets in the solar system, gives formulas for mean longitude and perihelion longitude, but not for mean anomaly and perihelion argument, which have to be derived from the longitudes. The "non-angle" longitudes are published as a matter of widely accepted convention.

Answer 3

The correct answer to the question posed appears to be that the usage is a matter of established convention. Perihelion longitude (lowercase omega with overbar) and mean longitude (L) are published in ephemeris references, but perihelion argument (lowercase omega) and mean anomaly (M) are not. However, perihelion longitude and mean longitude are both sums of angles in different planes, so neither one of them can, by itself, be the argument of a trigonometric function. Instead, the ascending node longitude (uppercase omega), an ecliptic plane angle, must be subtracted from the published perihelion longitude to yield the perihelion argument, a purely orbital plane angle, and the ascending node longitude must be subtracted from the published mean longitude to give the mean anomaly, also a purely orbital plane angle. The mean anomaly is used with the orbital eccentricity to give the eccentric anomaly, which is used with the eccentricity to give orbital plane coordinates, which are then used with the perihelion argument, the ascending node longitude, and the orbital inclination angle to determine the planet's ecliptic coordinates. The reference is Standish E, Williams J. Chapter 8 pp 338-341; Rohde J, Stollberg M. Chapter 9, pp 347-348; in Urban S, Seidelmann P (eds). Explanatory Supplement to the Astronomical Almanac, 3rd ed. 2013: Mill Valley, CA. University Science Books. The basic formulas for the relationships discussed in this answer follow.

Answer 4

Yes, it's mathematically meaningless to add angles in separate planes--after all, given one of these sums, how could we ever know how much of it came from each plane? The longitude of perifocus and the mean longitude are both examples of this meaningless sum, and perhaps their beginnings lies buried in the depths of history. It's perhaps only due to tradition (and listers of orbital elements such as NASA's JPL) that their use continues.

Suppose we call the sum of two angles in different planes a "nonsense angle" (because that's just what it is). As far as I have ever seen, one of the angles summed to get a nonsense angle is always the longitude of the ascending node, and I've only ever seen one nonsense angle subtracted from another nonsense angle, or perhaps the longitude of the ascending node subtracted from a nonsense angle. So in the subtraction "nonsense angle 2 - nonsense angle 1", the longitude of the ascending node cancels, and the right answer pops out because what is left over is all in the orbital plane, and thus well defined. It follows that the ill-defined nature of the nonsense angles doesn't lead to wrong answers, but I don't think that's a reason to keep using these angles.

I suggest that as soon as you see a nonsense angle, extract a mathematically well defined angle from it and use that. So, given the ill-defined longitude of the perifocus and the ill-defined mean longitude, convert them to meaningful angles as follows:

argument of perifocus = longitude of perifocus - longitude of ascending node ,

mean anomaly = mean longitude - longitude of perifocus .

Using the ill-defined angles benefits no one: they don't speed computations up, they don't make anything more elegant, and they don't make the subject easier to learn. It's time they were removed from the textbooks.