Details

As the computations in this Demonstration show, derivation of the big equation is relatively simple and fast when applied to a Hamiltonian of the form

where both and have the dimension of angular momentum. Per our ansatz for the phase space trajectory, the expansion parameter is then

.

All gravitational pseudopotentials need to be transformed to the form of equation (1). The effective Hamiltonian for a particular planet can be expanded around a point according to

Solar system data combined with the pseudopotential formulation [3, 4] uniquely determines the parameters and . As in [1], we then apply a canonical transformation

to put the Hamiltonian (2) into the form of (1) by determining and as many as necessary. Notice that transformation (3) always forces to equal , as in equation (1) and the parameter table. We also transform the equilibrium radius by the canonical transformation

.

Finally, we estimate energy from data by using the observed perihelion and aphelion values to compute a variable for the span between the extrema where . Then the equation

can be solved numerically for . This completes specification of the data transformation theory, which determines all parameters in the table, thus determining the annual precession rate.

Comparing the calculation of Fitzpatrick to the calculation presented here, the most striking difference is the number and type of required variables. Fitzpatrick requires only the sets of mean radii and planetary masses. In our estimation, these quantities go eventually into the formulation of the pseudopotentials. But we also need angular momentum and energy , which help to determine just how the orbit will deform away from a circular shape.

Taking a look at the comparison graph, we see that the energy-independent part of our approximation

does an acceptable job of matching the Fitzpatrick values, but the agreement is not exact for all planets. Mercury and Mars are noticeably different.

References

[1] B. Klee, "Plane Pendulum and Beyond by Phase Space Geometry." arxiv.org/abs/1605.09102.

[2] NASA. "Planetary Fact Sheet." (Nov 18, 2015) nssdc.gsfc.nasa.gov/planetary/factsheet.

[3] R. Fitzpatrick, An Introduction to Celestial Mechanics, New York: Cambridge University Press, 2012.

[4] R. Fitzpatrick, "Perihelion Precession of the Planets," Newtonian Dynamics, 2011 (Sep 28, 2016). farside.ph.utexas.edu/teaching/336k/Newtonhtml/node115.html.

[5] A. Einstein, "Erklärung der Perihelionbewegung der Merkur aus der allgemeinen Relativitätstheorie." Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften, 1915 pp. 831–839. (Sep 27, 2016) einsteinpapers.press.princeton.edu/vol6-doc/261. English translation: einsteinpapers.press.princeton.edu/vol6-trans/128?ajax.