() - 1409.3599v1.pdf

arXiv:1409.3599v1 [astro-ph.HE] 11 Sep 2014

S TROP HYSICAL J OURNAL

Preprint typeset using L

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STUDIES OF THE JET IN BL LACERTAE. II. SUPERLUMINAL ALFVÉN WA

VES

M. H. C

OHEN

, D. L. M

EIER

1,2

, T. G. A

RSHAKIAN

3,4

, E. C

LAUSEN

-B

ROWN

, D. C. H

OMAN

, T. H

OVATTA

7,1

, Y. Y. K

OVALEV

8,5

M.L. L

ISTER

, A. B. P

USHKAREV

10,11,5

, J. L. R

ICHARDS

, T. S

AVOLAINEN

5,7

(Received September 10, 2014)

ABSTRACT

Ridge lines on the pc-scale jet of the active galactic nucleu

s BL Lac display transverse patterns that move

superluminally downstream. The patterns are not ballistic

, but are analogous to waves on a whip. Their apparent

speeds

app

(units of

) range from 4.2 to 13.5, corresponding to

gal

wave

= 0

981

−

998 in the galaxy frame. We

show that the magnetic field in the jet is well-ordered with a s

trong transverse component, and assume that it

is helical and that the transverse patterns are longitudina

l Alfvén waves. The wave-induced transverse speed

of the jet is non-relativistic (

gal

∼

09) and in agreement with our assumption of low-amplitude wa

ves. In

2010 the wave activity subsided and the jet displayed a mild w

iggle that had a complex oscillatory behavior.

The waves are excited by changes in the position angle of the r

ecollimation shock, in analogy to exciting a

wave on a whip by shaking it. Simple models of the system are pr

esented; the preferred one assumes that the

sound speed in the plasma is

= 0

3 and this, combined with the measured speeds of the Alfvén wa

ve and a

component that is assumed to be an MHD slow wave, results in Lo

rentz factor of the jet

jet

∼

8, pitch angle

of the helix (in the jet frame)

∼

◦

, Alfvén speed

∼

86, and magnetosonic Mach number M

∼

This describes a plasma in which the magnetic field is dominan

t but not overwhelmingly so, and the field is in

a moderate helix.

Keywords:

BL Lacertae objects:individual (BL Lacertae) – galaxies:a

ctive – galaxies: jets – magnetohydrody-

namics (MHD) – waves

INTRODUCTION

This is the second in a series of papers in which we study

high-resolution images of BL Lacertae made at 15 GHz with

the VLBA, under the MOJAVE program (Monitoring of Jets

in Active Galactic Nuclei with VLBA Experiments, Lister et.

al., 2009). In Cohen et al. (2014, hereafter Paper I) we in-

vestigated a quasi-stationary jet component located 0.26 m

from the core, (0.34 pc, projected) and identified it as a reco

limation shock (RCS). Numerous bright radio features (com-

ponents) appear to emanate from this shock, or pass through

it. They propagate superluminally downstream, and their

tracks cluster around an axis that connects the core and the

RCS. This behavior is highly similar to the results of numer-

ical modeling (Lind et al. 1989; Meier 2012), in which MHD

waves or shocks are emitted by an RCS. In the simulations,

the jet has a magnetic field that dominates the dynamics, and

Department of Astronomy, California Institute of Technolo

gy,

Pasadena, CA 91125, USA; mhc@astro.caltech.edu

Jet Propulsion Laboratory, California Institute of Techno

logy,

Pasadena, CA 91109 USA

I. Physikalisches Institut, Universität zu Köln, Zülpiche

r Strasse 77,

50937 Köln, Germany

Byurakan Astrophysical Observatory, Byurakan 378433, Arm

enia and

Isaac Newton Institute of Chile, Armenian Branch

Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69

, 53121

Bonn, Germany

Department of Physics, Denison University, Granville, OH 4

3023

USA

Aalto University Metsähovi Radio Observatory, Metsähovin

tie 114,

02540 Kylmälä, Finland

Astro Space Center of Lebedev Physical Institute, Profsoyu

znaya

84/32, 117997 Moscow, Russia

Department of Physics, Purdue University, 525 Northwester

n Avenue,

West Lafayette, IN 47907, USA

Pulkovo Observatory, Pulkovskoe Chausee 65/1, 196140 St. P

eters-

burg, Russia

Crimean Astrophysical Observatory, 98409 Nauchny, Crimea

Ukraine

is in the form of a helix with a high pitch angle,

. In BL Lac

the motions of the components are similar to those in the nu-

merical models, and in addition the Electric Vector Positio

Angle (EVPA) is longitudinal; i.e., parallel to the jet axis

. For

a jet dominated by helical field, this indicates that the toro

idal

component dominates the poloidal component, a necessary

condition for the comparison of the observations with the nu

merical simulations. Hence, in Paper I, we assumed that the

superluminal components in BL Lac are compressions in the

beam established by slow- and/or fast- mode magnetosonic

waves or shocks traveling downstream on a helical field.

It has been common to assume that the EVPA is

perpendicular to the projection of the magnetic field

vector B that is in the synchrotron emission region.

This is correct in the frame of an optically-thin emis-

sion region, but may well be incorrect in the frame

of the observer if the beam is moving relativistically

(Blandford & Königl

1979;

Lyutikov, Pariev, & Gabuzda

2005). Lyutikov, Pariev, & Gabuzda (2005) show that if the

jet is cylindrical and not resolved transversely, and if the

B field has a helical form, then the EVPA will be either

longitudinal or perpendicular to the jet, depending on the

pitch angle. This is partly seen in the polarization survey

results of Lister & Homan (2005), where the BL Lac objects

tend to have longitudinal EVPA in the inner jet, whereas

the quasars have a broad distribution of EVPA, relative to

the jet direction. This suggests that in BL Lacs the field

may be helical, with pitch angles large enough to produce

longitudinal polarization, although strong transverse sh

ocks

in a largely tangled field are also a possibility (e.g. Hughes

2005). The wide distribution of EVPA values in quasars

suggests that oblique shocks, rather than helical structur

es,

might dominate the field order. However, a distribution of

helical pitch angles could also explain the EVPAs in quasars

if symmetry is broken between the near and far sides of the

OHEN ET AL

−

RA (mas)

−

Dec (mas)

(a)

2005-09-16

−

RA (mas)

−

Dec (mas)

(b)

2006-11-10

−

RA (mas)

−

Dec (mas)

(c)

1999-11-06

Figure 1.

15 GHz VLBA images of BL Lac with ridge line and components (th

e crosses). In (a) the components lie close to the ridge line.

In (b) the three outer

components are off the ridge line by up to 0.3 mas. In this case

the true ridge has a sharp bend and the algorithm has difficult

y in following it. In (c) the ridge has

a step near the core, and appears to bifurcate downstream. Th

e algorithm misses the step, and is unable to deal with the bif

urcation.

jet. It has been suggested (Meier 2013) that this difference

the magnetic field is fundamental to the generic differences

between quasars and BL Lacs.

BL Lacs often show a bend in the jet, and the literature

contains examples showing that in some cases the EVPA

stays longitudinal around the bend; e.g., 1803+784; Gabuzd

(1999), 1749+701; Gabuzda & Pushkarev (2001), and BL

Lac itself; O’Sullivan & Gabuzda (2009). In these examples

and in other BL Lacs studied in the MOJAVE program

, the

fractional polarization rises smoothly along the jet to val

ues

as high as 30%. The field must be well-ordered for the polar-

ization to be that high. We assume that the field is in a rather

tight helix (in the beam frame) and that the moving patterns

(the bends) are Alfvén waves propagating along the longitu-

dinal component of the field.

In a plasma dominated by the magnetic field, Alfvén

waves are transverse displacements of the field (and, per-

force, of the plasma), analogous to waves on a whip.

The tension is provided by the magnetic field (

∝

), and

the wave velocity is proportional to the square root of

the tension divided by the density.

Alfvén waves have

been employed in various astronomical contexts, includ-

ing the acceleration of cosmic rays (Fermi 1949), the solar

wind (Belcher, Davis & Smith 1969), the Jupiter-Io system

(Belcher 1987), turbulence in the ISM (Goldreich & Sridhar

1997), the bow shock of Mars (Edberg et al. 2010), and the

solar atmosphere(McIntosh et al. 2011). In our case they are

transverse waves on a relativistically-moving beam of plas

threaded with a helical magnetic field. The appropriate for-

mulas for the phase speeds of the MHD waves are given in

the Appendix of Paper I.

Changes in the bends of BL Lacs are also seen frequently.

Britzen et al. (2010a) showed that in 1.4 years the ridge line

in 0735+178 changed from having a “staircase" structure

http://www.physics.purdue.edu/astro/MOJAVE/

to being straight, and that there were prominent transverse

motions. Britzen et al. (2010b) also studied 1803+784 and

described various models that might explain the structure.

Perucho et al (2012) studied the ridge line in 0836+710 at sev

eral frequencies and over a range of epochs. They showed

that the ridge line corresponds to the maximum pressure in

the jet. They discussed the concept of transverse velocity,

and

concluded that their measured transverse motions are likel

to be caused by a “moving wave pattern"; this was elabo-

rated in Perucho (2013). In our work here on BL Lac we

also see transverse motions, but their patterns move longit

dinally and we identify them as Alfvén waves. We calculate

the resulting transverse velocity of the jet and show that it

non-relativistic.

It has been more customary to discuss the fast ra-

dio components in a relativistic jet in hydrodynamic

(HD) terms.

We note here only a few examples of

this.

The shock-in-jet model (Marscher & Gear 1985;

Marscher 2014) was used by Hughes, Aller, & Aller

(1989a, 1989b, 1991) to develop models of several sources,

including BL Lac (Hughes, Aller, & Aller 1989b) and

3C 279 (Hughes, Aller, & Aller 1991).

Lobanov & Zensus

(2001) recognized two threads of emission in 3C 273 that

they explained with Kelvin-Helmholtz (KH) instabilities,

and this was developed more by Perucho et al. (2006).

Hardee, Walker & Gómez (2005) discussed the patterns and

motions in 3C 120 in terms of helical instability modes. In

all these studies the magnetic field is needed of course for th

synchrotron radiation, but it also is explicitly used to exp

lain

observed polarization changes as due to compression of the

transverse components of magnetic field, by the HD shock.

But the magnetic field has no dynamical role in the jet. On

the contrary, in this paper, as in Paper I, we assume that the

dynamics in the jet are dominated by the magnetic field.

The plan for this paper is as follows. In Section 2 we briefly

describe the observations. The definition of the ridge line o

LFVÉN

AVES IN

BL L

ACERTAE

Figure 2.

Ridge lines for BL Lac, 1995.26 – 2012.94. Successive panels

are adjacent in time except that there is a 1-year gap between

panels (d) and (e).

Epochs are identified by color. In each panel the first occuren

ce of a color is further identified as the solid line, the next o

ccurence as a dashed line, and the third

occurence, when it exists, as a dotted line.

a jet is considered in Section 3, and the transverse waves and

their velocities, including the behavioral change in 2010,

are

presented and discussed in Section 4. Excitation of the wave

by changes in the P.A. of the RCS is considered in Section 5.

In Section 6 we identify the waves as Alfvén waves, discuss

their properties, and present some simple models of the sys-

tem.

For BL Lac

= 0

0686, and the linear scale is 1

29 pc mas

−

An apparent speed of 1 mas yr

−

corresponds to

app

= 4

20.

OBSERVATIONS

For this study of BL Lac we use 114 epochs of high-

resolution observations made with the VLBA at 15 GHz,

between 1995.27 and 2012.98.

Most of the observa-

tions (75/114) were made under the MOJAVE program

(Lister & Homan 2005), a few were taken from our earlier 2-

cm program on the VLBA (Kellermann et al. 1998), and the

rest were taken from the VLBA archive.

The data were all reduced by the MOJAVE team, using

standard calibration programs (Lister et al 2009). Followi

the reduction to fringe visibilities we calculated three ma

products at nearly every epoch: (1) an image, consisting of

a large number of “clean components" derived from the vis-

ibility function. The images in Figure 1 consist of the clean

components convolved with a “median restoring beam", de-

fined in Section 3. (2) a model consisting of a small number

of Gaussian “components" found by model-fitting in the vis-

ibility plane; each component has a centroid, an ellipticit

y, a

size (FWHM), and a flux density. The Gaussians are circu-

lar when possible. The centroids of the components for each

epoch are plotted on the images in Figure 1. (3) the ridge line

shown in Figure 1 and discussed in Section 3. The image,

the components, and the ridge line are not independent, but

each is advantageous when discussing different aspects of t

source. In most cases the ridge line runs down the smallest

gradient from the peak of the image, and the components lie

on the ridge line. However, when the jet has a sharp bend

the algorithm can fail, as in Figure 1c. This is discussed in

Section 3.

The components are assumed to represent regions of excess

brightness that persist across epochs, and are not merely it

ems

in a mathematical list that sums to the measured image; see

Paper I, also e.g., Lister et al (2009). We attempt to observe

BL Lac frequently, every three or four weeks, to ensure that

the components can be tracked unambiguously from epoch to

epoch. In this paper we only use components that have been

reliably measured four or more times, and have flux density

20 mJy.

The centroid locations are measured relative to the core,

which we take to be the bright spot at the north end of the

source; it usually is regarded as the optically-thick (

= 1)

region of the jet. In principle, the core can move on the sky.

We considered this in Paper I, and concluded that any motions

are less than 10

as in a few years, and they were ignored. Our

positional accuracy is conservatively estimated as

1 mas,

and in this paper we again ignore any possible core motions.

OHEN ET AL

The components move in a roughly radial direction, and

plots of

(

) as well as the sky (RA–Dec) tracks are shown

in Paper I and in Lister et al (2013).

The tracks cluster

around an axis at P

−

166

◦

and appear to emanate from a

strong quasi-stationary component, C7, that we identified a

an RCS in Paper I. The moving components have superlumi-

nal speeds; the fastest has

app

= 10

1 in units of the speed

of light. (Lister et al 2013)

THE RIDGE LINES

We are dealing with moving patterns on the jet of BL Lac,

and in order to quantify them we first need to define the ridge

line of a jet. At least four definitions have been used pre-

viously. Britzen et al. (2010b) used the line that connects

the components at a single epoch, in studying 1803+784.

Perucho et al (2012) investigated three methods of finding th

ridge line: at each radius making a transverse Gaussian fit an

connecting the maxima of the fits, using the geometrical cen-

ter, and using the line of maximum emission. They found no

significant differences among these procedures, for the cas

they studied, 0836+710. They showed that the intensity ridg

line is a robust structure, and that it corresponds to the pre

sure maximum in the jet.

To quantify a ridge line we start with the image as in

Figure 1, which is the convolution of the “clean compo-

nents" with a smoothing beam. Since we are comparing

ridge lines from different epochs, we have used a constant

“median beam" for smoothing, and not the individual (“na-

tive”) smoothing beams. The latter vary a little according

to the observing circumstances for each epoch, and their

use would effectively introduce “instrumental errors” int

o the

ridge lines. The median beam is a Gaussian with major axis =

89 mas (FWHM)

minor axis = 0

56 mas and P

−

◦

Each of the three parameters is the median of the correspond-

ing parameters for all the epochs.

The algorithm for the ridge line starts at the core, and at

successive steps (0.1 mas) down the image finds the midpoint,

where the integral of the intensity across the jet, along a ci

cular arc centered on the core, is equal on the two sides of

the arc. The successive midpoints are then smoothed with a

third-order spline.

Ridge lines are shown on the three images in Figure 1. In

Figure 1a the bends in the jet are gradual and the algorithm

works very well, as indeed would any of the methods men-

tioned above. In Figure 1b there are two sharp bends and our

algorithm makes a smooth line that misses the corners of the

bends. In this case connecting the components would be bet-

ter, if the modelling procedure actually put a component at

the corner. In Figure 1c the jet appears to bifurcate, and our

algorithm picks the west track. In this case a visual inspect

ion

of the image is required to see what is going on.

In fact there is another problem with Figure 1c. The image

has a step to the east (looking downstream) about 1 mas from

the core, where a short EW section connects two longer NS

sections. Since the restoring beam is nearly NS the details o

this step cannot be reconstructed. The calculated ridge lin

in Figure 1c does not reproduce the step, but makes a smooth

track.

Figure 2 shows nearly all the ridge lines that we consider

in this paper; a few are not shown because they occur very

close in time to another one. The epochs are identified by

color, and are further identified by the line type: in each pan

the first occurence of a color is shown solid, the second is

shown dashed, and the third, when it exists, is shown dotted.

Figure 3.

Ridge line for 2005

−

16 calculated (a) with native beams and

(b) with median beam. Solid line: using all the antennas, dot

ted line: omitting

SC and HN. In (a) the beam P.A.s differ by 17

◦

The core is shown as the solid dot at the origin of each panel,

and the semi-circle is drawn at

= 0

25 mas as a convenience.

The RCS is located close to the circle, but is not shown in the

figures. All the ridge lines are drawn relative to the core.

It is important to establish the reliability of the ridge lin

because our analysis rests on them, and some of the structure

that we interpret as waves are smaller than the synthesized

VLBA beam. We first note that as with all VLBI our sam-

pling of the (

) plane is sparse, and different samplings can

produce different ridge lines. To see how strong this effect

is, we emulated an observation with missing antennas by ana-

lyzing a data set with and without one and two antennas, and

we did this analysis both with the native restoring beams and

the median restoring beam described above. The results for

2005-09-16 are shown in Figure 3; they are similar to the re-

sults we obtained for two other epochs. In Figure 3a we show

two ridge lines, the solid one is calculated with the full dat

set and the dashed line is obtained when data from the SC and

HN antennas are not used. The latter calculation does not use

many of the baselines, including the longest ones. The chief

effect is a shift of the pattern downstream, by roughly 0.1 ma

This shift is not a statistical effect, but is mainly due to th

different smoothing beams that were used for the two cases.

We found that the differences in the ridge lines increased wi

increasing difference in the P.A.s of the smoothing beams. I

Figure 3a the difference in P.A of the smoothing beams is 17

◦

In Figure 3b we used the median beam. In this case the

curves are close with differences of typically 3

as out to 4

mas, where the surface brightness becomes low. Beyond 4

mas the differences rise to 50

as.

Another way to investigate the reliability of the ridge line

is to examine pairs of ridge lines measured independently bu

close together in time. The full data set contains 10 pairs

where the separation is no more than 10 days, and these are

all shown in Figure 4. They are calculated with the median

restoring beam. Note that the bottom three panels have a dif-

ferent vertical scale than the others. In general the compar

ison is very good within 4 mas of the core. Panel (i) con-

tains one ridge line that stops at

−

6 mas because the bright-

ness at the ridge becomes too low; this limit also can be seen

in a few places in the other figures. Panel (i) contains the

LFVÉN

AVES IN

BL L

ACERTAE

Figure 4.

Ridge lines for 10 close pairs. The axes are rotated from (RA,

Dec) by 9.5

◦

; North and East are indicated at top. The bottom 3 panels have

a different vertical scale than the others, and the coordina

te directions are

thereby changed by a small amount.

only pair that has a continuous offset, of about 30

−

as.

These data were taken during an exceptional flux outburst at

15 GHz in BL Lac, seen in the MOJAVE data (unpublished),

and roughly coincident with outbursts seen at shorter wave-

lengths (Raiteri et al. 2013). An extra coreshift leading to

position offset is expected with such an event (Kovalev et al

2008). In any event, this pair appears to be different from th

others, and we do not include it in the statistics.

Figure 5 shows the histogram of separations between the

paired ridge lines, after excluding those in panel (i) of Fig

ure 4. In forming the ridge lines a 3-pixel smoothing was

used, and for the histogram we have used every third point.

The median separation is 13

as. Thus the repeatability of

the ridge lines is accurate to about 13

as. The reliability also

depends on the effect discussed in connection with Figure 1,

separation [mas]

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Figure 5.

Histogram of separations between members of 9 close pairs of

ridge lines. The pairs are shown in Figure 4 but panel (i) is no

t included in

the histogram. See text.

that the ridge-finding algorithm can smooth around a corner,

and can be in error by perhaps 100

as. However, the error is

roughly constant over short time spans, as in Figure 4 panel

(e) where the sharp bend at

∼

5 mas is smoothed the same

in the two curves. This smoothing will have little effect on

calculations of wave velocity, which is the main quantitati

use of the ridge lines. We ignore the smoothing in this paper.

From this investigation we conclude that caution must be

taken in interpreting the ridge lines, especially when comp

ar-

ing ridge lines obtained at different epochs. The details of

the

restoring beam can have a noticeable effect on the ridge line

and to avoid misinterpretation the restoring beam should be

the same for all the ridge lines that are being intercompared

When considering these ridge lines it is important to keep

the geometry in mind: the jet has a small angle to the line-of-

sight (LOS), and the foreshortening is about factor of 10 (Pa

per I). Also, the projected images in Figure 1 can hide three-

dimensional motions. To work with skew and non-planar dis-

turbances, we use the coordinate systems shown in Figure 6.

East, North, and the LOS form the left-hand system (x,y,z)

and the jet lies at angle

from the LOS in the

sagittal plane

formed by the LOS and the mean jet axis. This plane is per-

pendicular to the sky plane and is at angle P.A. from the

axis.

The rotated system (

ξ,η,ζ

) is used to describe transverse mo-

tions:

is in the sagittal plane,

is perpendicular to it, and

is along the jet. By “transverse motion" we mean that a point

on the beam has a motion in the (

ξ,η

) plane:

. The com-

ponent

lies in the sagittal plane and its projection on the

sky is along the projection of the jet. The

component of the

transverse motion therefore is not visible, although a brig

superluminal feature moving in the

direction would be seen

as moving slowly along the jet. However, the

component

remains perpendicular to the LOS as

or P.A. changes, and

its full magnitude is always seen. Thus a measured transvers

motion is a lower limit. If the beam is relativistic then time

compression of the forward motion must be added; see Sec-

tion 4.3.

The term is taken from anatomy, where it refers to the plane th

at bisects

the frontal view of a figure with bilateral symmetry. It is als

o used in optics,

in discussions of astigmatism.

OHEN ET AL

x = East

y = North

Projection

of Jet

Sky Plane

Sagittal

Plane

Jet

z = LOS

Figure 6.

Coordinate system. The sagittal plane is defined as the plane

con-

taining the LOS and the mean jet axis; see text.

Some of the panels in Figure 2 show disturbances that ap-

pear to move down the jet, and at other epochs the jet is fairly

quiet. We now consider several of the disturbances in detail

starting with the structures seen in Figure 2, panel (b).

WAVES ON THE RIDGE LINES

Figure 7 is an expanded view of Figure 2, panel (b). It

includes ridge lines for 14 consecutive epochs over a period

about 1.6 yr. Beyond 1 mas the early epochs (solid lines) show

the jet bending to the SE. Later epochs show the bend farther

downstream, and at 2000.31 and later the jet bends to the SW

before bending SE. We anticipate a result from Section 4.2

and draw vector A at P

−

167

◦

across the tracks. The

intersections of vector A with the tracks are shown in the ins

in Figure 7. The velocity implied by the line in the inset is

close to 1 mas yr

−

app

≈

2. The pattern on the ridge

line is moving downstream at nearly constant velocity. We

consider three possible explanations for this.

1) We see the projection of a conical pattern due to a ballis-

tic flow from a swinging nozzle, like water from a hose. The

argument against this is that line B in Figure 7 is parallel to

vector A and approximately tangent to the western crest; thi

line is not radial from the core as it would be if it were a bal-

listic flow. In Figure 2 all the panels except (a), (b), and (e)

show clearly that the flow is constrained to lie in a cylinder,

not a cone.

2) The moving pattern is due to a helical kink instability

that is advected downstream with the flow. In the kink the

field is stretched out and becomes irregular and may no longer

be dominated by the toroidal component (Nakamura & Meier

2004; Mizuno, Hardee, & Nishikawa 2014), especially when

averaged over the VLBI resolution beam. This should reduce

the fractional polarization substantially, and could prod

uce an

EVPA normal to the wave crest in Figure 7 rather than lon-

gitudinal. But in BL Lac the fractional polarization remain

high and the field remains longitudinal, along the bend. In

Figure 8 we show the polarization image for one of the epochs

for the large wave shown later in Figure 10. Figure 8 is taken

from the MOJAVE website

; see also O’Sullivan & Gabuzda

(2009). In Figure 8 the linear polarization fraction

is indi-

cated by the color bar, and in the right-hand figure tick marks

show the EVPA corrected for the Galactic Faraday Rotation.

Figure 7.

Ridge lines for BL Lac at 15 GHz, for 14 epochs between 1999.37

and 2000.99. Below Dec =

−

2 mas, the displacement in space corresponds

to a displacement in time, and the inset shows the points wher

e the vector

A crosses the ridge lines – the ordinate is distance along the

vector A. The

velocity in the A direction is 0

92 mas yr

−

at P

−

167

◦

; the arrow itself

represents the propagation vector that is derived in the tex

t. The offset straight

line B is parallel to the propagation vector. It is approxima

tely tangent to the

wave crests, and so the wave has constant amplitude as it move

s to the SW.

The short arrow C shows a swing of the jet from west to east in ea

rly 2000;

see text Section 5. The point b shows the characteristic poin

t on the 2000.57

line where the slope changes; see text Section 4.2. Colors ar

e as in Figure 2.

The EVPA is nearly parallel to the jet out to about 5 mas,

and

is high on the ridge, up to

∼

30% except near the core

and in a slice at 2 mas where it drops to

= 15%. This drop

presumably is due to the blending of orthogonally polarized

components at the bend in the jet, where the EVPA changes.

We think it likely that the EVPA and fractional polarization

data preclude the identification of the structures seen in Fi

ure 7 as an advected kink instability.

3) The moving patterns are transverse MHD waves; i.e.,

Alfvén waves. For this to be possible the plasma must be

dynamically dominated by a helical magnetic field. This con-

dition for the jet of a BL Lac has been suggested many times,

see e.g., Gabuzda, Murray & Cronin (2004), Meier (2013).

Note that we implicitly assumed the helical, strong-field ca

in discussing the kink instability, in the preceding paragr

aph,

and we also assumed it in Paper I. Thus, we assume that the

moving pattern under vector A in Figure 7 is an Alfvén wave,

with velocity

∼

1 mas yr

−

In Figure 7 a second wave is seen between

= 1 and

= 2

mas, where the ridge lines for epochs 2000.31 and later bend

to the SW. The two waves in Figure 7 can be thought of as

one wave with a crest to the west. This wave is generated by

a swing of the nozzle to the west followed by a swing back to

the east about 2 years later, as discussed below in Section 5.

The 1999-2000 wave is displayed in a different form in Fig-

LFVÉN

AVES IN

BL L

ACERTAE

Figure 8.

Polarization image for BL Lac epoch 2005-09-23, one of those

forming the large wave in Figure 10. Linear polarization fra

ction

is indi-

cated by the color bar; at the core

≈

6%, in the slice at

∼

−

2 mas

drops to

15%, and on the ridge

remains near 30% from 2 to 4 mas. In the right-hand

image tick marks show the EVPA corrected for Galactic Farada

y Rotation;

the EVPA stays nearly parallel to the jet out to about 5 mas.

ure 9, which shows the ridge lines from 1999.37 to 2001.97.

Vertical spacing is proportional to epoch, and the axes have

been rotated by 13

◦

; arrows at top show North and East. Tick

marks on the right are 0.1 mas apart. The dots show the

points described later in Section 4.2, where the slope chang

es,

and the solid line A is a linear fit through the points, with

speed

= 0

05mas yr

−

. This wave is prominent un-

til 2000.99. In 2001.22 the structure has changed. There are

alternate possibilities to explain this new structure, B. I

t may

be a new wave, with the crests connected with line B (drawn

with the same slope as line A). In this case the wave must have

been excited somehow far from the RCS. The fit of line B to

the wave crests is poor and would be improved if acceleration

were included, but there is not enough data for that. Alterna

tively, structure B may simply be a relic of the trailing side

wave A, perhaps relativistically boosted by the changing ge

ometry (the bend) seen in Figure 2 panel (c). A third wave C

is shown by the dashed line that again is drawn with the same

slope.

Panel (c) of Figure 2 shows the ridge lines projected on the

sky for 2001 – 2002. Wave B from Figure 9 is seen as the

bump to the east at

= 2 mas, which moves downstream at

succeeding epochs. The projected axis of the jet is curved

at these epochs, and the possible acceleration noted above

for wave B may simply be a relativistic effect inherent in the

changing geometry.

Wave A in Figure 9 is barely visible in Figure 2 panel (a) as

a gentle bump in 1999.04, so it is first apparent in early 1999

at a distance

∼

1 mas from the core. This is reminiscent of

the behavior of the components discussed in Paper I; Figure 3

of that paper shows that most of the components first become

visible near

= 1 mas. Wave C also appears to start near

∼

−

4

−

3

−

2

−

1

0

Rotated Dec (mas)

2002

2001

2000

Epoch

Rotated RA (mas)

Figure 9.

Ridge lines for 1999.37-2001.97, plotted on axes rotated by

◦

North and East are indicated at the top. The ridge lines are sp

aced vertically

according to epoch, and the tick marks on the right-hand side

are spaced 0.1

mas apart. The solid line is a linear fit to the dots, which are t

he characteristic

points discussed in Section 4.2. The three lines are paralle

l and all have slope

0.92 mas yr

−

. See text.

mas.

In Figure 7 the short arrow C shows an eastward swing of

the inner jet between 2000.01 and 2000.31. This is seen in

Figure 9 in the ridge line for 2000.31, which shows a new

inner P.A. The effect of these P.A. swings on the beam is dis-

cussed in Section 5.

The different panels in Figure 2 show that the jet can be

bent, and even when relatively straight, can lie at differen

P.A.s. Hence there is no unique rotation angle for the ridge

lines in a plot such as that in Figure 9. The rotation angle use

in Figure 9 was found by the velocity algorithm described in

Section 4.2 for wave A.

Further examples of waves are shown in Figures 10 – 12,

omitting the extraneous ridge lines to avoid confusion. The

wave motions are indicated by the arrows, which are propa-

gation vectors derived in Section 4.2. Table 1 lists the deta

ils

for these waves.

is the measured proper motion,

app

the apparent motion in units of

gal

wave

the wave speed in the

coordinate frame of the galaxy, assuming

= 6

◦

, P.A. is the

direction of the propagation vector, and the amplitude is an

estimate that is not well-defined but is an indication of the

strength of the wave. The 2005 wave is the largest such fea-

ture seen in the data. Unfortunately, there was an 11-month

data gap prior to 2005.71, and the wave cannot be seen at ear-

lier times.

The amplitudes of the larger waves appear to be comparable

with the length scale, as indicated for example by the length

the diagonal part of the wave in Figure 7. However, this is an

illusion caused by the foreshortening. The angle to the LOS

≈

◦

and the foreshortening is approximately a factor of 10.

(Paper I). The amplitude is therefore only a few percent of

the length scale. The wavelength itself is not a well-defined

quantity, as the system is not periodic, at least not on time

scales up to 15 years. The length scales that we see in Figure 7

are controlled by the wobble in the nozzle at the RCS (see

Section 5).

OHEN ET AL

Figure 10.

Ridge lines for BL Lac at 15 GHz, for 5 epochs between 2005.7

and 2006.9. The propagation vector for Wave D is at P

−

180

◦

. Point

represents the advected beginning of the wave; see text.

Figure 11.

Ridge lines for BL Lac at 15 GHz, for 7 epochs between 2008.5

and 2008.9, showing Wave E with a propagation vector at P

−

175

◦

Figure 12.

Ridge lines for BL Lac at 15 GHz, for 6 epochs between 2009.3

and 2009.9, showing Wave F with propagation vector at P

−

166

◦

Figure 13 contains one frame of a movie of BL Lac showing

the jet motions and ridge line fits at 15 GHz. The full movie

is available in the electronic version of this paper or from t

MOJAVE web

4.1.

Different Jet Behavior in 2010-2013

In Figure 2 panels (g) and (h) we see that by 2010 the ear-

lier activity in the jet has subsided, and that after 2010.5 t

jet is well-aligned at P

−

170

◦

5 with a weak wiggle. But

the wiggle is not stationary. Figure 14 shows the ridge lines

plotted on axes rotated by 9

◦

5, and spaced proportionately to

epoch. Most of the ridge lines have a quasi-sinusoidal form,

with the wavelength increasing down the jet, and the ampli-

tude decreasing. Almost all the epochs show a negative peak

in the inner jet, with a minimum near Dec =

−

7 mas. This

is a quasi-standing feature, of variable amplitude. At most

epochs there is a positive peak near Dec =

−

6 mas. This

also is a quasi-standing feature, but less distinct than the

inner

one.

What is causing the quasi-standing features? The patterns

can hardly be true standing waves because that requires a re-

flection region. A rotating helix would project as a travelin

wave, as on a barber pole, so a simple barber-pole model is

excluded. Possible motions of the core are only about 10

(Paper I), so any registration errors due to core motion are

much smaller than the observed changes, which are up to

100

as. There is little indication of wave motion in Fig-

ure 14; at least, not at the speeds seen in Figure 2. Although

the transverse Alfvén waves appear to have stopped during

this period, the superluminal components, which we identi-

fied in Paper I as MHD acoustic waves, did not. Figure 2

http://www.astro.purdue.edu/MOJAVE/bllacpaper2.mpg

LFVÉN

AVES IN

BL L

ACERTAE

Table 1

Transverse Waves on the Jet of BL Lac.

Epoch

a p p

gal

wave

P.A.

Amplitude

(mas y

−

)

(deg)

(mas)

A 1999.37-2000.99

0.92

3.9

0.979

−

167

0.5

D 2005.71-2006.86

1.25

5.6

0.987

−

180

0.9

E 2008.33-2008.88

3.01

13.5

0.998

−

174

0.3

F 2009.33-2009.96

1.11

5.0

0.985

−

167

0.2

Notes. Columns are as follows: (1) Wave label, (2) Inclusive

range of epochs, (3) number of epochs, (4) apparent speed, (5

) error, (6)

apparent speed in units of

, (7) speed in galaxy frame, assuming

= 6

◦

, (8) P.A. of the wave, (9) error, (10) estimated amplitude.

Figure 13.

Movie of the BL Lac jet at 15 GHz. The total intensity image

is on the right, with a color bar indicating flux density. The c

ontour levels

begin at 7 mJy per beam, and increase by logarithmic factors o

f 2. The false

color scheme uses a square root transfer function, and is sat

urated at the core

position in order to highlight changes in the much fainter je

t. The core peak

brightness is highly variable; typically it is between 2 and

6 Jy/beam. The

projected linear scale is indicated by the 2 pc line at left. T

he movie frames

are linearly interpolated between the individual VLBA epoc

h images, which

have been registered to the fitted position of the core featur

e, and restored

with a median beam with FWHM dimensions of 0

57 mas, with a

major axis position angle at

−

◦

, as indicated in the lower left corner of

the frame. The fitted ridge line is shown as a dashed line in the

image, and

again as a solid line to the left of the image. These have also b

een linearly

interpolated between the individual VLBA epochs. The point

s of changing

slope (see Section 4.2) at individual VLBA epochs are shown a

s the small

symbols. At left the ridge lines are shown with different col

ors for the various

waves. The yellow

line on the right is advancing at the speed of light

(

app

= 1) and is included for reference.

in Paper I shows that they continued during this period, with

about the same frequency and speed as earlier.

It appears then, that during the “quiet" period 2010-2013,

the jet was essentially straight but with a set of weak quasi-

stationary patterns, with variable amplitude. The superlu

minal components, however, continued as before. A further

complication is that during the latter half of this period, f

rom

about 2011.4 to 2013.0, BL Lac was exceptionally active at

shorter wavelengths (Raiteri et al. 2013), from 1 mm through

gamma-rays. This behavior is not understood.

4.2.

Velocity of the Waves

We estimated the velocity of Wave A in Figure 7 in two in-

dependent ways. In the first we assume that there is a constant

propagation vector, and we shift and superpose the ridge lin

on a grid of (

, P.A.) where

is the speed of the wave and P.A.

is its propagation direction. If the ridge lines form a simpl

wave, then the solution is found when the lines lie on top of

each other. This is shown in Figure 15, where a reasonable fit

−

3

−

2

−

1

0

Rotated Dec (mas)

2013

2012

2011

Epoch

Rotated RA (mas)

Figure 14.

Ridge lines as in Figure 2 panels (g) and (h), plotted on axes

rotated by 9

◦

5 and with vertical spacing proportional to epoch. Tick mark

on right-hand side are 0.1 mas apart.

can be selected by eye. The result is

= 0

08 mas yr

−

at P

−

168

◦

. This solution is somewhat subjective

and the quoted errors do not have the usual statistical signi

fi-

cance.

As an alternative procedure to visually aligning the ridge

lines, we developed a method of identifying a characteristi

point on the wave, just downstream of the crest, where the

wave amplitude has begun to decrease. Define the slope of the

ridge line as

∆

in pixels, where in Figure 9,

and

are

rotated RA and Dec, and take the first downstream location

where the slope exceeds

05. This point is marked with the

dot b on the ridge line for 2000.57 in Figure 7. The

and

positions vs time for these locations are then fit independen

tly

using the same methods as described in Lister et al (2009) to

extract a vector proper motion for this characteristic poin

t on

the wave.

The two methods agree well and the analytic solution is

05 mas yr

−

at P

−

167

◦

5, and the apparent

speed is

app

= 3

2. The propagation vector is shown in

Figure 7 and the speed and direction of the wave are listed in

Table 1. The Table also includes

gal

wave

the speed of the wave in

the galaxy frame, assuming

= 6

◦

. This calculation assumes

that the ridge lines lie in a plane; i.e., are not twisted. Thi

s is

not neccessarily the case. Rather, since the inner jet, near

the

accretion disk, may wobble in 3 dimensions, (McKinney et al,

2013) it seems likely that the RCS will execute 3-dimensiona

motion and that the downstream jet will also. See Section 5.