A strong candidate for a source of gravitational waves is a highly magnetised , rapidly rotating neutron star ( magnetar ) deformed by internal magnetic stresses .
We calculate the mass quadrupole moment by perturbing a zeroth-order hydrostatic equilibrium by an axisymmetric magnetic field with a linked poloidal-toroidal structure .
In this work , we do not require the model star to obey a barotropic equation of state ( as a realistic neutron star is not barotropic ) , allowing us to explore the hydromagnetic equilibria with fewer constraints .
We derive the relation between the ratio of poloidal-to-total field energy \Lambda and ellipticity \epsilon and briefly compare our results to those obtained using the barotropic assumption .
Then , we present some examples of how our results can be applied to astrophysical contexts .
First , we show how our formulae , in conjunction with current gravitational wave ( non- ) detections of the Crab pulsar and the Cassiopeia A central compact object ( Cas A CCO ) , can be used to constrain the strength of the internal toroidal fields of those objects .
We find that , for the Crab pulsar ( whose canonical equatorial dipole field strength , inferred from spin down , is 4 \times 10 ^ { 8 } T ) to emit detectable gravitational radiation , the neutron star must have a strong toroidal field component , with maximum internal toroidal field strength B _ { \mathrm { tm } } = 7 \times 10 ^ { 12 } T ; for gravitational waves to be detected from the Cas A CCO at 300 Hz , B _ { \mathrm { tm } } \sim 10 ^ { 13 } T , whereas detection at 100 Hz would require B _ { \mathrm { tm } } \sim 10 ^ { 14 } T. Using our results , we also show how the gravitational wave signal emitted by a magnetar immediately after its birth ( assuming it is born rapidly rotating , with \Lambda \lesssim 0.2 ) makes such a newborn magnetar a stronger candidate for gravitational wave detection than , for example , an SGR giant flare .