Orbital paths of planets

Given any star or massive body, how would one calculate the orbital path that planets will take? Do we use the geodesic equation $$\frac{d^2x^\mu}{d \tau^2}+\Gamma^\mu_{\alpha \beta}\frac{dx^\alpha}{d \tau}\frac{dx^\beta}{d \tau}=0$$ or would we use the orbit ODE $$\frac{d^2u}{dϑ^2}+u=-\frac{m}{ℓ^2u^2}F\left(\frac{1}{2}\right)$$ $$u(0)=\frac{1}{r_{min}},\frac{1}{r_{max}}$$ $$u'(0)=0$$ or do I just use Kepler's laws?

– J.G.
Nov 19 '21 at 22:06
• It helps me in a sense yes, just to make sure I read everything right, $$F(u^{-1})=-mh^2u^2\left(\frac{d^2u}{d \theta^2}+u\right)$$ is the equation for any arbitrary orbit?
– aygx
Nov 19 '21 at 22:29
• For some $F$, yes.
– J.G.
Nov 19 '21 at 22:51
• How accurate do you want to be? Nov 19 '21 at 23:26
• @my2cts very close approximation or exact
– aygx
Nov 19 '21 at 23:32

In deciding what formalism to use, you need to balance competing considerations, such as exactness and simplicity, and choose the right one for your application.

Assuming you want to look a the orbital motion of small bodies around a star, ignoring (a) the effect of the small bodies on the star, (b) the effect of the small bodies on each other, and (c) the size and spin of the bodies, then the geodesic equation is the equation to use to get an exact solution using GR (humanity's best theory of gravity to date). If you make assumptions about the star being still, spherical, etc, you can analytically reduce the geodesic equation to a more manageable form.

However, you will get an excellent approximation to the GR solution using Newtonian gravity, provided (a) the gravitational fields are weak (in practice for orbital mechanics this means you aren't too close to a neutron star or black hole) and (b) the velocities of the objects is slow compared to the speed of light. You also can see differences between Newtonian theory and GR if you need a very high level of accuracy (this is true for GPS systems, for example, or when estimating the orbital motion of planets to state-of-the-art precision). The advantage of Newton's laws is that they are much simpler to work with than GR. I assume your "orbit ODE" was somehow derived from Newtonian mechanics; it would be fair to use if the conditions in this paragraph are satisfied.

Kepler's laws are not a complete theory, but give some relations that hold for gravitational systems with two bodies. They are derivable from Newton's laws. In principle the contain the same information Newton's laws for two bodies that are gravitationally bound (meaning neither one body will "escape" to infinity), so you can answer any question about a bound two body gravitational system with them, but they may not always be the most efficient tool to get the answer for every question. Kepler's laws are very useful if you want to know some basic information about the orbit, such as the size of the orbits given observed periods. They are more difficult to use if you want to know how things evolve in time, for example if you want to know the mean anomaly of an object on an elliptical orbit. They are very simple to state, compared to Newton's laws or the geodesic equation.

• Thanks for the clarification. If I take into account a,b,c I wouldnt use the geodesic equation instead I would use orbital mechanics or newtonian gravity?
– aygx
Nov 19 '21 at 23:24
• @aygx If you wanted to be more exact you always use GR, not Newtonian gravity (or orbital mechanics which is a special application of Newtonian gravity). You would need to use equations that govern the change of the gravitational field (Einstein's equations) and you would need to model the internal structure (the "stress-energy tensor") of the matter making up the bodies. But as you become more exact the equations become less practical. You can also come up with Newtonian gravity ways of incorporating some effects (especially using perturbation theory). Nov 20 '21 at 0:47
• It's difficult to recommend an approach without knowing what you want to do, since there are many approximations that work very well in different special situations, while the "general and exact" set of equations are usually impossible to solve exactly, and take much more expertise to work with. It takes some experience to know what methods are likely to work in a given problem. Nov 20 '21 at 0:49
• Im trying to find possible orbits of planets around Alpha Centauri, would the Newtonian gravity be the best due to the enourmous complexity of the EFEs?
– aygx
Nov 20 '21 at 1:35
• @aygx I see. Yes, use Newtonian gravity, GR is totally irrelevant for planets orbiting a star. As a first pass, it's probably fine to use Kepler's laws; I'm assuming you want to know things like "what is the size of the orbit given the observed period" and not "what is the position of each planet as a function of time." Nov 20 '21 at 1:37