Revisiting The Delaunay Variables of Celestial Mechanics

 

                             Perturbation diagram to for Delaunay variable analysis

We already saw the Delaunay variables play an important role in celestial mechanics, especially in perturbation theory.  This is concerned with how the presence of a third body (say m ₃  in the accompanying diagram) perturbs or disturbs the motion of m ₂.  Given the complexity, it is useful to revisit these concepts.  These Delaunay variables are defined by the equations:

ℓ  =   n(t  - T)

L =  Ö(m a)

G  =   L    Ö (1 -   e ²)

H =   G cos i

g   =  w

h  =   Ω

The symbols used are standard, e.g.  w   is the argument of the perihelion, W is the longitude of the ascending node, e is the eccentricity, a is the semi-major axis of the orbit,   and i is the inclination of the orbit.   Also, n is the mean motion (i.e. of the planet or other body per day),  T is the date of perihelion, t is any future date assigned,  and m  is the reduced mass, i.e.

m =   1/ (1/m ₂   + 1/ m ₃)

As with all similar problems in Hamiltonian mechanics, the approach begins with the applicable Hamiltonian for the system (see diagram).  In its most general form we write:

H  =   ½ å ³ _i=1  P _i ²    -  m / r  +  R

After modification and as a function of the Delaunay variables (L, ℓ):

H  (L,  ℓ  )  =   -  m ² / 2 L ²  -  k ²  m ₃    {1/  D (L,  ℓ  )  +   r ·  r ₃   /   r ₃  ² }

A particular set of differential equations will always be associated with the new Hamiltonian, e.g.

L’ =    - ¶ H  / ¶ ℓ        ℓ    =    ¶ H  / ¶ L

g ‘  =  ¶ H  / ¶ G          h’ =   ¶ H  / ¶ H

G’ =   -   ¶ H  / ¶ g     H’  =  -   ¶ H  / ¶ h

 A huge simplification arrives  if R = 0., which then reduces to the 2-body problem, which we've examined in previous blog posts.   Note that R in the second Hamiltonian is expressed:

R   =   k ²  m ₃    {1/  D (L,  ℓ  )  +   r ·  r ₃   /   r ₃  ² }

Now, for the case at hand (consult diagram), we may write:

D ²     =  r  ²   +   r ₃  ²    -   2 r ·  r ₃  cos S

 And:  r ·  r ₃  =   r   r ₃  cos S

Assume now that r ₃  is greater than r (perturbing an inner planet or body by an outer one):

1 /D      =  1/r [ 1 + (r / r ₃)  ²  -  2 (r / r ₃)  cos S] ^1/2

Then the above can be rewritten as the sum:

½ å ^¥ _{r= 0}  (r / r ₃) ⁱ  P _i 

=   1/  r ₃  [ P _o   +   (r / r ₃)  P ₁   + (r / r ₃)  ²  P ₂    +  ...]

The P _i   are functions of the angle S and are called Legendre polynomials.  In this case, the first three may be written:

P _o   =   1,      P ₁     =   cos S,    P ₂     =   ½ (3  cos ² S   -   1) 

Problem for ambitious readers:

Using the Legendre functions for the angle S just given rewrite the perturbation term R in terms of them.

Thence or otherwise, rewrite a Hamiltonian incorporating R and these functions.   From this, write out the form of at least one applicable differential equation incorporating Delaunay variables and explain your basic procedure (say for the Delaunay variables L, G,  ℓ and g.)

For those really wanting a challenge:  Estimate the magnitude of the error in the modified Hamiltonian H  in terms of k and   m ₃.  (You can take  m ₃ =  mass of Jupiter,  m ₂ = mass  of Earth, and a ₃ =  semi-major axis of Jupiter).   Make sure your reference Hamiltonian conforms with these.