Pulsar Halo Model Overview

• The pulsar halo model defines extended nonthermal regions produced when high-energy e± escape the compact pulsar wind nebula, yielding diffuse TeV γ rays.
• Models employ continuous spin-down injection and detailed spectral forms, using diffusion-loss equations and alternative transport prescriptions to match observations.
• Multiwavelength tests and population synthesis studies constrain key parameters like diffusion coefficients and magnetic effects, impacting local positron flux predictions.

Pulsar halo model denotes a family of astrophysical formulations for describing spatially extended environments associated with pulsars. In the high-energy literature, a pulsar halo is the extended nonthermal region outside the compact pulsar wind nebula (PWN) in which escaped relativistic $e^\pm$ propagate through the surrounding medium and produce diffuse emission, principally TeV $\gamma$ rays through inverse Compton scattering; in a separate population-synthesis usage, the term can also refer to canonical radio pulsars that have migrated into the Galactic halo after natal kicks (Liu, 2022, Rajwade et al., 2018). Most current usage concerns TeV halos, because they couple pulsar spin-down, pair injection, near-source transport, multiwavelength radiation, and the local positron problem into a single observable framework.

1. Terminology, source class, and physical setting

In the TeV-halo sense, the model distinguishes halos from PWNe both geometrically and dynamically. PWNe are the regions in which pulsar-driven outflows still dominate the local plasma and magnetic environment, whereas pulsar halos are the larger regions in which escaped pairs propagate in the ambient medium and radiate there (Liu, 2022). The same review organizes this evolution into a three-stage sequence and places the canonical halo systems in “Stage 3,” where the pulsar is outside, or dynamically decoupled from, the parent supernova remnant and the escaped pairs diffuse in the interstellar medium.

The observational prototypes are Geminga, Monogem / PSR B0656+14, and PSR J0622+3749. Their TeV halos extend to scales of at least $\sim 20$ pc and are much larger than the X-ray PWNe, which in Geminga are only $\sim 0.2$–$0.3$ pc (Liu, 2022). This scale separation is one of the principal empirical arguments that the emission is not simply from the compact nebula. Later work extends the same phenomenology to HESS J1831$-$098, which is well described by a pulsar-halo model although a simple 2D Gaussian cannot be rejected, and to PSR B1055$-$52, for which H.E.S.S. reports the first detection of extended TeV emission around the pulsar and interprets it as the third detected system of its kind (Sabri et al., 3 Oct 2025, Wach et al., 3 Oct 2025).

A separate and older population-synthesis use of “pulsar halo model” concerns ordinary, non-recycled radio pulsars in the gaseous circumgalactic medium. In that formulation, the halo is operationally defined as the region outside the Galactic disc and within the virial radius, with YMW16 used to identify the effective edge of the Galaxy along a given line of sight (Rajwade et al., 2018). The physical motivation is different, but the common theme is migration of pulsars or pulsar-powered particles into environments beyond the compact nebula and thin disc.

2. Injection, spin-down evolution, and source prescriptions

TeV-halo models generally treat the pulsar as a continuous point-like source of relativistic $e^\pm$. For Geminga, one representative source term is written as

$Q(E,\mathbi{r},t)=\left\{ \begin{aligned} & Q_0(E/E_{100})^{-p}\,\delta(\mathbi{r}-\mathbi{r}_s)\,[(t_s+t_{\rm sd})/(t+t_{\rm sd})]^2\,, & t>0 \ & 0\,, & t<0 \end{aligned} \right.$

with $t_s=342\ {\rm kyr}$, $\gamma$0, and $\gamma$1 in the Geminga analysis of superdiffusion (Wang et al., 2021). The same work emphasizes that $\gamma$2 TeV electrons cool in about $\gamma$3 kyr, much shorter than the pulsar age, so the current TeV halo is dominated by recently injected particles.

Other formulations retain continuous spin-down injection but replace the phenomenological power law by a spectrum derived from acceleration at the pulsar-wind termination shock. In the DSA-based Geminga model, the halo source term is

$\gamma$4

with $\gamma$5 supplied by the shock solution, $\gamma$6, $\gamma$7, and present spin-down power $\gamma$8 (Gao et al., 14 Jul 2025). In this class of models, the injection cutoff is not merely imposed phenomenologically; it is controlled by the finite-upstream DSA solution and by the diffusion coefficient in the acceleration zone.

Many multiwavelength halo calculations adopt a power law with exponential cutoff,

$\gamma$9

normalized by an efficiency $\sim 20$0 that converts a fraction of the pulsar rotational energy or spin-down power into escaping pairs. Representative values in the literature surveyed here include $\sim 20$1, $\sim 20$2 TeV, and $\sim 20$3 for a benchmark Geminga X-ray/TeV model (Krivonos et al., 15 Dec 2025), and $\sim 20$4, $\sim 20$5 TeV, and $\sim 20$6 for the LHAASO J0621+3755 / PSR J0622+3749 halo model (Adams et al., 3 Apr 2025). Taken together, these works suggest that continuous spin-down-modulated injection is standard, while the detailed spectral form remains a major source of model dependence.

3. Transport prescriptions

The common transport backbone is the diffusion-loss equation. In isotropic form it is written as

$\sim 20$7

or equivalently, in the Geminga superdiffusion study,

$\sim 20$8

where $\sim 20$9 recovers normal diffusion and $\sim 0.2$0 gives a Lévy-flight-like propagator with heavy tails (Wang et al., 2021). In that application, the HAWC surface-brightness profile of Geminga yields a monotonic increase of $\sim 0.2$1 as $\sim 0.2$2 decreases from $\sim 0.2$3 to $\sim 0.2$4, and models with $\sim 0.2$5 or $\sim 0.2$6, depending on the angular range used, are disfavored at $\sim 0.2$7 confidence level.

The standard isotropic slow-diffusion picture is often embedded in a two-zone construction. A generic form is

$\sim 0.2$8

or, in source-specific studies, a piecewise constant inner suppressed region plus normal Galactic diffusion outside it (Liu, 2022). This formulation is used to reconcile compact TeV morphologies with eventual positron escape to Earth, and it reappears in Monogem modeling, in candidate-halo analyses, and in CTA detectability forecasts (Li et al., 2024, Eckner, 2023).

Two alternatives are treated explicitly in the literature. The first is isotropic unsuppressed diffusion with a ballistic-to-diffusive transition. In the review formalism this is described with a generalized Jüttner propagator, finite mean free path $\sim 0.2$9, and asymptotic profiles $0.3$0 in the ballistic regime and $0.3$1 in the diffusive regime (Liu, 2022). The second is anisotropic diffusion,

$0.3$2

with $0.3$3, so compact projected halos can arise from slow perpendicular transport and favorable viewing geometry rather than from a genuinely small scalar diffusion coefficient (Liu, 2022).

The resulting controversy is not merely formal. The three frameworks—suppressed isotropic diffusion, ballistic-corrected standard diffusion, and anisotropic diffusion—can all be tuned to reproduce current TeV surface-brightness profiles, but they imply very different local transport coefficients, magnetic geometries, and positron escape properties (Wu et al., 2024).

4. Radiation, morphology, and representative systems

The emitted TeV $0.3$4 rays are modeled as inverse-Compton emission from the transported $0.3$5, with synchrotron emission from the same particles providing the X-ray counterpart. A useful pair of scaling relations quoted in the review literature is

$0.3$6

for IC on the CMB, and

$0.3$7

for synchrotron emission. The characteristic halo size is correspondingly estimated as

$0.3$8

which makes energy-dependent morphology a central observable (Liu, 2022).

Current modeling is strongly driven by multiwavelength tension. In Geminga, homogeneous isotropic slow diffusion with $0.3$9G tends to overpredict the X-ray flux by about an order of magnitude, which has motivated either $-$0G, radial inhomogeneity in the diffusion coefficient, or anisotropic transport (Liu, 2022). The dedicated X-ray diagnostic study argues that the three transport models predict distinct X-ray surface-brightness profiles even when their TeV profiles are similar, and identifies Monogem and PSR J0622+3749 as especially promising systems for large-field-of-view X-ray tests (Wu et al., 2024).

Several source-specific analyses now provide concrete parameter benchmarks:

System	Key model result	Source
Geminga	$-$1 or $-$2 disfavored at $-$3 CL in superdiffusion fits; normal diffusion gives the best fit	(Wang et al., 2021)
Geminga	$-$4; fit requires a rapid increase of $-$5 above $-$6 TeV	(Gao et al., 14 Jul 2025)
Monogem / PSR B0656+14	favored two-zone GALPROP models use $-$7, present-day SDZ sizes $-$8 or $-$9 pc, and predict a positron contribution smaller than $-$0	(Li et al., 2024)
LHAASO J0621+3755 / PSR J0622+3749	benchmark parameters $-$1, $-$2 TeV, $-$3, $-$4 pc, $-$5	(Adams et al., 3 Apr 2025)
HESS J1831$-$6098	halo model is plausible, with $-$7 relative to a Gaussian$-$8power-law model	(Sabri et al., 3 Oct 2025)
PSR B1055$-$952	H.E.S.S. reports $e^\pm$0 extension and $e^\pm$1 in the range $e^\pm$2–$e^\pm$3, far below the canonical ISM value	(Wach et al., 3 Oct 2025)

The broad implication is that suppressed diffusion around pulsars is repeatedly inferred, but the spectral shape and the size of the suppression region are not universal. LHAASO J0621+3755 is explicitly argued to require a harder injected spectrum and a smaller suppression zone than Geminga (Adams et al., 3 Apr 2025).

5. Population synthesis, diffuse emission, and the Galactic-halo usage

Population-level pulsar-halo models extrapolate the source-class picture from nearby prototypes to the Milky Way. One influential application argues that unresolved TeV halos explain the Milagro TeV excess: with a lepton injection spectrum $e^\pm$4 and $e^\pm$5 conversion of spin-down power into $e^\pm$6 above 1 GeV, the cumulative halo component overtakes hadronic diffuse emission above $e^\pm$7 GeV and exceeds the standard background by about $e^\pm$8 at $e^\pm$9 TeV and $Q(E,\mathbi{r},t)=\left\{ \begin{aligned} & Q_0(E/E_{100})^{-p}\,\delta(\mathbi{r}-\mathbi{r}_s)\,[(t_s+t_{\rm sd})/(t+t_{\rm sd})]^2\,, & t>0 \ & 0\,, & t<0 \end{aligned} \right.$0 at $Q(E,\mathbi{r},t)=\left\{ \begin{aligned} & Q_0(E/E_{100})^{-p}\,\delta(\mathbi{r}-\mathbi{r}_s)\,[(t_s+t_{\rm sd})/(t+t_{\rm sd})]^2\,, & t>0 \ & 0\,, & t<0 \end{aligned} \right.$1 TeV (Linden et al., 2017).

A later joint population synthesis of SNRs, PWNe, and halos embeds halos into an explicit PWN-to-halo evolutionary sequence in which halo injection begins when the pulsar exits the nebula. Under the assumption that all middle-aged pulsars in the bow-shock phase develop halos with Geminga-like or Monogem-like transport, the predicted number of detectable halos ranges from $Q(E,\mathbi{r},t)=\left\{ \begin{aligned} & Q_0(E/E_{100})^{-p}\,\delta(\mathbi{r}-\mathbi{r}_s)\,[(t_s+t_{\rm sd})/(t+t_{\rm sd})]^2\,, & t>0 \ & 0\,, & t<0 \end{aligned} \right.$2 to $Q(E,\mathbi{r},t)=\left\{ \begin{aligned} & Q_0(E/E_{100})^{-p}\,\delta(\mathbi{r}-\mathbi{r}_s)\,[(t_s+t_{\rm sd})/(t+t_{\rm sd})]^2\,, & t>0 \ & 0\,, & t<0 \end{aligned} \right.$3 of the number of detectable PWNe in H.E.S.S. and HAWC-like surveys, while CTA Galactic Plane Survey forecasts reach about $Q(E,\mathbi{r},t)=\left\{ \begin{aligned} & Q_0(E/E_{100})^{-p}\,\delta(\mathbi{r}-\mathbi{r}_s)\,[(t_s+t_{\rm sd})/(t+t_{\rm sd})]^2\,, & t>0 \ & 0\,, & t<0 \end{aligned} \right.$4 PWNe and up to $Q(E,\mathbi{r},t)=\left\{ \begin{aligned} & Q_0(E/E_{100})^{-p}\,\delta(\mathbi{r}-\mathbi{r}_s)\,[(t_s+t_{\rm sd})/(t+t_{\rm sd})]^2\,, & t>0 \ & 0\,, & t<0 \end{aligned} \right.$5 halos (Martin et al., 2022). A complementary CTA study, using full spatial-spectral likelihoods, finds that under optimistic Geminga-like assumptions the Galactic Plane Survey could detect about $Q(E,\mathbi{r},t)=\left\{ \begin{aligned} & Q_0(E/E_{100})^{-p}\,\delta(\mathbi{r}-\mathbi{r}_s)\,[(t_s+t_{\rm sd})/(t+t_{\rm sd})]^2\,, & t>0 \ & 0\,, & t<0 \end{aligned} \right.$6 halos, identify more than $Q(E,\mathbi{r},t)=\left\{ \begin{aligned} & Q_0(E/E_{100})^{-p}\,\delta(\mathbi{r}-\mathbi{r}_s)\,[(t_s+t_{\rm sd})/(t+t_{\rm sd})]^2\,, & t>0 \ & 0\,, & t<0 \end{aligned} \right.$7 through significant energy-dependent morphology, and characterize up to $Q(E,\mathbi{r},t)=\left\{ \begin{aligned} & Q_0(E/E_{100})^{-p}\,\delta(\mathbi{r}-\mathbi{r}_s)\,[(t_s+t_{\rm sd})/(t+t_{\rm sd})]^2\,, & t>0 \ & 0\,, & t<0 \end{aligned} \right.$8 in enough detail to study diffusion conditions (Eckner, 2023).

This extrapolation is contested. A phenomenological two-zone study of Geminga and B0656+14 argues that strong suppression by $Q(E,\mathbi{r},t)=\left\{ \begin{aligned} & Q_0(E/E_{100})^{-p}\,\delta(\mathbi{r}-\mathbi{r}_s)\,[(t_s+t_{\rm sd})/(t+t_{\rm sd})]^2\,, & t>0 \ & 0\,, & t<0 \end{aligned} \right.$9–$t_s=342\ {\rm kyr}$0 orders of magnitude at $t_s=342\ {\rm kyr}$1 TeV is required by HAWC and Fermi-LAT data, but that the suppression extent can be as small as $t_s=342\ {\rm kyr}$2 pc for both objects (Martin et al., 2022). The same paper concludes that if all nearby middle-aged pulsars develop such halos, their combined positron flux, including Geminga, would saturate the $t_s=342\ {\rm kyr}$3 GeV AMS-02 measurement for injection efficiencies much smaller than those inferred for the canonical halos. It therefore regards it as simpler to assume that most middle-aged pulsars do not develop halos, with a possible occurrence rate of only $t_s=342\ {\rm kyr}$4–$t_s=342\ {\rm kyr}$5 (Martin et al., 2022).

A distinct, non-TeV usage of “pulsar halo model” appears in radio pulsar population synthesis. In that framework, about $t_s=342\ {\rm kyr}$6 of all pulsars beaming toward Earth are in the Galactic halo if magnetic-field and inclination-angle evolution are ignored, but the fraction drops to $t_s=342\ {\rm kyr}$7 if both evolve as a falling exponential (Rajwade et al., 2018). The same study finds that current surveys are sensitive to only $t_s=342\ {\rm kyr}$8 of halo pulsars beaming toward us in the favorable non-evolving scenario, and that time-domain periodicity searches are more effective than single-pulse or Fourier-domain searches.

6. Open problems, observational tests, and computational frameworks

The leading unresolved problem is the origin of the transport environment itself. The review literature discusses self-generated turbulence, Bell instability, SNR-induced turbulence, ballistic-to-diffusive transition physics, and anisotropic diffusion in an ordered field, but does not identify a single accepted mechanism (Liu, 2022). This uncertainty propagates directly into the positron problem: in one-zone slow diffusion Geminga contributes very little at Earth, whereas mildly superdiffusive transport or finite-size two-zone models can produce a much larger local positron flux (Wang et al., 2021, Liu, 2022).

Current observational leverage is limited by morphology and background systematics. In the DSA-based Geminga test, only $t_s=342\ {\rm kyr}$9 is used as a hard fit constraint, the HAWC morphology bins are broad, and the strongest future discriminator is a rapidly increasing $\gamma$00 above $\gamma$01 TeV (Gao et al., 14 Jul 2025). In the X-ray band, wide-field imaging has become critical because narrow-field telescopes undersample degree-scale halos. The SRG/ART-XC Geminga study derives a $\gamma$02-confidence upper limit $\gamma$03G within a benchmark suppressed-diffusion model and finds, through simulation, that a 20-day exposure would yield a $\gamma$04 probability of detecting the halo for a $\gamma$05G field (Krivonos et al., 15 Dec 2025).

Large-field-of-view X-ray observations are also proposed as the clearest route to transport-model discrimination. The dedicated diagnostic study concludes that eROSITA and Einstein Probe, with sufficiently long exposures, can distinguish the isotropic suppressed-diffusion, ballistic-corrected, and anisotropic diffusion models through their X-ray surface-brightness profiles, especially for Monogem and PSR J0622+3749 (Wu et al., 2024). This suggests that the next decisive advance may come not from another TeV radial profile alone, but from combined TeV-plus-X-ray morphology.

The computational side of the subject has correspondingly broadened. PHECT, introduced as a lightweight computation tool for pulsar-halo emission, implements normal, normEx, normJ, superdiff, 2zoneNumr, 2zSNR, and aniso transport models, with finite-volume discretizations designed to remain stable on non-uniform grids and in the presence of discontinuous coefficients (Fang, 19 Aug 2025). Its existence reflects a general trend: pulsar halo modeling has moved from single-source phenomenology toward a modular numerical discipline in which alternative transport hypotheses can be confronted with increasingly precise $\gamma$06-ray and X-ray data.

Taken together, these developments define the contemporary pulsar halo model as a multi-regime framework rather than a single equation set. Its stable elements are continuous pulsar-powered $\gamma$07 injection, radiative cooling by IC and synchrotron, and degree-scale emission outside the compact PWN. Its unsettled elements are the transport operator, the size and physical origin of the slow-diffusion region, the role of magnetic anisotropy, and the population-level commonness of the phenomenon.