Barycenter: Objects orbiting each other

[Biker]

Hello, I was reading an article about how planets and the sun orbit a point in our solar system which is really close to the sun as known as barycenter or center of mass

To just make things clearer, Why not take the example of the moon and earth...?
I have searched about it but I didn't find my answer. So I thought my favorite forum will help me.

This is how I explained it. The Earth and the moon are attracted to each other so that makes the moon orbits the Earth in elliptical shape. However the gravitational force also affects the Earth so it makes it also move in an elliptical orbit because as it moves in orbit it attracts the Earth too.

But explaining it with this way doesn't seem to clarify why the orbit exactly at the center of mass not at any random point... and that is what I need help with.. Also if the way is wrong, Please explain it precisely.

Thanks in advance.

 

[Nugatory]

But explaining it with this way doesn't seem to clarify why the orbit exactly at the center of mass not at any random point... and that is what I need help with..

Are you OK with the idea that the barycenter is a point that isn't moving? It's the center, so stationary while the Earth and the moon are orbiting around it.

Then consider that there are no external forces acting on the earth-moon system. Therefore, the center of mass of the earth-moon system is not moving.

There can only be one point in the system that is not moving, so if the barycenter and the center of mass are both not moving, they must be one and the same.

 

[Dale]

But explaining it with this way doesn't seem to clarify why the orbit exactly at the center of mass not at any random point...

If they orbited any other point then, by definition, the center of mass would be accelerating. Per Newton's laws that can only happen if there is an external force on the system, and the gravitational force between them is internal.

 

[Biker]

Are you OK with the idea that the barycenter is a point that isn't moving? It's the center, so stationary while the Earth and the moon are orbiting around it.

Then consider that there are no external forces acting on the earth-moon system. Therefore, the center of mass of the earth-moon system is not moving.

There can only be one point in the system that is not moving, so if the barycenter and the center of mass are both not moving, they must be one and the same.

If they orbited any other point then, by definition, the center of mass would be accelerating. Per Newton's laws that can only happen if there is an external force on the system, and the gravitational force between them is internal.

You both seem to talk about moving points.. relative to what? Or do you mean changing?

Okay Nugatory, I get that the center of mass wouldn't change // move because it has only internal forces and the barycenter is a point that doesn't change assuming no changes happen to the system but why does the system must have only 1 point that is not moving?

 

[A.T.]

You both seem to talk about moving points..

Dale is correctly talking about acceleration of the center of mass, not movement.

why does the system must have only 1 point that is not moving?

The 2-body system has one center of mass, which doesn't accelerate in inertial frames. In some non-inertial frame it might be some other point.

 

[Dale]

You both seem to talk about moving points.. relative to what?

Relative to any Newtonian inertial reference frame. In any inertial frame Newton's laws hold, so the center of mass cannot accelerate.

 

[anorlunda]

why does the system must have only 1 point that is not moving?

That is simple geometry. Think of a spinning disk or a spinning sphere, only one point is not moving, the center of the circle or the center of the sphere.

 

[ogg]

It's simply wrong to claim that a center of mass (CM) is not moving. At the most general perspective, motion depends on the observer's frame of reference: that is, it is NOT a property of the object, but of the description of the object relative to a particular coordinate system (and therefore a particular origin). The CM of the Earth-Moon system is certainly moving; it is orbiting around the Sun. Likewise the barycenter (CM) of the Sun-Earth-Moon system, or the entire Solar System is also orbiting around the Milky Way's CM and that is moving relative to the CMB (cosmic microwave background) as well as moving towards "the Great Attractor". Orbital dynamics are really complicated, and I do not pretend to have a great understanding of the subject - I'm sure others here have much better grasp. The 'simplest' system is a two body system (ignoring the spin of a single object about its own CM). It isn't a coincidence that the "average" path of a system of n bound particles is the path of the CM. Or, putting it another way, the path of a system of n bound particles is well described by treating all of the mass as being present at a point which is the CM. (of course, this is only true when considered from a "sufficient distance", each individual particle has an effect on near-by objects, but that isn't relevant in a simple 2-body system (where there aren't any other objects)). Anyway, it IS true that if you use as an origin, a 2-body system's CM, then that is often well described as an "inertial" frame of reference (which means it isn't accelerating, isn't experiencing EXTERNAL forces acting on it). Since the Sun isn't too large or too near, the Earth-Moon CM can be approximately described as approximately inertial. You are right to ask why the CM doesn't have any movement associated with it. I can imagine a (different) Physics where the CM orbits some other point or points. Perhaps in a periodic orbit, elliptical or circular, or even one which never repeats but is confined onto the surface of an imaginary sphere or ovoid. Or perhaps in some chaotic path, never repeating itself, perhaps unpredictable even.

 

[Dale]

It's simply wrong to claim that a center of mass (CM) is not moving. At the most general perspective, motion depends on the observer's frame of reference

And you are free to choose the reference frame where the CM is not moving. I think "simply wrong" is vastly overstating the case, at worst it is "incompletely specified".

 

[ogg]

I don't think I've answered your question adequately. The two bodies each have a gravitational field, each is attracted to the other. In order for them to NOT collide and become a single body, there has to be angular momentum so that an orbit will be established. Of course the speed of the orbits of both depend on both of their gravities and on their momenta.

Consider a very different yet relevant question. Perhaps you've seen the toy which is composed of 5 steel balls (usually) hanging from a cradle on two strings each. When you lift one ball up on one end, and let it drop, it falls and hits the second ball which hits the third which hits the fourth which hits the 5th which then rises up in an arc coming to nearly the same height as you lifted the first ball, and then drops back and hits the 4th ball and the process is repeated in reverse, back and forth.

If you've never seen it, you should google a video. The amazing thing is how/why does this toy know to give all of the energy to the 5th ball? Why not give half the energy to the 4th and half to the 5th and have two balls rising into the air at the other end? There's plenty of on-line explanations you can find for this.

My explanation is that the system has to conserve not only the energy ½mv² but also the momentum mv that the first ball had right before impact with the 2nd. For 2 balls to move on the other end, you need their initial velocity to match the final velocity of the first ball ½mV² or solving for v, ½mV²= ½(2m)v² = mv². A little algebra gives us the speed (assuming both have the same speed) of v = √(V/2). OK?

Now we also know that momentum is conserved (it is a Law of Physics!) so mV = 2mv, so v = ½V. So what value must V have in order for v to be both ½ of it and √½ of it? There's only one solution. V=0. Which isn't really a solution, since it's telling us that the 1st ball must hit the 2nd at V=0. In other words, the only feasible solution is that the mass rising up from the other end is equal to the mass delivering the first impact.

One ball comes in, one ball leaves.In the case of a 2 body orbit, they will start with some velocity along the axis between them, and also have off-axis components. If the only velocity they have is on that axis, then they will either hit (and you've now got a different problem) or they will escape each other's gravity (and no orbit will happen). Because of the conservation laws (energy, angular momentum, linear momentum) they will eventually form a system of two circular orbits around their CM. (Ultimately, such a system isn't stable, and will eventually decay, but these second order effects are ignored here).

Since the system's angular momentum is conserved, it turns out that there must be a point about which the two objects rotate (the CM). The 2-body problem can be viewed as if gravitational attraction were a rod connecting the two objects. If you can imagine this "bar bell" (or baton) to be spinning, even if the two "bells" are of different sizes, can you perhaps intuit that the point (their axis of rotation) they are spinning around lies somewhere on the rod between them? (By spinning, I mean end-over-end.). So they orbit around a point, the point is on the line connecting them, and it must be "stationary". I still don't think I've fully addressed your question. I will shut up now, and defer to others.

 

[ogg]

Dale, I'm no physicist. BUT (and you knew there would be a "but", right?) If Physics is about stuff that doesn't depend on the observer's frame of reference, then stating that the CM isn't moving violates that principle. I guess by that I mean "simply wrong" can be interpreted in several ways. Of the INFINITE number of velocities in 3 dimensions we can choose for our coordinate system, there is one in which the CM isn't moving. So I'm claiming 1 ÷ ∞ = 0. Or so close to be indistinguishable from it. OTOH, I also acknowledge that for the beginning student, we often ignore that any ambiguity exists in choice of a coordinate system, that there is a "good" coordinate system, that we should tether it to "solid ground". Only problem is, if we do that, it won't work in orbital mechanics. In a problem that doesn't have any obvious frame of reference except one which disguises the problem (because the (correct) conclusion (that using the CM as the origin for the problem works pretty well) presumes the conclusion as a premise. Or it seems so for me.) We should, I think not use that frame.

 

[Dale]

If Physics is about stuff that doesn't depend on the observer's frame of reference

I wouldn't accept that premise. Among other things, I think physics is about energy, which is frame variant.

I just think that "simply wrong" is taking a fairly minor objection and substantially overstating it.

We should, I think not use that frame.

We would be a little masochistic not to use it when it is convenient.