Astronomy · Active galaxies and central black holes

Jets, lobes and radiation across the spectrum

Derive an apparent transverse motion from arrival-time geometry and connect jets, radiation and extended lobes without confusing observed pattern speed with local material speed.

Continue in the approved chapter order. Necessary vocabulary, quantities and assumptions are explained in the lesson.

Back to the chapter

After reading, you can

Distinguish a compact engine, a jet feature and an extended radio lobe.
Calculate the apparent transverse speed of a supplied approaching feature.
Explain why spectra, polarization and lobe structure require particle, field and dynamical models.

A jet and a lobe record different parts of a system

A jet is a relatively collimated outflow associated with a compact engine. A visible knot can be an emission feature within that flow. An extended lobe is a larger region affected by transported material, fields and energy, sometimes with bright structures where a jet interacts with its surroundings.

These components need not trace the same instant in the engine’s history. Extended emission can retain evidence of earlier activity, propagation and environmental interactions. A bright present nucleus is not automatically a measure of all the energy stored in its lobes.

Relativistic means that effects associated with motion near vacuum light speed c matter. Some jet observations require relativistic kinematics and directional emission. Not every active nucleus has a detected jet, and not every observed knot must move at exactly the bulk plasma speed.

Blandford, Meier and Readhead discuss jet launching, propagation and radiation across a range of scales. The observational framework supports testing those physical models; it does not supply one universal composition or launching efficiency for every active nucleus.

The observer’s clock measures a shortened separation

For an original kinematic exercise, consider one straight-moving emitting feature in a local flat-spacetime model. Its speed in the host rest frame is v = βc, with β = 0.95. Its velocity makes angle θ = 20° to the approaching line of sight. Assume a distant observer.

Compare two emission events separated by host-rest time Δt. The feature moves transverse distance βcΔt sinθ and approaches the observer by βcΔt cosθ. The second light signal therefore has a shorter remaining path to travel.

After applying the stipulated host-rest arrival-time convention, the arrival separation is Δt_arr = Δt(1 − βcosθ). Divide the transverse displacement by that arrival separation to define apparent transverse speed v_app = cβsinθ/(1 − βcosθ).

The two times in this ratio are not the same interval measured along the feature’s path. A large apparent speed can result from a shortened photon arrival interval while the feature’s local physical speed remains below c. No faster-than-light signal or material motion has been introduced.

Calculate the projection and timing together

For β = 0.95 and θ = 20°, βsinθ ≈ 0.324919136 and 1 − βcosθ ≈ 0.107292010. The transverse displacement is therefore about 0.324919136cΔt, received across only about 0.107292010Δt. Their ratio gives v_app/c ≈ 3.028362833.

Changing only the viewing angle changes the apparent speed. At θ = 0° the transverse displacement vanishes, so v_app = 0 even though the feature approaches rapidly. At θ = 90° the arrival shortening vanishes and v_app/c = β = 0.95.

For this fixed β, the maximum occurs at cosθ = β, approximately θ = 18.194872339°, with v_app/c ≈ 3.042434922. The apparent speed therefore need not keep increasing as the direction becomes more closely aligned with the observer.

The table and original plot concern apparent motion alone. They do not calculate a received flux boost, a cosmological distance or the speed of every particle in a real jet. Converting measured angular changes into a physical apparent speed requires the appropriate distance and clock conventions.

Original constant-speed approaching-feature calculation, β = 0.95. Angles are to the approaching line of sight. Values use the host-rest arrival-time convention and a distant observer; no actual jet measurements or radiative boosting factors are supplied.
θ	v_app/c
0°	0
5°	1.544304742
10°	2.560282840
15°	2.985027373
20°	3.028362833
30°	2.679439732
45°	2.046471876
60°	1.567093588
90°	0.950000000

Apparent speed and viewing angle

Scroll sideways to see the full diagram. Its explanation is below.

Original synthetic scalar plot for a straight emitting feature moving at the constant local speed v = 0.95 c toward a far observer in a local flat spacetime. This is not an observed jet. Theta is the angle between the feature velocity and the approaching line of sight, from 0 to 90 degrees. The linear horizontal axis starts at 0 degrees. The vertical axis is apparent transverse speed divided by vacuum light speed c, is dimensionless, and starts at 0 with a maximum axis value of 3.2. The blue solid curve is beta_app = beta sin(theta) / [1 - beta cos(theta)], with beta = 0.95 and trigonometric arguments converted to radians. An orange dashed horizontal reference marks apparent speed c. The curve starts at zero at 0 degrees, reaches its analytic maximum beta_app = 3.042434922 at theta = arccos(0.95) = 18.194872339 degrees, marked by a hollow orange diamond, then decreases to 0.95 at 90 degrees. A filled blue circle marks the separate 20-degree case, beta_app = 3.028362833. The table gives every original angle row plus the analytic maximum. The remaining original rows are 5 degrees: 1.544304742; 10: 2.560282840; 15: 2.985027373; 30: 2.679439732; 45: 2.046471876; and 60: 1.567093588. For an emission-event time interval Delta t in the host rest frame, the corresponding host-rest corrected arrival interval is Delta t times [1 - beta cos(theta)], and the transverse displacement is c Delta t beta sin(theta). At 20 degrees these factors are 0.107292010 and 0.324919136, respectively; their ratio gives the apparent speed. Thus the apparent transverse ratio can exceed c although the local feature speed is 0.95 c. The Lorentz factor for the declared speed is 3.202563076 and the kinematic Doppler factor at 20 degrees is 2.910281010; neither is a brightness measurement. The moving pattern need not have the same speed as the bulk plasma. No cosmological distance conversion, actual composition, launching mechanism, luminosity or spectral boosting law is supplied.

A jet image is also an emission model

Relativistic charged particles accelerated in magnetic fields can emit synchrotron radiation. The spectrum depends on the particle distribution, magnetic conditions and transfer. A characteristic synchrotron frequency is a spectral scale, not a claim that one electron emits only one exact frequency.

Polarization and spectral measurements can constrain field organization and emission mechanisms. Absorption, mixed components and propagation affect their interpretation, and polarization can also arise through scattering. One polarized detection is not a unique measurement of jet composition.

At other wavelengths, additional radiation processes can contribute. A consistent multi-band model must connect particle populations, fields, geometry and variability rather than declaring every high-energy photon to have one universal origin.

The NRAO synchrotron treatment and the jet review provide complementary physical descriptions. They support distinguishing radiated power from mechanical transport and stored energy, instead of equating a radio luminosity directly with a total jet energy budget.

Transported energy needs a dynamical interpretation

Jets can disturb surrounding gas and inflate regions visible through radio and X-ray evidence. Cavity and shock models can constrain transferred energy, with assumptions about geometry, pressure, material content and age. That inference is different from measuring a current nuclear band flux.

An observed knot may be a propagating pattern, shock or other brightness feature. Equating its apparent pattern speed with bulk material motion is an extra hypothesis. Combining motion with independent radiation constraints can test that hypothesis while preserving its assumptions.

Projection also affects apparent lengths, sidedness and the relation between a core and extended emission. A cartoon with arrows is useful only when it specifies the geometry and represented quantities. It must not depict material or signals emerging from inside an event horizon.

The next lesson examines central-mass evidence. It will use circular motions and reverberation products to show how inclination, tracer behaviour, line-width conventions and model uncertainty enter a quantitative mass inference.

Worked examples

Calculate an apparent speed without a local violation

What gives v_app ≈ 3.028c when the stipulated physical speed is only 0.95c?

Across Δt the transverse displacement is 0.95cΔt sin 20° ≈ 0.324919136cΔt.
The approaching motion reduces the arrival separation to Δt(1 − 0.95cos20°) ≈ 0.107292010Δt.
Dividing the displacement by this arrival separation gives v_app/c ≈ 3.028362833.

The apparent ratio uses a shortened photon arrival interval, while the supplied feature still moves physically at 0.95c.

Check a limiting direction

Why does the apparent transverse speed vanish for θ = 0°?

The direction is exactly along the line of sight.
Its transverse displacement contains sin 0° = 0.
The finite arrival factor is 1 − β = 0.05, so the transverse-speed ratio remains zero.

There is no transverse displacement to measure; alignment does not imply the largest apparent transverse motion.

Try the reasoning

Does v_app greater than c in this model mean a local signal outruns light?
Show the answer and reasoning
No.
The apparent ratio combines projected displacement with a shortened arrival-time interval.
Must an observed emission knot move at the same speed as all the underlying plasma?
Show the answer and reasoning
No.
A pattern or shock speed and bulk material speed require separate interpretation.
Does an extended lobe necessarily represent the engine’s current instantaneous power?
Show the answer and reasoning
No.
It records propagation, interaction and energy storage across a history.
Does one polarization measurement uniquely determine jet composition and total mechanical power?
Show the answer and reasoning
No.
Emission, field, transfer and dynamical models need additional compatible observations.

What to carry forward

A compact engine, moving feature and extended lobe probe different physical scales and histories.
Arrival-time geometry can produce apparent transverse speeds above c with subluminal physical motion.
Radiation and morphology constrain jets through conditional particle, field and environmental models.

References

Teaching resources and research supporting this lesson’s explanations and examples.