Doppler brightness and the speed of a moving pattern

This chapter assumes the arrival-time geometry of the preceding chapter. A prime marks a quantity in the emitting plasma's rest frame. We compare compact, optically thin, non-evolving knots with the same intrinsic normalization. Frequencies have units of cycles per second, and all Doppler factors are dimensionless.

Use the clock travelling with the emitter

Two emissions separated by proper time $\Delta\tau$ are separated by source time $\Gamma\Delta\tau$ . Arrival-time geometry then gives

\Delta T_s=\Gamma(1-s\beta\cos\theta)\Delta\tau =\frac{\Delta\tau}{\delta_s},

where

\delta_s=\frac{1}{\Gamma(1-s\beta\cos\theta)}.

The distinction between proper time and source time matters. The shortened source-frame interval alone gives only the factor $1-s\beta\cos\theta$ ; time dilation supplies the remaining Lorentz factor. The frequency transformation is $\nu=\delta_s\nu'$ . An approaching emitter need not be blueshifted at every angle: at a sideways view its Doppler factor is less than one.

Why a compact knot gives a spectral exponent

Let the rest-frame spectrum follow $F'_{\nu'}\propto\nu'^{-\alpha}$ . A positive spectral index means less flux density at higher frequency. We use $\alpha=0.75$ in the teaching example.

Specific intensity divided by frequency cubed is invariant under a Lorentz transformation. Thus the observed intensity at a fixed observed frequency samples rest-frame emission at $\nu/\delta$ and gains a cubic Doppler factor. Substituting the power law adds another factor $\delta^\alpha$ . For an unresolved compact knot with isotropic rest-frame emission, its flux density follows

F_\nu\propto\delta^{3+\alpha}.

The spectral index is not an extra time-compression effect. It accounts for comparing different rest-frame frequencies at one observing frequency. A steady continuous jet has a different emitting-volume problem and commonly a different exponent. Use the compact-knot expression for the compact-knot assumptions here.

For intrinsically equal sides, the isotropic-emission reference ratio is

R_{\rm iso}=\left(\frac{1+\beta\cos\theta}{1-\beta\cos\theta}\right)^{3+\alpha}.

The Lorentz factors cancel in the ratio, but they remain in each individual Doppler factor.

Compact-knot isotropic and axial-field intensity ratios vary differently with viewing angle. — Original analytic comparison at actual speed sixty percent of light speed. The upper curve pair shows the compact-knot beaming reference and the ordered axial-field result. A lower comparison at a sixty-degree view shows why changing field direction can change the physical brightness ratio without changing either knot's motion.

A magnetic field changes what the knot emits toward us

Synchrotron radiation comes from fast charged particles turning in a magnetic field. An isotropic power-law electron population in one uniform field still emits anisotropically. Let $\mathbf n'_s$ be the direction to the observer after transforming into a knot's rest frame, and let $\hat{\mathbf B}'_s$ be its unit magnetic-field direction. The directional factor is

A_s=\left|\mathbf n'_s\times\hat{\mathbf B}'_s\right|^{1+\alpha}.

The cross-product magnitude is the sine of the angle between those directions. It vanishes along the field and reaches one across it. The aberrated direction is

\begin{aligned} n'_{s,x}&=\delta_s\sin\theta,\quad n'_{s,y}=0,\\ n'_{s,z}&=\frac{\cos\theta-s\beta}{1-s\beta\cos\theta}. \end{aligned}

Aberration changes a photon's direction when we change reference frame. The two knots generally see the same distant observer in different directions. With unit intrinsic normalization, the physical ordered-field intensity and its two-side ratio are

\begin{aligned} I_s&=\delta_s^{3+\alpha} A_s,\\ R_{\rm ord}&=R_{\rm iso}\frac{A_+}{A_-}. \end{aligned}

For our pair of local magnetic patches, a pitch angle $\psi$ specifies $\hat{\mathbf B}'_s=(0,\sin\psi,s\cos\psi)$ . It measures tilt from each knot's outward axis. These are two uniform patches; assigning a pitch does not reconstruct a magnetic helix throughout a jet.

A transverse field in this geometry points along the source's $y$ axis. Both aberrated sightlines have zero $y$ component, so both angular factors equal one. This gives a clean case where the physical ratio equals the isotropic reference. An axial field instead has $A_s=(\delta_s\sin\theta)^{1+\alpha}$ , so its ratio gains another Doppler-factor dependence.

Work a brightness example

Take $\beta=0.6$ and $\theta=60^\circ$ . The Lorentz factor is $1.25$ , and the Doppler factors are about $1.143$ and $0.615$ . With the fixed spectral index, the isotropic compact-knot ratio is about $10.2$ . For the axial-field patches, the physical ratio is about $30.1$ .

Both calculations have identical apparent motions. A brighter approaching side can therefore result from a changed field direction as well as a changed speed or viewing angle. If the receding intensity vanishes in an analytic limit, the ratio is undefined; the individual intensities are the useful quantities to inspect.

A moving brightness peak need not move with the matter

A shock can travel through a stream of plasma. Its visible peak then has a pattern speed, whereas Doppler transformation of the radiation follows the velocity of the matter that emits it. Position slopes and brightness do not automatically constrain the same speed.

The two-knot model equates those speeds. Under that assumption, the same velocity can consistently enter both the arrival-time equation and the radiation transform. A real observation needs a reason for making that identification. If it fails, forcing both measurements into one speed can create an apparently precise but physically wrong solution.

Questions to take to a new jet

Hold actual speed and viewing angle fixed. Change only the field from axial to transverse. Which of the apparent speed, Doppler factor, physical intensity and electric-vector direction may change?
Someone uses the isotropic brightness-ratio formula to infer a viewing angle from an ordered-field source. What information is missing?
Does a small apparent speed guarantee a slow flow?

Answers

The apparent speed and Doppler factor stay fixed because they depend on motion and viewing geometry. Physical intensity and electric-vector direction may change with field orientation. The next chapter follows the electric vector explicitly.

The inference needs a rest-frame emission model, including field direction or an appropriate spatial average. It also needs intrinsic side symmetry and a justified relation between pattern and emitting-plasma speeds.

A small apparent speed can come from a slow pattern or from a nearly line-of-sight geometry. Even an exact zero sideways motion does not, by itself, exclude rapid motion along the line of sight.

Sources

Urry and Padovani (1995), Appendix B, B1–B4 and the paragraphs after B6, explicitly separates discrete sources, continuous jets and anisotropic ordered-field emission. Doppler enhancement assumptions.
Lyutikov, Pariev and Gabuzda, arXiv:astro-ph/0406144v1 (6 June 2004), Section 2, equations 2,4,13–15, for rest-frame synchrotron angular weighting and aberration. That paper's steady-flow exponent has its own geometry; the compact knots here use the discrete-source exponent. Pinned primary paper.

Try the prediction

Separate geometry from radiation

Predict which quantities change when the ordered magnetic pitch changes at fixed speed and viewing angle.
Pause and keep speed and viewing angle fixed. Move only the magnetic-pitch slider from zero to ninety degrees.
Compare the physical intensity ratio, the Doppler factor in the expanded data, and the sky tracks. At a transverse field the physical ratio matches the isotropic benchmark in this geometry.