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We discuss candidates for non-radial modes excited in a mass accreting andrapidly rotating neutron star to explain the coherent frequency identified inthe light curves of a millisecond X-ray pulsar XTE J1751-305. The spinfrequency of the pulsar is $\nu_{\rm spin}\cong435$Hz and the identifiedcoherent frequency is $\nu_{\rm osc}=0.5727595\times\nu_{\rm spin}$. Assumingthe frequency identified is that observed in the corotating frame of theneutron star, we find that the surface $r$-modes of $l^\prime=m=1$ and 2excited by $\epsilon$-mechanism due to helium burning in the thin shell cangive the frequency ratio $\kappa=\nu_{\rm osc}/\nu_{\rm spin}\simeq0.57$ at$\nu_{\rm spin}=435$Hz. As another candidate for the observed ratio $\kappa$,we also suggest a toroidal crustal mode that has penetrating amplitudes in thefluid core and is destabilized by gravitational wave emission.  Since the surface fluid layer is separated from the fluid core by a solidcrust, the amplitudes of an $r$-mode in the core, which is destabilized byemitting gravitational waves, can be by a large factor different from those inthe fluid ocean. We find that the amplification factor defined as $f_{\rmamp}=\alpha_{\rm surface}/\alpha_{\rm core}$ is as large as $f_{\rm amp}\sim10^2$ for the $l^\prime=m=2$ $r$-mode at $\nu_{\rm spin}=435$Hz for a$M=1.4M_\odot$ neutron star model. Because of this significant amplification ofthe $r$-mode amplitudes in the surface fluid layer, we suggest that, whenproper corrections to the $r$-mode frequency such as due to the generalrelativistic effects are taken into consideration, the core $r$-mode of$l^\prime=m=2$ can be a candidate for the detected frequency, without leadingto serious contradictions to, for example, the spin evolution of the underlyingneutron star.

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