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Abstract The initial spin periods of newborn magnetars are strongly associated with the origin of their strong magnetic fields, both of which can affect the electromagnetic radiation and gravitational waves (GWs) emitted at their birth. Combining the upper limit $$E_{\textrm{SNR}}\lesssim 10^{51}$$ E SNR ≲ 10 51 erg on the explosion energies of the supernova (SN) remnants around slowly-spinning magnetars with a detailed investigation on the evolution of newborn magnetars, we set constraints on the initial spin periods of magnetars born in weak SN explosions. Depending on the conversion efficiency $$\eta $$ η of the electromagnetic energy of these newborn magnetars into the kinetic energy of SN ejecta, the minimum initial spin periods of these newborn magnetars are $$P_{\textrm{i, min}}\simeq 5$$ P i, min ≃ 5 –6 ms for an ideal efficiency $$\eta =1$$ η = 1 , $$P_{\textrm{i, min}}\simeq 3$$ P i, min ≃ 3 –4 ms for a possible efficiency $$\eta =0.4$$ η = 0.4 , and $$P_{\textrm{i, min}}\simeq 1$$ P i, min ≃ 1 –2 ms for a relatively low efficiency $$\eta =0.1$$ η = 0.1 . Based on these constraints and adopting reasonable values for the physical parameters of the newborn magnetars, we find that their GW radiation at $$\nu _{\textrm{e,1}}=\nu $$ ν e,1 = ν may be undetectable by the Einstein Telescope (ET) since the maximum signal-to-noise ratio ( $$\mathrm{S/N}$$ S / N ) is only 2.41 even the sources are located at a very close distance of 5 Mpc, where $$\nu $$ ν are the spin frequencies of the magnetars. At such a distance, the GWs emitted at $$\nu _{\textrm{e,2}}=2\nu $$ ν e,2 = 2 ν from the newborn magnetars with dipole fields $$B_{\textrm{d}}=5\times 10^{14}$$ B d = 5 × 10 14 and $$10^{15}$$ 10 15 G may be detectable by the ET because $$\mathrm{S/N}$$ S / 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 $$\mathrm{S/N}$$ S / N . |
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