TIDE: Time-Dependent Leptonic PWN Model

• TIDE is a one-zone, time-dependent leptonic model for pulsar wind nebulae that couples pulsar spin-down evolution with cumulative particle injection to reproduce observed broadband spectra.
• The model dynamically evolves key properties such as the nebular magnetic field and radius while properly accounting for synchrotron and inverse Compton emissions.
• TIDE serves as a benchmark for global SED fitting and evolutionary studies, highlighting the trade-offs between simplified one-zone approaches and more detailed multi-zone transport models.

The Time-Dependent Leptonic PWN Model (TIDE) is a one-zone, time-dependent leptonic model for pulsar wind nebulae (PWNe) that evolves the nebular electron–positron population, magnetic field, radius, and broadband radiation from pulsar birth to the current epoch. In the formulation explicitly identified as TIDE, the nebula is treated as a spherically symmetric, homogeneous bubble powered by pulsar spin-down, with synchrotron and inverse Compton emission computed from a time-dependent lepton distribution $N(\gamma,t)$ and an evolving magnetic field $B(t)$ (Zhang, 16 Sep 2025). In later comparative work, TIDE is described as an independent 0D dynamical code whose principal domain is the integrated spectral energy distribution (SED) and the global dynamical evolution of a PWN, rather than its internal morphology (Kundu et al., 2024).

1. Definition and model class

TIDE belongs to the class of one-zone, homogeneous, time-dependent leptonic PWN models developed in the sequence of Martín et al. and Torres et al., and is used to connect pulsars, their nebulae, and multi-wavelength emission from young and middle-aged PWNe (Zhang, 16 Sep 2025). Its core assumptions are spherical symmetry, a single spatial zone, a uniform magnetic field $B(t)$, a spatially homogeneous particle distribution $N(\gamma,t)$, and radiative output dominated by electrons and positrons rather than hadrons (Zhang, 16 Sep 2025). In this sense it is a 0D spectral–dynamical model: the spatial volume evolves, but internal radial gradients are not resolved (Kundu et al., 2024).

This one-zone architecture distinguishes TIDE from later multi-zone transport codes that solve for $N_{\rm e}(r,E,t)$ in radius, energy, and time. Those spatially resolved models were explicitly introduced to calculate surface-brightness profiles and energy-dependent source sizes, whereas TIDE is designed for global SED fitting and dynamical inference (Rensburg et al., 2018). A plausible implication is that TIDE is best understood as a baseline evolutionary model for integrated PWN emission, against which more elaborate 1D transport models can be compared.

Within this one-zone class, closely related formulations appear in work aimed at predicting X-ray emission from evolved PWNe and in CTA-oriented studies of G0.9+0.1, both of which use a homogeneous sphere of radius $R_{\rm PWN}(t)$, a uniform $B(t)$, and a time-dependent electron spectrum to compute synchrotron and inverse Compton emission [(Mayer et al., 2012); (Fiori et al., 2020)]. The later dissertation that explicitly names TIDE places it in this same lineage and uses it as the central theoretical tool for systematic LAT analyses, detailed source modeling, and predictions at TeV and PeV energies (Zhang, 16 Sep 2025).

2. Transport equation, injection, and spin-down coupling

The dynamical core of TIDE is the time-dependent diffusion-loss or continuity equation for the lepton distribution. In the explicit TIDE formulation used for LHAASO J2226+6057, the equation is

$$\frac{\partial N(\gamma, t)}{\partial t} = Q(\gamma, t) - \frac{\partial}{\partial \gamma} \big[\dot{\gamma}(\gamma, t)\, N(\gamma, t)\big] - \frac{N(\gamma, t)}{\tau(\gamma, t)} ,$$

where $Q(\gamma,t)$ is the injection term, $\dot{\gamma}(\gamma,t)$ is the total energy-loss rate, and $B(t)$0 is the escape timescale (Sarkar et al., 2022). In the dissertation’s notation, the same equation is written as

$B(t)$1

with the same physical content (Zhang, 16 Sep 2025).

The injected pair spectrum is a broken power law. In one common TIDE form,

$B(t)$2

where $B(t)$3 is the break Lorentz factor and $B(t)$4 are the low- and high-energy indices (Zhang, 16 Sep 2025). In related one-zone implementations the same structure is written in energy rather than Lorentz factor, and in earlier time-dependent models a pure power-law injection was also used for representative evolved PWNe (Mayer et al., 2012). The normalization is fixed by the condition that a specified fraction of the pulsar spin-down power goes into particles:

$B(t)$5

in one notation (Sarkar et al., 2022), or equivalently

$B(t)$6

when the particle fraction is denoted by $B(t)$7 (Zhang, 16 Sep 2025).

The pulsar power is evolved with the standard braking-law form

$B(t)$8

where $B(t)$9 is the initial spin-down luminosity, $B(t)$0 the initial spin-down timescale, and $B(t)$1 the braking index (Zhang, 16 Sep 2025). Closely related one-zone models use the same law to control the time dependence of injection and nebular energetics [(Mayer et al., 2012); (Fiori et al., 2020)]. TIDE therefore couples pulsar evolution directly to particle injection, making the present-day spectrum a cumulative product of the full spin-down history rather than a steady-state snapshot.

The upper cutoff of the injected distribution is time dependent. In the dissertation, $B(t)$2 is set by the minimum of a synchrotron limit and a gyro-radius or confinement limit (Zhang, 16 Sep 2025). In the J2226+6057 study the equivalent maximum energy is controlled by synchrotron and gyroradius limits, with the gyroradius condition parameterized through a containment factor $B(t)$3 and a magnetic compression ratio $B(t)$4 (Sarkar et al., 2022). This suggests that TIDE does not treat the high-energy cutoff as a purely phenomenological fit parameter; it is tied to acceleration and confinement conditions at the termination shock.

3. Radius evolution, magnetic field evolution, and reverberation

A defining feature of TIDE, relative to purely spectral one-zone models, is that the PWN radius and magnetic field evolve dynamically with time. In the dissertation, the early free-expansion radius is written as

$B(t)$5

with $B(t)$6, $B(t)$7 the supernova explosion energy, and $B(t)$8 the ejecta velocity (Zhang, 16 Sep 2025). Related one-zone dynamical treatments use a thin-shell approximation for swept-up ejecta and solve the PWN expansion inside the supernova remnant, explicitly evolving $B(t)$9, $N(\gamma,t)$0, shell mass, and the internal pressure (Fiori et al., 2020).

The magnetic field is not fixed by hand in TIDE’s dynamical versions. Instead, the magnetic energy satisfies an energy-balance equation in which a fraction of the pulsar power feeds the field while expansion or compression produces adiabatic changes. In the dissertation this is written as

$N(\gamma,t)$1

with

$N(\gamma,t)$2

leading to

$N(\gamma,t)$3

(Zhang, 16 Sep 2025). A closely related derivation appears in the CTA study of G0.9+0.1, where the magnetic field is likewise evolved from the partition of spin-down power between particles and field energy (Fiori et al., 2020).

This dynamical framework also admits the reverse-shock interaction phase, or reverberation. In the dissertation’s source modeling, reverberation begins when the reverse shock reaches the nebula, causing compression, magnetic-field amplification, and adiabatic heating of particles (Zhang, 16 Sep 2025). The J2226+6057 study explicitly includes the dynamics of the PWN and SNR and their interaction through the reverse shock, and shows that as the age increases beyond the onset of reverberation the growing $N(\gamma,t)$4 enhances synchrotron cooling and softens the high-energy particle and photon spectra (Sarkar et al., 2022). This is an important point of physical interpretation: reverberation can boost nebular emissivity, but it can also destroy the hard electron tail required for the hardest TeV–PeV spectra.

4. Radiative processes and construction of the broadband SED

TIDE computes synchrotron and inverse Compton emission from the evolved lepton distribution. In the dissertation, the resulting synchrotron spectrum covers radio through X-rays, while the inverse Compton spectrum extends through the GeV, TeV, and in some applications PeV bands (Zhang, 16 Sep 2025). The target photon fields are parameterized as the cosmic microwave background plus far-infrared and near-infrared or optical components, typically treated as blackbody or modified blackbody fields with specified temperatures and energy densities (Zhang, 16 Sep 2025, Sarkar et al., 2022).

The full one-zone diffusion-loss model developed in 2012 explicitly included synchrotron, inverse Compton on the CMB and IR/optical fields, self-synchrotron Compton, and bremsstrahlung, all without radiative approximations (Martin et al., 2012). Its treatment of inverse Compton used the full Klein–Nishina cross section rather than a Thomson-limit approximation (Martin et al., 2012). Later TIDE-related implementations continued to use synchrotron and IC as the dominant channels, with SSC and bremsstrahlung retained where relevant (Zhang, 16 Sep 2025).

The SED is constructed from the instantaneous $N(\gamma,t)$5, $N(\gamma,t)$6, and photon fields at the current age, then converted to flux using the source distance (Zhang, 16 Sep 2025). Because the model is time dependent, the observed SED is interpreted as the output of the cumulative injection and cooling history rather than as a steady equilibrium (Mayer et al., 2012). This is especially important for evolved PWNe, where the synchrotron component may fade strongly while IC emission remains bright because of long-lived accumulated electrons and declining magnetic field strength (Mayer et al., 2012).

A recurring methodological conclusion is that simplified transport approximations can bias evolutionary inferences even when they fit the current SED. The 2012 full diffusion-loss study showed that models neglecting escape or replacing continuous losses by catastrophic removal can reproduce the Crab at one epoch, yet diverge strongly from the full model in their time evolution of both the lepton spectrum and the multi-wavelength emission (Martin et al., 2012). This directly motivates TIDE’s use as a genuinely time-dependent transport code rather than as a sequence of independent static SED fits.

5. Relation to spatially resolved and hybrid extensions

TIDE is not a spatially resolved transport code. In comparative work on Kes 75 and G21.5, it is explicitly described as a one-zone leptonic, time-dependent 0D model used to describe the SED and dynamical evolution of a PWN, while a separate 1D code is used to calculate relativistic particle injection, transport, and emission as particles traverse the nebula (Kundu et al., 2024). The 1D model solves a Fokker–Planck-type equation in $N(\gamma,t)$7, $N(\gamma,t)$8, and $N(\gamma,t)$9, includes advection, diffusion, adiabatic losses, and radiative losses, and predicts surface-brightness profiles, photon-index gradients, and energy-dependent sizes (Kundu et al., 2024).

This distinction matters physically. Multi-zone models were built specifically because one-zone models can fit integrated spectra but cannot predict morphology (Rensburg et al., 2018). In the spatially dependent modeling of G0.9+0.1, simultaneously fitting the SED and the energy-dependent source size was shown to yield more stringent constraints on transport and magnetic-field parameters than SED-only fitting (Rensburg et al., 2018). The comparison with TIDE in the 2024 study found similar values for overlapping global parameters between the 1D code and the independent 0D dynamical code, but also emphasized that the two models differ in their magnetic-field descriptions and in their ability to treat spatial transport (Kundu et al., 2024).

A further extension is the hybrid TIDE+L framework. In the 2026 population-synthesis study, TIDE is combined with Lagrangian hydrodynamic modules that evolve the supernova remnant structure and its interaction with the PWN, with TIDE used during free expansion and the Lagrangian treatment taking over once the reverse shock reaches the nebula (Torres et al., 6 Jun 2026). This hybrid framework was introduced to model reverberation more realistically, including reverse-shock-driven compression, shell thickening, delayed compression, and multiple reflections (Torres et al., 6 Jun 2026). A plausible implication is that TIDE provides the spectral–dynamical backbone, while TIDE+L addresses the specific limitations of thin-shell reverberation treatments.

6. Applications, parameter inference, and limitations

TIDE has been used as a benchmark-fitting and survey-support tool. In the dissertation, it was validated on the Crab Nebula, 3C 58, and G11.2−0.3, where it reproduced broadband data and yielded fitted ages, magnetic fractions, break energies, and present-day fields consistent with the adopted one-zone framework (Zhang, 16 Sep 2025). The same work then used TIDE in a systematic LAT search for MeV–GeV PWNe, in detailed modeling of individual sources, and in predictions for potential TeV-emitting PWNe (Zhang, 16 Sep 2025). In the CTA study of G0.9+0.1, a one-zone time-dependent leptonic model was used to show that future high-quality gamma-ray spectra could constrain the cutoff energy and the magnetization of the nebula, and that a pure power-law spectrum would rule out that specific model realization (Fiori et al., 2020).

The model has also been applied to ultra-high-energy gamma-ray sources. In the case of LHAASO J2226+6057, a TIDE-style fit to the full multi-wavelength SED produced a large present-day PWN radius and a magnetic field of order $N_{\rm e}(r,E,t)$0, while the preferred IC target field was effectively the CMB rather than strong local FIR or NIR fields (Sarkar et al., 2022). The study explicitly identified the resultant large radius and low magnetic field as caveats for the physical connection between the pulsar and the PeV source (Sarkar et al., 2022). The dissertation generalized this point: standard one-zone leptonic fits to some UHE sources can match the SED only by requiring nebular magnetic fields comparable to the interstellar field, which was presented as a central limitation of pure leptonic one-zone scenarios (Zhang, 16 Sep 2025).

Several common misconceptions can therefore be addressed directly. One is that a good fit to the current SED guarantees a reliable evolutionary model; the diffusion-loss study shows that approximate transport prescriptions may fit one epoch while failing badly in time evolution (Martin et al., 2012). A second is that one-zone models are sufficient once broadband spectra are available; spatially resolved studies show that morphology adds independent constraints on diffusion, advection, and $N_{\rm e}(r,E,t)$1 that a 0D model cannot supply (Rensburg et al., 2018). A third is that reverberation necessarily solves hard gamma-ray problems; in the J2226+6057 modeling, stronger reverberation increased synchrotron cooling and worsened the fit to the hard VHE–PeV spectrum (Sarkar et al., 2022).

Taken together, these results place TIDE in a well-defined position within PWN theory. It is a time-dependent, one-zone, leptonic spectral–dynamical model that links pulsar spin-down, particle injection, magnetic-field evolution, radius evolution, and broadband emission in a single framework (Zhang, 16 Sep 2025). It is robust enough for benchmark SED fitting, evolutionary studies, and population-level searches, but its simplifying assumptions become consequential when spatially resolved X-ray or TeV data, detailed reverse-shock physics, or extreme UHE interpretations are required (Kundu et al., 2024, Torres et al., 6 Jun 2026).