[TML] Planets in binary systems

[Jerry W Barrington]

On 3/2/08 5:35 AM, "Timothy Little" <tim at little-possums.net> wrote:

> On Sun, Mar 02, 2008 at 03:06:39AM -0500, Jerry W Barrington wrote:
>> 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.
> 
> There is an equation that can be solved for the positions, but it's a
> quintic.  They tend to be nasty to solve exactly.
> 
> However, it really doesn't matter a great deal.  The distance of the
> L1 & L2 points doesn't really change a great deal from the simpler
> approximation for the Hill radius R (M2 / (3 M1))^(1/3), where R is
> the distance between the stars.
> 
> Even at the extreme case where M1 = M2, the true L2 point is almost
> identical to that given by the formula.  L1 is a bit out, since it
> should be at the barycentre.  It is wrong by about 30%, which isn't
> too bad for applying a formula to a situation way outside its
> assumptions.

Yeah, I looked at the Hill sphere.  For equal masses, it puts L1, as figured
from either star, about 69% of the way to the other star, when it should be
exactly 50%.  That seems a pretty major error to me.  I've pretty much
concluded from my research that L1 will always be closer than L2 or L3, so
just being able to find that one would be sufficient.  Since it's to be done
by computer, I don't mind the math being a little ugly.