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The standard model particles can be gauged in an anomaly free way by threepossible 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 mediatebetween the Sun and the planets and change the perihelion precession ofplanetary orbits. It is well known that a deviation from the $1/r^2$ Newtonianforce can give rise to a perihelion advancement in the planetary orbit, forinstance, as in the well known case of Einstein's gravity which was tested fromthe observation of the perihelion advancement of the Mercury. We consider thelong range Yukawa potential which arises between the Sun and the planets if themass 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 dueto the mediation of such $U(1)_{L_e-L_{\mu,\tau}}$ gauge bosons. The perihelionadvancement for Yukawa potential is proportional to the square of the semimajor axis of the orbit for small $M_{Z^{\prime}}$, unlike GR, where it islargest for the nearest planet. However for higher values of $M_{Z^{\prime}}$,an exponential suppression of the perihelion advancement occurs. We take theobservational limits for all planets for which the perihelion advancement ismeasured and we obtain the upper bound on the gauge boson coupling $g$ for allthe planets. The Mars gives the stronger bound on $g$ for the mass range $\leq10^{-19}\rm{eV}$ and we obtain the exclusion plot. This mass range of gaugeboson can be a possible candidate of fuzzy dark matter whose effect cantherefore be observed in the precession measurement of the planetary orbits.

Constraints on long range force from perihelion precession of planets in a gauged $L_e-L_{\mu,\tau}$ scenario