We calculate light curves produced by r -modes with small azimuthal wavenumbers , m , propagating in the surface fluid ocean of rotating neutron stars .
We include relativistic effects due to rapid rotation , and propagate photons from the stellar surface to a distant observer using the Schwarzschild metric .
The wave motions of the surface r -modes are confined to the equatorial region of the star , and the surface pattern of the temperature variation can be either symmetric ( for even modes ) or anti-symmetric ( for odd modes ) with respect to the equator .
Since for the surface r -modes the oscillation frequency in the corotating frame of the star is much smaller than the rotation frequency , \Omega , we employ the approximation in which the oscillation frequency in the inertial frame , \sigma , is given by \sigma = - m \Omega .
We find that the even , m = 1 r -mode produces the largest light variations .
The dominant Fourier component in the light curves of these modes is the fundamental having \sigma = - \Omega , and the first harmonic component having \sigma = -2 \Omega is always negligible in comparison .
The dominant Fourier component of the even , m = 2 r -modes is the first harmonic .
Although the odd r -modes produce smaller amplitude light variations compared with the even modes , the light curves of the former have a stronger first harmonic component .
If both m = 1 and 2 r -modes are excited simultaneously , a rich variety of light curves is possible , including those having an appreciable first harmonic component .
We show that the phase difference , \delta - \delta _ { E } , between the bolometric light curve and that at a particular photon energy can possibly be used as a probe of the stellar compactness , R / M , where R and M are the radius and mass of the star .