# Crazy solar system question

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• sqljunkey
There's no clear answer to this question. It would depend on the assumptions you make about how the gravitational potential energy is distributed among the planets and the Sun. However, a rough estimate might be that the Sun would gain a negligible amount of mass.In summary, the sun's energy would not change if all the planets were removed from the solar system.

#### sqljunkey

If I take away the planets in the solar system and leave the sun alone, would the sun have less energy than before?

etotheipi
sqljunkey said:
would the sun have less energy than before?

How would you measure the sun's energy?

DaveC426913
watts/m^2 at some distance r.

sqljunkey said:
watts/m^2 at some distance r.

Then the answer to your question is no. The sun's energy by this definition would be the same.

etotheipi
sqljunkey said:
watts/m^2 at some distance r.
That isn't a measure of energy, it's a measure of flux. Depending on how you measure it removing the planets (which might temporarily eclipse the Sun if you are only measuring flux over a small area near the ecliptic plane) could change your results. I still agree with Peter's answer, though, since I wouldn't regard this as changing the Sun's energy. And you could remove any effect anyway by integrating over a sphere enclosing the Sun.

I'm not clear on what question exactly you are asking. I rather suspect that you aren't either - perhaps if you give some context for the question we might be able to comment more helpfully.

etotheipi
Well Ibix I was trying to understand the internal structure of a star or planet when you change gravity. A fun one was trying to imagine what would happen to the Earth if all gravity disappeared.

I was thinking that the sun is affected to some degree by the gravity of the other planets orbiting it, like our sea is affected by the moon.

And then when peter asked me how I would measure it, I didn't know, if I had a test particle to measure the gravity around the sun before and after I remove all the planets one can argue that the test particle's orbit was being influenced before by the orbits of the other planets.

So I came up with watts per meter squared, which would change if anything happened with the rate of fusion on the sun. :P

sqljunkey said:
I was trying to understand the internal structure of a star or planet when you change gravity.

"When you change gravity" is much too vague. You need to have some specific model in mind. But unless it's based on one of the alternative theories of gravity that have been proposed over the years, and you can find a reference for the alternative theory of gravity you want to use as a basis for discussion, it's going to run afoul of PF's rules on personal speculation.

sqljunkey said:
I was thinking that the sun is affected to some degree by the gravity of the other planets orbiting it, like our sea is affected by the moon.

In principle the planets do have tidal effects on the Sun, but (a) those effects are much, much too small to matter, and (b) those effects wouldn't be expected change the Sun's energy output anyway; they would just change the Sun's shape slightly.

Energy density in GR is well-defined, it's part of the stress-energy tensor. A meaningful notion of total energy is possible under some circumstances in GR, but is an advanced topic. People tend to think it should be as simple in GR as it is in Newtonian mechanics, but it's nowhere near so straightforwards.

The total extrasolar mass in the solar system is thought to be less than 0.2%. Any effect is going to be very small.

sqljunkey said:
Well Ibix I was trying to understand the internal structure of a star or planet when you change gravity. A fun one was trying to imagine what would happen to the Earth if all gravity disappeared
As others have noted "changing gravity" is a very broad notion. You can certainly ask about the effect of a change in gravity due to, for example, a close pass with another star. That seems to be the kind of thing you have in mind when you are talking about the effect of tides on the Sun's luminosity (too small to measure for the case of the planets, but a close encounter with another star would be another matter). The question's probably better asked in Astronomy and Astrophysics, though, because it's about solar physics not relativity.

However you also talk about "if gravity disappeared", which is a whole different question. To get an answer that isn't pure science fiction you need an actual mathematical model of gravity disappearing. That's certainly not something you can build into relativity. So I think this isn't an answerable question outside of the SF forum.

I don't know if anyone has calculated a rough estimate of the average total gravitational potential energy of the solar system (or at least only those interactions that include the sun) but if you were to throw all of the planets out to infinity then part of that should manifest itself as an increase in the mass of the sun... but surely it would be very tiny?

etotheipi said:
but surely it would be very tiny?
Suppose the ancillary 0.2% mass (see above) is at the ~at the distance of jupiter. And jupiter's orbit is ##10^3## solar radii
So the energy change would be roughly ##10^{-6}## or less

etotheipi
etotheipi said:
if you were to throw all of the planets out to infinity then part of that should manifest itself as an increase in the mass of the sun

No, it wouldn't. Such a process would involve adding energy to the planets (enough for them to escape), not the sun. It would not change the mass of the sun at all.

etotheipi
PeterDonis said:
No, it wouldn't. Such a process would involve adding energy to the planets (enough for them to escape), not the sun. It would not change the mass of the sun at all.

I will take your word for it. I just remember that in one of my Physics exams we were asked to determine the increase in mass of the Earth-ball system if we perform work on the system to increase its gravitational potential energy. Maybe that is a lie

etotheipi said:
I will take your word for it.

You shouldn't. You should try to figure out where the disconnect is.

etotheipi said:
I just remember that in one of my Physics exams we were asked to determine the increase in mass of the Earth-ball system if we perform work on the system to increase its gravitational potential energy. Maybe that is a lie

Ok, let's compare these two scenarios.

Scenario #1: We have the sun and a planet, and we have a test body orbiting the sun well inside the orbit of the planet in order to measure the mass of the sun (by tracking the orbital parameters of the test body, correcting for any perturbations due to the planet, and applying Kepler's third law). We do work on the planet to give it just enough energy to escape to infinity. The mass of the sun, as measured by the test body, is unchanged.

Scenario #2: We have the Earth and a ball resting on its surface, and we have a test body orbiting the Earth-ball system at a much higher altitude than the ball will ever reach, in order to measure the mass of the Earth-ball system (using the Kepler's third law method described above). We do work on the Earth-ball system to increase the altitude of the ball relative to the Earth. The mass of the Earth-ball system increases by exactly the amount of work that we did. We attribute this increase to an increase in the gravitational potential energy of the system.

Do you see the difference? (Note that in scenario #2, the reason the mass of the system increases is that the test object is orbiting outside the whole system both before and after the work is done; but in scenario #1, it is impossible to have a test object that orbits outside the "whole system" both before and after, since the scenario says the planets escape to infinity.)

etotheipi
Interesting, thanks for explaining! Yes I think I can see the disconnect you're getting at though it is slightly unintuitive for me.

In the scenario where the test body is well outside the system, it makes sense that doing work on the configuration and increasing its rest energy would result in a higher mass "seen" by the test body. In the other scenario where the test body is between the two bodies, it's a little bit harder for me to interpret. The rest energy of the system still seems to be greater than before, but the test body won't "see" an increase in mass.

I don't want to hijack the OPs thread, though I wondered if you have any favourite texts on this topic? I'm using Morin but he doesn't really talk about this too much.

PeterDonis said:
Scenario #2:

So how would your argument in scenario #2 play out if the "ball" were the same size as earth? I can't see it.

hutchphd said:
how would your argument in scenario #2 play out if the "ball" were the same size as earth?

You would have Earth and Twin-Earth in some bound state (for example, orbiting each other), and the test object orbiting the Earth-Twin Earth system much farther out, so it stays well outside the system throughout the scenario. Then you would do work on Earth-Twin Earth to increase the total energy of the system (by increasing the orbital energy). The test object's orbital parameters would change accordingly, indicating an increase in the total mass of the system.

hutchphd and etotheipi
etotheipi said:
The rest energy of the system

In scenario #1, you can't consider the sun and planet to be a "system", because the planet gets moved out to infinity, so there is no way to observe the "system" from outside. So "the rest energy of the system" is a meaningless concept in scenario #1. Scenarios where a "system" is assigned an energy always implicitly assume that there is something outside the "system" that can observe its overall properties.

etotheipi
etotheipi said:
I wondered if you have any favourite texts on this topic?

I don't have any good suggestions, unfortunately.

PeterDonis said:
In scenario #1, you can't consider the sun and planet to be a "system", because the planet gets moved out to infinity, so there is no way to observe the "system" from outside.

Fair enough, I see. Is it wrong to talk about the potential energy of two bodies that are separated "at infinity"? Usually we would just call this zero, or at least$$U = -\lim_{r \to \infty} \frac{Gm_1 m_2}{r}$$but it does beg the question of how we can then consider it a system if such a concept requires an external observer?

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etotheipi said:
Is it wrong to talk about the potential energy of two bodies that are separated "at infinity"?

In many circumstances it works fine. But it has limitations, which need to be recognized so you don't trip over them.

etotheipi
PeterDonis said:
In many circumstances it works fine. But it has limitations, which need to be recognized so you don't trip over them.

Awesome, thanks for taking the time to straighten this out! ☺

etotheipi said:
I don't know if anyone has calculated a rough estimate of the average total gravitational potential energy of the solar system (or at least only those interactions that include the sun) but if you were to throw all of the planets out to infinity then part of that should manifest itself as an increase in the mass of the sun... but surely it would be very tiny?

General relativity is a non-linear theory. As a result of this, most of the valid definitions of mass in GR only apply to the mass of an entire system, they do not break down the total system mass into different parts.

Note that I say "defintitions", plural. GR has more than one definition for the mass of a system, and it's usually (though not always) calculated from the metric far away from the system, at one of the various infinities. The "big three" are Komar mass, ADM mass, and Bondi mass.

With such a metric formulation, one can find the mass of the system of the sun+planets, and one can find the mass of the sun without the planets, but one can't find the mass of "just the sun" as part of the sun+planet system.

This is in contrast to Newtonian theory, which might assign a mass to the sun, a mass to the planets, and a mass to the field. However, there are known technical issues with finding the "mass of the gravitational field". The math doesn't work - it's an expression that on the surface looks like it should make sense, but the mathematics doesn't pan out.

The mathematical issues arise around the question of covariance. The notion of convariance says that a physial quantity, in order to be meaningful, should be independent of human conventions, such as the choice of a particular set of coordinates.

etotheipi
etotheipi said:
I just remember that in one of my Physics exams we were asked to determine the increase in mass of the Earth-ball system if we perform work on the system to increase its gravitational potential energy.
And if you move the planets out further away from the sun you likewise increase the mass of the solar system. But you do not increase the mass of the sun.

etotheipi
Why can't you measure mass/energy in a certain region of space in GR? I thought because the manifold is Riemann and has a metric on it, I can take measurements. I can cut the manifold in cubes and measure the cubes and these measurements would be independent of coordinate system. I can know how much curvature one of these cubes have compared to the other. Unless you are saying all of these cubes are curved the same into infinity.

sqljunkey said:
Why can't you measure mass/energy in a certain region of space in GR?

You can measure non-gravitational energy locally; just measure the stress-energy tensor. What you can't measure is "energy stored in the gravitational field". That's because there is no tensor that represents that.

When you externally measure the mass of an object like a planet or star, or a system of objects, however, you are not making a local measurement. You are essentially measuring a global property of some spacetime region. That global property includes contributions from things like "energy stored in the gravitational field" that can't be measured locally. Or, equivalently, in a curved spacetime you can't just add up all the local contributions from things that can be measured locally--the stress-energy tensor--to get the global property; the local contributions get adjusted because of the curvature of the spacetime. "Energy stored in the gravitational field" is just an intuitive way of describing the adjustments you have to make.

vanhees71
Fair enough... Well I'm going to add, even though I know it is wrong, that a region of space has to have a set energy because energy is quantized.

sqljunkey said:
I'm going to add, even though I know it is wrong, that a region of space has to have a set energy because energy is quantized.

I'm not sure why you felt you had to post this, but it's a good way to get your thread closed. Which I have just done.

vanhees71