estimating outward orbital migration

[jedidia]

here's the problem: I have a complete stelar evolution set up, but of course it's not only the star that changes, but the planets change with it. Among other things, the planets experience significant outward migration as the mass of the central body decreases.

So I have the complete orbital elements of my planets, the mass of the star it once had and the mass of the star at its current stage. How would I best go about guessing the new orbital elements (a guess of the semimajor axis should be enough, as the eccentricity should not be modified too harshly if the star looses it's mass gradualy).

I COULD iterate through several stages, of course, but that will eat too many CPU cycles. The rather complex generation model of Orbiter Galaxy, combined with Stargen, is slow enough as it is. So all I really need is a quick and dirty estimate of the semi-major axis, it doesn't have to be too precise.

 

[chad_s]

You could assume that the orbital energy of the planets remain constant:

E = -G(M+m)/(2a)

Where M is the mass of the sun that changes, m is the mass of the planet, and a is the semimajor axis.

So:

(M_1+m)/(2a_1)=(M_2+m)/(2a_2)

 

[jedidia]

Thanks a lot. I did some more diging and arrived at the same conclusion, I just wasn't sure if specific orbital energy remains constant. It seems logical to me, but orbital mechanics work a bit counterintuitive at times. So, if specific energy stays the same, (as it seems according to your post), I have my solution.

 

[Linguofreak]

Thanks a lot. I did some more diging and arrived at the same conclusion, I just wasn't sure if specific orbital energy remains constant.

Well, unless you have some sort of source or sink, it'll be conserved like any other energy.

 

[jedidia]

Ummm... something isn't right, though. When trying to implement, I noticed that specific orbital energy is always smaller than zero for an eliptical orbit, zero for a parabolic orbit and larger than zero for a hyperbolic orbit. However, it might well be that a planet escapes when the star looses mass, so it's specific orbital energy wouldn't be the same anymore (it would be larger than zero, in any case).

It's probably the "specific" that gives me trouble here, I'll have to look at it a bit more...

edit: small correction... the orbit wouldn't get hyperbolic unless some other star interferes if the mass-loss occurs gradually... That's the reason why I'm getting planets at 4e10 AU's out. Unless I define a sensible sphere of influence, the math still sys that the planet is in Orbit. On the other hand, it seems like my AGB giants would loose pretty much ALL their planets. Is that possible?

second edit: forgett what I just wrote. converting meters back to AU before feeding it to stargen hepls a lot to get more believable results... :lol:

---------- Post added at 08:49 PM ---------- Previous post was at 04:48 PM ----------

(M_1+m)/(2a_1)=(M_2+m)/(2a_2)

After some further analysis, I have concluded that this formula can't be right (I noticed it after watching my debugger for a while, and when I took the thing apart it became obvious that it doesn't work, because in this constellation the smaller M_2 leads to a smaller a_2, while a should increase the smaller the mass is. I'll have to look a bit further into it, but it doesn't seem to be as easy as this, unfortunately.

---------- Post added 05-10-10 at 05:45 PM ---------- Previous post was 05-09-10 at 08:49 PM ----------

Hmmm... I'll need a bit of help here, or I will loose myself in assumptions that might not be correct...

Specific orbital energy is the sum of the body's kinetic energy and its potential energy, right?

Now, if the mass of the central body decreases, obviously the planet looses potential energy. The radius will have to increase to keep it constant (which is obviously exactly what I want).

I ASSUME that kinetic energy is also relative to the central body. That would stay the same, unless the body looses speed, which obviously won't happen, since there's nothing there to loose it on. So the kinetic energy would lead to a bigger radius, restoring the potential energy of the body. From this train of thought, I would expect specific orbital energy to stay the same, since kinetic energy never changes, and potential energy is restored by increasing the radius. Hence, the calculation of the new radius via specific energy SHOULD work.

which leaves me completely baffeled why the above formula doesn't work. Either my assumptions here are just wrong (verifying that is the major purpouse of this post) or the formula has been oversimplified so that that specific property got lost along the way (allthough I cannot really explain how this could happen).

However, if kinetic energy never changes, and above assumptions are correct, that would mean that I could calculate the new radius SOLELY on the basis of potential energy. Right or wrong?

 

[Sky IsNoLimit]

Hello, maybe I can help here:

1. The orbital energy is NOT constant in this case! Imagine the Sun doesn't loose mass gradually but instantaneously. The kinetic energy of the planet would not change at once but the potential energy would, because of the now lower gravitational field. As the orbital energy is the sum of kinetic and potential energy, orbital energy is not preserved here.

2. But there is another constant: angular momentum, L = r X p. L is not dependent on the mass of the central body!

We start with the general assumption that centripetal force equals gravitational force. For the simplified case of a circular orbit and that the mass of the planet is much smaller than the mass of the sun, we have:

v^2 / r = G M m / r^2

Multiplying both sides with r^2 m^2 gives:

v^2 r^2 m^2 = G M m^3 r

Because for a circular orbit L = r m v, we get:

L^2 = G M m^3 r

Because angular momentum is constant here, L1 (orbit around star with mass M1) must equal L2 (orbit around star with mass M2), and so we get:

G M1 m^3 r1 = G M2 m^3 r2

And finally:

r1/r2 = M2/M1

So at least for a circular orbit, planetary orbital radius increases linearly with decreasing mass of the star. Hope this helps!

------------------------------
P.S.

Orbital energy is E = - G M m / 2a

and not - G (M+m) / 2a

 

[jedidia]

So at least for a circular orbit, planetary orbital radius increases linearly with decreasing mass of the star. Hope this helps!

if it can be sayd about the radius in a circular orbit, shouldn't the statement also be aplicable to the semi-major axis of an eliptical orbit (at least assuming that the mass-loss happens at an apsis. Since the mass-loss will be gradual anyways, it won't make that much of a difference).

1. The orbital energy is NOT constant in this case!

Yeah, since the mass of the star is a part of the formula, it can't possibly remain constant. An interesting question would be, where the heck does that energy go? It's not really relevant to the problem at hand, but it would still be interesting to know. I mean, it has to go SOMEWHERE, doesn't it?

P.S.

Orbital energy is E = - G M m / 2a

and not - G (M+m) / 2a

Are you positively sure about this? the standard gravitational Parameter is GM + Gm, which makes G(M+m). Multiplying seems a bit strange here...

 

[Sky IsNoLimit]

Are you positively sure about this?

Just look here, wikipedia article about specific orbital energy:
http://en.wikipedia.org/wiki/Specific_orbital_energy

Since specific energy is defined as the amount of energy per unit mass, the total orbital energy of an orbiting body is its specific orbital energy multiplied by its total mass.

And so, with e being the specific orbital energy:

E = e m = - (G M / 2a ) m

if it can be sayd about the radius in a circular orbit, shouldn't the statement also be aplicable to the semi-major axis of an eliptical orbit (at least assuming that the mass-loss happens at an apsis. Since the mass-loss will be gradual anyways, it won't make that much of a difference).

For a slow and continuous mass removal I think so too. At least every orbit that has the same angular momentum in principal, should be valid. The exact shape of the orbit would be determined by the way and how fast the star looses its mass. (I'm just speculating here, maybe some kind of simulation would clarify this).

 

[jedidia]

Just look here, wikipedia article about specific orbital energy:
http://en.wikipedia.org/wiki/Specific_orbital_energy

well, that's where I looked all the time, and it says E = -GM/2a. GM is defined as the standard gravitational parameter, which, when you look it up, is GM + Gm:

http://en.wikipedia.org/wiki/Standard_gravitational_parameter

However, I had another thought: Some of the potential energy of the body "vanishes", wherever it goes (I guess if you define Gravity as negative energy, as some people do, you could say the sum is still zero... it sounds a bit too hypothetical for me, however, and I'm rather ill versed in the matter), the total orbital energy decreases by that amount (some of the kinetic energy gets converteded back to potential energy, but that exchange happens all the time in an orbit anyways). That together with the new mass should in theory give me back the new semimajor axis. Will have to try it out...

 

[Sky IsNoLimit]

Ah, ok now I understand. I thought that we use the assumption that M >> m, where the mass of the planet is neglectable compared to the mass of the sun. If you want to analyse the more general case where two bodies are orbiting each other then the standard gravitational parameter would indeed be G(M+m). So I would say finally we have two cases here. The second case is more precise but also more complicated and it's up to you if you want to follow that road:

1. M >> m (m neglectable):
e = - G M / 2a
E = - G m M / 2a

2. m not neglectable:
e = - G (M+m) / 2a
E = - G m (M+m) / 2a

with e, E the specific and total orbital energy of the orbiting body respectively.



Now to the other question: Is the law of conservation of energy still valid (where did all the potential energy go?)

Let's assume we have the first case here and that M >> m and the mass of the star at the beginning is M+dm. So the potential energy of the planet at a distance r from the center of the sun would be:

U1 = - G (M+dm) m / r

After the sun has ejected the mass dm to outer space the potential energy of the planet is:

U2 = - G M m / r

Please note that U2 > U1, so the planet is actually gaining not losing energy!
And the difference in energy is:

dE = U2 - U1 = [- (G M m) + (G (M+dm) m)] / r = (-G M m + G M m + G m dm) / r = G m dm / r

This is exactly the potential energy of dm with distance r relative to the planet. Or in other words exactly the energy that the ejected mass dm needs to overcome the gravitational field of the planet. So finally one can say that by ejecting mass to outer space, the sun transfers energy to the orbiting planet of the amount G m dm / r.

 

[jedidia]

with e, E the specific and total orbital energy of the orbiting body respectively.

errr... TOTAL orbital energy? could you explain what that is? I can't find the term anywhere.

 

[Sky IsNoLimit]

Total orbital energy is the sum of the kinetic and potential energy of an orbiting body whereas specific orbital energy is the sum of kinetic and potential energy per unit mass. Total orbital energy or just orbital energy for short is measured in J and specific orbital energy is measured in J/kg. So specific orbital energy is total orbital energy divided by the orbiting bodys mass.

Here is an example, total orbital energy of the earth:

E = - G M m / 2a (see above) = - 2.649E33 J

with
G = 6.670E-11 Nm^2/kg^-2
M = mass of the sun = 1.989E30 kg
m = mass of the earth = 5.974E24 kg
a = 1 AU = 1.496E11 m

And the specific orbital energy of the earth:

e = E/m = - G M / 2a = - 443.4 MJ/kg (quite a lot!)

 

[jedidia]

whereas specific orbital energy is the sum of kinetic and potential energy per unit mass.

ouch. Should have noticed that sooner. Thanks a lot, it's all becoming rather clear now why I was still getting weird results.