I'm looking for a mathematical justification of the longitude of periapsis as it makes no mathematical sense to me as a notion. By definition, it is the sum of the argument of periapsis and the longitude of the ascending node, each of which makes sense on their own. I understand these latter two notions geometrically. But I don't know what the longitude of periapsis is supposed to represent. The longitude of the ascending node is an actual angle measured within the ecliptic, while the argument of periapsis is an actual angle measured within the given bodies orbital plane. These two planes are skewed with respect to one another by an inclination angle. Thus, you cannot mathematically just add the two angles together; it geometrically has no representation. At least not without making certain unmentioned assumptions which cannot possibly be true for all bodies. I cannot make sense of why it seems to be the preferred orbital parameter.

This question is not a duplicate of What is exactly the "longitude of the perigee". The two-sentence afterthought regarding this topic therein doesn't adequately respond to the spirit of my question. I'm looking for a mathematical justification for its use, including why its regarded as more accurate than splitting up the angles into their separate components.

  • $\begingroup$ Not a duplicate. Read the respective questions; dont just blindly compare titles. $\endgroup$ Nov 12, 2018 at 19:47
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    $\begingroup$ The answer to your question however is exactly the same as the answer I already wrote to that other question. Don't just read the question, read the answers. $\endgroup$ Nov 12, 2018 at 19:51
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    $\begingroup$ Did you read my answer? In particular, did you read the last two paragraphs of my answer, which address exactly the issues you have raised? $\endgroup$ Nov 12, 2018 at 20:13
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    $\begingroup$ Rather than editorialize about the baselessness of this concept in your question, why don't you ask why the concept, which you admit you do not understand, is useful? By the way, it is not baseless and it is useful. It wouldn't be used if it wasn't useful. $\endgroup$ Nov 12, 2018 at 20:48
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    $\begingroup$ Cogito, I find your aggressive argumentativeness off-putting. If you're finding that your questions aren't getting traction and SE isn't responding favourably to you, it might be worthwhile to review how you engage with us. For example, the core of your question is an interesting one, but the content that accompanies it comes across as an angry rant, which disinclines people to help. $\endgroup$ Nov 13, 2018 at 3:36

2 Answers 2


I agree that the common explanation of the longitude of periapsis, as the question gave it, appears at first sight to make no sense -- because it constructs a sum of two angles in two different planes, and that seems at first sight to lack geometrical meaning.

In fact the fault, if any, is in the explanation rather than in the geometry. It is not so difficult to show a geometrically sensible construction for the longitude of periapsis, without adding any angles in different planes. But before showing this, it may be worth answering the question 'Why do it at all, what's the point?'.

At this point the present answer links up with the answer given by James K. One of the uses of such a construction came to attention in the early days of astrodynamics. When programmed orbit calculations are to be done for artificial space objects, the orbits to be calculated can in principle have any values whatever of the inclination and eccentricity, in principle down to zero. When either or both of these orbital quantities approaches zero, then the position of the node and/or the periapsis, or else computations involving them, can become in practice numerically ill-defined, or anyhow show some undesired numerical behavior. As a result of the difficulty, searches were made a long time ago now, for alternative systems of orbital elements that do not generate undesired numerical behavior at zero or near-zero values of inclination or eccentricity.

The resulting formulae for orbital equations, usable for low-inclination, low-eccentricity orbits, typically made use of devices that included the equivalent of replacing the anomaly of an orbiting object by its longitude from a defined initial point, and replacing the argument of periapsis by a corresponding longitude from the initial point. The development of orbital equations in terms of such quantities can usefully be generated, or at least illustrated, by the following construction that also provides an in-plane account of the longitude of periapsis, the subject of the question.

Sources for the geometry and formulae connected with this kind of development include R M L Baker (1967) 'Astrodynamics', especially Appendix C, and applications are shown for example in A E Roy, 'Orbital Motion' (especially 4th ed., 2005) (or google for other sources of the same title), especially chapter 8, Special Perturbations, section 8.5, for perturbational equations and their derivation.

Baker's 'Astrodynamics' illustrates the definition for these purposes of an alternative ('F G W') coordinate system. See the figure below.

F G W coordinate diagram

One starts with ordinary rectangular coordinate axes defined by points X, Y, Z on a unit sphere with center O, where the xy plane is for example the ecliptic plane, and OX is in a chosen initial direction, e.g. towards the equinox of date. Define an orbital plane through F N G, intersecting the xy-plane at node N with inclination angle $i$ . Angle XON is then the longitude $\Omega$ of node N. Define axis OW perpendicular to orbital plane F N G. Let angle FON = XON, and let angle FOG = 90°. Then make F the initial point of coordinate system FGW. In this coordinate system, the longitude of the orbital periapsis starting from the initial point is FOP, which is the sum, as usual, of $\Omega$ the longitude of the node FON and $\omega$ the argument NOP of periapsis, but now they are measured in the same plane -- which was the thing to be illustrated.

This FGW system can be conveniently oriented by the coordinates of point W, the pole of the orbital plane, which has coordinates (in terms of the x,y,z axes and the quantities $i$ and $\Omega$) of

$ W_x = + \sin (\Omega) . \sin (i) $ ,
$ W_y = - \cos (\Omega) . \sin (i) $ ,
$ W_z = \cos (i) $ .

Then initial point F can be shown (with no difficulty but at a little length) to have coordinates relative to initial axes x, y, z, of

$ F_x = 1 - W_x^2 / (1 + W_z) $ ,
$ F_y = - W_x W_y / (1 + W_z) $ ,
$ F_z = - W_x $ .

Quantities Wx, Wy, Wz are sometimes identified as $h_x/h_m, h_y/h_m, h_z/h_m,$ and can be called direction cosines made by the orbital angular momentum vector (in direction OW) with axes x, y, z.

You might also find some of the formulae useful in Preliminary orbit-determination method having no co-planar singularity by R M L Baker and N H Jacoby, Celestial Mechanics 15 (1977) 137-160.

  • $\begingroup$ Thank you. That was somewhat enlightening. I am going to have to go over this a few times though. Not sure how adequately it answers my question but its a good start. I appreciate the effort. $\endgroup$ Nov 13, 2018 at 22:04
  • $\begingroup$ I really do appreciate it. This is probably the first time Ive been the recipient of a legitimate and intelligent answer on any SE site. I usually just get hateful comments non-conducive to learning, as evidenced in this very question, mind you. Its pretty standard. $\endgroup$ Nov 13, 2018 at 22:13

If you are dealing with planets, or bodies with a pretty low inclination, it might make sense.

If the inclination of all the planets was exactly zero the longitude of perihelion would be an actual physical angle in space. You would be able to compare the relative position of perihelion of all the planets.

Now actually the planets are slightly inclined but you can still a general idea of the relative position of perihelion from the longitude of perihelion, whereas the argument of perihelion can't be compared between planets, as it is not measured from the same direction.

There is a second slight convenience in that the longitude of perihelion can be defined for a planet with zero inclination, whereas the argument of perihelion isn't. For an orbiter with a very small inclination, it may be possible to determine the longitude of perihelion with much greater accuracy than the argument of perihelion.

Take, for example an asteroid with an inclination of 0.1 degrees +-0.05. The very small inclination means that the longitude of the ascending node is not well known. A slight error in inclination could make a big error in the ascending node, and consequently the argument of periapsis is also not known accurately. But the longitude of periapsis can be measured, and a small error in the inclination won't significantly affect it.

So when dealing with bodies that orbit close to the reference plane, it may make sense to use the longitude. But for objects that are far from the reference plane it may be better to use the argument.


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