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

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 $$E_{\textrm{SNR}}\lesssim 10^{51}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mtext>SNR</mml:mtext> </mml:msub> <mml:mo>≲</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>51</mml:mn> </mml:msup> </mml:mrow> </mml:math> 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 $$\eta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math> 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 $$P_{\textrm{i, min}}\simeq 5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow> <mml:mtext>i, min</mml:mtext> </mml:mrow> </mml:msub> <mml:mo>≃</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:math> –6 ms for an ideal efficiency $$\eta =1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , $$P_{\textrm{i, min}}\simeq 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow> <mml:mtext>i, min</mml:mtext> </mml:mrow> </mml:msub> <mml:mo>≃</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> –4 ms for a possible efficiency $$\eta =0.4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.4</mml:mn> </mml:mrow> </mml:math> , and $$P_{\textrm{i, min}}\simeq 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow> <mml:mtext>i, min</mml:mtext> </mml:mrow> </mml:msub> <mml:mo>≃</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> –2 ms for a relatively low efficiency $$\eta =0.1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.1</mml:mn> </mml:mrow> </mml:math> . Based on these constraints and adopting reasonable values for the physical parameters of the newborn magnetars, we find that their GW radiation at $$\nu _{\textrm{e,1}}=\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ν</mml:mi> <mml:mtext>e,1</mml:mtext> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>ν</mml:mi> </mml:mrow> </mml:math> may be undetectable by the Einstein Telescope (ET) since the maximum signal-to-noise ratio ( $$\mathrm{S/N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>/</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> ) is only 2.41 even the sources are located at a very close distance of 5 Mpc, where $$\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ν</mml:mi> </mml:math> are the spin frequencies of the magnetars. At such a distance, the GWs emitted at $$\nu _{\textrm{e,2}}=2\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ν</mml:mi> <mml:mtext>e,2</mml:mtext> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>ν</mml:mi> </mml:mrow> </mml:math> from the newborn magnetars with dipole fields $$B_{\textrm{d}}=5\times 10^{14}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mtext>d</mml:mtext> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>14</mml:mn> </mml:msup> </mml:mrow> </mml:math> and $$10^{15}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>15</mml:mn> </mml:msup> </mml:math> G may be detectable by the ET because $$\mathrm{S/N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>/</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> 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 $$\mathrm{S/N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>/</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> .