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 $$r = dfrac{p}{1+ecos(theta)} qquad (star) $$ 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.