Relaxed Dynamics of the Trojan Asteroids

Details Relaxed Dynamics of the Trojan Asteroids Relaxed Dynamics of the Trojan Asteroids

Jun 2006

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2006dda....37.0203c&link_type=abstract

American Astronomical Society, DDA meeting #37, #2.03; Bulletin of the American Astronomical Society, Vol. 38, p.667

Mathematics

Probability

Scientific paper

A statistical theory of gravitationally relaxed linearized Newtonian dynamics for small bodies near L4 and L5 in the stable case is developed.
It is well known that if n3 is the angular velocity of the principle bodies and if the z axis is perpendicular to their orbit plane then the z motion equations are S3'=n3C3,C3'=-n3S3 where '=d/dt,S3=n3z,C3=z'.
For the x,y plane there are 2 modes of motion with angular velocities n1,n2 obeying n1The mass specific energy for the 3 modes is ei=1/2 Ci2+1/2 Si2, and is constant implying that the phase point of a small body moves steadily in a circle in the (Ci,Si) plane.
If gravitational relaxation holds then a kinematic form of 6 dimensional Eherenfest phase space may be used for each small body: {C1,S1,C2,S2,C3,S3}. It will be shown that the appropriate specific energy e on which a particle probability distribution depends is the total of the mode specific energies, e=e1+e2+e3 which is not the specific Jacobi integral to 2nd order.
Most probable statistics and microcanonical statistics imply that the distribution is given as a function of N, the total population of small bodies and a dimensionless specific energy ɛ, ɛ=3e/ where is the average energy. Then to lowest order in N the probability distribution ψ is dψ=k(Ne-ɛ-1)ɛ2dɛ for ɛ≤ɛc, and dψ=0 for ɛc ≤ɛ where ɛc is the specific energy cutoff given by ɛc=ln(N) and where k is the distribution normalization constant.
A fitting of the Minor Planet Center data for the Jovian Trojans to this distribution will be given at the 2006 DDA meeting.

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