How to calculate the force of gravitational attraction in co-orbiting planets?

[Robittybob1]

How do we calculate the force of gravitational attraction between co-orbiting planets?

 

Take 2 planets both the same radius from the Sun and have them start off at the L3 lagrangian position with respect to each other. (That is directly opposite each other on different sides of the Sun.)

Now if for some reason one or the other is perturbed and and they get off the L3 Lagrangian Spot and hence over time they get pulled around the orbital circumference toward each other, but at the same time both are still orbiting the Sun.

 

This is definitely not the same as two bodies separated by a distance 2 times the radius, because that was where they start from and they didn't move toward each other until they were disturbed.

 

Has anyone got any idea how we could solve:

1. how long it would take to come around and

2. what is the closing speed when they finally meet up with each other?

 

Help will be most appreciated. Are there any clues please.

 

http://www.popsci.com/science/article/2011-02/two-planets-discovered-sharing-same-orbit

 

What keeps the planets from attracting to each other?

 

This is ultimately what I want to know: "Did Earth Once Share Its Orbit with a Mars-Sized Planet? New Kepler Data Suggests "Yes""

http://www.dailygalaxy.com/my_weblog/2011/02/-did-earth-once-share-its-orbit-with-another-mars-sized-planet-new-kepler-data-suggests-yes.html

 

This YT just about enabled me to understand Lagrange Points

"Lagrange Points - Sixty Symbols"

Now that was amazing for up till then I was lost in space!

Edited April 7, 2015 by Robittybob1

[Robittybob1]

Wow what a topic! I hope someone comes and discusses it with me.

It has huge ramifications. Was Theia once at the unstable L3 position? Pointless having it a a stable spot (L4 or L5) as it could still be there if that was the case.

There isn't much on this on the web so it will need original calculations to work through it, but the L3 position seems to have a simplicity that we might be able to cope with.

[pavelcherepan]

Robittybob, why are you starting the same kind of discussion as the one you already have?

 

The original question is about gravitational attraction so let's not stray away from subject, shall we? Does this look familiar?

 

[latex]F_g = G \frac {Mm}{r^2} [/latex]

 

Something's wrong with my LaTex skills. At L3 you get 13.34*10¹⁵ Newtons roughly.

Edited April 7, 2015 by pavelcherepan

[hypervalent_iodine]

!

Moderator Note

Robittybob1, you already have a thread covering everything following your first question, so I'm going to ask that this thread be confined to that question alone. If this starts to stray too much into the other stuff, this will be closed.

[Robittybob1]

....

The original question is about gravitational attraction so let's not stray away from subject, shall we? Does this look familiar?

 

[latex]F_g = G \frac {Mm}{r^2} [/latex]

 

Something's wrong with my LaTex skills. At L3 you get 13.34*10¹⁵ Newtons roughly.

That can't be the whole answer. Initially the forces are balanced, and there is no closing speed. What are the distances you were using? We will try and keep this thread just learning how to calculate the three questions in the OP only.

1. How do we calculate the force of gravitational attraction between co-orbiting planets?

"Has anyone got any idea how we could solve:

2. how long it would take to come around and

3. what is the closing speed when they finally meet up with each other?"

 

Edited April 7, 2015 by Robittybob1

[pavelcherepan]

That can't be the whole answer. We will try and keep this thread just learning how to calculate the two questions in the OP only.

 

You asked about gravitational attraction, right? So for the Earth-size body and Mars-size body at 3*10¹¹ meters away F_g will be 13.34*10¹⁵ Newtons. What other answer do you want? Also, I believe you've been explicitly told that his thread is only for the first question in the OP.

Edited April 7, 2015 by pavelcherepan

[Strange]

How do we calculate the force of gravitational attraction between co-orbiting planets?

 

Take 2 planets both the same radius from the Sun and have them start off at the L3 lagrangian position with respect to each other. (That is directly opposite each other on different sides of the Sun.)

Now if for some reason one or the other is perturbed and and they get off the L3 Lagrangian Spot and hence over time they get pulled around the orbital circumference toward each other, but at the same time both are still orbiting the Sun.

 

This is a three-body problem and so, as far as I know, the only way of answering it is via numerical methods / simulation. There you go: a nice little programming exercise for you. All you need is the equation in post #3 (and a bit of basic vector arithmetic).

[Robittybob1]

 

This is a three-body problem and so, as far as I know, the only way of answering it is via numerical methods / simulation. There you go: a nice little programming exercise for you. All you need is the equation in post #3 (and a bit of basic vector arithmetic).

I think you are on the right track. I'll see if I can get a paper discussing this to get an idea where I need to go next.

[imatfaal]

 

This is a three-body problem and so, as far as I know, the only way of answering it is via numerical methods / simulation. There you go: a nice little programming exercise for you. All you need is the equation in post #3 (and a bit of basic vector arithmetic).

 

Strange - agree completely. BTW - last time I was telling RBob this I tried looking for a list of the three body problems that were amenable to an analytic rather than numerical/simulation solution. I know that there is a decent number of situations which have been "solved" - anybody ever seen a list?

[Robittybob1]

 

You asked about gravitational attraction, right? So for the Earth-size body and Mars-size body at 3*10¹¹ meters away F_g will be 13.34*10¹⁵ Newtons. What other answer do you want? Also, I believe you've been explicitly told that his thread is only for the first question in the OP.

Assuming you dome the math right this extra force has to be balanced by a shorter orbital period and a shorten orbital radius as a consequence.

[Robittybob1]

Assuming you dome the math right this extra force has to be balanced by a shorter orbital period and a shorten orbital radius as a consequence.

If there was a planet the size of "Theia" at Earth's L3 point how much closer to the Sun would that make the Earth orbit?

[Enthalpy]

An additional planet at L3 wouldn't directly make the orbit bigger. The size of orbits (except if resonating with a heavier body ilke Jupiter) is arbitrary; it just relates with the orbital period.

 

So you could keep the same orbit radius and have a shorter year. Imagine two Earths (6e24 kg, 3ppm of Sun's 2e30 kg) exactly in L3 configuration (2*150Gm instead of 1*) : a stronger centrifugal force mRw² compensates it if the angular speed w increases by 375ppb. This means one turn difference after 2.7 million years.

 

Or you can keep the year and have a bigger orbital radius (it acts squared on the attraction but linearly on the centrifugal force). Or any case between or outside these two situations.

 

One interesting point is that planets of different masses would have their years shortened by different amounts, as the same additional force accelerates the lighter one more strongly. The shorter year value suggests a very rough approximation of the divergence time of the configuration: millions of years, not thousands nor billions.

 

Though, the model of two planets exactly in an L3 configuration is obviously wrong. The proper way is to model how perturbations evolve over time, not what the ideal equilibrium is. It can be approximated if the additional body is much lighter than the Sun and the Earth, for positions near the Lagrange ones.

 

Wiki doesn't address it

http://en.wikipedia.org/wiki/Lagrange_point#Stability

but we can say from observation that L4 and L5 points are stable (same orbit but 60° before or after) because small bodies often orbit there (at Earth, Jupiter, moons of Jupiter and Saturn...) while L1, L2, L3 are not as we see nothing there.

 

That's why I suggest that Nasa puts at our Moon's L4 or L5 the asteroid brought back, rather than on a lunar orbit. Though, tidal forces by the Sun are big at the Moon's orbit, so the comparison isn't obvious.

[swansont]

Another way to look at that is the mass of Theia is much less than that of the sun, and the sun converts an earth mass into radiation and emits it over the course of ~50 million years. And being twice as far away, it's the equivalent of only 1/4 of any mass change of the sun.

[Robittybob1]

An additional planet at L3 wouldn't directly make the orbit bigger. The size of orbits (except if resonating with a heavier body ilke Jupiter) is arbitrary; it just relates with the orbital period.

 

So you could keep the same orbit radius and have a shorter year. Imagine two Earths (6e24 kg, 3ppm of Sun's 2e30 kg) exactly in L3 configuration (2*150Gm instead of 1*) : a stronger centrifugal force mRw² compensates it if the angular speed w increases by 375ppb. This means one turn difference after 2.7 million years.

 

Or you can keep the year and have a bigger orbital radius (it acts squared on the attraction but linearly on the centrifugal force). Or any case between or outside these two situations.

 

One interesting point is that planets of different masses would have their years shortened by different amounts, as the same additional force accelerates the lighter one more strongly. The shorter year value suggests a very rough approximation of the divergence time of the configuration: millions of years, not thousands nor billions.

 

Though, the model of two planets exactly in an L3 configuration is obviously wrong. The proper way is to model how perturbations evolve over time, not what the ideal equilibrium is. It can be approximated if the additional body is much lighter than the Sun and the Earth, for positions near the Lagrange ones.

 

Wiki doesn't address it

http://en.wikipedia.org/wiki/Lagrange_point#Stability

but we can say from observation that L4 and L5 points are stable (same orbit but 60° before or after) because small bodies often orbit there (at Earth, Jupiter, moons of Jupiter and Saturn...) while L1, L2, L3 are not as we see nothing there.

 

That's why I suggest that Nasa puts at our Moon's L4 or L5 the asteroid brought back, rather than on a lunar orbit. Though, tidal forces by the Sun are big at the Moon's orbit, so the comparison isn't obvious.

There were a couple of papers that might have given me clues but drat they weren't ones free to read.

Now I did a division of the Sun's mass by the Earth's mass and it was a surprising 333,000 times more massive. So will it a piddly planet with the mass of Theia have much effect?

But instability is what we need in the end for Theia has to come around to impact the Earth. So instability is really good.

It is the time that it will take that is important, for that time still gives Theia the opportunity to pick up sufficient mass.

How can we estimate the time this would catch-up could take take?

[pavelcherepan]

How can we estimate the time this would catch-up could take take?

 

I guess it would be really hard to estimate because we don't know or can estimate the initial mass of Theia as it sets off towards Earth we don't know and can't really estimate the rate of accretion and without these it's just a guessing game.

[Janus]

There were a couple of papers that might have given me clues but drat they weren't ones free to read.

Now I did a division of the Sun's mass by the Earth's mass and it was a surprising 333,000 times more massive. So will it a piddly planet with the mass of Theia have much effect?

But instability is what we need in the end for Theia has to come around to impact the Earth. So instability is really good.

It is the time that it will take that is important, for that time still gives Theia the opportunity to pick up sufficient mass.

How can we estimate the time this would catch-up could take take?

The problem is that Theia is not going to just come around the Sun and hit the Earth. It's a lot more complicated than that.

 

Here's why. will start by assuming that Theia is coming around the Sun in such a way that it is catching up to the Earth in its orbit. (we could do it the other way around too, it doesn't matter)

 

As theai begins to catch up to Earth, Earth's gravity starts to pull it forward, Theia's gravity pulls back on the Earth. This adds some energy to Theia's orbital energy and subtracts some from Earth's.

As a result, Theia begin to drift into a higher orbit and Earth into a lower orbit. A lower orbit however is faster than a higher orbit, and the closing speed between Theia and Earth actually starts to decrease.

This continues until the difference in the orbital velocities actually has Theia orbitng the Sun slower than the Earth does and Earth will start pulling away from Theia.

Eventually, the faster orbiting Earth will start to catch up to Theia from behind. Now the gravity pull between them pull Earth in to the higher slower orbit and Theia into the lower faster one. The above process repeats.

What you end up with is a situation where the two planets take turns chasing each other around the Sun but never end up catching each other.

 

I've been running a simulation since last night of this situation, it has been running for 25,000 simulator years, repeating this pattern roughly every 550 yrs with no collision yet. How long this will continue is anyone's guess and there is no good way to estimate it.

[Robittybob1]

1. Janus - I'm so happy to hear you can run the simulation. Are they crossing their orbits in that simulation? If Theia's mass was different would the results be different?

The other thing to consider would be the drag planets would experience at times they are speed up, the protoplanetary disc would not be fully cleared at this stage. "Giant Impact" happened after about 30 million years didn't it? So can you run the simulation for at least that long please?

Edited April 8, 2015 by Robittybob1

[Janus]

• Janus - I'm so happy to hear you can run the simulation. Are they crossing their orbits in that simulation? If Theia's mass was different would the results be different?
• The other thing to consider would be the drag planets would experience at times they are speed up, the protoplanetary disc would not be fully cleared at this stage. "Giant Impact" happened after about 30 million years didn't it? So can you run the simulation for at least that long please?

Yes in a sense they cross orbits in that they trade places as to which is closer to the Sun. But they don't cross orbits in the the sense of there being a chance of collision.

 

This type of orbital interaction is sometimes called a Horseshoe orbit because when viewed from the perspective of the one of the bodies, the other's apparent motion traces out a horse shoe shape like the one below.

 

 

The thickness of the "horseshoe" is a bit exaggerated for the Theia-Earth scenario, but this is essentially what Theia's orbit would look like from the perspective of Earth's orbit. The points where Theia crosses Earth orbit are at the ends of the Horseshoe.

Differences in the masses just determines which body changes it orbital the most. If the bodies are equal in size, they do so equally. If one is more massive than the less massive body does most of the orbital distance change.

 

Sorry, but since it has taken ~18 hrs of straight running of the simulator to do 30,000 yrs worth, at this rate it would take 750 days running it 24/7 to do 30,000,000 yrs worth.

 

I can up the simulation rate, but the simulator does this by calculating in larger time steps. This has the effect of introducing errors into the simulation and it can behave very chaotically.

 

However, if you want to tie up your own computer for two years, here's the site from which you can download the simulator and run the problem yourself.

 

http://www.orbitsimulator.com/gravity/articles/what.html

[Robittybob1]

Yes in a sense they cross orbits in that they trade places as to which is closer to the Sun. But they don't cross orbits in the the sense of there being a chance of collision.

 

This type of orbital interaction is sometimes called a Horseshoe orbit because when viewed from the perspective of the one of the bodies, the other's apparent motion traces out a horse shoe shape like the one below.

 

 

The thickness of the "horseshoe" is a bit exaggerated for the Theia-Earth scenario, but this is essentially what Theia's orbit would look like from the perspective of Earth's orbit. The points where Theia crosses Earth orbit are at the ends of the Horseshoe.

Differences in the masses just determines which body changes it orbital the most. If the bodies are equal in size, they do so equally. If one is more massive than the less massive body does most of the orbital distance change.

 

Sorry, but since it has taken ~18 hrs of straight running of the simulator to do 30,000 yrs worth, at this rate it would take 750 days running it 24/7 to do 30,000,000 yrs worth.

 

I can up the simulation rate, but the simulator does this by calculating in larger time steps. This has the effect of introducing errors into the simulation and it can behave very chaotically.

 

However, if you want to tie up your own computer for two years, here's the site from which you can download the simulator and run the problem yourself.

 

http://www.orbitsimulator.com/gravity/articles/what.html

not sure if I'd be allowed to do that!

 

Where did you get that diagram from? Is that horseshoe shaped loop representing the path of the planet originally in the L3 position?

The motions look wrong to me. Are you getting something similar on the simulator?

 

Looks like someone else has done something similar judging from the headlines

http://www.nature.com/news/puzzle-of-moon-s-origin-resolved-1.17279

 

The planets would have closely resembled each other because of their similar distance from the Sun, meaning that they would have formed from the same kind of orbiting proto-planetary material. “The Earth and the Moon are not twins born from the same planet, but they are sisters in the sense that they grew up in the same environment,” says Perets.

Looking into horseshoe orbits I found this http://physics.stackexchange.com/questions/8340/how-do-horseshoe-orbits-work

 

 

Some years ago, a paper was published showing that as the Earth accumulated its mass during the formation of the Solar System, a nearby object in a horseshoe orbit relative to Earth, like this asteroid, would almost inevitably also form and grow, given all the debris floating about, but would later become unstable and crash into the Earth.

I wasn't organised enough to note the reference, and I'm not sure how well that theory squares with the Mars-sized body collision, which seems to be the accepted explanation these days; but if this horseshoe mechanism was involved, which seems quite plausible, then for Earth-like planets in other solar systems a moon, with a similar mass ratio and distance of ours, might be not a rarity but almost inevitable.

So I might have to get a computer to run that simulation.

On both sides of the orbit there is a place where the smaller planet is accelerated faster than the larger one so the smaller one would be thrown to a higher orbit and slow down and the heavier one slow down and go to the lower orbit.

 

But But But! On the side where they orbiting toward any gravitational attraction toward each other will slow the smaller (goes in and speeds up in a lower orbit) and speeds up the heavier (goes out and slows in a higher orbit), but on the other limb of the horseshoe orbit the lighter will be accelerated (goes out and slows in a higher orbit) and the heavier will be slowed (goes in and speeds up in a lower orbit).

Those situations where an acceleration will slow something always confuse me.

Edited April 8, 2015 by Robittybob1

[Robittybob1]

...

This type of orbital interaction is sometimes called a Horseshoe orbit because when viewed from the perspective of the one of the bodies, the other's apparent motion traces out a horse shoe shape like the one below.

 

 

Here is an animation of what happens in the horseshoe orbits. It is hard to comprehend what is actually happening in fact.

[Robittybob1]

Another article about the Moon formation in a similar vein to yesterday's article.

http://www.irishexaminer.com/examviral/science-world/is-this-the-final-piece-to-the-puzzle-of-how-the-moon-was-formed-323087.html

 

 

This new material contained a lot of Tungsten, but relatively little of this was of a lighter isotope known as Tungsten-182. Taking these two observations together, you would expect that Earth would have less Tungsten-182 than the moon. That’s exactly what Walker and his team found.

“The small, but significant, difference in the Tungsten isotopic composition between Earth and the moon perfectly corresponds to the different amounts of material gathered by Earth and the moon post-impact,” Walker said. “This means that, right after the moon formed, it had exactly the same isotopic composition as Earth’s mantle.”

It sounds insignificant, but that pretty much proves one of the earlier theories – that the materials mixed together. The finding supports the idea that the mass of material created by the impact, which later formed the moon, must have mixed together thoroughly before the moon coalesced and cooled. This would explain both the overall similarities in isotopic fingerprints and the slight differences in Tungsten-182.

If it was formed from the same stuff where was it positioned? Was Theia at L3 Lagrangian Point or in one of these horseshoe orbits?

Janus - in your simulation did you start Theia off at the L3 point from stationary?

Edited April 9, 2015 by Robittybob1

[Enthalpy]

Why do you bring L3 again and again? This point isn't stable.

 

The paper you link tells two planets have been oberved in an L4+L5 position, which is stable.

 

And anyway, why should colliding planets have been on the same orbit?

[Robittybob1]

Why do you bring L3 again and again? This point isn't stable.

 

The paper you link tells two planets have been oberved in an L4+L5 position, which is stable.

 

And anyway, why should colliding planets have been on the same orbit?

The point is that the whole idea of Theia involved an originally unstable planet, one that drifted close to the Earth. Looking at the horseshoe orbit an L3 planet could get into there are two points on that loop that get very close to the principle planet (the one that would be equivalent to the Earth). It wouldn't take too much more perturbation to simulate a collision.

They don't have to be on the same orbit but to be very close isotopically, if they are made out of the same annular region of the protoplanetary disc they would be mixed thoroughly before accretion IMO and supported by the two recent articles.

I had proposed this concept years ago and it is possible to alter (vary) the iron content in a situation like that as well.

Edited April 10, 2015 by Robittybob1

[Robittybob1]

......

They don't have to be on the same orbit but to be very close isotopically, if they are made out of the same annular region of the protoplanetary disc they would be mixed thoroughly before accretion IMO and supported by the two recent articles......

 

I have just re-read the post above that I wrote yesterday and I find the word "isotopically" in my post that I don't know what it means and it doesn't appear to be a spelling mistake. That seems to be a miracle if that is the right word used in the right context. Where and how did that word appear in my post?

In fact that part sentence "they don't have to be on the same orbit but to be very close isotopically," is not what I understand or intended to write, so not only a word is inserted but a whole thought! What is going on?

 

What does isotopically mean?

Edited April 10, 2015 by Robittybob1

[Janus]

I have just re-read post that I wrote yesterday and I find a word "isotopically" in my post that I don't know what it means and it doesn't appear to be a spelling mistake. That seems to be a miracle if that is the right word used in the right context. Where and how did that word appear in my post?

In fact that part sentence "they don't have to be on the same orbit but to be very close isotopically," is not what I understand, so not only a word is inserted but a whole thought! What is going on?

 

What does isotopically mean?

Isotopically refers to the make up of the bodies in terms of isotopes of elements.

 

To be isotopically similar means to be made up of the same ratios of isotopes.

 

Now, if you had actually read the article you quoted earlier, you will have noted that it said that Theia was expected to be iostopically different from the Earth. The problem was that this did not seem to jive with the similarity in isotopic make up the Earth and Moon.

 

It then goes on to say that there were two possible reasons for this that still included the Theia hypothesis. One was that coincidentally, Theia had the same make-up of the Earth. The other was that after the collision, the material from the collision mixed thoroughly together before coalescing.

 

It finally notes a small difference in the measured amounts of one type of isotope between the Moon and Earth and explains that this small difference is what you should expect if the second of the two options were correct.

 

IOW, the gist of the article is that the similarity between Moon and Earth can be explained without assuming that the make-up of Earth and Theia were similar. Theia and the Earth did not have to originally form the same distance from the Sun.

 

Additionally, in the other post you said this:

 

The point is that the whole idea of Theia involved an originally unstable planet, one that drifted close to the Earth. Looking at the horseshoe orbit an L3 planet could get into there are two points on that loop that get very close to the principle planet (the one that would be equivalent to the Earth). It wouldn't take too much more perturbation to simulate a collision.

Up until a couple of days ago, you didn't even know that such a thing as a horseshoe orbit existed, and even stated some the aspects of orbital mechanics (the behavior of objects under acceleration) confused you. Now you take it on yourself to assume you know how much of a perturbation it would take to change that orbit into an Earth collision one. You simply do not have the prerequisite knowledge or skill set to make such a statement. If you did, you wouldn't had to start this topic with the question you did because you would have been able to work it out for yourself. .