The Astrophysical Journal, 696:567–573, 2009 May 1 doi:10.1088/0004-637X/696/1/567
C
2009. The American Astronomical Society. All rights reserved. Printed in the U.S.A.
DISPERSION OF MAGNETIC FIELDS IN MOLECULAR CLOUDS. I
Roger H. Hildebrand
1,2
, Larry Kirby
1
, Jessie L. Dotson
3
, Martin Houde
4
, and John E. Vaillancourt
5
1
Department of Astronomy and Astrophysics and Enrico Fermi Institute, The University of Chicago, Chicago, IL 60637, USA
2
Department of Physics, The University of Chicago, Chicago, IL 60637, USA
3
NASA Ames Research Center, Moffett Field, CA 94035, USA
4
Department of Physics and Astronomy, The University of Western Ontario, London, ON N6A 3K7, Canada
5
Division of Physics, Mathematics, & Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Received 2008 November 5; accepted 2009 February 5; published 2009 April 16
ABSTRACT
We describe a method for determining the dispersion of magnetic field vectors about large-scale fields in turbulent
molecular clouds. The method is designed to avoid inaccurate estimates of magnetohydrodynamic or turbulent
dispersion—and help avoiding inaccurate estimates of field strengths—due to a large-scale, nonturbulent field
structure when using the well known method of Chandrasekhar and Fermi. Our method also provides accurate,
independent estimates of the turbulent to large-scale magnetic field strength ratio. We discuss applications to the
molecular clouds OMC-1, M17, and DR21(Main).
Key words: ISM: clouds – ISM: magnetic fields – polarization – turbulence
1. INTRODUCTION
Chandrasekhar & Fermi (1953) used the dispersion of
starlight polarization vectors about contours of Galactic lat-
itude (Hiltner 1949)—together with estimates of gas density
and line-of-sight velocity dispersion—to determine the strength
of the magnetic field in the arms of the Galaxy. The same
technique, “the Chandrasekhar–Fermi (CF) method,” has been
applied, with modifications, to estimate field strengths in the
relatively dense medium of molecular clouds at varying temper-
ature, wavelengths, sensitivities, and resolutions (e.g., Lai et al.
2001, 2002, 2003; Crutcher et al. 2004; Houde 2004;Girart
et al. 2006; Curran & Chrysostomou 2007).
The basis for deriving field strengths from dispersion mea-
surements is the same for observations of Galactic arms or
molecular clouds: in either case dispersion decreases as the
field strengthens. But in the case of the Galactic arms, the dis-
persion is due to magnetohydrodynamic (MHD) waves; the dis-
placements are perpendicular to the direction of propagation.
In the case of turbulent dispersion in molecular clouds, there is
no preferred direction. The turbulent component can be in any
orientation.
Moreover, in dense clouds, the field may have structure due
to effects such as differential rotation, gravitational collapse,
or expanding H ii regions, i.e., structure not accounted for by
the basic CF analysis. Consequently, dispersion measured about
mean fields, assumed straight, may be much larger than should
be attributed to MHD waves or turbulence. Dispersion measured
about model large-scale fields (Schleuning 1998; Lai et al. 2002;
Girart et al. 2006) that give approximate fits to a polarization
map will result in better estimates but still give inaccurate values
of the turbulent component, since they are unlikely to perfectly
match the true morphology of the large-scale magnetic field.
In this paper, we describe a method for determining magnetic
field dispersion about local structured fields, without assuming
any model for the large-scale field. This method also provides
accurate, independent estimates of the turbulent to large-scale
magnetic field strength ratio.
We begin (Section 2) by discussing difficulties one must
overcome in order to infer turbulent structure from polarization
maps, regardless of large-scale effects. In Section 3 we present
the method, and in Section 4 we give applications to the
molecular clouds OMC-1, M17, and DR21(Main). Detailed
derivations resulting in the relations and functions used in the
aforementioned sections, as well as the data analysis, will be
found in the appendices at the end of the paper.
2. DIFFICULTIES IN DERIVING THE TURBULENT
STRUCTURE FROM POLARIZED EMISSION
Turbulent velocities of gas motion within and between clumps
of material along the line of sight can often be inferred from the
widths and centers of molecular lines (e.g., Kleiner & Dickman
1984, 1985, 1987). But dust polarization measurements of dis-
persion in magnetic field direction do not separate contributions
from either volume elements located along the line of sight or
across the area subtended by the telescope beam. Hence, the
measured angular dispersion tends to be a smoothed version of
the true dispersion (Myers & Goodman 1991; Wiebe & Watson
2004). Nonetheless, a corresponding average of the dispersion
remains and is measurable; for a given object observations will
thus reveal a higher degree of dispersions when they are realized
at an accordingly higher spatial resolution.
A potentially fruitful line of attack for estimating magnetic
field strengths relies on comparisons of observed and simulated
maps of the net polarization (e.g., Ostriker et al. 2001
; Heitsch
et al. 2001; Falceta-Gon¸calves et al. 2008). If the simulations
are computed for the resolution, column density, and other
characteristics of the cloud under study, and if they are computed
for several models of the key variables (e.g., field strength and
turbulent fraction), then one can find the model giving the best
fit to the observations. A valid simulation must also take into
account temperatures (Vaillancourt 2002) and grain alignment
efficiencies in different environments (Hoang & Lazarian 2008).
The comparisons are facilitated if both the observations and the
simulations are presented in tables of Stokes parameters, so that
each can be analyzed in the same way. The various modifications
of the CF method that have been used to relate net dispersion
to field strength (e.g., Ostriker et al. 2001; Padoan et al. 2001;
Heitsch et al. 2001; Kudoh & Basu 2003; Houde 2004)are,in
effect, first-order substitutes for simulations.
But a meaningful comparison between simulations and ob-
servations can only be achieved if a reliable estimate of the
567
568 HILDEBRAND ET AL. Vol. 696
spatially averaged angular dispersion can be secured experi-
mentally. It would therefore be advantageous if a more general
method, which does not depend on any assumption concerning
the morphology of the large-scale magnetic field, was devised.
The method we describe in the following section allows for
the evaluation of the plane-of-the-sky turbulent angular disper-
sion in molecular clouds while avoiding inaccurate estimates of
the turbulence and corresponding inaccurate estimates of field
strengths due to distortions in polarization position angles by
large-scale nonturbulent effects. This method can lead to valid
estimates of magnetic field strengths only under conditions such
that the CF method can properly be applied: a smooth, low noise,
polarization map, precise measured densities and gas velocities
that are moderately uniform, and an adequate accounting of the
integration process implicit to polarization measurements. This
latter aspect will be addressed in a subsequent paper.
3. A FUNCTION TO DESCRIBE DISPERSION ABOUT
LARGE-SCALE FIELDS
Consider a map precisely showing the angle Φ(x)ofthe(two-
dimensional) plane-of-the-sky projected magnetic field vector
B(x) at many points in a molecular cloud. We obtain a measure
of the difference in angle, ΔΦ() ≡ Φ(x) − Φ(x + ), between
the N() pairs of vectors separated by displacements ,also
restricted to the plane of the sky, through the following function:
ΔΦ
2
(
)
1/2
≡
1
N
(
)
N
(
)
i=1
[
Φ
(
x
)
− Φ
(
x +
)
]
2
1/2
, (1)
where ··· denotes an average and =||. The square of
Equation (1) is also often referred to as a “structure function”
(of the second order in this case; see Falceta-Gon¸calves et al.
2008;Frisch1995), but for our applications we shall refer to it
as the “dispersion function” and assume that it is isotropic (i.e.,
it only depends on the magnitude of the displacement, , and not
its orientation). We seek to determine how this quantity varies
as a function of .
To do so, we will assume that the magnetic field B(x)
is composed of a large-scale structured field, B
0
(x), and a
turbulent (or random) component, B
t
(x), which are statistically
independent. We also limit ourselves to cases where δ< d,
where δ is the correlation length characterizing B
t
(x) and d is
the typical length scale for variations in B
0
(x).
Focusing on B
0
(x) we would expect its contribution to the
dispersion function to increase (since ΔΦ
2
() is positive
definite) almost linearly starting at = 0 and for small
displacements d, as would be expected from the Taylor
expansion of any smoothly varying quantity. We denote by m
the slope characterizing this linear behavior. We also expect a
contribution from the turbulent component of the magnetic field
B
t
(x). This contribution will vary from zero as → 0 (when
the two magnetic field vectors are co-aligned) to a maximum
average value when the displacement exceeds the correlation
length δ characterizing B
t
(x). More precisely, we expect that
the turbulent contribution to the angular dispersion will be a
constant, which we denote by b, as long as >δ. These two
contributions must be combined quadratically, since the large-
scale and turbulent fields are statistically independent, to yield
ΔΦ
2
()b
2
+ m
2
2
, (2)
when δ< d.
Figure 1. Dispersion: idealized plots of the angular dispersion function,
ΔΦ
2
()
1/2
, between pairs of magnetic field vectors separated by displacements
, for values of d, with d the typical length scale for variations in the large-
scale magnetic field (see Section 3). Curve A: no measurement uncertainty;
no turbulence. Curve B: with measurement uncertainty, σ
M
. Curve C: with
turbulence. Curves D and E: accounting for correlation in polarization angles at
displacements smaller than the larger of the telescope beam (1.22λ/D;curve
D) or the turbulent correlation length δ (curve E).
A more formal and rigorous derivation of Equation (2)is
established in Appendix A under the further assumptions of
homogeneity and isotropy in the magnetic field strength over
space. Although these assumptions are unlikely to be realized
across molecular clouds, this level of idealization is necessary
to allow us to gain insights into, and some quantitative measure
of, the importance of the turbulent component of the magnetic
field in molecular clouds.
In reality, the measured dispersion function from a polariza-
tion map will also include a contribution, σ
M
(), due to mea-
surement uncertainties on the polarization angles Φ(x) that must
be added (quadratically) to Equation (2). The square of the total
measured dispersion function then becomes
ΔΦ
2
()
tot
b
2
+ m
2
2
+ σ
2
M
(
)
, (3)
when δ< d. The function ΔΦ
2
()
tot
, not ΔΦ
2
(),is
the one calculated from a polarization map (from an averaging
process similar to Equation (1), and will thus contain separate
components due to the large-scale structure (i.e., m), the
turbulent dispersion about the large-scale field (i.e., b,the
quantity we wish to measure), and measurement uncertainties
(i.e., σ
M
()).
If there were no turbulence and no measurement uncertainties,
then, for d, the measured dispersion function would
be a straight line with zero intercept, ΔΦ()
2
1/2
tot
= m (see
Figure 1, curve A). Taking the measurement uncertainty, σ
M
(),
into account, the line would be displaced upward as specified by
Equation (3) (curve B, where σ
M
was assumed to be independent
of ). Likewise when we next consider turbulence, the curve
will again be displaced upward in the same manner (curve C)
except at values of below the angular resolution scale at which
the observations were made (curve D), or below the turbulent
correlation scale δ (curve E). Theoretical and observational
estimates of δ for molecular clouds are on the order of 1 mpc
(Lazarian et al. 2004; Li & Houde 2008, respectively), well
below the size of the telescope beam with which the observations
presented in this paper were obtained. Although it has not yet
been feasible to resolve δ,itis now feasible to determine the
turbulent dispersion at scales comparable to the approximately
linear portion of ΔΦ()
2
1/2
tot
.
Note that σ
M
() can be accurately determined through the
uncertainties on the measured polarization angles of each pair
of points used in the calculation of ΔΦ()
2
tot
, and by then
No. 1, 2009 DISPERSION OF MAGNETIC FIELDS IN MOLECULAR CLOUDS. I 569
subtracting its square to obtain ΔΦ()
2
. As the number and
precision of the vectors improve, Equation (2) can be fitted to
the data for δ< d, and the intercept at = 0 provides us
with the turbulent contribution, b
2
, to the square of the angular
dispersion.
The CF method for evaluating strength of the plane-of-the-
sky component of the large-scale magnetic field (Chandrasekhar
& Fermi 1953) implies that
δB
B
0
σ
(
v
)
V
A
, (4)
where δB stands for the variation in the magnetic field about
the large-scale field B
0
, σ (v) is the one-dimensional velocity
dispersion of the gas (of mass density ρ) coupled to the magnetic
field, and
V
A
=
B
0
√
4πρ
(5)
is the Alfv
´
en speed. It is further assumed that the dispersion,
σ
Φ
, in the polarization angles Φ(x) across a map is given by
σ
Φ
δB
B
0
. (6)
The combination of Equations (4)–(6) allows for the aforemen-
tioned determination of the plane-of-the-sky component of the
large-scale magnetic field strength as a function of ρ, σ (v) (de-
termined from the width of appropriate spectral line profiles),
and σ
Φ
(determined from polarization measurements).
It is shown with Equation (A24) in Appendix A that the ratio
of the turbulent to large-scale magnetic field strength is given
by
B
2
t
1/2
B
0
=
b
√
2 − b
2
. (7)
It is therefore apparent that we should make the correspondence
B
2
t
1/2
→ δB and that
B
0
(2 − b
2
)4πρ
σ
(
v
)
b
8πρ
σ
(
v
)
b
, (8)
where the last equation applies when B
t
B
0
. The fact that the
turbulent dispersion, b, is to be divided by approximately
√
2
before being inserted the CF equation is readily understood by
the fact that (neglecting the contribution of the large-scale field)
ΔΦ
2
()=[Φ(x) − Φ(x + )]
2
= 2(Φ
2
−Φ
2
)
= 2σ
2
Φ
,
when >δ. Since we also know that ΔΦ
2
()=b
2
at these
scales, we then find that b
2
= 2σ
2
Φ
, which is consistent with
Equations (6) and (7).
It should be noted that the combination of Equations (7) and
(8) allows, in principle, for the determination of both the large-
scale and turbulent magnetic fields’ strength from polarization
and spectroscopy data.
Figure 2. Angular dispersion function, ΔΦ
2
()
1/2
, for M17, DR21(Main), and
OMC-1. The turbulent contribution to the total angular dispersion is determined
by the zero intercept of the fit to the data at = 0. The measurement uncertainties
were removed prior to operating the fits to the corresponding data sets. The
results are given in Table 1.
4. APPLICATIONS TO THE MOLECULAR CLOUDS
OMC-1, M17, AND DR21(MAIN)
Using data from the Hertz polarimeter (Dowell et al. 1998)
at the Caltech Submillimeter Observatory at 350 μm, we have
measured dispersion functions for the molecular clouds OMC-
1, M17, and DR21(Main). These data are discussed in detail in
Houde et al. (2004) for OMC-1, Houde et al. (2002) for M17,
and Kirby (2009) for DR21(Main). Figure 2 shows the results
for all sources. More details on the data analysis will be found
in Appendix B.
For each object, we show ΔΦ
2
()
1/2
over the cloud along
with the best fit from Equation (2) using the first three data points
to ensure that d as much as possible. The measurement
uncertainties were removed prior to operating the fits to the
corresponding data sets. The turbulent contribution to the total
angular dispersion is determined by the zero intercept of the
fit to the data at = 0. The net turbulent component, b,is
0.18 ± 0.01 rad (10
◦
.4 ± 0
◦
.6), 0.12 ± 0.02 rad (6
◦
.8 ± 1
◦
.3), and
0.15±0.01 rad (8
◦
.3±0
◦
.3) for M17, DR21(Main), and OMC-1,
respectively.
Although large variations in density within the observed
regions prevent a reliable estimate in the field strength at
precise locations, it is still possible to give some average
570 HILDEBRAND ET AL. Vol. 696
Tab le 1
Results for the Dispersion, the Turbulent-to-Mean Magnetic Field Strength
Ratio, the Line Widths, and the Mean Field Strength
Object b
a
B
2
t
1/2
/B
0
b
σ (v) B
0
c
(deg) (km s
−1
) (mG)
OMC-1 8.3 ±0.30.10 ±0.01 1.85 3.8
M17 10.4 ±0.60.13 ±0.01 1.66 2.9
DR21(Main) 6.8 ±1.30.08 ±0.02 4.09 10.6
Notes.
a
Turbulent dispersion (i.e., the dispersion limit as → 0).
b
Calculated with Equation (7).
c
Calculated with Equation (8), assumes a density of 10
5
cm
−3
and a mean
molecular weight of 2.3. These estimates are not precise to better than a factor
of a few. The process of signal integration through the thickness of the cloud
and across the telescope beam inherent to the polarization measurements has
also not been taken into account.
value for the large-scale and turbulent field strengths. To do so
we use representative line width measurements from H
13
CO
+
J = 3 → 2 detections within the three clouds. For OMC-
1 and M17 we have used the corresponding measurements
published in Houde et al. (2000; more precisely, an average of
the variances obtained at the two positions listed for M17), while
for DR21(Main) we haveused previously unpublished data. This
molecular species is well suited for this as the effective density
needed for line detection with the aforementioned transition
(n
eff
∼ 10
5
cm
−3
; see Evans 1999) is close to the densities
at which dust continuum emission is detected at the measured
wavelength. Also, the corresponding spectral lines are likely to
be optically thin (like the dust continuum) and an ion molecule
such as this one is better coupled to the magnetic field (and
the dust) than corresponding neutral species (e.g., H
13
CN for
the same rotational transition) over the whole turbulent energy
density spectrum (Li & Houde 2008). Therefore, using a density
of 10
5
cm
−3
and a mean molecular weight of 2.3 we obtain
the results shown in Table 1. As a simple comparison, the
values of dispersion shown in the table are approximately three
times lower than would be obtained if one naively calculated
the dispersions about the global mean field (i.e., the field
direction defined by the mean of all polarization vectors in the
corresponding map). More precisely, we get dispersions of 27
◦
.2,
21
◦
.0, and 26
◦
.8 about the global mean field orientation for M17,
DR21(Main), and OMC-1, respectively.
We wish to emphasize the fact that the quoted values for B
0
could not be precise to better than a factor of a few due to a lack of
precise gas density numbers. Moreover, the values for the large-
scale magnetic field strength we derived are up to an order of
magnitude higher than those obtained with other observational
means (e.g., the results of Crutcher et al. 1999 for OMC-1
and M17 using CN Zeeman measurements). These high values
are in part the result of the smaller angular dispersions obtained
using our technique as compared with more common methods
used when applying the CF equation (e.g., model fits to large-
scale fields). One must keep in mind, however, that the process
of signal integration through the thickness of the cloud and
across the telescope beam that is inherent to polarization
measurements has not been taken into account. We will show
in a subsequent publication how this situation is rectified when
these considerations (and others) are carefully taken into account
(Myers & Goodman 1991; Ostriker et al. 2001; Wiebe & Watson
2004). Nevertheless, the turbulent to large-scale magnetic field
strength ratio is precisely evaluated through our Equation (7).
5. SUMMARY
We have described a method to estimate plane-of-the-sky
turbulent dispersion in molecular clouds while avoiding inac-
curate estimates of the turbulence and corresponding inaccurate
estimates of field strengths due to distortions in polarization
position angles by large-scale nonturbulent effects. The method
does not depend on any model of the large-scale field. We plot
a “dispersion function,” the mean absolute difference in angle
between pairs of vectors as a function of their displacement
, and show that this function increases approximately linearly
for displacements greater than the instrument resolution, greater
than the correlation length, δ, and less than the typical length
scale, d, for variations in the large-scale magnetic field (Section
4). We emphasize that this method can lead to valid estimates
of magnetic field strengths only under conditions such that the
CF method can be properly applied: a smooth, low-noise, po-
larization map, precise measured densities and gas velocities
that are moderately uniform, and an adequate accounting of the
integration process implicit to polarization measurements. This
method, however, provides accurate estimates of the turbulent
to large-scale magnetic field strength ratio.
Although the resolution of the instruments now available is
not adequate to directly determine the correlation length, δ,
one can still determine the dispersion in the fields at scales
where δ< d for the angular dispersion function. We have
successfully done this for the OMC-1, M17, and DR21(Main)
molecular clouds.
We thank Shantanu Basu for helpful discussions. This work
has been supported in part by NSF grants AST 05-05230, AST
02-41356, and AST 05-05124. L.K. acknowledges support from
the Department of Astronomy and Astrophysics of the Univer-
sity of Chicago. M.H.’s research is funded through the NSERC
Discovery Grant, Canada Research Chair, Canada Foundation
for Innovation, Ontario Innovation Trust, and Western’s Aca-
demic Development Fund programs. J.E.V. acknowledges sup-
port from the CSO, which is funded through NSF AST 05-
40882.
APPENDIX A
DISPERSION RELATION DERIVATION
A.1. Analysis in Three Dimensions
Let us define the total magnetic field B(x) as being composed
of a deterministic, B
0
(x), and a turbulent (or random), B
t
(x),
component such that
B(x) =B
0
(
x
)
+ B
t
(
x
)
. (A1)
These quantities have the following averages at points x and
y:
B
0
(
x
)
= B
0
(
x
)
B
0
(
x
)
· B
0
(
y
)
= B
0
(
x
)
· B
0
(
y
)
B
t
(
x
)
= 0
B
0
(
x
)
· B
t
(
y
)
=
B
0
(
x
)
·
B
t
(
y
)
= 0. (A2)
We will further assume homogeneity in the field strength over
space. That is,
B
2
0
(
x
)
=
B
2
0
(
y
)
= B
2
0
B
2
t
(
x
)
=
B
2
t
(
y
)
=
B
2
t
. (A3)
No. 1, 2009 DISPERSION OF MAGNETIC FIELDS IN MOLECULAR CLOUDS. I 571
Let us now consider the quantity
cos
[
ΔΦ
3D
(
)
]
≡
B
(
x
)
· B
(
x +
)
[B
2
(
x
)
B
2
(
x +
)
]
1/2
. (A4)
The quantity ΔΦ
3D
(
)
is the angle difference between two
magnetic field (or polarization) vectors separated by a distance
, the averageof its square is the function that we wish to evaluate
through polarization measurements (albeit in two dimensions,
see Section A.2). Using Equations (A1) and (A2), we find that
the numerator of Equation (A4) (i.e., the autocorrelation of the
total magnetic field; see Frisch 1995) becomes
B
(
x
)
· B
(
x +
)
= B
2
0
+
B
0
(
x
)
·
∞
n=1
n
n!
(
e
·∇
)
n
B
0
(
x
)
+
B
t
(
x
)
· B
t
(
x +
)
, (A5)
where we used the Taylor expansion
B
0
(
x +
)
= B
0
(
x
)
+
∞
n=1
n
n!
(
e
·∇
)
n
B
0
(
x
)
, (A6)
with e
being the unit vector in the direction of .
If we introduce d the scale length characterizing (large-scale)
variations in B
0
and we consider situations where =||d,
then we would expect that only the first term in the summation on
the right-hand side of Equation (A6) would need to be retained.
If we define ϕ
i
as the angle between the gradient of the i-
component (i.e., i = x, y, z)ofB
0
and e
, then when averaging
over a large polarization map we have
B
0,i
(x)[(e
·∇)B
0,i
(x)]=B
0,i
(x)|∇B
0,i
|cos(ϕ
i
). (A7)
But since e
is equally likely to be oriented in any direction
over the whole map we have cos(ϕ
i
)=0 and the first-
order term of the Taylor expansion (i.e., Equation (A7)) cancels
out. It therefore follows that the first nonvanishing term in the
summation on the right-hand side of Equation (A6) is of second
order with
B
0
(
x
)
·
∞
n=1
1
n!
(
·∇
)
n
B
0
(
x
)
1
2
B
0
(
x
)
·
(
e
·∇
)
2
B
0
(
x
)
2
, (A8)
when d. If we also assume stationarity for the turbulent
magnetic field, then we define the autocorrelation of the turbu-
lent field as
B
t
· B
t
()
≡
B
t
(
x
)
· B
t
(
x +
)
, (A9)
which, if we now define δ as the correlation length for B
t
(
x
)
,
has the following limits:
B
t
· B
t
(
)
=
B
2
t
, when → 0
0, when >δ
, (A10)
since the turbulent field is assumed uncorrelated over separa-
tions exceeding δ and
B
t
= 0 from the third of Equation (A2).
Inserting Equations (A8) and (A9) into Equation (A5)wehave
B
(
x
)
· B
(
x +
)
B
2
0
(
x
)
+
1
2
B
0
(
x
)
·
(
e
·∇
)
2
B
0
(
x
)
2
+
B
t
· B
t
(
)
, (A11)
when d.
Using the assumed homogeneity in the fields’ strength (i.e.,
Equation (A3)) the denominator of Equation (A4) can readily
be simplified to
[B
2
(
x
)
B
2
(
x +
)
]
1/2
=B
2
=
B
2
0
+ B
2
t
+2
(
B
0
· B
t
)
,
which, with the fourth of Equation (A2), becomes
[B
2
(x)B
2
(x + )]
1/2
= B
2
0
+
B
2
t
. (A12)
If we further assume isotropy over space (i.e., ΔΦ
3D
() =
ΔΦ
3D
()) and insert Equations (A11) and (A12) into Equation
(A4), we have
cos
[
ΔΦ
3D
(
)
]
1
−
B
2
t
−
B
t
· B
t
(
)
−
1
2
B
0
(
x
)
·
(
e
·∇
)
2
B
0
(
x
)
2
B
2
0
+
B
2
t
,
(A13)
when d. For cases where ΔΦ
3D
(
)
is small Equation (A13)
simplifies to
ΔΦ
2
3D
(
)
2
B
2
t
−
B
t
· B
t
(
)
B
2
0
+
B
2
t
−
B
0
(
x
)
·
(
e
·∇
)
2
B
0
(
x
)
B
2
0
+
B
2
t
2
, (A14)
still when d.
Examining Equation (A10) we recover the behavior of the
turbulent contribution to
ΔΦ
2
3D
(
)
(i.e., the first term on the
right-hand side of Equation (A14)) described in Section 3 that
goes from 0 when → 0 to a constant, which we now define as
b
2
3D
, when >δ. The data sets analyzed in this paper are such
that >δin all cases. We therefore find that the dispersion
function is of the form
ΔΦ
2
3D
(
)
b
2
3D
+ m
2
3D
2
, (A15)
with
b
2
3D
=
2
B
2
t
B
2
0
+
B
2
t
,
when δ< d. Once again, we identify b
3D
with the
constant contribution stemming from the turbulent field to
the total angular dispersion, while the larger scale contribution
due to variations in the large-scale field B
0
is accounted for by
the presence of a term proportional to
2
in Equation (A15).
A.2. Analysis in Two Dimensions
The analysis presented above can still be used when we
limit ourselves to two dimensions. This is needed in order
to enable comparisons with polarization measurements, which
only probe the plane-of-the-sky component, B
, of the magnetic
field. Defining e
⊥
as the unit vector directed along the line of
sight, we have for the total magnetic field
B
= B −
(
B · e
⊥
)
e
⊥
, (A16)
and similar relations for B
0
and B
t
.
We need to evaluate, among others, the following autocorre-
lation:
B
(
x
)
· B
(
x +
)
=
B
(
x
)
· B
(
x +
)
−
[
B
(
x
)
· e
⊥
][
B
(
x +
)
· e
⊥
]
, (A17)
