The orbital period and system parameters of the recurrent nova T Pyx

T Pyx is a luminous recurrent nova that accretes at a much higher rate than is expected for its photometrically determined orbital period of about 1.8 hours. We here provide the first spectroscopic confirmation of the orbital period, P =1.829 hours (f=13.118368(11) c/d), based on time-resolved optical spectroscopy obtained at the VLT and the Magellan telescopes. We also derive an upper limit of the velocity semi-amplitude of the white dwarf, K1 = 17.9 +/- 1.6 km/s, and estimate a mass ratio of q = 0.20 +/- 0.03. If the mass of the donor star is estimated using the period-density relation and theoretical main-sequence mass-radius relation for a slightly inflated donor star, we find M2 = 0.14 +/- 0.03 Msun. This implies a mass of the primary white dwarf M1 = 0.7 +/- 0.2 Msun. If the white-dwarf mass is > 1 Msun, as classical nova models imply, the donor mass must be even higher. We therefore rule out the possibility that T Pyx has evolved beyond the period minimum for cataclysmic variables. We find that the system inclination is constrained to be approximately 10 degrees, confirming the expectation that T Pyx is a low-inclination system. We also discuss some of the evolutionary implications of the emerging physical picture of T Pyx. In particular, we show that epochs of enhanced mass transfer (like the present) may accelerate or even dominate the overall evolution of the system, even if they are relatively short-lived. We also point out that such phases may be relevant to the evolution of cataclysmic variables more generally.