1. Celestial Mechanics VI The N-body Problem: Equations of motion and general integrals The Virial Theorem Planetary motion: The perturbing function Numerical integration Examples of orbital evolution

2. N-body problem The motion of N (>2) bodies under the influence of each others’ gravity The masses of some bodies may in some cases be considered negligible Solar System case: One object is much more massive than the others

3. Useful applications Accurate ephemeris calculation Space probe navigation with gravity assist Prediction of NEO close encounters Tracing long-term dynamical transfer of comets or asteroids Assessing the stability of the Solar System

4. Equations of Motion This holds in any inertial frame of reference If the system is isolated, the barycenter defines such a frame The system of equations for all the massive bodies is coupled The phase space is 6N-dimensional Acceleration Sum of forces per unit mass

5. General Integrals Definition: Functions of the phase space coordinates that are constant along all trajectories (conserved quantities during any motion) In the general case of an isolated system, only 10 are known: – Six define the uniform motion of the barycenter – Three define the total angular momentum vector – One defines the total energy of motion

6. From eqs. of motion to general integrals Motion of barycenter: the sum of the forces acting on all bodies is zero  the acceleration of the barycenter is zero Angular Momentum: the sum of radius vector  force for all bodies is zero; this is the time derivative of the total A.M. Energy: the sum of velocity force on all bodies  the time derivative of the sum of kinetic and potential energies of all bodies is zero

7. The Virial Theorem Another important property of an N-body system, obtainable from the eqs. of motion Consider the moment of inertia about the origo, and take its time derivatives:

8. The Virial Theorem, ctd. If T is the total kinetic energy of the system and  is the total potential energy, we have: For a system in equilibrium, J must be on average constant 

9. Motion in heliocentric frame Identify body nr. 1 as the Sun (m 1 dominates) Subtract the Sun’s inertial acceleration from that of an arbitrary body:

10. Heliocentric eq. of motion Separate Sun from other bodies and simplify notations!

11. A Note on Calculation Cost All bodies influence each other! With 10 3 asteroids and 8 planets, we would have 1008 coupled differential equations. But asteroids do not affect the planets or each other measurably – waste of time. Only apply this equation for the planets and treat each asteroid as a massless particle in the combined planetary gravity field!

12. Massless particle Calculate planetary r j (t) and use it as “source function” when solving this equation for each asteroid, individually. Thus we have 8 coupled, and 1000 uncoupled differential equations.

13. Perturbations Central force function Perturbing function

14. Relativistic Equation of Motion

15. Numerical integration The equations of motion in cartesian coordinates may be integrated numerically using different methods Integrators in common use are often - but not always - of standard type as described e.g. in Numerical Recipes Symplectic integrators are specially devised to conserve the “Hamiltonian”, i.e., dissipation-free, character of the motion

16. Numerical integration, ctd. Automatic time step control is important, especially when close encounters occur The time step will be determined by the quickest- moving body (i.e., typically, Mercury in case all the planets are included) In particular when treating objects in the outer Solar System, it is common to “throw the terrestrial planets into the Sun” - i.e., add their masses to that of the Sun and otherwise neglect them; this allows to use longer steps!

17. Short-term integration results The motion of the asteroid is sometimes accelerated, sometimes decelerated by the planets (primarily Jupiter) This leads to quasi- periodic behaviour of the “osculating” (current) orbital elements on the time scale of the orbital motion Idea for studying long-term motion: average out the short-period fluctuations, use “mean elements”

18. Motion on different time scales Osculating elements: the elements of a two- body orbit (conic section) that has the same position and velocity as that of the object at the given time (= epoch of osculation) Mean elements: average values of the osculating elements over a short period of time (for near-resonant orbits the geometry is more repeatable and the period should be longer) Over a much longer interval, the mean elements may stay constant, oscillate or evolve secularly

19. Mid-term integration results Over an interval of ~10 4 yr, the semi-major axis stays constant, but the behaviour of the other elements is uncertain

20. Long-term integration results The semi-major axis stays on the average constant; the eccentricity and inclination oscillate, and the argument of perihelion + longitude of the ascending node show secular variations with well- determined frequencies This behaviour is typical in the general case

21. Special cases Sometimes, very often near the main mean motion resonances in the asteroid belt, the regular behaviour is interrupted chaotically by jumps to higher eccentricity The reason is typically an overlap of resonances, so that the character of motion at one resonance is changed by effects of the other The orbits become planet-crossing, and close encounters clear the Kirkwood Gaps

22. Close encounters A close encounter with a planet leads to hyperbolic deflection of the velocity w.r.t. the planet, changing the magnitude of the object’s heliocentric velocity The angle of deflection is given by: and U is given by: T=Tisserand parameter velocity diagram b = impact parameter

23. Encounter-driven chaos For an object undergoing repeated close encounters, even the slightest difference of initial conditions leads to completely different evolutions, since the outcome of an encounter is very sensitive to the detailed geometry (“butterfly effect”) Therefore, orbits of short- period comets and Centaurs are chaotic on short time scales example due to H.F.Levison