  A gravity theory called scalar-tensor-vector gravity (STVG) has been recently
developed and succeeded in solar system, astrophysical and cosmological scales
without dark matter [J. W. Moffat, J. Cosmol. Astropart. Phys. 03, 004 (2006)].
However, two assumptions have been used: (i) $B(r)=A^{-1}(r)$, where $B(r)$ and
$A(r)$ are $g_{00}$ and $g_{rr}$ in the Schwarzschild coordinates (static and
spherically symmetric); (ii) scalar field $G=Const.$ in the solar system. These
two assumptions actually imply that the standard parametrized post-Newtonian
parameter $\gamma=1$. In this paper, we relax these two assumptions and study
STVG further by using the post-Newtonian (PN) approximation approach. With
abandoning the assumptions, we find $\gamma\neq1$ in general cases of STVG.
Then, a version of modified STVG (MSTVG) is proposed through introducing a
coupling function of scalar field G: $\theta(G)$. We have derived the metric
and equations of motion (EOM) in 1PN for general matter without specific
equation of state and $N$ point masses firstly. Subsequently, the secular
periastron precession $\dot{\omega}$ of binary pulsars in harmonic coordinates
is given. After discussing two PPN parameters ($\gamma$ and $\beta$) and two
Yukawa parameters ($\alpha$ and $\lambda$), we use $\dot{\omega}$ of four
binary pulsars data (PSR B1913+16, PSR B1534+12, PSR J0737-3039 and PSR
B2127+11C) to constrain the Yukawa parameters for MSTVG:
$\lambda=(3.97\pm0.01)\times10^{8}$m and $\alpha=(2.40\pm0.02)\times10^{-8}$ if
we fix $|2\gamma-\beta-1|=0$.
