Read a helical-jet polarization paper one frame at a time

Open Lyutikov, Pariev and Gabuzda, Polarization and structure of relativistic parsec-scale AGN jets, arXiv:astro-ph/0406144v1, dated 6 June 2004. This is the forty-five-page preprint. Read Section 2 on printed pages 7–11 and Section 3.1 on pages 11–13. Figure 7 is on printed page 38 and Figure 9 on page 40. A later journal layout uses different page and figure numbering; use the pinned PDF beside this guide.

Begin with the material being modelled

The paper starts with optically thin synchrotron radiation from a cylindrical jet with a large-scale helical field. “Helical” means that the field has both an axial component and a component circling the axis. A prime denotes the local plasma rest frame. The field pitch angle $\psi'$ measures the ratio of those components there; $\psi$ without a prime is the corresponding laboratory-frame angle.

The thin-shell example confines emission to a narrow cylindrical surface and moves it parallel to the axis. Integrating around that shell predicts the unresolved transverse polarization. A resolved cross-section asks a different question and is treated later. The assumptions omit tangled fields and propagation effects such as Faraday rotation; the paper's curves are model predictions, not a set of measured polarization points.

Before looking at a curve, write down four labels: the frame of the field, the observer angle, the electron index and the spatial average. Those labels will prevent most plausible misreadings.

Figure 7 compares two different directions

Find panels c and d on printed page 38. Both use a rest-frame helical pitch of forty-five degrees and a viewing angle of sixty degrees. Panel c has Lorentz factor one; panel d has Lorentz factor two. Large arrowheads identify projected background magnetic fields, and small arrowheads identify the electric vector of the radiation.

In the stationary panel the two directions are perpendicular. In the moving panel they generally are not. Equations 6–8 in Section 2 explain the difference: the electric and magnetic fields of the wave undergo a relativistic transformation. The projected background magnetic field alone does not fix an observed perpendicular electric line.

Use this as a reading test. Cover the small arrowheads in panel d and try to reconstruct them by rotating every large arrow through a right angle. Compare your prediction with the actual panel. The failure comes from changing reference frames, even though the local synchrotron rule was perpendicular in the plasma frame.

Figure 9 has a signed polarization axis

Now turn to printed page 40. The cylindrical shells have Lorentz factor ten and electron energy index $p=2.4$. In the paper's spectral convention, flux density varies as frequency raised to a negative index of about $-0.7$. Our previous chapters instead wrote a positive $\alpha$ after an explicit minus sign. Both the notation and the numerical slope differ: that paper uses a magnitude of 0.7, whereas the teaching knots use 0.75 and electron index 2.5. Read the paper’s curves with its own index 2.4.

The vertical axis is the signed ratio $Q/I$. Section 2 explains why axial symmetry makes the integrated $U$ vanish in this example. Positive values mean an electric vector parallel to the projected jet; negative values mean a perpendicular electric vector. The usual nonnegative fractional linear polarization is the absolute value of this signed ratio here. A negative plotted value is an orientation label, not a negative intensity.

In panels a and b, the horizontal axis is observer viewing angle in degrees. Panel a labels curves by rest-frame pitch; panel b labels them by laboratory pitch. They are not two independent sets of magnetic structures. The field transformation gives

At Lorentz factor ten, a rest-frame pitch of forty-five degrees corresponds to a laboratory pitch of about eighty-four point three degrees. Conversely, a laboratory pitch of forty-five degrees corresponds to a rest-frame pitch of about five point seven degrees. Compare equal physical fields after applying this conversion, not curves with equal printed angle labels.

Panels c and d put rest-frame and laboratory pitch on their respective horizontal axes. Curves now correspond to fixed observer angles. Read the labels again before comparing panels. Where a curve crosses zero, the signed preferred orientation changes, and the net linear polarization vanishes at the crossing. An EVPA assigned exactly at that zero has no defined direction.

Ordered local fields can give a small total fraction

Every small emitting element may be highly polarized. Around a helical shell, its electric-vector direction changes with azimuth. Adding the Stokes intensities around the shell can cancel much of that polarization. That extra spatial sum explains why the integrated curves need not sit at the uniform-field maximum.

This is also the boundary of the two-knot exercise: each knot there contains one uniform patch, so it follows a local electric vector and a fixed nonzero-emission polarization fraction. It provides the transformation used inside an extended calculation. It does not perform the shell integration of Figure 9.

Section 5, on printed page 19, extends the paper to an axial speed that changes with radius. There is then no single rest frame for the entire cross-section. Local radiation must be transformed and weighted before contributions are added. The dependence on emissivity and velocity structure is why an unresolved EVPA is not a unique magnetic-topology measurement.

A multi-band delay answers a different question

For an observational comparison, open Agudo and collaborators, arXiv:1011.6454v1, dated 30 November 2010. Read Sections 3.3 and 3.5 and Figure 4 on PDF page 4. Its horizontal axis is lag in days; the red points are a discrete correlation function between gamma-ray and one-millimetre light curves of OJ287. The peak near minus eighty days means gamma rays lead under the authors' sign convention. The dotted thresholds come from five thousand simulated light-curve pairs for each chosen pair of stochastic spectral slopes. Their reported 99.7 percent level is conditional on those variability models. Section 3.5 says the peak is dominated by event B; event A's different timing is not a second visible correlation peak. These are selected reading steps, not a recalculation of the light curves.

A lag relates variations received in different bands. It can include the evolution of an emitting region as well as travel-time geometry. Multiplying eighty days by light speed does not directly give the source separation, and the lag alone does not select a helical magnetic field. The field paper predicts polarization from a specified geometry; the timing paper tests associations among observed variations.

Reading questions

1. In Figure 9, why is a negative ordinate compatible with a positive measured intensity?
2. Why can a forty-five-degree curve label refer to different physical fields in panels a and b?
3. What two checks should precede interpreting the lag peak in Agudo Figure 4 as a physical clock?

Sources

• M. Lyutikov, V. I. Pariev and D. C. Gabuzda, arXiv:astro-ph/0406144v1, 6 June 2004: Section 2, Section 3.1, Section 5; Figures 7 and 9, printed pages 38 and 40. Exact theoretical source.
• I. Agudo et al., Location of gamma-ray Flare Emission in the Jet of the BL Lacertae Object OJ287 more than 14 pc from the Central Engine, arXiv:1011.6454v1, 30 November 2010, Sections 3.3 and 3.5, Figure 4 on PDF page 4; published as ApJ Letters 726, L13 (2011). Exact timing source.