Almog Yalinewich
Stability of the 3 body problem
Possible Fates of Star + 2 Planets
Stable evolution
Plunge into the star
Unbinding from star
"Collision"
Hill Stability
Why not use an N-body integrator?
Numerical Errors
Time
Large Survey
Rebound
Introducing Angular Momentum Deficit
Hill stability guaranteed if
Schematic
Numerical Experiments
Numerical Experiments cont'd
Derivation
Sundman Inequality
Let us consider an ensemble of particles
(masses, radii and velocities)
Kinetic energy
Moment of inertia
Angular Momentum
Sundman Inequality 2
Absorb mass
Kinetic energy
Moment of inertia
Angular Momentum
Sundman Inequality 3
Cauchy Schwartz Inequality
Sundman Inequality 4
Cauchy Schwartz Inequality
Sundman Inequality 5
Sundman Inequality 6
Angular Momentum Deficit
Maximum value of right hand side
Angular Momentum Deficit 2
Angular Momentum Deficit 3
Stability is guaranteed if
Recalling the definitions
Angular Momentum Deficit 4
Conclusion
Hill stability criterion for
arbitrary
and