# How do relativity explain elliptical orbits of planets?

## Main Question or Discussion Point

How do relativity explain elliptical orbits of planets?

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D H
Staff Emeritus
You don't need to learn about relativity to understand why planets have elliptical orbits. Any central force that follows an inverse square law will result in orbits that are conic sections (circles, ellipses, parabolas, and hyperbolas). This is Newtonian mechanics, not relativistic mechanics.

It typically takes quite a bit of time before some new scientific theory that turns some branch of science upside down become accepted. Newtonian mechanics and relativity are two exceptions to this rule. Newtonian mechanics was accepted quickly in part because it explained what we already knew: planets follow elliptical orbits.

The planets do not truly follow elliptical orbits because they interact gravitationally with one another as well as with the Sun. For example, Mercury's orbit is not perfectly elliptical (it "precesses") because of the influence of Jupiter, Saturn, and all of the other planets. Newtonian mechanics could only explain a part of this precession. By the end of the 19th century, it was clear that the Newtonian explanation of the precession did not agree with the observed value. Something was missing. One reason general relativity was quickly accepted in the scientific community was that it provided that missing something.

robphy
Homework Helper
Gold Member
To add to D H's post...
in the classical limit, the Schwarzschild solution in GR leads to the Newtonian explanation of planetary motion.

So relativity can't explain the elliptical orbits of the planets?
Didn't relativity replace newtonian gravity?
How can you explain elliptical orbits without gravity?

ZapperZ
Staff Emeritus
So relativity can't explain the elliptical orbits of the planets?
Didn't relativity replace newtonian gravity?
How can you explain elliptical orbits without gravity?
Did you not read, or not understand, D H's post?

"elliptical orbit" is a solution of any inverse-square central force problem. If you can use Newtonian classical law to get such a solution, then you can also get it from general relativity, because Newtonian mechanics can be derived from relativity.

BTW, do you also have issues with circular orbits? If you don't, then I don't understand why you are asking about elliptical orbits, since circular orbits is only a special case of elliptical orbits.

Zz.

I read D H's post, but I don't think it answered my question.
In my mind I find it easy to picture a circular orbit. If i think about the classic diagram with a 2d surface pressed down by the sun and the earth following a line in this depression of spacetime, it seem to result in a circular orbit. But in reality its elliptical and i just wondered why.
I don't really care about the mathematics, but more about the physical cause, if that make any sense.
Thanks for trying to help me.

D H
Staff Emeritus
Your question is incorrect. (You don't need relativity to explain elliptical orbits.)

All that is needed to explain elliptical orbits (or hyperbolic orbits) is good old Newtonian mechanics. That inverse square central force systems result in conic sections falls right out of the math. Gravity is an inverse square central force in Newtonian mechanics.

General relativity explains why orbits are NOT elliptical. See: precession of the perihelion of Mercury.

How can my question be incorrect? It's my question.
Either relativity can explain elliptical orbits or it can't. Which is it?
I'm sure that you could explain it with newtonian physics but that was not what is asked. I don't find it relevant that we don't need relativity to explain it. I want to know if relativity can explain it or not.
I'm sorry if this is all basic stuff for you guys, but i find relativity hard to understand. I'm just trying to learn a little more about this.

I think I answered your question. But anyway, here goes: Circular orbits are solutions to the geodesic equations for objects moving in the Schwarzschild spacetime (around a star, heavy body, for example). Elliptical orbits are only approximate solutions, and the true solutions resemble precessing elliptical orbits.

robphy
Homework Helper
Gold Member
Amplifying my first response above...
Suppose that Newton had not formulated his theory of gravity...
Einstein's GR with the Schwarzschild solution would have derived (in the classical limit when its so-called "relativistic-corrections" are ignored) what would have been Newton's theory of gravity and its interpretation of elliptical orbits [to explain Kepler's observations].

Fredrik
Staff Emeritus
Gold Member

Slightly longer answer: Newton's law of gravity says that the gravitational potential around a star as a function of the distance "r" to the star has the form 1/r. A little math shows that this leads to exactly elliptical orbits. General relativity says that the potential has the form 1/r + a/r2+b/r3+..., where the a, b, etc, are pretty small numbers (which can also be calculated). So GR says that the orbits are very close to being elliptical, but they aren't exactly elliptical. This prediction of GR has been confirmed by astronomers.

Guys, thanks for trying to help me, but I think i may be misunderstood a little.
Descriptive math is fine, but that was not what i was looking for. I'm sure you would agree that planets don't move because we have a few equations that are good at describing reality. There must be a physical cause. Right?
I'm guessing that the cause is gravity and now i want to know how the relativity version of gravity explain elliptical orbits. Maybe, this is impossible for relativity or no one has though it up yet, but I would still like to know that.
Maybe if someone would explain the cause for a normal circular orbit in relativity and then we could try to work our way to elliptical orbits. I would appreciate any help. Thanks.

As mentioned above, the Schwarzschild solution for a static, spherically symmetric spacetime can be approximated in the weak-field region -- that is, far away from the gravitating mass' Schwarzschild radius.

One particularly important aspect of the Schwarzschild solution is that as one moves closer to a massive body, one experiences stronger and stronger gravitational time dilation. It is this increase (or gradient, technically) in this strength of gravitational time dilation that results in greater acceleration as one approaches closer to the massive body.

Simplifying the gravitational source as a pressureless perfect fluid at rest, we consider solely energy density (in units of Joules per metre cubed). Since we're assuming a spherically symmetric body, it may be approximated as a point object thanks to Newton's shell theorem. And so we just lump the whole thing together as:

$${\it T_{\rm 00}} = \frac{E}{1^3}{\;}{\rm J^1}{\rm m}^{-3}$$

Curvature of spacetime due to a single point source can be "fudged" as:

$${\it G_{\rm 00}} = \frac{8\pi G}{c^4} {\it T_{\rm 00}}{\;} {\rm m}^{-2}$$

The Schwarzschild radius of a spherically symmetric body of energy (again, approximated as a point source), or also in terms of mass-energy:

$${\rm R_{\rm S}} = \frac{{\it G_{\rm 00}} \cdot 1^3}{4\pi} = \frac{2GM}{c^2}{\;}{\rm m}^{1}$$

Here is the formula that describes how one's rate of time diminishes. At the Schwarzschild radius it is seen that one's rate of time is zero and so velocity is practically that of light. However, we'll assume that we're far away from the Schwarzschild radius, and that one's rate of time is practically identical to that in the absence of a gravitational source (Newton didn't know that time was variable):

$$\tau = {\rm t}{\;}\sqrt{1 - \frac{\rm R_{\rm S}}{\rm r}} \approx 1{\;}{\rm s^1}$$

The derivative of the previous gravitational time dilation formula, which describes the gradient of time dilation, or how fast the rate of time changes with a change in distance, simplifies to:

$$\frac{\partial \tau}{\partial \rm r} \approx \frac{\rm R_{\rm S}}{2 {\rm r^2}}{\;}{\rm m^{-1}}$$

Which gives acceleration based on distance (the second version is Newton's):

$${\rm a} = \frac{\partial \tau}{\partial \rm r} {c^2} = {\frac{GM}{\rm r^2}}{\;}{\rm m^1 s^{-2}}$$

Which gives orbit velocity based on distance (the second version is Newton's):

$${v} = \sqrt{{\rm a} \rm r} = \sqrt{\frac{GM}{\rm r}}{\;}{\rm m^1 s^{-1}}$$

The direct answer to your questions is then: the gradient of gravitational time dilation. For another example of how time dilation is related to velocity, see special relativity. It's not quite the same principle, but you'll get the idea that one's rate of time reduces due to velocity. In general relativity it's almost the other way around... One's velocity increases because one's rate of time reduces.

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I read D H's post, but I don't think it answered my question.
In my mind I find it easy to picture a circular orbit. If i think about the classic diagram with a 2d surface pressed down by the sun and the earth following a line in this depression of spacetime, it seem to result in a circular orbit. But in reality its elliptical and i just wondered why.
I can see why you're having problems trying to think in terms of the 'marbles on a rubber sheet' model because, despite the fact that it's used so often, it's actually a very poor model of what's going on in general relativity
I don't really care about the mathematics, but more about the physical cause, if that make any sense.
I think you're likely to be disappointed. Going back to the Newtonian case, if it were possible to say 'there is an inverse square force therefore the orbits are obviously ellipses' then why did Halley have to pester Newton for a mathematical demonstration of this fact.

Dale
Mentor
I'm guessing that the cause is gravity and now i want to know how the relativity version of gravity explain elliptical orbits. Maybe, this is impossible for relativity or no one has though it up yet, but I would still like to know that.
You are already aware of the explanation. GR explanation for gravitation is curved spacetime.

An orbiting body travels through the curved spacetime along a special kind of geometric path called a geodesic. Because spacetime is curved these geodesics are also curved. In fact, these geodesics are elipses in the classical limit.

Fredrik
Staff Emeritus
Gold Member
Descriptive math is fine, but that was not what i was looking for. I'm sure you would agree that planets don't move because we have a few equations that are good at describing reality. There must be a physical cause. Right?
I would say that planets move because reality (in particular gravity) actually behaves as described by those equations. The rest is just math. I understand that you want an easy-to-visualize geometric picture of what's going on, but I don't think that's possible. That doesn't mean that GR doesn't explain elliptical orbits. It does, in exactly the way that we described.

Edit: What DaleSpam just said is probably as close as you're ever going to get to the kind of explanation that you seem to be looking for.

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You are already aware of the explanation. GR explanation for gravitation is curved spacetime.

An orbiting body travels through the curved spacetime along a special kind of geometric path called a geodesic. Because spacetime is curved these geodesics are also curved. In fact, these geodesics are elipses in the classical limit.
Thanks. I hope you would help me clarify some points.
So, would you say that the cause for orbits is that the planets follow geodesic? which is the shortest path between points?
Why are the geodesics ellipses?
When i try to picture it in my mind circles looks like a shorter path than ellipses, but maybe I'm wrong.

Hi niin,
I think your question has been answered as well as science can answer it. Elliptical orbits are a reality and are solutions of the Euler-Lagrange equations for a body in a gravitational field. This means that elliptical orbits are just as 'short' and graceful as circular ones.

M

General relativity explains why orbits are NOT elliptical. See: precession of the perihelion of Mercury.
True, but neither are they under Newton's laws. It looks pretty elliptical if one if the masses is negligible but in the general case two masses in orbit do not follow an elliptical path under Newton's laws.

This means that elliptical orbits are just as 'short' and graceful as circular ones.
So would you say that a comets orbit which is very elongated ellipse (i think), is just at short as a circular one from...lets say a satellite?

Edit: I wonder if this is a fair way to ask this. maybe it's not, because I'm comparing two different objects...forget this question if it is.

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niin,
So would you say that a comets orbit which is very elongated ellipse (i think), is just as short as a circular one from...lets say a satellite?
It's a fair question. A comet has a great deal more energy than an artificial satellite, but it is still effectively 'falling' around the earth.

MeJennifer,
I think you're right. I once wrote a simple coarse grained simulation of Newtonian gravity and got elliptical orbits. I found I was using a retarded position in my code, so inadvertently adding the finite propagation speed of gravity. This is included in GR I expect, hence the elliptical orbits.

So it looks as if relativity does explain elliptical orbits..

... I'm sure you would agree that planets don't move because we have a few equations that are good at describing reality. There must be a physical cause. Right?
I'm guessing that the cause is gravity and now i want to know how the relativity version of gravity explain elliptical orbits. Maybe, this is impossible for relativity or no one has though it up yet, but I would still like to know that.
The reason for closed orbits must be analysed before the shape is analysed.
(I assume that it is obvious why it moves in a plane ;>} )
The reason for closed orbit is simply because the radial oscillations and angular motion have exactly the same period for a Newtonian force.
The radial and angular motion have a 1/1 resonance.

However, other forces may behave completely differently.
They may show a rational resonance n/m with n and m integer.
Or they most probably have no resonance at all.

Actually the real motion of a planet is NOT periodic.
That's -as is well known- the first main prediction of general relativity.
Therefore, it is not really elliptic either.

The Einstein field equations lead to the motion equation and fully explain the Newtonian limit and the Newtonian force of gravity. Therefore they alos explain the -nearly- closed and -nearly- elliptical orbits.

If this is not enough, they we are back to an old debate: what is meant by "an explanation?".

But read that too: http://www2.hawaii.edu/~zxu/feynman.pdf [Broken]

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Dale
Mentor
Thanks. I hope you would help me clarify some points.
So, would you say that the cause for orbits is that the planets follow geodesic?
Yes. Same with any other object in freefall under the influence of gravitation alone.

which is the shortest path between points?
Actually, in spacetime geodesics are the longest path between events. Remember, an event has four dimensions, 3 space and 1 time. So it is not just the longest distance between points A and B, but the longest distance between point A at t0 and point B at t1. This is a very important point to notice. One reason that the "rubber sheet" analogy is so poor is that it only demonstrates curvature in space. In GR it is not just space that is curved, it is spacetime that is curved. And since v<<c in typical scenarios the curvature in the time part is actually very important.

Why are the geodesics ellipses?
If you want an answer for that you need the math. That is just how the differential geometry works out.

When i try to picture it in my mind circles looks like a shorter path than ellipses, but maybe I'm wrong.
When you include the time dimension both circles (helixes) and ellipses (distorted helixes) are geodesics in the same spacetime depending on the initial or boundary conditions.

HallsofIvy