We know that the orbit of the Earth around the Sun is not a perfect circle, but instead, quite elliptical, not to say “eccentric”. But how eccenctrically would the Earth travel around the Sun if the Sun’s gravitional field were an inversely exponential gravity field, a so-called Yukawa potential, meaning the potential engergy of a body with mass m at a distance R from another body with mass M follows this equation:
Epot = -k/R * e -R/a
(with k = G*M*m and G as the Gravitational Constant and a determining the slope of the decay of the potential energy towards larger distances which we assume here as 1.5* 10^15 m which means at its actual distance from the Sun the Earh would have a potential energy 0.999900005 of its actual potential energy – not much of a difference really)
instead of the good old and familiarily inversely linear Newtonian one
Epot = -k/R ?
Well, now, who are we to claim that a perfectly linear gravitional field is the only one possible, like a God-given fixed rule, just because it is so convenient and easy to calculate with? No, we are not that smug.
A not really shocking or Earth-shattering answer to this question is provided fully mathematically and in theory here – but, mind you, it is not a bedtime story for Joe the plummer. But if are sufficiently nerdy you might give it a shot and have fun if you can, always in the save assurance that the Earth would not be flung away into space:
In order to give some background information why this paper seems very difficutl to understand and to make head or tails of: It is a solution of an optional (ungraded) problem set in the Coursera MOOC ‘From Big Bang to Dark Energy’ which simply assumes that the question is fresh in the mind of the reader and the used equations and entities are equally familiar. Hence, the paper might appear all Greek to the reader who is not familiar with the task this paper solves – and those are practically all readers who come here. So my apologies for the hardship. Íf I have enough time I going to revamp the paper so that it becomes understandable more easily for the general audience (still, an audience sufficiently familiar with the physics and the mathematics. That I cannot avoid).