Synchrotron and Thermal Emission Models

An image changes when the spacetime bends different rays, when the fluid changes, and when the electron distribution converts that fluid into different emission and absorption coefficients. These effects are not identifiable from one rendered image. This page treats thermal and nonthermal synchrotron prescriptions as explicit forward-model components and builds comparisons that keep geometry, plasma closure, transfer, and camera choices separable.

Required background. Affine and Variational Geodesics supplies the null paths along which radiation is transported.

Helpful background. Formulations of General Relativity supplies numerical spacetime and frame data; Differential Equations for Relativity supplies coefficient integration and transfer; Black-Hole Astrophysics supplies accretion and radiation physics; and Computational Relativity supplies the common numerical-evidence standard.

Electron distributions define the coefficients

A general-relativistic fluid simulation typically evolves bulk density, internal energy, velocity, and magnetic field, not the radiating electron distribution. An emission model must therefore specify electron number density, temperature or distribution function, pitch-angle treatment, and the relation between simulation and physical units.

For thermal electrons, define the dimensionless temperature

$$\Theta_e=\frac{k_{\mathrm B}T_e}{m_e c^2},$$

temporarily restoring $c$ and $k_{\mathrm B}$ for clarity. The synchrotron emissivity $j_\nu$ and absorptivity $\alpha_\nu$ depend on $\nu$, electron density, magnetic-field strength, pitch angle, and $\Theta_e$. In local thermal equilibrium their ratio must satisfy Kirchhoff’s law,

$$\frac{j_\nu}{\alpha_\nu}=B_\nu(T_e),$$

within the validity of the coefficient approximation. This relation is an independent check, not a replacement for evaluating the coefficient accurately.

A nonthermal power law adds an index, normalization, and lower and upper Lorentz-factor cutoffs. A hybrid distribution adds still more choices. Pandya and collaborators compare exact coefficient integrals with practical fits for several distributions in Pandya et al. 2016, §§2–4, especially §4.3–§4.5, Eqs. (24)–(28). A fit must be rejected outside its stated frequency, temperature, and distribution domain even if it remains numerically smooth.

An electron distribution with a checkable normalization

For isotropic thermal electrons, integrate over momentum directions to write the Maxwell–Jüttner number distribution as

$$\frac{dn_e}{d\gamma} =\frac{n_e\,\gamma^2\sqrt{1-\gamma^{-2}}} {\Theta_e K_2(1/\Theta_e)} e^{-\gamma/\Theta_e}, \qquad \gamma\ge1.$$

Here $\gamma$ is the electron Lorentz factor and $K_2$ is a modified Bessel function. Its integral is $n_e$; treating it as a probability density without dividing by $n_e$ would count density twice in an emissivity. This is the angular integral of Pandya et al. 2016, §3.1, Eq. (15), PDF.

A simpler numerical test uses $dn_e/d\gamma=C\gamma^{-p}$ on $[\gamma_{\min},\gamma_{\max}]$, with $p\gt1$ and zero outside. Direct integration gives

$$C=\frac{n_e(p-1)} {\gamma_{\min}^{1-p}-\gamma_{\max}^{1-p}}.$$

For $p=2$, $\gamma_{\min}=10$, and $\gamma_{\max}=100$, $C=n_e/0.09$. Omitting the upper cutoff changes the normalization and high-frequency emission even if the power-law index is unchanged.

In a homogeneous thermal cell, combine Kirchhoff’s law with the slab solution: $I_\nu=B_\nu(T_e)(1-e^{-\tau})$ for zero incident intensity. At fixed temperature and magnetic field, doubling electron density doubles both $j_\nu$ and $\alpha_\nu$, leaving the source function unchanged. Thin emission nearly doubles; thick emission saturates at $B_\nu$. This is a useful density-scaling check before tracing a curved ray.

Separating geometry from emissivity

Use a crossed comparison rather than one model per spacetime. Hold the metric, fluid snapshot, camera, and transfer controls fixed while varying the electron prescription. Then hold one electron prescription and camera fixed while varying the spacetime or ray geometry on a physically matched fluid model. The first comparison measures emission-model sensitivity; the second is closer to a geometric sensitivity, although the fluid matching remains a modeling assumption.

Scaling a dimensionless simulation to physical density and magnetic field changes optical depth and characteristic synchrotron frequency. Normalizing every model to the same total flux can remove one difference while introducing a fitted nuisance parameter. Report both the pre-normalization prediction and the normalization rule. Never describe a feature as a spacetime signature if an allowed electron-heating or nonthermal-tail variation produces a comparable change.

The Event Horizon Telescope modeling study illustrates how electron prescriptions and simulation families enter a forward-model comparison in Event Horizon Telescope Collaboration 2019, §§3.1–3.3 and §§6–7, especially §7.2. Those astrophysical conclusions depend on the named model families and data treatment; they are not universal validation of a synchrotron formula.

Versioned tests and selections

The computational dataset should contain two separately versioned objects: a table of coefficient-integral benchmarks over $(\Theta_e,\nu/\nu_c,\theta_B)$ with a checksum, and a frozen metric/fluid snapshot with exact repository revision and units. If an official 2019 M87 public product is used for an instrument-level comparison, identify its release by EHT data DOI 10.25739/g85n-f134 and record the associated calibration version. No observational likelihood is evaluated on this page. The evidence cutoff is 2026-08-18.

Calibration records camera scale and handedness, observing frequency, flux and angular units, tetrad normalization, density and magnetic-field scaling, and the exact transfer-coefficient convention. Selections fix spacetime and snapshot, inclination, frequency band, field of view, pixel mask, time averaging, magnetization or density floors, excluded cells, electron distribution, and any quality cuts before images are inspected.

Compare coefficient fits with direct numerical integrals first. Then compare images by a declared normalized mean-squared error, spectra by a frequency-weighted residual, and integrated flux before and after any normalization. Image moments or ring measures may be reported only with their extraction algorithm and uncertainty. These metrics describe deterministic forward-model differences; likelihoods and priors belong only to an actual inference analysis.

Refinement and systematic variations

Refine coefficient quadrature or approximation accuracy, geodesic and transfer tolerance, fluid interpolation, image pixels, frequency grid, and snapshot cadence separately. The coefficient test should recover the published approximation domain and reveal where error grows. The transfer test should recover optically thin and thick limits. The image test should stabilize in both pixel norms and integrated spectra.

Systematic variations include thermal versus nonthermal distributions, electron-temperature closure, pitch-angle averaging, density and magnetization cuts, optical depth, time sampling, and flux normalization. Coordinate and tetrad changes probe the ray/transfer implementation rather than the plasma model. Report which variation dominates each synthetic observable.

The environment record pins coefficient implementation, metric and snapshot checksums, unit conversion, camera and transfer software, precision, libraries, and revision. A clean reproduction regenerates coefficient tables, runs the fixed crossed comparison, emits configuration files and raw images or spectra, and calculates the declared residuals. This is a reproducibility contract, not a claim that an absent program has run.

Reject the finite claim if coefficient errors leave their validity range unnoticed, Kirchhoff’s relation fails where applicable, refinement plateaus above the target, image conclusions reverse under an allowed electron prescription, or post hoc flux calibration hides the dominant difference. The principal failure mode is attribution: a numerically converged emissivity choice remains a model assumption.

Check your understanding

Model A uses Kerr spin $a_*=0.9$ with thermal electrons; model B uses $a_*=0$ with a nonthermal tail. Their 230 GHz ring diameters differ by $6\%$. Can the difference be attributed to spin?

Solution

No. Geometry and electron distribution changed together. Build the crossed comparison: both spins with the same thermal prescription and both with the same nonthermal prescription, keeping fluid matching, camera, units, transfer, and selections fixed. Report coefficient and numerical uncertainty plus the interaction between spin and emission model. Only a difference stable across defensible electron prescriptions supports a geometry-sensitive statement.

Continue with Polarized Ray Tracing.

References

• Event Horizon Telescope Collaboration. 2019. “First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring.” The Astrophysical Journal Letters 875, L5. arXiv.
• Event Horizon Telescope Collaboration. 2019. First M87 EHT Results: Calibrated Data. CyVerse Data Commons. Dataset DOI.
• Pandya, Alex, Zhaowei Zhang, Mani Chandra, and Charles F. Gammie. 2016. “Polarized Synchrotron Emissivities and Absorptivities for Relativistic Thermal, Power-Law, and Kappa Distribution Functions.” The Astrophysical Journal 822, 34. arXiv.