MAGTHOMSCATT: Polarized X-Ray Transport in Neutron Stars

• MAGTHOMSCATT is a Monte Carlo radiative-transfer code that simulates polarized X-ray transport in magnetized neutron-star atmospheres via magnetic Thomson scattering.
• It accommodates arbitrary magnetic field orientations and tracks the full complex electric field vector to capture linear, circular, and elliptical polarization.
• The extended framework incorporates general relativistic light bending and vacuum birefringence, enabling predictions of intensity pulse profiles and high linear polarization in magnetars.

Searching arXiv for MAGTHOMSCATT and related neutron-star polarized radiative transfer papers. MAGTHOMSCATT is a Monte Carlo radiative-transfer code for polarized X-ray transport in magnetized neutron-star atmospheres. It was developed to model how soft X-rays emerging from magnetized neutron-star surfaces acquire strong anisotropy and polarization through repeated magnetic Thomson scatterings, while accommodating arbitrary local magnetic-field orientations and explicitly tracking the full photon electric field vector. In its atmospheric form, the code treats fully ionized atmospheric slabs threaded by a magnetic field $\boldsymbol{B}$, spanning regimes from effectively non-magnetic Thomson scattering to the magnetar domain. In its later extension, MAGTHOMSCATT also includes general relativistic light bending and parallel transport in the Schwarzschild metric, together with magnetospheric vacuum birefringence, so that intensity pulse profiles, phase-resolved linear polarization degree, polarization angle, and circular polarization can be predicted at infinity (Thi et al., 1 Aug 2025, Thi et al., 28 Dec 2025).

1. Physical problem and domain of applicability

MAGTHOMSCATT addresses polarized radiative transfer in optically thick, fully ionized neutron-star surface layers, with particular emphasis on the angular dependence and polarization of soft X-ray emission. The motivating problem is that magnetic fields alter the photon-electron scattering cross sections, making the transport anisotropic and polarization dependent, while the observable pulse profiles additionally depend on field geometry, hot-spot extent, and line-of-sight orientation (Thi et al., 1 Aug 2025).

The code is designed to complement traditional radiative-transfer solvers, which work best in special geometries, especially near the magnetic pole. MAGTHOMSCATT instead handles arbitrary local field orientations, including mid-latitude and equatorial surface regions where the field is not aligned with the surface normal. This makes it suitable for modeling extended polar caps and for connecting atmospheric microphysics to rotationally modulated observables (Thi et al., 1 Aug 2025).

The basic atmospheric model is a thin slab of thickness $h$, uniform in electron density $n_e$, with local field orientation

$$\theta_{\rm B}=\arccos(\hat{\boldsymbol{n}}\cdot \hat{\boldsymbol{B}}),$$

where $\hat{\boldsymbol{n}}$ is the slab normal. In the later stellar implementation, the emitting regions are idealized as spherical polar caps or hot spots on the stellar surface, and photons subsequently propagate through the exterior spacetime and magnetosphere (Thi et al., 28 Dec 2025).

2. Magnetic Thomson scattering and polarization structure

The transport physics is governed by magnetic Thomson scattering. In a magnetized plasma, the two linear polarization states are $\parallel$, with electric field in the plane containing $\hat{\boldsymbol{k}}$ and $\boldsymbol{B}$, and $\perp$, with electric field perpendicular to that plane. The key control parameter is the frequency ratio

$\frac{\omega}{\omega_{\rm B}},$

with $h$0 (Thi et al., 1 Aug 2025).

In the sub-cyclotron domain $h$1, the cross section is strongly modified through

$h$2

and the azimuth-integrated differential cross sections are

$h$3

$h$4

$h$5

where $h$6, $h$7, and $h$8 (Thi et al., 1 Aug 2025).

A central geometric concept is the magnetic scattering cone, with opening angle

$h$9

For $n_e$0, the $n_e$1 channel dominates except within this narrow cone. Inside the cone, all polarization channels become comparable and the emergent radiation can be less polarized; outside it, the radiation is strongly beamed and mostly $n_e$2-polarized. This underlies the pronounced angular anisotropy of magnetar-atmosphere emission (Thi et al., 1 Aug 2025).

The code treats several field regimes. In the non-magnetic or super-cyclotron limit, $n_e$3, the cross section approaches

$n_e$4

so the field has negligible influence on scattering. In the intermediate sub-cyclotron regime, $n_e$5, both intensity and polarization evolve strongly with frequency ratio. In the magnetar regime, $n_e$6, radiation becomes almost entirely $n_e$7-mode outside the magnetic scattering cone, intensity is strongly beamed along $n_e$8, and circular polarization is negligible in the deep sub-cyclotron limit (Thi et al., 1 Aug 2025).

3. Monte Carlo formulation and electric-field-vector transport

MAGTHOMSCATT uses a photon-by-photon Monte Carlo scheme in which photons are injected at the bottom of the slab and propagated through repeated scatterings. The free path is sampled from Poisson statistics as

$n_e$9

with $$\theta_{\rm B}=\arccos(\hat{\boldsymbol{n}}\cdot \hat{\boldsymbol{B}}),$$0 uniform random, after which the position is updated according to

$$\theta_{\rm B}=\arccos(\hat{\boldsymbol{n}}\cdot \hat{\boldsymbol{B}}),$$1

where $$\theta_{\rm B}=\arccos(\hat{\boldsymbol{n}}\cdot \hat{\boldsymbol{B}}),$$2. The post-scattering direction is then chosen by accept-reject sampling from the polarization-dependent differential cross section, and the photon complex electric field vector is updated (Thi et al., 1 Aug 2025).

The distinctive methodological feature is that MAGTHOMSCATT does not only track Stokes parameters. It tracks the full complex electric vector, allowing it to capture linear, circular, and elliptical polarization. At injection,

$$\theta_{\rm B}=\arccos(\hat{\boldsymbol{n}}\cdot \hat{\boldsymbol{B}}),$$3

with

$$\theta_{\rm B}=\arccos(\hat{\boldsymbol{n}}\cdot \hat{\boldsymbol{B}}),$$4

The injected amplitudes are tied to Stokes variables by

$$\theta_{\rm B}=\arccos(\hat{\boldsymbol{n}}\cdot \hat{\boldsymbol{B}}),$$5

with

$$\theta_{\rm B}=\arccos(\hat{\boldsymbol{n}}\cdot \hat{\boldsymbol{B}}),$$6

$$\theta_{\rm B}=\arccos(\hat{\boldsymbol{n}}\cdot \hat{\boldsymbol{B}}),$$7

where $$\theta_{\rm B}=\arccos(\hat{\boldsymbol{n}}\cdot \hat{\boldsymbol{B}}),$$8 (Thi et al., 1 Aug 2025).

After atmospheric escape, the Stokes parameters are computed in the slab-normal coordinate system,

$$\theta_{\rm B}=\arccos(\hat{\boldsymbol{n}}\cdot \hat{\boldsymbol{B}}),$$9

Because of azimuthal symmetry, one often finds $\hat{\boldsymbol{n}}$0 after integration around the zenith in the one-dimensional angular distributions (Thi et al., 1 Aug 2025).

The later stellar implementation introduces an anisotropic and polarized injection protocol, denoted AP, and a magnetar-oriented efficiency variant, AP$\hat{\boldsymbol{n}}$1. The AP injection distribution is

$\hat{\boldsymbol{n}}$2

with

$\hat{\boldsymbol{n}}$3

This refined injection protocol is described as more efficient, especially for magnetar-like low-frequency-ratio transport (Thi et al., 28 Dec 2025).

4. Numerical behavior, asymptotics, and validation

A major result is that MAGTHOMSCATT reproduces classical non-magnetic radiative transfer in the high-$\hat{\boldsymbol{n}}$4 limit. For $\hat{\boldsymbol{n}}$5, the intensity and polarization angular distributions agree excellently with Chandrasekhar’s classical radiative-transfer solution for Thomson scattering and with Sunyaev & Titarchuk (1985). The reported agreement in both intensity profiles and linear polarization degree functions as a validation of the Monte Carlo implementation and of its angular and polarization handling (Thi et al., 1 Aug 2025).

In the non-magnetic limit, the paper provides analytical fits for the angular dependence. For $\hat{\boldsymbol{n}}$6,

$\hat{\boldsymbol{n}}$7

$\hat{\boldsymbol{n}}$8

with

$\hat{\boldsymbol{n}}$9

where $\parallel$0. These fits are intended for studies of millisecond pulsars and magnetic white dwarfs (Thi et al., 1 Aug 2025).

The code also exhibits a transport-regime transition that the paper characterizes as Markovianity. After enough scatterings, the photon population loses memory of the injection anisotropy and polarization. For top-escaping photons, the distributions and medians converge to a common form after roughly

$\parallel$1

and in the example setup the average number of scatterings for surface escape is about $\parallel$2. A plausible implication is that once this regime is reached, the emergent angular and polarization distributions are controlled more by the slab transport than by the initial injection details (Thi et al., 1 Aug 2025).

An important practical asymptotic result concerns the deep magnetar regime. The pulse profiles become effectively invariant once

$\parallel$3

The paper reports that pulse profiles at $\parallel$4 and $\parallel$5 are nearly identical, with differences only at the $\parallel$6 level in many cases, while even comparison with $\parallel$7 yields only modest differences, typically $\parallel$8. This circumvents the need for simulations at even higher magnetic field strengths, which are inherently slower (Thi et al., 1 Aug 2025).

The average-scattering scaling is reported as

$\parallel$9

with orientation-dependent coefficients for sufficiently large $\hat{\boldsymbol{k}}$0: $\hat{\boldsymbol{k}}$1

$\hat{\boldsymbol{k}}$2

$\hat{\boldsymbol{k}}$3

$\hat{\boldsymbol{k}}$4

At small $\hat{\boldsymbol{k}}$5, a linear correction is needed, especially near the equator (Thi et al., 1 Aug 2025).

5. Extension to stellar geometry, general relativity, and vacuum birefringence

In its extended form, MAGTHOMSCATT treats three coupled stages: magnetic Thomson transport in the atmosphere, general relativistic propagation in the Schwarzschild metric, and magnetospheric vacuum birefringence. Surface emission is sampled over azimuthally symmetric polar caps according to

$\hat{\boldsymbol{k}}$6

with $\hat{\boldsymbol{k}}$7. For the north cap, $\hat{\boldsymbol{k}}$8; for the south cap, $\hat{\boldsymbol{k}}$9; and $\boldsymbol{B}$0 (Thi et al., 28 Dec 2025).

After leaving the atmosphere, photons propagate in the Schwarzschild metric. The implementation includes light bending, which increases surface visibility and reduces pulse fraction, and parallel transport of the electric field vector along the curved ray. The paper uses the Poutanen (2020) approximation for light bending with $\boldsymbol{B}$1, $\boldsymbol{B}$2 km, and compactness $\boldsymbol{B}$3. The observer basis is

$\boldsymbol{B}$4

and at large distances $\boldsymbol{B}$5 (Thi et al., 28 Dec 2025).

The magnetospheric extension introduces vacuum birefringence. For $\boldsymbol{B}$6, the refractive-index difference is

$\boldsymbol{B}$7

where $\boldsymbol{B}$8 G and $\boldsymbol{B}$9 is the angle between photon momentum and magnetic field. The mode amplitudes in the O/X basis evolve according to

$\perp$0

where $\perp$1 is the angle between the projected magnetic field and the sky basis (Thi et al., 28 Dec 2025).

An equivalent Poincaré-sphere form is

$\perp$2

with $\perp$3. This expresses the polarization as precession around the birefringence vector (Thi et al., 28 Dec 2025).

A key concept is the recoupling radius, or polarization-limiting radius, defined through the Heyl & Shaviv criterion

$\perp$4

In the weak-field, far-field dipole limit,

$\perp$5

or numerically

$\perp$6

For magnetars, this places recoupling far outside the star, often tens to hundreds of stellar radii (Thi et al., 28 Dec 2025).

6. Astrophysical implications, source applications, and limitations

For magnetars, the extension shows that vacuum birefringence increases the observable linear polarization degree by reducing geometrical cancellation among photons emitted from different surface zones. The paper states that, with vacuum birefringence included, the predicted linear polarization degree can become very high, often above $\perp$7. By contrast, circular polarization is generally small in the high-field magnetar regime; if the phase lag $\perp$8, vacuum birefringence does not change $\perp$9, and under randomized phase lag, $\frac{\omega}{\omega_{\rm B}},$0 averages to zero (Thi et al., 28 Dec 2025).

The polarization angle in the magnetar regime follows the rotating-vector-model geometry. For a dipole,

$\frac{\omega}{\omega_{\rm B}},$1

where $\frac{\omega}{\omega_{\rm B}},$2 is the angle between magnetic and spin axes, $\frac{\omega}{\omega_{\rm B}},$3 the viewing angle, and $\frac{\omega}{\omega_{\rm B}},$4 the rotational phase. If the surface signal is nearly pure O-mode, then

$\frac{\omega}{\omega_{\rm B}},$5

so the observed polarization angle is essentially $\frac{\omega}{\omega_{\rm B}},$6 (Thi et al., 28 Dec 2025).

The code is intended for magnetars, neutron stars with weaker surface fields, millisecond pulsars, and magnetic white dwarfs. In the atmospheric-only study, the non-magnetic analytical fits are explicitly suggested for millisecond pulsars and magnetic white dwarfs. In the extended study, the main target is thermal X-ray emission from magnetar atmospheres, including pulse profiles from extended hot spots and constraints on geometric parameters such as $\frac{\omega}{\omega_{\rm B}},$7, $\frac{\omega}{\omega_{\rm B}},$8, and $\frac{\omega}{\omega_{\rm B}},$9 (Thi et al., 1 Aug 2025, Thi et al., 28 Dec 2025).

The paper applies the extended framework to 1RXS J170849.0−400910 using XMM-Newton pulse-profile data in $h$00–$h$01 keV and a model of two identical antipodal polar caps. The reported best fit to the full data set is

$h$02

while excising phases $h$03–$h$04 gives

$h$05

This suggests, as the paper states, that the full profile may contain longitudinal asymmetry not captured by uniform antipodal caps (Thi et al., 28 Dec 2025).

The same framework is also applied to RX J0822.0−4300. For this lower-field source, the recoupling radius is reported as $h$06, and vacuum birefringence increases the maximum linear polarization degree only from about $h$07 to $h$08. This suggests a weaker birefringence signature than in magnetars (Thi et al., 28 Dec 2025).

The principal limitation identified in the extension is the absence of vacuum resonance and mode conversion in the atmosphere. The paper describes this as the most important missing ingredient because plasma birefringence and vacuum birefringence can cancel deep in the atmosphere, allowing photons to convert between the two linear modes and thereby depolarize the emergent radiation. Additional stated limitations include the absence of detailed free-free opacity in deeper atmospheric layers, simplified hotspot geometry restricted to uniform antipodal caps, the lack of complex non-axisymmetric surface temperature patterns, relatively abrupt recoupling treatment, and the absence of a fully self-consistent curved-spacetime polarization evolution through the recoupling region for lower-field stars (Thi et al., 28 Dec 2025).

Taken together, these studies define MAGTHOMSCATT as a transport framework that links atmospheric magnetic Thomson scattering, polarization-state evolution via the complex electric field vector, general relativistic propagation, and, in its extended form, quantum-electrodynamical vacuum birefringence. Its central technical result is that it provides a self-consistent route from neutron-star surface microphysics to observable intensity and polarization pulse profiles, while its central practical result is that the deep magnetar regime can be represented efficiently once $h$09 (Thi et al., 1 Aug 2025, Thi et al., 28 Dec 2025).