Next: Lagrange Points
Up: The Three-Body Problem
Previous: Tisserand Criterion
Let us transform to a non-inertial frame of reference rotating with angular
velocity
about an axis normal
to the orbital plane of masses
and
, and passing through their center of mass.
It
follows that masses
and
appear stationary in this new reference frame.
Let us define a Cartesian coordinate system
in the rotating frame of reference which is
such that masses
and
always lie on the
-axis, and the
-axis
is parallel to the previously defined
-axis. It follows that masses
and
have the fixed position vectors
and
in our new coordinate system. Finally, let the position vector of
mass
be
--see Figure 48.
Figure 48:
The co-rotating frame.
 |
According to Chapter 7, the equation of motion of mass
in the rotating
reference frame takes the form
 |
(1050) |
where
, and
Here, the second term on the left-hand side of Equation (1050) is the Coriolis acceleration,
whereas the final term on the right-hand side is the centrifugal acceleration. The components of Equation (1050)
reduce to
which yield
where
 |
(1059) |
is the sum of the gravitational and centrifugal potentials.
Now, it follows from Equations (1056)-(1058) that
Summing the above three equations, we obtain
![\begin{displaymath}
\frac{d}{dt}\left[\frac{1}{2}\left(\dot{x}^2+\dot{y}^2+\dot{z}^2\right) + U\right] = 0.
\end{displaymath}](img2555.png) |
(1063) |
In other words,
 |
(1064) |
is a constant of the motion, where
. In fact,
is the
Jacobi integral introduced in Section 13.3 [it is easily demonstrated that Equations (1039) and
(1064) are identical].
Note, finally, that
the mass
is restricted to regions in which
 |
(1065) |
since
is a positive definite quantity.
Next: Lagrange Points
Up: The Three-Body Problem
Previous: Tisserand Criterion
Richard Fitzpatrick
2011-03-31