The 2-body Problem
The general problem of the orbits of 2 bodies with similar
masses, e.g. the Earth-Moon system or a double-star is both
of direct and indirect importance to us. Indirectly, because
it can be reduced to a 1-body problem for the relative
motion, and so, we can economically consider both at the
same time. Begin by reviewing the standard decoupling in
center-of-mass and relative motions.
Equations of motion:
m1
r r
r˙˙
r1 - r2
m1r1 = -Gm1m2
,
r3
r
r1
r r
r
r
-r
m2˙r˙2 = -Gm1m2 2 3 1 ,
r
m2
r2
origin
r
r
Add these equations: m1 ˙r˙1 + m 2 ˙r˙ = 0
r
r
r
Integrate: m1 v1 + m 2 v 2 = a = cons tan t vector
r
r
r
r
Integrate again: m1 r1 + m 2 r 2 = at + b
Let M = m1 + m 2 and
r
r m1 rr1 + m 2 rr2
r r
R=
, then M R = at + b .
M
I.e. Uniform Center-of-Mass Motion.
Relative Motion:
Consider a new (moving) coordinate system with its origin at the CM.
r
r
-m 2 r
R = 0, r 1 =
r .
m1 2
r
m1 r ˆ
r1
r˙˙ -Gm 2 Ê r
Then, r 1 =
r ˜ = -GM 3 ,
Á r1 +
m2 1 ¯
r3 Ë
r
r
r2
r˙˙
r 2 = -GM 3 .
r
r -GM r
Subtract to get - ˙r˙ =
r.
r3
If this is written in the form:
r r
r˙ = v ,
r -GM r
v˙ = 3 r ,
r
then it is seen to be a system of 6 first-order ordinary differential equations.
Thus, we expect 6 constants of integration. Three of these can be associated
with the angular momentum vector.
r r
Pr oof : r x ˙r˙ = 0
d r r˙
r r r r
but,
( r x r ) = r˙ x r˙ + r x ˙r˙ = 0
dt
r r r
\
r x r˙ = h = specific angular momentum.