Why are there six orbital elements?

Question

I have knew for a long time that there were six orbital elements but I never knew why it was six and not any other arbitrary number? Does it have do with the fact that there is 3 components for both the position and velocity vectors?

Answer 1

$$\newcommand{\F}{\mathbf{\vec{F}}}\newcommand{\p}{\mathbf{\vec{p}}}\newcommand{\q}{\mathbf{\vec{q}}}\newcommand{\v}{\mathbf{\vec{v}}}\newcommand{\r}{\mathbf{\vec{r}}}\newcommand{\a}{\mathbf{\vec{a}}}\newcommand{\x}{\mathbf{\vec{x}}}\newcommand{\R}{\mathbf{\vec{R}}}$$Every set of orbital elements has six numbers, because six numbers is all that are necessary to categorize the complete behavior of this very simple (two body) system. The description of the system using the difference between the point masses has three numbers for position and three numbers for velocity, and those six turn out to be enough without being too many. The same thing goes for methods of initial orbit determination. Angles-only methods like Gauss's, Laplace's, and Gooding's require three measurements because each angle is two numbers, and three times two is six. If you have a radar and measure range and range rate, you also need three observations. If you can observe whole vectors -- that is, two angles and also range -- at once, you only need two observations, though Gibbs's and Herrick's methods use three for improved accuracy and reduced sensitivity to error. You could also do it by writing down as many different expressions relating orbital quantities as you can think of, and then finding the rank of that system of equations, which should tell you there are exactly six independent facts which constitute all the information contained in that set, no matter how large you tried to make it.

But, why is six the right number? Why not five, or nine, or more?

The answer boils down to "because $\F = m\a$". The interesting places I see to go from there are "why not F = something else?" and "but how does that give six?"

"What else might F equal" treads perilously close to philosophy, but there are some interesting speculations in this thread on Physics SE, along the lines of "weak field theories are one-term Taylor expansions of arbitrarily more complicated things". Physicists of varying specialties have varying ways of describing the dimensions of the problem being studied. My personal favorite is the one that says we know we live in three dimensions because the period of the upper rows of the periodic table is eight, which is two times (one plus three), where two is for spin, one is for zero angular momentum, and three is the dimension of the rotation group. The most awkward is the approach taken by string theorists, who decades ago calculated that their representation predicts wildly unphysical behavior in all but certain specific dimensionalities of spacetime. Unfortunately, all of these numbers (used to be 26, then went down to 10, then came up to 11) are much bigger than four (three position plus time), so there's been a lot of effort put into inventing ways the others might be hidden from our view (Calabi-Yau / Ricci-flat Kähler manifolds of complex dimension 3 are fascinating objects to study, but we don't yet know whether that math describes the physics of our universe).

Also note, we already know $\F$ doesn't equal $m\a$. For example, the rocket equation comes from a different form, which has $m\a$ as a special case. Writing linear momentum as $\p$, a more thorough version of F is

$$\F = \frac{d\p}{dt} = \frac{d}{dt}\left(m\v\right) = m\a + \frac{dm}{dt}\v$$

which is why the main variables in rocket performance are the rate at which mass is expended, and the speed with which the mass is ejected.

There's an aspect of orbital mechanics that closely resembles the string theoretic approach: we have six because we actually started with twelve, and then subtracted six. That is, a system of N particles ought to be described by 6N numbers, but somehow, if you count up all the variables you need to keep track of N stars and planets and moons and asteroids, etc., we somehow manage to deal with only 6(N-1) numbers. What happened to the other six? They are the position and velocity of the center of mass of the system -- the barycenter -- and can be anything at all. We don't care what they are, and we don't have to, because orbital mechanics is all about how things move with respect to the barycenter. The Sun is moving around the galaxy, and the galaxy is moving around the universe, but we can ignore those in what we do because the form of the equations we have to solve is such that we can just add or subtract those velocities without changing anything, in exactly the same way we can study the motion of Earth's Moon while considering the Sun as merely a small additive perturbation. We can study each planet's moons' motion with respect to their planetary barycenter, each of which is then hooked up to the solar system barycenter, in a hierarchical arrangement we can take as far as we like. Note that in general relativity, we can't do this, because we can't add velocities that way anymore.

This "12 - 6 = the 6 we have" business is worked out in detail here. The short version is, we start with two particles, each of which has a position, $\x_1$ and $\x_2$, and its motion solves $\ddot{\x_i}=\F_i/m_i$ in three dimensions of space, giving six degrees of freedom (DOF) for each. We then decide to change variables, defining the center of mass $$\R = \frac{m_1\x_1 + m_2\x_2}{m_1 + m_2} $$, and the distance between the two particles $\r = \x_2-\x_1$. Orbital mechanics, astrodynamics, or whatever else one calls what we do, is then the choice to specialize in what happens to $\r$ (6 DOF) while ignoring utterly what happens to $\R$: those 6 DOF are someone else's problem.

Every new particle we introduce to the system adds six more degrees of freedom, because its motion is given by a second degree differential equation in three variables, any solution to which therefore involves six constants of integration. That is, for a vector of generalized coordinates $\q$, we write $\F=m\a$ as $\ddot{\q} = f(\q, \dot{\q}, t)$, which leads into Lagrange's form of classical mechanics. Alternately, by writing velocity in terms of momentum, rather than the other way around, and also incorporating in momentum things like the magnetic vector potential (the A of which B is the curl), one arrives at Hamilton's form of classical mechanics, which can be reached from Lagrange's by Legendre transforms. These topics are not taught even to all physics majors anymore, but they are both powerful and beautiful (see, for example, Noether's theorem, Liouville's theorem, and the Hamilton-Jacobi equation), and historically orbital mechanics provided much of the inspiration to develop them. Mathematicians then went farther on their own, and recently physicists have adopted their terminology, so on Physics SE one must be prepared to encounter terms like "tangent vector bundle", "form", and "fibre", which is why I wrote this.

Finally, having introduced all of that, I can now truly answer the question, "why do TLEs use such strange coordinates?" It is because they are an attempt to solve Hamiltonian mechanics in the traditional way: by making coordinate transformations to change variables into whatever they need to be to make the Hamiltonian constant, and thus trivial to solve because all the variables are then constant in time. I kid you not. Brouwer and Kozai were both starting from Delaunay's action-angle variables, adding more perturbations to the equations, and then applying the machinery of classical Hamiltonian mechanics, trying to find a way to parameterize orbits with numbers that were nearly constant in time, to facilitate making predictions that would be good for longer times. The reason we see SGP4 as unsophisticated is that in order to make something that would run reasonably fast on computers in 1961, they had to make huge compromises, ignoring most of the theory that was developed at the time -- that's why it's called Simplified General Perturbations, version 4 -- and mostly stop paying attention to a theory that has continued developing in more complicated and more accurate ways to the present day.