As mentioned in the previous section, if a magnetized WD rotates with a misalignment between its magnetic field and rotation axes (similar configuration to a pulsar), it can emit a continuous GW. We already provided a detailed discussion on GWs emitted from WDs with different magnetic field geometries and strengths in GR (Kalita & Mukhopadhyay 2019b; Kalita et al. 2020). Figure 7 shows an illustrative diagram of a magnetized WD where the magnetic field is along the z′-axis and rotation is along the z-axis, with χ being the angle between these two axes. We calculate the amplitude of GW using the set of Equations (34) assuming the difference in radii of the WD between those along x- and z-axes to be 0.01%, i.e., 





, due to the presence of a very weak magnetic field and slow rotation. The choice of weak fields and slow rotation assures that the underlying WD mass–radius solutions do not practically differ from the solutions based on the f(R) gravity without magnetic fields and rotation. In future, we plan to check rigorously by solving the set of equations, if indeed such ϵ is possible in the presence of weak magnetic fields and rotation keeping the mass and radius practically intact. As we will show below, however, the chosen ϵ appears to be the minimally required value to have any appreciable effect. Nevertheless, there are examples of weakly magnetized WD pulsars, which can be explained even in the GR framework, e.g., AE Aquarii (Bookbinder & Lamb 1987) and AR Scorpii (Marsh et al. 2016), where magnetic fields hardly affect their mass–radius relations. Figure 8 shows the PSD as a function of frequency for various detectors along with 





 over 5 s integration time for various f(R)-gravity-induced WD pulsars with different i assuming χ = 90° and r = 100 pc. It is evident that while DECIGO and BBO can immediately detect such weakly magnetized super-Chandrasekhar WDs, the Einstein Telescope can detect them in 





 minutes with S/N ≈ 5 (see Figure 5(b)). However, for ALIA and LISA, the corresponding integration time respectively turns out to be 





 days and 





 yr3

3
Note that even if the threshold S/N for detection increases slightly (say, from 5 to 20), many of these sources can still be detected in a few seconds to a few days of integration time depending on the type of the detectors.
. Hence, it is also possible to detect such weakly magnetized WDs using ALIA, whereas for LISA it is quite impossible. Figure 5(b) depicts 





 for these WDs with different integration times to show that S/N increases if the integration time increases so that various detectors can detect them eventually. For such a system, the GW luminosity is given by (Zimmermann & Szedenits 1979)
43





It is expected that a source can emit electromagnetic radiation in the presence of a magnetic field, and it is the dipole radiation in the case of a WD pulsar. However, because of the presence of a weak magnetic field, the dipole radiation emitted from such a WD is minimal, and the corresponding dipole luminosity is negligible as compared to LGW. Hence, the spin-down timescale is mostly governed by LGW, given by (Kalita et al. 2020)
44





Figure 9 shows the variation of LGW and P with respect to M for various WDs with χ = 90°. The maximum LGW in the case of a WD is ∼1037 erg s−1. The empirical relations of LGW and P, in various branches, are same as in the previous case provided in Table 1. It is also clear from the figure that the massive WD pulsars are short-lived as compared to the lighter ones.
Citation: Marsh et al. 2016
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