Excitation of a non-radial mode in a millisecond X-ray pulsar XTE J1751-305

We discuss non-radial modes in mass-accreting and rapidly rotating neutron stars for the coherent frequency detected in a millisecond X-ray pulsar XTE J1751-305. The spin frequency of the pulsar is ν_spin ≅ 435Hz and the identified frequency is ν_osc = 0.572 7595 × ν_spin. Assuming that the frequency detected is that in the corotating frame of the star, we examine r and g modes in the surface layer of accreting matter composed mostly of helium, inertial and r modes in the fluid core, and toroidal modes in the solid crust. We find that the r modes of ľ = m = 1 and 2 excited by ε-mechanism in the surface layer can give the ratio κ = ν_osc/ν_spin ≃ 0.57 at ν_spin = 435Hz, where m and ľ are the azimuthal wavenumber and the harmonic degree of the modes. We also suggest a toroidal crust mode and a core r mode destabilized by gravitational wave emission for the observed ratio κ. We find that the amplitude of the core r mode of ľ = m = 2 can be amplified at the surface layer by a large factor f_amp ∼ 10² at νspin = 435Hz for a M = 1.4M⊙ neutron-star model. This amplification, however, may not be large enough for the r-mode amplitude to be consistent with an estimation byMahmoodifar & Strohmayer (2013).