Solar System barycenter - Orbit of planets

[Andrew1955]

Hi

As far as I know the Earth orbits around the Sun Earth barycenter while the Sun orbits the Solar System BaryCenter formed by the changing center of mass of the Solar system. So even while the Sun orbits the SSBC and Earth orbits the Sun-Earth BC it would not be true to say the Earth orbits the SSBC.

So for example ISS orbits Earth rather than Earth moon BC.

I have though got myself into an almighty argument about this topic and I feel I need a fresh pair of scientifically minded eyes on the subject.

All thoughts welcomed!

Andrew

 

[marcus]

I think the positions and motions of all the bodies affect the shape of the gravitational field.
they contribute to the dynamic changing geometry encompassing the solar system,
a web of distances,angles, "straight" lines in the sense that great circles on the Earth surface are the shortest distances between points, so-called geodesics.
I think that the path of a planet in orbit is a geodesic defined by whatever the geometry is, and all the bodies contribute to forming that.

 

[marcus]

Talking about the Earth and Sun each orbiting the barycenter of the Earth-Sun system is an approximation. Often just studying a two-body subsystem of the whole is an extremely useful approximation but it is probably not the best way to think of it in reality. Might be good for calculating stuff though. Let's see what other people say in response to your question. You said "All thoughts welcome" and those are my thoughts, not in any sense authoritative or conclusive.

 

[Drakkith]

As far as I know the Earth orbits around the Sun Earth barycenter while the Sun orbits the Solar System BaryCenter formed by the changing center of mass of the Solar system. So even while the Sun orbits the SSBC and Earth orbits the Sun-Earth BC it would not be true to say the Earth orbits the SSBC.

My understanding was that the Earth orbits the Earth-Moon barycenter, which itself orbits the solar system's barycenter.

 

[Andrew1955]

>>My understanding was that the Earth orbits the Earth-Moon barycenter,

Yes, I should have mentioned that.

So for example the ISS orbits Earth rather than the barycenter of the Earth and moon

>> which itself orbits the solar system's barycenter.

I don't think so. The barycenter created by a theoretical object orbiting the ISS would still orbit Earth rather than the Earth moon bc

 

[Andrew1955]

Actually it appears to be precise about the topic, we need to say that at each instant of time the Moon orbits the center of the Earth while simultaneously the Earth is accelerated towards the centre of the moon. Therefore when we look at the results of this behaviour after it has happened we find the Earth and Moon appear to orbit the center of mass of the two objects.

 

[Bandersnatch]

When asking whether A orbits B you need to be precise about what frame of reference are you using. It's also good to stick to one meaning of orbit.

ISS orbits the Earth (or rather the ISS-Earth barycentre) in the frame of reference centred on the barycentre of the Earth-ISS system. Note, that it means that it is easy in this reference frame to describe the motion of ISS using 2-body solutions (i.e. Keplerian orbits).

If you were to switch to the reference frame of the Earth-Moon (and, implicitly, ISS as well) barycentre, you'd find out that ISS follows a spiralling path around it that can no longer be described in terms of Keplerian conic sections. But it can be treated as a combination of ISS orbiting Earth-ISS barycentre, plus the Earth-ISS barycentre orbiting the Earth(with ISS)-Moon barycentre.
As long as when you say 'orbit', you're thinking of Keplerian orbits and not just of the fact that something goes around some point in whatever fashion, you can't say that in this FoR ISS orbits just the Earth, and you can't say it orbits just the E-M barycentre.

To visualise this, imagine attaching thrusters to ISS and raising its orbit. When you begin, you're likely to say it's orbiting Earth (you use the Earth-centric FoR), and if you raise it e.g. far beyond the orbit of the Moon you'd be inclined to say that it now orbits the E-M barycentre. Notice how it was a smooth process. While magnitudes of forces acting on ISS varied, there was no sudden qualitative jump. There was never a moment when the Moon 'turned on' its influence on ISS. The only thing that changed is your choice of a FoR to describe motion in a more convenient way.Keep in mind, though, even these piecemeal Keplerian orbits are going to be just approximations of the actual paths of the objects, due to perturbations from the objects you disregard at any given stage. Since there are always more than 2 objects outside idealised thought experiments, perfect Keplerian orbits don't exist.Back to the initial question. When you say that the Sun orbits the CoM of the solar system, you obviously don't mean Keplerian orbits. The path our star follows in this reference frame is an irregular, looping pattern that doesn't admit analytical solutions.
There's no difference if you were to say that Earth 'orbits' the CoM, as it also follows an irregular path around it, affected by all the bodies in the system.

When you say that Earth orbits the Earth-Sun barycentre, you choose a different reference frame and decide to treat all the other influences as perturbations in the hope of drastically simplifying the calculations (to a 2-body orbit) and allowing for at least an approximate solution without having to resort to numerical simulations.

In the same way you could say that the Sun orbits Sun-Jupiter barycentre, as this planet's gravitational pull on our star is the strongest, and treat all other planets as perturbers.

 

[rootone]

I would assume that the the solar system does not have a fixed center of mass.
The center of mass will vary depending on the positions of planets in their orbit at any given time, with the gas giants contributing mostly to shifting it.
It won't correspond with the center of the Sun's core, at least only rarely would it do so, although it is probably always at some point within the body of the Sun.

 

[Bandersnatch]

I would assume that the the solar system does not have a fixed center of mass.

When you say that you're using some unspecified reference frame not coincident with the CoM. In the reference frame of CoM it is fixed by definition.

 

[rootone]

Hmm yes I guess so.
The Solar system's center of mass clearly cannot be moving in relation to itself.
However the center of mass will always be changing it's position in relation to every other gravitationally significant solar system object, including the geometric center of the Sun core.
Gets difficult to visualise considering that center of mass at a given time is (probably) always going to be *somewhere* between the Sun's core and the surface.

 

[Bandersnatch]

Gets difficult to visualise considering that center of mass at a given time is (probably) always going to be *somewhere* between the Sun's core and the surface.

There's always this picture from the Wikipedia article on the Sun:

 

[Andrew1955]

Thanks for the answers so far.

It seems to me that the force of Jupiter's gravity at the surface of the Sun is absolutely tiny, and if we calculate where the gravitational center of the Solar system is it is very near the center of the Sun. The barycenter therefore misleads a person into thinking Jupiter has some great ability to cause changes upon the Sun, and also causes a person to think that the Earth must orbit the center of mass of the solar system.

Can somebody guide me towards working out:

1. What the pull of Jupiter's gravity is at the Suns surface and

2. How i can calculate a 'center of gravity' of the two object system.

 

[Drakkith]

It seems to me that the force of Jupiter's gravity at the surface of the Sun is absolutely tiny, and if we calculate where the gravitational center of the Solar system is it is very near the center of the Sun.

If by gravitational center you mean the center of mass, then I don't see how you're coming to that conclusion since the barycenter is the center of mass of the system.

2. How i can calculate a 'center of gravity' of the two object system.

From wiki: http://en.wikipedia.org/wiki/Barycenter

The distance from the center of a body (thought of as a point-mass) to the barycenter in a simple two-body case can be calculated as follows:

where :

r1 is the distance from body 1 to the barycenter
rtot is the distance between the two bodies
m1 and m2 are the masses of the two bodies.

 

[Bandersnatch]

1. What the pull of Jupiter's gravity is at the Suns surface and

How comfortable are you with algebra and around equations in general? Do you know how to use Newton's law of gravity? $F_g=GMm/R^2$
It's just a matter of plugging in the numbers for masses and distance you can find on the wikipedia.
Note: while you're at it, calculate the same for the Sun on Earth or the Sun on Jupiter, or Jupiter on Earth (at their closest) for comparison.
It might be more meaningful to calculate acceleration than force, though. Take $F_g=ma$ and go from there.

2. How i can calculate a 'center of gravity' of the two object system.

Notice the meaning of the equation Drakkith supplied: the heavier body is as much closer to the CoM than the lighter one as it is heavier.

edit: jesus, forgot the gravitational constant there[/size]

 

[Andrew1955]

How comfortable are you with algebra and around equations in general? Do you know how to use Newton's law of gravity? $F_g=GMm/R^2$
It's just a matter of plugging in the numbers for masses and distance you can find on the wikipedia.
Note: while you're at it, calculate the same for the Sun on Earth or the Sun on Jupiter, or Jupiter on Earth (at their closest) for comparison.
It might be more meaningful to calculate acceleration than force, though. Take $F_g=ma$ and go from there.

Notice the meaning of the equation Drakkith supplied: the heavier body is as much closer to the CoM than the lighter one as it is heavier.

edit: jesus, forgot the gravitational constant there

Thanks i have learned about that equation and could use it. But if i want to calculate the suns gravity at Jupiter it appears you are asking me to use the same equation?

 

[Bandersnatch]

Thanks i have learned about that equation and could use it. But if i want to calculate the suns gravity at Jupiter it appears you are asking me to use the same equation?

Yes, and it means that the force the Sun exerts on Jupiter is the same as Jupiter exerts on the Sun. That's why it'd be more meaningful to calculate the acceleration (just divide the force by the mass of whichever body).

 

[Andrew1955]

If by gravitational center you mean the center of mass, then I don't see how you're coming to that conclusion since the barycenter is the center of mass of the system.
From wiki: http://en.wikipedia.org/wiki/Barycenter

The distance from the center of a body (thought of as a point-mass) to the barycenter in a simple two-body case can be calculated as follows:

where :

r1 is the distance from body 1 to the barycenter
rtot is the distance between the two bodies
m1 and m2 are the masses of the two bodies.

It could be I am totally mixed up here but here is the nuts and bolts of the situation

Leif svalgaard a famous solar scientist has said the Earth orbits around the Sun and this is known to great precision using cm accurate results from the JPL lab where ephemerides are available to 7 decimal places.

Another group say he is lying! They say the Earth must be orbiting the SSBC.

If we calculate the BC for the Sun Jupiter system we get a result of 742,723km from the center of the sun as Earth's orbital center

If we now place Jupiter at Mars we find the BC is 3 times nearer the center of the Sun and yet the force upon Earth by Jupiter is much greater

Similarly if we place an Earth size orbit 4 light years from the Sun and assume it will orbit the Sun the BC is now 113,000,000km from the Sun.

So as far as i can see there is no relationship at all between the SSBC and where an object will orbit the solar system - apart from the Sun that is.

We also know that the gravitational difference across a satellite creates a torque and for larger distances the center of gravity is inside the center of mass.

As i say i could be totally muddled up......

 

[Andrew1955]

Yes, and it means that the force the Sun exerts on Jupiter is the same as Jupiter exerts on the Sun. That's why it'd be more meaningful to calculate the acceleration (just divide the force by the mass of whichever body).

Thanks i had sort of figured that out earlier but I was struggling with the idea even while seeing F was M times A

For some reason my head begins exploding when i think about anything other than arithmetic

I had a look at this calculator earlier so it should be simple to get this result

http://astro.unl.edu/classaction/animations/renaissance/gravcalc.html

 

[Bandersnatch]

You need to step back and say precisely what you mean when you make a statement such as 'A orbits B'.

 

[Andrew1955]

Thanks. I did read your earlier Frame of Reference text and it was helpful to me.

In the first instance I am considering what part of the Solar system the Earth is being accelerated towards, where it seems the center of mass is not helping me to know the answer when gravity is inversely proportional to the square of the distance. Surely the Earth orbits the Sun Earth BC with perturbations, just like the ISS orbits the ISS Earth BC with perturbations?

According to my calculations at the surface of the Sun, the suns gravity pulling downwards is 131 million times more powerful than jupiters gravity pulling upwards

 

[D H]

You need to step back and say precisely what you mean when you make a statement such as 'A orbits B'.

This is key.

In the one body problem, an object of negligible mass orbits a massive body. The central body doesn't move because there is essentially no acceleration toward the test body (the object of negligible mass). In the limit that that mass of the test body goes to zero, the central body undergoes zero acceleration toward the test body.

The standard treatment of the two body problem is to reduce the problem to the one body problem. So long as neither body has zero mass, it doesn't matter which body one picks as the central mass. In other words, there's nothing wrong with saying the Earth orbits the Sun, or that Sun orbits the Earth. With more work, one can find that both objects "orbit" the center of mass of the two objects. Which is "right"? The answer is that insisting one point of view is "right" and the others are "wrong" is wrong. All frames of reference are equally valid.

One thing that can be said in favor of a center of mass (barycentric) frame is that the equations of motion take on their simplest form in this frame. That does not mean that this is the only valid point of view. That all frames of reference are equally valid also applies in the N-body problem.

In the two body problem, the acceleration vector of each body points toward the other body, and hence toward the system barycenter. This is no longer the case in the N-body problem. What this means with regard to the term "orbit", I'll leave up to the original poster.

 

[Drakkith]

If we calculate the BC for the Sun Jupiter system we get a result of 742,723km from the center of the sun as Earth's orbital center

If we now place Jupiter at Mars we find the BC is 3 times nearer the center of the Sun and yet the force upon Earth by Jupiter is much greater

Similarly if we place an Earth size orbit 4 light years from the Sun and assume it will orbit the Sun the BC is now 113,000,000km from the Sun.

So as far as i can see there is no relationship at all between the SSBC and where an object will orbit the solar system - apart from the Sun that is.

Think about what a center of mass is. If I have a spherical ball of clay, the center of mass of the ball is directly in the center. If I then split the ball in half and move the pieces apart, the center of mass will be halfway between the two pieces. The further apart they move, the larger the distance between each piece and the center of mass becomes, even though it's always halfway between them.

Similarly, as you move Jupiter closer to the Sun, both bodies become closer to the center of mass, and the ratio of the two distances is the same as is was when Jupiter was at its original position.

In the two body problem, the acceleration vector of each body points toward the other body, and hence toward the system barycenter. This is no longer the case in the N-body problem. What this means with regard to the term "orbit", I'll leave up to the original poster.

So would three equally massive objects in a circular orbit about a common center of mass, each 120 degrees apart along the orbital path, be an good example?

 

[D H]

In the two body problem, the acceleration vector of each body points toward the other body, and hence toward the system barycenter. This is no longer the case in the N-body problem. What this means with regard to the term "orbit", I'll leave up to the original poster.

So would three equally massive objects in a circular orbit about a common center of mass, each 120 degrees apart along the orbital path, be an good example?

That's a special case. (And an unstable one, to boot.) The general case is that objects don't accelerate toward the barycenter in the N-body problem. There are special circumstances where this is the case, but they constitute a space of measure zero. (So in practice, this never happens.)

 

[Drakkith]

That's a special case. (And an unstable one, to boot.) The general case is that objects don't accelerate toward the barycenter in the N-body problem. There are special circumstances where this is the case, but they constitute a space of measure zero. (So in practice, this never happens.)

I've been playing around in Universe Sandbox (you can buy it on Steam for $9.99 and I highly recommend it) and trying to see what happens when you set up various N-body situations. As far as I can tell, the view I had in post 4, that the Earth and Moon orbit around their barycenter which itself orbits around the Sun-Earth-Moon barycenter, holds. By that I mean that the Earth and the Moon have a complicated set of acceleration vectors which causes them to orbit around a point between them, and that point itself can be thought of as orbiting the Sun, which creates another barycenter for all 3 objects.

Does that view make sense?

Thanks for your post, by the way. It took a little bit of time, but now I understand what you mean by saying that objects don't always accelerate towards the system barycenter in an N-body problem. For example, as the Moon swings around in its orbit around the Earth, it has a component of acceleration that points away from the Sun while its closer to the Sun than Earth is. This of course is counterbalanced by an additional acceleration component pointing towards the Sun during the other half of its orbit when it is further from the Sun than the Earth is.

 

[Andrew1955]

Sticking my neck out here a bit, I believe in the two body example it is also true that objects do not accelerate towards the barycenter. They are accelerating towards each other.

Likewise in an N-body case objects placed near the BC will rapidly move towards the largest mass.

In the case of the solar system the Sun orbits the SSBC while the planets orbit the Sun. Earth and the Sun experience very similar pulls from Jupiter while Earth orbits the Sun, so both are falling by similar amounts towards Jupiter. The idea the SSBC is a center of gravity seems to be a misuse of terms??

 

[D H]

As far as I can tell, the view I had in post 4, that the Earth and Moon orbit around their barycenter which itself orbits around the Sun-Earth-Moon barycenter, holds. By that I mean that the Earth and the Moon have a complicated set of acceleration vectors which causes them to orbit around a point between them, and that point itself can be thought of as orbiting the Sun, which creates another barycenter for all 3 objects.

That is approximately true, and the approximation is quite good. It's not good enough for a high-precision ephemeris, but that's not what you're doing.

Sticking my neck out here a bit, I believe in the two body example it is also true that objects do not accelerate towards the barycenter. They are accelerating towards each other.

In the two body problem, accelerating toward one another and accelerating toward the barycenter is one and the same.

------------------------------------------------------------

Below is a graph of the distance between Venus and the Sun and distance between Venus and the solar system barycenter, from January 1970 to December 2014. The graph is based on the JPL DE430, which was generated using a barycentric frame. You can make what inferences you want.

 

[Andrew1955]

Playing around with this calculator to compare true solar headings to "barycentric" headings http://astro.unl.edu/classaction/animations/renaissance/gravcalc.html

I found the Earth has a similar gravitational influence at the Sun as Saturn. Earth and Venus combined are greater than Saturn. As expected tiny near planets outweigh the "barycentric" importance of large very distant planets, where size and distance matters for greater barycenter importance but distance reduces gravitational influence very quickly

 

[tony873004]

Earth should have a similar force as Saturn. Saturn is about 100 times as massive and it is about 10 times farther. 100/10^2 = 1.
But even so, Earth doesn't do much to the solar system barycenter, while Saturn does a lot. You can try it here:

http://orbitsimulator.com/BA/ssbc.html

Delete everything but Earth, and you probably find it hard to see any Sun movement at all.
Refresh your browser to start over. If the screen gets too cluttered, tap “c” on your keyboard to clear the Sun’s trail.

 

[Andrew1955]

Saturn is pulling for say 14 years though while the Earth and venus when aligned pull for say 5 months. If the minor planets were lined up oppositely to Saturn, would they not be overwhelming Saturns pull for a few weeks? I can see i am nit picking though

 

[Bandersnatch]

The point was that the magnitude of the force a planet exerts on the sun does not translate to the position of barycentre. Another solar-mass star sufficiently far away will exert the same pull on the Sun as the Earth does, while the barycentre will be in an obviously different position than for the Earth-Sun system.

 

[Drakkith]

Well, I made an animation in Construct 2 (a video game creation program) of the Earth orbiting the Sun using my physics equations I've just learned in class along with the gravitational force equation. I've also included a barycenter. I'd provide a link, but unfortunately I need a website to publish it on, as it's not just a single image file that I can just upload somewhere.

There are some oddities with it, however. The Earth likes to fall into the Sun and then go zooming off into interstellar space after about a half dozen orbits. Probably something to do with the fact that the program only updates everything 60 times a second.

 

[Andrew1955]

Well, I made an animation in Construct 2 (a video game creation program) of the Earth orbiting the Sun using my physics equations I've just learned in class along with the gravitational force equation. I've also included a barycenter. I'd provide a link, but unfortunately I need a website to publish it on, as it's not just a single image file that I can just upload somewhere.

There are some oddities with it, however. The Earth likes to fall into the Sun and then go zooming off into interstellar space after about a half dozen orbits. Probably something to do with the fact that the program only updates everything 60 times a second.

What do you mean by "I've also included a barycenter"? The barycenter is only a mathematical point. The relevant forces upon the Earth only come from the objects in the solar system. No object is being drawn to the SSBC as if it were a real center of gravity that could influence objects orbiting around that mathematical point.

 

[Drakkith]

What do you mean by "I've also included a barycenter"?

I mean I calculated the position of the barycenter and placed a dot there.

 

[Andrew1955]

I mean I calculated the position of the barycenter and placed a dot there.

Thanks, Can you confirm you agree with my original post?

https://www.physicsforums.com/threads/solar-system-barycenter-orbit-of-planets.817673/

 

[Drakkith]

Thanks, Can you confirm you agree with my original post?

https://www.physicsforums.com/threads/solar-system-barycenter-orbit-of-planets.817673/

As others have stated in this thread, it depends on what you mean by 'orbits'. I agree that your way is certainly one way of describing it.

 

[Andrew1955]

My understanding was that the Earth orbits the Earth-Moon barycenter, which itself orbits the solar system's barycenter.

Andrew1955 said:
>>As far as I know the Earth orbits around the Sun Earth barycenter while the Sun orbits the Solar System BaryCenter formed by the changing center of mass of the Solar >>system. So even while the Sun orbits the SSBC and Earth orbits the Sun-Earth BC it would not be true to say the Earth orbits the SSBC.My understanding was that the Earth orbits the Earth-Moon barycenter, which itself orbits the solar system's barycenter.

Drakkith,

Are you still making the same distinction you are describing in that post?

Ie You stressed the Earth-Moon barycenter orbits the SSBC, compared to my view the Earth moon barycenter orbits the Sun Earth barycenter?

Can you expand a bit on what you are getting at please if so?

Directly related to that, you also said this in response to me:

>>Andrew1955 said: ↑
It seems to me that the force of Jupiter's gravity at the surface of the Sun is absolutely tiny, and if we calculate where the gravitational center of the Solar system is it is very near the center of the Sun.If by gravitational center you mean the center of mass, then I don't see how you're coming to that conclusion since the barycenter is the center of mass of the system

The point i was making was the center of mass of the solar system is not a gravitational center for the purposes of calculating the Earths orbit. The gravitational center is much closer to the center of the Sun than the SSBC is for Earths orbit.

Thanks

 

[Drakkith]

Drakkith,

Are you still making the same distinction you are describing in that post?

Ie You stressed the Earth-Moon barycenter orbits the SSBC, compared to my view the Earth moon barycenter orbits the Sun Earth barycenter?

I was mistaken in my earlier post. The Earth-Moon BC is NOT orbiting the SSBC 'directly', but the Sun-Earth-Moon BC. But we don't usually use barycentric coordinates with respect to the SSBC, so it's still correct to say that the Earth orbits the Sun.

Can you expand a bit on what you are getting at please if so?

Sure. Here's my understanding. The ISS obviously orbits the Earth. However, the Earth itself orbits the Earth-Moon barycenter, and as such the ISS will 'orbit' around both points. In other words, the orbit of the ISS, as viewed from an inertial coordinate system, will have two components: the motion around the Earth, and the motion of the Earth-ISS around the Earth-Moon barycenter. Assuming the ISS had a perfectly circular orbit around the Earth, its motion according to our coordinate system will be a circle whose center point (the center of the Earth) moves in an elliptical pattern around another point in space which is the Earth-Moon barycenter (or more accurately the Earth-ISS-Moon barycenter) This barycenter in turn moves around the Sun-Earth-Moon-ISS barycenter, and the Sun moves around the solar system barycenter.

The point i was making was the center of mass of the solar system is not a gravitational center for the purposes of calculating the Earths orbit. The gravitational center is much closer to the center of the Sun than the SSBC is for Earths orbit.

I agree for the usual definition of orbit, which is the motion around the Sun. For that we usually use a coordinate system centered on the Sun, which does not include the SSBC as far as I know. I think this is a heliocentric coordinate system instead of a barycentric coordinate system, but I'm not certain.

All the above is my current understanding of the subject anyhow. I can't say with any certainty that it is correct.

 

[phinds]

... I need a website to publish it on, as it's not just a single image file that I can just upload somewhere.

I've got plenty of space on my hosted server accounts. What kind of server facilities do you need? I think both the ones I use run Apache and have a pretty standard set of available utilities (none of which I use, other than the FTP upload capability)

 

[Drakkith]

I've got plenty of space on my hosted server accounts. What kind of server facilities do you need? I think both the ones I use run Apache and have a pretty standard set of available utilities (none of which I use, other than the FTP upload capability)

Don't worry about it. If I make something a little more presentable I might hit you up on that offer.

 

[Andrew1955]

The Earth-Moon BC is NOT orbiting the SSBC 'directly'

Thanks the words 'directly' and 'indirectly' might provide a solution for me.

For my purposes I can say;

1. The planets only indirectly orbit the SSBC.

2. The planets directly orbit the Sun with minor perturbations.

 

[ianchristie]

A big part of the confusion here is that word "orbits". Another big part is the constant chopping and changing of frames of reference.

At any moment a body (star, planet, moon ...) will move with an instantaneous velocity and be subject to an acceleration due to the local gravitational field. (This is the Newtonian description and these properties are relative to some frame of reference.) That local gravitational field will be the sum of the gravitational fields of all the other masses in the universe. Depending on the point of view we wish to discuss we ignore some or most of these contributions. For example, we usually ignore everything outside the MIlky Way Galaxy except when we are interested in the future collision with the Andromeda Galaxy.

If we ignore all contributions (big and small) except those of two bodies which we wish to discuss we get a two-body problem which is easy to describe mathematically. If one of those two bodies is much more massive than the other, say Sun & Earth, or Earth & ISS, we can simplify things even more by taking the centre of the more massive body as the origin for our frame of reference. This makes the more massive body stationary in the chosen frame of reference and the path followed by the less massive body in this frame of reference becomes a Keplerian conic section. We often use the word "orbit" to describe this sort of path.

For bodies where the difference in mass is not so great (say Alpha Centauri A and B, or Pluto and Charon), a better frame of reference is one where the origin for our frame of reference is a point between the two masses with a distance from the centre of each in inverse proportion to the relative masses of each, i.e. the barycentre, centre of mass or centre of gravity of the system. In this frame of reference the paths of both bodies are Keplerian conics. Once we chose a frame of reference which is not fixed relative to either of these frames of reference the paths of the bodies are not Keplerian conics. Note that these are simplifications, we have ignored most of the universe and General Relativity!

If you need a more accurate description of the path of a body relative to some frame of reference then the nature of your description depends on which bits of the simplification you dispense with: do you allow for Jupiter? other planets? the Moon? the Sun? the Milky Way?. Do you use General Relativity or Newton? What do you use for a frame of reference? Do you need to account for the solar wind? Light pressure?

One of the ways of dealing with the need for more accuracy than the simple two-body solution is to use the two body solution and the barycentre frame of reference and then "add in" an instantaneous adjustment to account for whatever extra bits we wish to consider. These "add ins" are often referred to as perturbations.

If you go back to just considering the intantaneous acceleration of a body at two consecutive moments and calculate the position of the intersection of the two acceleration vectors, the point of intersection can be described as the point about which the body is "orbiting" at that time.

The idea of an "orbit" is a simplified description of a complex path.

 

[Andrew1955]

Ian, Perhaps I should have made it clearer in my original post that:

1. One discussion group is saying the Earth system directly orbits the SSBC where, in this argument, the SSBC is the principal focus of the orbit, and the Sun Earth distance changes accordingly by relatively large amounts.

2. Another discussion group, including me, is saying the principal focus of the Earth system orbit is the Sun-Earth BC, where the effect of the other planets creates only very minor changes to the Sun Earth distance.

3. Neither 1 or 2 are correct but 2 is far more correct than 1, and this reality is known to the number 2 discussion group.

(4. I later learned that Newton said “The focus of the orbit of the Earth [is] in the common centre of gravity of Venus, Mercury and the Sun.” – Mathematical Principles of Natural Philosophy Vol III pp28 )

Yes, I know this is a physics forum discussion, but the actual topic I am interested in is not really very complicated at all.

 

[ianchristie]

Andrew;
Orbital mechanics is not my speciality so there is a limit to my understanding here. However, as I said, the problem is that word "orbits", whether it is qualified by the adverb "directly" or not. "Orbit" is a human description of the path of motion not a property of the body in question. The body has time, position, velocity, acceleration, and mass, nothing else. These are in all measured in some frame of reference. Which frame you choose effects the description you derive from your calculations of how these properties change with time. When you use the term "focus" to describe an "orbit" you are thinking of an idealised Keplerian elliptical orbit. You have simplified the system to a two body system and are using either the centre of mass of the more massive object as the origin of your frame of reference or you are using the centre of mass of the two body system as a whole. As soon as you start talking about a dynamic n-body system and its barycentre you need to specify the frame of reference you are using to locate that barycentre. Diagrams showing the "changing position" of the Solar system barycentre are almost always drawn to show its position in heliocentric co-ordinates. In this schema the Sun does not move and the barycentre oscillates all over the place. A better way of describing this would be to use the Solar system barycentre as the origin of the co-ordinate system and show how the Sun wobbles around it.

Consider the direction of the acceleration vector. The acceleration vector always points towards the barycentre of the system. For a two body system this lies on the line between the two centres of the bodies. For an n-body system it may be in open space somewhere. For the Solar system it is sometimes inside the Sun but sometimes above it. It is not always on the line between the centre of the Earth and the Centre of the Sun.

So taking your four points:-
1 The Earth is always accelerating towards the SSBC. If we define a frame of reference with origin at the SSBC, the Earth's path relative to that point at a given instant in time will be an ellipse with one focus at that point. At another instant in time it will also be an ellipse, but not necessarily the same ellipse. (Using Newtonian gravity.) If we define a frame of reference with origin anywhere else and moving relative to the SSBC the Earth's path will be some wobbly line. For example if we use galactic co-ordinates the Earth's path is a wiggly spiral of some sort. The Sun Earth distance changes. According to Wikipedia's sources the variation is about 5 million kilometres. The distance from the Earth to the SSBC varies. The distance from the Sun to the SSBC also varies. The net result of the two variations is that 5 million kilometres. The diameter of the Sun is about 1.4 million kilometres and the furthest possible point from the centre of the Sun to the SSBC is about 1.9 million kilometres. You can see that most of the variation in Sun Earth distance is not due to the distance of the Sun from the SSBC.
In my opinion that 5 million kilometres is not "relatively large". If you draw an accurate ellipse to the scale of the Earth's orbit it looks like a circle to the naked eye.
The expression "directly orbits" is meaningless.

2. To say that the "orbit" of the Earth has a "principal focus" is semantics. These terms have meaning only in a simplified two body model. This is not reality.

3. This statement has no real meaning. 1 and 2 are both simplifications.

4. The system described by Newton is still a simplification.

Have a browse through the Wikipedia article on "The n-body problem." this is the best description of the problem I can find. Ignore all the maths. Just skim through and note all the mentions of words like "intractable" and "approximation". The essence of the article is that all solutions are simplifications of some sort and the Keplerian/Newtonian concepts of ellipses etc are gross simplifications.

The net result of this is that both groups are wrong, you are both using simplifications, neither group is specifying frames of reference for their claims, the things you are referring to are human concepts, not properties of the bodies in question and arguing over who is less wrong is pointless.

Cheers!
Ian

 

[Bandersnatch]

Consider the direction of the acceleration vector. The acceleration vector always points towards the barycentre of the system.

This isn't true outside special cases. I believe it has been already pointed out by others in this thread.

 

[ianchristie]

What am I missing here? I can't conceive how the acceleration can point to any place other than the solar system barycentre. The acceleration vector is in the direction of the net force surely? And the net force always points to the solar system barycentre? (Simplifying to the extent that we ignore masses outside the Solar system.) And what are those "special cases"? How can the force of gravity act in a direction other than along the line towards the net position of the masses doing the attracting?

 

[Andrew1955]

What am I missing here? I can't conceive how the acceleration can point to any place other than the solar system barycentre. The acceleration vector is in the direction of the net force surely? And the net force always points to the solar system barycentre? (Simplifying to the extent that we ignore masses outside the Solar system.) And what are those "special cases"? How can the force of gravity act in a direction other than along the line towards the net position of the masses doing the attracting?

The barycenter is a mathematical point established with reference to distance and mass. A large distance from the Sun for say Neptune which is only 17 times the mass of the Earth means the mathematical point is positioned large distances from the Sun by Neptune

In reality distant objects in the solar system have almost no ability at all to create changes upon Earths orbit because gravity is inversely proportional to the square of the distance *and* both the Sun and the Earth are both almost equally influenced by the planets gravity.

Barycenter wise the planets beyond Jupiter have a similar barycentric influence as Jupiter, but in reality their combined gravitational influence is very much weaker

Relative gravitational influence upon the Sun taking Earth as unity.

m/r2
====
Hg=0.3672
V=1.559
E=1
Ma=0.04
J=11.733
S=1.0363
N=0.0189
U=0.0393

 

[Bandersnatch]

What am I missing here? I can't conceive how the acceleration can point to any place other than the solar system barycentre. The acceleration vector is in the direction of the net force surely? And the net force always points to the solar system barycentre? (Simplifying to the extent that we ignore masses outside the Solar system.)

When you drop an apple, does the acceleration vector point towards the SSBC or somewhere else? It's easy to calculate the acceleration the apple experiences from the Sun's attraction and from Earth's, and see which is greater.

And what are those "special cases"? How can the force of gravity act in a direction other than along the line towards the net position of the masses doing the attracting?

By special cases I mean those situations where it does actually act towards the barycentre. Drakkith provided one such setup in post #22. A two-body problem is another such case (but it's indistinguishable from saying that the forces act towards the two bodies), which was mentioned by DH in post #26 in discussion with Andrew.

The one thing to take away from this is that there is no single spot in space towards all forces acting on all planets point, or even net force acting one planet always points to. When you add all the force vectors acting on any particular body in the N-body system, you end up with a vector that is constantly varying in time and pointing all over the place.@Andrew1955: proposition 1 and 2 have an easily verifiable difference in predictions - the Sun-Earth distance. You can change the bodies in question to the Sun-Venus system, and use DH's graph in post #26 to resolve this.

As long as the point of contention can be rephrased to: if we were to model planetary motion using keplerian conic sections, which point at the focus results in motion that resembles actual motion more: SSBC or Sun-planet BC? Then proposition 2 is a better one.

 

[ianchristie]

Ah... yes. I see what I was missing. The barycentre position is a function of the inverse of the distance and the mass, the gravitational force is a function of the inverse of the square of the distance and the mass. So the net force and acceleration vectors point in the same direction but it is not towards the SSBC except in the two body case and only coincidentally then. I stand corrected.

So .. is there a name for the location towards which a given body accelerates under the gravitational influence of n-1 other bodies? "Centre of Gravity" sounds right but has a different meaning. Can we call it the "Momentary Centre of Revolution"? The implication here is that from moment to moment the direction of the acceleration vector changes and the position of the intersections of these vectors changes from moment to moment also. So for the Earth we have a Momentary Centre of Revolution - Earth or MCoRE for short!

I did say "Orbital mechanics is not my specialty so there is a limit to my understanding here."

Given that correction about the point towards which the Earth is accelerating most of what I have said previously about semantics and simplifications still seems to hold, with the exception of a couple of bits about the variation in the Sun Earth distance.

Lets see if I can do better this time. Going back to the four points in Andrew's post #42 of last week:-
1. This point of view is wrong. This group is probably working under the same misunderstanding as I was. The SSBC is not the point to which the acceleration and force vectors point. If you substitute MCoRE for SSBC in my statements above a lot of what I said follows with the exception of the actual distances from Sun and Earth to MCoRE and also note that the MCoR will be in a different position for every body in the Solar System. Hence that dropped apple accelerates towards the Earth not the SSBC. (I do like a simple reductio ad absurdum to show me how thick I have been!)

2. To say that the "orbit" of the Earth has a "principal focus" is semantics. These terms have meaning only in a simplified two body model. This is an approximation, not reality. Whether or not you think it is a "rough" approximation depends on your definition of "rough". See my previous comments about simplifications in post #41. Note that if we were talking about Mercury, not Earth, it would be more obviously "rough" and we would need to use General Relativity to get a "smooth" approximation.

3. Viewpoint 1 is just plain wrong, (like I was), 2 is an approximation. Does that make it "wrong" or just "less correct"? That all depends on how precise you want to be. It might be better to say viewpoint 2 is accurate within certain limits of precision. Those limits of precision must be greater than the effects of the other planets.

4. The arrangement described by Newton is still a simplification.

Bandersnatch's third last paragraph in #47 above is a fair summary. Note that it does not use the words "focus" or "orbit". (And now I know more orbital mechanics than previously.)

An aside: In my role as a science educator I often come across similar debates amongst students which ultimately derive from a combination of misconceptions, (like mine), poorly defined terms, poorly defined frames of reference and lots of semantics. The semantics are often like those here with terms being used which are human descriptions not properties of objects. On occasion I come across debates in similar places to this where the initial post stems from these confusions. One of the difficulties which responders often have is detecting those misconceptions and confusions and addressing them. Responders often go into long and convoluted explanations which add layers of complexity but which never dig to the base of the misconceptions and they often add their own semantic confusion. (And then there are the loonies who add their own peculiar personal confusions which are occasionally entertaining but often just a rabbit hole of distraction. This discussion appears to have escaped their attention so far.)

There is good research to show that unless misconceptions are addressed explicitly real understanding will not take hold if it conflicts with the misconception. Given a choice between new, correct idea and old, wrong idea, students throw away the new one. This discussion has been the first time I have been made aware of a misconception of my own which has been a block to understanding.

Have I missed anything pertinent to the original post?

 

[Andrew1955]

Ah... yes. I see what I was missing. The barycentre position is a function of the inverse of the distance and the mass, the gravitational force is a function of the inverse of the square of the distance and the mass. So the net force and acceleration vectors point in the same direction but it is not towards the SSBC except in the two body case and only coincidentally then. I stand corrected.

So .. is there a name for the location towards which a given body accelerates under the gravitational influence of n-1 other bodies? "Centre of Gravity" sounds right but has a different meaning. Can we call it the "Momentary Centre of Revolution"? The implication here is that from moment to moment the direction of the acceleration vector changes and the position of the intersections of these vectors changes from moment to moment also. So for the Earth we have a Momentary Centre of Revolution - Earth or MCoRE for short!

I did say "Orbital mechanics is not my specialty so there is a limit to my understanding here."

Given that correction about the point towards which the Earth is accelerating most of what I have said previously about semantics and simplifications still seems to hold, with the exception of a couple of bits about the variation in the Sun Earth distance.

Lets see if I can do better this time. Going back to the four points in Andrew's post #42 of last week:-
1. This point of view is wrong. This group is probably working under the same misunderstanding as I was. The SSBC is not the point to which the acceleration and force vectors point. If you substitute MCoRE for SSBC in my statements above a lot of what I said follows with the exception of the actual distances from Sun and Earth to MCoRE and also note that the MCoR will be in a different position for every body in the Solar System. Hence that dropped apple accelerates towards the Earth not the SSBC. (I do like a simple reductio ad absurdum to show me how thick I have been!)

2. To say that the "orbit" of the Earth has a "principal focus" is semantics. These terms have meaning only in a simplified two body model. This is an approximation, not reality. Whether or not you think it is a "rough" approximation depends on your definition of "rough". See my previous comments about simplifications in post #41. Note that if we were talking about Mercury, not Earth, it would be more obviously "rough" and we would need to use General Relativity to get a "smooth" approximation.

3. Viewpoint 1 is just plain wrong, (like I was), 2 is an approximation. Does that make it "wrong" or just "less correct"? That all depends on how precise you want to be. It might be better to say viewpoint 2 is accurate within certain limits of precision. Those limits of precision must be greater than the effects of the other planets.

4. The arrangement described by Newton is still a simplification.

Bandersnatch's third last paragraph in #47 above is a fair summary. Note that it does not use the words "focus" or "orbit". (And now I know more orbital mechanics than previously.)

An aside: In my role as a science educator I often come across similar debates amongst students which ultimately derive from a combination of misconceptions, (like mine), poorly defined terms, poorly defined frames of reference and lots of semantics. The semantics are often like those here with terms being used which are human descriptions not properties of objects. On occasion I come across debates in similar places to this where the initial post stems from these confusions. One of the difficulties which responders often have is detecting those misconceptions and confusions and addressing them. Responders often go into long and convoluted explanations which add layers of complexity but which never dig to the base of the misconceptions and they often add their own semantic confusion. (And then there are the loonies who add their own peculiar personal confusions which are occasionally entertaining but often just a rabbit hole of distraction. This discussion appears to have escaped their attention so far.)

There is good research to show that unless misconceptions are addressed explicitly real understanding will not take hold if it conflicts with the misconception. Given a choice between new, correct idea and old, wrong idea, students throw away the new one. This discussion has been the first time I have been made aware of a misconception of my own which has been a block to understanding.

Have I missed anything pertinent to the original post?

>>2. To say that the "orbit" of the Earth has a "principal focus" is semantics. These terms have meaning only in a simplified two body model. This is an approximation, not reality.

By principal focus I meant it is the lead focus. Eg principal Architect or Principal (teacher). In the two body model there is only one focus and the word 'principal' would be redundant. I made it perfectly clear that 2 was an approximation. I made it perfectly clear it is not reality.

1. May not be absolutely wrong if barycenter can mean center of gravity rather than CoM. In my discussion elsewhere, One subgroup were claiming CoM was a center of gravity and another subgroup were saying all objects gravitate in the direction of CoM - even though they agreed there is no center of gravity at CoM.

 

[D H]

So .. is there a name for the location towards which a given body accelerates under the gravitational influence of n-1 other bodies? "Centre of Gravity" sounds right but has a different meaning. Can we call it the "Momentary Centre of Revolution"?

No, there isn't. You can easily find the direction in which an object is accelerating, but attaching a radius to that makes no sense in the N-body problem.

An aside: In my role as a science educator I often come across similar debates amongst students which ultimately derive from a combination of misconceptions, (like mine), poorly defined terms, poorly defined frames of reference and lots of semantics.

That is exactly what is going on here.

From a 16th century perspective (and maybe even a 21st century backyard astronomy perspective), saying that planets have elliptical orbits about the Sun is perfectly fine. From a 21st century perspective where we send probes to Mercury, Venus, the Moon, Mars, Jupiter, Saturn, and Pluto (going to Uranus and Neptune is so last century), that perspective is anything but fine. One needs to drop the notion of elliptical orbits.

By principal focus I meant it is the lead focus.

What does that mean? You haven't changed a thing. By saying "focus" you are implicitly assuming elliptical orbits. That's a 16th century notion, one that is still good enough for backyard astronomy. That notion falls apart when it comes to sending spacecraft to other planets or pointing telescopes with arc second or better pointing requirements at another planet.