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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