# Newtonian deflection of light and energy conservation

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?

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!

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

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).