Monte Carlo Simulations of Polarized Radiative Transfer in Neutron Star Atmospheres

TL;DR

The paper advances polarized radiative transfer modeling for neutron star atmospheres by enhancing MAGTHOMSCATT, a Monte Carlo code that tracks the full electric-field evolution of photons in strong magnetic fields. It analyzes emergent intensity and polarization across magnetic-to-nonmagnetic domains, quantifying diffusion in atmospheric slabs, and demonstrates that convergence to the Markovian regime occurs after roughly $5\times10^5$ scatterings, enabling reliable pulse-profile predictions. The authors validate the magnetar-domain behavior, show strong beaming and near-total linear polarization outside the magnetic scattering cone, and provide analytical fits for the non-magnetic regime to support fast modeling of MSPs and magnetic white dwarfs. They also introduce an anisotropic/polarized injection protocol (AP) and a refined AP* scheme that substantially accelerate simulations, with up to ~9x speedups, facilitating practical fitting of observational data and constraining neutron-star geometries and hot-spot properties.

Abstract

Soft X-ray emission from neutron stars affords powerful diagnostic tools for uncovering their surface and interior properties, as well as their geometric configurations. In the atmospheres of neutron stars, the presence of magnetic fields alters the photon-electron scattering cross sections, resulting in non-trivial angular dependence of intensity and polarization of the emergent signals. This paper presents recent developments of our Monte Carlo simulation, MAGTHOMSCATT, which tracks the complex electric field vector for each photon during its transport. Our analysis encompasses the anisotropy and polarization characteristics of X-ray emission for field strengths ranging from non-magnetic to extremely magnetized regimes that are germane to magnetars. In the very low field domain, we reproduced the numerical solution to the radiative transfer equation for non-magnetic Thomson scattering, and provided analytical fits for the angular dependence of the intensity and the polarization degree. These fits can be useful for studies of millisecond pulsars and magnetic white dwarfs. By implementing a refined injection protocol, we show that, in the magnetar regime, the simulated intensity and polarization pulse profiles of emission from extended surface regions becomes invariant with respect to the ratio of photon ($ω$) and electron cyclotron ($ω_{\rm B}$) frequencies once $ω/ ω_{\rm B} \lesssim 0.01$. This circumvents the need for simulations pertinent to really high magnetic field strengths, which are inherently slower. Our approach will be employed elsewhere to model observational data to constrain neutron star geometric parameters and properties of emitting hot spots on their surfaces.

Figures (10)

• Figure 1: Left: 3D trajectories of four photons in the MAGTHOMSCATT simulation of an atmospheric slab of height $h = \tau_T / (n_e\sigma_{\hbox{T}} )$ at the magnetic pole ($\boldsymbol{B} \parallel \boldsymbol{n}$) . The $z$-axis (zenith) is oriented along the direction of the normal vector $\boldsymbol{n}$. The green, red, blue, and olive trajectories correspond to photons that scatter 1068475, 614756, 1299544, and 681861 times, respectively, before exiting through the upper boundary of the slab. Right: projection of these trajectories onto the $x-z$ plane. The arrows indicate the photon propagation directions after emerging from the upper boundary into the magnetosphere. The simulation was performed at a frequency ratio of $\omega / \omega_{\hbox{B}} = 0.03$ and an effective optical depth of $\tau_{\rm eff} = 6$, as defined in Equation (\ref{['eq:tau_eff']}). At injection, the photons with red and green trajectories have $\perp$-polarization mode, while those with blue and olive trajectories were injected in $\parallel$ mode. Upon emergence, they are all in $\parallel$-polarization mode.
• Figure 2: The phase space diagram of vertical position versus scattering number. $68\%$ (solid regions) and $99\%$ (dashed regions) confidence intervals (CIs) of vertical positions $z/h$ for $\mathcal{N} = 10^{5}$ photons recorded at the top (blue) and bottom (red-brown) of the slab as a function of the scattering number $n_{\rm scat}$. Six restrictive ranges of zenith angles were considered: $\theta_{zi} \in [0^{\circ}, 5^{\circ}]$ (panel a), $\theta_{zi} \in[10^{\circ}, 15^{\circ}]$ (panel b), $\theta_{zi} \in[20^{\circ}, 25^{\circ}]$ (panel c), $\theta_{zi} \in[40^{\circ}, 45^{\circ}]$ (panel d), $\theta_{zi} \in[60^{\circ}, 65^{\circ}]$ (panel e), and $\theta_{zi} \in[80^{\circ}, 85^{\circ}]$ (panel f). The simulations were performed at $\theta_{\rm B} = 0^{\circ}$ (i.e., at the magnetic pole), $\omega / \omega_{\hbox{B}} = 0.03$, and $\tau_{\rm eff} = 6$.
• Figure 3: Top panels: Distributions of $z/h$ of photons exiting the slab from the top at $n_{\rm scat}$ = 10 (panel a), $10^3$ (panel b), and $5 \times 10^5$ (panel c) for $\theta_{zi} \in [0^{\circ}, 5^{\circ}]$ (red), $\theta_{zi} \in [10^{\circ}, 15^{\circ}]$ (blue), and $\theta_{zi} \in [80^{\circ}, 85^{\circ}]$ (green). Bottom panel: median values of $z/h$ for top photons injected in six $\theta_{zi}$ ranges. The simulations were performed at $\theta_{\rm B} = 0^{\circ}$, $\omega / \omega_{\hbox{B}} = 0.03$, and $\tau_{\rm eff} = 6$.
• Figure 4: Distributions of photons exiting the top of an atmospheric slab positioned at the magnetic pole ($\theta_{\rm B} = 0^{\circ}$). These density maps are functions of the emerging zenith angle $\theta_z$ (x-axes) and the injected zenith angle $\theta_{zi}$ (panel a), logarithmic scattering numbers $\log_{10}{n_{\rm scat}}$ (panel b), Stokes parameter $Q$ (panel c), and Stokes parameter $V$ (panel d). The maps are for $\omega / \omega_{\hbox{B}} = 0.03$, appropriate for surface X-ray emission from a young pulsar. The colorbar on the right of each panel indicates the number of photons in each bin, so that the sum of these over all pixels equals the total number of photons recorded is $\mathcal{N}_{\rm rec} = 5\times 10^5$. The pixel size is 1 degree for both $\, \theta_z\,$ and $\, \theta_{zi}\,$
• Figure 5: Distributions of photons exiting the top of the slab as a function of the emerging zenith angle $\theta_z$ and: the injected zenith angle $\theta_{zi}$ (top row) and logarithmic scattering numbers $\log_{10}{n_{\rm scat}}$ (bottom row) for $\omega / \omega_{\hbox{B}} = 0.03$. Four columns correspond to four different field orientations, $\theta_{\rm B} = 0^{\circ}, 30^{\circ}, 60^{\circ}, 90^{\circ}$, corresponding to surface locales spanning the pole to the equator. The total number of photons recorded $\mathcal{N}_{\rm rec}$ and the value of the effective optical depth $\tau_{\rm eff}$ for each $\theta_{\rm B}$ are indicated in the bottom panel of each column. The colorbar on the right of each panel indicates the number of photons in each bin.