Gravitational wave radiation from the magnetar-driven supernovae

A long-lived magnetar can be formed by the core collapse of a massive star, which can produce a transient event, such as a gamma-ray burst (GRB) and a supernova (SN). The rotational energy of the magnetar is a reservoir for the afterglow emission of SN and GRB with diverse evolutionary features, such as short GRB having extended emission or GRB with X-ray plateau (Usov, 1992; Dai & Lu, 1998a, b; Zhang & Mészáros, 2001; Metzger et al., 2011; Gompertz et al., 2013; Rowlinson et al., 2013; Metzger & Piro, 2014; Lü & Zhang, 2014; Sarin et al., 2020; Li & Dai, 2021; Xie et al., 2022a, b). The high kinetic energy or high luminosity SNe are generally interpreted as an extra injection of spin-down energy from the magnetar engine (Kasen & Bildsten, 2010; Woosley, 2010; Inserra et al., 2013; Metzger et al., 2015; Kashiyama et al., 2016; Kasen et al., 2016; Wang et al., 2016; Moriya et al., 2017; Nicholl et al., 2017; Omand & Sarin, 2024).

Many observations show that some broad-lined Type Ic (Ic-BL) SNe and superluminous SNe (SLSNe) have peak luminosities and radiant energies tens of times higher than those of typical SNe (Gal-Yam, 2012; Nicholl, 2021). Various models have been applied in the literature to interpret the striking kinetic energy and luminosity. One popular model involves the energy injection from a rapidly spinning magnetar (Woosley, 2010; Kasen & Bildsten, 2010). The basic picture is that winds from a magnetar driven by magnetic dipole radiation interact with the SNe ejecta and deposit their spin energy into the SN ejecta. Such a magnetar-driven model has successfully explained the light curves of SNe Ic-BL (Greiner et al., 2015; Cano et al., 2016; Wang et al., 2016) and SLSNe (i.e, (Dessart et al., 2012; Chatzopoulos et al., 2013; Inserra et al., 2013; Nicholl et al., 2013; Howell et al., 2013; Nicholl et al., 2017; Yu et al., 2017; Liu et al., 2017; Villar et al., 2018; Wang et al., 2019)), as well as the nebular spectral observations of hydrogen-poor SLSNe (Jerkstrand et al., 2017; Quimby et al., 2018; Nicholl et al., 2019). Other models for powering the energetic SNe involve the radioactive decay of massive amounts of ⁵⁶Ni (Barkat et al., 1967; Heger & Woosley, 2002; Gal-Yam et al., 2009), the jets launched by the center engine for SNe Ic-BL (Lazzati et al., 2012; Gilkis et al., 2016; Barnes et al., 2018; Eisenberg et al., 2022; Corsi et al., 2023), the interaction of SN ejecta with the circumstellar medium (Blinnikov & Sorokina, 2010; Chevalier & Irwin, 2011; Ginzburg & Balberg, 2012; Moriya & Maeda, 2012), and an accreting black hole (i.e., (MacFadyen et al., 2001; Dexter & Kasen, 2013; Li et al., 2020)).

In the magnetar-driven SNe scenario, a magnetar with a strong magnetic field of $B\sim 10^{15}\mbox{ G}$, a rapid rotation of initial spin period $P_{0}\sim 1\mbox{ ms}$, and high temperatures is born after the core collapse of massive star (Duncan & Thompson, 1992). The rapid rotation of the magnetar can induce a large deformation due to the magnetic pressure or fluid oscillations (Owen et al., 1998; Lai & Shapiro, 1995; Bonazzola & Gourgoulhon, 1996; Andersson, 1998; Lindblom et al., 1998; Cutler, 2002; Stella et al., 2005; Haskell et al., 2008; Alford & Schwenzer, 2015; Lasky et al., 2017), implying that the magnetar could emit the remarkable GWs. The GW quadrupole radiation and magnetic dipole (MD) radiation torque could act together for magnetar spin-down (Shapiro & Teukolsky, 1983; Zhang & Mészáros, 2001). The GW energy loss would therefore affect the spin evolution and also electromagnetic outflows of the magnetar. In other words, if a deformed magnetar powers an SN, the GW-energy loss could result in less rotational energy supplied to the SNe, and then lead to a change in the shape of the light curve, indicated by a lower peak luminosity and a slower decay rate. A magnetar-driven system should produce a unique luminosity evolution, different from only MD radiation losses.

Non-axisymmetrically deformed magnetars are potential continuous GW sources for the Advanced Laser Interferometer gravitational wave Observatory (aLIGO) detector (Aasi et al., 2015a) and the Advanced Virgo detector (Acernese et al., 2015). The aLIGO has searched for continuous GWs from various types of isolated neutron stars (NSs), such as known pulsars (Abbott et al., 2019a), post-burst magnetars (Abbott et al., 2019b, 2022), accreting NSs and young NSs in SN remnants (Abbott et al., 2019c, d). However, limited by the sensitivity at the designed frequencies, no GWs have yet been detected from these newly formed compact objects.

Detecting GWs from magnetar-driven SNe could help us constrain the structure and composition of NSs and provide theoretical traction for future ground-based GW telescopes. We explore in this paper possible indications of the GW radiation in SNe electromagnetic light curves. We develop a model for magnetar-driven SNe by considering the GW radiation and the effect on the bolometric luminosity evolution of SNe. From the SN luminosity curves, we can constrain the characteristics of magnetars, i.e., the initial spin period $P_{0}$, the dipole magnetic field strength $B$, and the ellipticity of the magnetar $\epsilon$, as well as the parameters of SN explosions. Through the best-fitting for parameters of the magnetar obtained by the Markov chain Monte Carlo (MCMC) method, we derive the GW characteristic amplitude of the magnetar and then evaluate the GW detectability of future ground-based GW telescopes. This paper is organized as follows. We investigate the effect of the GW energy-less on the bolometric luminosity evolution of SNe under the assumption of the magnetar model in section 2. In section 3, we take some SNs as examples and fit their SN observation data with the magnetar model. In Section 4, we analyze the detectability of GWs from magnetars that power SNe. The discussion and conclusions are given in Section 5.

Some authors (Inserra et al., 2013; Nicholl et al., 2017) have used magnetar models to explain SNe light curves and constrain the parameters of magnetars and the ejecta mass. However, the effect of GW radiation is usually neglected.

A rapidly spinning magnetar can be produced after the core collapse of a massive star. The rotational energy of the magnetar provides a huge energy reservoir for SNe, which greatly enriches the features of the SNe optical emission. However, not all of the rotational energy is used to heat the SN ejecta, part of which may be lost via the significant GW radiation because of non-axisymmetric deformation. Thus, the total rotational energy of the magnetar, $E_{\rm rot}=I\Omega^{2}/2$, is released through both GW radiation and MD radiation, and the spin-down evolution can then be written as (Shapiro & Teukolsky, 1983; Zhang & Mészáros, 2001)

where $I$ is the moment of inertia of a neutron star, $\Omega$ and $\dot{\Omega}$ are the spin velocity and its time derivative. The electromagnetic luminosity $L_{\rm EM}$ generated by the magnetar spin-down is given by Spitkovsky (2006))

Here $B$ is the magnetic dipole field strength on the neutron star surface, $\theta$ is the angle between the magnetic axis and the spin axis, and $R$ and $c$ are the radius of the NS and the speed of light, respectively. In this paper, we adopt the typical parameters for a neutron star with a mass of $M=1.4\,M_{\sun}$, a radius of $R=10$ km, and the moment of $I=1\times 10^{45}{\rm gcm^{2}}$, and also assume that the neutron star has evolved to an orthogonal rotator ($\theta=\pi/2$).

Various causes for the NS deformation have been proposed in the previous literature. First, super-strong magnetic fields exist inside the NS due to differential rotation and magnetic dissipation instability. The anisotropic pressure generated by strong magnetic fields then deforms the NS into an ellipsoid (Bonazzola & Gourgoulhon, 1996; Cutler, 2002; Stella et al., 2005). It has been argued that magnetic-induced deformation plays an important role in the early evolution of magnetars (Corsi & Mészáros, 2009; Dall’Osso et al., 2015; Lasky et al., 2017; de Araujo et al., 2016). Second, the NS accretion can also lead to the deformation (Haskell et al., 2015; Zhong et al., 2019; Sur & Haskell, 2021). The accreted material flows towards the poles of the NS, compresses and obscures the local magnetic field, and thus forms “mountains” on the surface of the NSs. Third, the rapid spin of young NSs undergoes unstable oscillatory modes that couple to the gravitational field (Andersson, 1998; Lindblom et al., 1998; Alford & Schwenzer, 2015). The spinning magnetars with non-axisymmetric deformations possess quadrupole moments, which would emit significant GW as being (Usov, 1992; Zhang & Mészáros, 2001))

Here $G$ is the gravitational constant, and $\epsilon$ is the ellipticity of the star. Considering the Eq.1, one can find the spin velocity $\Omega$ evolving:

where $\alpha\equiv B^{2}R^{6}/6c^{3}I$, $\beta\equiv 32GI\varepsilon^{2}/5c^{5}$. The evolution of spin velocity $\Omega$ depends on the combination of MD radiation and GW radiation. Here are three possible kinds of spin-down evolution of magnetars: (1) If the magnetar spins down only due to MD radiation and the contribution of GW radiation is negligible, the electromagnetic luminosity evolves as $L_{\rm EM}=L_{\rm EM,0}(1+t/\tau_{\rm em})^{-2}$, where $\tau_{\rm em}=\frac{1}{2\alpha\Omega_{0}^{2}}$ is the timescale of MD radiation. The electromagnetic luminosity shows a plateau at $t<\tau_{\rm em}$, then decays as $L_{\rm EM}\propto t^{-2}$ for $t>\tau_{\rm em}$. (2) If the GW radiation dominates the spin-down, the electromagnetic luminosity can be expressed as $L_{\rm EM}=L_{\rm EM,0}(1+t/\tau_{\rm gw})^{-1}$, where $\tau_{\rm gw}=\frac{1}{2\beta\Omega_{0}^{4}}$ is the timescale of GW radiation. The electromagnetic luminosity also exhibits a plateau before decaying as $L_{\rm EM}\propto t^{-1}$ for $t>\tau_{\rm gw}$. (3) If both the GW radiation and MD radiation act together for the magnetar spin-down, the electromagnetic luminosity would show a change from $L_{\rm EM}\propto t^{-1}$ to $L_{\rm EM}\propto t^{-2}$ after the plateau phase. The radiation efficiency of GW and MD depends on the spin frequency (i.e., GW for $\Omega^{6}$ vs MD for $\Omega^{4}$), so the GW radiation should dominate the early evolution of the electromagnetic luminosity and the MD dominates the later evolution, causing the diversity of magnetar-driven SNe.

With the above basic ideas, we simulate how magnetars interact with SNe ejecta in different spin-down processes and how the different SNe light curves can be found. Our models are built based on the magnetar-driven SNe theory proposed by Kasen & Bildsten (2010)(see also Kashiyama et al. (2016); Ho (2016); Omand & Sarin (2024) and Kasen et al. (2016)).

A relativistic wind generated by the MD radiation from the magnetar goes outward and heats the SN ejecta. Approximating the ejecta at time $t$ in the form of a uniform sphere with a mass of $M_{\rm ej}$ and a radius of $R_{\rm ej}$, with a velocity of $v_{\rm sc}$ in the volume of $V\approx 4\pi R_{\rm ej}^{3}/3$ and the density of $\rho=M_{\rm ej}/V$, one can describe the evolution of the internal energy as being (Kasen et al., 2016)

The first term on the right is the electromagnetic luminosity injected by the magnetar due to spin-down. Here $\xi$ is the efficiency of thermalization of electromagnetic energy into radiation. The second term is the thermal radiation luminosity of SNe. The third term is the energy loss due to the expansion of the ejecta, $P$ is the radiation pressure that is related to the internal energy by $P=E_{int}/3V$, ${\partial V}/{\partial t}=4\pi R_{\rm ej}^{2}v_{\rm sc}$, and then $-P({\partial V}/{\partial t})=E_{\rm int}/t$. The $\xi$ evolves with time and is related to the gamma-ray leakage of the ejecta. Hence, we denote $\xi$ as

where

is parameter of gamma-ray leakage (Wang et al., 2015), $\kappa_{\gamma}$ represents the opacity of gamma-ray. The radiated luminosity of SN is given by (Kasen & Bildsten, 2010; Kotera et al., 2013)

where the optical depth of the ejecta is $\tau=3\kappa M_{\rm ej}/{4\pi R_{\rm ej}^{2}}$, and the radiative diffusion timescale is

here $\kappa$ is the opacity of ejecta. The expansion velocity of the ejecta is given by (Kasen & Bildsten, 2010)

where $E_{sn}$ is the initial kinetic energy of ejecta, $E_{\rm EM}$ is the energy of MD radiation integrated $L_{\rm EM}$ over time.

Kasen & Bildsten (2010) and Woosley (2010) were the first to develop a simplified model of magnetar-driven SNe and showed that the magnetar models can reproduce the timescales and brightness of SNe within reasonable parameters. Modeling the light curves of magnetar-driven SNe has to choose the magnetar spin evolution channel and get the solution of the internal energy evolution and the velocity of ejecta. Fig 1 demonstrates the magnetar-driven SNe light curves for three different spin-down modes, the magnetar MD radiation spin-down (blue line), the GW radiation spin-down (green line), and both (orange line), calculated for the initial parameters such as the magnetic field strength of $B=1\times 10^{14}\mbox{ G}$, the initial spin period of $P_{0}=1\mbox{ ms}$, the ellipticity of $\epsilon=2\times 10^{-3}$, the opacity of $\kappa$ = 0.1 cm² g^-1, and the gamma-ray opacity of ${\kappa_{\gamma}}$ = 0.1 cm² g^-1, the initial explosion energy of $E_{\rm sn}=10^{51}\mbox{ erg}$, and the ejecta mass of $1\,M_{\sun}$ (solid lines) and $3\,M_{\sun}$ (dashed lines).

In magnetar-driven SNe scenarios, the light curve evolution after its peak luminosity can be described as $L_{rad}\propto t^{-\alpha}$. The value of $\alpha$ depends on the declining index of the electromagnetic luminosity injected by the magnetar. Often $\alpha=2$ is taken for a magnetar to spin down due to only the MD radiation, and $\alpha=1$ for the GW radiation spin-down, or some values in between for the combined cases.

The evolution of the bolometric luminosity of the SN differs due to the ellipticity of a deformed NS, as shown in Fig 2. A larger ellipticity would cause a lower peak luminosity of the SN. The greater the loss of the GW energy, the smaller the amount of electromagnetic energy provided to the SN for a smaller luminosity peak. In other words, the low-luminosity SNe is more likely to have a previously unknown GW radiation. When the MD radiation is involved for NSs with strong magnetic fields, only a large ellipticity can cause a significant decrease of the bolometric luminosity peak. The GW energy loss causes a dwarf peak of the SNe light curve and a slower decay rate of the SNe light curve tail. Note, however, that the observed $t^{-2}$ tail in some SN light curves cannot be directly considered as a signature of the presence of a magnetar engine, since the ⁵⁶Co radioactive decay could exhibit a similar tail (Inserra et al., 2013).

We construct a magnetar model containing three spin-down modes and obtain the best-fitting parameters and the posterior parameter distributions by employing the emcee python package and the MCMC method (Foreman-Mackey et al., 2013). The model contains the following seven free parameters: dipole magnetic field strength $B$, initial spin period $P_{0}$, NS ellipticity $\epsilon$, ejecta mass $M_{\rm ej}$, initial SN explosion energy $E_{\rm sn}$, opacity $\kappa$ and gamma-ray opacity $\kappa_{\gamma}$. The prior distribution of the parameters is set in the form of log-uniform or uniform distributions to ensure the parameters are in sufficiently large intervals, as seen in Table 1. The values of ${\rm\chi^{2}}/{\rm dof}$ are adjusted for the best-fitting model of the SN light curves.

Three criteria are adopted for the choice of observation data. First, there must be sufficient observational data during the rise and decay of the light curve, and the observational data needs to extend beyond $100$ days. Second, the later phase of the light curve exhibits a slow decay tail with a decay rate between $-1$ to $-2$. This feature is consistent with what we expected from magnetar models, where enlarged decay rates for more rotational energy loss due to the GW radiation. Finally, SNe with the complex evolved behaviors in the decay phase are excluded from our sample. We here work on two cases, SN 2009bb and SN 2007ru.

The current aLIGO and Virgo detectors have discovered more than 90 cases of compact star mergers with high confidence, which have opened a new era of GW astronomy. In addition to the merger of NSs and/or black holes, the continuous GWs from isolated deformed NSs are also important targets for aLIGO and Virgo detectors. Post-collapse magnetars are considered to be the most promising class of NSs for detecting continuous GWs due to their extremely strong magnetic fields and fast spinning. If the magnetars are the central engines of SNe, they would emit observable GWs due to non-axisymmetric deformation.

The GW strain from spinning magnetar with a mass quadrupole moment is (Corsi & Mészáros, 2009; Howell et al., 2011; Dall’Osso et al., 2015)

Here $d$ is the distance from the source. For long-lived GW transient sources, the corresponding optimal matched filter signal-to-noise ratio is (Corsi & Mészáros, 2009)

where $h_{c}=fh(t)\sqrt{dt/df}$ is the GW amplitude, $h_{rms}=\sqrt{fS_{h}(f)}$ is the noise curve of detector, $S_{h}(f)$ is the power spectral density of the detector noise. Using Eq 12 to solve for $h_{c}$, one can obtain the GW amplitude of the magnetar as being (Corsi & Mészáros, 2009)

where $f=\Omega/\pi$ is the frequency of the GW signal.

According to Eq 13 we can see that the $h_{c}$ is not only related to the evolution of the stellar spin frequency but also depends on the distance to a source. By fitting the SNe luminosity curve with the magnetar model, we can derive the evolution of the spin frequency and hence the $h_{c}$. The left side of Fig 4 shows the frequency evolution of the predicted GW for the SN 2007ru and SN 2009bb. From Eq 4 and Eq 13 it can be seen that the fractional uncertainty in the magnetar parameters will be directly converted to the uncertainty on the GW frequency and $h_{c}$. The corner plots show that the ellipticity has a large uncertainty, which increases the uncertainties of the inferred GW frequency and $h_{c}$. We plot the $h_{c}$ curves of two SNe on the right side of Fig 4, together with the sensitivity curves of aLIGO and Einstein detector (ET) (Abbott et al., 2018). The uncertainties of the magnetar parameters have a small effect on the evolution of the $h_{c}$ curve. One can find that the GWs from SN 2009bb and SN 2007ru can reach the detection threshold of the aLIGO and ET detectors. On the other hand, for magnetars not yet detected by ground-based detectors for GWs, we can give an upper limit on the $\epsilon$ by comparing the sensitivity of GW detectors and $h_{c}$. Note that our estimate of the detectability of the GW signal is based on the assumption of an optimally matched filter, which may not be feasible for the detection of the proposed signals. The $h_{c}$ curves in Fig 4 represent the most optimistic upper limit for GW detection.

A rapidly spinning magnetar can be formed after the SN explosion, emitting continuous GWs due to the magnetic-induced deformation or fluid oscillation excited by fast spinning. We explore the GW radiation in the magnetar-driven SNe model and try to find evidence of GW radiation in the SN light curves. Both MD radiation and GW radiation can act for the spin-down of magnetars. The GW energy loss can lead to a reduced MD radiative energy to heat the SN, causing a reduced peak luminosity of the SN and a fast slowing down, and also a low decay rate of light curve tails. We made magnetar models with GW emission and reproduced the light curves of SN 2009bb and SN 2007ru. We derived the parameters of deformed magnetars by fitting the observed light curves of SNe, and also estimated the detectability of the GWs from deformed magnetars. Our results suggest that the GWs from SN 2009bb and SN 2007ru can reach the detection threshold of the aLIGO and ET detectors. If GWs associated with SNe are detected in the future, this would be a smoking gun that magnetars can power SNe.

The effect of long-lived magnetar remnants on SN has been explored in detail (e.g. Inserra et al., 2013; Kasen et al., 2016). The rotational energy reservoir of magnetars has been used to explain many different features of SN, such as the high peak luminosity, the large ejecta kinetic energy, and the tail decaying with $t^{-2}$. However, in some cases, SNe with such a tail may be attributed to the magnetar plus ⁵⁶Ni model. The magnetar model explains the early fast rise and peak luminosity, while the ⁵⁶Ni model explains the late tails, depending on if sufficiently massive Ni is synthesized. It may be difficult to synthesize more than $0.2M_{\sun}$ of ⁵⁶Ni (Nishimura et al., 2015; Suwa & Tominaga, 2015) in the SN with a magnetar. Our simulations suggest that a magnetar model plus ⁵⁶Ni model does have some difficulty in explaining the overly flat decay of the tail (slope much less than $-2$), especially for decays longer than 300 days because the decay rate of ⁵⁶Co gradually deviates over time.

By modeling the SN bolometric luminosity curve, we derive the ellipticity of the magnetar to be $\epsilon\sim 10^{-3}$. The current GW detection has set an upper limit of the ellipticity of $\sim 10^{-4}-10^{-6}$ for pulsar (Abbott et al., 2019a), and $\sim 10^{-3}$ for young NSs (Aasi et al., 2015b, 2016); while the electromagnetic observations from the GRB X-ray plateau constrain the ellipticity of magnetars to $\sim 10^{-3}-10^{-4}$ (Lasky & Glampedakis, 2016; Xie et al., 2022a). Note, however, that new-born magnetars may have greater non-axisymmetric deformation than old pulsars, and thus may produce stronger GW emission.

The ellipticity of a magnetar inferred from two SNe are consistent with those inferred from the GRB observations, implying that both GRB and some SNe Ic-BL may have similar central engines. The deformation of NSs may be induced by anisotropic stresses caused by the internal magnetic field (Cutler, 2002; Mastrano et al., 2011; Lander, 2014). If NS distortion is induced by the pure toroidal magnetic field ($B_{\rm t}$), the relation between the $\epsilon$ and $B_{\rm t}$ can be described as $\epsilon\approx 1.6\times 10^{-4}({B_{\rm t}}/{10^{16}\,{\rm G}})^{2}$ (Cutler, 2002; Stella et al., 2005; Dall’Osso et al., 2009). It can be seen that the $B_{t}$ ought to reach $\sim 2\times 10^{16}\mbox{ G}$ for the $\epsilon$ of $\sim 10^{-3}$, which implies that a stronger toroidal field strength is required, at least $1-2$ orders of magnitude larger than the dipole field ($B\sim 10^{14}-10^{15}\mbox{ G}$). Theoretically, the toroidal magnetic field of magnetar can be amplified to $\sim 10^{16}-10^{17}\mbox{ G}$ due to the $\alpha-\Omega$ dynamo mechanism (Duncan & Thompson, 1992). This conclusion is supported by previous works on constraining the internal field strength of soft gamma-ray repeating bursts (see, (Ioka, 2001; Corsi & Owen, 2011)). Other channels than magnetic-induced NS deformation, such as secular bar-mode instability (also known as unstable f-mode oscillation), can also play an important role in nascent magnetars. Lasky & Glampedakis (2016) proposed that the value of the maximum ellipticity induced by f-mode depends on the saturation energy of the mode. The maximum energy of the f-mode is $E_{\rm mode}\sim 10^{-6}Mc^{2}$, which corresponds to the effective ellipticity of $\sim 2\times 10^{-3}$. In addition, NS accretion can also induce distortion. The material around the accretion disc will flow into the poles of the NS due to the gravitational effect to form mountains. In this scenario, the magnitude of the ellipticity value of the NS depends on the mass of the accreting material (Zhong et al., 2019). Sur & Haskell (2021) suggested that for accreting NSs located at $1\mbox{ Mpc}$, the GW amplitudes at kHz frequencies can reach the detection threshold of aLIGO.

Even though aLIGO has not detected continuous GWs from isolated NSs, we are optimistic about the prospect of detecting GWs from SNe. If continuous GWs from the NSw can be detected by future GW detectors, then we can constrain the ellipticity $\epsilon$ of the NS by Eq 12. Furthermore, the duration and frequency of continuous GWs can be obtained from their waveforms. According to Eq 2, 3 and 13, the mass-radius relation can be constrained and then combined with the inferred ellipticity $\epsilon$ to provide valuable information for constraining the NS deformation after the SN explosion.

We thank the anonymous referee for helpful comments to improve this paper. The authors are supported by the National Natural Science Foundation of China (Nos. 11988101, 11833009, 12073080, 12233011, 11933010, 11921003 and 12303050) and by the Chinese Academy of Sciences via the Key Research Program of Frontier Sciences (No. QYZDJ-SSW-SYS024), China Postdoctoral Science Foundation (grant No. 2023M743397), and the Fundamental Research Funds for the Central Universities.