A Self-Consistent Framework for Synchrotron Equipartition Analysis

Technical Summary: A Self-Consistent Framework for Synchrotron Equipartition Analysis

Problem Statement
Determining the energy, size, and velocity of synchrotron-emitting outflows is critical for testing models of their formation and evolution. However, these quantities are often poorly constrained by observations alone. While equipartition analysis provides a robust framework for estimating these properties by assuming energy balance between relativistic particles and magnetic fields, prior works have typically applied necessary physical corrections—such as self-absorption effects, contributions from hot protons, and deviations from strict equipartition—independently. This ad hoc application of corrections leads to internal inconsistencies because the underlying physical parameters are coupled. Furthermore, previous formulations often relied on approximations (e.g., $\Gamma \gg 1$, small viewing angles, or fixed numerical prefactors) that limit their applicability across Newtonian and relativistic regimes.

Methodology
The authors derive a self-consistent equipartition framework that simultaneously accounts for the interdependence of various correction factors for both Newtonian outflows and on- and off-axis relativistic jets. The methodology involves:

1. Unified Derivation: The framework solves for the bulk Lorentz factor ($\Gamma$) and minimum electron Lorentz factor ($\gamma_m$) numerically without assuming $\Gamma \gg 1$ or small viewing angles. It consistently treats the energy of electrons radiating at the peak frequency ($\nu_m$), whether that frequency is directly observed or hidden by synchrotron self-absorption (SSA).
2. Hot Proton and Non-Radiating Components: The authors introduce a parameter $\xi$ to account for energy in non-radiating components (e.g., hot protons) relative to radiating electrons and magnetic fields. Unlike previous approximations that assumed $\xi \approx 1 + \epsilon_e^{-1}$, this formulation holds for any values of electron ($\epsilon_e$) and magnetic ($\epsilon_B$) energy fractions.
3. Deviations from Equipartition (DFE): The framework defines a dimensionless parameter $\epsilon$ to quantify deviations from equipartition. Crucially, the authors demonstrate that $\epsilon$ depends on the ratio $\epsilon_B / (1 - \epsilon_B)$ rather than the traditional $\epsilon_B / \epsilon_e$, due to the coupling with hot proton corrections.
4. Numerical Implementation: The authors implement these equations in a publicly available Python code. They solve for the 4-velocity ($u = \beta\Gamma$) by finding the roots of a constraint function derived from the relation between shock velocity, observer time, and viewing angle, utilizing Brent's method for reliable root-finding.
5. Case Studies: The framework is applied to four distinct astrophysical sources: Tidal Disruption Events (TDEs) ASASSN-19bt and AT2019dsg, the fast X-ray transient EP240414a, and the Active Galactic Nucleus (AGN) J0231-0433.

Key Contributions

• Self-Consistency: The primary contribution is the derivation of equipartition equations where correction factors (SSA, hot protons, DFE, and the numerical constant $C$) are not applied multiplicatively as independent factors but are solved simultaneously within a single system of equations.
• Refined Numerical Prefactors: The authors replace the approximate constant $C \sim 3$ used in previous works with precise, $p$-dependent expressions derived from matching Rayleigh-Jeans and synchrotron peak fluxes.
• Generalized Geometry: The method removes the assumption of small viewing angles and ultra-relativistic speeds, allowing for a continuous treatment of Newtonian and relativistic outflows.
• Open Source Tool: The authors provide a public code repository to facilitate the adoption of this consistent framework by the broader community.

Results
Applying the self-consistent framework to the selected sources yields significant quantitative differences compared to previous analyses that treated corrections independently:

• ASASSN-19bt: For off-axis relativistic models, the inferred energy increases by a factor of $\sim 5$ (specifically 5.17) compared to the analysis by Christy et al. (2024), which used the Barniol Duran et al. (2013) formalism. The radius increases by $\sim 1.24$. In the Newtonian case, the magnetic field estimate is significantly lower (factor of 2.13), while the electron number density is higher.
• AT2019dsg: When all corrections are included, the inferred energy for the Newtonian case increases by a factor of $\sim 70$ compared to Matsumoto & Piran (2023), and by a factor of $\sim 4$ for off-axis cases. The authors note that the self-consistent treatment of hot protons and DFE accounts for a factor of $\sim 4$ of this increase.
• EP240414a: The inferred minimum energy is $\sim 6$ times larger than the value reported by Bright et al. (2025), primarily driven by hot proton and $\nu_m$ electron corrections.
• J0231-0433: For this AGN, the authors constrained the deviation parameter $\epsilon_B$ using measured angular sizes. They found that while the equipartition radius overestimates the physical size, incorporating DFE and hot proton corrections results in an energy estimate roughly 7 times higher than the minimum equipartition estimate, with a magnetic field strength about an order of magnitude larger than previous non-equipartition estimates.

Significance
The paper claims that the interdependence of correction factors can increase energy estimates by a factor of $\sim 5$, suggesting that the energies of other synchrotron sources in the literature may be similarly underestimated. The authors conclude that simultaneously incorporating these correction factors is essential for determining accurate outflow properties and constraining launch mechanisms. While these differences may not alter the qualitative conclusions of previous studies, they are significant at the level of modern observational precision. The work provides a flexible and robust tool for the consistent interpretation of radio observations across a wide range of astrophysical sources, from Newtonian outflows to relativistic jets.