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The Astrophysical Journal, 764:117 (16pp), 2013 February 20

1021
1020

Blanc et al.

aCO (Mó pc-2 (K km s-1)-1)• ßNÞďKĂøàĊĐ:ŁķīĹľêetc.ł• £àĊ!Ü÷ĜŁňł
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REPORTS
colder dust in the cloud halos. It contributes
only 0.6 and 0.9% of the CMB and warmer
dust emissions Efor a local electron spectrum
Ie 0 708 (E/1 GeV)j3.3 mj2 sj1 srj1 GeVj1(21) and two dust components with n1.67 Bn(9.4 K) and n2.70 Bn (16.2 K) emissivities
scaled to match typical fluxes of 1.5 MJy/sr at
500 GHz (FIRAS) and 15 MJy/sr at 3000 GHz(IRAS)^. The lack of correlation with the
408 MHz synchrotron map shows that GeV
cosmic rays are not efficiently trapped in thesecloud halos. Their distribution couples to gas
density (19) and spiral arms (17), but their diffusion properties imply coupling lengths muchlarger than the cloud sizes: 1.76 T 0.20 kpc for
nuclei (19) and several hundred parsecs for

CO dark gas

the highest energy electrons with large radiative losses (22). Measurements of qHI indeed yield equivalent cosmic-ray densities
within 1 kpc from the Sun (17). Because thelocal dust mass is only 1% of the gas one,hadronic interactions with grains are negligible. The g rays therefore trace the dark gas
forming these cloud halos, a gas invisible in
HI, CO, and free-free emission.
Assuming the same cosmic-ray flux in the
diffuse dark and HI gas, the g-ray intensities
give dark-gas column densities, NHdark (Fig. 4and fig. S2). The E(B-V) and I94 tracers, total
and residual maps, give consistent NHdark column densities within 30% over most regions.
Fitting cloud complexes individually may lead• COý|ĂĞčøH2Ĥĩý?0• H2 : dark : HI = 12 – 51% : 6 – 59% : 29 – 67%(Grenier+ 2005)• [CII]Ï¹ÃP¾čĤĩď77%ń
íFTĤĩď20%(Langer+ 2014)(Grenier+ 2005)• z ~ 0.05ďàĊė©Å (Wong+ 2017)

Fig. 1. Map, in Galactic coordinates centered on l 0 70-, of the excess dust reddening found above thatlinearly correlated with the integrated HI and CO line intensities. The 94-GHz emission map shows thesame excesses. CO intensities above 4 K km sj1 are overlaid in cyan. The dust excesses form extended
halos around all CO clouds, the bright ones as well as the fainter CO cloudlets that are not overlaid.
This dust spreads in a dark gas, not seen in HI and CO but detected in g rays.

Table 1. Best fits to the g-ray data at 5- e kbk e 80- and 10- e kbk e 80-, forthe complete and residual (in bold and italics) dust maps. The log-likelihood
ratios 2.ln(l), measuring the fit improvement, are given with respect to the
N(HI)þW(CO) gas model and are distributed as a c2 with one degree of

2.ln(l)qHI(10j26 sj1
srj1)

qCO
(10j6 cmj2 srj1
Kj1 kmj1)

0
241.8
1298.01556.7241.81298.01556.71.62 T 0.01
1.63 T 0.01
1.64 T 0.01
1.65 T 0.01
1.49 T 0.02
0.77 T 0.02
0.71 T 0.04

5.65 T 0.10
5.54 T 0.09
5.82 T 0.09
5.91 T 0.09
4.84 T 0.11
1.63 T 0.15
0.53 T 0.54

0
284.4
957.9
1048.61.74 T 0.02
1.74 T 0.02
1.75 T 0.02
1.76 T 0.02

5.31 T 0.12
5.28 T 0.12
5.70 T 0.12
5.60 T 0.12

qEBV
(10j6 magj1
cmj2 sj1 srj1)

freedom (12). The errors are statistical, not including systematic uncertainties[10% in the EGRET exposure (10), 20% in the true N(HI) (32) and dust maps(5), and 20% in the ‘‘dust-free’’ qCO because of cloud-to-cloud variations in
the dust residuals toward the CO clouds].

qI94(10j4 mKj1
cmj2 sj1 srj1)

qI3000(10j7 MJyj1
cmj2 sj1)

qIC (10j1)

qSOU

IE
(10j6 cmj2
sj1 srj1)

7.46 T 0.06
6.54 T 0.09
6.39 T 0.07
6.43 T 0.07
6.54 T 0.09
6.39 T 0.07
6.43 T 0.07

1.01 T 0.01
1.00 T 0.01
0.96 T 0.01
0.96 T 0.01
1.00 T 0.01
0.96 T 0.01
0.96 T 0.01

13.21 T 0.09
13.93 T 0.10
13.96 T 0.09
13.91 T 0.09
13.74 T 0.10
14.30 T 0.09
14.45 T 0.09

7.90 T 0.09
7.07 T 0.10
7.25 T 0.09
7.17 T 0.09

0.98 T 0.01
0.97 T 0.01
0.91 T 0.01
0.91 T 0.01

12.40 T 0.10
13.03 T 0.10
12.84 T 0.10
12.83 T 0.10

5- e kbk e 802.83 T 0.20
4.27 T 0.12
57.5 T 1.5
2.83 T 0.21

Ëk«ďõ§ö
4.27 T 0.10

57.5 T 3.7

10- e kbk e 80-

4.04 T 0.25
4.32 T 0.14
54.4 T 1.7• 0.6  z  1.5 Đ 12CO J=1–0ďÃ(Ň p_ ý'ĔĜ=Ėĝqv
www.sciencemag.org

SCIENCE16

A. Katsianis et al.

10-1
log10 SFRD [MO• yr-1 Mpc-3]

SFE density(Mó yr-1 Mpc-3)

1293

VOL 307 25 FEBRUARY 2005

10-2
10-3

-1

-2

Reddy et al. (2008, UV + IR)
Ly et al. (2011, Ha)
Bouwens et al. (2012, UV)-3

Cucciati et al. (2012, UV)

Ch24_eA_EDW

Sobral et al. (2013, Ha)

Ch24_NF

Magnelli et al. (2013, IR)

Kr24_eA_sW

Gruppioni et al. (2013, IR)

Ch24_eA_nW

Madau & Dickinson (2014, Compilation)

Ch24_eA_MDW

Bouwens et al. (2015, UV)

10-4
0 1 2 3 4 5 6 7
redshift(Katsianis+ 2017)
Parsa et al. (2015, UV)

-4
0

1

2

3

4

5

6

7

z

Figure 7. The evolution of the cosmic star formation rate density from z ∼ 7 to z ∼ 0 in cosmological hydrodynamic simulations and observations.
The orange filled circles represent the results of Reddy et al. (2008) . The red filled reversed triangles, circles and diamonds represent the CSFRD from theintegration of the SFRFs implied by the IR luminosity functions of Magnelli et al. (2013) and Gruppioni et al. (2013), respectively. The dark green filled
diamonds and green filled circles use the Hα data of Ly et al. (2011) and Sobral et al. (2013), respectively. The blue filled triangles, open diamonds and filledsquares represent the CSFRD obtained from UV luminosity functions of Cucciati et al. (2012), Bouwens et al. (2012), Bouwens et al. (2015) and Parsa et al.
(2015), respectively. The black dotted line represents the compilation study of Madau & Dickinson (2014). The luminosity limit is set to Lmin = 0.03 L⋆ as
in Madau & Dickinson (2014).

simulated SFRFs are in good agreement with the UV and Hα studies for objects with −1.0 ! log(SF R/( M⊙ yr −1 )) ! 1.0, andwith IR data for log(SF R/( M⊙ yr −1 )) " 1.0. The variable energy driven winds that efficiently decrease the SFR of low mass objects are quite successful at reproducing the shallow SFRFs implied
by UV data, and also have good agreement with the constraints
from IR studies for the high star forming and dusty objects.

6 THE SIMULATED AND OBSERVED COSMIC STARFORMATION RATE DENSITY

The evolution of the cosmic star formation rate density of the Universe is commonly used to test theoretical models, since it represents a fundamental constraint on the growth of stellar mass ingalaxies over time. In the above Sections we saw that the SFRFs
can commonly be described by the Schechter (1976) functional
form. The integration of the Schechter (1976) fit gives the total
Cosmic Star Formation Rate Density (CSFRD) of the Universe at
MNRAS 000, 000–000 (0000)

Downloaded from http://science.sciencemag.org/ on February 5, 2017

to the N(HI)þW(CO)þI3000 model mostly
comes from the Cygnus, Ophiuchus, and Orion
A and B clouds. Little g-ray emission is associated with the I3000 map on larger scales.
The g-ray fit largely improves (Fig. 3 andfig. S1) when using the colder dust tracer I94
and the dust column densities as traced by theE(B-V) reddening Esee the 2ln(l) values inTable 1^.
The qHI emissivity found at 5- e kbk e 80is consistent with the solar circle average(TSD) of (1.60 T 0.02) " 10j26 sj1 srj1 (16)and 15% lower than those obtained near
major local clouds when neglecting the IC
emission (17). The extragalactic intensity agrees
with that measured with the EGRET interstellar model: (14.5 T 0.5) " 10j6 cmj2 sj1
srj1 (18, 19). It is larger than the (11.10 T
0.12 T 2.2) " 10j6 cmj2 sj1 srj1 GALPROP
estimate (20) because of the 30% lower IC
intensity than that in GALPROP (9). Provided
cosmic rays uniformly pervade the HI and H2phases, the N(HI)þW(CO) model yields anaverage Xg 0 qCO/(2qHI) 0 (1.74 T 0.03) " 1020
molecules (mol.) cmj2 Kj1 kmj1 s in thesolar neighborhood, fully consistent with thefactor X 0 (1.8 T 0.3) " 1020 mol. cmj2 Kj1
kmj1 s obtained with the temperature-correctedI¶3000 map and a smoothed I¶3000/N(HI) ratio
to measure the total gas (3). It is close to the
Galactic averages of (1.9 T 0.2) " 1020 mol.
cmj2 Kj1 kmj1 s (16) and (1.56 T 0.05) "1020 mol. cmj2 Kj1 kmj1 s (19) derived
from g rays and different gas models.
The considerable improvement in the gray fits when adding the I94 or E(B-V) dust
tracers demonstrates that g rays are producedin the cold dust halos, in excess of those bornin the HI and CO gas or in the warmer dustin these directions. Splitting the sky in threelongitude intervals did not change the result,
so one particular region does not drive the fit.
This bright emission, comparable to the entire
IIC map, is not due to IC emission from the$~uÚáĠĈćÄ¯• CO J=1–0 à pä5ďÕ·CIÏ¹ďQ3eŁHerschelĊĐ36”@500µmý~4”Ďł
• SÂÝĠüăćeà àď8ºÛėģĲŀà, ĠģĲŀ• ÃĊª4ĠÆĒĝď>ĤĩĠbĝďa•ďpä5ďhJ à ĨĴĸĺĎĳŀĦ
T = 20 K
z=0
100 K
10 K
1K

1.0

No. 6]

Molecular Gas in M 33

1173

Table 1. Wide-area CO observatons of M 33.

Transition

Telescope

Area

Resolution
spatial velocity(0 )(km s!1 )Noise level

J = 1–0BIMA"
FCRAO 14 m!
BIMIA + FCRAO 14 m + NRO 45 m"
NRO 45 m#

1100
960
172
900

13
50
20
19

2
1
2.6
2.5

J = 2–1

IRAM 30 mk

650

12

• CO J=1–0 @NRO 45m• 1.1 mm @ASTE

OàM 33ďÕ·Ã(arcmin2 )• Ìéņ840 kpc• W^ņSA(s)cd

240
53
60
130

2.6

" Engargiola et al. (2003).
! Heyer et al. (2004).

" Rosolowsky et al. (2007).
# This work.

k Gratier et al. (2010).

1174

(mK)

20–50

T. Tosaki et al.

[Vol. 63,

The observations were performed using the OTF mapping
technique (Sawada et al. 2008). The “scanning noise” wasremoved by combining the horizontal and perpendicular scans
using basket weaving. Data reductions were made with the
OTF reduction software package NOSTAR, which was implemented by the NRO. Although the beam size of the telescopewas 1500, the spatial resolution of the final map was 19:003 due
to a grid spacing of 7:005.
The rms-noise levels in the velocity channel map witha velocity range of 2.6 km s!1 and in the total integrated intensity map were 130 mK and 1.6 K km s!1 , respectively, on themain beam temperature scale. The noise level in the integrated intensity map corresponds to 9.6 # 104 Mˇ in beam
size (80 pc) by applying the Galactic CO-to-H2 conversion
factor, 3 # 1020 cm!2 (K km s!1 )!1 (Wilson & Scoville 1990).
This value corresponds to a limiting mass surface density of
15 Mˇ pc!2 , including a correction of He, and is much lower
than the surface density of GMCs in the Milky Way, 40–
100 Mˇ pc!2 (Heyer et al. 2009).

Dec. (J2000) +30˚

1144

50’

3. Results[Vol. 63,

31 10

00

Declination (J2000.0)

50’

40’
30’

S. Komugi et al.

30 50

40

30’

30

30”
1 kpc
10’ 1 kpc
34m 33m
34m
32m
hR.A. (J2000) 01
R.A. (J2000) 01h
20

30”

1 kpc

10

01 35 30

00

34 30

00

33 30

00

32 30

00

Right ascension (J2000.0)

3.1. Total Integrated Intensity Map and Velocity Field

Fig. 5. 1.1 mm contours overlayed on H˛ image from Hoopes and Walterbos (2000). Contours are drawn at 9.0, 13, 18, and 25 mJy beam!1 .

Figure 2 shows the total integrated intensity map andFig. 2. Total integrated intensity map (left) and intensity-weighted mean radial velocity map (right) of 12 CO (J = 1–0) emission obtained with the NRO
12
0 ! 300 , andintensity-weighted
velocitymap
ofresolution
CO is(J1900=
the effective
spatial, corresponding to 7.3 kpc ! 7.3 kpc and 81 pc at the distance of M 33
45 m telescope. The map mean
size is 30radial(D
840 kpc), respectively.
The contour
interval
is 0.5clumps
K km s"1 withon the Taa#typicalscale, corresponding to 1 !.
1–0)=emission.
This figure
revealsmany
size of $ 100 pc, similar to that of GMCs in the Milky
Way (Scoville & Sanders 1987), rather than to that of Giant
Molecular Associations (GMAs: Rand & Kulkarni 1990).
There are chains of such “GMCs” in the northern and southernspiral arms. Note that most of the emission consists of such
GMCs, and no diffuse emission is seen in the map above thelimiting mass surface density of 15 Mˇ pc!2 . We could see
no concentration of CO at the galactic center in this map,
unlike most molecule-rich galaxies (Sakamoto et al. 1999).
These properties are consistent with maps obtained by previous
studies (Rosolowsky et al. 2007).
The velocity field of the 12 CO (J = 1–0) emission in figure 2
is similar to that of the H I emission (Deul & van der Hulst
1987), except for a few GMCs (e.g., GMCs located in thesouthern arm). The velocity field is globally dominated by(Tosaki+ 2011; Komugi+ 2011)

Fig. 1. Observed area (dashed box, 300 # 300 ) superposed on a three-color composite optical image of M 33 obtained with the Subaru telescope: courtesy of V. Vansevicius, S. Okamoto, and N. Arimoto. The
optical image consists of three colors: blue (B), green (V ), and red(H˛). The H˛ image is not continuum-subtracted.

width of 1332 km s!1 with a velocity resolution of 2.6 km s!1
at 115.271204 GHz. We monitored an SiO maser source,
IRC+30021, with the five-point observation method using
a 43 GHz SIS receiver (S40) every 1 h to check the pointing
accuracy. Note that the final map was drawn by using data withpointing errors within ˙ 7:00 5, obtained under weather conditions of lower wind speed than 10 m s!1 , so as to avoid any
systematic intensity loss due to pointing errors. The total observation time was 136 h.

OàM 33ďÕ·Ã• pä5ďTďà!Ü

Fig. 3. Total integrated intensity map of 12 CO (J = 1–0) emission (contours) superposed on H I emission (left, Rosolowsky et al. 2007) and star-formation rate (right) derived from the H˛ luminosity (Hoopes & Walterbos 2000) with extinction-correction by using MIPS 24 "m data (Rieke et al. 2004).
The contour interval and lowest contour are the same as in figure 2.
No. 5]

15

1147

20

# 160 "m and 1100 "m can still suffer from variations in ˇ(see subsection 4.5), it gives a more reasonable representationof the colder31 10component and its characteristic temperature, Tc ,compared to attempts using shorter wavelengths.
4.3. Color Temperature00
The color temperature was derived using"!
30 50"mF160
1100 !ˇ B160 .T /R%;=
F1100 "m160
B1100 .T /1.4 2.0 2.6 3.2
log l (µm)

Fig. 7. Two-component fit to the observed global SED of M 33. The
green dotted curve is the warm dust component with temperature
Tw = 52 ˙ 7 K, and the blue dash-dotted curve is the cold dust component with temperature Tc = 18 ˙ 3 K, using the observed 1.1 mm fluxof 10 Jy. The red solid curve is the sum of the two components. Errors
in the infrared fluxes are typically 20%, and are taken from Hippelein
et al. (2003) and Hinz et al. (2004). For the 1.1 mm flux, explanation oferrors are given in section 3.

25

50

S. Komugi et al.

24

observed 1.1 mm
20extended flux
corrected
16

Tc (K)

total (b=2.0)[Vol. 63,1148

Cold Dust in M 33

10

Declination (J2000.0)

4
3
2
1

Tc (K)
10 15 20 25

S. Komugi et al.

Dec. (J2000) +30˚

log Flux (Jy)

1146

(3)

which is obtained
from equation (2) at two wavelengths,
40
for a single temperature component. Equation (3) was
solved numerically to obtain the correspondence between theobserved flux30 ratio and the color temperature, and interpolated to the observed ratios. Uncertainties were estimated from the flux uncertainties in the 1.1 mm flux of 45%
20(section 2) and
160 "m of 16% (subsection 4.1), which added Fig. 9. Cold dust temperature gradient. Red circles are the temperain quadrature results in an flux ratio uncertainty estimate of tures derived from observed flux ratios. Green crosses are temperatures
48%. At temperaturesof 10 K, 18 K, and 20 K, the flux after corrections for extended flux loss, explained in subsection 4.4.
10
ratio, F160 "m =F1100 "m , is 20, 120, and 160, corresponding Errorbars were determined from flux uncertainties in the observations
to color temperature
uncertaintiesof
2.8 K, (see subsection 4.3).
01 35 30
00
34 30
00
33 30 0.9 K,
00 2.332 30 K, and00
Right ascension (J2000.0)
respectively.
In the actual derivation, the region corresponding to theobserved AzTEC field was cut out from the 160 "m image, For each of the arcminute-sized clumps in the map, theand further re-gridded to the same pixel size (600 pixel!1 ). Thetemperature seems to be higher at the edges compared to theSpitzer field of view did not include the northeastern and south-inner regions. This may be caused by the data-reduction
western edge regions, and have been excluded from tempera-procedure, since the flux retrieval-fraction of extended features
ture analysis.
decrease with increasing source size. At large scales, theThe two images were then divided pixel-by-pixel, using only
those pixels where 1.1 mm fluxes exceeded 6.0 mJy beam!11.1 mm flux is underestimated, giving a higher temperature.
(the characteristic 1 # noise level of the map, but $ 1.5 # for Figure 9 shows the radial temperature gradient, constructed00
most cases. See figure 2.) and 160 "m flux exceeded the 1 #from figure 8 averaged over radial bins of 90 width, which
noise level of 0.9 MJy str!1 , to ensure that the flux ratios areis chosen to mitigate the effect of temperature variations at
not affected by noise (virtually no cuts were made for 160 "mthe edges of arcminute structures. The declining temperature
data using this criterion; all regions where the map satisfiedgradient is clearly apparent. Only a few cold dust temperathe 1.1 mm criterion had # 10 # in the Spitzer map). The fluxture gradients have previously been found (e.g., Kramer et al.
ratios were then converted into temperature using the numer-2010). It is important to note that this gradient is found even

12

30
10

34m
32m
R.A. (J2000) 01h

0

2
4
6
radius (kpc)

Fig. 8. Cold dust temperature map of M 33. Units are in Tc (K).

Krügel 1993; Krügel & Siebenmorgen 1994; Lis & Menten
1998; Dunne et al. 2000; Dunne & Eales 2001; Hill et al.
2006; Kramer et al. 2010), we keep ˇ = 2.0 constant hereafter.
Our derived global cold dust temperature, Tc , should beconsidered to be a slight overestimate, since extended features
were subtracted in the data reduction (see subsection 2.1). If weused the corrected 1.1 mm flux estimate of 20 Jy, we obtained
a cold dust temperature of 16 ˙ 2 K. This temperature and thecorrected 1.1 mm flux is consistent with extrapolations of the
far-IR SED to 1.1 mm in previous measurements (Hippelein
et al. 2003) and the recent value derived from Herschel(Kramer et al. 2010).
Using the observed 1.1 mm flux of 10 Jy and cold dust

(Komugi+ 2011)

observations (Hemore extended flconsidered to be
ically plausible mby accounting fo
ratio R in equat
shown as green dture decreases inis still apparent.

4.5. Variation i

The propertie
optical properties
grain emissivity
analysis, the gravary within a ga
the case where ais not valid, andis expected theo2002), and a wi
been observed in
Variations inture. In particu
mimic a tempera
figure 9 to be ent
ature at 20 K an
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HerschelÀpďÃ• MÛà»ďOàM 33
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Gratier, P. et al.: The Gratier,
Molecular
P.Gas
et al.:
Mass
TheofMolecular
M33
Gas Mass of M33

A&A proofs: manuscript no. M33_Xfactor

0(Gratier+ 2017)2
Fig. 1. Dust surface density
Fig. 1.[ g/,Dustcmsurface] mapsdensityof M33[ g/,
atcm
25"2 ]resolution:
maps of M33(left)atfor
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constant (left)= 2, for(right)a constantradially variable= 2, (right) radially variable(2-1.3) as derived in Tabatabaei(2-1.3) as etderivedal. (2014).
in Tabatabaei
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tively H2 dominated,strongly H i dominat
inner to outer region
be expected given th
a decline in CO emis
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The lines withou
the H2 mass at mode
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CO has not formed oclouds, or unexpecte

gram of dust (Kruegel
gram& Siebenmorgenof dust (Kruegel1994),& Siebenmorgenor 350 = ture1994),
is computedor 350 =withturea radiallyis computed
varyingwith. Thea radiallyvaluesvaryingof. The values of
4.7 cm2 g 1 at 350 µm.4.7 cm2 g 1 at 350 µm.
are below 2 in M33 (Tabatabaei
are below 2etinal.
M33
2014)
(Tabatabaei
so the temperaet al. 2014) so the temperatures
are traditionhigher. Sincetures
the Planck
are higher.
function
SinceBthefunction B⌫,T ( ) increases
It is now clear that theIt dustis nowemissivity
clear thatindex,
the dust
traditionemissivity
index,⌫,T (Planck) increases
T, the dustwith density
T, the indust
Figuresurface1 (right
densitypanel)
in Figure 1 (right panel)
ally designated , is notallynecessarily
designated2 as, ishasnotgenerally
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2 as withhas generally
beensurface
proofs:manuscript
no. M33_Xfactor
lower,haveparticularly
in lower,
the A&A
outerparticularly
disk wherein theisouterlower.
disk where is lower.
4. Leroy-Sandstrom
assumed. In particularassumed.
Tabatabaei
In particularet al. (2014)
Tabatabaei
have shown
et al. is(2014)
shown isthisfrom
work,thewe onlyIndiscuss
this work,hydrogen
we only
content
discuss
andhydrogendocontent and dothat is variable andthat
lower isinvariable
M33 ( and= 2 lower
1.3 infrom
M33
the( = 2 In 1.3
4.1. Prior discussion o
not include
helium.
notheliumincludeis present
helium. in
Asbothheliumtheisatomicpresent
in both theatomiccenter to the outer disk).
centerHowever,to the outer
withoutdisk).
being
However,able towithout
being able
to As
tivelyH2 dominated,slightly
H i dominated (1 2 kpc), andandmolecular
phasesandin molecularequal proportion,phases in
this
equal
doesproportion,
notH i dominatedthis does
calibrate the value  calibrateat the wavelength
the valueof interest,at the wavelength
it is dif- ofinterest,
it is difstrongly
withnot
weak CO emission.
From theThe GDRis likely
aﬀect forthe thecalculations.
As inthemany
calculations.
other works,
As in
we
many
use totheother
we theuse theficult to be sure of theficultconstant
to be sure(0.4 ofabove
the constantfor the dust
(0.4 above
dust aﬀect
inner
outerworks,
regions,
XCO factor increases,
as forcould
at least
metallic
GDR to =refer
the GDRhydrogento referto dustto themasshydrogenratio.
todust
mass
ratio.
opacity) as extrapolations
opacity)
have
as generally
extrapolations
assumedhave generally= 2. term
assumed
2. toterm
be expected given that there is a metallicity
gradient
and issolar
metallicity
If the intrinsicof Ifthethedust
intrinsic
grains isoflessthethandust2,grainsthen is less than 2, thena decline in CO emission (Gratier et al. 2010b)
plundandet cloudal. 2009,
using = 2 will result
using
in an =
underestimate
2 will result ofin thean underestimate
temperof the tempertemperature (Gratier et al. 2012).
dard hydrogen-to-du
3. Dust-derived
Dust-derived
CO intensity
H2 versus CO
intensity0
ature and thus an overestimate
ature and thusof theandustoverestimate
mass (compareof the dust
mass (compareH23.versus
The lines without a Kdark
systematicallyoverestimate
2007,
Table 3), thethe two panels of Fig.the1).twoIn thispanelscontext,
of Fig.a 1).
more
In accurate
this context,
a moreapproachaccurate is A
Hpre-existingmass
at moderateandbothfits over- meta
+ gas-phase
A simpleto simple
take theapproachpre-existing
is to map
taketheofthethemap of theH2high ICO andHe
2H
2
but more complex means
but moreof deriving
complexthemeans
dust surface
of deriving
den-thecolumn
dust surface
estimate) at fromhigh IBraine
reason tocontrib
be negligible
densitydenbasedcolumn
on Herscheldensityandbased
H i data
on HerschelfromBraineandNH(Hi 2data
CO . There is no physical0
sity has been tested.sity
Tabatabaei
has beenettested.
al. (2014)
Tabatabaei
find a link
et al. et(2014)
find awherelink Net(Hal.
a constant
oﬀset
(Kand
that
plund,
M there(H) = 0.7
al. (2010)
is estimated
where fromN (H2the) isexpect
dust
estimatedandfrom thedust2 ) (2010)
dark ) but it appears
between the galactocentric
betweendistancethe galactocentricandin M33
distance(theirandH i emissionin M33 (theiris (H)
gastheirwhose
is detected butnoting
is not theseengas-phaseinas N (H2H) i=emission(N (H) as NN(H(Hi))/2,asin
N (H dusti))/2,emission
as in their2 ) = (NFig. 3). This (r)is used
Fig. to3).derive
This dust(r)istemperatures
used to deriveover
dust Figure
temperatures
over
CO – this could be optically thick H i, moleculargas wherehydrogengas-to-dus
4.
Figure 4.
the disk of M33.
the disk of M33.
CO
has
not
formedorisphotodissociated,lowdensityH
0.0142Z)/(0.0142(
In this case, the variables
In thisare
case,
XCO
theand,variables
potentially,
are XCOa and, potentially, a
2
gas
0
0
clouds,or2 unexpectedly
large 2quantities of ionizedgas.
In a similar way as inIn
Brainea similar
et al.way(2010),
as in weBrainethenettakeal. (2010),
we thentake CO-darkto this number.
For G
CO-darkgas column
density designatedgas columnKdensity
designatedKdark. Figuredark . Figure
the gas-phase-metalpixels with H i columnpixels
densitywithmeasurements
H i column densityand dust
measurements
tem- showsand
tem-plots
thedustscattershows
for athesamplescatterof plots
threeforradiala samplebins –of three radial bins –metals are in the duperatures but no CO emission
peraturesandbutcomputeno CO emission
the mediananddustcomputethe <median0 kpc
r < 1dust
kpc, 10kpc
kpc << rr << 21kpc,
kpc, and1 kpc4 kpc< r << r2 <kpc, and 4 kpc < r <proofs: manuscript
no. M33_Xfactor
4. Leroy-Sandstrom
methodbust; for a solar com
cross-section ( dust ) cross-sectionper H-atom: ( dust
SA&A
H-atom:S⌫ /(B
kpc.
These
radii
5 kpc.
progressively
These radiithe
showtransition
progressively
from the transition
fromdust )=per⌫ /(B⌫,T NH ),dust 5=⌫,T N
H ), show
50 ± 10% of the meta
where S⌫ is the dustwhereemissionS⌫ isandtheB⌫,T
dusttheemission
Planckandblack
B⌫,Tanthe
blackISM,
H2Planckdominated
anto
H2approximate
dominated ISM,
H i–Htoapproximate
be-discussion
H i–H2 equality
be2 equality
4.1. Prioron the gas-to-dust
ratio (GDR)
tivelyHdominated,slightlyHidominated
(1
2
kpc),
and
What about lowe
body emissivity for abodyfrequency
emissivity⌫ andforaatemperature
frequency ⌫ Tand
. tweena temperature. tweenradii 1 2andT 2kpc,
to theradii
H i1dominatedand 2kpc, toouter
the regions.
H i dominated outer regions.
stronglyHidominated
with
weak
CO
emission.
From
the
The
GDR
is
likelywell-constrainedbythemetallicity,
condenses
from the gAt submillimeter wavelengths
At submillimeterthe dust wavelengths
emission is opticallythe dust emission
is opticallyThick
red lines show Thickthe binning
red linesof theshowscatter-plot
the binningin of the scatter-plot in
at
least
for
metallicities
reasonably
closetosolar.
The nova
inner
to
outer
regions,theX
factorincreases,
as
could
2010)
and superthin. This yields a cross-section
thin. This per
yieldsH-atoma cross-section
which naturally
per H-atomwhich
naturallyCO
0.5 K wide intervals. The
0.5 K
cloud
wideofintervals.
points areThe
fit cloudby twooflines,
points are fit by two lines,
solar
metallicity
is
about
Z
=
0.0142bymass(Asbe expected
given
that
there
is
a
metallicity
gradient
andexpects
that
when tvaries with radius, muchvarieslikewiththe radius,
metallicity
much(Magrinilike the etmetallicity
al. one (Magrini
etNal.
assuming(H)
one=
assuming
X⇥
I
N(H(light
)
=
red
X
line)
⇥
I
and
(light
red
line)
and
2
CO
CO
2
CO
CO
0
0plundandal. As2009,
Sectionby3.1.2). Assuming
the stanin⇥COal.
2010b)
gas-phase
metal frac2009). Using dust (r),2009).
we calculate
Using dust
the (r),totalweHcalculate(i.e., cold,
the N
total
cold,
(H2a)Hdecline=(i.e.,XCO
ICONemission+(HK= (GratierX
inCO
green.
⇥ Iet
As+described
Kdark
in et
green.
bycloud
described
2 )darkCO
dardhydrogen-to-dust(Draine& Li exc
temperature(Gratier
et ratioal.
metallicities,
neutral hydrogen gas:neutral
Hi + Hhydrogengas:density.
H i + The
H2 ) column
dust Dickman
density.
The
et al.dust(1986)
Dickmana XCO
et 2012).
al.is(1986)
an average
a XCO
over
ratio
many
is an average overmass
manyratio of 100 low2 ) column
0
2007,
Table
the total gas/dust
mass ratio
M (H be+ high
without
abeK
GDRisshould
opacity is NH dust =opacity⌃dustis=NSH⌫ /B
, and
=S
dust
the lines
dust
diﬀerent
cloudsso itcannotdiﬀerent
cloudsexpectedso ittocannot
characterize
beoverestimate
expectedall 3),
to characterizealldust
dustthe⌫ /B⌫,T
dark systematically
⌫,T =
( )⌃
( ) , andTheHe+
gas-phase
metals)/M(dust),
assuming
H
and
He
the
H
mass
at
moderate
and
high
I
and
both
fits
overthe diﬃculty toin formi
2 above,
CO points.
surface density ⌃dustsurface= S⌫ /(Bdensity⌃dust
For=350
S⌫ /(Bas ⌫,T
above,
clouds,350orasallof our dataclouds,points.
or all of our data⌫,T ( ) ).
( ) ). Forbenegligible
contributors
to
the
dust
mass.
From
Asestimate
N
(H
)
at
high
I
.
There
is
no
physicalreasontoquickly such that ev
CO2 showsthe dust surface density
the candustbesurfacecomputeddensityfor canall points
be computedinforFigureall points
2 shows
in 2 theFigure
relationshipbetweenthe relationshipthe dust- between the dust0
plund,
M there(H) = 0.7154 and M (He) = 0.2703,
and thede- dust m
expectadiﬀerconstantoﬀset) butdensity
it appears
not reduce
M33, as shown in Fig.M33,
1 (right
as shown
panel),
in suchFig. that
1 (right
the panel),diﬀer- suchderivedthatHthedensityderivedandH(Kradialand interIthatdark
2 column
2 Icolumn
CO for three
CO for three radial internoting
thegas-phase-metal-fractionas Zgas , we define the
isthe
gas
whose
dust
emissionis represent
detectedbutinneris respecnot
seen
in regions, respecence with respect to Fig.
ence1with(left respectpanel) is
tothatFig. the1 (lefttemperapanel) isvals
thatchosentemperato
represent
valsthechoseninnertoandouter regions,
the
and
outer
hydrogengas-to-dust mass ratio as GDR = (0.7154 +CO – this could be optically thick H i, moleculargas where
and appl
CO
CO not formed or is photodissociated,
CO
0.0142just Method
under 40%
CO haslow3density
H2
Article number, page
ArticleofZ15
number,page Z3gasof )).
15 helium adds 4.2.
gas )/(0.0142(1to this number.
For GDR = 100, the typical Developed
Galactic value,
clouds, or unexpectedly large quantities of ionizedgas.
in LeroyCOCOCO
the gas-phase-metal-fraction
Zgas = 0.49 and
51%toofthe
theHERAappliedmetals are in the dust phase. This value is etreasonably
al. (2013),rothe ide
4. Leroy-Sandstrom method
bust; for a solar composition, if GDR = pressed
100 ± 20as thenthe sum
50
±
10%
of
the
metals
are
in
the
gas
phase.
molecular componen
4.1. Prior discussion on the gas-to-dust ratio (GDR)
What about lower metallicity environments?
Sincefromdust the io
tribution
The GDR is likely well-constrained by themetallicity,
condenses
from the gas in AGB stellar winds(Gielen etal. apsumption
is likely
2. Fit of dust-derived
N
(H
)
as
a
function
of
I
for
data
in
at least for metallicities Fig.
reasonably
close 2010)
to solar.
The
2
CO (Matsuura et
and supernova remnantsal. 2011), one
studies.
radialbetween0
and1
kpc(top),
1
2
kpc(middle),solar metallicity is about
Z intervals= 0.0142bymass(Asexpects that when there is less dust and less metals, theand 3.1.2).
4 5 kpc(bottom).
No cutin intensity
applied.
plund et al. 2009, Section
Assuming
thestangas-phasemetal
fractionhas
willbeentend
to be higher.
very⇥ ⌃dus⌃gas = At
GDRThe color
indicates
the density
points andthe thickredenvironments, thedard hydrogen-to-dust mass
ratio scaleof 100(Draine& Li ofexcept
low
metallicities,
for verydense= mp ⇥ [N (H i)histogram shows the N (H2 ) data averaged in bins of 0.5 Kkm/s.
2007, Table 3), the total gas/dust mass ratio
M (H +
GDRisshould
be higher than the relation given above due to
The thin green line shows an aﬃne fit between N (H2 ) and ICO ;= ⌃H i + ↵CO ⇥He + gas-phase metals)/Mthe(dust),
assumingfit
Hresults
and Hetothediﬃculty
in forming
dust grains
andredmantles suﬃciently
corresponding
are printed
in green.
The thinbe negligible contributorsline
tois thedustfit mass.
From
Asquickly
thatprocesses
do is a s
a linearwithout
ansuchoﬀset;
the evaporation
correspondingorfitdestruction
results
where ↵CO
plund, M (H) = 0.7154 andM (He)in red.
= 0.2703,
andAveragede- dust
not reduce
themass(Rémy-Ruyer
2014).
are printedBluecross:
valueof theplotted data.et al.
ICO
to ⌃H2 . Equatin
noting the gas-phase-metal-fraction as Zgas , we define thehydrogen gas-to-dust mass
rationumber,as GDR+
Articlepage =
4 of(0.7154
15
and application to M33
0.0142 Zgas )/(0.0142(1 Zgas )). helium adds 4.2.
just Method
under 40%to this number. For GDR = 100, the typical Developed
Galactic value,in Leroy et al. (2011) and later extended andthe gas-phase-metal-fraction Zgas = 0.49 and
51%toofthe
theHERACLES/KINGFISH data in Sandstrom
appliedmetals are in the dust phase. This value is etreasonably
al. (2013),rothe idea is that the dust emission can be exbust; for a solar composition, if GDR = pressed
100 ± 20as thenthe sum of the emission from the atomic and
50 ± 10% of the metals are in the gas phase.molecular components, implicitly assuming that the conWhat about lower metallicity environments?
Sincefromdust the ionized gas is negligible. The latter astribution
condenses from the gas in AGB stellar winds(Gielen etal. appropriate and is also common to othersumption
is likely

HerschelÀpďÃ

N(H2) (1021 cm-2)• CO-H27djďà!Ü
3
2

R = 0 – 1 kpc

R = 1 – 2 kpc

R = 4 – 5 kpc

X = 2.54 î 0.05
X = 0.91 î 0.05
K = 4.95 î 0.12

X = 3.98 î 0.05
X = 2.02 î 0.04
K = 4.05 î 0.07

X = 6.83 î 0.11
X = 2.82 î 0.07
K = 4.17 î 0.04

1
0

-1

0

5 10 15 0
5 10 15 0
5 10 15-1-1
ICO (K km s )ICO (K km s )ICO (K km s-1)

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HerschelÀpĎĚĝÐàďÃ• NGC 3627

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N3627_MOM0

-0.02

-0.03

26972

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34694

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500 µm (36”) 350 µm (25”) 250 µm (18”) CO J=1–0 (17”.4)

CO J=1–0 (~3”)(http://alma.mtk.nao.ac.jp/)ÐàďSED
E–S0a

Sa–Sb Sbc–Scd Sd–Im–Sm–BCD

normalized flux density

A. Boselli et al.: FIR colours and SEDs of nearby galaxies observed with Herschel

M87
M84
M86

NGC4435
NGC4476

M58
M90
NGC4491

NGC4388
NGC3982

IC3476IC3311
NGC4294

M100
M99

NGC4532
NGC4438
IC3583

NGC3448
VIIIZw182

Tdust  20 K

Hubble±ĎĈĉ
Tdust ýn
ŁE–S0aĐÒĎíøł
1 102 104l (µm)

1 102 104l (µm)

1 102 104l (µm)

1 102 104l (µm)

Fig. 2. The UV to radio centimetre observed SEDs of a subsample of galaxies, in 4 diﬀerent panels according to their morphological type. Theinteracting systems NGC 3448 (Arp 205) and NGC 4438 (Arp 120) are included in the Sd-BCD-Pec sample. SPIRE data are marked in red. The
continuum line in the optical domain is the medium resolution integrated spectra with emission lines in star-forming systems taken from Gavazzi
et al. (2004). The dashed line shows the combination of two modified black bodies (B(ν, T )νβ with β = 2) of temperature T = 20 and 40 K
respectively normalized to the 100 and 60 µm flux densities. The dotted line in the radio domain indicates the best-fit synchrotron emission.

have colour indices indicating that the cold dust has a higher
temperature than in star-forming systems. As for the interpretation of the FIR properties of the early-type galaxies in the
SINGS galaxy sample (Draine et al. 2007), higher dust-weighted
mean starlight intensities can explain the high FIR colour temperatures of E/S0 galaxies. However, the relative importance of
X-ray heating (Wolfire et al. 1995), stochastic heating, heatingshows some examples of UV to radio centimetre SEDs of galaxies according to their morphological type. Figure 2 shows thatin the elliptical galaxies M 87 and M 84, the sub-mm domain
is dominated by synchrotron emission. M 87 is a powerful radiogalaxy (Virgo A), where synchrotron dominates down to ∼10 µm(Baes et al. 2010). M 84 is a moderately active radio galaxy witha luminosity at 20 cm of 2 × 1023 W Hz−1 . In spirals, the SPIRE(Boselli+ 2010)àª4ċ>êW_LETTER

doi:10.1038/nature10551

The alignment of molecular cloud magnetic fields
with the spiral arms in M33• M 33ď>êďÃŁCO J=2–1ł
– >êďª4Đ½Ďù
LETTERGMC 1

31° 00′

3

1

5

–7

4

–7 1 h 34 min 09.3 s +7
+7

–14

–28

+7 GMC 4

30° 30′30° 25′

–7

GMC 6
+20

+10
1 h 33 min 59.0 s

30˚30’

1 h 34 min 00 s

GMC 5

–28

+10
30° 46′ 19.7″

Declination

2

30° 40′30° 35′

–14

–7

6

30° 45′GMC 3+730° 46′ 47.7″

30˚45’

30° 40′ 43.9″

30° 55′30° 50′

GMC 2
+7

+14

30° 33′ 44.7″

–10

–10

–20

30° 20′–7

01 h
35 min00 s

34 33
R.A. (J2000)
01 h
34 min
30 s

01 h
01 h
34 min33 min00 s
30 s
Right ascension (J2000)

01 h
33 min00 s

01 h
32 min30 s+7

1 h 34 min 13.7 s

–7

+20

+10

Figure 1 | The optical spiral arms and the locations of the six most massive
Figure 2 | CO (2–1) maps and polarization vectors. The contours are 90%,
GMCs in M33. The background is an optical image of M33 (ref. 21) andthe… 10% of the peak intensity of each cloud (from dark to light grey). The
80%
vectors show the 3.6-cm synchrotron polarization21, with the telescope beam
red vectors show polarization detections, for which the ratio of the polarization
size shown in the lower left corner. The vector length is proportional to level
the to its uncertainty is greater than three and the error in direction is less thanintensity of polarized emission. A sketch of the optical arms27 is shown with
10u. The thick grey vectors show the tangents of the local optical arms27,30 for
thick purple solid lines. The contours show the structures between 3699 and 4899
derived from scale decomposition28 of the 500-mm Herschel data29 (the lowest
isconsistent with the scenario in which the B fields in the GMCs arecontour level has 30% of the peak intensity, and the following levels increase
compressed within the spiral arms, and the fields can exert tension
linearly with steps of 10% peak intensity). The GMC locations (plus symbols)
are numbered 1 to 6. The optical arms related to GMCs 1, 5 and 6 are clear.
forces (because of d; see Supplementary Fig. 1) strong enough to resist
GMC 3 is between two arms. GMCs 2 and 4 are on the extensions (purple
cloud rotation (‘magnetic braking’4,9). If mass has been accreted todashed lines) from two different solid lines. GMC 4 has a short straight
extension. For GMC 2, we adopt the southern arm defined by ref. 30 as form
the a cloud from the accumulation length scale (hundreds of parsecs)
ina shearing galactic disk, cloud rotation is inevitable owing to conextension, which traces the 500-mm clumps well. The arms are traced slightly
differently in the literature, and this observational uncertainty will contributetoservationof angular momentum (Supplementary Fig. 1 and figure 6 ofd, the dispersion of polarization-arm offset (Supplementary Information).
10

Max-Planck Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany.

2 4 NO V E M B E R 2 0 1 1 | VO L 4 7 9 | N AT U R E | 4 9 9©2011 Macmillan Publishers Limited. All rights reserved8

1 h 34 min
31.5 s

1 h 34 min
34.5 s–10–20

Right ascension

GMC 1, 2, 4, 5 and 6. For the inter-arm GMC 3, the grey vector shows the meanof the two tangents of the nearby arms at the positions closest to GMC 3. The
ellipses indicate the Submillimeter Array synthetic beams; all detections are
spatially independent. The coordinate (right ascension, declination) offsets are
in arcseconds.

(Li & Henning 2011)ref. 2), unless the momentum is consumed by another mechanism suchas magnetic tension. The fact that the GMCs in M33 show significantly
smaller angular momenta than predicted by the Toomre instability
criterion9 supports our observations. Figure 3 is also consistent with
0.9

30

0.8

GMC 1
GMC 2
GMC 3
GMC 4
GMC 5
GMC 6

25%

25

0.7
0.6

20

N

6

55%

0.4

CIÏ¹ċ>Ĥĩ
4

0.5

15

Likelihood35

δ (degrees)

Dec. (J2000)

31˚00’

30° 49′ 07.0″

combinations of s and d are able to produce the observed confidence
level (Fig. 4). Random fields or field-arm offsets are very unlikely. Only
when s 5 17u–22u with d , 8u do the simulations give a confidence
level similar to that observed. This indicates that the mean field direcRESEARCH
tions are well-defined and highly correlated with the spiral arms, which

30° 48′ 57.3″

he formation of molecular clouds, which serve as stellar nurseries
n galaxies, is poorly understood. A class of cloud formation models
uggests that a large-scale galactic magnetic field is irrelevant at thecale of individual clouds, because the turbulence and rotation of aoud may randomize the orientation of its magnetic field1,2.
lternatively, galactic fields could be strong enough to impose theirirection upon individual clouds3,4, thereby regulating cloud accumulation and fragmentation5, and affecting the rate and efficiency
f star formation6. Our location in the disk of the Galaxy makes anssessment of the situation difficult. Here we report observations ofhe magnetic field orientation of six giant molecular cloud comlexes in the nearby, almost face-on, galaxy M33. The fields are
ligned with the spiral arms, suggesting that the large-scale fieldn M33 anchors the clouds.
At a distance of about 900 kiloparsecs (kpc) from us7, M33 is ourearest face-on galaxy with pronounced optical spiral arms. To resolve
typical giant molecular cloud (GMC) with a size of tens to a hundred
arsecs, we used the Submillimeter Array8, which offers a linear spatial
esolution of about 15 pc at 230 GHz (the frequency of the CO J 5 2–1
ansition) at the distance of M33 using the array’s most compact
onfiguration. To observe the strongest CO line emission, we picked
he six most massive GMCs from M33 (ref. 9). It is clear to which spiralrms the GMCs are related, except for GMC3, which is situatedetween two optical arms (Fig. 1).
We determined the orientations of the GMC magnetic fields (Belds) from the polarization of CO emission lines, which should bether perpendicular or parallel to the local B-field direction projectedn the sky (the Goldreich–Kylafis effect10). Although there are other-field tracers11 that do not have this 90u ambiguity, CO is much morebundant and allows current radio telescopes to perform extragalactic
oud observations. Despite the 90u ambiguity, such a B-field obseration is still valuable12. An intrinsically random field distribution,s occurs when the turbulence is super-Alfvenic13 (that is, whenurbulent energy dominates B-field energy), will still be random withhis ambiguity. On the other hand, an intrinsically single-peaked
Gaussian-like field distribution, in the presence of sub-Alfvenic
urbulence13 (that is, when B-field energy dominates turbulent energy),
ill either remain single-peaked, or split into two peaks approximately
0u apart (‘double peaks’). From the total distribution of the offsets
etween the CO polarization of the M33 GMCs and the local arm
irections (Fig. 2), the trend of double peaks is clearly visible
Fig. 3). The distribution can be fitted by a double-Gaussian function
ith peaks at 21.9u 6 4.7u and 91.1u 6 3.7u and a standard deviation
f 20.7u 6 2.6u. This result is barely affected if the inter-arm GMC3 is
xcluded.
The angle dispersion (S) of the CO polarization and its offset (D)om the arm directions are determined by the dispersion of B-field
rientations (s), the dispersion of the offsets between cloud mean fields
nd spiral arms (d), the observational errors (e , 10u), and theGoldreich–Kylafis effect. By performing Monte Carlo simulations
see the Supplementary Information), we can estimate the likelihoodf observing S and D within certain ranges, and determine which

Declination (J2000)

Hua-bai Li1 & Thomas Henning1

0.3

10

85%
0.2

2

5

0

–40

80
100
120
–20
0
20
40
60
Offset of CO polarization from local arm direction (degrees)

140

Figure 3 | Distribution of the CO polarization-arm offsets. The offsets arefrom the difference between the red and grey vectors in Fig. 2. N is the number
of offsets falling in each bin. Contributions from different GMCs are indicatedby colour. The distribution can be fitted by a double-Gaussian function with astandard deviation of 20.7u 6 2.6u and peaks at 21.9u 6 4.7u and 91.1u 6 3.7u
(peaks shown by vertical black lines). The directions of synchrotron
polarization from the regions near the GMCs (within one beam size, as shownin Fig. 1) are also shown as dashed lines, with the same colour code as the
GMCs.

0

0

0.1

5

10

15
20
α (degrees)

25

30

Figure 4 | Likelihood of obtaining simulated angle dispersions S and offsets
D of CO polarization within the observed 90% confidence intervals. The
90% confidence intervals from the data in Fig. 3 are 15.1u , S , 26.4 and| D | , 10.7u. The likelihood is estimated by Monte Carlo simulations(Supplementary Information) with various combinations of s (B-field
dispersion) and d (dispersion of the offsets between mean fields and arms).
Only s < 20u with d , 8u can give a similar confidence level within theintervals.

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