[TML] Planets in binary systems

[Jerry W Barrington]

On 3/2/08 3:53 AM, "shadow at shadowgard.com" <shadow at shadowgard.com> wrote:

> On 2 Mar 2008 at 3:06, Jerry W Barrington wrote:
> 
>> Planets can orbit far from a pair of stars, or close to either of the stars.
>> "Close" is often quoted as about 1/3 of the way to the other star.  A useful
>> rule of thumb, but it assumes the stars are equal mass.  Obviously, if one
>> is much less massive than the other, it's planets must be much closer in.
>> This is why Jupiter can't have moons 1.7 AU out!  (Even if there weren't
>> other planets affecting things)
>> 
>> I think a more accurate figure would be 1/2 of the way to the nearest
>> Lagrange point, at their closest approach.  Unfortunately, I can't seem to
>> find simple equations for the locations of of L1/2/3 for arbitrary masses.
>> :(
> 
> Nope, Lagrange points are very special cases.
> 
> "Simplest" is to merely figure out the distance where the gravity of
> the star and that of the planet balance. You won't find "real"
> satellites (and not much "captured" stuff either) outside that point.

This is either L1 (inner) or L2 (outer) of the planet relative to the star.
Useful for limiting the range of moons/satellites, but not my issue at the
moment.  See below.

> Most notable exception is the Earth/Moon system. But then again
> careful analysis shows that the moon doesn't really orbit the earth,
> it just shares an orbit with Earth.
> 
>> I tried looking at the Hill sphere too, but all the online info I can find
>> on Hill and Lagrange assume one mass much greater than the other.  Any
>> suggestions where I could find more general equations?  Or even an easy
>> derivation?  :)
> 
> There *are* no *solvable* general equations for the three body
> problem.
> 
> The Lagrange point solution is a special case and *requires* those
> substantial mass differences to work.

I think you may have read too much into my question.  I'm not looking to
place planets at the L points, nor solve the 3-body problem.  I merely want
to find L1/2/3 for any 2 *stars*.  These are just those balance points due
to the masses of the stars.  L4/5 are easy, just equilateral triangles, but
I'm not interested in those.

See, if a planet is orbiting 1 star of a pair, and it's orbit exceeds L1
between the stars, it can flip to the other star.  That's not stable.
Likewise, if it exceeds L2/3, it can flip to orbiting both, or escape.
Again, unstable.  Since even approaching those distances risks instability,
I'll cut those distances to about 1/2 or 1/3, as a safety margin.

And yes, the Lagrange points always exist, they are, as you said, where the
forces balance.  They just aren't stable orbits for many configurations.  I
just don't want them for that purpose.  I just want equations showing the
L1/2/3 points for 2 stars given there mass in Sols and distance in AUs.  Or
could be done in kg and km and convert it, whatever.