  With the plethora of detailed results from heliospheric missions and at the
advent of the first mission dedicated IBEX, we have entered the era of
precision heliospheric studies. Interpretation of these data require precision
modeling, with second-order effects quantitatively taken into account. We study
the influence of the non-flat shape of the solar Ly-alpha line on the
distribution of neutral interstellar H in the inner heliosphere. Based on
available data, we (i) construct a model of evolution for the solar Ly-alpha
line profile with solar activity, (ii) modify an existing test-particle code
used to calculate the distribution of neutral interstellar H in the inner
heliosphere so that it takes the dependence of radiation pressure on radial
velocity into account, and (iii) compare the results of the old and new
version. Discrepancies between the classical and Doppler models appear between
~5 and ~3 AU and increase towards the Sun from a few percent to a factor of 1.5
at 1 AU. The classical model overestimates the density everywhere except for a
~60-degr cone around the downwind direction, where a density deficit appears.
The magnitude of the discrepancies appreciably depends on the phase of the
solar cycle, but only weakly on the parameters of the gas at the termination
shock. For in situ measurements of neutral atoms performed at ~1 AU, the
Doppler correction will need to be taken into account, because the
modifications include both the magnitude and direction of the local flux by a
few km/s and degrees, respectively, which, when unaccounted for, would
introduce an error of a few km/s and degrees in determination of the magnitude
and direction of the bulk velocity vector at the termination shock.
