  A new iterative method for calculating the travel time of a photon as a
function of the spatial positions of the emitter and the receiver in the field
of a static, spherically symmetric body is presented. The components of the
metric are assumed to be expressible in power series in $m/r$, with $m$ being
half the Schwarzschild radius of the central body and $r$ a radial coordinate.
The procedure exclusively works for a light ray which may be described as a
perturbation in powers of $G$ of a Minkowskian null geodesic, with $G$ being
the Newtonian gravitational constant. It is shown that the expansion of the
travel time of a photon along such a ray only involves elementary integrals
whatever the order of approximation. An expansion of the impact parameter in
power series of $G$ is also obtained. The method is applied to explicitly
calculate the perturbation expansions of the light travel time and the impact
parameter up to the third order. The full expressions yielding the terms of
order $G^3$ are new. The expression of the travel time confirms the existence
of a third-order enhanced term when the emitter and the receiver are in
conjunction relative to the central body. This term is shown to be necessary
for determining the post-Newtonian parameter $\gamma$ at a level of accuracy of
$10^{-8}$ with light rays grazing the Sun.
