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Comprehensive Analysis of the Transient X-ray Pulsar MAXI J1409-619

We probe the properties of the transient X-ray pulsar MAXI J1409$-$619 through \textit{RXTE} and \textit{Swift} follow up observations of the outburst in 2010. We are able to phase connect the pulse arrival times for the 25 days episode during the outburst. We suggest that either an orbital model (with $P_{\rm{orb}} \simeq 14.7(4)$ days) or a noise process due to random torque fluctuations (with $S_r \approx 1.3 \times 10^{-18}$ Hz$^2$ s$^{-2}$ Hz$^{-1}$) is plausible to describe the residuals of the timing solution. The frequency derivatives indicate a positive torque-luminosity correlation, that implies a temporary accretion disc formation during the outburst. We also discover several quasi-periodic oscillations (QPOs) in company with their harmonics whose centroid frequencies decrease as the source flux decays. The variation of pulsed fraction and spectral power law index of the source with X-ray flux is interpreted as the sign of transition from a critical to a sub-critical accretion regime at the critical luminosity within the range of $6\times 10^{37}$ erg s$^{-1}$ to $1.2\times 10^{38}$ ergs s$^{-1}$. Using pulse-phase-resolved spectroscopy, we show that the phases with higher flux tend to have lower photon indices, indicating that the polar regions produce spectrally harder emission.