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Trajectory of a Test Mass in a Roche Potential

​
launching parameters
angular position on the donor star
0
initial velocity
1
radial position of the launching
0.2
Roche lobe shape
mass ratio q =
M
1
/
M
2
= donor/accretor
1
Roche lobe scale
mass of the accretor
M
2
1
orbital pulsation
0.1
This Demonstration describes the trajectory of a test mass in the Roche potential of a binary system with masses
M
1
(the star on the left) and
M
2
(the compact object on the right). The frame of reference, corotating with the binary system, is noninertial, and the effective potential takes into account both gravitational and centrifugal forces. One of the equipotentials is a distorted figure-eight shape crossing at the Lagrange point
L
1
. This defines Roche lobes for the two stars, with their shapes determined by the ratio of masses
q=
M
1
/
M
2
. If the star overflows its Roche lobe, mass flows from the donor star to the accreting compact object through the Lagrange point
L
1
. The test mass is launched from the green stellar surface as long as its radius is smaller than that of the Roche lobe. The entire Roche potential is mapped in the graphic, with deeper values shown in darker purple.
Since the accretor, possibly a compact object such as a white dwarf or a black hole, is very small on the scale of the diagram, its position is indicated by a black dot, with negligible radial extension.
Since the motion of the test mass is conservative (with no dissipation), its energy must remain constant. Thus, any deviation from its initial value has to be attributed to numerical errors in computation of the trajectory. The relative amplitude of those errors with time can then be monitored in the bottom plot, with
t
the time and
Δ
E
m
/
E
m
the relative deviation from the initial energy. Also, the angular momentum
L
of the test mass is plotted on the lower left.

References

[1] Wikipedia. "Roche Lobe." (Sep 16, 2015) en.wikipedia.org/wiki/Roche_lobe.

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