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

We discuss candidates for non-radial modes excited in a mass accreting and rapidly rotating neutron star to explain the coherent frequency identified in the light curves of a millisecond X-ray pulsar XTE J1751-305. The spin frequency of the pulsar is $ν_{\rm spin}\cong435$Hz and the identified coherent frequency is $ν_{\rm osc}=0.5727595\timesν_{\rm spin}$. Assuming the frequency identified is that observed in the corotating frame of the neutron star, we find that the surface $r$-modes of $l^\prime=m=1$ and 2 excited by $ε$-mechanism due to helium burning in the thin shell can give the frequency ratio $κ=ν_{\rm osc}/ν_{\rm spin}\simeq0.57$ at $ν_{\rm spin}=435$Hz. As another candidate for the observed ratio $κ$, we also suggest a toroidal crustal mode that has penetrating amplitudes in the fluid core and is destabilized by gravitational wave emission. Since the surface fluid layer is separated from the fluid core by a solid crust, the amplitudes of an $r$-mode in the core, which is destabilized by emitting gravitational waves, can be by a large factor different from those in the fluid ocean. We find that the amplification factor defined as $f_{\rm amp}=α_{\rm surface}/α_{\rm core}$ is as large as $f_{\rm amp}\sim 10^2$ for the $l^\prime=m=2$ $r$-mode at $ν_{\rm spin}=435$Hz for a $M=1.4M_\odot$ neutron star model. Because of this significant amplification of the $r$-mode amplitudes in the surface fluid layer, we suggest that, when proper corrections to the $r$-mode frequency such as due to the general relativistic effects are taken into consideration, the core $r$-mode of $l^\prime=m=2$ can be a candidate for the detected frequency, without leading to serious contradictions to, for example, the spin evolution of the underlying neutron star.