Energy-dependent nebula extent and spatially resolved spectra of the pulsar wind nebula 3C 58
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MNRAS 498, 1911–1919 (2020) doi:10.1093/mnras/staa2338
Energy-dependent nebula extent and spatially resolved spectra of the
pulsar wind nebula 3C 58
Fang-Wu Lu,1,2 Quan-Gui Gao1,2 and Li Zhang2‹
1 Department of Physics, Yuxi Normal University, Yuxi 653100, China
2 Department of Astronomy, Key Laboratory of Astroparticle Physics of Yunnan Province, Yunnan University, Kunming 650091, China
Accepted 2020 August 2. Received 2020 July 28; in original form 2020 June 7
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ABSTRACT
3C 58 is a pulsar wind nebula (PWN) that shows an interesting energy-dependent nebula extent and spatial variations of the
photon index and surface brightness in the X-ray band. These observations provide useful information with which to study the
spatially dependent radiative cooling of electrons and the energy-dependent transport mechanisms within the nebula. In this
paper, the energy-dependent nebula extent and spatially resolved spectra of this PWN are investigated in the framework of a
spatially dependent particle transport model. The observations of the nebula, including the photon spectral energy distribution,
spatial variations of the X-ray spectrum, and measurements of the nebula extent, can be naturally explained in this model.
Our results show that the energy-dependent nebula extent favours an advection–diffusion scenario with advection-dominated
transport, and the variations of the nebula extent with energy in the X-ray band can be attributed to the cooling losses of high-
energy electrons affected by synchrotron burn-off. Particle diffusion plays an important role in modifying the spatial variations
of the photon index and surface brightness in the X-ray band. The radial extents of the nebula at radio, GeV and TeV energies
are predicted by the model, indicating that the nebula extent of 3C 58 varies with energy in these bands. The analyses show that
the dependence of the adiabatic cooling rate and synchrotron radiation on the spectral index of injected particles is important for
changing the nebula extent at different energies.
Key words: radiation mechanisms: non-thermal – pulsars: general – ISM: individual objects: 3C 58 – X-rays: general.
Scheuer 1992; Bocchino et al. 2001; Slane et al. 2004, 2008). IRAS
1 I N T RO D U C T I O N
observations indicated a break in the spectrum between the radio
As one of the largest very-high-energy (VHE) γ -ray source classes and IR bands, implying that the injected particles do not conform
in the Galaxy, pulsar wind nebulae (PWNe) are created by the to a single power-law distribution (e.g. Green & Scheuer 1992). X-
interaction of pulsar winds with the surrounding media. These ray observations revealed a non-thermal spectrum from the nebula
pulsar winds are composed of high-energy particles and magnetic that varies with radial distance (e.g. Slane et al. 2004; Gotthelf,
fluxes and are powered by the central rotation-powered pulsars (e.g. Helfand & Newburgh 2007; An 2019), and that the power-law photon
Goldreich & Julian 1969; Kennel & Coroniti 1984; Reynolds & index changes from 1.9 in the centre to 2.8 at the edge of the PWN
Chevalier 1984). Observations indicate that in some PWNe the in the energy range 3–10 keV (Gotthelf et al. 2007). In particular,
photon spectra vary with radial distance, and the morphologies according to NuSTAR measurements, the nebula of 3C 58 has an
depend on the energy in the X-ray and/or TeV γ -ray bands. They interesting energy-dependent morphology; that is, the nebula size
thus provide useful information on the energy-dependent transport decreases with increasing energy in the X-ray band (An 2019). γ -ray
and spatially dependent radiative cooling mechanisms of the particles emission from the nebula has been detected by Fermi-LAT (Abdo
within the nebula. et al. 2009; Ackermann et al. 2013; Ajello et al. 2017) and the MAGIC
3C 58 is a composite supernova remnant (SNR) powered by the telescopes (Anderhub et al. 2010; Aleksić et al. 2014): the spectrum
central pulsar PSR J0205 + 6449 (Murray et al. 2002). It was in the energy range 0.1–316 GeV was found to be well described
classified as a PWN owing to its flat radio spectrum and very bright by a power-law function with a photon index of 1.61 ± 0.21, and
core (Weiler & Panagia 1978). Radio observations revealed that 3C the photon index in the energy range 0.4–10 TeV was derived to be
58 has a filled-centre elliptical morphology with a compact size 2.4 ± 0.2stat ± 0.2sys . These high-quality observations make 3C 58
of 6 arcmin × 9 arcmin (Reynolds & Aller 1988; Bietenholz, a particularly important laboratory for the study of the physics of a
Kassim & Weiler 2001; Bietenholz 2006), and the spectral index PWN.
of the flat radio spectrum has been estimated to be −0.10 ± 0.02 The non-thermal photon emission from 3C 58 has been exten-
(Wilson & Weiler 1976; Green 1986). Similar morphologies have sively studied in the literature. It is generally believed that the
been observed in the infrared (IR) and X-ray bands (Green & radio, IR and X-ray emission from the nebula is produced by the
synchrotron radiation of relativistic electrons within the PWN, and
the GeV and TeV γ -rays are generated through inverse Compton (IC)
E-mail: lizhang@ynu.edu.cn scattering off various soft background photon fields (e.g. Tanaka &
C Crown copyright 2020.
This article contains public sector information licensed under the Open Government Licence v3.0
(http://www.nationalarchives.gov.uk/doc/open-government-licence/version/3/).1912 F.-W. Lu, Q.-G. Gao and L. Zhang
Takahara 2013; Torres, Cillis & Martı́n Rodriguez 2013; Lu et al. conditions, the spectrum of the injected particles can be fitted as a
2017b; Li et al. 2018; van Rensburg et al. 2020). Furthermore, it broken power law (Sironi & Spitkovsky 2011). Therefore, in this
is often suggested that the spatial variations of the non-thermal paper a broken power law with different indices and some break-
X-ray spectrum are affected by the spatially dependent diffusion energy inject spectrum is adopted, which has been widely used to
transport and synchrotron radiative cooling of electrons within the reproduce the broadband photon emission from Crab, G21.5 – 0.9,
nebula (e.g. Porth et al. 2016; Lu et al. 2017b; van Rensburg et al. 3C 58, G54.1 + 0.3, G0.9 + 0.1, MSH 15-52, Kes 75, HESS J1356 –
2020). An (2019) attributed the energy-dependent morphology of 645 and other PWNe (e.g. Zhang, Chen & Fang 2008; Tanaka &
3C 58 measured in the X-ray band to the synchrotron burn-off Takahara 2010, 2013; Bucciantini, Arons & Amato 2011; Martı́n,
effect according to synchrotron radiation scenarios, but the energy- Torres & Rea 2012; Torres et al. 2014; Gelfand, Slane & Temim
dependent transport mechanisms of the particles within the nebula are 2015). In this case, the distribution of the injected particles can be
still unclear. Thus, in this paper, the non-thermal radiative properties expressed as
of 3C 58 are revisited in the framework of the spatially dependent γ −α
model proposed by Lu, Gao & Zhang (2017a), and the investigations ( γb ) 1 for γ < γb
Qinj (γ , t) = Q0 (t) , (2)
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are focused on the energy-dependent extent of the nebula. ( γγb )−α2 for γ ≥ γb
The paper is organized as follows. A brief description of the
spatially dependent model and the modelling results of 3C 58 are where γ b is the break energy, α 1 and α 2 are the spectral indices,
presented in Section 2. In Section 3, the effects of multiple mecha- and Q0 (t) is the normalization constant that is determined by (1 −
∞
nisms on the energy-dependent extent of the PWN are investigated. η)L(t) = 0 γ me c2 Qinj (γ , t)dγ , with η the fraction of the spin-
Finally, the conclusions and a discussion are given in Section 4. down luminosity L(t) converted into magnetic flux.
For a pure dipole radiative pulsar, the braking index has a value of
n = 3, and the temporal evolution of the spin-down luminosity can
2 M O D E L A N D R E S U LT S be given by
− n+1
2.1 Model description t n−1
L(t) = L0 1+ , (3)
In this section, the spatially dependent model proposed by Lu et al. τ0
(2017a) is briefly reviewed. Assuming that the PWN is spherically where L0 is the initial luminosity and τ 0 is the initial spin-down
symmetric, the evolution of electrons within the nebula can be time-scale of the pulsar, which is calculated by τ 0 = 2τ c /(n − 1)
described by the Fokker–Planck transport equation − Tage , with τ c the characteristic age and Tage the age of the pulsar.
∂ne ∂ 2 ne 1 ∂ ∂ne 1 ∂ 2 The inner boundary condition located at the TS can be given by (e.g.
= κ 2 + 2 (r 2 κ) − V − 2 r V ne Ng & Gleeson 1975)
∂t ∂r r ∂r ∂r r ∂r
∂ ∂ne Qinj (γ , t)
+ [γ̇ ne ] + Qinj . (1) V0 ne − κ(r0 , γ , t) = , (4)
∂γ ∂r 4π r02
In the above equation, ne = ne (r, γ , t) is the number of electrons per
as a result of the fact that the number of particles that flow through
energy and volume; κ = κ(r, γ , t) denotes the diffusion coefficient;
the shock must be equal to the number of injected particles. To
V = V(r) is the bulk velocity of electrons; and γ̇ = γ̇ (r, γ , t) is
simulate the particles escaping from the edge of the PWN, a free-
the summation of energy losses due to adiabatic expansion γ̇ad ,
escape condition is imposed at the outer boundary in the model for
synchrotron radiation γ̇syn and IC scattering γ̇IC . The function Qinj
simplicity. It should be noted that, in a realistic scenario, only some
is the distribution of particles injected from the termination shock
of the particles can escape from the edge of the PWN owing to
(TS).
the confinement of the ambient medium; therefore, a partial escape
According to Kennel & Coroniti (1984), the bulk velocity of the
boundary should be used. However, it was found that the choice of
particles is described by V(r) = V0 v(r), with V0 the velocity at the
the condition at the outer boundary, for example a free-escape or
TS and v(r) the radial profile of the velocity. The spatially dependent
no-escape condition, does not affect the solutions (e.g. Vorster &
magnetic field in the PWN is assumed to be B(r, t) = B0 (t)[r0 /rv(r)],
Moraal 2013; Porth et al. 2016).
where B0 (t) is the magnetic field at the TS, and r0 = r0 (t) is the radius
As the injection electrons propagate away from the centre of the
of the shock (Lu et al. 2017a). The dependences of particle diffusion
PWN, there are two important physical processes responsible for the
on radius and on energy are assumed to be modelled as κ r ∝ 1/B(r, t)
broadband photon emission from the nebula, namely synchrotron
and κ γ ∝ γ δ , respectively (e.g. Caballero-Lopez et al. 2004; Aguilar
radiation and IC scattering. For synchrotron radiation, the emission
et al. 2016). Thus, the particle diffusion coefficient can be expressed
coefficient can be given by (Rybicki & Lightman 1979; Blumenthal &
as κ(r, γ , t) = κ0 (t)(Ee /1 TeV)δ [rv(r)/r0 ], where Ee = γ me c2 is the
Gould 1970)
energy of electrons, and the constant δ is taken to be 0.333 (e.g. Lu,
∞
Gao & Zhang 2020).
Qsyn (r, ν, t) = ne (r, γ , t)Psyn (ν, γ , B(r, t))dγ , (5)
Observations of PWNe indicate that the particles injected into 0
the nebula at the TS can be divided into two components: a low-
with Psyn (ν, γ , B(r, t)) the power emitted by a single relativistic
energy component that is responsible for producing the radio and
electron, which is expressed as
GeV emission, and a high-energy component that produces the X-
√ 3
ray and TeV emission (e.g. Gaensler & Slane 2006; de Jager & 3e B(r, t) ν
Djannati-Ataı̈ 2009). Theoretical studies predicted that the shape of Psyn (r, ν, γ ) = 2
F , (6)
hνme c νc (γ , B(r, t))
the particle population downstream of the relativistic TS can be well
described by a Maxwellian with a high-energy power-law tail (e.g. where ν c (γ , B(r, t)) = 3eB(r, t)γ 2 /(4π me c) is the critical frequency,
∞
Spitkovsky 2008; Sironi & Spitkovsky 2009). Furthermore, recent F (y) = y y K5/3 (z)dz with y = ν/ν c , and K5/3 denotes the modified
particle-in-cell simulations have shown that, under certain physical Bessel function of order 5/3.
MNRAS 498, 1911–1919 (2020)Energy-dependent nebula extent of 3C 58 1913
Table 1. Values of the parameters for the spatially dependent model used to Fig. 1 shows a comparison of the modelling result with the
reproduce the observations of 3C 58. observed photon spectral energy distribution (SED) of 3C 58. In the
calculation, the soft photon fields involved in the IC scattering are as
Symbol Value Symbol Value follows: the CMB with temperature TCMB = 2.7 K has an energy
Esn 1.0 × 1051 erg L0 2.5 × 1038 erg s−1 density of UCMB = 0.26 eV cm−3 ; the FIR has TFIR = 25 K and
Mej 10.0 M Tage 2 kyr UFIR = 0.6 eV cm−3 ; and the NIR has TNIR = 2800 K and UNIR =
ρ ism 0.1 cm−3 η 7.3 × 10−3 3.2 eV cm−3 . From the modelling results, the current magnetic field
P 65.7 ms γb 6.5 × 104 of the nebula at the TS is derived to be 15.9 μG, and the spatial
Ṗ 1.93 × 10−13 s s−1 γ max 1.0 × 109 average has a value of 26.8 μG. The current diffusion coefficient is
n 3 α1 1.0 estimated to be 7.1 × 1025 cm2 s−1 at the shock and has a spatial
τc 5.4 kyr α2 3.03 average of 4.4 × 1025 cm2 s−1 at an electron energy of 1 TeV. It can
d 2.0 kpc κ0 1.3 × 1021 cm2 s−1 be seen from Fig. 1 that the photon spectra in the radio, IR and X-
ray bands are produced by synchrotron radiation of the relativistic
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The emission coefficient of the IC scattering can be expressed as electrons within the nebula, while the GeV and TeV γ -rays are
(Blumenthal & Gould 1970) generated mostly from IC scattering off the FIR and NIR soft photons.
∞ With the parameters listed in Table 1, the current radius of the
Qics (r, x, t) = ne (r, γ , t)Pics (r, γ , ν, t)dγ , (7) nebula of 3C 58 is estimated to be 2.24 pc, and the radius of the
0 reverse shock has a value of 6.23 pc according to the dynamical
with Pics (r, γ , ν, t) the scattered photon spectrum per electron, which models of the SNR and PWN presented by Truelove & McKee
is given by (1999) and Bucciantini et al. (2011), indicating that the nebula is
∞ inside the freely expanding part of the supernova; in other words,
3 σT c nγ (r, νi , t)
Pics (r, γ , ν, t) = f (q, )dνi , (8) the PWN is expanding inside the cold ejecta. In this case, there are
4 γ2 0 νi no interactions between the PWN and the reverse shock; therefore,
where ν i represents the frequency of the scattered photon and nγ (r, the central nebula is independent of the evolution of the SNR shell.
ν i , t) is the distribution of the target photon fields, including the cos- In the calculation, following Bucciantini et al. (2011), the number
mic microwave background (CMB), the galactic far-infrared (FIR) density of the surrounding ISM of 3C 58 is adopted to be 0.1 cm−3 .
background, and the near-infrared (NIR) and optical photon field However, we note that the evolution of the reverse shock is strongly
due to the stars, as well as to the synchrotron photons. The function dependent on the density of the ambient ISM; that is, the radius of
f(q, ), taking the Klein–Nishina cross-section into account, can be the reverse shock decreases with increasing ambient medium density
expressed as (see equation 83 of Truelove & McKee 1999). Moreover, with the
age of 3C 58 estimated to be 2 kyr, we worked out that, when the
1 ( q)2
f (q, ) = 2q ln q + (1 − q) (1 + 2q) + , (9) value of the density of the ambient ISM is less than 9.0 cm−3 , the
2 (1 + q) ambient medium density will not be important for the evolution of
with the central nebula. In this scenario, the radius of the PWN is less than
hν the radius of the reverse shock, and the nebula continues to expand
q= and = 4γ hνi /me c2 . (10) inside the cold ejecta. Thus, although the value of the density of the
(γ me c2 − hν)
ISM is adopted to be 0.1 cm−3 in the framework of our model, the
According to the kinematics of the IC scattering, the parameter q is adoption of other values that are less than 9.0 cm−3 will not affect
limited to the range 1/4γ 2 < q < 1. the model results.
In combination with the dynamical evolution model of PWNe To illustrate the contribution of electrons with different energies to
proposed by Bucciantini et al. (2011), the transport equation (1) the photon emission from the nebula, the photon spectra for electrons
of electrons can be solved numerically. Then, the electron spectra in the energy ranges of 0.001−0.1 TeV, 0.1−4.0 TeV and 4.0−300.0
and corresponding non-thermal photon spectra of the nebula at TeV are calculated in the framework of our model. As shown in Fig. 1,
different positions can be obtained. Note that the parameters in- lower-energy electrons are responsible for producing the synchrotron
volved in the dynamical model, namely the number density of the radio spectrum; medium-energy electrons are responsible for the
surrounding interstellar material (ISM) and the mass and energy of synchrotron IR emission; and higher-energy electrons emit X-ray
the progenitor supernova ejecta, are denoted by ρ ism , Mej and Esn , synchrotron photons. Note that the break energy of the injected
respectively. particles has a value of 0.033 TeV according to the modelling results.
Therefore, electrons in the energy band 0.001−0.1 TeV, which are
responsible for producing the synchrotron radio spectrum, can be
2.2 Modelling results
divided into two populations: a lower-energy population with a
The model described above is now applied to 3C 58. Observations spectral index of 1.0 in the energy range 0.001–0.033 TeV; and a
indicate that the central pulsar of 3C 58 (PSR J0205 + 6449) higher-energy population with a spectral index of 3.03 in the energy
has a rotational period of P = 65.7 ms, a period derivative of range 0.033 − 0.1 TeV. Knowledge of the contribution of electrons
Ṗ = 1.93 × 10−13 s s−1 , and a characteristic age of τc = 5.4 kyr of different energies to radiation is important for studying the energy-
(Murray et al. 2002; Livingstone et al. 2009). According to recent dependent extent of PWNe (e.g. HESS Collaboration 2020).
H I measurements, the distance to the PWN is estimated to be The modelling results of the radial variations of both the photon
d = 2 kpc (Kothes 2013). The age of the system is usually suggested index and the surface brightness in the energy range 3–10 keV of
to be between 0.8 and 7 kyr from the radio expansion of the PWN 3C 58 are shown in Fig. 2, where the solid lines are the results
(Bietenholz 2006). The values of the parameters for the spatially calculated by our model with both advection and diffusion, and the
dependent model used to reproduce the observed data of 3C 58 are dashed lines represent the results for a pure advection scenario. In
listed in Table 1. the pure advection model, particle transport within the nebula is
MNRAS 498, 1911–1919 (2020)1914 F.-W. Lu, Q.-G. Gao and L. Zhang
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Figure 1. Modelling results of the photon SED of 3C 58 with the parameters listed in Table 1. The observations are taken from Green (1986), Morsi & Reich
(1987), Salter et al. (1989), Planck Collaboration et al. (2016) (radio), Slane et al. (2008) (IR), Torii et al. (2000), An (2019) (X-rays), Abdo et al. (2013),
Ackermann et al. (2013) and Aleksić et al. (2014) (γ -rays). The black solid line represents the total flux. Also shown are the photon emission of synchrotron
radiation (long-dashed line) and IC scattering with the CMB (dotted line), FIR (dash–dotted line), NIR(dash−dot–dotted line), and the synchrotron photons
(short-dashed line) calculated with our model. The synchrotron and IC scattering for the electrons in the energy bands of 0.001−0.1 TeV (orange line), 0.1−4.0
TeV (violet line) and 4.0−300.0 TeV (magenta line) are also shown to illustrate the contribution of electrons of different energies to the radiation.
Figure 2. Comparisons of modelled radial variations of the photon index (left panel) and surface brightness (right panel) with the X-ray data of 3C 58 in the
energy range of 3−10 keV. The data are taken from Gotthelf et al. (2007). The solid and dashed lines are the results calculated with the advection-diffusion
and the pure advection scenario, respectively. The black and red dots are the observations of the European Photon Imaging Gamera (EPIC) pn and EPIC MOS
imagers respectively.
assumed to be advection-dominated and the diffusion is neglected. because these observations cover the entire emission region of the
For the synchrotron radiation scenarios, the increase in the photon nebula.
index of the X-ray spectrum with radius is usually attributed to the Fig. 3 shows the half-power radii (HPRs) of 3C 58 at various
cooling influence of the synchrotron burn-off of the high-energy energies calculated in our model with the parameters listed in Table 1,
electrons (e.g. Slane et al. 2004; Gotthelf et al. 2007; An 2019). where the HPR is defined as the radius of the inner emission region
However, it can be seen from Fig. 2 that particle diffusion also that accounts for half of the total flux of the entire nebula. The
plays a very important role in modifying the photon index and observed data of the energy-dependent radius of the nebula in the
surface brightness of the X-ray spectrum with radius. Note that, X-ray band are from An (2019). It can be seen from Fig. 3 that
for the X-ray data, the XMM–Newton observations in the band the HPRs of the nebula vary with energy in the different bands in
3–10 keV given by Gotthelf et al. (2007) are taken as a benchmark, our model. According to the published explanations, the variation
MNRAS 498, 1911–1919 (2020)Energy-dependent nebula extent of 3C 58 1915
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Figure 3. Variation of the energy-dependent radial extent of 3C 58. The Figure 4. Calculated results of the radial extent of 3C 58 for scenarios A
observations of the HPRs (red squares)in the X-ray band are taken from An (blue line), B (orange dashed line), C (magenta dashed line) and D (green
(2019). line). The solid black line is the result in the benchmark case, as shown in
Fig. 3. The red squares are the observed data obtained from An (2019).
of the nebula extent with energy in the X-ray band is due to the
synchrotron burn-off effect of the high-energy electrons; that is, Scenario D: Same as scenario A, but the adiabatic loss is taken
the higher-energy electrons, which are responsible for producing into account and a spatially dependent magnetic field is used.
the higher-energy synchrotron X-ray photons, suffer more severe With the parameters listed in Table 1, the HPRs of the nebula of 3C
radiation losses as they propagate away from the centre of the nebula, 58 at different energies are calculated for the above four scenarios.
resulting in the propagation distances of higher-energy electrons The results are shown in Fig. 4.
being shorter than those of the lower-energy populations (e.g. Slane
et al. 2004; An 2019). The same parent electrons are responsible 3.1 Influence of adiabatic loss
for producing the TeV γ -ray spectrum via IC scattering, and this
leads to the nebula extent also varying with energy in the TeV γ - It can be seen from Fig. 4 that the HPRs are independent of the
ray band. These scenarios could be used to explain the dependence energy in scenario A, implying that spatially dependent advection has
of the nebula extent on the energy in the X-ray and TeV γ -ray no direct impact on the energy-dependent nebula extent. In scenario
bands. For the variations of the HPRs at the radio and GeV γ - B, with the adiabatic loss considered, the modelling results reveal
ray energies shown in Fig. 3, however, a more detailed investigation, that the HPRs vary with energy in the radio and hard X-ray bands,
which includes the influence of particle transport mechanisms such as and two bumps can be seen in the GeV band. These results can be
advection and diffusion, is needed. Thus, in Section 3, a more detailed explained by the following analyses.
discussion will be presented to demonstrate the influences of multiple In a steady-state system with a constant bulk velocity V0 , the
mechanisms on the energy-dependent nebula extent of 3C 58. particle transport equation (1) in scenario B can be simplified to
∂fe 2γ ∂fe
− =0, (11)
∂r 3r ∂γ
3 E N E R G Y- D E P E N D E N T N E B U L A E X T E N T
where the function fe = ne (r, γ )/γ 2 . Solving the above equation with
In this section, the energy-dependent nebula extent of 3C 58 is the characteristic method and using the inner boundary condition
investigated for various scenarios. To compare the results, the model presented in equation (4), the evolution of particles within the nebula
result shown in Fig. 3 is used as a benchmark to discuss the influences can be given by
of the various mechanisms, such as synchrotron loss versus adiabatic 2(α1 +2)
expansion and advection versus diffusion, on the energy-dependent Q0 ( γγb )−α1 ( rr0 )− 3
extent of the nebula. ne (r, γ ) = 2(α +2) , (12)
4πr02 V0 ( γ )−α2 ( r )− 23
Because the bulk velocity, magnetic field and diffusion are consid- γb r0
ered to be spatially dependent in our model, we examine the effects of for γ < γ b and γ ≥ γ b in the two equations, respectively. The
these quantities on the nebula extent. For the sake of discussion, four above analytical solution shows that the injected electrons with a
scenarios for the particle transport and radiation within the nebula softer spectrum will suffer more severe adiabatic loss as they are
are considered, as follows. transported within the nebula. Thus, the decline of the number density
Scenario A: The bulk velocity is spatially dependent, and a of electrons with a softer injection spectrum along distance will be
spatially independent magnetic field within the nebula is adopted, but faster than that with a harder injection spectrum during the evolution
the particle diffusion and the energy losses resulting from adiabatic of the PWN. Furthermore, the HPRs of the photon spectra that are
expansion and radiation are neglected. produced by synchrotron radiation of the electrons with a harder
Scenario B: Same as scenario A, but the adiabatic loss is taken injection spectrum should be larger than those with a softer injection
into account. spectrum. However, it is important to note that temporal evolution
Scenario C: Same as scenario A, but the magnetic field in the and a more complicated bulk velocity of the particles are included in
nebula is spatially dependent. our model. Therefore, although the analytical solution can roughly
MNRAS 498, 1911–1919 (2020)1916 F.-W. Lu, Q.-G. Gao and L. Zhang
account for the variation of the nebula extent with the spectral index
of the injected electrons, the changes of the radial extent in scenario
B could be considerably more complicated.
In the radio band, the calculated result shows that the HPRs of the
nebula decrease with increasing energy as a result of adiabatic loss.
The radio photon spectrum is mostly produced by the synchrotron
radiation of the electrons in the energy range 0.001–0.1 TeV, which
includes two electron populations as detailed in Section 2.2. The
major contribution of the lower- and higher-energy populations of
electrons with different spectral indices to the radio spectra at low
and high energies, resulting in the HPRs vary with energy in the
radio band. These two electron populations are also responsible
for emitting the GeV γ -rays via the process of IC scattering and
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lead to variation of the HPRs with energy in the GeV band as
shown in Fig. 4, where the two bumps in the photon energy
are caused by the two soft photon fields in our model. The left
bump is caused by IC scattering off the soft NIR photons, while
the right bump is caused by IC scattering off the FIR photons (see
Fig. 1). Figure 5. Radial profiles of the emissivity of synchrotron radiation at spectral
The modelling results also reveal that the HPRs of the nebula indices of α = 1.0 (solid line) and α = 3.03 (dashed line).
vary with energy in the hard X-ray and VHE γ -ray bands in
scenario B. During the evolution of the PWN, although electrons are of the nebula than those with a harder injection spectrum. Thus, the
continuously injected from the TS, owing to cooling, an energetic HPRs in the radio band could be larger than those at IR and X-ray
electron experiences adiabatic loss and becomes a lower-energy energies, because the distribution of injected particles is assumed to
electron. Thus, the number of VHE electrons, which are responsible be a broken power law with different spectral indices for the low-
for producing the hard X-rays, will be lower at a larger radius and high-energy electrons in the model. Moreover, the changes of the
than the number of low-energy electrons. Therefore, the decrease HPRs with energy in the radio band are the result of the presence of
of the current number density of VHE electrons along distance is two energy populations with different spectral indices in the parent
faster. Thereby, the HPRs of the nebula will decrease with increasing electrons of the synchrotron radio spectrum.
energy in the hard X-rays, as is shown in Fig. 4. Because the X- According to the emissivity of IC scattering given in equation (7),
rays and TeV γ -rays are generated from the same parent electrons the magnetic field has no direct impact on the IC scattering spectrum.
via different radiative processes, and because a uniform magnetic Thus, the HPRs are independent of energy in the GeV and TeV energy
field is adopted in scenario B, the variations of the HPRs with bands, which are the same as those obtained for scenario A. However,
energy in the TeV γ -ray band could be same as those shown in the the emissivity of synchrotron radiation given in equation (5) indicates
X-ray. that the critical frequency of a synchrotron photon also depends
on the magnetic field, which leads to changes of the nebula HPRs
with energy in the very hard X-rays, as shown in Fig. 4, because
3.2 Influence of the magnetic field
the spatially dependent magnetic field is taken into account in
In scenario C, a spatially dependent magnetic field is taken into scenario C.
account to investigate the effect of the magnetic field on the energy- In scenario D, both the adiabatic loss and the spatially dependent
dependent extent of the nebula. It can be seen from Fig. 4 that the magnetic field are taken into account. It can be seen that the nebula
HPRs in the radio band are different from those in the IR and X-ray HPRs vary with energy, as shown in Fig. 4, as a result of the adiabatic
bands, while the radii at the GeV and TeV energies are the same cooling of electrons and the influence of the spatially dependent
as those obtained from scenario A. The differences of the HPRs in magnetic field. Comparisons of the results of scenario D with those
the radio, IR and X-ray bands arise because the photon spectra in in the benchmark case show that the changes of the HPRs with
these bands are produced by synchrotron radiation, which depends energy in the X-ray and TeV γ -ray bands in the benchmark case
strongly on the magnetic field. are more significant than those in scenario D, because diffusion
According to Blumenthal & Gould (1970), the emission coefficient transport and the synchrotron radiative cooling of electrons are
of synchrotron radiation given by equation (5) can be approximated involved in the benchmark case. Therefore, in the following, the
as effects of synchrotron loss and particle diffusion on the energy-
dependent extent of 3C 58 will be studied in the framework of our
Qsyn (r, ν, t) ∝ [B(r, t)](α+1)/2 , (13)
spatially dependent model.
where α is the spectral index of the injection electrons. The ap-
proximated analytical solution indicates that the radial profile of the
3.3 Synchrotron loss versus adiabatic expansion
synchrotron emissivity is determined mostly by the dependence of
the magnetic field on the radius within the PWN. To illustrate the To illustrate the importance of diffusion relative to advection and
influence of a spatially dependent magnetic field on the nebula extent, of synchrotron loss relative to adiabatic expansion, the ratios of
the radial profiles of the emissivity of synchrotron radiation with the time-scales of these quantities at the positions of 15 and 195
spectral indices of α = 1.0 and α = 3.03 are shown in Fig. 5, where arcsec are shown in Fig. 6, where τ ad , τ syn , τ adv and τ dif are the
the radial profile of the magnetic field is obtained from Kennel & time-scales of adiabatic expansion, synchrotron loss, advection and
Coroniti (1984). It can be seen that the electrons with a softer injection diffusion, respectively. When τ syn /τ ad 1, the cooling is dominated
spectrum will produce more synchrotron photons in the inner regions by adiabatic expansion, while it is dominated by synchrotron loss
MNRAS 498, 1911–1919 (2020)Energy-dependent nebula extent of 3C 58 1917
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Figure 8. Calculated results of the radial extent of 3C 58 in the pure diffusion
Figure 6. Ratios of τ syn /τ ad (solid lines) and τ dif /τ adv (dashed lines) at the
(blue line) and pure advection (magenta line) scenarios. The solid black line
positions of 15 arcsec (blue lines) and 195 arcsec (red lines). The vertical
is the result in the benchmark case, as shown in Fig. 3. The red squares are
magenta dot–dashed lines indicate electron energies of 0.001, 0.1, 4.0 and
the observations taken from An (2019).
300.0 TeV, from left to right.
The cooling mechanism that affects the energy-dependent nebula
extent for the X-ray emission from 3C 58 is consistent with those
suggested by Slane et al. (2004) and An (2019) based on Chandra
and NuSTAR data, respectively; that is, the changes of the nebula
extent with energy in the X-ray band are caused by the synchrotron
burn-off effect.
3.4 Advection versus diffusion
The particles injected from the shock will propagate away from
the centre of the nebula, because of the transport of advection
and diffusion (e.g. Gaensler & Slane 2006). Furthermore, particle
transport is expected to be dominated by diffusion once the pressure
inside the nebula has reduced to the pressure of the ambient material
during the evolution of the PWN, because this means that the particles
are no longer strictly confined within the nebula and begin to diffuse
into the ambient medium. In turn, if the particle pressure inside the
nebula remains greater than that of the ambient material out to large
distances, the transport may be dominated by the bulk particle flow,
Figure 7. Calculated results of the radial extent of 3C 58 in the pure as a result of the significant confinement of the ambient medium.
synchrotron loss (blue line) and pure adiabatic loss (magenta line) scenarios. To investigate the influences of various particle transport mecha-
The solid black line is the result in the benchmark case, as shown in Fig. 3. nisms on the energy-dependent nebula extent of 3C 58, we calculated
The red squares are the observations taken from An (2019). the HPRs for the scenarios of pure advection and pure diffusion,
and the results are shown in Fig. 8. In the pure advection scenario,
for τ syn /τ ad 1. Furthermore, when τ dif /τ adv 1, the particle diffusion is neglected, and in the pure diffusion scenario, advection is
transport is advection-dominated, but it is diffusion-dominated when not taken into account. It can be seen that the variations of the nebula
τ dif /τ adv 1. The results reveal that the transport and cooling of the extent with energy are mostly affected by the transport mechanism
lower-energy population of electrons are dominated by advection and of particle advection, and that diffusion transport has only a small
adiabatic expansion, respectively, and that diffusion and synchrotron effect on the HPRs of the X-ray and TeV γ -ray spectra. This is
loss dominate for the higher-energy particles. because particle transport within the nebula of 3C 58 is dominated
Fig. 7 shows the HPRs of 3C 58 calculated for the scenarios by advection in the framework of our spatially dependent model,
of pure synchrotron and adiabatic losses, respectively. In the pure and, compared with advection, diffusion is important only for the
synchrotron loss scenario, adiabatic expansion is neglected, while transport of VHE electrons (see Fig. 6).
the cooling arising from synchrotron loss is not considered in the
pure adiabatic loss scenario. The results indicate that the variations
4 S U M M A RY A N D D I S C U S S I O N
of the HPRs with energy in the radio, IR and GeV bands are most
affected by the cooling mechanisms of adiabatic loss, while in the The observed features of 3C 58, including the photon emission from
X-ray and TeV γ -ray bands they can be attributed to the synchrotron the nebula and the energy-dependent morphology, are investigated in
radiative cooling effect, because the higher-energy electrons suffer the framework of the spatially dependent model proposed by Lu et al.
more severe radiation loss as they are transported within the nebula. (2017a). In this paper, the photon SED of the PWN is calculated first,
MNRAS 498, 1911–1919 (2020)1918 F.-W. Lu, Q.-G. Gao and L. Zhang
then the radial variations of the photon index and surface brightness filamentary structure is present throughout the body of the central
of the X-ray spectrum are obtained, and finally the HPRs of the nebula (e.g. Wilson & Weiler 1976; Reynolds & Aller 1988; Slane
nebula at different energies are found. et al. 2004; Bietenholz 2006), revealing that a disrupted toroidal
The observed photon SED, radial variations of the X-ray spectrum, and an even more complicated magnetic field should be taken into
and decreases of the HPRs with energy in the X-ray band of 3C 58 can account to study the detailed radiative properties of the PWN.
be naturally explained by the spatially dependent particle transport Moreover, the images reveal that the PWN of 3C 58 is not quite
model. The changes of HPRs with energy in the X-ray band can symmetric and the pulsar is located closer to the western edge of
be attributed to the advection-dominated transport and to cooling the nebula, as a result of the proper motion of the pulsar or a higher
effects dominated by synchrotron loss. On the one hand, the cooling medium density in the western direction (e.g. Slane et al. 2004).
mechanisms accounting for the nebula extent of 3C 58 in the X- Thus, although our model can account for most of the observations
ray band are consistent with the results suggested by Slane et al. of 3C 58, a more complicated magnetic field and the proper motion of
(2004) and An (2019). On the other hand, the energy-dependent the pulsar should be included in the model to investigate the transport
nebula extent of 3C 58 described by advection-dominated transport and radiative cooling mechanisms within the nebula more accurately.
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within the nebula is consistent with the results of HESS J1825-137 In summary, the energy-dependent extent of 3C 58 favours
obtained by the HESS Collaboration et al. (2019), according to the advection-dominated transport within the nebula, and the greatest
HESS observed data. It is important to note that, although particle effect of particle diffusion is to modify the spatial variations of the
transport within the nebula is advection-dominated and the diffusion X-ray spectrum. The influences of cooling mechanisms on the nebula
has only a small effect on the energy variations of the HPRs of the extent are dominated by synchrotron loss in the X-ray and TeV γ -
nebula, particle diffusion is very important for modifying the radial rays, while adiabatic loss dominates in the radio, IR and GeV bands.
variations of the photon index and the surface brightness of the X-ray The variations and differences of the nebula extent in the radio and X-
spectrum (see Fig. 2). ray bands may be important for studying the distribution of injection
Observations show that the nebula of 3C 58 has a similar particles and further investigating particle acceleration within the
morphology in the radio, IR and X-ray energy bands, with a compact PWN. The HPRs of 3C 58 in the radio, IR, GeV and TeV γ -ray bands
size of 6 arcmin × 9 arcmin (e.g. Reynolds & Aller 1988; Bietenholz have not been reported in the literature. Therefore, more high-quality
et al. 2001; Bocchino et al. 2001; Slane et al. 2004, 2008). However, data are required to test the predicted results of the energy-dependent
according to IRAS observations, the photon spectrum shows a break nebula extent of 3C 58 and to draw a firm conclusion.
between the radio and IR bands, indicating that a break in the
spectrum of the injected particles could be required (e.g. Green &
AC K N OW L E D G E M E N T S
Scheuer 1992; Slane et al. 2008). Moreover, in combination with the
Spitzer and Chandra X-ray observed data, Slane et al. (2008) found We would like to thank the anonymous referee for very valuable
that the fluxes in the mid-IR band are consistent with those predicted comments. This work is partially supported by the National Key
by the extrapolation of the soft X-ray spectrum. This means that the R & D Program of China under grant no. 2018YFA0404204, by the
mid-IR photons and X-rays are produced by synchrotron radiation of National Natural Science Foundation of China (NSFC 11803027,
the same parent electrons. Thus, a break in the power-law distribution U1738211), and by Yunnan local colleges applied basic research
of the injected particles for 3C 58 should be adopted, which leads projects (2017FH001-17, 2018FH001-015, 2019FH001-012).
to the HPRs of the nebula in the radio bands being different from
those in the IR and X-ray bands, because the adiabatic cooling rate
DATA AVA I L A B I L I T Y
strongly depends on the spectral index of the injected particles (see
equation 12). No new data were generated or analysed in support of this research.
The nebula HPRs in the radio, IR, GeV and TeV bands predicted
by our model are shown in Fig. 3, and indicate that the nebula extent
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