On the timing properties of SAX J1808.4-3658 during its 2015 outburst

We present a timing analysis of the 2015 outburst of the accreting millisecond X-ray pulsar SAX J1808.4-3658, using non-simultaneous XMM-Newton and NuStar observations. We estimate the pulsar spin frequency and update the system orbital solution. Combining the average spin frequency from the previous observed, we confirm the long-term spin down at an average rate $dot{ u}_{text{SD}}=1.5(2)times 10^{-15}$ Hz s$^{-1}$. We also discuss possible corrections to the spin down rate accounting for mass accretion onto the compact object when the system is X-ray active. Finally, combining the updated ephemerides with those of the previous outbursts, we find a long-term orbital evolution compatible with a binary expansion at a mean rate $dot{P}_{orb}=3.6(4)times 10^{-12}$ s s$^{-1}$, in agreement with previously reported values. This fast evolution is incompatible with an evolution driven by angular momentum losses caused by gravitational radiation under the hypothesis of conservative mass transfer. We discuss the observed orbital expansion in terms of non-conservative mass transfer and gravitational quadrupole coupling mechanism. We find that the latter can explain, under certain conditions, small fluctuations (of the order of few seconds) of the orbital period around a global parabolic trend. At the same time, a non-conservative mass transfer is required to explain the observed fast orbital evolution, which likely reflects ejection of a large fraction of mass from the inner Lagrangian point caused by the irradiation of the donor by the magneto-dipole rotator during quiescence (radio-ejection model). This strong outflow may power tidal dissipation in the companion star and be responsible of the gravitational quadrupole change oscillations.