Newtonian deflection of light and energy conservation

Question

I was looking at some calculations of the deflection of light by the sun. However, there is something that I cannot understand. They usually express a relation $$r = \dfrac{p}{1+e\cos(\theta)} \qquad (\star) $$ as if it were massive, knowing that nothing in this expression depends on the mass (but it depends on the massive energy $\epsilon = E/m$).

Here's the "energy" conservation for a massless particle (it has the dimension of energy over a mass) : $$ \epsilon = \frac 1 2 c^2 - \frac \kappa r $$ with $\kappa = GM_s$, where $M_s$ denotes the sun's mass.

I believe that in this paper¹, they assume that $(\star)$ holds:

A light ray, from a distant star, under the Sun’s gravitational force field describes the usual central force hyperbolic orbit.

So here's my problem: since $c$ is a constant, how can $\epsilon$ be a constant knowing that $r$ is not?

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Warning: I choose to right as $e$ the eccentricity and $\epsilon$ the quantity conserved through the movement. However, in the paper I linked above, they denote $\epsilon$ the eccentricity!

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¹ D. S.L. Soares, "Newtonian gravitational deflection of light revisited", arXiv:physics/0508030.

Answer 1

In these calculations, the light is modeled as an ordinary particle whose speed happens to be $c$ either at infinity or at the point of closest approach. The speed isn't (and can't be) constant on the whole orbit. The paper that you linked sets the speed to $c$ at closest approach (see the text after equations 5 and 7).

There's really no adequate Newtonian theory of light, so the concept of computing the Newtonian gravitational deflection of light doesn't make a lot of sense. The alternatives to treating light as an ordinary particle are treating it as a particle moving at infinite speed (which would lead to a prediction of no deflection) or as a wave in a luminiferous aether (in which case I suppose the prediction would depend on how gravity affects the aether).

Answer 2

As I vaguely remember, Newtonian gravity does not have any affect on light. Einstein's General Relativity was the first prediction about gravity bending light, and this was confirmed a few years later by seeing the effect in a telescope during a full solar eclipse.