Evolution of High-energy Electron Distribution in Pulsar Wind Nebulae

Pulsar wind nebulae (PWNe) are dynamic and energetic structures in the cosmos driven by the rotational energy released from pulsars. These nebulae consist of bubbles filled with magnetized relativistic particles, primarily electrons and positrons, which interact with the surrounding medium to emit a wide range of electromagnetic radiation from radio waves to TeV gamma-rays (e.g. Slane 2017; Amato 2024). Particle acceleration occurs at the termination shock within the PWN (Kennel & Coroniti 1984), creating a complex system where particle transport and radiation processes shape their observed properties. PWNe are considered as potential sources of high-energy electrons and positrons of cosmic rays (e.g. Fiori et al. 2022). The observational results from the Large High Altitude Air Shower Observatory (LHAASO) provide strong evidence for the particle acceleration capability of PWNe. The high-energy gamma-ray photons above 1 PeV detected from Crab Nebula not only confirm that PWNe are efficient particle accelerators in the Galaxy, but also make it to be the first officially recognized PeV-scale leptonic sources (Collaboration*† et al. 2021). Moreover, most of the very-high-energy gamma-ray sources observed by LHAASO are associated with pulsars (Cao et al. 2024).

Theoretical models, including pure leptonic scenarios, have been proposed to explain the observed multiband nonthermal photon spectra emitted by PWNe (e.g. Kennel & Coroniti 1984; de Jager et al. 2008; Zhang et al. 2008; Gelfand et al. 2009; Fang & Zhang 2010; Tanaka & Takahara 2010; Bucciantini et al. 2011; Martín et al. 2012; Torres et al. 2014; Zhu et al. 2018, 2021, 2023). These models often involve two distinct particle populations that are accelerated within the PWN, such as the pulsar wind termination shock (Rees & Gunn 1974; Kennel & Coroniti 1984) or the nebula’s equatorial region (e.g. Atoyan & Aharonian 1996; Lyutikov et al. 2019). While synchrotron radiation is believed to generate radio to X-ray photons, and the GeV to TeV gamma-rays are typically produced through inverse Compton scattering processes involving ultrarelativistic electrons and positrons. Additionally, a possible hadronic origin for gamma-ray emission has been suggested Atoyan & Aharonian (1996); Bednarek & Protheroe (1997); Amato et al. (2003); Bednarek & Bartosik (2003), highlighting the complexity of particle interactions within PWNe.

Overall, PWNe serve as important astrophysical laboratories for studying the dynamics of relativistic particles, magnetic fields, and shock physics in extreme environments. By analyzing the broadband spectral energy distributions and evolution of PWNe, We can deepen the understanding of cosmic ray production and propagation mechanisms. Unlike the currently prevailing time-dependent models accounting of the nebula evolution and electron losses (e.g. Torres et al. 2014; Zhu et al. 2018), here we used the simplest phenomenological model (time-independent one-zone model) to fit the multiband spectra of 16 PWNe. Attempting to reveal the evolution of electrons in PWNe through the correlation between model parameters and age. In Section 2, we briefly describe our selected sample sources and models. Section 3 shows the results of the spectral fitting and their implications. Summary and discussion are drawn in Section 4.

Figure 1-17 illustrate the best-fit SEDs with the one-dimensional (1D) posterior probability distribution and two-dimensional (2D) confidence contours of parameters for 17 PWNe. The thick black line represents the total SED, while the green solid line depicts the synchrotron emission. The dashed, dash–dotted and dotted lines correspond to the ICS components with CMB and other background photon fields, respectively. References for the spectra and background photon field parameters can be found in the figure caption.

Table 1 presents the fundamental physical quantities alongside the best-fit values of the model parameters. It is evident that all spectral fits exhibit a reduced $\chi^{2}<2.0$, except for Crab, and HESS J1303-631. For PWNe with butterfly-shaped X-ray data, the minimum, maximum, and geometric mean energy values within the energy range will be utilized for participation in the multi-band fitting, such as Kes 75, HESS J1640$-$465, G54.1$+$0.3, G0.9$+$0.1, N157B, HESS J1356$-$645, VER J2227$+$608 and HESS J1418$-$609. The energy spectrum of the Crab nebula softens from 0.3 at radio wavelength to 0.6 in the optical range, and there is also thermal radiation in the infrared band (Reynolds et al. 2017), resulting in a poor fit with a single power law.

The TeV data from HESS observation and the GeV data detected by Fermi-LAT cannot be well connected. The TeV data of HESS observation (H. E. S. S. Collaboration et al. 2012) from HESS J1303-631 exhibit uneven fine structures in detail and do not match the GeV data detected by Fermi-LAT (Zhou et al. 2023), resulting in a relatively large chi-square value ($\chi^{2}=3.99$) for the fitting results. Generally, the multi-wavelength spectra can be well-fitted by one-zone model described by a broken power law with a high-energy cutoff.

The left panel of Figure 18 illustrates the high-energy electron distributions (a) and SEDs (c) of those 17 PWNe with the best-fit parameters. To demonstrate the evolution of PWNe, the right panel depicts the high-energy electron distributions normalized at 10 TeV and SEDs normalized at 100 GeV. It is observed that the high-energy electron spectra gradually harden with the ages of PWNe. Indeed, younger PWNe exhibit softer spectra in the gamma-ray band, while older PWNe show harder GeV spectra with a high-energy cutoff in the TeV band, which is opposite to the evolution of supernova remnants (Zeng et al. 2019).

Figure 19(a) and 19(b) display the correlation between the low-end and high-end energy spectral indices and the ages of PWNe. The results indicate that the low-end spectral index remains relatively constant with age, mainly between 1.0 and 1.5, while the high-end spectral index decreases with age. The origin of these electron spectra within PWNe is still under debate. Atoyan & Aharonian (1996) suggested the existence of two distinct populations of relativistic electrons within PWNe. The low-energy electrons with an index of $\sim 1.5$ are believed to be generated within the light cylinder of the pulsar, reflecting the history of the pulsar and the nebula, and primarily contributing to radio photons by synchrotron radiation, thus called “radio electrons”. Therefore, this process has not significantly changed over time, resulting in the constancy of the low-end spectral index. On the other hand, the high-energy electrons are freshly accelerated ultra-high-energy electrons with $E>E_{b}$, mainly produced at the wind termination shock location beyond the light cylinder, via Fermi acceleration, and form the high-energy part with an index $\geq 2$, also called as the “wind electrons” component. When the termination shock propagates outward, the magnetic field $B(r)$ will also decay (also see Figure 19(c)), and the cooling of high-energy electrons will slow down rapidly, resulting in a hardening of the energy spectrum. Many time-dependent models considering such magnetic field evolution exhibit the characteristic of hardening in the high-end energy spectrum ((e.g. Torres et al. 2014; Zhu et al. 2018)). Additionally, the radio electron possibly originates from relativistic magnetic reconnection, and its spectral slope is in the range of $1.0-2.0$ (e.g. Sironi & Spitkovsky 2011, 2012, 2014). The present of Arad et al. (2021) suggests a sub-population with a hard spectrum (radio photon spectral indices of $\alpha=0.01\pm 0.06$) near the termination shock and softer spectra elsewhere, possibly due to a recent evacuation of the shock surroundings. Note that the spectral index of the source VER J2227$+$608 deviates significantly from the clustering distribution of the sample sources, as the fitting data provided in this paper mainly come from the tail region of the radio morphology, which is generally considered to be a supernova remnant.

Figure 19(c), (d) and (e) illustrates the correlations between the average magnetic field, the break energy, and the cutoff energy of the electron distribution and the age of PWNe. It is observed that the magnetic field gradually decreases with the evolution of PWNe. Two theoretical lines are provided in the figure based on the dynamical evolution of PWNe (the conversion of the pulsar’s rotational energy into magnetic energy)(Torres et al. 2014): when $T_{age}\ll\tau_{0}$(where $\tau_{0}$ is the initial spin-down timescale), the magnetic field follows $B\propto t^{-1.3}$, and when $T_{age}\gg\tau_{0}$, $B\propto t^{-1.8}$. From Figure 19(d), it can be seen that the break energy of the electron distribution shows no significant correlation with the age of PWNe. In traditional time-dependent evolution models, the magnitude of the break energy of the injected spectral lies between 10 GeV and 1.0 TeV (e.g Torres et al. 2014; Zhu et al. 2018), which is basely consistent with the break energy in our non-time-dependent model, although its physical mechanism remains unclear (Bucciantini et al. 2011; Tanaka & Takahara 2011; Torres et al. 2014; Zhu et al. 2018). If the magnetic energy is entirely transferred to particles, either in the pulsar wind or at the termination shock, the break energy is believed to be proportional to the product of the Lorentz factor $\gamma_{w}$ and the wind magnetization strength $\sigma_{w}$ (Lyutikov et al. 2019; Luo et al. 2020), i.e., $E_{b}\propto\gamma_{w}\sigma_{w}m_{\pm e}c^{2}$. Although most studies suggest $\gamma_{w}$ ranges between $10^{2}$ and $10^{6}$, there is significant variation in magnetization strength. Theoretical models of pulsar magnetosphere and wind predict $\sigma_{w}$ to be much greater than 1.0 (e.g. Arons 2012), while simple one-dimensional models of PWNe require $\sigma_{w}$ to be much less than 1.0 (e.g. Kennel & Coroniti 1984), a discrepancy known as the ”$\sigma$-problem”. Figure 19(e) also shows no apparent correlation between the cutoff energy and the age of PWNe. This lack of correlation may stem from the fact that the fitting of most PWNe only provides lower limits. However, the decrease in the cutoff energy for the two oldest PWNe suggests a potential negative correlation between them.

Figure 19(f) and (g) depicts the correlation between the total electron energy $W_{e}$, and the total magnetic energy $W_{B}=\frac{B^{2}}{8\pi}\frac{4}{3}\pi R_{\rm PWN}^{3}$ within PWNe and their respective ages. Except for PWN N157B in the Large Magellanic Cloud and PWN CTA 1, the total electron energy of most PWNe ranges from $3\times 10^{47}-10^{49}$ erg, show no significant variation with the evolution of PWNe. However, the total magnetic energy appears to be inversely correlated with age. As the evolution of PWNe progresses, there is a conversion of magnetic energy into electron energy, resulting in an overall decrease in magnetic energy.

Figure 19(h) and (j) illustrates the correlation between the synchrotron cooling timescale at break energy (h), and at the cutoff energy (j) and the ages of PWNe. A strong positive correlation is observed between the synchrotron cooling timescale and the ages of the corresponding PWNe, likely attributable to the relationship between magnetic fields and PWN ages. The red solid line in the figure represents when the cooling timescale equals the age, and the blue dashed line is a linear fit without error. It is notable that except for the Crab Nebula PWN, the cooling timescale of the break energies exceeds the age of PWNe and a slope between them greater than 1.0, indicating a gradual increase in the cooling timescale as the magnetic field weakens. The cooling timescale of the cutoff energy timescales are mostly smaller than or equal to the ages of the corresponding PWNe, implying that the termination shock can efficient accelerate electrons to higher energies. Simultaneously, with the evolution of PWNe, the wind speed and magnetic field of PWNe weaken, resulting in a decrease in the maximum energy attained by accelerated electrons.

Multiwavelength observations of PWNe provide a unique opportunity to study evolution of high-energy electron distribution within PWNes, which is closely related to the acceleration, radiation, escape, and transport of high-energy particles inside them (e.g Zhang et al. 2008; Ishizaki et al. 2018; H. E. S. S. Collaboration et al. 2019; Lu et al. 2020; Zhu et al. 2021, 2023), and the the origin of Galactic electrons and positrons (e.g Di Mauro et al. 2014; Manconi et al. 2020; Di Mauro et al. 2021) and the contributions to the gamma-ray sky (e.g Pagliaroli et al. 2021; Martin et al. 2022). In this paper, we select 16 PWNe as sample sources with observational data in at least three bands between radio, X-ray, GeV, and TeV. We fit the SED of each PWN source using a simplified time-independent single-zone model, and then reveal the evolution of high-energy electron distributions in PWNe by analyzing the correlation of the model parameters. Below we give a brief summary of our conclusion.

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  The electron distribution in PWNe: Due to observational limitations, multi-band information for most PWNe is still somewhat lacking, with the exception of the Crab Nebula. Currently, a simplified double power law with a super-exponential cutoff can be used to describe the electron distribution inside observed PWNe.

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  The evolution of the electron distribution: With the evolution of PWNe, the electron spectrum becomes harder at higher energies, transitioning from approximately 3.5 to around 2.5 for the power-law index, while maintaining a power-law index around 1.0-1.5 at low energies. The break energy of the electron distribution is not correlated with the age of the PWN, and there is no significant correlation between the cutoff energy and the age of the PWN.

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  The evolution of the magnetic field: The mean magnetic field decreases with the age of the respective PWNe, resulting in a positive correlation between the synchrotron cooling timescales of the break energy and the cutoff energy of the electron distribution with the age of the PWN.

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  The total energy: The total energy of electrons inside PWNe remains relatively constant at approximately $2.0\times 10^{48}$ erg, while the total magnetic energy decreases with the age of the corresponding PWNe.