RADIO TRANSIENTS FROM THE ACCRETION-INDUCED COLLAPSE OF WHITE DWARFS

As an accreting white dwarf (WD) grows toward the Chandrasekhar limit, a well-known potential outcome is ignition of its nuclear fuel, leading to a Type Ia supernova (SN Ia; Hillebrandt & Niemeyer 2000). However, in some cases (e.g., mass transfer onto O/Ne/Mg WDs and C/O WD mergers; Canal & Schatzman 1976; Nomoto & Kondo 1991) electron capture can rob the core of its degeneracy pressure support leading to formation of a neutron star (NS). This "accretion-induced collapse" (AIC) has been invoked to explain millisecond pulsars (e.g., Bhattacharya & van den Heuvel 1991), subsets of gamma-ray bursts (e.g., Dar et al. 1992; Metzger et al. 2008b), magnetars (e.g., Usov 1992), and may be a source of r-process nucleosynthesis (Hartmann et al. 1985; Fryer et al. 1999).

Despite its potential importance, there has been no reported detection of an AIC event. To start with, the expected AIC rate is no more than ≈1% of that of SNe Ia (Yungelson & Livio 1998). Next, relative to Type I and Type II SNe, the ejecta mass is expected to be small (≲ 10⁻¹ M_☉), produce little ⁵⁶Ni (≲ 10⁻² M_☉), and move at high velocity (≈0.1c). The resulting optical transient is thus considerably fainter than a typical SN (5 mag or more) and lasts ∼1 day (Metzger et al. 2009b; Darbha et al. 2010).

Rapid rotation should accompany AIC due to the accretion of mass and angular momentum. Furthermore, a strong magnetic field may be amplified through flux freezing during collapse and via dynamo action (Duncan & Thompson 1992). Therefore, a plausible outcome of AIC is creation of a quickly spinning magnetar (Usov 1992; King et al. 2001; Levan et al. 2006). The WD collapse unbinds material (Dessart et al. 2006), and the remnant disk loses mass via outflows driven by neutrino heating, turbulent viscosity, and recombination of free nuclei into helium (Lee & Ramirez-Ruiz 2007; Metzger et al. 2008a, 2009a; Lee et al. 2009). This leads to M_ej ≈ 10⁻³–10⁻¹ M_☉ of ejecta with velocity v_ej ≈ 0.1c. Such a configuration will also follow a double NS merger if the total binary mass is below the maximum NS mass.

Here we investigate the detectability of the pulsar wind nebula (PWN) powered by a newly formed magnetar following AIC. In Section 2 we describe a model for how the PWN expands into the surrounding ejecta. In Section 3 we estimate the radio spectrum and light curve, showing that the PWN is a transient radio source for a few months. We estimate the detection rate in Section 4 and discuss the detectability of such events with soon-to-be-commissioned high-speed radio mapping machines in Section 5.

Following the implosion of the WD and disk outflows, the ejecta expands with velocity v_ej ≈ 0.1c and kinetic energy E_ej ≈ M_ejv²_ej/2 ≈ 10⁵⁰ erg. The ejecta plows into the interstellar medium (ISM) with particle density n₀, which can vary greatly from ∼10⁻⁶ cm⁻³ for events outside of their host galaxy to ∼1 cm⁻³ in denser environments. At early times, the ejecta maintains constant velocity with radius R_ej ≈ v_ejt. The forward shock is merely a distance ≈(4/3)^1/3R_ej ahead of R_ej, and the reverse shock has barely developed behind it. The pressure behind the forward shock and down to the reverse shock is roughly constant and given by the strong shock limit ≈(4/3)m_pn₀v²_ej.

The magnetar injects energy in the form of magnetic fields and relativistic particles at a rate

$$\begin{eqnarray} &&L(t) = L_0/(1+t/\tau)^p, \end{eqnarray} \tag{ 1 }$$

which powers a PWN with radius R_p. We assume dipole spin-down¹ and thus p = 2. For magnetic moment μ, initial spin frequency Ω₀ = 2π/P₀, and moment of inertia I = 0.35M_*R²_* (Lattimer & Prakash 2001),

$$\begin{eqnarray} &&L_0 = \mu ^2\Omega _0^4/6c^3 = 1.2\times 10^{47}\mu _{33}^2P_3^{-4}\,{\rm erg\, s^{-1}}, \end{eqnarray} \tag{ 2 }$$

and

$$\begin{eqnarray} &&\tau = 6Ic^3/\mu ^2\Omega _0^2 = 4.8\times 10^4\mu _{33}^{-2}P_3^2\,{\rm s}, \end{eqnarray} \tag{ 3 }$$

where μ₃₃ = μ/10³³ G cm³, P₃ = P₀/3 ms, and we use M_* = 1.4 M_☉ and R_* = 12 km.

The spin-down energy E_spin ≈ L₀τ ≈ 10⁵² erg exceeds E_ej, so quickly E_spin dominates in setting the ejecta speed. Once the ejecta has swept up a mass comparable to its own, it would decelerate as it enters the Sedov–Taylor phase on a timescale (McKee & Truelove 1995):

$$\begin{eqnarray} &&\tau _{\rm ST} = 0.5E_{\rm spin}^{-1/2}M_{\rm ej}^{5/6}(m_pn_0)^{-1/3}\approx 2 E_{52}^{-1/2}M_{-2}^{5/6} n_0^{-1/2} {\rm yr}, \end{eqnarray} \tag{ 4 }$$

where E₅₂ = E_spin/10⁵² erg and M₋₂ = M_ej/10⁻² M_☉. But since τ_ST is much longer than the times of interest for this work, the ejecta is always in an ejecta-dominated (or free-expansion) phase. Deeper inside, a wind termination shock forms at radius R_t, where the ram pressure of the pulsar wind equals the PWN pressure P (Gaensler & Slane 2006),

$$\begin{eqnarray} &&R_t \approx (L/4\pi c P)^{1/2}, \end{eqnarray} \tag{ 5 }$$

where R_t ≪ R_p. The general picture developed from galactic PWNe is that particles are accelerated at or near R_t. This seeds the PWN with relativistic electrons out to R_p.

A central issue is the content of the PWN. As a pulsar wind flows from the light cylinder, it is inferred to have a large magnetization (σ ∼ 10⁴, where σ is the ratio of Poynting flux to particle energy flux; Arons 2002). However, multiple lines of evidence, including the expansion velocities and high-energy modeling of PWNe, require that the magnetization decreases significantly (σ ≪ 1) by the time the wind reaches R_t. In the region where the radiation arises, R_t < r < R_p, we define η_e and η_B as the fraction of the magnetar luminosity that goes into electrons and magnetic fields, respectively, with typical values of η_e ≈ 0.999 and η_B ≈ 10⁻³.

Following Reynolds & Chevalier (1984), we assume the energy density of electrons and magnetic fields evolve independently and obey a relativistic equation of state. Thus,

$$\begin{eqnarray} &&\frac{d}{dt}\big(4\pi P_eR_p^4 \big) = (\eta _eL-\Lambda)R_p, \end{eqnarray} \tag{ 6 }$$

where Λ is the radiative loss rate, and

$$\begin{eqnarray} &&\frac{d}{dt}\big(4\pi P_BR_p^4 \big) = \eta _B LR_p, \end{eqnarray} \tag{ 7 }$$

where P_B = B²/8π. Note that in the adiabatic limit η_BL ≈ 0, this predicts P_B∝R⁻⁴_p and B∝R⁻²_p, as expected from flux freezing. Momentum conservation is given by

$$\begin{eqnarray} &&M_s\frac{d^2R_p}{dt^2} = 4\pi R_p^2 \left[P - \rho _{\rm ej}\left(v_p - \frac{R_p}{t} \right)^2 \right], \end{eqnarray} \tag{ 8 }$$

where v_p = dR_p/dt and P = P_e + P_B is the total pressure.

The PWN sweeps up ejecta and creates a shell with mass

$$\begin{eqnarray} M_s = \left\lbrace \begin{array}{@{}l l}M_{\rm ej}(R_p/v_{\rm ej}t)^3, &\ R_p<v_{\rm ej}t\\ M_{\rm ej}, &\ R_p\ge v_{\rm ej}t\\ \end{array}.\right. \end{eqnarray} \tag{ 9 }$$

Once R_p ≈ v_ejt, the PWN reaches material that has been shock-heated by its interaction with the ISM. Since we generally find P ≫ (4/3)m_pn₀v²_ej, we do not expect the shock-heated pressure to have a large dynamical effect on the PWN. This is in contrast to PWNe growing within an SN remnant, where the large pressure behind the reverse shock (on a timescale t ≫ τ_ST) can cause significant compression and magnetic field amplification (Reynolds & Chevalier 1984).

Equations (6), (7), and (8) with dR_p/dt = v_p provide four first-order differential equations for the four dependent variables P_e, P_B, R_p, and v_p, respectively. For simplicity, we calculate the PWN evolution using Λ ≈ 0, so that Equations (6) and (7) are combined using P = P_e + P_B and η_e + η_B = 1. When L is constant, the analytic result is (Chevalier 1977)

$$\begin{eqnarray} &&R_p \approx 1.3\times 10^{16} L_{47}^{1/5} E_{50}^{3/10} M_{-2}^{-1/2} t_6^{6/5}\,{\rm cm}, \end{eqnarray} \tag{ 10 }$$

and

$$\begin{eqnarray} &&B \approx 6\eta _{B,-3}^{1/5} L_{47}^{1/5} E_{50}^{-9/20} M_{-2}^{3/4} t_6^{-13/10}\,{\rm G}, \end{eqnarray} \tag{ 11 }$$

where L₄₇ = L/10⁴⁷ erg s⁻¹, t₆ = t/10⁶ s, and η_{B, −3} = η_B/10⁻³. This gives some idea of the rough values expected, although for our detailed calculations, R_p and B deviate slightly from these scalings when t ≳ τ.

Rayleigh–Taylor instabilities can act at R_p due to the low-density PWN pushing up against high-density ejecta. The growth time for a large density contrast is τ_RT ≈ (g_effk)^−1/2, where g_eff is the effective gravitational acceleration at the boundary and k the wavenumber. For R_p given by Equation (10), g_eff ≈ d²R_p/dt² ≈ (6/25)R_p/t² and the growth rate for k ≈ n/R_p is τ_RT ≈ (25/6n)^1/2t. For sufficiently small wavelengths (large n), τ_RT ≲ t and instability results. This is not surprising since similar systems, like the Crab Nebula, have morphologies strongly impacted by instabilities. But, for the current analysis we do not include this complication.

A power-law spectrum of relativistic electrons n(E) = KE^−s is accelerated near the termination shock at R_t and fills the PWN out to R_p, where n(E) is in units of electrons erg⁻¹ cm⁻³. For our fiducial models, we use s = 1.5, motivated by Galactic PWNe (Bucciantini et al. 2011). The power-law parameters within the PWN can change via cooling and injection of new electrons (e.g., Gelfand et al. 2009). For the present work, we estimate the synchrotron spectrum at any time by fixing P_e and P_B from our dynamical calculations.

Our discussion of synchrotron emission largely follows the work of Pacholczyk (1970). Synchrotron emission is self-absorbed below a frequency

$$\begin{eqnarray} &&\nu _{\rm SA} = 2c_1(R_pc_6)^{2/(s+4)}K^{2/(s+4)}(B\sin \theta)^{(s+2)/(s+4)}, \end{eqnarray} \tag{ 12 }$$

where c₁ = 6.27 × 10¹⁸ in cgs units, c₆ depends on s and can be found in Appendix 2 of Pacholczyk (1970), and θ is the pitch angle. Throughout we assume an average of sin θ = (2/3)^1/2. In the optically thick limit, the flux is

$$\begin{eqnarray} &&F_{\nu } = \frac{\pi R_p^2}{D^2}\frac{c_5}{c_6}(B\sin \theta)^{-1/2}\left(\frac{\nu }{2c_1} \right)^{5/2}, \end{eqnarray} \tag{ 13 }$$

where D is the distance and c₅ is another constant. In the optically thin limit,

$$\begin{eqnarray} &&F_{\nu } = \frac{4\pi R_p^3}{3D^2}c_5K(B\sin \theta)^{(s+1)/2}\left(\frac{\nu }{2c_1} \right)^{-(s-1)/2}, \end{eqnarray} \tag{ 14 }$$

which assumes an emission filling factor of the order of unity. We use a simple interpolation between these two limits for the total emission spectrum F_ν. The location of the spectrum's peak at ν_SA∝B^{(s + 2)/(s + 4)} shifts to lower frequencies as B decreases during expansion (Equation (11)). The peak flux scales ∝B^{(2s + 3)/(s + 4)}, and thus also decreases with time.

The synchrotron cooling time scales as $E/|\dot{E}|\propto \nu ^{-1/2}$. Therefore, there is a ν_c above which synchrotron cooling beats adiabatic expansion (Reynolds & Chevalier 1984):

$$\begin{eqnarray} &&\nu _c = \frac{c_1}{c_2^2B^3\sin ^3\theta }\left(\frac{v_p}{R_p}\right)^2, \end{eqnarray} \tag{ 15 }$$

where c₂ = 2.37 × 10⁻³ in cgs units. Electrons that evolve adiabatically maintain their spectrum, while electrons that cool from synchrotron emission steepen, therefore

$$\begin{eqnarray} n(E) = \left\lbrace \begin{array}{@{}l l}KE^{-s}, &\ E<E_c\\ KE_cE^{-(s+1)}, &\ E\ge E_c\\ \end{array},\right. \end{eqnarray} \tag{ 16 }$$

where E_c is the energy of electrons emitting a frequency ν_c. The prefactor K is set by the total energy density of electrons U_e = 3P_e, found by integrating

$$\begin{eqnarray} U_e &=& \int _{E_{\rm min}}^{E_{\rm max}}n(E)EdE, \end{eqnarray} \tag{ 17 }$$

where E_min and E_max are the minimum and maximum energies of electron spectrum, respectively. For s = 1.5 and the limit E_max ≫ E_c ≫ E_min, U_e ≈ 4KE^{2 − s}_c. The energy density is roughly set by E_c since the spectrum steepens for E > E_c.

We next consider potential free–free absorption. The absorption coefficient is (Rybicki & Lightman 1979)

$$\begin{eqnarray} &&\alpha _{\rm ff} \approx 1.9\times 10^{-2}T^{-3/2}Z^2 n_e n_i \nu ^{-2}g_{\rm ff}\,{\rm cm^{-1}}, \end{eqnarray} \tag{ 18 }$$

for hν ≪ k_BT, where T is the temperature, Z is the average charge per ion, n_e and n_i are the electron and ion densities, respectively, g_ff ∼ 1 is the Gaunt factor, and all quantities are in cgs units. Just behind the forward shocks (see Section 2), the temperature is high (≫10⁹ K), and free–free absorption is negligible. But, at the front edge of the ejecta, with density ρ_ej and temperature T, pressure continuity across the contact discontinuity requires ρ_ejk_BT/m_p ∼ n₀m_pv²_s. From this we estimate that T ∼ 10⁵–10⁸ K at t ∼ 10⁶–10⁷ s. For Z/A ≈ 1/2 (where A is the average atomic number), we estimate Z²n_en_i ∼ (ρ_ej/m_p)². The observed flux is thus $F_{\nu ,\rm obs} \approx F_\nu e^{-\tau _{\rm ff}}$, where τ_ff ≈ α_ffΔR_p and ΔR_p ≈ 0.1R_p is the shell thickness.

In the top panel of Figure 1, we plot the evolution of the key frequencies ν_SA (Equation (12)), ν_c (Equation (15)), and ν_ff, where the latter is defined by τ_ff ≈ 1. From this one can follow the spectrum peak (at ν_SA) and determine when it is detectable. For example, 10 GHz emission peaks at ≈6 × 10⁶ s. In the bottom panel, we explore the light curves at 1.4 GHz as we vary T and η_B. Typical timescales are ∼ months with a peak of ∼3–10 mJy. Free–free absorption is mostly negligible unless T ≲ 10⁴ K. The peak flux and time of peak are sensitive to η_B.

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In Figure 2 we plot the peak flux at 1.4 GHz for a range of dipole field strengths and initial spin periods. The general trend is that faster spins and larger fields result in a more luminous radio source. This reverses in the top left corner where τ is sufficiently short that L(t) has decreased significantly by the time of peak. Over most of this parameter space, the magnetar winds are not expected to be strong enough to generate a collimated outflow (Bucciantini et al. 2012).

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AIC can result from both channels popularly discussed for SNe Ia (single degenerate and double degenerate; see Section 1). Thus it makes sense to normalize the AIC rate to that of the SN Ia rate. The Lick Observatory Supernova Search finds a rate of (3.01 ± 0.062) × 10⁻⁵ Ia Mpc⁻³ yr⁻¹ (Li et al. 2011), which corresponds to (4.0–7.1) × 10⁻³ Ia yr⁻¹ for the Milky Way. Using population synthesis, Yungelson & Livio (1998) find AIC rates of 8 × 10⁻⁷–8 × 10⁻⁵ AIC yr⁻¹ for the Milky Way, depending on assumptions about the common-envelope phase and mass transfer. Their upper bound is similar to the constraint obtained from observed abundances of neutron-rich isotopes (Hartmann et al. 1985; Fryer et al. 1999).

For a typical peak 1.4 GHz flux of F_{p, 100} ∼ 5 mJy at a distance of 100 Mpc, and an AIC rate that is a fraction f of the SN Ia rate, the detection rate of radio transients from AIC is

$$\begin{eqnarray} &&{\rm Rate} ({>}F_{p,100}) \approx 14 \left(\frac{f}{10^{-2}} \right)\left(\frac{F_{p,100}}{5\,{\rm mJy}} \right)^{-3/2} {\rm yr^{-1}}, \end{eqnarray} \tag{ 19 }$$

where f ∼ 10⁻⁴–10⁻². For a ∼3 month duration emitting above ∼1 mJy, we expect ∼4(f/10⁻²) AICs above threshold at a given time. In contrast, merely a few AICs are estimated to be detected as kilonovae per year (Metzger et al. 2009b), and this number could be much less if a lack of differential rotation during WD accretion (Piro 2008) inhibits disk formation upon collapse (Abdikamalov et al. 2010).

Similar radio emission is possible if some NS mergers produce a massive NS. Indeed, there is tantalizing evidence that the NSs in black widow system have masses well above 2 M_☉ (van Kerkwijk et al. 2011; Romani et al. 2012). If so, NS coalescence can contribute meaningfully to the magnetar channel. This would then provide an electromagnetic counterpart following months after the gravitational wave emission during coalescence² (Metzger & Berger 2012; Nissanke et al. 2012). In fact, the distance at which the radio emission can be detected is similar to that probed by the next generation of "advanced" ground-based laser interferometers. The NS merger rate is comparable to or greater than the AIC rate. For example, Kim et al. (2005) estimate a Galactic rate of ∼(0.1–3) × 10⁻⁴ yr⁻¹. The detection frequency is thus similar to Equation (19) with f = 10⁻², with the caveat that it may be much less if most NS mergers produce black holes.

This is a timely topic to discuss given the renaissance that is now occurring in decimetric radio astronomy. Refurbished (Jansky Very Large Array (VLA)) and new facilities promise high mapping speeds by using small-diameter antennas to realize a given total area (e.g., MeerKAT), through the use of focal plane arrays instead of a single feed to gain massive multiplex advantage (APERTIF), or both (ASKAP).

We offer a plausible project aimed at detection of AICs that can be undertaken over the next few years. For a 7σ detection³ of a 1 mJy point source, the Jansky VLA mapping speed in the 1.4 GHz band (for a 500 MHz bandwidth) is ≈86η deg² hr⁻¹, where η is the efficiency of time spent integrating on the sky (as opposed to slewing antennas or observing calibrator sources). For "on-the-fly" mapping, η = 0.9 (S. Myers, private communication) and a 250 hr allocation results in observing ≈19, 000 deg² (comparable in area and depth to Faint Images of the Radio Sky at Twenty centimeters (FIRST); Becker et al. 1995). Another strategy would be to use the same allocation for several epochs focused on fields selected by their richness in nearby galaxies (as was done for the Palomar Transient Factory (PTF) key project "Investigation of Transients in the Local Universe"; Kasliwal 2011). A systematic optical mapping of the sky should be undertaken a week or more prior to the radio observations and continued for a week after (for instance, with PTF; Law et al. 2009; Rau et al. 2009).

The Jansky VLA radio survey would yield roughly two AICs. The signature of these events would be unique: they would be coincident with galaxies in the nearby Universe, would not be accompanied by any optical SNe of the sort that have detected so far, and would often occur outside of the nuclear regions of the host galaxy. For especially interesting radio detections, it may be worth conducting follow-up in the infrared to rule out the presence of an extinguished, radio-bright SN (although the steeper spectral index for the SNe may also be a useful discriminant; e.g., Chevalier 1998). The issue of associated X-ray emission is less certain. It is possible that young magnetars shine in the X-rays, but if the ejecta is dominated by heavy elements then the classical X-ray emission is suppressed. A background active galactic nucleus or a foreground active star would be revealed by the distinctive spectra of the astrometric coincident quiescent source.

The future is quite bright. The soon-to-be-commissioned APERTIF (Verheijen et al. 2008) mapper could do the same survey 30% faster or alternatively redo the survey every few months. In the future, ASKAP and MeerKAT mappers could undertake the survey five times faster.

We thank Roger Chevalier, Peter Goldreich, Gregg Hallinan, Mansi Kasliwal, Keiichi Maeda, Brian Metzger, Christian Ott, E. Sterl Phinney, and Eliot Quataert. A.L.P. is supported through NSF grants AST-1212170, PHY-1151197, and PHY-1068881, NASA ATP grant NNX11AC37G, and the Sherman Fairchild Foundation. S.R.K.'s research is in part supported by NSF.