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We report on a high-precision timing analysis and an astrophysical study ofthe binary millisecond pulsar, PSR J1909$-$3744, motivated by the accumulationof data with well improved quality over the past decade. Using 15 years ofobservations with the Nan\c{c}ay Radio Telescope, we achieve a timing precisionof approximately 100 ns. We verify our timing results by using both broad-bandand sub-band template matching methods to create the pulse time-of-arrivals.Compared with previous studies, we improve the measurement precision of secularchanges in orbital period and projected semi-major axis. We show that thesevariations are both dominated by the relative motion between the pulsar systemand the solar system barycenter. Additionally, we identified four possiblesolutions to the ascending node of the pulsar orbit, and measured a precisekinetic distance of the system. Using our timing measurements and publishedoptical observations, we investigate the binary history of this system usingthe stellar evolution code MESA, and discuss solutions based on detailed WDcooling at the edge of the WD age dichotomy paradigm. We determine the 3-Dvelocity of the system and show that it has been undergoing a highly eccentricorbit around the centre of our Galaxy. Furthermore, we set up a constraint overdipolar gravitational radiation with the system, which is complementary toprevious studies given the mass of the pulsar. We also obtain a new limit onthe parameterised post-Newtonian parameter, $\alpha_1<2.1 \times 10^{-5}$ at 95% confidence level, which is fractionally better than previous best publishedvalue and achieved with a more concrete method.

A revisit of PSR J1909$-$3744 with 15-year high-precision timing