Undergrad Astronomy in a Simple Solar System

[PeterDonis]

Does concave just mean "no retrograde motion"?

No. It means the net acceleration is always towards the Sun, never away from the Sun.

 

[PeterDonis]

I wonder what the limiting case for this is - i.e. what's the lowest Earth orbit that's still concave wrt the Sun?

For an object directly on a line between the Earth and the Sun, the accelerations due to the Sun's and Earth's gravity are equal and opposite at about 257,000 km from the Earth (the Moon's orbit is at an average distance of about 400,000 km). So that altitude is the limiting altitude for an orbit to always be concave towards the Sun.

 

[sophiecentaur]

For an object directly on a line between the Earth and the Sun, the accelerations due to the Sun's and Earth's gravity are equal and opposite at about 257,000 km from the Earth (the Moon's orbit is at an average distance of about 400,000 km). So that altitude is the limiting altitude for an orbit to always be concave towards the Sun.

So not a lot lower than the Moon's orbit then? Below that, the orbital speed of the satellite around the Earth would be greater than its orbital speed round the Sun. (?)

 

[PeterDonis]

So not a lot lower than the Moon's orbit then?

Yes.

Below that, the orbital speed of the satellite around the Earth would be greater than its orbital speed round the Sun. (?)

No, that's not what I said. Please re-read what you quoted from my post, and note that "orbital speed" does not appear at all. I stated what the criterion was explicitly in that quote, and it has nothing to do with orbital speed.

 

[Ibix]

So not a lot lower than the Moon's orbit then? Below that, the orbital speed of the satellite around the Earth would be greater than its orbital speed round the Sun. (?)

No, that's the calculation I did. A bit of Googling turns up this paper, which models a moon moving in an Earth-centered circle superimposed on the Earth's Sun-centered circle. It shows that the instantaneous radius of curvature of the orbit can change sign for close-in orbits, but does not for the Moon's real orbit.

 

[DrStupid]

Actually, the physics says the Moon orbits the Sun, not the Earth. More precisely, the Sun's gravitational force on the Moon is always stronger than the Earth's, so the Moon's orbit is always concave towards the Sun.

The force doesn't matter. You need to look at the energy. Moon is gravitationally bound to Earth and not only to the Sun. It would still orbit Earth when the Sun would be removed.

 

[PeterDonis]

The force doesn't matter.

It does if you're looking at the shape of the orbit and whether or not it is always concave towards the Sun.

Moon is gravitationally bound to Earth and not only to the Sun.

In other words, it is bound to both. Yes, I never said it wasn't.

It would still orbit Earth when the Sun would be removed.

And it would still orbit the Sun if the Earth were removed. None of that contradicts what I said.

 

[DrStupid]

In other words, it is bound to both. Yes, I never said it wasn't.

You claimed that Moon does not orbit Earth.

 

[PeterDonis]

You claimed that Moon does not orbit Earth.

In the sense that its orbit is not always concave towards the Earth, but is always concave towards the Sun. I said nothing about the Moon not being "bound" to the Earth.

 

[DrStupid]

In the sense that its orbit is not always concave towards the Earth, but is always concave towards the Sun.

In the rest frame of Earth it is not always concave towards the Sun, but is always concave towards the Earth. And no, there is no reason to prefer the rest frame of Sun. The world lines of both, Earth and Sun are geodesics.

Long story short: You are wrong. The Moon orbits Earth - even according to your own definition.

 

[PeterDonis]

In the rest frame of Earth

Which is not inertial over the relevant time scale.

there is no reason to prefer the rest frame of Sun

Yes, there is: that it is inertial over the relevant time scale (a month), while the rest frame of the Earth is not.

More precisely, the inertial frame in which the barycenter of the solar system is at rest is the frame in which the orbit of the Moon has the property I stated, and it is inertial over the relevant time scale. Strictly speaking, the rest frame of the Sun is not, because of the Sun's movement relative to the barycenter, but it is still much closer to being inertial over a time scale of a month than the Earth's rest frame, because the Sun is so much closer to the barycenter than the Earth is (so the non-inertial terms are much, much smaller).

The Moon orbits Earth - even according to your own definition.

No, it doesn't, because my definition requires an inertial frame.

 

[PeterDonis]

The world lines of both, Earth and Sun are geodesics.

This implies a GR viewpoint, rather than the Newtonian viewpoint I have been taking up to now. From a GR viewpoint, I'm not sure what the relevant criterion would be and whether, or how, it would depend on our choice of frame. The first criterion that occurs to me is to look for (approximate) Killing vector fields in the spacetime geometry. The second is to compare the metric perturbations due to the Sun and Earth in the vicinity of the Moon.

 

[DrStupid]

Which is not inertial over the relevant time scale.

Earth is free falling around the Sun. Thus, the rest frame of Earth is a local inertial frame (you don't need GR to see that). There are tidal forces from the Sun but for the Moon they are orders of magnitude below the tidal forces from Earth and therefore negligible in good approximation.

No, it doesn't, because my definition requires an inertial frame.

As the rest frame of Earth is sufficiently inertial for the Moon you are still wrong.

 

[sophiecentaur]

In the sense that its orbit is not always concave towards the Earth

It is always that way from my viewpoint. It's not a good idea to try to describe what things 'really' are because it's all relative.

Imagine you are in a craft, orbiting the Moon. How would you describe things - assuming you knew nothing about modern astronomy? And what would the orbits of the ISS look like from there`?

 

[PeterDonis]

Earth is free falling around the Sun. Thus, the rest frame of Earth is a local inertial frame (you don't need GR to see that)

Wrong. A local inertial frame is local in time as well as in space in a curved spacetime. That is true even if the worldline of an object at rest in the frame is a geodesic. A local inertial frame centered on Earth now and a local inertial frame centered on Earth a month from now are not the same frame; they are moving in different directions. A local inertial frame centered on the barycenter of the solar system does not have that problem; the time scale for it to be "moving in a different direction" to a large enough degree to matter is millions of years.

You could set up Fermi normal coordinates centered on the Earth's worldline, and those would cover the Earth now and a month from now (or indeed as far into the future as you like); however, those would not extend far enough in space to cover the Sun, so it is impossible to describe an "orbit" for the Sun in such a frame.

 

[PeterDonis]

It is always that way from my viewpoint.

I was describing a "global" viewpoint using a frame centered on the barycenter of the solar system. In my exchanges with @DrStupid I have been giving reasons why that frame is the right one for describing orbits, at least if we want to include the Sun as well as the Earth and Moon in our description.

Of course "right" here is ultimately a matter of preference; nothing forces you to use such a frame. And if you insist on using a different frame, yes, the trajectories of objects will look different (but that assumes that the frame covers a given object's trajectory; note my remarks about the Sun not being covered by Fermi normal coordinates centered on the Earth). But the reasons I am giving for why the frame I suggest is the "right" one are not arbitrary; they are based on physical facts about the particular situation being described.

Imagine you are in a craft, orbiting the Moon. How would you describe things - assuming you knew nothing about modern astronomy?

Nothing requires you to use a frame in which you, or the particular object you are orbiting (or think you are orbiting), is at rest. So in the case you describe, you could use the frame I have been suggesting, centered on the barycenter of the solar system, just fine.

 

[DrStupid]

You could set up Fermi normal coordinates centered on the Earth's worldline, and those would cover the Earth now and a month from now (or indeed as far into the future as you like); however, those would not extend far enough in space to cover the Sun, so it is impossible to describe an "orbit" for the Sun in such a frame.

I'm not talking about the Sun but about Moon orbiting Earth. it seems I can't prevent you from claiming it doesn't, but don't wonder if you are quite alone with your opinion.

 

[PeterDonis]

I'm not talking about the Sun but about Moon orbiting Earth.

No, you're talking about the Moon not orbiting the Sun--about its path being concave towards the Earth, not the Sun. That is the claim you are making in opposition to my claim that the Moon's path is always concave towards the Sun. In order to even assess that, you must be using a frame that covers the Sun as well as the Moon and the Earth.

 

[CallMeDirac]

Summary:: Is there a simple way of proving that the Earth moves around the Sun and not vice versa?

It seems to me that Galileo and his successors benefitted from there being other bodies in the solar system other than the Earth and the Sun to prove that the Earth (and other bodies) orbited the Sun, and not the other way round.

In an imagined solar system where the Earth has no moon, and there are no other planets, asteroids, comets, etc, is there a (relatively) simple way of proving that the Earth orbits the Sun, and not the other way round?

best regards ... Stef

If you use gravity. (specifically quantum because sub-atomic gravity and gravitons are far too confusing) you can point out that either the entirety of physics is a lie or the suns gravity pulls the Earth into orbit and not vice versa.

(the sun does get moved slightly by other celestial objects in solar systems but for this it will work)

 

[sophiecentaur]

Of course "right" here is ultimately a matter of preference;

I'm not talking about the Sun but about Moon orbiting Earth.

Your are both on a hiding to nothing here. I think we all 'know what you both mean' because you are talking about different reference frames. We all assume that the Moon orbits the Earth except when the apparent paradox of the shape of the Moon's solar orbit is brought into the discussion. Is there more to it than that?
My comment about a possible lunar satellite also applies. In this case we could actually see that paradoxical behaviour in a back garden telescope. Now that would be an interesting point of discussion - when it arises.

PS Perhaps someone may point out that the situation could be unlikely or even impossible, with the two masses being so nearly equal(?).

 

[andrew s 1905]

I suspect we are conflating the technical and everyday definitions of orbit. When I observe the Galilean moons of Jupiter I am with Galileo in that they appear to orbit it.
Regards Andrew

 

[PeterDonis]

We all assume that the Moon orbits the Earth except when the apparent paradox of the shape of the Moon's solar orbit is brought into the discussion.

I would phrase it differently. I would say we all assume that the Moon orbits the Earth if the Earth and the Moon are the only bodies we are considering. We assume that because the Earth is the larger body. (Note that even here a more precise description would say the Earth and Moon both orbit the barycenter of the Earth-Moon system.)

My point is that if we include the Sun as a third body, we can no longer just assume that the Moon orbits the Earth; we have to decide what "orbits" means and what criterion to apply to decide whether the Moon is "orbiting" the Earth or the Sun. And I have been arguing for a criterion that arises from the physics of the scenario which, when applied, says that the Moon is actually orbiting the Sun (with the Earth supplying a perturbation to that orbit).

When I observe the Galilean moons of Jupiter I am with Galileo in that they appear to orbit it.

And in this particular case, the criterion I have described agrees: it says that the Galilean moons of Jupiter do indeed orbit Jupiter, even when we include the Sun in our considerations.

My point is that the same is not true for the Moon.

 

[andrew s 1905]

I am not disputing your technical definition but given the baricenter of the Earth Moon system is inside the Earth a Galileo observing from Mars might conclude the moon orbits the Earth at least informally. Regards Andrew

 

[PeterDonis]

a Galileo observing from Mars might conclude the moon orbits the earth

Sure, if he's only looking at the Earth and the Moon. Once he includes the Sun in his model he might see things differently.

 

[sophiecentaur]

I would phrase it differently. I would say we all assume that the Moon orbits the Earth if the Earth and the Moon are the only bodies we are considering.

I think that's just being controversial - you'd have to absolutely bend over backwards to hold that view genuinely.

There are some Horseshoe orbits that could possibly be interpreted that way. But have any of those actually been seen, apart from on computer sims.

 

[PeterDonis]

I think that's just being controversial

Not at all. Have you read and considered the arguments I have made in this thread? If not, please do so. If you have, do you have any counter arguments that address the substance of what I said?

 

[PeterDonis]

There are some Horseshoe orbits that could possibly be interpreted that way.

The orbit of the Moon in the frame I have been describing is not a "horseshoe orbit". It is always concave towards the Sun, as I have said.

 

[DrStupid]

No, you're talking about the Moon not orbiting the Sun

I never claimed something like that. The discussion will lead to nothing. I'm out.

 

[PeterDonis]

I never claimed something like that.

You said:

In the rest frame of Earth it is not always concave towards the Sun, but is always concave towards the Earth.

(Emphasis mine.)

This is equivalent to saying that, in the rest frame of the Earth, the Moon orbits the Earth and not the Sun, by the definition of "orbit" that I was using (and which you accepted for this discussion). And such a claim cannot even be made unless you can construct an inertial "rest frame of Earth" that covers the Sun for the required time period. Which, as I have explained, you can't.

 

[sophiecentaur]

by the definition of "orbit" that I was using

etc.

This is the point of my issue with what you have written. The flow of your logic is quite OK but your initial definition of Orbit is so far from normal experience that it's not surprising that your linear argument leads you to something that is not part of our experience. As far as Earth people are concerned, the Orbit of the Moon around the Earth would not change substantially, were the Sun to disappear. You and I know that the Moon is sometimes nearer and sometimes further away from the Sun but that it always 'goes the same way' around the Earth. The geometry of its path through space can be interpreted in many different ways - as with all astronomical objects.
You have a model that is, perhaps, self consistent but that doesn't make it anything other than an interesting ('controversial') sideline. It looks like some sort of paradoxical behaviour but that's all.
As @DrStupid says, this conversation is not going anywhere. I think I'm out too. You are neither right nor wrong.