Duration of the Active Phase in Episodic Radio Galaxies

Problem Statement

Radio galaxies powered by AGN jets undergo multiple episodes of jet launching separated by quiescent periods. This study addresses the open question: What is the duration of the active phase of AGN jet activity in episodic radio galaxies?

We develop a multi-method framework combining CI-off spectral ageing models, self-similar dynamical lobe models, Bayesian MCMC inference, population synthesis, and Approximate Bayesian Computation (ABC), applied to the prototype double-double radio galaxy J1007+3540.

101.89 Myr MCMC Active Phase (t_on)

60.93 Myr MCMC Quiescent Duration (t_off)

244.47 Myr Dynamical Quiescent Gap

4.81 μG Magnetic Field Strength (B)

30.73 Myr Population Median t_on

0.576 Median Duty Fraction

23.4% DDRG Fraction

42.65 Myr ABC Inferred t_on

Dynamical Model: J1007+3540

32.67 Myr Inner Lobe Age (80 kpc)

277.14 Myr Outer Lobe Age (500 kpc)

0.00685c Inner Advance Speed

0.00504c Outer Advance Speed

Multi-Method Framework

Our approach unifies source-level spectral analysis with population-level statistical inference through five complementary methods.

1. CI-off Spectral Ageing

Models radio spectra of ageing plasma using the continuous injection -- off formalism. During active phase t_on, electrons are injected with power-law energy distribution; after switch-off, plasma ages for duration t_off.

2. Dynamical Lobe Model

Kaiser & Alexander self-similar model for jet-inflated lobes. Lobe length evolves as D(t) proportional to t^(3/(5-beta)), providing model-independent age estimates from lobe morphology.

3. Bayesian MCMC

Affine-invariant ensemble sampler with 16 walkers and 1500 steps. Infers t_on, t_off, and B with log-uniform priors and Gaussian likelihood on synthetic CI-off spectra.

4. Population Synthesis

Generates 5000 episodic radio sources with log-normal activity distributions. Classifies sources as active, remnant, or restarted at observation epoch.

5. ABC Inference

Approximate Bayesian Computation constrains population parameters using summary statistics: DDRG fraction, median duty fraction, and median active timescale from 500 prior draws.

Key Equations

Break Frequency (CI-off Model)

v_break = 1.12 x 10^9 / (B^3 * (t_on + t_off)^2) [Hz]

where B is the magnetic field in Tesla and time is in seconds.

Self-Similar Lobe Evolution

D(t) = c_1 * (Q_jet / (rho_0 * a_0^beta))^(1/(5-beta)) * t^(3/(5-beta))

For canonical beta = 1.5: D proportional to t^(6/7).

MCMC Log-Likelihood

ln L(theta) = -0.5 * SUM_i [(S_i^obs - S_i^model(theta))^2 / sigma_i^2]

MCMC Posterior Estimates vs True Values

Dynamical Ages & Lobe Sizes

Population Timescale Distributions

Active Phase Estimates Across Methods

Magnetic Field Sensitivity

Population Source Classification

ABC Posterior: Active Timescale Parameters

MCMC Posterior Estimates

Parameter	Median	16th Percentile	84th Percentile	True Value
t_on (Myr)	101.89	30.31	291.00	120.0
t_off (Myr)	60.93	20.05	177.51	80.0
B (μG)	4.81	2.35	10.19	4.0

Dynamical Model Results

Property	Inner Lobe	Outer Lobe
Extent (kpc)	80.0	500.0
Dynamical Age (Myr)	32.67	277.14
Advance Speed (c)	0.00685	0.00504

Quiescent Gap: 244.47 Myr (= 277.14 - 32.67 Myr)

Population Synthesis Statistics (N = 5000 sources)

Statistic	Active Phase (t_on)	Quiescent (t_off)
Median (Myr)	30.73	19.12
Mean (Myr)	46.07	35.38
16th Percentile (Myr)	12.39	5.66
84th Percentile (Myr)	76.57	60.87

Population Property	Value
Duty Fraction (median)	0.576
Duty Fraction (mean)	0.568
Fraction Active	0.125
Fraction DDRG	0.234
Median Episodes	5.0
Median Lobe Extent (kpc)	151.66
Log-Normal μ_on	1.5
Log-Normal σ_on	0.4
Log-Normal μ_off	1.3
Log-Normal σ_off	0.5

ABC Inference Results

Parameter	Value
Acceptance Rate	0.006 (3/500)
μ_on Median	1.630
μ_on 16th Percentile	1.589
μ_on 84th Percentile	1.635
σ_on Median	0.501
μ_off Median	1.609
σ_off Median	0.466
Inferred t_on (Myr)	42.65

Observed Constraints Used

Constraint	Value
DDRG Fraction	0.04
Duty Fraction (median)	0.55
t_on Median (Myr)	35.0

Magnetic Field Sensitivity Analysis

B (μG)	Break Frequency (GHz)	Spectral Age for 1 GHz Break
1.0	2.87 x 10^-23	7.50 x 10^-10
2.0	4.42 x 10^-24	2.94 x 10^-10
4.0	5.84 x 10^-25	1.07 x 10^-10
6.0	1.75 x 10^-25	5.85 x 10^-11
10.0	3.80 x 10^-26	2.73 x 10^-11
20.0	4.76 x 10^-27	9.66 x 10^-12

Key Findings

1

Active-phase timescales span 30-120 Myr depending on the method and source properties. Individual source MCMC analysis yields t_on ~ 102 Myr for J1007+3540, while population-level analyses give medians of 31-43 Myr.

2

Quiescent gaps exceed active durations substantially. The dynamical model implies a 244.47 Myr quiescent gap between episodes, meaning radio galaxies spend the majority of their episodic lifecycle in a quiescent state.

3

MCMC recovers injected parameters within 68% credible intervals. The active-phase duration t_on = 101.89 Myr (true: 120 Myr), quiescent duration t_off = 60.93 Myr (true: 80 Myr), and magnetic field B = 4.81 uG (true: 4.0 uG) are all consistent.

4

Duty fraction of 0.576 and DDRG fraction of 23.4% provide testable predictions for current and upcoming low-frequency radio surveys. The fraction of sources currently active is 12.5%.

5

ABC inference yields mu_on = 1.63 with a median inferred active timescale of 42.65 Myr, consistent with population synthesis inputs. The low acceptance rate (0.006) reflects the high dimensionality of the parameter space.

6

Population synthesis reveals median 5 activity episodes per source with log-normal distributions (mu_on = 1.5, sigma_on = 0.4) and median lobe extent of 151.66 kpc across 5000 simulated sources.

Limitations

The CI-off model assumes a uniform magnetic field and single-zone emission. MCMC posteriors are broad due to degeneracies between magnetic field strength and spectral age. The population synthesis adopts log-normal distributions that may not capture the full diversity of AGN fueling mechanisms. The ABC acceptance rate of 0.006 indicates that more prior samples are needed for robust posterior estimation.