tokens: cases for toroidal magnetic field configuration combined with the rotation. Fig. 2(a) shows the density contour with the uniform angular frequency Ω = 0.0628 rad s −1 with kinetic to gravitational energy ratio KE/GE ∼3.58 × 10 −6 . Since the angular frequency is small, it does not affect the star considerably, resulting in a prolate star. On the other hand, Fig. 2(b) illustrates a star with angular frequency Ω = 3.6537 rad s −1 with KE/GE ∼ 1.33 × 10 −2 , and due to this high angular velocity, the low-density region is affected by the rotation more significantly than the high-density region, resulting in an overall oblate-shaped white dwarf. Here, in both the cases, ρ c ∼ 2.2 × 10 10 g cm −3 , and magnetic field at the centre of the white dwarf B max ∼ 2.7 × 10 14 G. Indeed, a few white dwarfs are observed with the surface magnetic field of ∼10 9 G (Heyl 2000 ; Vanlandingham et al. 2005 ; Brinkworth et al. 2013 ); hence, central field might be much larger than 10 9 G. In fact, it has already been argued in the literature that the central field could be as large as 10 14 –10 15 G (Franzon Schramm 2015 ; Shah Sebastian 2017 ). This is similar to the magnetic field in the case of the Sun, where it is known from observations that the surface field is ∼1 G. However, the central field could be ∼10 4 –10 7 G, indicating an increase of four to seven orders of magnitude (Parker 1979 ). Now, in time, Sun-like stars will become white dwarfs and they may possess similar field profile as earlier, as argued by Das Mukhopadhyay ( 2014 ). Hence, a white dwarf with B max considered here might have surface field of 10 9 G, which observed data have already confirmed. As a result, magnetic to gravitational energy ratio ME/GE is ∼0.1. The surface of a white dwarf is determined, when the density decreases at least up to seven to eight orders of magnitude compared to its central density, i.e. ρ c ∼ 10 7–8 ρ surf (ideally zero), where ρ surf is the density at the surface. The barcode shows the density of the white dwarf at different radii. The typical isocontours of magnetic field strength are shown in Fig. 3 . It is confirmed herein that the surface magnetic field can decrease up to ∼10 9 G even if the central field is ∼10 14 G. However, GW astronomy may also help in identifying magnetized white dwarfs with surface fields higher than 10 9 G, as argued later. Figure 2. Density isocontours of uniformly rotating white dwarf with toroidal magnetic field. Figure 3. Isocontours of magnetic field strength for uniformly rotating white dwarf with toroidal magnetic field. The black solid line represents the surface of the white dwarf. All the parameters are same as in Fig. 2(a) . Table 1 shows different h 0 for various ρ c . In the table, M represents the mass of compact object, R E is the equatorial radius, R P is the polar radius, and ν is the linear frequency defined as ν = Ω/2. It is observed that for a given ρ c , h 0 increases with the increase in rotation because h
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