The Millennium Arecibo 21-cm Absorption Line Survey. IV. Statistics of Magnetic Field, Column Density, and Turbulence

0.82 MB·English

by  Carl Heiles

THE MILLENNIUM ARECIBO 21-CM ABSORPTION LINE SURVEY. IV.
STATISTICS OF MAGNETIC FIELD, COLUMN DENSITY, AND
TURBULENCE
February 2, 2008
Carl Heiles
5 Astronomy Department, University of California, Berkeley, CA 94720-3411;
0
[email protected]
0
2
T.H. Troland
n
a
J Department of Physics and Astronomy, University of Kentucky, Lexington, KY;

1 [email protected]
2

1
ABSTRACT
v
2
8
4 We discuss observations of the magnetic field, column density, and turbulence in
1
the Cold Neutral Medium (CNM). The observed quantities are only indirectly related
0
5 to the intrinsic astronomical ones. We relate the observed and intrinsic quantities by
0
relating their univariateand bivariate probability distribution functions(pdfs). We find
/
h
that observations of the line-of-sight component of magnetic field do not constrain the
p
- pdf of the total field Btot very well, but do constrain the median value of Btot.
o
In the CNM, we find a well-defined median magnetic field 6.0±1.8 µG. The CNM
r
t
s magnetic field dominates thermal motions. Turbulence and magnetism are in approxi-
a
: mate equipartition. We find the probability distribution of column density N (HI)
v ⊥
in the sheets closely follows N (HI) 1 over a range of two orders of magnitude,
i −
X ⊥
0.026 . N (HI) . 2.6×1020 cm 2. The bivariate distributions are not well enough
−
r ⊥
a determined to constrain structural models of CNM sheets.
Contents
1 INTRODUCTION 3
2 SOME IMPORTANT WORDS ON NOTATION 4
3 THEORETICAL CONVERSIONS OF INTRINSIC ASTRONOMICAL PROB-
ABILITY DENSITY FUNCTIONS TO OBSERVED ONES 5
3.1 Conversion of the univariate distributions . . . . . . . . . . . . . . . . . . . . . . . . 6
– 2 –
3.1.1 Conversion of the intrinsic φ(B ) to the observed ψ(B ) . . . . . . . . . 6
tot los
3.1.2 Conversion of the intrinsic φ(N⊥) to the observed ψ(Nlos) for sheets . . . . 8
3.1.3 Conversion of the intrinsic φ(V ) to the observed ψ(V ) for turbu-
turb turb,los
lence perpendicular to B . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
tot
3.2 Conversionoftheintrinsicbivariatedistributionφ(Btot,N⊥)totheobservedψ(Blos,Nlos) 11
3.2.1 Case of B perpendicular to the sheet . . . . . . . . . . . . . . . . . . . . . 11
tot
3.2.2 Case of B parallel to the sheet . . . . . . . . . . . . . . . . . . . . . . . . . 13
tot
3.2.3 Discussion of Figure 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Conversion of the intrinsic to observed bivariate distributions involving turbulent
velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3.1 Conversion of the intrinsic φ(B ,V ) to the observed ψ(B ,V ) 14
tot turb los turb,los
3.3.2 Conversion of the intrinsic φ(N⊥,Vturb) to the observed ψ(Nlos,Vturb,los)
for the perpendicular model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3.3 Conversion of the intrinsic φ(N⊥,Vturb) to the observed φ(Nlos,Vturb,los)
for the parallel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 DERIVATIONOFINTRINSICASTRONOMICALUNIVARIATEPDFsFROM
OBSERVED HISTOGRAMS 15
4.1 Derivation of the intrinsic φ(B ) from the histogram of observed B . . . . . . . 15
tot obs
4.1.1 Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.1.2 A Single Uncertainty δB for Purposes of Convolution . . . . . . . . . . 19
noise
4.2 Derivation of the intrinsic φ(N⊥) from the histogram of observed Nlos . . . . . . . 20
4.2.1 A Preliminary: latitude dependence of N⊥ . . . . . . . . . . . . . . . . . . . 20
4.2.2 Derivation of φ(N⊥) assuming the sheets are randomly oriented . . . . . . . 22
4.3 Derivation of the intrinsic φ(V ) from the histogram of observed V . . . . 25
turb turb,los
5 DERIVATION OF PREDICTED BIVARIATE OBSERVED DISTRIBUTIONS
AND COMPARISON WITH DATA 25
5.1 φ(Btot,N⊥) and ψ(Bobs,Nlos) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
– 3 –
5.1.1 Case of B Perpendicular to the Sheet . . . . . . . . . . . . . . . . . . . . . 27
tot
5.1.2 Case of B Parallel to the Sheet . . . . . . . . . . . . . . . . . . . . . . . . 27
tot
5.2 φ(V ,B ) and ψ(V ,B ) . . . . . . . . . . . . . . . . . . . . . . . . . 30
turb tot turb,los los
5.3 φ(Vturb,N⊥) and ψ(Vturb,los,Nlos) . . . . . . . . . . . . . . . . . . . . . . . . . 30
6 WHICH SHEET MODEL FITS BETTER? 30
7 ASTROPHYSICAL DISCUSSION: MAGNETIC FIELDS 33
7.1 Observational issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7.2 Astrophysical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
8 ASTROPHYSICAL DISCUSSION: COLUMN DENSITIES 35
9 ASTROPHYSICAL DISCUSSION: TURBULENT VELOCITIES 36
10 FINALCOMMENTS:TURBULENTANDMAGNETICENERGYEQUIPAR-
TITION 39
1. INTRODUCTION
Beginning February 1999 we used the Arecibo telescope 1 to begin a series of Zeeman-splitting
measurements of the 21-cm line in absorption against continuum radio sources. Heiles & Troland
(2004; Paper III and references therein) discussed technical details of the observing technique
and data reduction and presented observational results, including magnetic field measurements.
As described in Paper III, our analysis of the data identified Gaussian components in the HI
absorption spectra. We assume that each component samples a single, isothermal, sheet-like region
of CNM. For 69 such components, we are able to estimate (1) the line-of-sight magnetic field
strength B , often subject to significant error, (2) the line-of-sight column density N(HI) ,
los los
and (3) the contribution of turbulence V to the line-of-sight velocity dispersion. Systematic
turb,los
instrumental errors are small and the uncertainties are Gaussian distributed. Therefore, our survey
has yielded a statistically well-defined ensemble of observed values for each of these three CNM
quantities.
From these data, we seek to answer several astrophysically significant questions. For example,
1TheAreciboObservatoryispartoftheNationalAstronomyandIonosphereCenter,whichisoperatedbyCornell
University undera cooperative agreement with the National Science Foundation.
– 4 –
what are the probability distribution functions (pdfs) of magnetic field strengths, column densities
and turbulent energies in the CNM? Are any of these quantities statistically related to each other?
If so, can we determine if the magnetic field lies preferentially in the planes of CNM sheets or
perpendicular to them? What is the ratio of magnetic to turbulent energy in the CNM? As we will
see below, the data provide answers to some, but not all, of these questions.
We observe quantities that are only indirectly related to the intrinsic astronomical ones. For
the magnetic field, we observe the line of sight component B , not the total field B . For the
los tot
column density N(HI), we must account for the fact that the Cold Neutral Medium is in sheets,
not spheres; our observed column densities N are always larger than the intrinsic column density
los
normaltothesheetN . Toproperlyinterpretourresultsandtoaddressthequestions listedabove,
⊥
wemustconsiderhowtheintrinsicandobservedquantities arerelatedinastatistical sample. These
statistical transformations,andtheresultsofapplyingthemtoourdata, arethefocusofthecurrent
paper.
Wetreatbothunivariatedistributionsandbivariatedistributions. Theunivariatedistributions
are most interesting because we obtain definitive results; in contrast, the possible correlations that
could be revealed by bivariate distributions are obscured by noise and inadequate numbers of data.
Accordingly, the reader who is interested in astrophysical results can concentrate on the sections
dealing with univariate distributions of magnetic field, column density, and turbulent velocity. We
recommendbeginningwith§2, whichintroducesthenotationandconceptsofintrinsicandobserved
quantities. We develop the theoretical relationships between observed and intrinsic univariate
distributions in §3, and use these to obtain the actual intrinsic univariate distributions in §3.1.
Finally, we discuss astrophysical implications in §7, §8, §9, and §10.
2. SOME IMPORTANT WORDS ON NOTATION
We needtointroduceimportantdistinguishingnomenclaturebecauseof theanisotropic nature
of magnetic-field-related quantities. First, we make the crucial distinction between line-of-sight
(los) quantities, such as B , and the intrinsic astronomical ones such as B . We will often refer
los tot
to the latter with one the shorter terms “intrinsic” or “astronomical”. The important point is to
distinguish between observed and intrinsic quantities.
The line-of-sight component is the quantity to which the telescope responds, and we designate
these with the subscript los. There is an additional complication for the magnetic field, which is
the presence of significant instrumental noise. Thus, we must distinguish between the line-of-sight
component of the field, designated by B , and the actual measured value, designated by B . We
los obs
define
B ≡ the line−of−sight component of B ; (1a)
los tot
– 5 –
B ≡ the observed field strength . (1b)
obs
The essential difference between B and B is
los obs
B = B +δB (1c)
obs los noise
where δB is the uncertainty contributed by random measurement noise.
noise
Noise is small enough to neglect for column density and velocity, so theobserved quantities are
essentially identical to the los ones; we will use the subscript los for the observed quantities, with
the implicit assumption that their uncertainties from noise are negligible. For column density, we
always refer to HI so we often write N instead of N(HI). We usually consider sheets, for which the
column density perpendicular to the face is N(HI) and for which the apparent column density
⊥
along the line of sight is
N(HI)
Nlos ≡ N(HI)los = N(HI)obs = ⊥ , (2)
cosθ
where θ is the angle between the sheet’s normal vector and the line of sight. Similarly, for velocity
linewidths, we use the symbol V to indicate the line-of-sight component of the turbulent
turb,los
velocity.
Finally, the probability density functions (pdf) of intrinsic quantities, such as B , differ from
tot
those of the line-of-sight quantities. We will always use the symbol φ for the pdfs of intrinsic
quantities, and thesymbolψ for line-of-sight (los) or the observed ones. Unless otherwisespecified,
units are always as follows: magnetic field B in µGauss; column density N in HI atoms 1020 cm 2;
−
velocity V in km s 1.
−
3. THEORETICAL CONVERSIONS OF INTRINSIC ASTRONOMICAL
PROBABILITY DENSITY FUNCTIONS TO OBSERVED ONES
We begin with some light theory by considering elementary transformations of magnetic field
and column density in statistical distributions. First, we consider how the intrinsic probability
density function (pdf) of B , defined as φ(B ), converts to the observed histogram or pdf of
tot tot
observed B , defined as ψ(B ), under the assumption that fields are randomly oriented with
los los
respect to the observer. Note the important distinction: B is the total field strength; B is the
tot los
observed line-of-sight component, which is always smaller.
Next we incorporate one of the fundamental results of Heiles & Troland (2003; Paper II),
namely that the CNM components are thin sheets. We define N = N(HI) as the HI column
⊥ ⊥
densityperpendiculartothesheet;theobservedquantityisN = N(HI) ,whichisalwayslarger.
los los
As with the magnetic field, we consider how the pdf φ(N ) converts to the observed histogram
⊥
or pdf ψ(N ), again under the assumption that the sheets are randomly oriented with respect
los
– 6 –
to the observer. We also consider the statistical transformation of the distribution of the actual
nonthermal velocity dispersionV to that of the observed one V under the assumption that
turb turb,los
turbulence is perpendicular to the magnetic field.
We then tackle the bivariate distributions. First we consider how the bivariate distribution
φ(B ,N ) converts to the observed one ψ(B ,N ). We then assume two extreme models, one
tot los los
⊥
with the fields always perpendicular to the sheets and one with fields parallel to the sheets, and
assuming random orientations derive the observed bivariate distributions ψ(B ,N ) for the two
los los
cases. We illustrate and discuss the result by considering delta-function distributions of B and
tot
N , and we also apply the transformation of the observed ψ(B ,N ) to its intrinsic counterpart
los los
⊥
φ(B ,N ). Finally, we examine the bivariate distributions involving the pairs (V ,N ) and
tot turb,los los
⊥
(V ,B ), which produces little in the way of useful results.
turb,los los
3.1. Conversion of the univariate distributions
3.1.1. Conversion of the intrinsic φ(B ) to the observed ψ(B )
tot los
We first consider the simple case in which all clouds have the same B , which is randomly
tot
oriented with respect to the observer. The line of sight component B is
los
B = B cosθ , (3)
los tot
where θ is the angle between the field direction and the line of sight. θ can run from 0 to π, but
the intervals from 0 to π and π to π are identical except for a change of sign in B . It’s simpler
2 2 los
and no less general to consider the smaller interval θ from 0 to π so that we can ignore the slight
2
complications of the sign change. In this case, the pdf of θ is the familiar
φ (θ)= sinθ (4)
θ
and we wish to know the pdf of B , which is given by (see Trumpler and Weaver 1953 for a
los
discussion of these conversions)
d[θ(B )]
los
ψ(B ) =φ [θ(B )] , (5)
los θ los
dB
(cid:12) los (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
which gives (cid:12) (cid:12)
1 if 0 ≤ B ≤ B
ψ(Blos) = Btot los tot (6)
0 otherwise
(
– 7 –
In other words, B is uniformly distributed between the maximum possible extremes 0 and B
los tot
(actually ±B ). This leads to the well-known results that in a large statistical sample for which a
tot
constant B is viewed at random angles, both the median and the mean observed field strengths
tot
are half the total field strength and that B2 = Bt2ot. More generally, observed fields are always
los 3
smaller than the actual total fields, and with significant probability they range all the way down
to zero.
Now suppose B has an arbitrary pdf φ(B ). The univariate pdfφ(θ) becomes the bivariate
tot tot
pdf φ(B ,θ), and we assume B is independent of the observer’s location so that φ(B ,θ) =
tot tot tot
φ (B )φ (θ). Notethatweintroducesubscriptsonthedifferentφ’stodistinguishthem, instead
Btot tot θ
of designating them with different Greek letters. To obtain ψ(B ) we again follow the standard
los
techniques; it’s easy to integrate over θ and obtain
φ(B )
∞ tot
ψ(B )= dB , (7)
los tot
B
Z[Blos>Btotmin] tot
where the symbol [B > B ] means the larger of the two quantities. The presence of B in
los totmin tot
the denominator means that smaller ranges of B are emphasized. This is commensurate with the
los
equation 6’s uniform pdf for a single field value. We note that this can be regarded as an integral
equation for φ(B ) and it is straightforward to invert.
tot
Figure1illustratesthesolutionofequation7forfourfunctionalformsofφ(B )plottedagainst
tot
B
| | , where the subscript 1/2 denotes the median value. These forms include the following:
|B1/2|
1. φ a Kronecker delta function (DELTA FCN), φ (B ) = δ(B −B ), yielding ψ a flat
Btot tot tot tot,0
function (as discussed immediately above, equation 6);
2. φ a flat distribution (FLAT FCN) between 0 ≤ |B | ≤ B , yielding ψ ∝ ln B0 ;
tot 0 Blos
(cid:16) (cid:17)
3. φ a weighted Gaussian (EXP FCN),
φ(Btot)= πB22 B2Bt2o2t e−(Bt2ot/2B02) , (8)
s 0 0
yielding ψ a Gaussian with dispersion B .
0
B2
4. φ a Gaussian (GAUSS FCN) with dispersion B , yielding ψ ∝ E los , where E is the
0 1 2B02 1
exponential integral of order 1. (cid:16) (cid:17)
All four φ(B ) are plotted with respect to Btot , so the medians of all lie at unity on the
tot Btot,1/2
x-axis. However, the means differ. Similarly, the medians and means of the associated ψ(B )
los
differ from each other. These relationships between median and mean are summarized in Table 1.
– 8 –
Fig. 1.— Top panel: The intrinsic φ(B ) for four representative functional forms. Bottom panel:
tot
their line-of-sight counterparts ψ(B ). The vertical scales are arbitrary.
los
The medians and means for ψ(B ) are all about half those for hB i, which is a direct result of
los tot
the weighting by B 1 in equation 7.
t−ot
Figure 1 is disappointing from the observer’s standpoint, because the observed distributions
ψ(B ) donot differ very much. These differences become smaller—inconsequential, in fact—when
los
one includes measurement noise, as we shall discuss in detail for our data in §4.1. Unfortunately,
giventheinevitableerrorsinanyobservationthatissensitivetoB ,itseemspracticallyimpossible
los
to distinguish among different functional forms for φ(B ). Nevertheless, the average value of B
tot los
is close to half the average value of B for a wide range of intrinsic pdfs of the latter; this also
tot
applies to the medians, but less accurately. Therefore, this rule of thumb may be used to estimate
the median or average B from an ensemble of measurements of B .
tot los
3.1.2. Conversion of the intrinsic φ(N⊥) to the observed ψ(Nlos) for sheets
HereweassumetheCNMisdistributedinsheetshavingHIcolumndensityN inthedirection
⊥
perpendicular to the sheet. If the normal vector to the sheet is oriented at angle θ with respect to
– 9 –
the line of sight, we have
N
Nlos = ⊥ . (9)
cosθ
To find ψ(N ) we follow the same procedures as above in §3.1.1. For a single value of N we
los
⊥
obtain
N⊥ if N ≥ N
ψ(Nlos) = Nl2os los ⊥ (10)
( 0 otherwise
For a single N , N has a long tail extending to infinity. The median value of N is N =
los los los,1/2
⊥
2N , reflecting the increased observed column for tilted sheets. The mean value of N (hN i) is
los los
⊥
not defined because the integral diverges logarithmically; of course, this doesn’t occur in the real
world, where sheets don’t extend to infinity. For example, if all sheets have aspect ratio 5 :1, then
hN i = 1.6N and N =1.7N .
los los,1/2
⊥ ⊥
For an arbitrary pdf φ(N ) and infinite slabs, we obtain
⊥
1 [Nlos<N⊥max]
ψ(N ) = N φ(N ) dN . (11)
los Nl2os Z0 ⊥ ⊥ ⊥
Asabove, this can beregarded asan integral equation forφ(N )anditis almostas straightforward
⊥
to invert.
3.1.3. Conversion of the intrinsic φ(V ) to the observed ψ(V ) for turbulence
turb turb,los
perpendicular to B
tot
Theobservedvelocitywidthcomesfromtwosources,thermalandnonthermal. Wecanseparate
these because we have independent measurements of the kinetic temperature. Thus, we can derive
Table 1. Medians and Means of B and B for Representative pdfs
tot los
φ(B B hB i B hB i
tot tot,1/2 tot los,1/2 los
DELTA FCN 1.00 1.00 0.50 0.50
FLAT FCN 1.00 1.00 0.40 0.52
GAUSS FCN 1.00 1.18 0.38 0.59
EXP FCN 1.00 1.04 0.44 0.51
– 10 –
the line-of-sight (los) nonthermal (“turbulent”) line width and its associated energy density. How
this relates to the total turbulent energy density depends on whether the turbulence is 1d, 2d, or
3d.
Here we assume that this turbulence is 2d, i.e. we assume that it is restricted to motions per-
pendicular to the mean magnetic field. We make this assumption because we show below that the
typical turbulentMach numberis equal to 3.7. For velocities that areparallel to themagnetic field,
such turbulence would produce strong shocks that would damp very rapidly. However, velocities
that are perpendicular to the magnetic field can be as high as the Alfv´en velocity without pro-
ducing shocks; this is the basis for considering turbulent motions perpendicular to the mean field.
We caution, however, that numerical simulations of magnetohydrodynamical turbulence find that
magnetic fields do not ameliorate turbulent dissipation (MacLow et al. 1998), so our assumption
might not have any basis in physical reality; in this case the turbulence is isotropic (3d) and the
observed distribution ψ(V ) is equal to the intrinsic one φ(V ).
turb,los turb
Suppose that the magnetic field is oriented at angle θ with respect to the line of sight, as
in §3.1.1. Suppose that turbulent motions are perpendicular to the fieldlines and isotropic in the
azimuthal directions around the fieldlines, and along one direction perpendicular to B have width
V ; this makes the full turbulent width 21/2V . Then line-of-sight width is
turb turb
V = V sinθ . (12)
turb,los turb
To find ψ(V ) we again follow the same procedures as in §3.1.1 and obtain
turb,los
1/2
Vturb,los V2 −V2 − if V < V
ψ(Vturb,los)= Vturb turb turb,los turb,los turb (13)

0 h i otherwise


Given a value for V , ψ(V ) → ∞ as V → V , but the cumulative distribution is
turb turb,los turb turb,los
well-defined with
1/2
V 2
turb,los
cum(V )= 1− 1− (14)
turb,los
V
" (cid:18) turb (cid:19) #
which gives the median V = 0.87V median and mean hV i = 0.79V . These
turb,los,1/2 turb turb,los turb
high values reflect the large fraction of sheets tilted to the line of sight, where V is large.
turb,los
For an arbitrary pdf φ(V ) we obtain
turb
V
ψ(Vturb,los)= ∞ tVurb,los Vt2urb−Vt2urb,los −1/2φVturb(Vturb)dVturb (15)
ZVturb,los turb
(cid:2) (cid:3)