Parameter Reference

This page provides a comprehensive reference for all parameters used in VegasAfterglow, including their physical meanings, typical ranges, and units.

Physical Parameters

Jet Structure Parameters

Parameter	Symbol	Units	Typical Range	Description
E_iso	\(E_{\rm iso}\)	erg	\(10^{50} - 10^{54}\)	Isotropic-equivalent kinetic energy of the jet
Gamma0	\(\Gamma_0\)	dimensionless	\(10 - 1000\)	Initial bulk Lorentz factor of the jet
theta_c	\(\theta_c\)	radians	\(0.01 - 0.5\)	Half-opening angle of the jet core
theta_v	\(\theta_v\)	radians	\(0 - \pi/2\)	Viewing angle (angle between jet axis and line of sight)
duration	\(T_{\rm dur}\)	seconds	\(0.1 - 1000\)	Duration of energy injection (affects reverse shock)
k	\(k\)	dimensionless	\(1 - 10\)	Power-law index for structured jets (PowerLawJet only)
sigma0	\(\sigma_0\)	dimensionless	\(0.001 - 10\)	Initial magnetization parameter

Ambient Medium Parameters

Parameter	Symbol	Units	Typical Range	Description
n_ism	\(n_{\rm ISM}\)	cm⁻³	\(10^{-4} - 10^{3}\)	Number density of uniform ISM
A_star	\(A_*\)	dimensionless	\(10^{-3} - 10\)	Wind parameter: \(\rho = A_* \times 5 \times 10^{11} r^{-2}\) g/cm³

Observer Parameters

Parameter	Symbol	Units	Typical Range	Description
lumi_dist	\(d_L\)	cm	\(10^{26} - 10^{29}\)	Luminosity distance to the source
z	\(z\)	dimensionless	\(0.01 - 10\)	Cosmological redshift

Radiation Microphysics Parameters

Parameter	Symbol	Units	Typical Range	Description
eps_e	\(\epsilon_e\)	dimensionless	\(10^{-3} - 0.5\)	Fraction of shock energy in relativistic electrons
eps_B	\(\epsilon_B\)	dimensionless	\(10^{-6} - 0.5\)	Fraction of shock energy in magnetic field
p	\(p\)	dimensionless	\(2.01 - 3.5\)	Power-law index of electron energy distribution
xi_e	\(\xi_e\)	dimensionless	\(10^{-3} - 1\)	Electron acceleration efficiency

Energy Injection Parameters (Magnetar)

Parameter	Symbol	Units	Typical Range	Description
L_0	\(L_0\)	erg/s	\(10^{44} - 10^{48}\)	Initial luminosity of magnetar spin-down
t_0	\(t_0\)	seconds	\(10 - 10^4\)	Characteristic spin-down timescale
q	\(q\)	dimensionless	\(1 - 6\)	Power-law index of spin-down: \(L(t) = L_0(1+t/t_0)^{-q}\)

Two-Component Jet Parameters

Parameter	Symbol	Units	Typical Range	Description
theta_n	\(\theta_n\)	radians	\(0.01 - 0.2\)	Half-opening angle of narrow component
E_iso_n	\(E_{\rm iso,n}\)	erg	\(10^{51} - 10^{54}\)	Isotropic energy of narrow component
Gamma0_n	\(\Gamma_{0,n}\)	dimensionless	\(100 - 1000\)	Initial Lorentz factor of narrow component
theta_w	\(\theta_w\)	radians	\(0.1 - 0.5\)	Half-opening angle of wide component
E_iso_w	\(E_{\rm iso,w}\)	erg	\(10^{50} - 10^{53}\)	Isotropic energy of wide component
Gamma0_w	\(\Gamma_{0,w}\)	dimensionless	\(10 - 300\)	Initial Lorentz factor of wide component

Computational Parameters

Model Resolution

Parameter	Units	Description
phi_ppd	points/degree	Angular resolution in azimuthal direction
theta_ppd	points/degree	Angular resolution in polar direction
t_ppd	points/decade	Temporal resolution (logarithmic spacing)

MCMC Parameters

Parameter	Typical Value	Description
total_steps	1000-50000	Total number of MCMC steps per walker
burn_frac	0.2-0.5	Fraction of steps to discard as burn-in
thin	1-10	Thinning factor (keep every nth sample)
n_walkers	2×n_params to 10×n_params	Number of ensemble walkers

Parameter Scaling Types

Scale Type	Description and Usage
Scale.LOG	Sample in log₁₀ space. Use for parameters spanning multiple orders of magnitude (energies, densities, microphysics parameters)
Scale.LINEAR	Sample in linear space. Use for parameters with limited ranges (angles, power-law indices)
Scale.FIXED	Keep parameter fixed at initial value. Use when you don’t want to vary a parameter

Parameter Relationships and Constraints

Physical Constraints

Energy Conservation:

• \(E_{\rm iso}\) should be consistent with the kinetic energy available from the central engine

• For structured jets: \(E_{\rm iso} = \int E(\theta) d\Omega\) over the jet solid angle

Causality:

• Light travel time sets minimum variability timescale: \(\delta t \geq R/c\Gamma^2\)

• Jet opening angle and Lorentz factor: \(\theta_c \gtrsim 1/\Gamma_0\) for causal contact

Microphysics:

• Energy fractions: \(\epsilon_e + \epsilon_B \leq 1\) (though often \(\ll 1\))

• Electron power-law index: \(p > 2\) for finite energy in fast-cooling regime

Typical Parameter Combinations

Short GRB (Neutron Star Merger):

ParamDef("E_iso",     5e50,  5e52,  Scale.LOG),
ParamDef("Gamma0",    50,    500,   Scale.LOG),
ParamDef("theta_c",   0.05,  0.3,   Scale.LINEAR),
ParamDef("theta_v",   0.1,   0.5,   Scale.LINEAR),
ParamDef("eps_e",     0.01,  0.3,   Scale.LOG),
ParamDef("eps_B",     1e-4,  0.1,   Scale.LOG),
ParamDef("p",         2.1,   2.8,   Scale.LINEAR),

Long GRB (Collapsar):

ParamDef("E_iso",     1e52,  1e54,  Scale.LOG),
ParamDef("Gamma0",    100,   1000,  Scale.LOG),
ParamDef("theta_c",   0.02,  0.2,   Scale.LINEAR),
ParamDef("theta_v",   0.0,   0.1,   Scale.LINEAR),
ParamDef("eps_e",     0.1,   0.5,   Scale.LOG),
ParamDef("eps_B",     1e-3,  0.1,   Scale.LOG),
ParamDef("p",         2.2,   2.6,   Scale.LINEAR),

Kilonova-associated GRB:

ParamDef("E_iso",     1e49,  1e52,  Scale.LOG),
ParamDef("Gamma0",    10,    300,   Scale.LOG),
ParamDef("theta_c",   0.1,   0.5,   Scale.LINEAR),
ParamDef("theta_v",   0.2,   0.8,   Scale.LINEAR),
ParamDef("eps_e",     0.01,  0.3,   Scale.LOG),
ParamDef("eps_B",     1e-5,  1e-2,  Scale.LOG),
ParamDef("p",         2.1,   2.8,   Scale.LINEAR),

Unit Conversions

Common unit conversions for convenience:

Distance:

• 1 Mpc = 3.086 × 10²⁴ cm

• 1 kpc = 3.086 × 10²¹ cm

• Luminosity distance: \(d_L = (1+z) \times d_A\) (angular diameter distance)

Energy:

• 1 BeV = 1.602 × 10⁻³ erg

• 1 keV = 1.602 × 10⁻⁹ erg

• Solar rest mass energy: \(M_\odot c^2 = 1.8 \times 10^{54}\) erg

Angles:

• 1 degree = π/180 ≈ 0.01745 radians

• 1 arcminute = π/10800 ≈ 2.91 × 10⁻⁴ radians

• 1 arcsecond = π/648000 ≈ 4.85 × 10⁻⁶ radians

Frequencies:

• X-ray (1 keV): ν ≈ 2.4 × 10¹⁷ Hz

• Optical (V-band): ν ≈ 5.5 × 10¹⁴ Hz

• Radio (1 GHz): ν = 10⁹ Hz

Parameter Degeneracies

Understanding parameter correlations helps in MCMC fitting:

Strong Correlations:

• \(E_{\rm iso}\) ↔ \(n_{\rm ISM}\): Higher energy can compensate for lower density

• \(\epsilon_e\) ↔ \(\epsilon_B\): Microphysics parameters are often correlated

• \(\theta_c\) ↔ \(\theta_v\): Jet geometry parameters affect observed flux similarly

Frequency-dependent Constraints:

• Radio data: Most sensitive to \(\epsilon_B\), \(n_{\rm ISM}\)

• Optical data: Constrains \(\epsilon_e\), \(p\), \(E_{\rm iso}\)

• X-ray data: Sensitive to \(\Gamma_0\), high-frequency cutoffs

Time-dependent Constraints:

• Early times (< 1 day): Constrain \(\Gamma_0\), \(\epsilon_e\)

• Jet break time: Determines \(\theta_c\), \(E_{\rm iso}\)

• Late times (> 100 days): Sensitive to \(n_{\rm ISM}\), \(p\)

For more detailed information on parameter estimation strategies, see the Examples page.