On the initial spin periods of magnetars born in weak supernova explosions and their gravitational wave radiation

The initial spin periods of newborn magnetars are $\mathbf{\text{strongly}}\mathbf{\text{associated}}\mathbf{\text{with}}\mathbf{\text{the}}\mathbf{\text{origin}}\mathbf{\text{of}}\mathbf{\text{their}}\mathbf{\text{strong}}\mathbf{\text{magnetic}}\mathbf{\text{fields,}}\mathbf{\text{both}}\mathbf{\text{of}}\mathbf{\text{which}}\mathbf{\text{can}}\mathbf{\text{affect}}\mathbf{\text{the}}\mathbf{\text{electromagnetic}}\mathbf{\text{radiation}}\mathbf{\text{and}}\mathbf{\text{gravitational}}\mathbf{\text{waves}}\mathbf{\text{(GWs)}}\mathbf{\text{emitted}}\mathbf{\text{at}}\mathbf{\text{their}}\mathbf{\text{birth.}}$ Combining the upper limit $E_{\mathrm{S}\mathrm{N}\mathrm{R}}\lesssim {10}^{51}$ erg on the explosion energies of $\mathbf{\text{the}}\mathbf{\text{supernova}}\mathbf{\text{(SN)}}\mathbf{\text{remnants}}\mathbf{\text{around}}\mathbf{\text{slowly-spinning}}\mathbf{\text{magnetars}}$ with a detailed investigation on the evolution of newborn magnetars, we set constraints on the initial spin periods of magnetars $\mathbf{\text{born}}\mathbf{\text{in}}\mathbf{\text{weak}}\mathbf{\text{SN}}\mathbf{\text{explosions}}$. Depending on the conversion efficiency $\eta$ of the electromagnetic energy of $\mathbf{\text{these}}$ newborn magnetars into the kinetic energy of SN ejecta, the minimum initial spin periods of $\mathbf{\text{these}}$ newborn magnetars are $P_{\mathrm{i},\mathrm{m}\mathrm{i}\mathrm{n}}\simeq 5-6$ ms for an ideal efficiency $\eta =1$, $P_{\mathrm{i},\mathrm{m}\mathrm{i}\mathrm{n}}\simeq 3-4$ ms for a possible efficiency $\eta =0.4$, and $P_{\mathrm{i},\mathrm{m}\mathrm{i}\mathrm{n}}\simeq 1-2$ ms for a relatively low efficiency $\eta =0.1$. \textbf{Based on these constraints and adopting reasonable values for the physical parameters of the newborn magnetars, we find that their GW radiation at ${\nu }_{\mathrm{e},1}=\nu$ may be undetectable by the Einstein Telescope (ET) since the maximum signal-to-noise ratio ($\mathrm{S}\mathrm{/}\mathrm{N}$) is only 2.41 even the sources are located at a very close distance of 5 Mpc, where $\nu$ are the spin frequencies of the magnetars. At such a distance, the GWs emitted at ${\nu }_{\mathrm{e},2}=2\nu$ from the newborn magnetars with dipole fields $B_{\mathrm{d}}=5\times {10}^{14}$ and ${10}^{15}$ G may be detectable by the ET because $\mathrm{S}\mathrm{/}\mathrm{N}$ are 10.01 and 19.85, respectively. However, if these newborn magnetars are located at $20$ Mpc away in the Virgo supercluster, no GWs could be detected by the ET due to low $\mathrm{S}\mathrm{/}\mathrm{N}$.