Monday, September 27, 2021

Delaunay Variable Problem Solution

First obtain the perturbation term R in terms of Legendre functions:

R=  k 2  3    [ 1/  r 3 +  ½  2/ r 3 3 - 3/2  2/ r 3 3  cos 2 S

If we take  m =  mass of Jupiter,  m = mass  of Earth, and a =  semi-major axis of Jupiter we can calculate the first order perturbations in L, G,  ℓ and g using the reference Hamiltonian:

=

-  m 2 / 2 L 2  -  2  3    [1/  r 3 +  ½  2/ r 3 3 - 3/2  2/ r 3 3  cos 2 S

We thereby obtain a functional Hamiltonian:

(L, G,  ℓ,  g, m 2 , m 3  ,  a 3  ,  t)

And can write out the differential equations to solve the problem.  One such equation would be:

dL/ dt =      /  ℓ

Integration yielding:

L  -  L o =   ò t  o      F (ℓ)   dt

Where F (ℓ)  =  F(L, G,  ℓ ,  g, constants, t)

We  then substitute for each of the variables: L, G etc. Earth and Jupiter values, and also:

ℓ   -  ℓ  o  ,   g   -   g  o  ,   etc.  leaving everything else constant and taking the specific integral in each case.  Do this for L, G,  ℓ and g

Using the  mass values for Jupiter and Earth  expressed in terms  of solar (m )

m =  1/ (1047.355 m ☉ )

m =  1/( 32930 m ☉ )

=  5.2 AU

On computation using the preceding, we get an error in the reference Hamiltonian:

=  2  3    10   -2

For an error magnitude e  »   0.012