Dark Matter Heating of Compact Stars Beyond Capture: A Relativistic Framework for Energy Deposition by Particle Beams

Compact astrophysical objects, such as neutron stars and white dwarfs, can act as detectors of energetic particle fluxes originating from astrophysical accelerators. While most existing capture and heating calculations assume isotropic very low energetic incident fluxes from the halo dark matter, many realistic sources produce highly directional beams or jets, for which gravitational focusing, trajectory multiplicity, and local energy deposition must be treated consistently. In this work, we develop a general relativistic formalism to compute the local density, capture probability, and energy deposition of particles arriving as directed beams onto compact objects. The framework is based on the mapping of an asymptotic particle flux to local densities through geodesic congruences, allowing for gravitational focusing, multi-stream regions, and optical depth effects to be incorporated in a unified way. The formalism applies to arbitrary particle species and interaction models, and separates capture from through-going energy deposition in a frame-consistent manner. As an explicit application, we consider relativistic particle beams generated in astrophysical jets and evaluate their interaction with two compact objects samples: a white dwarf and a neutron star. In particular, we illustrate the framework using boosted dark matter produced in a list of 324 blazars as a representative case study, computing the resulting fluxes and the associated heating in the selected stars. Additional regimes such as the interaction roof and geometric limit are discussed, highlighting the conditions under which compact objects can efficiently convert incident beam energy into observable heating.

💡 Research Summary

The paper presents a comprehensive relativistic framework for calculating the capture and heating of compact stars—specifically neutron stars (NS) and white dwarfs (WD)—by directed, high‑energy particle beams. Traditional studies of dark‑matter (DM) capture in stars assume an isotropic, low‑energy halo flux, which is inadequate for realistic astrophysical sources such as jets, blazars, or cosmic‑ray‑boosted DM that produce highly collimated, relativistic beams. The authors therefore develop a method that maps an asymptotic flux defined at infinity onto local particle densities inside the star by following families of geodesics (geodesic congruences). Each geodesic is characterized by its impact parameter and conserved energy, allowing the calculation of gravitational focusing (through the Jacobian dV₀/dVᵣ), multi‑stream regions where several trajectories intersect the same spatial point, and optical‑depth effects that account for the probability of scattering along the path.

The core of the formalism rewrites the differential capture rate in a way that separates three physical ingredients: (i) the rate at which particles enter a given volume element, (ii) the time spent by a particle in that element (determined by the local velocity ω(r) obtained from the geodesic equations), and (iii) the probability Ω₋ᵥₑ(ω) that a single scattering reduces the particle’s speed below the local escape velocity, thereby binding it gravitationally. This expression reduces to the classic Gould‑Press capture formula when the flux is isotropic and the particles are non‑relativistic, but it remains valid for arbitrary directionality, relativistic energies, and any interaction model (scalar, vector, etc.). In addition to capture, the framework explicitly computes through‑going energy deposition: particles that traverse the star without being captured still lose kinetic energy through elastic, deep‑inelastic, or resonant scattering, and this lost energy is deposited as heat. The deposition rate scales with the product of the local number density, the scattering cross‑section, the optical depth τ, and the average energy loss per interaction ΔE.

Two limiting regimes are identified. The “interaction roof” corresponds to τ≈1, where essentially every particle experiences at least one scattering event; the “geometric limit” corresponds to τ≫1 and impact parameters below the capture radius, so that all incident particles become gravitationally bound. Realistic scenarios lie between these extremes, with the actual heating efficiency determined by the star’s mass, radius, equation of state, and the beam’s energy spectrum.

To illustrate the formalism, the authors consider boosted dark matter (BDM) produced in the jets of 324 observed blazars (BBDM). For each blazar they use measured jet parameters to estimate the flux of relativistic DM particles that have been accelerated by interactions with protons in the jet. The resulting BDM spectrum spans from MeV to multi‑TeV energies. This flux is then propagated through the relativistic spacetime of a benchmark WD (M≈0.81 M⊙, R≈1.08 R⊕) and a benchmark NS (M≈1.4 M⊙, R≈12.1 km) using the geodesic‑congruence method. The calculations reveal that gravitational focusing dramatically enhances the local DM density near the NS surface—by factors of 10–100 compared with the WD—while the WD’s larger radius captures a larger geometric cross‑section. However, the actual heating power depends sensitively on the DM‑nucleon scattering cross‑section and the energy loss per collision; for typical vector‑mediated interactions the NS heating can reach ≳10³⁰ erg s⁻¹, potentially observable as a modest surface temperature increase, whereas the WD heating remains below current detection thresholds unless the cross‑section is near existing experimental limits.

The paper’s strengths lie in its generality (applicable to any particle species, interaction model, and beam geometry), its rigorous treatment of relativistic trajectories, and its clear separation of capture versus non‑capture heating channels. Limitations include the assumption of static, spherically symmetric stars (ignoring rotation, magnetic fields, and possible anisotropies), and the reliance on a single interaction model for the illustrative BDM case. The authors suggest extensions such as incorporating rotating metrics (Kerr‑like spacetimes), magnetic field effects on charged beams, multi‑beam superposition from several astrophysical sources, and linking the predicted heating to observable signatures (e.g., X‑ray or infrared excesses). Overall, the work provides a versatile toolkit for future studies of high‑energy particle beams interacting with compact objects and opens a new avenue for indirect dark‑matter detection via stellar heating.