  The standard model particles can be gauged in an anomaly free way by three
possible gauge symmetries namely ${L_e-L_\mu}$, ${L_e-L_\tau}$, and
${L_\mu-L_\tau}$. Of these, ${L_e-L_\mu}$ and ${L_e-L_\tau}$ forces can mediate
between the Sun and the planets and change the perihelion precession of
planetary orbits. It is well known that a deviation from the $1/r^2$ Newtonian
force can give rise to a perihelion advancement in the planetary orbit, for
instance, as in the well known case of Einstein's gravity which was tested from
the observation of the perihelion advancement of the Mercury. We consider the
long range Yukawa potential which arises between the Sun and the planets if the
mass of the gauge boson is $M_{Z^{\prime}}\leq \mathcal{O}(10^{-19})\rm {eV}$.
We derive the formula of perihelion advancement for Yukawa type fifth force due
to the mediation of such $U(1)_{L_e-L_{\mu,\tau}}$ gauge bosons. The perihelion
advancement for Yukawa potential is proportional to the square of the semi
major axis of the orbit for small $M_{Z^{\prime}}$, unlike GR, where it is
largest for the nearest planet. However for higher values of $M_{Z^{\prime}}$,
an exponential suppression of the perihelion advancement occurs. We take the
observational limits for all planets for which the perihelion advancement is
measured and we obtain the upper bound on the gauge boson coupling $g$ for all
the planets. The Mars gives the stronger bound on $g$ for the mass range $\leq
10^{-19}\rm{eV}$ and we obtain the exclusion plot. This mass range of gauge
boson can be a possible candidate of fuzzy dark matter whose effect can
therefore be observed in the precession measurement of the planetary orbits.
