Transcript Advanced Celestial Mechanics

Advanced Celestial
Mechanics
Questions III
Question 21
Starting from the expression for binary
energy change in three-body scattering,
derive the energy gain expression
for a binary in a stream of small bodies
Question 22
Starting from expressions for energy gain and energy loss in the threebody scattering,
derive
and state Heggie’s law.
Question 23
Starting from the expression for energy change in
three-body scattering for hard binaries
show that the evolution of the semi-major axis of
the binary is given by
Describe the significance of the different factors.
Question 24
Show that the change of inverse semi-major
axis of a comet, after passing Jupiter, is
where e-vector points to the pericentre of
the orbit relative to Jupiter, and v0-vector
is the velocity of Jupiter. Speed of comet
at large distance from Jupiter is u and
impact parameter b.
Question 25
Starting from the dimensionless energy
change of a comet passing by Jupiter
derive the crossection for energy change U
Question 26
Show that the cross-section for binary
energy to change by
is
if the geometrical cross-section is
and
Question 27
Show that the rate of three-body scatterings in a
star cluster is
and the rate of energy transfer in those scatterings
Calculate the average energy change
Use
and
.
Question 28
Show that the three-body perturbing function
is
using
in
Question 29
The twice averaged perturbing function is
Using
Derive,
where
Question 30
Lagrangian equations of motion in the three-body problem
are
Together with
they describe long-term orbital evolution. Outline this
evolution.
Question 31
Introduce a perturbing acceleration f:
Derive
making use of
Question 32
Show that the semi-major axis changes
under perturbing force f as
make use of
Question 33
The three-body perturbing force is
Make use of
Show that averaged over mean anomaly M
Question 34
Show that averaged over mean anomaly M
where
Question 35
Making use of
Show that
Question 36
Energy change in a single three-body encounter is
given by
Derive first approximation for the stability boundary
and justify qualitatively the better formulae
xxx