The electromagnetic and gravitational-wave radiations of X-ray transient CDF-S XT2

The X-ray data of CDF-S XT2 observed by Chandra within energy band 0.3–10 keV are taken from Xue et al. (2019). We perform a temporal fit to the light curve with a smooth broken power law model, which is expressed as

$$\begin{eqnarray}&&L={L}_{0}{\left[{\left(\displaystyle \frac{t}{{t}_{b}}\right)}^{\omega {\alpha }_{1}}+{\left(\displaystyle \frac{t}{{t}_{b}}\right)}^{\omega {\alpha }_{2}}\right]}^{-1/\omega },\end{eqnarray} \tag{ 1 }$$

where t_b (2525 ± 242) s is the break time, L_b = L₀ ⋅ 2^−1/ω = (1.28 ± 0.16) × 10⁴⁵ erg s⁻¹ is the luminosity at the break time t_b , α₁ = (0.09 ± 0.11) and α₂ = (2.43 ± 0.19) are decay indices before and after the break, respectively. The ω describes the sharpness of the break. The larger the ω parameter, the sharper the break, and ω = 3 is fixed for the light curve fitting. An IDL routine named "mpfitfun.pro" is employed for our fitting (Markwardt 2009). This routine performs a Levenberg-Marquardt least-squares fit to the data for a given model to optimize the model parameters. The light curve fit is shown in Figure 1.

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X-ray transient CDF-S XT2 associated with a galaxy at redshift z = 0.738 lies in the outskirts of its star-forming host galaxy with a moderate offset from the galaxy center, and there is no significant source-like gamma-ray emission signal above background. The observed properties are similar to those of other typical short GRBs, but in off-axis observed (Xue et al. 2019). On the other hand, the estimated event rate density of this event is similar to double NS merger rate density inferred from the detection of GW170817, suggesting that the progenitor of this event is likely from double NS merger (Xue et al. 2019). Moreover, the observed X-ray plateau of CDF-S XT2 is consistent with wind dissipation of magnetar central engine, and it indicates that the remnants of such double NS mergers should be either supra-massive NS or stable NS. However, the decay slope after the plateau emission (t^−2.43) is slightly larger than the theoretical value of spin-down (t⁻²) in EM dominated by losing its rotation energy. Here, we assume that the feature of X-ray emission is caused by a supra-massive magnetar central engine for surviving thousands of seconds to collapse into a black hole.

In order to compare the properties of CDF-S XT2 with other short GRBs with an internal plateau, Figure 2 shows the correlation between break luminosity (L_b ) and collapse time (τ_col = t_b/(1 + z)), as well as the distributions of L_b and τ_col. We find that the CDF-S XT2 falls into the 2σ deviation in L_b – τ_col diagram, suggesting that the other short GRBs with internal plateau samples shared a similar central engine with the CDF-S XT2. However, the distributions of luminosity and collapse time of the CDF-S XT2 is much lower and longer than other short GRBs with internal plateau samples, respectively. It may be caused by the directions of observations (i.e., on and off-axis with short GRBs and the CDF-S XT2), or having different populations of magnetars.

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If we believe that a supramassive NS is a potential candidate central engine of CDF-S XT2, then one interesting question is: what is the energy loss channel of the rotating magnetar, dominated by magnetic dipole or GW radiation? We will discuss more details for the rotation energy loss of magnetar dominated by EM or GW radiations.

The energy reservoir of a millisecond magnetar is the total rotation energy, which reads as

$$\begin{eqnarray}&&{E}_{{\rm{rot}}}=\displaystyle \frac{1}{2}I{\Omega }^{2}\simeq 2\times {10}^{52}\,{\rm{erg}}\,{M}_{1.4}{R}_{6}^{2}{P}_{-3}^{-2},\end{eqnarray} \tag{ 2 }$$

where I is the moment of inertia, Ω, P, R, and M are the angular frequency, rotating period, radius, and mass of the neutron star, respectively. The convention Q = 10^x Q_x in cgs units is adopted. A magnetar spinning down loses its rotational energy via both magnetic dipole torques (L_EM) and GW (L_GW) radiations (Zhang & Mészáros 2001; Fan et al. 2013; Giacomazzo & Perna 2013; Lasky & Glampedakis 2016; Lü et al. 2018),

$$\begin{eqnarray}\begin{array}{ll}-\displaystyle \frac{d{E}_{{\rm{rot}}}}{dt} & =-I\Omega \dot{\Omega }={L}_{{\rm{total}}}={L}_{{\rm{EM}}}+{L}_{{\rm{GW}}}\\ & =\displaystyle \frac{{B}_{{\rm{p}}}^{2}{R}^{6}{\Omega }^{4}}{6{c}^{3}}+\displaystyle \frac{32G{I}^{2}{\epsilon }^{2}{\Omega }^{6}}{5{c}^{5}},\end{array}\end{eqnarray} \tag{ 3 }$$

where B_p is the surface magnetic field at the pole and = 2(I_xx – I_yy )/(I_xx + I_yy ) is the ellipticity describing how large of the neutron star deformation. $\dot{\Omega }$ is the time derivative of the angular frequency. One can find that for a magnetar with given R and I, its L_EM depends on B_p and Ω and L_GW depends on and Ω.

If the rotation energy loss of magnetar is dominated by EM emission, one has

$$\begin{eqnarray}&&{L}_{{\rm{EM}}}\simeq -I\Omega \dot{\Omega }=\displaystyle \frac{\eta {B}_{{\rm{p}}}^{2}{R}^{6}{\Omega }^{4}}{6{c}^{3}},\end{eqnarray} \tag{ 4 }$$

where η is the efficiency of converting the magnetar wind energy into X-ray radiation. The characteristic spin-down luminosity (L_EM,sd) and time scale (τ_EM,sd) of magnetar can be given as

$$\begin{eqnarray}\begin{array}{ll}{L}_{{\rm{EM}},{\rm{sd}}} & =\displaystyle \frac{\eta {B}_{p}^{2}{R}^{6}{\Omega }_{0}^{4}}{6{c}^{3}}\\ & \simeq 1.0\times {10}^{46}\,{\rm{erg}}\,{{\rm{s}}}^{-1}({\eta }_{-3}{B}_{p,15}^{2}{P}_{0,-3}^{-4}{R}_{6}^{6}),\end{array}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}\begin{array}{ll}{\tau }_{{\rm{EM}},{\rm{sd}}} & =\displaystyle \frac{3{c}^{3}I}{{B}_{p}^{2}{R}^{6}{\Omega }_{0}^{2}}\\ & \simeq 2.05\times {10}^{3}\,{\rm{s}}\,({I}_{45}{B}_{p,15}^{-2}{P}_{0,-3}^{2}{R}_{6}^{-6}),\end{array}\end{eqnarray} \tag{ 6 }$$

where Ω₀ and P₀ are initial angular frequency and period at t = 0, respectively.

Within the magnetar central engine scenario, the observed plateau luminosity is closed to L_b, which is roughly equal to L_EM,sd, and τ_EM,sd > τ_col. One can derive the magnetar parameters B_p and P₀,

$$\begin{eqnarray}&&{B}_{{\rm{p}},15}=2.05({\eta }_{-3}^{1/2}{I}_{45}{R}_{6}^{-3}{L}_{{\rm{EM}},{\rm{sd}},46}^{-1/2}{\tau }_{{\rm{EM}},{\rm{sd}},3}^{-1})\,{\rm{G}},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{P}_{0,-3}=1.42({\eta }_{-3}^{1/2}{I}_{45}^{1/2}{L}_{{\rm{EM}},{\rm{sd}},46}^{-1/2}{\tau }_{{\rm{EM}},{\rm{sd}},3}^{-1/2})\,{\rm{s}}.\end{eqnarray} \tag{ 8 }$$

As radiation efficiency η depends strongly on the injected luminosity and wind saturation Lorentz factor (Xiao & Dai 2019). By adopting the lower limit of τ_EM,sd, we derive the upper limits of P₀ and B_p with different η values. One has P₀ < 3.4 × 10⁻³ s and B_p < 4 × 10¹⁵ G for η = 0.001, P₀ < 10.6 × 10⁻³ s and B_p < 1.2 × 10¹⁶ G for η = 0.01, and P₀ < 33.8 × 10⁻³ s and B_p < 4 × 10¹⁶ G for η = 0.1. Figure 3 shows the B_p – P₀ diagram for X-ray transient CDF-S XT2 with different η values, and compares with other short GRBs with an internal plateau sample taken from Lü et al. (2015). It seems that small P₀ required by supra-massive magnetar is needed for lower radiation efficiency, and estimated B_p of CDF-S XT2 is lower than other typical short GRB samples for smaller P₀. It may be either off-axis observations or a different population of CDF-S XT2 by comparing with short GRBs.

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The inferred collapsing time can be used to constrain the neutron star EOS (Lasky et al. 2014; Ravi & Lasky 2014; Lü et al. 2015). The basic formalism is as follows.

The standard dipole spin-down formula gives (Shapiro & Teukolsky 1983)

$$\begin{eqnarray}\begin{array}{lll}P(t) & = & {P}_{0}{(1+\displaystyle \frac{4{\pi }^{2}}{3{c}^{3}}\displaystyle \frac{{B}_{p}^{2}{R}^{6}}{I{P}_{0}^{2}}t)}^{1/2}\\ & = & {P}_{0}{(1+\displaystyle \frac{t}{{\tau }_{{\rm{EM}},{\rm{sd}}}})}^{1/2}.\end{array}\end{eqnarray} \tag{ 9 }$$

The maximum NS mass for a non-rotating NS (M_TOV) can be derived for given EOS of NS. The maximum gravitational mass (M_max) depends on spin period, read as (Lyford et al. 2003)

$$\begin{eqnarray}&&{M}_{{\rm{\max }}}={M}_{{\rm{TOV}}}(1+\hat{\alpha }{P}^{\hat{\beta }}),\end{eqnarray} \tag{ 10 }$$

where $\hat{\alpha }$, $\hat{\beta }$, and M_TOV depend on the EOS of NS.

As the neutron star spins down, the centrifugal force can no longer sustain the star, and the NS will collapse into a black hole. By using Equations (9) and (10), one can derive the collapse time as function of M_p,

$$\begin{eqnarray}\begin{array}{ll}{t}_{{\rm{col}}} & =\displaystyle \frac{3{c}^{3}I}{4{\pi }^{2}{B}_{{\rm{p}}}^{2}{R}^{6}}[{(\displaystyle \frac{{M}_{{\rm{p}}}-{M}_{{\rm{TOV}}}}{\hat{\alpha }{M}_{{\rm{TOV}}}})}^{2/\hat{\beta }}-{P}_{0}^{2}]\\ & =\displaystyle \frac{{\tau }_{{\rm{EM}},{\rm{sd}}}}{{P}_{0}^{2}}[{(\displaystyle \frac{{M}_{{\rm{p}}}-{M}_{{\rm{TOV}}}}{\hat{\alpha }{M}_{{\rm{TOV}}}})}^{2/\hat{\beta }}-{P}_{0}^{2}].\end{array}\end{eqnarray} \tag{ 11 }$$

Here, we consider 12 EOSs that are reported in the literature (Lasky et al. 2014; Ravi & Lasky 2014; Li et al. 2016; Ai et al. 2018). The basic parameters of those EOSs are shown in Table 1.

Table 1. The Basic Parameters of EOS of NS

	MTOV	R	I	$\hat{\alpha }$	$\hat{\beta }$		hc(f)
	(M⊙)	(km)	(1045 g cm2)	$({10}^{-10}\,{s}^{-\hat{\beta }})$		(10−3)	(10−25)
BCPM	1.98	9.94	2.86	3.39	–2.65	1.5	3.02
SLy	2.05	9.99	1.91	1.60	–2.75	1.8	2.47
BSk20	2.17	10.17	3.50	3.39	–2.68	1.3	3.34
Shen	2.18	12.40	4.68	4.69	–2.74	1.2	3.87
APR	2.20	10.0	2.13	0.303	–2.95	1.7	2.61
BSk21	2.28	11.08	4.37	2.81	–2.75	1.2	3.74
GM1	2.37	12.05	3.33	1.58	–2.84	1.4	3.26
DD2	2.42	11.89	5.43	1.37	–2.88	1.1	4.16
DDME2	2.48	12.09	5.85	1.966	–2.84	1.0	4.32
AB-N	2.67	12.90	4.30	0.112	–3.22	1.2	3.71
AB-L	2.71	13.70	4.70	2.92	–2.82	1.2	3.87
NL3ωρ	2.75	12.99	7.89	1.706	–2.88	0.89	5.02

As noted, one can infer B_p , P₀, and t_col from the observations by adopting η = 0.001. Following the method of Lasky et al. (2014), a tight mass distribution of the our Galactic binary NS population is adopted (e.g., Valentim et al. 2011; Kiziltan et al. 2013), and one can infer the expected distribution of proto-magnetar masses, which is found to be ${M}_{p}={2.46}_{-0.15}^{0.13}\,{M}_{\odot }$. For X-ray transient CDF-S XT2, the lower limit of τ_{EM, sd} = t_col is derived. Figure 4 presents the collapse time (t_col) as a function of protomagnetar mass (M_p ) for CDF-S XT2 with different EOS. Our results show that the GM1, DD2, and DDME2 models give an M_p band falling within the 2σ region of the protomagnetar mass distribution, so that the correct EOS should be close to those three models. The maximum mass for non-rotating NS in those three models are M_TOV = 2.37 M_⊙, 2.42 M_⊙, and 2.48 M_⊙, respectively.

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Within GW dominated scenario, one has (Lü et al. 2018)

$$\begin{eqnarray}&&{L}_{{\rm{GW}}}\simeq -I\Omega \dot{\Omega }=\displaystyle \frac{32G{I}^{2}{\epsilon }^{2}{\Omega }^{6}}{5{c}^{5}}.\end{eqnarray} \tag{ 12 }$$

The characteristic spin-down luminosity (L_GW,sd) and time scale (τ_GW,sd) of NS can be given as

$$\begin{eqnarray}\begin{array}{ll}{L}_{{\rm{GW}},{\rm{sd}}} & =\displaystyle \frac{32G{I}^{2}{\epsilon }^{2}{\Omega }_{0}^{6}}{5{c}^{5}}\\ & \simeq 1.08\times {10}^{48}\,{\rm{erg}}\,{{\rm{s}}}^{-1}({I}_{45}^{2}{\epsilon }_{-3}^{2}{P}_{0,-3}^{-6}),\end{array}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}\begin{array}{ll}{\tau }_{{\rm{GW}},{\rm{sd}}} & =\displaystyle \frac{5{c}^{5}}{128GI{\epsilon }^{2}{\Omega }_{0}^{4}}\\ & \simeq 9.1\times {10}^{3}\,{\rm{s}}\,({I}_{45}^{-1}{\epsilon }_{-3}^{-2}{P}_{0,-3}^{4}).\end{array}\end{eqnarray} \tag{ 14 }$$

The supra-massive NS of CDF-S XT2 has collapsed into the black hole before it is spin-down, so that one has τ_GW,sd > τ_col. Combining with Equation (14), the upper limit of ellipticity () can be expressed as

$$\begin{eqnarray}&&\epsilon \lt 2.5\times {10}^{-3}{I}_{45}^{-1/2}{P}_{0,-3}^{2}.\end{eqnarray} \tag{ 15 }$$

The maximum value of for different EOS with P₀ = 1 ms are shown in Table 1. We find that those values are in the range of [0.32 – 1.3] × 10⁻³. This upper limit value is larger than the maximum elastic quadrupole deformation of conventional neutron stars, but is comparable to the upper limit derived for crystalline color-superconducting quark matter (Lin 2007; Johnson-McDaniel & Owen 2013).

If most of the rotation energy is released via GW radiation with a frequency f, the GW strain for a rotating neutron star at distance D_L can be expressed as

$$\begin{eqnarray}&&h(t)=\displaystyle \frac{4GI\epsilon }{{D}_{{\rm{L}}}{c}^{4}}\Omega {(t)}^{2}.\end{eqnarray} \tag{ 16 }$$

The signal-to-noise ratio of optimal matched filter can be expressed as

$$\begin{eqnarray}&&{\rho }^{2}=\displaystyle {\int }_{{f}_{1}}^{{f}_{2}}\displaystyle \frac{{\mathop{f}\limits^{\sim }}^{2}(f)}{{S}_{h}(f)}df,\end{eqnarray} \tag{ 17 }$$

where f₁ and f₂ are the initial and final GW frequencies, respectively. $\mathop{h}\limits^{\sim }(f)$ is the Fourier transform of h(t), namely $\mathop{h}\limits^{\sim }(f)=h(t)\sqrt{dt/df}$. S_h (f) is the noise power spectral density of the detector (Lasky & Glampedakis 2016). The characteristic amplitude of GW from a rotating NS can be estimated as (Corsi & Mészáros 2009; Hild et al. 2011; Lasky & Glampedakis 2016; Lü et al. 2017)

$$\begin{eqnarray}\begin{array}{ll}{h}_{{\rm{c}}} & =fh(t)\sqrt{\displaystyle \frac{dt}{df}}=\displaystyle \frac{f}{{D}_{{\rm{L}}}}\sqrt{\displaystyle \frac{5GI}{2{c}^{3}f}}\\ & \approx 8.22\times {10}^{-24}{\left(\displaystyle \frac{I}{{10}^{45}{{\rm{gcm}}}^{2}}\displaystyle \frac{f}{1{\rm{kHz}}}\right)}^{1/2}\\ & \,\times {\left(\displaystyle \frac{{D}_{{\rm{L}}}}{100{\rm{Mpc}}}\right)}^{-1}\,.\end{array}\end{eqnarray} \tag{ 18 }$$

For X-ray transient CDF-S XT2, its redshift z = 0.738 corresponds to D_L ∼ 4480 Mpc. By adopting the frequency range of GW from f = 120 Hz to 1000 Hz, one can estimate the maximum value of the strain h_c for different EOS of NS. The estimated values of h_c are reported in Table 1. The maximum value of the strain h_c for NL3ωρ is about 5 × 10⁻²⁵, which is about one order of magnitude smaller than the aLIGO sensitivity, and also less than more sensitive Einstein Telescope (ET; see Fig. 5). It means that even if the merger remnant of double NS of this transient is a millisecond massive NS, the post merger GW signal is undetectable when the rotation energy of the NS is taken away by the GW radiation.

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