STUDIES OF THE JET IN BL LACERTAE. II. SUPERLUMINAL ALFVÉN WAVES
M. H. Cohen
1
, D. L. Meier
1
,
2
, T. G. Arshakian
3
,
4
, E. Clausen-Brown
5
, D. C. Homan
6
,T.Hovatta
1
,
7
, Y. Y. Kovalev
5
,
8
,
M. L. Lister
9
, A. B. Pushkarev
5
,
10
,
11
, J. L. Richards
9
, and T. Savolainen
5
,
7
1
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA;
mhc@astro.caltech.edu
2
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
3
I. Physikalisches Institut, Universität zu Köln, Zülpicher Strasse 77, D-50937 Köln, Germany
4
Byurakan Astrophysical Observatory, Byurakan 378433, Armenia and Isaac Newton Institute of Chile, Armenian Branch
5
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany
6
Department of Physics, Denison University, Granville, OH 43023, USA
7
Aalto University Metsähovi Radio Observatory, Metsähovintie 114, FI-02540 Kylmälä, Finland
8
Astro Space Center of Lebedev Physical Institute, Profsoyuznaya 84/32, 117997 Moscow, Russia
9
Department of Physics, Purdue University, 525 Northwestern Avenue, West Lafayette, IN 47907, USA
10
Pulkovo Observatory, Pulkovskoe Chausee 65/1, 196140 St. Petersburg, Russia
11
Crimean Astrophysical Observatory, 98409 Nauchny, Crimea, Russia
Received 2014 September 8; accepted 2015 January 14; published 2015 April 6
ABSTRACT
We study the kinematics of ridge lines on the parsec-scale jet of the active galactic nucleus BL Lacertae. We show
that the ridge lines display transverse patterns that move superluminally downstream, and that the moving patterns
are analogous to waves on a whip. Their apparent speeds
β
app
(
units of
c
)
range from 3.9 to 13.5, corresponding to
b
=-
0.981 0.998
wave
gal
in the galaxy frame. We show that the magnetic
fi
eld in the jet is well ordered with a strong
transverse component, and assume that it is helical and that the transverse patterns are Alfvén waves propagating
downstream on the longitudinal component of the magnetic
fi
eld. The wave-induced transverse speed of the jet is
non-relativistic
(
b
0.09
tr
gal
)
. In 2010 the wave activity subsided and the jet then displayed a mild wiggle that had
a complex oscillatory behavior. The Alfvén waves appear to be excited by changes in the position angle of the
recollimation shock, in analogy to exciting a wave on a whip by shaking the handle. A simple model of the system
with plasma sound speed
β
s
=
0.3 and apparent speed of a slow MHD wave
β
app,
S
=
4 yields Lorentz factor of
the beam
Γ
beam
∼
4.5, pitch angle of the helix
(
in the beam frame
)
α
∼
67
°
, Alfvén speed
β
A
∼
0.64, and
magnetosonic Mach number
M
ms
∼
4.7. This describes a plasma in which the magnetic
fi
eld is dominant and in a
rather tight helix, and Alfvén waves are responsible for the moving transverse patterns.
Key words:
BL Lacertae objects: individual
(
BL Lac
)
–
galaxies: jets
–
magnetohydrodynamics
(
MHD
)
–
waves
Supporting material:
animation
1. INTRODUCTION
This is the second in a series of papers in which we study
high-resolution images of BL Lacertae
(
BL Lac
)
made at
15 GHz with the Very Long Baseline Array
(
VLBA
)
, under the
Monitoring of Jets in Active Galactic Nuclei with VLBA
Experiments
(
MOJAVE
)
program
(
Lister et al.
2009
)
.In
Cohen et al.
(
2014
, hereafter Paper I
)
we investigated a quasi-
stationary bright radio feature
(
component
)
in the jet located
0.26 mas from the core,
(
0.34 pc, projected
)
and identi
fi
ed it as
a recollimation shock
(
RCS
)
. Numerous components 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 numerical modeling
(
Lind
et al.
1989
; Meier
2012
, p. 717
)
, in which MHD waves or
shocks are emitted by an RCS. In the simulations, the jet has a
magnetic
fi
eld that dominates the dynamics, and is in the form
of a helix with a high pitch angle,
α
. In BL Lac,
the motions of
the components are similar to those in the numerical models,
and in addition the Electric Vector Position Angle
(
EVPA
)
is
longitudinal,
i.e., parallel to the jet axis. For a jet dominated by
helical
fi
eld, this indicates that the toroidal component is
substantial
(
f
B
B
1
pol
)
, a necessary condition for the
comparison of the observations with the numerical 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
fi
eld.
It has been common to assume that the EVPA is
perpendicular to the projection of the magnetic
fi
eld vector
B
that is in the synchrotron emission region. This is correct in the
frame of an optically thin emission region, but may well be
incorrect in the frame of the observer if the beam is moving
relativistically
(
Blandford & Königl
1979
; Lyutikov
et al.
2005
)
. Lyutikov et al.
(
2005
)
show that if the jet is
cylindrical and not resolved transversely, and if the B
fi
eld 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 Lac objects
the
fi
eld may be helical, with pitch
angles large enough to produce longitudinal EVPA, although
strong transverse shocks in a largely tangled
fi
eld are also a
possibility
(
e.g., Hughes
2005
)
. The wide distribution of EVPA
values in quasars suggests that oblique shocks, rather than
helical structures, might dominate the
fi
eld 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 jet. It has been suggested
(
Meier
2013
)
that this
difference in the magnetic
fi
eld is fundamental to the generic
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803:3
(
16pp
)
, 2015 April 10
doi:10.1088/0004-637X/803/1/3
© 2015. The American Astronomical Society. All rights reserved.
1
differences between quasars and BL Lac objects
and, by
inference, between Fanaroff & Riley
(
1974
)
class II and I
sources, respectively
(
Fanaroff & Riley
1974
)
.
BL Lac objects
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,
Gabuzda
1999;
1749 + 701,
Gabuzda & Pushkarev
2001
;
and BL Lac itself,
O
’
Sullivan & Gabuzda
2009
)
.In
these examples the fractional polarization
p
rises smoothly
along the jet to values as high as
p
=
30%. The
fi
eld must be
well
ordered for the polarization to be that high. In this paper,
we assume that the
fi
eld is in a rather tight helix
(
in the beam
frame
)
and that the moving patterns
(
the transverse distur-
bances
)
are Alfvén waves propagating along the longitudinal
component of the
fi
eld.
In a plasma dominated by the magnetic
fi
eld, Alfvén waves
are transverse displacements of the
fi
eld
(
and, perforce, of the
plasma
)
, analogous to waves on a whip. The tension is
provided by the magnetic
fi
eld
μ
B
(
)
2
, and the wave velocity is
proportional to the square root of the tension divided by the
(
relativistic
)
mass density. Alfvén waves have been employed
in various astronomical contexts, including the acceleration of
cosmic rays
(
Fermi
1949
)
, the solar wind
(
Belcher et al.
1969
)
,
the Jupiter-Io system
(
Goldreich & Lynden-Bell
1969
)
,
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 plasma threaded with a
helical magnetic
fi
eld. The appropriate formulas for the phase
speeds of the MHD waves are given in the appendix of Paper
I
.
Changes in the ridge lines of BL Lac objects
are also seen
frequently. Britzen et al.
(
2010a
)
showed that in 1.4 yr the BL
Lac object 0735 + 178 changed from having a
“
staircase
”
structure 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 several 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 likely to be caused by a
“
moving wave pattern
”
; this was
elaborated in Perucho
(
2013
)
. In our work here on BL Lac we
also see transverse motions, but their patterns move long-
itudinally and we identify them as Alfvén waves. We calculate
the resulting transverse velocity of the wave motion and show
that it is non-relativistic.
It has been more customary to discuss the fast radio
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 et al.
(
1989a
,
1989b
,
1991
)
to develop models of
several sources, including BL Lac
(
Hughes et al.
1989b
)
and
3C 279
(
Hughes et al.
1991
)
. Lobanov & Zensus
(
2001
)
recognized two threads of emission in 3C 273 that they
explained with Kelvin
–
Helmholtz instabilities, and this was
developed more by Perucho et al.
(
2006
)
. Hardee et al.
(
2005
)
discussed the patterns and motions in 3C 120 in terms of helical
instability modes. In all these studies the magnetic
fi
eld is
needed of course for the synchrotron radiation, but it also is
explicitly used to explain observed polarization changes as due
to compression of the transverse components of magnetic
fi
eld
by the HD shock. But in these works the magnetic
fi
eld 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
fi
eld.
The plan for this paper is as follows. In Section
2,
we brie
fl
y
describe the observations. The de
fi
nition of the ridge line of 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 waves
by changes in the position angle
(
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 simple
models of the system.
For BL Lac,
z
=
0.0686, and the linear scale is
1.29 pc mas
−
1
. An apparent speed of 1 mas yr
−
1
corresponds
to
β
app
=
4.20.
2. 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
12
(
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
)
. Following
the reduction to fringe visibilities we calculated three main
products at nearly every epoch:
1. An image, consisting of a large number of
“
clean delta
functions
”
produced by the algorithm used for deconvo-
lution, convolved with a
“
median restoring beam,
”
de
fi
ned in Section
3
.
2. A model, consisting of a set of Gaussian
“
components
”
found by model-
fi
tting in the visibility plane; each
component has a centroid, an ellipticity, a size
(
FWHM
)
,
and a
fl
ux density. The Gaussians are circular when
possible. The total set of components sums to the image,
but in this paper we only use components that have been
reliably measured at four or more epochs, have
fl
ux
density
>
20 mJy
, and can be tracked unambiguously
from epoch to epoch. A typical epoch shows four to six
of these
“
robust
”
components. The RCS is a permanent
component and, together with the core, usually produces
more than half of the total
fl
ux density from the jet. The
centroids of the robust components for each epoch are
plotted on the images in Figure
1
.
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
(
τ
=
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
μ
as in a few years, and
they were ignored. Our positional accuracy is conserva-
tively estimated as
±
0.1 mas, and in this paper we again
ignore any possible core motions.
3. The ridge line, shown in Figure
1
and discussed in
Section
3
.
The image, the components, and the ridge line are not
12
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independent, but each is advantageous when discussing
different aspects of the source. In most cases the ridge
line runs down the smallest gradient from the peak of the
image, and the centroids of the components lie on the
ridge line. However, when the jet has a sharp bend the
algorithm can fail, as in Figure
1
(
c
)
. This is discussed in
Section
3
.
The components move in a roughly radial direction, and plots
of
r
(
t
)
as well as the sky
(
R.A.
–
decl.
)
tracks are shown in
Paper
I
and in Lister et al.
(
2013
)
. The tracks cluster around an
axis at P.A.
=
−
166
°
and appear to emanate from a strong
quasi-stationary component, C7, that we identi
fi
ed as an RCS
in Paper
I
. The moving components have superluminal speeds;
the fastest has
β
app
=
10
±
1 in units of the speed of light
(
Lister et al.
2013
)
.
3. THE RIDGE LINES
We are dealing with moving patterns on the jet of BL Lac,
and in order to quantify them we
fi
rst need to de
fi
ne the ridge
line of a jet. At least four de
fi
nitions have been used previously.
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
fi
nding the
ridge line: at each radius making a transverse Gaussian
fi
t and
connecting the maxima of the
fi
ts, using the geometrical center,
and using the line of maximum emission. They found no
signi
fi
cant differences among these procedures
for the case
they studied, 0836 + 710. They showed that the intensity ridge
line is a robust structure, and that it corresponds to the pressure
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 delta
functions
”
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
(
“
native
”
)
smoothing beams. The latter vary a little according
to the observing circumstances for each epoch, and their use
would effectively introduce
“
instrumental errors
”
into the ridge
lines. The median beam is a Gaussian with major
axis
=
0.89 mas
(
FWHM
)
, minor axis
=
0.56 mas, and
=-
◦
P
.A.
8.6.
Each of the three parameters is the median of
the corresponding 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
fi
nds the midpoint,
where the integral of the intensity across the jet, along a circular
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
1
(
a
)
the bends in the jet are gradual and the algorithm
works very well, as indeed would any of the methods
mentioned above. In Figure
1
(
b
)
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
better, if the modeling procedure actually put components at
the corners. In Figure
1
(
c
)
the jet appears to bifurcate, and our
algorithm picks the west track. In this case a visual inspection
of the image is required to see what is going on.
In fact, there is another problem with Figure
1
(
c
)
. The image
has a step to the east
(
looking upstream
)
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 of
this step cannot be reconstructed. The calculated ridge line in
Figure
1
(
c
)
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. In all cases the RCS is located close to
the semi-circle, drawn 0.25 mas from the core. Successive
Figure 1.
15 GHz VLBA images of BL Lac with ridge line and components
(
the 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 dif
fi
culty in following it. In
(
c
)
the ridge has a step
near the core, and appears to bifurcate downstream. The algorithm misses the step, and is unable to deal with the bifurcation.
3
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panels are adjacent in time, although there is a one-year gap in
the data between panels
(
d
)
and
(
e
)
. The only other substantial
data gap is seen in panel
(
a
)
from 1998.18 to 1999.04. In
Figure
2,
the epochs are set nearly equally among the panels,
with the separations picked to emphasize the various waves that
are discussed below.
It is important to establish the reliability of the ridge lines
because our analysis rests on them, and some of the structures
that we interpret as waves are smaller than the synthesized
VLBA beam. We
fi
rst note that as with all VLBI our sampling of
the
(
u
,
v
)
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 analyzing 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 September
16 are shown in Figure
3
; they are similar to the results we
obtained for two other epochs. In Figure
3
(
a
)
we show two ridge
lines, the solid one is calculated with the full data set and the
dashed line is obtained when data from the SC and HN antennas
are omitted. 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 mas. This shift is not a
statistical effect, but is mainly due to the different smoothing
beams that were used for the two cases. We found that the
differences in the ridge lines increased with increasing difference
in the P.A.s of the smoothing beams. In Figure
3
(
a
)
the
difference in P.A of the smoothing beams is 17
°
.
In Figure
3
(
b
)
we used the median beam. In this case the
curves are close with differences of typically 3
μ
as out to
4 mas, where the surface brightness becomes low. Beyond
4 mas the differences rise to 50
μ
as.
Another way to investigate the reliability of the ridge lines is
to examine pairs of ridge lines measured independently but
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 different
vertical scale than the others. In general the comparison is very
good within 4 mas of the core. Panel
(
i
)
contains one ridge line
that stops at 3.6 mas because the brightness at the ridge
becomes too low; this limit also can be seen in a few places in
the other
fi
gures. Panel
(
i
)
contains the only pair that has a
continuous offset, 30
–
50
μ
as. These data were taken during an
exceptional
fl
ux outburst at 15 GHz in BL Lac, seen in the
MOJAVE data
(
unpublished
)
, and roughly coincident with
outbursts seen at shorter wavelengths
(
Raiteri et al.
2013
)
.An
extra coreshift leading to a position offset is expected with such
an event
(
Kovalev et al.
2008
; Pushkarev et al.
2012
)
. In any
event, this pair appears to be different from the 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 Figure
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
μ
as. Thus the repeatability of the ridge lines is
Figure 2.
Ridge lines for BL Lac
1995.26
–
2012.94. Successive panels are adjacent in time. Epochs are identi
fi
ed by color. In each panel the
fi
rst occurence of a color
is further identi
fi
ed as the solid line, the next occurence as a dashed line, and the third occurence, when it exists, as a dotted line. The core is shown as the solid dot, and
the semi-circle is drawn 0.25 mas from the core. In all cases the RCS is close to the circle.
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accurate to about 13
μ
as. The reliability also depends on the
effect discussed in connection with Figure
1
, that the ridge-
fi
nding algorithm can smooth around a corner, and can be in
error by perhaps 100
μ
as. However, the error is roughly
constant over short time spans, as in Figure
4
panel
(
e
)
where
the sharp bend at
∼
1.5 mas is smoothed the same in the two
curves. This smoothing will have little effect on calculations of
wave velocity, which is our main quantitative 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 comparing
ridge lines obtained at different epochs, or with different
frequencies. The details of the restoring beam can have a
noticeable effect on the ridge line, and to avoid misinterpreta-
tion 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 a factor of 10
(
Paper
I
)
.
Also, the projected images in Figure
1
can hide three-
dimensional motions. To work with skew and non-planar
disturbances, 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
θ
from the LOS in the
sagittal plane
13
formed by the LOS and the mean jet axis. This plane is
Figure 3.
Ridge line for 2005 September 16 calculated
(
a
)
with native beams
and
(
b
)
with median beam. Solid line: using all the antennas;
dotted line:
omitting SC and HN. In
(
a
)
the beam P.A.s differ by 17
°
.
Figure 4.
Ridge lines for 10 pairs that each occur close in time. The axes are
rotated from
(
R.A., decl.
)
by 9
◦
.5; north and east are indicated at the top. The
bottom three panels have a different vertical scale than the others, and the
coordinate directions are thereby changed by a small amount.
Figure 5.
Histogram of separations between members of nine close pairs of
ridge lines. The pairs are shown in Figure
4,
but panel
(
i
)
is not included in the
histogram. See the text.
Figure 6.
Coordinate system. The sagittal plane is de
fi
ned as the plane
containing the LOS and the mean jet axis; see the text.
13
The term is taken from anatomy, where it refers to the plane that bisects the
frontal view of a
fi
gure with bilateral symmetry. It is also used in optics
in
discussions of astigmatism.
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perpendicular to the sky plane and is at angle P.A. from the
y
axis. The rotated system
(
ξ
,
η
,
ζ
)
is used to describe transverse
motions:
ξ
is in the sagittal plane,
η
is perpendicular to it, and
ζ
is along the jet. By
“
transverse motion
”
we mean that a point
on the beam has a motion in the
(
ξ
,
η
)
plane:
xh
vv
,
. The
component
v
ξ
lies in the sagittal plane and its projection on the
sky is along the projection of the jet. This component therefore
is not visible, although a bright feature moving in the
ξ
direction might be seen as moving slowly along the jet.
However, the
v
η
component remains perpendicular to the LOS
as
θ
or P.A. changes, and its full magnitude is always seen.
Thus, a measured transverse motion is a lower limit. If the
beam is relativistic then time compression of the forward
motion must be added; see Section
4.3
.
Some of the panels in Figure
2
show disturbances that appear
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
)
.
4. 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 of
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.A.
=
−
167
°
across the tracks. The
intersections of vector
A
with the tracks are shown in the
inset in Figure
7
. The velocity implied by the line in the inset is
close to 1 mas yr
−
1
or
b
»
4
app
. The pattern on the ridge line is
moving superluminally downstream at nearly constant velocity.
We consider three possible explanations for this.
1. We see the projection of a conical pattern due to a
ballistic
fl
ow 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; this feature of the ridge lines is not radial
from the core as it would be if it were a ballistic
fl
ow. In
Figure
2
, all the panels except
(
a
)
,
(
b
)
, and
(
e
)
show
clearly that the excursions of the ridge lines are
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
fl
ow. In the kink the
fi
eld would be stretched out and become largely parallel
to the observed bends in the jet that, in this case, seem to
be transverse waves
(
Nakamura & Meier
2004
; Mizuno
et al.
2014
)
. This would produce an EVPA normal to the
wave crest in Figure
7
rather than longitudinal. But in BL
Lac,
the EVPA tends to be longitudinal, even along the
bends. In Figure
8,
we show the polarization image for
2005 September 23, taken from the MOJAVE websi-
te.
12
Similar polarization images for BL Lac, at several
wavelengths, are shown in O
’
Sullivan & Gabuzda
(
2009
,
Figure 19
)
for epoch 2006 July 2. Both of these epochs
are part of the large Wave D shown later in Figure
10
.In
these polarization images the EVPA is nearly parallel to
the jet out to about 5 mas and
p
is high on the ridge,
indicating that the magnetic
fi
eld remains in a relatively
tightly coiled helix around the bend and is not nearly
parallel to the axis, as it should be for an advected kink
instability.
Figure 7.
Ridge lines for BL Lac
at 15 GHz
for 14 epochs between 1999.37
and 2000.99. Below
r
=
−
2 mas, the displacement in space corresponds to a
displacement in time, and the inset shows the points where 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
−
1
at P.A.
=
−
167
°
; the arrow itself represents the
propagation vector that is derived in the text. The offset straight line B is
parallel to the propagation vector. It is approximately tangent to the wave
crests, and so the wave has a constant amplitude as it moves to the SW. The
short arrow C shows a swing of the jet from west to east in early 2000; see
Section
5
. The point
b
shows the characteristic point on the 2000.57 line where
the slope changes; see Section
4.2
. Colors are the same as in Figure
2
.
Figure 8.
Polarization image for BL Lac epoch 2005 September 23, one of
those forming the large wave in Figure
10
. Linear polarization fraction
p
is
indicated by the color bar; at the core
»
p
6%
, in the slice at
∼
−
2 mas
p
drops
to 15%, and on the ridge
p
remains near 30% from 2 to 4 mas. On the right-
hand side, image tick marks show the EVPA corrected for Galactic Faraday
Rotation; the EVPA stays nearly parallel to the jet out to about 5 mas.
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Wave D is the largest wave in the BL Lac data, and
seems to have the cleanest longitudinal polarization. At
other epochs the EVPA tends to be longitudinal, but can
be off by up to 40
°
. We have only one epoch of
polarization data for Wave A, but that one does show an
EVPA that is tightly longitudinal in the bend. Thus we
believe that the EVPA results preclude the identi
fi
cation
of the structures seen in Figure
7
as due to a 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
fi
eld. This
condition for the jet of a BL Lac object has been
suggested many times; see, e.g., Gabuzda et al.
(
2004
)
and
Meier
(
2013
)
. Note that we implicitly assumed the
helical, strong-
fi
eld case in discussin
g the kink instabil-
ity, in the preceding paragraph, 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
∼
1masyr
−
1
.
In Figure
7
a second wave is seen between
r
=
1 and
r
=
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 two years
later, as discussed below in Section
5
.
The 1999
–
2000 wave is displayed in a different form in
Figure
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 changes, and the
solid line A is a linear
fi
t through the points, with speed
v
=
0.92
±
0.05 mas yr
−
1
. This wave is prominent until
2000.99. In 2001.22 the structure has changed. There are
alternate possibilities to explain this new structure, B. It 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
fi
t 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. Alternatively,
structure B may simply be a relic of the trailing side of wave A,
perhaps relativistically boosted by the changing geometry
(
the
bend
)
seen in Figure
2
(
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
r
=
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
(
a
)
as a
gentle bump in 1999.04, so it is
fi
rst apparent in early 1999 at a
distance
r
∼
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
fi
rst become
visible near
r
=
1 mas. Wave C also appears to start near
r
∼
1 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 discussed
in Section
5
.
The different panels in Figure
2
show that the jet can be bent,
and even when relatively straight, can lie at different 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 used 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
Figure 9.
Ridge lines for 1999.37
–
2001.97, plotted on axes rotated by 13
°
.
North and east are indicated at the top. The ridge lines are spaced 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
fi
t to the dots, which are the
characteristic points discussed in Section
4.2
. The three lines are parallel and all
have a slope of 0.92 mas yr
−
1
. See the text.
Table 1
Transverse Waves on the Jet of BL Lac
Epoch
Nv
β
app,
T
b
wave
gal
P.A.
Amplitude
(
mas yr
−
1
)(
deg
)(
mas
)
A 1999.37
–
2000.99
14
0.92
±
.05
3.9
0.979
−
167.0
±
1.4
0.5
D 2005.71
–
2006.86
5
1.25
±
.11
5.6
0.987
−
180.2
±
1.1
0.9
E 2008.33
–
2008.88
8
3.01
±
.16
13.5
0.998
−
174.2
±
0.7
0.3
F 2009.33
–
2009.96
6
1.11
±
.19
5.0
0.985
−
167.1
±
2.4
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
c
,
(
7
)
speed in galaxy frame, assuming
θ
=
6
°
,
(
8
)
P.A. of the wave,
(
9
)
error,
(
10
)
estimated amplitude.
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propagation vectors derived in Section
4.2
. Table
1
lists the
details for these waves.
v
is the measured proper motion,
β
app
is
the apparent speed in units of
c
,
b
wave
gal
the wave speed in the
coordinate frame of the galaxy, assuming
θ
=
6
°
, and P.A. is
the projected direction of the propagation vector. The
amplitude is an estimate of the projected distance
(
in mas
)
across the wave, perpendicular to the propagation vector. Wave
D is the largest such feature seen in the data. Unfortunately,
there was an 11 month data gap prior to 2005.71, and the wave
cannot be seen at earlier times.
The amplitudes of the larger waves appear to be comparable
with the wavelength, as suggested for example by the
inclination angle
ψ
shown in Figure
10
:
y
»
36
. But this is
an illusion caused by the foreshortening, which is approxi-
mately a factor of 10
(
Paper
I
)
, so the deprojected value of
ψ
is
about 5
°
. Note that this is a lower limit, since the transverse
motion can have a component in the
(
ξ
,
ζ
)
plane in Figure
6
.
Figure
13
contains one frame of a movie of BL Lac, showing
the jet motions and ridge line
fi
ts at 15 GHz. The full movie is
available in the electronic journal.
4.1. Different Jet Behavior in 2010
–
2013
In Figure
2,
panels
(
g
)
and
(
h
)
we see that by 2010 the
earlier transverse wave activity in the jet has subsided, and
that after 2010.5 the jet is well aligned at
=-
◦
P
.A.
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. Almost all the epochs show a
negative peak in the inner jet, with a minimum near
r
=
−
0.7 mas. This is a quasi-standing feature, of variable
amplitude. At most epochs there is a positive peak near
r
=
−
1.6 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
fl
ection region. A rotating helix would project as a traveling
wave, as on a barber pole, so a simple barber-pole model is
excluded. Possible motions of the core are only about 10
μ
as
(
Paper
I
)
, so any registration errors due to core motion are
much smaller than the observed changes, which are up to
100
μ
as. There is little indication of wave motion in Figure
14
,
at least not at the speeds seen in Figure
2
. It appears then, that
during the period 2010
–
2013, the jet was essentially straight
but with a set of weak quasi-stationary patterns, with variable
amplitude.
4.2. Velocity of the Waves
We estimated the velocity of Wave A in Figure
7
in two
independent ways. In the
fi
rst we assume that there is a constant
propagation vector, and we shift and superpose the ridge lines
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.A.
=
−
180
°
.
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.A.
=
−
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.A.
=
−
166
°
.
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on a grid of
(
v
, P.A.
)
where
v
is the speed of the wave and P.A.
is its propagation direction. If the ridge lines form a simple
wave, then the solution is found when the lines lie on top of
each other. This is shown in Figure
15
, where a reasonable
fi
t
can be selected by eye. The result is
v
=
0.98
±
0.08 mas yr
−
1
at P.A.
=
−
168
°±
4
°
. 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 characteristic
point on the wave, just downstream of the crest, where the
wave amplitude has begun to decrease. De
fi
ne the slope of the
ridge line as
Δ
x
/
Δ
y
in pixels, where in Figure
9
,
x
and
y
are
rotated R.A. and decl., and take the
fi
rst downstream location
where the slope exceeds
±
0.05. This point is marked with the
dot
b
on the ridge line for 2000.57 in Figure
7
. The
x
and
y
positions versus time for these locations are then
fi
t
independently using the same methods as described in Lister
et al.
(
2009
)
to extract a vector proper motion for this
characteristic point on the wave.
The two methods agree well and the analytic solution is
v
=
0.92
±
0.05 mas yr
−
1
at
=-
◦◦
P
.A.
167.0 0.5
, and the
apparent speed is
β
app
=
3.9
±
0.2. 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
b
wave
gal
the speed of the
wave in the galaxy frame, assuming
θ
=
6
°
. This calculation
assumes that the ridge lines lie in a plane; i.e., are not twisted.
This is not neccessarily the case. Rather, since the inner jet,
near the accretion disk, may wobble in three
dimensions,
(
McKinney et al.,
2013
)
it seems likely that the RCS may
execute three-dimensional motion and that the downstream jet
will also. See Section
5
.
Note that the P.A. of the
fi
rst and last propagation vectors
in Table
1
(
−
167
◦
.0,
−
167
◦
.1
)
is the same
(
to within the
uncertainties
)
as the P.A. of the axis
(
−
166
◦
.6
)
de
fi
ned in
Paper
I
as the line connecting t
he core with the mean
position of the RCS. In the context to be developed later, the
jet acts as a whip being shaken rapidly at the RCS,
and tension in the whip continually pulls it toward the mean
PA.
In Table
1
the speeds for the
fi
rst, second, and fourth waves
are all similar at
β
app
∼
5, but Wave E
(
2008
)
is three times
faster. Wave E has
b
»
13.5
app,E
, which is comparable to the
speed for the fastest component in BL Lac,
b
»
10
app
, although
the components speeds vary widely, from
b
»
2
app
to
b
»
10
app
(
Lister et al.
2013
)
. Wave E is also distinguished
by its polarization; the EVPA is transverse not longitudinal like
the others. We defer further discussion of Wave E to another
paper.
4.3. Transverse Velocity
The ridge waves are relativistic transverse waves with
apparent speeds
β
app
from 3.9 to 13.5 times the speed of light,
and we assume that they have a small amplitude. From the
usual formula for apparent speed,
b
bq
bq
=
-
sin
1cos
(1)
app,wave
wave
gal
wave
gal
and taking values of
β
app,
T
from Table
1
and using
θ
=
6
°
,we
fi
nd
b
=-
0.979 0.998
wave
gal
for the speed of the waves in the
frame of the host galaxy. We now discuss the jet motion in
terms of the coordinate system
(
ξ
,
η
,
ζ
)
shown in Figure
6
.
Consider a transverse motion that is in the
(
η
,
ζ
)
plane. Let
the beam contain a co-moving beacon that is at the origin and
emits a pulse at time
t
′
=
0, where
t
′
is in the coordinate frame
of the galaxy. When
t
′
=
1 yr the signal from the origin will
have traveled one light-year
down the
z
axis, toward the
observer. Also at
t
′
=
1 the beacon has moved from the origin
to the point
hz b b
=
(
,) ( ,
)
tr beam
where
β
tr
is the transverse
speed, and
β
beam
is the longitudinal speed of the beam, both in
the frame of the galaxy. At this point the beacon emits a second
signal that also travels at the speed of light. In the
z
-direction,
this signal trails the
fi
rst one by
(
bq
-
1cos
beam
gal
)
yr. The
apparent transverse speed of the beacon in the direction
perpendicular to the jet, in the galaxy frame, is then
b
b
bq
=
-
()
1cos
(2)
app,tr
tr
beam
gal
and is to be differentiated from the apparent speed
β
app
commonly used in studies of superluminal motion, which is the
apparent speed
along
the jet. Note the close relation between
Equations
(
1
)
and
(
2
)
. Equation
(
2
)
can be inverted to
fi
nd
β
tr
,
a lower limit to the transverse speed.
For Wave A in Figure
7
we obtain an estimate for the
transverse speed at
r
∼
2 mas by taking the transverse motion
as 0.5 mas and the time interval as
(
2000.57
−
1999.41
)
yr,
Figure 13.
Movie of the BL Lac jet at 15 GHz. The total intensity image is on
the right, with a color bar indicating
fl
ux density. The contour levels begin at
7 mJy/beam, and increase by logarithmic factors of two. The false color scheme
uses a square root transfer function, and is saturated at the core position in order
to highlight changes in the much fainter jet. 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 on the left. The movie frames are linearly interpolated
between the individual VLBA epoch images, which have been registered to the
fi
tted position of the core feature, and restored with a median beam with
FWHM dimensions of 0.89 × 0.57 mas, with a major axis position angle at
−
8
◦
.6, as indicated in the lower left corner of the frame. The
fi
tted 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 been linearly interpolated between the individual
VLBA epochs. The points of changing slope
(
see Section
4.2
)
at individual
VLBA epochs are shown as the small symbols. At left the ridge lines are shown
with different colors for the various waves. The yellow
v
=
c
line on the right
is advancing at the speed of light
(
β
app
=
1
)
and is included for reference. The
entire movie is available in the online journal.
(
An animation of this
fi
gure is available.
)
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)
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giving
»
v
0.43
tr
mas yr
−
1
and
β
app,tr
=
1.9 and, from
Equation
(
2
)
with
θ
=
6
°
and
Γ
beam
=
3.5
(
Paper
I
)
,
b
~
0.09
tr
gal
. This is a model-dependent rough value, but it
shows that the transverse speed is non-relativistic. This is
necessary for consistency, since the derivation of the relativistic
form of the MHD wave speeds shown in Paper
I
assumes that
the velocity perturbation is small.
5. EXCITATION OF THE WAVES
We suggested in Paper
I
that Component 7 is an RCS, and
that the fast components emanate from it. If this is correct, then
the RCS should be a nozzle and its orientation should dictate
the direction of the jet. In this Section, we investigate this
possibility. We
fi
rst note that it is not possible to make a
detailed mapping between the P.A. of the RCS and the later
wave shape, for two reasons. First, the algorithm for the ridge
line smooths over 3 pixels
(
0.3 mas
)
, and thus smooths over
any sharp features in the advected pattern. The second reason is
more speculative. Our conjecture is that the wave is launched
by plasma
fl
owing through the nozzle and moving close to
ballistically until its direction is changed by a swing in the P.A.
of the nozzle. But magnetic tension in the jet continually pulls
it toward the axis, and this means that it will bend, and that
small-scale features will be stretched out and made smooth.
We start by comparing the P.A. of the RCS with the P.A. of
the downstream ridge line at
r
=
1 mas. Figure
16
shows the P.
A. of the RCS measured at 15 GHz and at 43 GHz. The latter is
calculated from data kindly provided by the Boston University
VLBI group. We used the result found in Paper
I
, that the
15 GHz core is a blend of the
fi
rst two 43 GHz components and
that the 15 GHz component 7 is the RCS, as is the third 43 GHz
component. We calculated the centroid of the
fi
rst two 43 GHz
components, to
fi
nd an approximate position for the 15 GHz
core, and then calculated the P.A. of the 43 GHz RCS from that
centroid. The result is shown in Figure
16
. We eliminated one
discrepant point at 43 GHz, which was separated by about 20
°
from nearby 43 GHz points, and one discrepant point at
15 GHz. The correspondence between the two frequencies is
generally good, especially after 2005.0 where the agreement is
typically within 3
°
. This further justi
fi
es our claim
(
Paper
I
)
that the the location of this component is independent of
frequency, and that it is an RCS.
Figure
16
also contains the P.A. of the 15 GHz ridge line,
close to
r
=
1.0 mas. Between 2005.0 and 2010.0 the ridge line
P.A. lags the RCS PA, by roughly 0.6
–
1.5 yr. After 2010, the
PA of both the RCS and the ridge line stabilizes, and the
subsequent variations, with rms amplitude about 3
°
, may
mainly be noise. Prior to 2005.0 the variations are faster and
more frequent and the lag is erratic. In places there appears to
be no lag, but around 2000.0 and again around 2004.0 it is
about 0.5 yr. Thus it appears that the swinging in P.A. of the
RCS is coupled to the transverse motions of the ridge line.
When the RCS is swinging rapidly and strongly, as before
2005, then so also is the ridge at 1 mas, with an irregular lag in
P.A. that sometimes is about a half a year, and at other times is
negligible. But when the RCS is swinging more slowly, as after
2005, then the ridge at 1 mas is also swinging slowly, with a lag
of about a year, and after 2010.0 they both are stable, with only
small motions that may be dominated by measurement errors.
We suggest that the large transverse waves on the ridge are
excited by the swinging in P.A. of the RCS. Consider Wave A,
seen in Figure
7
. Its crest lies near line B and moves
downstream at 0.92 mas yr
−
1
. In 1999.37 the crest is at about
r
=
1.2 mas and at 0.92 mas yr
−
1
would have been at the RCS
(
r
=
0.25 mas
)
around 1998.3. This is in a data gap at 15 GHz,
but at 43 GHz there was a peak in P.A. in middle
or late 1998.
Given that in 1999 the time lag between the RCS and the ridge
at 1 mas apparently was much less than 1 yr, the association
between the peak in the RCS P.A. in 1998 and the crest of
Wave A is plausible. The fall in P.A. in 1999 and 2000 is seen
as the short arrow C in Figure
7
, and it corresponds to the
upstream side of Wave A. The downstream side is the advected
rise in P.A. of the RCS from mid-1997 to the peak in middle or
late 1998. The P.A. of the RCS fell from mid-1996 to mid-
1997, and we might expect to
fi
nd a corresponding crest to the
east on Wave A, about 1 mas downstream of the main crest to
the west. In fact, several of the earliest ridge lines in Figure
7
do show a minor crest to the east at about
r
=
3.2 mas, which is
2 mas or 2 yr at 0.92 mas yr
−
1
, downstream of the main crest to
the west. A substantial acceleration in the wave speed would be
needed for this to match. In any event, we cannot speculate
usefully on this because it takes place beyond 3 mas, where
there is a general bend to the east at all epochs. We conclude
that a plausible association can be made between the large
swing west then east of the RCS between 1998.0 and 2000.1,
and Wave A, which
is later seen on the ridge line.
A similar connection can be made for Wave D, seen in
Figure
10
in 2005
–
2006. It can plausibly be attributed to the
large swing of the RCS to the east that began in 2004 and
continued into 2005. This wave does not have a crest as Wave
A does, but a crude analysis can be made as follows. Assume
that point
a
on the 2005.71 ridge line is the advected beginning
of the wave. With a speed of 1.25 mas yr
−
1
(
Table
1
)
, this
means that the swing to the east began around 2003.5. This
date is indicated on the abscissa in Figure
16
. Apart from one
high point at 2004.1 the P.A. of the RCS falls gradually from
2003.1 until late 2004, when it must fall abruptly to meet the
fi
rst point after the data gap in 2005. This also is seen in
Figure
10
; the
fi
rst four epochs have ridge lines that lie together
and are straight at
»-
P
.A.
180
out to
>
1
mas. This is
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 marks on right-
hand side are 0.1 mas apart.
10
The Astrophysical Journal,
803:3
(
16pp
)
, 2015 April 10
Cohen et al.