Radiation Models¶

blastwave implements six synchrotron radiation models, selected via the model parameter. All models compute the specific intensity \(I_\nu\) at each point on the blast wave surface, which is then integrated over the equal arrival time surface (EATS) or forward-mapped grid to produce observer-frame flux densities.

Analytic synchrotron (`sync`)¶

The default model. Implements the Sari, Piran & Narayan (1998) piecewise power-law synchrotron spectrum.

Characteristic quantities:

Minimum electron Lorentz factor:

\[ \gamma_m = \frac{p-2}{p-1} \, \varepsilon_e \, \frac{m_p}{m_e} \, (\Gamma - 1) \]

Cooling Lorentz factor:

\[ \gamma_c = \frac{6\pi \, m_e \, c}{\sigma_T \, B^2 \, t} \]

Magnetic field from post-shock energy density:

\[ B = \sqrt{8\pi \, \varepsilon_B \, e_\mathrm{density}} \]

where \(e_\mathrm{density}\) is the post-shock internal energy density.

Characteristic frequencies \(\nu_m\) and \(\nu_c\):

\[ \nu_{m,c} = \frac{3e}{4\pi \, m_e \, c} \, B \, \gamma_{m,c}^2 \]

Peak spectral power:

\[ P_{\nu,\mathrm{max}} = \frac{\sqrt{3} \, e^3 \, B \, n_\mathrm{blast}}{m_e \, c^2} \]

Spectral segments:

The spectrum is divided at \(\nu_m\) and \(\nu_c\) into power-law segments. For slow cooling (\(\gamma_m < \gamma_c\)):

Frequency range	Slope
\(\nu < \nu_m\)	\(\nu^{1/3}\)
\(\nu_m < \nu < \nu_c\)	\(\nu^{-(p-1)/2}\)
\(\nu > \nu_c\)	\(\nu^{-p/2}\)

Fast cooling (\(\gamma_c < \gamma_m\)) swaps the roles of \(\nu_m\) and \(\nu_c\) with an intermediate \(\nu^{-1/2}\) segment.

flux = blastwave.FluxDensity_tophat(t, nu, P, model="sync")

Deep Newtonian phase (`sync_dnp`)¶

Identical to sync but handles the trans-relativistic regime where \(\gamma_m\) would drop below unity. When \(\gamma_m < 1\), clamps \(\gamma_m = 1\) and adjusts the spectral normalization. Use this for late-time afterglows or mildly relativistic blast waves.

flux = blastwave.FluxDensity_tophat(t, nu, P, model="sync_dnp")

Synchrotron self-absorption (`sync_ssa`)¶

Extends the sync model with self-absorption at low frequencies, essential for radio light curves.

Approach: Computes the self-absorption Lorentz factor \(\gamma_a\) by finding where the optically thin intensity equals the Rayleigh-Jeans blackbody limit:

\[ I_\mathrm{thin}(\nu_a) = I_\mathrm{BB}(\nu_a) = \frac{2 k_B T_\mathrm{eff}}{c^2} \nu^2 \]

where the effective temperature \(T_\mathrm{eff} \propto (\gamma_\mathrm{eff} - 1) \, m_e c^2 / 3\) corresponds to the electrons radiating at frequency \(\nu\).

The final intensity is:

\[ I_\nu = \min(I_\mathrm{thin}, I_\mathrm{BB}) \]

This produces the characteristic self-absorbed radio spectrum: \(I_\nu \propto \nu^2\) below \(\nu_a\), transitioning to the optically thin power-law above.

The code handles six spectral regimes determined by the ordering of \(\gamma_a\), \(\gamma_m\), \(\gamma_c\), covering all slow/fast cooling configurations.

flux = blastwave.FluxDensity_tophat(t, nu, P, model="sync_ssa")

Synchrotron self-Compton (`sync_ssc`)¶

Adds inverse Compton (IC) scattering to the synchrotron model.

Thomson Y parameter:

\[ Y = \frac{1}{2}\left(\sqrt{1 + 4b} - 1\right), \quad b = \frac{\eta_e \, \varepsilon_e}{\varepsilon_B} \]

where \(\eta_e\) is the radiative efficiency (depends on the cooling regime). The Y parameter modifies the cooling break:

\[ \gamma_c = \frac{6\pi \, m_e \, c}{\sigma_T \, B^2 \, t \, (1 + Y)} \]

The solver iterates (~100 steps) until \(\gamma_c\) and \(Y\) converge self-consistently.

Klein-Nishina corrections (ssc_kn=1 in P): When the scattered photon energy approaches \(m_e c^2\) in the electron rest frame, the Thomson cross-section must be replaced by the Klein-Nishina cross-section. The code computes piecewise power-law segments for \(Y(\gamma)\) with breaks at the KN transition Lorentz factors.

IC flux contribution: Approximated as \(Y_T \times\) synchrotron evaluated at the upscattered frequency \(\nu_\mathrm{IC} = \nu / (\frac{4}{3} \gamma_m^2)\).

flux = blastwave.FluxDensity_tophat(t, nu, P, model="sync_ssc")

# With Klein-Nishina corrections
P["ssc_kn"] = 1
flux = blastwave.FluxDensity_tophat(t, nu, P, model="sync_ssc")

Thermal synchrotron (`sync_thermal`)¶

Implements the Margalit & Quataert (2021, MQ21) thermal + non-thermal electron model. This is important for mildly relativistic shocks where a significant fraction of the post-shock electron energy remains in a thermal (Maxwellian) distribution rather than being accelerated into a power-law tail.

Dimensionless electron temperature:

\[ \Theta = \varepsilon_T \cdot \frac{9 \mu \, m_p}{32 \mu_e \, m_e} \cdot \beta^2 \]

where \(\varepsilon_T\) is the electron thermalization efficiency, solved via a quadratic to account for relativistic corrections.

Thermal emissivity (MQ21 Eq. 10): Uses the Mahadevan et al. (1996) fitting function \(I'(x)\) for the thermal synchrotron kernel, with a fast-cooling correction factor.

Non-thermal (power-law) emissivity (MQ21 Eq. 14): Standard power-law electrons with modified minimum Lorentz factor \(\gamma_m = 1 + a(\Theta) \cdot \Theta\) that depends on temperature, and energy fraction \(\delta = \varepsilon_e / \varepsilon_T\).

Self-absorption: Both thermal and power-law absorption coefficients are computed:

\[ \tau = (\alpha_\mathrm{th} + \alpha_\mathrm{pl}) \cdot dr \]

\[ I_\nu = \frac{j_\mathrm{th} + j_\mathrm{pl}}{\alpha_\mathrm{th} + \alpha_\mathrm{pl}} \left(1 - e^{-\tau}\right) \]

Parameters:

Key	Default	Description
`eps_T`	1.0	Electron thermalization efficiency
`delta`	`eps_e/eps_T`	Power-law energy fraction
`full_volume`	0	Set to 1.0 for FM25 full-volume post-shock
`k`	0.0	CSM density power-law index (for full-volume mode)

P = {**P, "eps_T": 1.0, "delta": 0.1}
flux = jet.FluxDensity(t, nu, P, model="sync_thermal")

Full-volume post-shock (Ferguson & Margalit 2025)¶

By default, sync_thermal uses a thin-shell approximation. The full_volume flag activates the FM25 extension where emission comes from the entire post-shock volume:

Post-shock fluid Lorentz factor \(\Gamma_\mathrm{fluid}\) from Rankine-Hugoniot jump conditions
Volume-integrated shell thickness \(dr = R \cdot (1 - \xi_\mathrm{shell})\)
Downstream density and energy density rescaled for uniform post-shock conditions

This is important for trans-relativistic shocks (\(\beta\Gamma \sim 0.1\)--10) where the thin-shell approximation underpredicts optically thin flux by up to ~1 dex.

P = {**P, "eps_T": 1.0, "full_volume": 1.0, "k": 0.0}
flux = jet.FluxDensity(t, nu, P, model="sync_thermal")

Numeric electron distribution (`numeric`)¶

Uses a Chang-Cooper implicit finite-difference scheme to evolve the full electron energy distribution \(N(\gamma)\) and compute synchrotron emissivity + self-absorption from the numerical solution, rather than analytic approximations.

See Numeric Model for details.

flux = blastwave.FluxDensity_tophat(t, nu, P, model="numeric")

Model comparison¶

Model	SSA	IC	Thermal	Numeric	Speed	Use case
`sync`					Fastest	Standard GRB X-ray/optical
`sync_dnp`					Fast	Late-time / trans-relativistic
`sync_ssa`	Yes				Fast	Radio light curves
`sync_ssc`		Yes			Moderate	High-energy / IC-dominated
`sync_thermal`	Yes		Yes		Moderate	NS mergers, FBOTs, mildly relativistic
`numeric`	Yes			Yes	Slow	Non-standard distributions, pair production

References¶

Sari, R., Piran, T., & Narayan, R. (1998). ApJL, 497, L17.
Wijers, R. A. M. J. & Galama, T. J. (1999). ApJ, 523, 177.
Margalit, B. & Quataert, E. (2021). ApJ. arXiv:2111.00012
Ferguson, R. & Margalit, B. (2025). arXiv:2509.16313 | GitHub
Dermer, C. D. (2009). High Energy Radiation from Black Holes. Princeton.

Radiation Models¶

Analytic synchrotron (sync)¶

Deep Newtonian phase (sync_dnp)¶

Synchrotron self-absorption (sync_ssa)¶

Synchrotron self-Compton (sync_ssc)¶

Thermal synchrotron (sync_thermal)¶