Pulsars are fast spinning neutron stars that lose their rotational energy via various processes such as gravitational and magnetic radiation, particle acceleration and mass loss processes. This dissipation can be quantified by a spin-down equation that measures the rate of change of the frequency as a function of the rotational frequency itself. We explore the pulsar spin-down and consider the spin-down equation upto the seventh order in frequency. This seventh order term accounts for energy loss due to the gravitational radiation caused by a current type quadrupole in the pulsar due to r-modes. We derive the rotational frequency due to the r-modes and find a solution in terms of the Lambert function. We also present an analytic exact solution for the period from the spindown equation and numerically verify this for the Crab pulsar. This analysis will be relevant for the detection of continuous gravitational waves by 3G ground based and space based gravitational wave detectors.
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