A BROADBAND EMISSION MODEL OF MAGNETAR WIND NEBULAE

Rotation-powered pulsars (RPPs) convert most of their rotational energy into pulsar winds and create pulsar wind nebulae (PWNe) around them (e.g., Pacini & Salvati 1973; Rees & Gunn 1974). More than 70 PWNe have been detected so far and most of them are around energetic RPPs whose spin-down powers L_spin are ≳10³⁵ erg s⁻¹ (e.g., Kargaltsev et al. 2013). However, some less energetic, isolated pulsars (L_spin ≳ 10³³ erg s⁻¹) are also associated with bow shock structures, called bow shock PWNe, as a consequence of the wind from the pulsars (e.g., Kargaltsev et al. 2015). Both the rotational energy lost by pulsar winds and the formation of PWNe are considered universal features of isolated pulsars.

Magnetars are a class of isolated pulsars and have an inferred surface magnetic field strength above the quantum critical field ${B}_{{\rm{QED}}}\equiv {m}_{{\rm{e}}}^{2}{c}^{3}/e{\hslash }\,\approx \,4.4\times {10}^{13}\,{\rm{G}}$ (e.g., Mereghetti 2008). Their large persistent X-ray luminosity exceeds the spin-down power L_spin and bursting activities in soft γ-rays indicate that they are magnetically powered objects (Thompson & Duncan 1995, 1996). On the other hand, some magnetars are also pulsating in radio as RPPs (e.g., Camilo et al. 2006). This is evidence that plenty of electron–positron pairs are produced in the magnetar magnetosphere (Medin & Lai 2010; Beloborodov 2013) against the photon splitting process in the intense magnetic field (Baring & Harding 1998). In addition, the literature discusses that the pulsed radio emission from magnetars is rotation-powered (Rea et al. 2012; Szary et al. 2015). Besides the magnetic energy responsible for bright X-rays and bursting activities, magnetars also release their rotational energy L_spin as the angular momentum extraction by the magnetar wind.

Detection of magnetar wind nebulae (MWNe) is one of the best ways to confirm the presence of a magnetar wind. Although some studies report detections of MWN-like extended emission around AXP 1E 1547.0-5408 (Vink & Bamba 2009) and Swift J1834.9-0846 (Younes et al. 2012), others claim that the extended emission is consistent with dust-scattering halos of their past magnetar activities (Olausen et al. 2011; Esposito et al. 2013). However, we have obtained upper limits on MWN emissions, which are still important to give a constraint on the presence of MWNe. By combining deep observations of magnetars with spectral studies of PWNe, we are able to constrain the pair-production multiplicity κ (c.f., de Jager 2007; Bucciantini et al. 2011) and magnetization σ (c.f., Martin et al. 2014) of the magnetar winds, and also the spin-down evolution of the central magnetars (c.f., de Jager 2008). It is especially interesting to explore their differences from RPPs; for example, σ- and κ-problems are known for young RPPs (c.f., Kirk et al. 2009; Arons 2012; Tanaka & Takahara 2013a).

Studies of MWNe have another interesting aspect. Long-lasting activities of gamma-ray bursts (<10⁵ s) are often interpreted as caused by the spin-down activity of a magnetar born with a period of milliseconds (e.g., Zhang & Mészáros 2001) although other models suggest alternative causes (e.g., Kisaka & Ioka 2015). In addition, millisecond magnetars are also proposed to be the engines of gamma-ray bursts (e.g., Usov 1992) and luminous supernovae (c.f., Kashiyama et al. 2015). On the other hand, population synthesis studies of magnetars show difficulty in compatiblility with the Galactic magnetar population with such millisecond magnetars although population studies do not easily to account for initial spin periods of less than 10 ms (Rea et al. 2015). Recently, the very early phase (<10⁶ s) an embryonic MWN's system and explosion ejecta around a newly born millisecond magnetar has been studied to find evidence of millisecond magnetars (e.g., Metzger et al. 2014; Murase et al. 2015). Here, we study the spectrum of MWNe at an age of ∼kyr, where it may have some signatures of millisecond magnetars.

In Section 2, we describe our model of MWN spectral evolution. We consider the persistent magnetar outflow as well as the pulsar wind. The model is similar to past studies of young PWNe around RPPs (Tanaka & Takahara 2010, 2011, 2013b, hereafter TT10, TT11 and TT13b respectively). In Section 3, we apply the model to the MWN around 1E 1547.0-5408. 1E 1547.0-5408 is the only object for which we find the flux upper limit in the published literature and is the most promising object for detecting MWNe among all magnetars because of its large L_spin and its small distance. In Section 4, the results are discussed with regards to  the connection between an RPPs and magnetars. We especially compare the MWN around 1E 1547.0-5408 with PWN Kes 75 around PSR J1846-0258, which is a RPP with large surface magnetic field and has experienced magnetar-like bursts in 2006 (Gavriil et al. 2008). We also discuss the existence of millisecond magnetars and conclude this paper.

Here, based on the presumption that magnetars also create wind nebulae like young RPPs, a one-zone spectral model of MWNe is introduced. The energy distribution of accelerated electrons and positrons inside MWNe N(γ, t) is found from the continuity equation,

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}N(\gamma ,t)+\displaystyle \frac{\partial }{\partial \gamma }(\dot{\gamma }(\gamma ,t)N(\gamma ,t))={Q}_{{\rm{inj}}}(\gamma ,t),\end{eqnarray} \tag{ 1 }$$

where γ is the Lorentz factor of the relativistic electrons and positrons, Q_inj(γ, t) is the injection from the central magnetar, and $\dot{\gamma }(\gamma ,t)$ includes adiabatic ${\dot{\gamma }}_{{\rm{ad}}}(\gamma ,t)$, synchrotron ${\dot{\gamma }}_{{\rm{syn}}}(\gamma ,t)$, and inverse Compton scattering ${\dot{\gamma }}_{{\rm{IC}}}(\gamma )$ coolings. Although most concepts are shared with our and other past studies of PWN spectra (Gelfand et al. 2009; Bucciantini et al. 2011; Martin et al. 2014; Torres et al. 2014), we introduce some modifications for the application to MWNe that have much less observational information than young PWNe.

The radiation processes are synchrotron radiation and the inverse Compton scattering off the interstellar radiation field (IC/ISRF). We ignore the synchrotron self-Compton (SSC) process because SSC is always subdominant to IC/ISRF for young PWNe other than the Crab Nebula (e.g., Torres et al. 2013). The ISRF has three components: the cosmic microwave background radiation (CMB), dust infrared (IR), and optical starlights (OPT). The CMB has the blackbody spectrum of the temperature T_CMB = 2.7 K, and the others are assumed to have the modified blackbody spectra which are characterized by energy densities u_ISRF and temperatures T_ISRF. Below, we use (u_IR, T_IR) = (1.0 eV cm⁻³, 40 K) and (u_OPT, T_OPT) = (2.0 eV cm⁻³, 4000 K) for the model's application to 1E 1547.0-5408.

2.1. Spin-down Evolution

The spin-down of pulsars is customarily described as the differential equation $\dot{{\rm{\Omega }}}=-k{{\rm{\Omega }}}^{n}$, where Ω and $\dot{{\rm{\Omega }}}$ are the current angular frequency and its derivative, respectively. We apply the same equation to magnetars. To specify the spin-down behavior Ω(t) from the differential equation, we need one initial condition and two constants, n and k. We take the three corresponding quantities as the braking index n, the initial period P₀ = 2π/Ω₀, and the initial dipole-magnetic field ${B}_{0}\equiv 3.2\times {10}^{19}\,{\rm{G}}\sqrt{{P}_{0}{\dot{P}}_{0}}$. Assuming that the moment of inertia of a magnetar is I = 10⁴⁵ g cm², the evolution of the spin-down power ${L}_{{\rm{spin}}}(t)=I{\rm{\Omega }}(t)\dot{{\rm{\Omega }}}(t)$ is expressed as

$$\begin{eqnarray}&&{L}_{{\rm{spin}}}(t)={L}_{0}{\left(1+\displaystyle \frac{t}{{t}_{0}}\right)}^{-\tfrac{n+1}{n-1}},\end{eqnarray} \tag{ 2 }$$

where the initial spin-down power ${L}_{0}=I{{\rm{\Omega }}}_{0}{\dot{{\rm{\Omega }}}}_{0}$, and the initial spin-down time ${t}_{0}={{\rm{\Omega }}}_{0}/(1-n){\dot{{\rm{\Omega }}}}_{0}$ relate P₀ and B₀ as

$$\begin{eqnarray}&&{L}_{0}=3.9\times {10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1}{\left(\displaystyle \frac{{B}_{0}}{{10}^{14}{\rm{G}}}\right)}^{2}{\left(\displaystyle \frac{{P}_{0}}{10{\rm{ms}}}\right)}^{-4},\end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray}&&{t}_{0}=\displaystyle \frac{3.2}{(n-1)}\times {10}^{-4}\,{\rm{kyr}}{\left(\displaystyle \frac{{B}_{0}}{{10}^{14}{\rm{G}}}\right)}^{-2}{\left(\displaystyle \frac{{P}_{0}}{10{\rm{ms}}}\right)}^{2},\end{eqnarray} \tag{ 4 }$$

respectively. We also obtain the simple relation between the initial spin-down time t₀, the age ${t}_{{\rm{age}}}$ , and the characteristic age t_c as

$$\begin{eqnarray}&&{t}_{{\rm{c}}}=\displaystyle \frac{n-1}{2}({t}_{{\rm{age}}}+{t}_{0}),\end{eqnarray} \tag{ 5 }$$

where ${t}_{{\rm{c}}}\equiv P/2\dot{P}$ is obtained from P and $\dot{P}$ at an age of ${t}_{{\rm{age}}}$.

For magnetars of P₀ ≲ 10 ms and B₀ ≳ 10¹⁴ G, because observed magnetars have an age of ${t}_{{\rm{age}}}$ ≳ kyr ≫ t₀, Equations (2) and (5) are simplified as

$$\begin{eqnarray}&&{L}_{{\rm{spin}}}(t)\,\approx \,L{\left(\displaystyle \frac{t}{{t}_{{\rm{age}}}}\right)}^{-\tfrac{n+1}{n-1}},\,{t}_{{\rm{age}}}\,\approx \,\displaystyle \frac{2{t}_{{\rm{c}}}}{n-1}.\end{eqnarray} \tag{ 6 }$$

The current spin-down power $L=4{\pi }^{2}I\dot{P}/{P}^{3}$ and the characteristic age ${t}_{{\rm{c}}}=P/2\dot{P}$ are obtained from the observed values of P and $\dot{P}$. Only the braking index n is a parameter of the magnetar spin-down within this approximation. We take n = 3 (${t}_{{\rm{age}}}$ ≈ t_c) as a fiducial value, although the observed values of some pulsars have variation (e.g., Table 1 of Espinoza et al. 2011).

Table 1.  Summary of The Parameters for the Calculations in Figures1 and 2

Symbol	Kes 75	Model 1	Model 2	Model 3	Model 4
Fixed Parameters
d(kpc)	6.0			4.5
R(pc)	0.29			0.98
P(s)	0.326			2.07
$\dot{P}$(10−12 s s−1)	7.08			47.7
uIR(eV cm−3)	1.2			1.0
uOPT(eV cm−3)	2.0			2.0
γmin(103)	<5.0			1.0
γmax(109)	>1.0			1.9
p1	−1.6			−1.5
αR	1.0			1.0
αB	−1.5			−1.5
Adopted Parameters
B(μG)	20	3	25	3	3
γb(105)	20	10	0.1	100	40
p2	−2.5	−2.6	−2.6	−3.0	−3.0
n	2.65	3	3	3	2
Dependent Parameters
γcool/γmax	⋯	2.1	0.030	2.1	1.0
${t}_{{\rm{age}}}$(kyr)	0.7	0.69	0.69	0.69	1.4
ts(year)	⋯	3.0	12	3.0	7.6
P0(ms)	126	<137	<278	<137	<11
η(10−3)	0.05	<0.04	<11	<0.04	<0.00028

Note. The parameters of Kes 75 is taken from TT11.

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2.2. Expansion and Magnetic Field Evolution

We assume that an MWN is an expanding, uniform sphere whose radius is expressed as

$$\begin{eqnarray}&&{R}_{{\rm{MWN}}}(t)\,\approx \,R{\left(\displaystyle \frac{t}{{t}_{{\rm{age}}}}\right)}^{{\alpha }_{{\rm{R}}}},\end{eqnarray} \tag{ 7 }$$

where R is the present radius of the MWN and is estimated from observations. Equation (7) with the index α_R ≳ 1 expresses an early phase of expansion evolution (c.f., Reynolds & Chevalier 1984; van der Swaluw et al. 2001; Chevalier 2005; Gelfand et al. 2009). As discussed in the Appendix(see Figure 3), our model spectrum is insensitive to α_R. We adopt α_R = 1.0 for the application to 1E 1547.0-5408.

Mean magnetic field strength inside an MWN should evolve with time because of the magnetic energy injection from central magnetars and because of the expansion of the MWN. For simplicity, we also adopt a power-law dependence on time for magnetic field evolution, i.e.,

$$\begin{eqnarray}&&{B}_{{\rm{MWN}}}(t)\,\approx \,B{\left(\displaystyle \frac{t}{{t}_{{\rm{age}}}}\right)}^{{\alpha }_{{\rm{B}}}},\end{eqnarray} \tag{ 8 }$$

where B is the current magnetic field strength of the MWN and is a parameter. The current magnetic field strength ranges from 3 μG ≲ B ≲ 80 μG for young PWNe (c.f., TT13b; Torres et al. 2014). Although the index α_B ∼ −2 varies between models (e.g., TT10; Torres et al. 2014), our model spectrum is insensitive to α_B (see the Appendix and Figure 3). We will take α_B = −1.5 for the model's application to 1E 1547.0-5408.

In past studies, the magnetic energy fraction η has been used to determine the magnetic field strength B of young PWNe, where the magnetic power of the central pulsars is expressed as η L_spin. TT11 and Martin et al. (2014) studied the value of η for each object because η is closely related to the wind magnetization parameter σ (the ratio of Poynting flux to particle energy flux of the pulsar wind (e.g., Kennel & Coroniti 1984)). On the other hand, in the current model, we estimate the magnetic fraction η from B and R by dividing the total energy injected by the magnetar,

$$\begin{eqnarray}&&{E}_{{\rm{spin}}}(t)={\int }_{0}^{t}{L}_{{\rm{spin}}}(t^{\prime} ){dt}^{\prime} \,\approx \,\displaystyle \frac{2{\pi }^{2}I}{{P}_{0}^{2}}\,\mathrm{for}\,{P}_{0}\ll P,\end{eqnarray} \tag{ 9 }$$

by the magnetic energy inside the MWN,

$$\begin{eqnarray}&&{E}_{{\rm{B}}}(t)=\displaystyle \frac{{B}^{2}{R}^{3}}{6}{\left(\displaystyle \frac{t}{{t}_{{\rm{age}}}}\right)}^{2{\alpha }_{{\rm{B}}}+3{\alpha }_{{\rm{R}}}}.\end{eqnarray} \tag{ 10 }$$

i.e., $\eta \equiv {E}_{{\rm{B}}}(t)/{E}_{{\rm{spin}}}(t)$. For ${P}_{0}\ll P$ and $({\alpha }_{{\rm{R}}},{\alpha }_{{\rm{B}}})=(1,-1.5)$ (i.e., 2α_B + 3α_R = 0), η is almost constant with time and is compatible with our past studies (TT10, 11, 13b).

From Equations (7) and (8), the cooling term in Equation (1) is written as

$$\begin{eqnarray}&&\dot{\gamma }(\gamma ,t)=-\displaystyle \frac{{\alpha }_{{\rm{R}}}}{t}\gamma -\displaystyle \frac{{\gamma }^{2}}{{t}_{{\rm{syn}}}}{\left(\displaystyle \frac{t}{{t}_{{\rm{age}}}}\right)}^{2{\alpha }_{{\rm{B}}}}-\displaystyle \sum _{i}\displaystyle \frac{{u}_{i}}{{u}_{{\rm{B}}}{t}_{{\rm{syn}}}}\displaystyle \frac{{\gamma }^{2}{\gamma }_{{\rm{K}},i}^{2}}{{\gamma }^{2}+{\gamma }_{{\rm{K}},i}^{2}},\end{eqnarray} \tag{ 11 }$$

where i = CMB, IR, OPT, and ${\gamma }_{{\rm{K}},i}=3\sqrt{5}{m}_{{\rm{e}}}{c}^{2}/8\pi {k}_{{\rm{B}}}{T}_{i}$. The last term of Equation (11) is an approximated form of ${\dot{\gamma }}_{{\rm{IC}}}$ given by Schlickeiser & Ruppel (2010) and we introduce the current synchrotron timescale

$$\begin{eqnarray}&&{t}_{{\rm{syn}}}\equiv \displaystyle \frac{3{m}_{{\rm{e}}}c}{4{\sigma }_{{\rm{T}}}^{}{u}_{{\rm{B}}}}\,\approx \,2.5\times {10}^{8}\,{\rm{kyr}}{\left(\displaystyle \frac{B}{10\mu {\rm{G}}}\right)}^{-2},\end{eqnarray} \tag{ 12 }$$

and u_B ≡ B²/8π, respectively. The cooling Lorentz factor ${\gamma }_{{\rm{cool}}}\equiv {\alpha }_{{\rm{R}}}{t}_{{\rm{syn}}}/{t}_{{\rm{age}}}$ will be used to characterize the typical Lorentz factor dividing the dominant cooling process into ${\dot{\gamma }}_{{\rm{ad}}}$ or ${\dot{\gamma }}_{{\rm{syn}}}$ at t = ${t}_{{\rm{age}}}$.

2.3. Particle Injection

Magnetar wind plasma is accelerated and injected into an MWN. From past studies of PWNe, we consider that almost all of the spin-down power is converted into the energy of accelerated electron–positron plasma, i.e., the magnetic power is a small fraction of L_spin(t) ($\eta \ll 1$). We adopt a broken power-law injection spectrum of accelerated particles characterized by five parameters γ_min, γ_b, γ_max, p₁, and p₂ which are the minimum, break, and maximum Lorentz factors and the power-law indices at the low and high energy parts respectively. The injection term of Equation (1) is expressed as

$$\begin{eqnarray}&&{Q}_{{\rm{inj}}}(\gamma ,t)=\chi \displaystyle \frac{L}{{\gamma }_{{\rm{b}}}^{2}{m}_{{\rm{e}}}{c}^{2}}{\left(\displaystyle \frac{t}{{t}_{{\rm{age}}}}\right)}^{-\tfrac{n+1}{n-1}}H(t-{t}_{{\rm{s}}})R(\gamma )\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&\chi \equiv {\gamma }_{{\rm{b}}}^{2}{\left({\int }_{1}^{\infty }d\gamma \gamma R(\gamma )\right)}^{-1},\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}R(\gamma )\equiv \left\{\begin{array}{ll}{(\gamma /{\gamma }_{{\rm{b}}})}^{{p}_{1}} & \mathrm{for}\,{\gamma }_{\min }\,\leqslant \,\gamma \,\leqslant \,{\gamma }_{{\rm{b}}},\\ {(\gamma /{\gamma }_{{\rm{b}}})}^{{p}_{2}} & \mathrm{for}\,{\gamma }_{{\rm{b}}}\,\leqslant \,\gamma \,\leqslant \,{\gamma }_{\max },\end{array}\right.\end{eqnarray} \tag{ 15 }$$

where H(x) is the Heaviside's step function and χ is a value of order unity for typical sets of parameters ${\gamma }_{{\rm{\min }}}\ll {\gamma }_{{\rm{b}}}\ll {\gamma }_{{\rm{\max }}}$ and p₂ < −2 < p₁. Q_inj(γ, t) is normalized to satisfy ${L}_{{\rm{spin}}}(t)=\int d\gamma {Q}_{{\rm{inj}}}(\gamma ,t)\gamma {m}_{{\rm{e}}}{c}^{2}$.

In Equation (13), we introduced an additional parameter, the start time t_s (<${t}_{{\rm{age}}}$), because we cannot set t = 0 in Equations (11) and (13). However, it is possible to set a reasonable finite value of t_s from ${t}_{{\rm{age}}}$ and B (Equation (24)), and we only limit our interest to the particle energy range of γ ≥ γ_min. As shown in the Appendix, the current particle spectrum N(γ ≥ γ_min, ${t}_{{\rm{age}}}$) is not affected by any particles injected before t = t_s. On the other hand, we require t_s ≫ t₀ for Equation (6) to be valid even at t = t_s. Combining with Equation (4), the condition t_s > t₀ constrains the initial period of the magnetar. From $L\propto \dot{P}{P}^{-3}\propto {t}^{-(n+1)/(n-1)}$, we obtain $P\propto {t}^{1/(n-1)}$ in the same approximation, i.e.,

$$\begin{eqnarray}&&P({t}_{{\rm{s}}})=P{\left(\displaystyle \frac{{t}_{{\rm{s}}}}{{t}_{{\rm{age}}}}\right)}^{\tfrac{1}{n-1}}.\end{eqnarray} \tag{ 16 }$$

P(t_s) gives the upper limit on P₀ within our formulation.

For typical values of young PWNe, ${\gamma }_{{\rm{\min }}}$, γ_max, and p₁ have little influence on emissions in radio, X-rays, and TeV γ-rays, while γ_b and p₂ remain as the injection spectrum parameters. For the method's application to 1E 1547.0-5408 in the next section, we adopt γ_min = 10³ because the characteristic frequencies of synchrotron radiation and inverse Compton scattering off the ISRF ${\nu }_{{\rm{IC/ISRF}}}$ in the Thomson limit are given by (e.g., Blumenthal & Gould 1970; Rybicki & Lightman 1979)

$$\begin{eqnarray}&&{\nu }_{{\rm{syn}}}\,\approx \,1.2\times {10}^{7}\,{\rm{Hz}}{\left(\displaystyle \frac{\gamma }{{10}^{3}}\right)}^{2}\left(\displaystyle \frac{B}{10\,\mu {\rm{G}}}\right),\end{eqnarray} \tag{ 17 }$$

and

$$\begin{eqnarray}&&{\nu }_{{\rm{IC/ISRF}}}\,\approx \,3.0\times {10}^{20}\,{\rm{Hz}}{\left(\displaystyle \frac{\gamma }{{10}^{3}}\right)}^{2}\left(\displaystyle \frac{{T}_{{\rm{ISRF}}}}{4000\,{\rm{K}}}\right).\end{eqnarray} \tag{ 18 }$$

We set p₁ = −1.5, which is the typical value to reproduce radio observations of young PWNe (TT13b). Finally, we assume that the maximum energy of the accelerated particles satisfies γ_maxm_ec² = eΦ_cap (Bucciantini et al. 2011), i.e.,

$$\begin{eqnarray}&&{\gamma }_{{\rm{\max }}}\,\approx \,1.3\times {10}^{9}\left(\displaystyle \frac{{B}_{{\rm{NS}}}}{{10}^{14}\,{\rm{G}}}\right){\left(\displaystyle \frac{P}{1{\rm{s}}}\right)}^{-2},\end{eqnarray} \tag{ 19 }$$

where the potential difference at the polar cap is estimated as ${{\rm{\Phi }}}_{{\rm{cap}}}={{\rm{\Omega }}}^{2}{B}_{{\rm{NS}}}{R}_{{\rm{NS}}}^{3}/2{c}^{2}$ (e.g., Ruderman & Sutherland 1975). Equation (19) essentially corresponds to the Hillas condition γ_maxm_ec² = eB(r)r (Hillas 1984) on the assumption that the magnetic field is dipolar inside the light cylinder R_LC = c/Ω and is toroidal beyond R_LC, i.e., the magnetic field strength at an acceleration point B(r) = B_NS(R_NS/R_LC)³(R_LC/r) (Goldreich & Julian 1969).

1E 1547.0-5408 is an X-ray source discovered in 1980 (Lamb & Markert 1981) and was recognized by Gelfand & Gaensler (2007) as an anomalous X-ray pulsar located inside SNR G327.24-0.13. Camilo et al. (2007) discovered radio pulsations of $P=2.07\,{\rm{s}}$, which is the smallest among magnetars. For the period derivative, we adopt the long-term average value $\dot{P}\,\approx \,4.77\times {10}^{-11}\,{\rm{s}}\,{{\rm{s}}}^{-1}$ (Dib et al. 2012; Olausen & Kaspi 2014), which is different from the $\dot{P}$ that was obtained by Camilo et al. (2007, 2008). The braking index is not well-determined because $\dot{P}$ of 1E 1547.0-5408 shows temporal variations, which is likely associated with flaring activities (Camilo et al. 2008; Dib et al. 2012). A current surface magnetic field strength of B_NS = 3.2 × 10¹⁴ G is typical while the spin-down power of L_spin = 2.1 × 10³⁵ erg s⁻¹ is the largest among magnetars. The potential difference of the polar cap gives γ_max = 1.9 × 10⁹ (Equation (19)). We adopt a distance to 1E 1547.0-5408 of d ∼ 4.5 kpc from the possible association of SNR G327.24-0.13 with a nearby star-forming region (Gelfand & Gaensler 2007) and from the flux decline of the dust–scattering X-ray rings (Tiengo et al. 2010), although, the dispersion measure of ∼830 cm⁻³ pc (Camilo et al. 2007) and the neutral hydrogen column density of N_H ∼ 3 × 10²² cm⁻² (Gelfand & Gaensler 2007; Vink & Bamba 2009) are relatively large compared with sources of similar distances. The characteristic age of t_c = 0.69 kyr indicates that 1E 1547.0-5408 is one of the youngest pulsars ever observed and is consistent with the small angular size ∼4' of SNR G327.24-0.13 corresponding to ∼5.2 pc in diameter (Gelfand & Gaensler 2007).

Vink & Bamba (2009) reported the discovery of an extended X-ray emission around 1E 1547.0-5408 from archival Chandra and XMM-Newton data in 2006 with the source in quiescence. They interpreted the extended emission as a PWN whose angular radius is ∼45''(∼0.98 pc in radius) and has a 2–10 keV flux of F_VB09 = (1.5 ± 0.3) × 10⁻¹³ erg s⁻¹ cm⁻². However, both its soft spectrum (photon index of Γ_X ∼ 3.5) and large X-ray efficiency of η_X = (4πd²F_VB09)/L_spin ∼ 1.7 × 10⁻³ are not typical among young PWNe. Olausen et al. (2011) reanalyzed the XMM-Newton observation in 2006 with data from 2007, 2009, and 2010. They also found an extended emission but its flux differed between observations. As already reported by Tiengo et al. (2010) for the 2009 event, the extended emission data on the 2007, 2009, and 2010 data was interpreted as a dust–scattering halo. For the 2006 data, Olausen et al. (2011) did not rule out the presence of a faint PWN and gave the stronger upper limit on the 2–10 keV flux ≲4.7 × 10⁻¹⁴ erg s⁻¹ cm⁻² ≡ F_O11 than F_VB09. Although 1E 1547.0-5408 has been extensively observed in radio, infrared, hard X-rays, and GeV γ-rays (e.g., Olausen & Kaspi 2014), we do not find any other flux information about its extended emission.

We summarize the calculations' parameters (Figures1 and 2) together with Kes 75's parameter in Table 1 obtained by TT11. Since Kes 75 shows some exceptional features among other young PWNe around RPPs, we expect that the wind from the high B-field pulsar (HBP) PSR J1846-0258 is similar to that of magnetars rather than RPPs (e.g., Safi-Harb 2013). γ_min and p₁ are fixed parameters and so we do not obtain information about the pair multiplicity of the wind (c.f., TT10; de Jager 2007; Bucciantini et al. 2011). The dependent parameters, such as the ratio γ_cool/γ_max, the age on the system ${t}_{{\rm{age}}}$, the start time t_s, and the upper limits of the initial period P₀ (Equation (16)), and the magnetic energy fraction η (Equations (9) and (10)) are also tabulated.

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3.1. Results

Considering the plasma outflows from the magnetar, we build a broadband emission model of MWNe based on the one-zone model of young PWNe around RPPs (TT10, TT11, TT13b). The model is simplified for application to MWNe that have less observational information than PWNe. We apply the model to the MWN around 1E 1547.0-5408, which is the most promising object for detecting MWNe among all the known magnetars because of its large L_spin and its small distance to the Earth.

Because of poor current observational constraints on the MWN around 1E 1547.0-5408, various combinations of parameters are allowed. However, here, we compare Models 1 and 2 based on past studies of young PWNe, especially focusing on Kes 75 around PSR J1846-0258, which is the only HBP showing a magnetar-like behavior (Gavriil et al. 2008). Model 1 has η and γ_b similar to Kes 75, while Model 2 has a similar magnetic field strength B. It is known that the mean magnetic field B is very different for each young PWN 3 ≲ B ≲ 80 μG (e.g., TT13b; Torres et al. 2014). On the other hand, TT11 and TT13b showed that η ∼ 10⁻³ is common among young PWNe, except for Kes 75, by using almost the same model as this study, i.e., adopting (α_R, α_B) = (1.0, −1.5) in Equations (7), (8), and (10). Kes 75 has an exceptionally small η compared to other young PWNe for our model and also for other models, although the values of η are different among models (Bucciantini et al. 2011; Torres et al. 2014). We favor Model 1 over Model 2 because we expect that magnetars have the wind property η and γ_b similar to HBPs, where γ_b is related to the bulk Lorentz factor of the wind just upstream the termination shock (e.g., Kennel & Coroniti 1984; Bucciantini et al. 2011).

Models 3 and 4 are cases for the soft injection spectrum p₂ = −3 and the large break Lorentz factor (γ_b ≫ 10⁶), although such γ_b and p₂ are not typical among young PWNe ( TT13b). Both models are interesting because they predict a larger TeV γ-ray flux than Model 1 without increasing the local ISRF (u_IR, u_OPT). For example, the γ-ray flux increases if the photons from the nearby star-forming region contribute to the local ISRF ( Torres et al. 2014). Model 3 is brighter than Model 2 in TeV γ-rays because the injection of the high energy particles ${Q}_{{\rm{inj}}}(\gamma \gt {\gamma }_{{\rm{b}}})\propto {\gamma }_{{\rm{b}}}^{-{p}_{2}-2}$ is an increasing function of γ_b. Model 4 predicts larger TeV γ-ray flux than Model 3 because the braking index changes P(t_s) (Equation (16)), i.e., a larger amount of the rotational energy is injected for Model 4 than Model 3. The models will be distinguished by future CTA observations. Note that we postulate B = 3 μG and the X-ray flux to be at the level of F_O11 for Models 3 and 4. On the other hand, the soft spectrum p₂ = −3 with B ≫ 3 μG and 10⁴ ≲ γ_b ≪ 10⁶ predicts dark MWNe that are difficult to detect with current or near future instruments in both X-rays and TeV γ-rays. In other words, a non-detection of the MWN around 1E 1547.0-5408 does not simply reject the existence of magnetar winds.

Contrary to studies of young PWNe, the current model gives only the upper limits on the initial period of the magnetar P₀ (e.g., de Jager 2008). The upper limits P(t_s) for each model are tabulated in Table 1. From population synthesis studies of isolated neutron stars (e.g., Faucher-Giguère & Kaspi 2006; Popov et al. 2010; Igoshev & Popov 2013) and from some estimated initial periods of individual pulsars (e.g., TT11; TT13b; Popov & Turolla 2012), the condition P₀ < 100 ms is usually satisfied for most pulsars. We consider that the simplified spin-down (Equation (6)) is a reasonable approximation for MWNe. This is different from the cases of young PWNe whose initial spin-down time t₀ is mostly closer to the age of the systems (e.g., TT11; TT13b). Nevertheless, if P₀ > P(t_s), we should use Equation (2). The use of Equation (2) simply reduces the total injection of the rotational energy compared to Equation (6) and the MWN becomes dark. This is another possible reason for the non-detection of MWN around 1E 1547.0-5408.

Our simplified model of an MWN spectrum does not allow us to test the existence of millisecond magnetars because P(t_s) is much larger than milliseconds for 1E 1547.0-5408. Severe adiabatic and synchrotron cooling effects reject the nonthermal emission from the particles injected at t < t_s unless considering a reacceleration of these particles, for example. However, the magnetic field inside the MWN is found from the combination of P₀ and η. From Equations (9) and (10) with (α_R, α_B) = (1, −1.5), we obtain

$$\begin{eqnarray}&&{B}_{{\rm{MWN}}}\,\approx \,210\,\mu {\rm{G}}{\left(\displaystyle \frac{\eta }{{10}^{-5}}\right)}^{\tfrac{1}{2}}{\left(\displaystyle \frac{{R}_{{\rm{MWN}}}}{0.98{\rm{pc}}}\right)}^{-\tfrac{3}{2}}{\left(\displaystyle \frac{{P}_{0}}{1{\rm{ms}}}\right)}^{-1}.\end{eqnarray} \tag{ 20 }$$

Adopting $\eta \sim {10}^{-5}$ for the wind from magnetars and HBPs, the mean magnetic field ≈200 μG for P₀ ≈ 1 ms does not conflict with the upper limit in X-rays when the parameters of the injection spectrum are p₂ = −3 and γ_b = 10⁴, for example. Note that this estimate of B_MWN strongly depends on the choice of (α_R, α_B) and is the reason why we do not use η as a parameter in the present model.

Last, we discuss particle injections accompanied by flaring activities of magnetars. The transient radio nebula was detected following the giant flares of SGR 1806-20 (Cameron et al. 2005; Gaensler et al. 2005) and also of SGR 1900+40 (Frail et al. 1999). Although the isotropic energy of the strongest giant flare from SGR 1806-20 reaches ∼10⁴⁷ erg (Hurley et al. 2005; Palmer et al. 2005; Terasawa et al. 2005), it is still an exceptionally large event. Considering that a typical giant flare has the isotropic energy ranging ∼10⁴⁵-10⁴⁷ erg and has the event rate of once every 50–100 year per source (Woods & Thompson 2006, pp. 547–586), the total energy associated with giant flares is ≲10⁴⁸ erg for ${t}_{{\rm{age}}}$ ∼ kyr. On the other hand, the total rotational energy injected from t_s reaches 2π²I/P²(t_s) ≈ 10⁴⁸ erg for Model 1. We consider that the particle energy injection associated with giant flares does not dominate over persistent particle injection by the magnetar wind for 1E 1547.0-5408.

Magnetars also show bursting activities less luminous than giant flares but they are much more frequent. Although the cumulative energy distributions of magnetar bursts are a power-law distribution ${dN}/{dE}\propto {E}^{-\beta }$ with β < 2 (Cheng et al. 1996; Göǧüs et al. 1999, 2000), it is argued that the persistent X-ray emissions of magnetars L_X (>L_spin) might be the accumulation of unresolved small bursts (e.g., Enoto et al. 2012). If there are particle injections associated with these short bursts, observed L_X larger than L_spin may be interpreted using the wind-loss model (Harding et al. 1999; Thompson et al. 2000, 2002). In their models, the wind luminosity L_wind is allowed to be significantly larger than the power estimated from the observed period and its derivative $I{\rm{\Omega }}\dot{{\rm{\Omega }}}={L}_{{\rm{spin}}}$. Note that because the L_X of 1E 1547.0-5408 in quiescence phase is less than L_spin, the episodic particle wind model by Harding et al. (1999) should be applied. We simply expect that the particle luminosity and also the radiation from the MWN are proportional to L_wind, and then we require B ≲ 3 μG, p₂ < −2.6 and/or γ_b < 10⁶ for the calculated X-ray flux to be below the upper limit. In addition, a large γ-ray flux is also expected in this case.

S.J.T. would like to thank S. Kisaka, T. Enoto, K. Asano, and T. Terasawa for useful discussion. S.J.T. would also like to thank the anonymous referee for his/her very helpful comments. This work is supported by JSPS Research Fellowships for Young Scientists (S. T., 2510447).

The model presented in Section 2 has many parameters. Here, we show that ${\alpha }_{{\rm{R}}},{\alpha }_{{\rm{B}}}$ and t_s can be chosen in order not to affect the calculated current spectra. For simplicity, we ignore ${\dot{\gamma }}_{{\rm{IC}}}$ in Equation (11) because ${\dot{\gamma }}_{{\rm{IC}}}$ is not the dominant cooling process. It is convenient to measure the times t, t_s, and t_syn with ${t}_{{\rm{age}}}$ (τ, τ_s, and τ_syn) and the particle number N(γ, t) with $\chi {{Lt}}_{{\rm{age}}}/{\gamma }_{{\rm{b}}}^{2}{m}_{{\rm{e}}}{c}^{2}$ ($\tilde{N}(\gamma ,\tau )$). These normalizations are unique for a given central magnetar (${t}_{{\rm{age}}}$, L) and for a given set of (γ_min, γ_b, γ_max, p₁, p₂). Equation (1) becomes

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial \tau }\tilde{N}(\gamma ,\tau )+\displaystyle \frac{\partial }{\partial \gamma }\left[\left(\displaystyle \frac{d\gamma }{d\tau }\right)\tilde{N}(\gamma ,\tau )\right]={\tau }^{-\tfrac{n+1}{n-1}}H(\tau -{\tau }_{{\rm{s}}})R(\gamma ),\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d\gamma }{d\tau }\equiv -\displaystyle \frac{{\alpha }_{{\rm{R}}}}{\tau }\gamma -\displaystyle \frac{{\tau }^{2{\alpha }_{{\rm{B}}}}}{{\tau }_{{\rm{syn}}}}{\gamma }^{2}.\end{eqnarray} \tag{ 22 }$$

The normalized Equation (21) has only four parameters: τ_s, τ_syn, α_R, and α_B. One significant parameter is τ_syn, which corresponds to the cooling Lorentz factor γ_cool determined by the magnetic field strength B. Below, we study the role of three other parameters.

Limiting our interest to the particles whose Lorentz factor is larger than γ_min at τ = 1, the start time τ_s is determined as follows. Here, we consider the cooling evolution of an accelerated particle whose Lorentz factor is γ_s at a time τ_s. Equation (22) represents the cooling evolution and has the analytic solution (Pacini & Salvati 1973)

$$\begin{eqnarray}\gamma (\tau ) & = & {\gamma }_{{\rm{s}}}{\left(\displaystyle \frac{{\tau }_{{\rm{s}}}}{\tau }\right)}^{{\alpha }_{{\rm{R}}}}{\left(\displaystyle \frac{{\gamma }_{{\rm{s}}}{\tau }_{{\rm{s}}}^{{\alpha }_{{\rm{R}}}}}{\xi {\tau }_{{\rm{syn}}}}({\tau }_{{\rm{s}}}^{-\xi }-{\tau }^{-\xi })+1\right)}^{-1}\\ & \approx & \xi {\tau }_{{\rm{syn}}}{\tau }^{-{\alpha }_{{\rm{R}}}}{({\tau }_{{\rm{s}}}^{-\xi }-{\tau }^{-\xi })}^{-1}\,\mathrm{for}\,{\gamma }_{{\rm{s}}}\to \infty ,\end{eqnarray} \tag{ 23 }$$

where ξ ≡ α_R − 2α_B − 1 > 0, for example, ξ = 3 and 5.5 for (α_R, α_B) = (1.0, −1.5) and (1.5, −2.5), respectively. From the last expression in Equation (23), we find that all the particles injected at τ = τ_s have Lorentz factor less than ${\gamma }_{\infty }({\tau }_{{\rm{s}}})\,\equiv \,\xi {\tau }_{{\rm{syn}}}/({\tau }_{{\rm{s}}}^{-\xi }-1)$ at τ = 1. For our purpose, we obtain the start time τ_s by solving ${\gamma }_{\infty }({\tau }_{{\rm{s}}})={\gamma }_{{\rm{\min }}}$, i.e.,

$$\begin{eqnarray}&&{\tau }_{{\rm{s}}}={\left(1+\displaystyle \frac{\xi {\tau }_{{\rm{syn}}}}{{\gamma }_{{\rm{\min }}}}\right)}^{-\tfrac{1}{\xi }}\approx {\left(\displaystyle \frac{{\gamma }_{{\rm{\min }}}}{\xi {\tau }_{{\rm{syn}}}}\right)}^{\tfrac{1}{\xi }}\,\mathrm{for}\,{\tau }_{{\rm{s}}}\ll 1,\\ \approx \left\{\begin{array}{ll}1.5\times {10}^{-2}{({\tau }_{{\rm{syn}}}/{10}^{8})}^{-1/3}{({\gamma }_{{\rm{\min }}}/{10}^{3})}^{1/3} & \mathrm{for}\,\xi =3.0,\\ 9.0\times {10}^{-2}{({\tau }_{{\rm{syn}}}/{10}^{8})}^{-2/11}{({\gamma }_{{\rm{\min }}}/{10}^{3})}^{2/11} & \mathrm{for}\,\xi =5.5.\end{array}\right.\end{eqnarray} \tag{ 24 }$$

From the last expressions, τ_s weakly depends on α_R, α_B, τ_syn, and also γ_min. Adopting τ_s given by Equation (24), signatures of the particles injected before τ_s do not appear in $\tilde{N}(\gamma ,\tau )$ at τ = 1 in the range of γ ≥ γ_min.

The particle spectrum $\tilde{N}(\gamma ,\tau )$ at τ = 1 is not sensitive to (α_R, α_B) for a given τ_syn. In Figure 3, we study the dependence of $\tilde{N}(\gamma ,1)$ on α_R and α_B, where we set (γ_min, γ_b, γ_max, p₁, p₂) = (10³, 10⁵, 10⁸, −1.5, −2.5) and n = 3. Thick upper lines are (α_B, τ_syn) = (−1.5, 10⁸) and we study the dependence on α_R for three different values 1.0 (dotted), 1.2 (dashed), and 1.5 (solid). Thin lower lines are (α_R, τ_syn) = (1.0, 10⁵) and we study the dependence on α_B for three different values −1.5 (dotted), −2.0 (dashed), and −2.5 (solid). We find no significant difference of $\tilde{N}(\gamma ,\tau =1)$ for the different values of both α_R and α_B, respectively. Note that particles of γ ≳ 10⁵ are responsible for the X-ray and TeV γ-ray emission. We adopt (α_R, α_B) = (1.0, −1.5) in this paper.

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