Draft version October 12, 2013
Preprint typeset using LATEX style emulateapj v. 11/10/09
ON THE INFLUENCE OF DYNAMICS IN SPIRAL GALAXIES ON DUST-TO-GAS RATIOS MEASURED BY
KINGFISH
Sharon E. Meidt et al.
(Dated: October 12, 2013)
Draft version October 12, 2013
ABSTRACT
We explore the possibility of galaxy-scale radial and azimuthal variations in the dust-to-gas
ratio (DGR) that arise in the presence of a spiral potential perturbation. Where the coupling—
proportional to the relative velocity of the gas and dust—is weak, the DGR can change significantly
from its unperturbed value DGR0 . We show that the spiral arms are the primary sites of the dynamical decoupling of gas and dust, capable of introducing drift velocities as large as ∼ 1 km s−1 .
The reduction of the DGR with respect to DGR0 increases with progressively larger distance from
the corotation radius (CR) of the pattern. As a consequence of dissipation in the gas, torquing by
the stellar spiral, which drives radially inward gas motions inside corotation and radially outward
motions outside corotation, evacuates the zone around the CR of gas but leaves the collisionless
(pressureless) dust behind. This leads to a reduction of the DGR in the spiral arms and a recognizable pattern of peaks and troughs in radial profiles of the DGR arranged over the radial extent
of the pattern. As an additional consequence of the grain-size dependence of the drift velocity, we
further predict 1) radial segregation of dust by size and 2) radial migration of the largest grains with
sizes >0.4 µm out to large galactocentric distance. The extent of this migration can be significant
in the presence of multiple, extended non-axisymmetric structures and lead to a much lower DGR
in the outermost gas disks of galaxies. We explore the dependence of these DGR signatures on dust
grain size, magnitude of streaming motions (perturbation strength) and relative thicknesses of the
dust and gas disk. We test these predictions in 2 nearby galaxies from the KINGFISH sample and
discuss competing effects on observations of the DGR.
1. INTRODUCTION
The advent of high spatial resolution and sensitivity
FIR imaging has revealed several surprising features in
the dust distributions of nearby galaxies. In the last 15
years it is clear that cold dust exists at large galactocentric radii, in some cases far beyond the edge of the
optical disk (Nelson, Zaritsky, Cutri 1998; Bianchi, Alton & Davies 1999; Munoz-Mateos et al. 2009). This
dust does not appear to be directly associated with any
evolved stellar source and is instead often distributed
closely with the extended atomic gas reservoir (i.e. Alton et al. 2000; Thomas et al. 2004). As the outer gas is
likely recently accreted and unenriched, the origin and
the nature of outer dust is unclear.
Radial profiles of the dust-to-gas ratio in normal spiral galaxies are consistently measured to decrease with
radius, indicating that, at least to first order, ISM enrichment is closely tied to other global galaxy properties. But such radial profiles are rarely smooth and
often show bumps and wiggles (Munoz-Mateos et al.
2009; Pohlen et al. 2010) in addition to truncations
(Groves et al. in prep). Two-dimensional maps of the
dust-to-gas ratio additionally show structure (Aniano
et al. 2012; Mentuch-Cooper et al. 2012), suggesting
the possibility of imperfect coupling between the dust
and gas on kpc-scales in galaxies.
One explanation for these features is the existence of
a dynamical mechanism that works to decouple the gas
and dust. Another scenario might be preferential cre-
ation and destruction of dust grains in spiral arms, or
local modifications to the timescales for these processes,
which are otherwise estimated at much longer than an
orbital period (Draine 2003). The properties of the dust
(mass, temperature, emissivity) inferred from observations may also be uncertain. If the emissivity of dust
increases at high gas surface density, then most current
techniques to model the mass in dust will lead to local
overestimation of this quantity. Together with uncertainties in the assumed mass and distribution of gas,
which is sensitive to the assumed CO-to-H2 conversion
factor (XCO ), these sorts of uncertainties may introduce unphysical variation in the measured dust-to-gas
ratio.
In this paper we investigate the possibility of a dynamical influence on the coupling between dust and gas
and the degree to which this alters the dust-to-gas ratio. When and where gas and dust dynamically decouple must be understood in order to reliably identify
changes in the properties of dust grains from environment to environment and thus to interpret signatures
of grain formation and destruction in the ISM from observations. From a dynamical perspective, the study of
the coupling of dust and gas on large-scales in galaxies
is also a necessary step for the use of gas kinematics to
trace the dynamics of dust.
The idea that dust and gas in spirals may spatially
separate is not uncommon when interpreted from within
the density wave framework. But in our own MW, the
dust and gas along various sight lines appear to be well
2
mixed (although distance ambiguities might be at play).
This has motivated theoretical arguments in favor of
coupling, most of which implicitly assume a simple, unperturbed potential. However, this is not predictive on
larger scales in spiral galaxies.
In the presence of a non-axisymmetric perturbation to
the underlying gravitational potential, such as a bar or
spiral arms, the gas and dust react strongly and nearly
identically, except for the fact that the dust, which feels
an additional force of friction coupling it to the dissipative gas, is pressureless. Consequently, perturbations
that torque the gas and dust—and set up a series of
dynamical resonances—can lead to certain preferred locations in the disk where the gas and dust can decouple.
When the coupling—proportional to the relative velocity between the dust and gas—is weak, the DGR can
change significantly from its unperturbed value. This
has been found in numerical simulations of dusty disks
in the presence of a spiral potential perturbation; strong
spatial variations in the dust-to-gas ratio (DGR) develop as a result of variations in the frictional coupling
between the two components in models of both protoplanetary disks and in the disks of spiral galaxies
(Noh et al. 1991; Theis & Orlova 2004; Vorobyova &
Shchekino 2006).
The largest variations in the DGR are predicted in
and around spiral arms (Theis & Orlova 2004) and there
may even be evolution in the radial variation of the
DGR compared to its unperturbed level. Vorobyova &
Shchekino (2006) interpreted the results of their simulations to suggest that the coupling is the least effective
near CR, in particular, allowing the dust to decouple
from the gas in this region and migrate radially outward. But the exact degree of decoupling and how it
depends on the properties of the dust and the nature of
the spiral pattern remain unconstrained.
While numerical solution of the equations of motion
offer the most realistic picture of the coupling between
the gas and dust, with some simplifying assumptions,
we can make analytical predictions for the dust-to-gas
ratio in terms of observationally accessible quantities.
In this paper we solve the Euler equations of motion for
the gas and dust in the presence of a spiral perturbation
in the tight-winding limit (§ 2). We find that a non-zero
drift velocity between gas and large dust grains can be
introduced and, as a result, the two components exhibit
different density responses to the potential.
This dynamical effect produces radial and azimuthal
signatures in the DGR profile, which are distinct from
most other possible sources of variation in the DGR,
such as the CO-to-H2 conversion factor (XCO ) or dust
emissivity and temperature, as discussed in §§ 4 and
5. In the regime appropriate for standard ISM dust
grain sizes and chemistries, the velocities of the gas and
dust will be most strongly decoupled in the spiral arms,
leading to a reduction in DGR there on the order of 1030%. In addition to azimuthal variation in the DGR,
radial variation is introduced along the arm, as the gas
and dust remain well-coupled in the zone around CR.
Here, the DGR remains at its unperturbed value.
In § 6, we present a test of these predictions, consid-
ering radial profiles of the DGR in two spiral galaxies
from KINGFISH (Key Insights on Nearby Galaxies: a
Far-Infrared Survey with Herschel; PI: Kennicutt). The
high spatial resolution KINGFISH observations offer an
unparalleled view of how dust is distributed in spiral
galaxies, clearly resolving key bar and spiral features.
We use the Aniano et al. (2012) realizations of the
predictions from the Draine & Li (2007) dust model for
the dust mass surface densities in these galaxies. These
are compared to maps of the total gas surface density
constructed from archival THINGS and HERACLES
data. We interpret the measured DGR according to our
predictions and quantify the significance of the decoupling signature. We conclude by assessing the relative
importance of competing effects on measurements of the
DGR, including changing dust emissivity and variation
in the CO-to-H2 conversion factor.
2. A MODEL FOR DUST AND GAS MOTIONS
In this section we present an expression for
dynamically-induced variation in the DGR in terms of
observationally accessable quantities. We first derive
a relation between the motions of the dust and gas,
which defines the expected drift velocity, and then
use the perturbed continuity equation to relate these
motions to the observed surface densities.
2.1. Perturbed equations of motion
The Euler equations for the motions of the gas and
dust can be written as
∂~vg
∇Pg
+ (~vg · ∇)~vg +
+ ∇Φ = 0
(1)
∂t
Σg
and
∂~vd
+ (~vd · ∇)~vd + ∇Φ = Fc
(2)
∂t
respectively. Here we assume that the gas is collisional
and feels a pressure Pg while the dust is pressureless.
The dust and gas respond analogously to the potential Φ, but the dust feels an additional frictional force
due to coupling with gas, denoted here by the force
term Fc =A(~vd − ~vg ), where the inverse of the coefficient A represents the coupling timescale (and contains
the physics of the dust-gas interaction). We ignore the
influence of the dust on the gas and omit the equivalent term Fc /(Σg /Σd ) for the gas under the assumption
that the ratio of the gas surface density Σg to the dust
surface density Σd is high (Σd <<Σg ) as observed in
nearby galaxies (Σd /Σg ∼0.001; Draine ?; Sandstrom et
al. 2013).
Solutions to the equations of motion for a potential
perturbation with m-fold symmetry have dependence of
the form e−i(mφ−ωt) , where ω=mΩp is the angular rotation rate of the perturbation. Thus for perturbations
p(r, φ) = p1 (r)e−i(mΩ−ω)t , the perturbed equations become
d
i(ω − mΩ)vd1r − 2Ωvd1φ =
Φ1 − A(vd1r − vg1r ) (3)
dr
3
i(ω − mΩ)vg1r − 2Ωvg1φ =
d
(Φ1 + hg1 )
dr
(4)
are
vdrif t,r = −vg1r Cdg
vdrif t,φ = −vg1φ Cdg
for the r component and
Ω
im
vd1r + i(ω − mΩ)vd1φ =
Φ1 − A(vd1φ − vg1φ ) (5)
2
r
Ω
im
vg1r + i(ω − mΩ)vg1φ =
(Φ1 + hg1 )
(6)
2
r
for the φ component, with the standard replacement
of the gas pressure with the specific enthalpy hg1 =
vs2 Σg1 /Σg0 where vs is the sounds speed in the gas.
It is straightforward to solve for vg1r and vd1r in the
tight-winding limit |kr|>>1 where Φ1 ∝ eif (r) with
k=df (r)/dr. In this case, we can drop terms of order
(Φ1 + hg1 )/r relative to those with d(Φ1 + hg1 )/dr; i.e.
Binney & Tremaine 1987). To first order the perturbed
equations of motion for the gas imply
∆g vg1r = −k(ω − mΩ)(Φ1 + hg1 )
∆g vg1φ = −ik2Ω(Φ1 + hg1 )
(7)
(8)
where ∆g =Ω2 − (ω − mΩ)2 .
The motions of the dust due to the potential perturbation can be formally expressed in terms of the analogous gas response. With ∆=Ω2 + (i(ω − mΩ) + A)2
and using the continuity equation (see eq. 17 below) to
relate the perturbed gas density to the perturbed gas
velocity this yields:
iA
∆vd1r = vg1r ∆ + k 2 vs2 1 +
m(Ω − Ωp )
2 2
∆vd1φ = vg1φ ∆ + k vs
(9)
Now taking the real part of the two velocity components we find:
2
2
2 2 α + β tan θ
(10)
vdr = vgr 1 + k vs
∆∗ ∆
2 α
vdφ = vgφ 1 + k 2 vs2
(11)
∆∗ ∆
where θ=(mΩ − ω)t and
∆ ∗ ∆ = Ω2 + A2 − m2 (Ω − Ωp )2
with
Cdg =
k 2 vs2
A2 − Ω2 + m2 (Ω − Ωp )2
∆∗ ∆
(14)
which simplifies considerably to
k 2 vs2
(15)
A2
taking the limit of short coupling timescale (A>>Ω).
According to eq. (13), the kinematic coupling between gas and dust remains stronger in galaxies with
weak streaming motions (weak potential perturbations)
than in galaxies with the strongest streaming motions.
The frictional coupling is moreover weakest when the
gas motions are small. Within an individual galaxy,
the frictional coupling between the two components will
locally weaken at the corotation radius where motions
approach zero (described further below). The motions
of the dust and gas may even reverse, when Cdg >1.1
The magnitude of the drift velocity also very clearly
depends on the coupling timescale A−1 in comparison
−1
to (kvs ) , the timescale with which the material in
the rotating frame interacts with the (spiral) perturbation. The factor Cdg represents the square of a ratio
of these two timescales. If A−1 is short compared to
the spiral interaction time (Cdg is small), the velocities
of the gas and dust remain very well coupled. But if
the gas and dust frequently come in to contact with the
spiral potential perturbation, then frictional drag may
not have the chance to couple the dust and gas back together before exiting the spiral potential. As described
in the next section the kinematic decoupling (non-zero
drift velocity) can introduce measurable variation in the
dust-to-gas ratio. The magnitude of the decoupling depends on the coupling timescale, which we consider in
detail in § 3.1.
Cdg ≈
2.3. The dust-to-gas ratio
The perturbed continuity equation
α2 = Ω2 − A2 − m2 (Ω − Ωp )2
β 2 = Ω2 + A2 − m2 (Ω − Ωp )2
(12)
(13)
A
m(Ω − Ωp )
2
+ 4A2 m2 (Ω − Ωp )2
2.2. Velocity differential between the dust and gas
By expressing the dust motions in terms of the gas
motions we see that the magnitude of the velocity differential between the dust and gas, or the drift velocity, is
largest where gas motions are high, i.e., at the location
of the spiral arm. Indeed, at the location of maximum
perturbation (the spiral arm; i.e. ignoring the component of vd1r that is non-zero at an offset π/m from the
spiral) the drift velocities between the two components
1 d
(rΣ0 v1r ) + imΣ0 v1φ = 0 (16)
r dr
relates the perturbed gas and dust surface densities to
the perturbed velocities. Since v1r and v1φ are of the
same order for both the gas and dust, the third term
above can be dropped relative to the second whereby
the continuity equation for either the gas or dust becomes
m(Ω − Ωp )Σ1 + kΣ0 v1r = 0
(17)
i(ω − mΩ)Σ1 +
1 If the coupling timescale becomes long enough (as might be
expected for very large grains and planetesimals in protoplanetary
disks) the velocities of the gas and dust can be opposite over a
finite radial zone.
4
Now with the total surface density Σ = Σ1 + Σ0 , and
expressing Σg1 /Σg0 in terms of vg1r (eq. 17), this yields:


Σd0 
kvg1r
Cdg
Σd

=
1+
(18)
Σg
Σg0
m(Ω − Ωp ) 1 − kvg1r
m(Ω−Ωp )
using that vdrif t,r = −vg1r Cdg as in eq. (13) and ignoring the creation/destruction of the dust or gas, which
we expect to be minimal on the timescales considered
here.
Note that because vg,1 also approaches zero at corotation, there is no discontinuity when Ω = Ωp . Indeed,
according to eq. (17) kvg,1 follows m(Ω − Ωp ) to within
a factor Σg,1 /Σg,0 so that
−1 !
Σd
Σd0
Σg1
Σg1
=
1−
Cdg 1 +
(19)
Σg
Σg0
Σg0
Σg0
Alternatively, we can write
Σg1
dgr ≈ dgr0 1 − Cdg
Σg0
vdrif t,r Σg1
dgr ≈ dgr0 1 +
vg1r Σg0
(20)
expanding the numerator in the second term in eq. (19)
to lowest order in Σg1 /Σg0 (or kvgr /m(Ω − Ωp )).
At fixed vdrif t /vgr these two expressions relate
enhanced gas arm-interarm contrast to a reduced
DGR with respect to the unperturbed value. Where
Σg1 /Σg0 ≈0.4, the DGR will be reduced by 20% when
the drift velocity is half the radial gas velocity. Given
typical gas streaming velocities 10-15 km s−1 , this
amount of reduction in the DGR requires large drift velocities, such as expected for the largest grains in nearby
galaxies (see § 3.1).
3. DYNAMICAL DECOUPLING OF GAS AND
DUST
3.1. Dependence on the coupling timescale
In the previous section we identified a dynamical decoupling of gas and dust that leads to specific signatures in the dust-to-gas ratio described in detail in §
4. The degree of dynamical decoupling depends on
the timescale of the dust-gas coupling relative to the
timescale with which the two ISM components interact
with the bar/spiral perturbation. The latter mainly depends on the nature and geometry of the spiral, which
varies from galaxy to galaxy. As outlined in Appendix
A, the spiral interaction timescale is a fraction of the
dynamical time τdyn /fsg , with fsg set by the stellar-togas mass surface density ratio.
The coupling timescale depends on the nature of the
dust and gas disks—their relative scale heights, the size
and charge of dust grains, and the fractional ionization
of the gas (in addition to the gas density and temperature and the grain density). In the systems where we expect well-defined non-axisymmetric structure (and thus
active dynamical decoupling) we also expect a large
fraction of the ISM to be in the coldest, densest molec-
ular phase, with low fractional ionization (i.e. as in
prototypical star-forming galaxies). Our fiducial coupling timescale is therefore set by neutral gas drag. But
several additional sources of drag can modify the coupling timescale in, e.g., the WNM or for large, strongly
charged grains, as discussed below.
3.1.1. Neutral gas drag
For dust disks much thinner than gas disks, the coupling timescale due to neutral gas drag is the dynamical timescale, assuming that the gas is in vertical hydrostatic equilibrium and vertical mixing occurs in a
sound crossing time H/vs ∼ Ω−1 where H is the vertical scale height. (Further reduction by an amount
(Hg /r)n , where 1≤n≤2, is invoked for different types
of mixing; Noh et al. (1991).)
In this case,
2 Σg1
dgr ≈ dgr0 1 − fsg
Σg0
Σg1
≈ dgr0 1 − k 2 H 2
(21)
Σg0
Thus, if the spiral is tightly wrapped and the distance
to the adjacent arm is smaller than the vertical scale
height, the ISM interacts with the perturbation quicker
than the vertical mixing time, resulting in a large reduction in the DGR on the arm.
In the more realistic limit where the dust and gas
disks have similar scale heights, the dust-gas collision
timescale sets the gas drag timescale:
√
md
4 πρd aQs
−1
A =
=
(22)
σng mg vs
3Σg κ
as evaluated in Appendix B.
Adopting realistic grain properties appropriate for
nearby galaxy disks, namely a dust density ρ=3.5 g
cm−3 typical for silicates and grain sizes 50 Å.a.0.25
µm, A−1 is much shorter than the dynamical timescale
Ω−1 ≈κ−1 .
For this regime,
2
√
4 πρd aQs
Cdg = k 2 vs2
3Σg κ
√
2
4 πρd a
2 2
=k H
3Σg
2
√
4 πρd a Qs
= fsg
(23)
3Σg Q
and
"
dgr ≈ dgr0
1−
#!
2
√
Σg1
4 πρd a
2fsg
3Σg
Σg0
(24)
conservatively assuming Qs /Q≈2, although this could
be as low as 1.2 (Note that this reduces to the thin dust
2 Q is measured in the range 1.5-2.5 in nearby galaxies (van
s
der Kruit & Freeman). Assuming that star-forming gas has Q∼1,
then Qs /Q∼1-2.
5
0.5
3.1.2. Coulumb drag
In the inner disk and spiral arms, which are typically dominated by molecular gas with low fractional
ionization, we expect little deviation from the coupling
timescale due to only neutral gas drag. But when the
ionization fraction in the ISM approaches (or exceeds)
Xe = 10−4 , such as expected in the CNM and WNM
traced by atomic gas, then the Coloumb drag experienced by the dust becomes comparable to the neutral
gas drag. For dust grains with charge Ze and electrostatic potential U the coupling coefficient increases by
an amount
2 " 1/2 #
∆A
3
1
eU
kT
kT
ln
= 1 + Xe
A
2
kT
2ae
eU
πne
(25)
where A is the coupling coefficient due only to neutral
gas drag, T is the gas temperature and ne is the electron
number density (Draine 2003).
At low molecular-to-atomic gas ratios, such as in the
low-density inter-arm and outer disk regions, the higher
Xe increases the Coulomb drag. This acts to offset the
reduced neutral gas drag characteristic at these lower
gas densities, keeping the coupling timescale roughly
fixed to the value typical at higher density.
0.4
3.1.3. Trends with grain size
Figure 1 illustrates the grain-size dependence of the
ratio of the spiral interaction timescale to the dust-gas
coupling timescale in cold dense molecular gas (assuming constant gas surface density and negligible Xe ).
The smallest grains a<0.1µm (quickly) reach similar
speeds to the gas so that a strong coupling between the
dust and gas is maintained.3 Variation in the DGR
due to these grains is insignificant, well below 10% of
the unperturbed ratio. But the velocity differential
can be quite large for larger grains, for which A−1 decreases (relative to small grains) and becomes comparable to (kvs )−1 . In galaxies with stellar-to-gas mass
surface density ratios fsg =10 and moderately strong
perturbations for which Σg1 /Σg0 =0.4, eq. (23) implies
vdrif t =0.1vgr for grains with sizes a≈ 0.1 µm. Given
typical streaming velocities 10-15 km s−1 , we therefore
expect drift velocities on the order of 1 km s−1 .
Since roughly half of the dust mass resides in
grains with a<0.15µm and half is in grains with
0.15<a<0.5µm (Draine 2003), a significant reduction
in the DGR from its unperturbed value can be introduced in the presence of such large drift velocities. For
the largest of the plentiful grains Cdg >0.5, whereby the
DGR will be reduced by &30%, assuming Σg1 /Σg0 =0.4.
3.2. Impact of magnetic fields
Partial suppression of the dust drift is induced by
Lorentz forces exerted on charged dust by magnetic
3 A tight coupling between gas and dust is implied by their
similar scale heights and lengths (Hunt; Garcia-Burillo).
Cdg
disk case for extremely large grains a∼5 µm.)
fsg =10
0.3
0.2
0.1
0.0
0.0
fsg =1
0.1
0.2
0.3
a HΜmL
0.4
0.5
Fig. 1.— Variation of Cdg – the square of the ratio of the coupling and spiral-arm interaction timescales – with grain size a
for different stellar-to-gas mass ratios fsg from large (black) to
small (gray) assuming a constant Σg =20 M pc−2 , Qs /Q=2 and
a dust grain density appropriate for silicates. For large grains
(a>0.1 µm), the expected increase in the dust-gas drift velocity vdrif t = Cdg vg1 and corresponding reduction in the DGR
∝ Cdg Σg1 /Σg0 increases with grain size at fixed gas fraction.
Grains with sizes a<0.1µm remain very well coupled to the gas.
This scenario represents dust-gas coupling in molecular gas via
neutral gas drag with negligible Coulomb drag, or equivalently, to
dust-gas coupling in lower (by a factor of 2) density CNM/WNM
with Xe ∼ 10−4 .
fields. These forces act to suppress motion perpendicu
1/2
lar to the field B by an amount 1 + (ωB τA )2
where
the neutral gas drag timescale τA = A−1 and the cyclotron frequency ωB =<Z> eB/md c for grains of mass
md and charge <Z>e. (Draine 2003; Weingartner &
Draine 2001). In molecular gas with n=300 cm−3 and
T=25 K, the cyclotron period 2π/ωB is about 0.1τA for
grains with sizes 0.1 µm in a 3µG magnetic field. Consequently, a factor of 10 reduction in the component of the
drift motion perpendicular to the field can be expected
in this case. For slightly larger grains (a=0.3 µm) this
reduction is only a factor 2.5. In denser, colder gas
the suppression is less and smaller still for grains with
little (or no) charge, as for the largest of the grains.4
The drift velocity will also be smaller in a completely
isotropic field, for which drift is reduced by one-third
of the amount due to a regular (or ordered) field of the
same total magnitude.
In nearby galaxies, polarization of radio continuum
emission suggests that the magnetic field is primarily
regular between spiral arms but contains a larger, dominant isotropic component in the arm (Beck et al. 1994).
We therefore expect at most a factor of ∼3, and as little
as ∼ 20%, reduction in each the radial and azimuthal
drift velocities in the spiral arms for grains with sizes
0.1-0.3 µm, given a total isotropic field of 3 µG. In
the lower-density inter-arm, however, radial drift motions may be preferentially suppressed. This may be
4 At fixed charge, larger grains experience less suppression of
their drift motions than the smallest grains. But since large grains
can acquire large positive charge in warmer gas with higher electron density (Weingartner & Draine 2001), their drift may still be
efficiently suppressed, particularly in the radial direction in the
inter-arm.
6
0.5
model Sg1 Sg0
nearly complete, since ωB τA >>1 at the low gas densities and high temperatures characteristic of the CNM
and WNM.
Depending on the nature, orientation and strength
of the magnetic field, the Lorentz forces that lead to
partial or complete suppression of the dust radial drift
(depending on the grain properties) will also decrease
the magnitude of the perturbation to the DGR due to
dynamical effects, alone. Again, though, for large grains
(0.1<a<0.5 µm, which contribute roughly half of the
dust mass) in molecular spiral arms, this effect should
be at or below µm 20%.
Variation from arm to interarm will exist as long as
the coupling between gas and dust remains modest;
with a short coupling timescale, the dust response becomes identical to that of the gas, and the two ISM components trace spirals with similar surface density contrasts. As discussed in § 3.1, the largest grains, which
couple less effectively to the gas than smaller grains,
will exhibit the strongest variation from arm to interarm relative to the gas distribution. This introduces
a measurable difference in the dust-to-gas ratio when
these large grains contribute to the bulk of the dust
mass.
4.2. Radial variation in the DGR: the corotation dust
surplus
In the previous section we showed that the dust-to-gas
ratio is reduced in spiral arms as a result of the non-zero
drift velocity between the two ISM components (i.e. the
perturbed velocities of the dust and gas are different).
Only at the corotation radius will the drift velocity go
to zero, given that the velocities of the two components
DGRDGR0 -1
vg1
0.3
0.2
CR1
0
1
2
3
4
Radius HkpcL
0.0
a=0.05
-0.2
a=0.15
5
6
5
6
5
6
-0.4
CR1
-0.6
-0.8
0
1
0.0
DGRDGR0 -1
CR2
0.1
0.0
4. MODEL PREDICTIONS
4.1. Azimuthal variation in the DGR: the spiral arm
dust deficit
In the presence of a non-axisymmetric perturbation
and modest dust-gas coupling, the slightly different natures of gas and dust are exposed; whereas gas is dissipative, dust is pressure-less and responds to the potential in a manner more similar to (collisionless) stars
than the gas. Like stars, the dust distribution is therefore characterized by less arm-interarm contrast than
the gas, leading to a recognizable pattern of azimuthal
variation in the dust-to-gas ratio.
This is clear when we consider that, at the peak in
radial gas motions (i.e. in the spiral arm), the dust velocity is lower than the gas velocity by an amount proportional to Cdg . According to eq. (17), this translates
to a smaller perturbed dust surface density (relative to
the unperturbed value) than exhibited by the gas. As
a result, the dust-to-gas ratio is reduced on the spiral
arm ridge, as parameterized by eq. (20).
In the interarm, the reverse is true and the perturbed
dust-to-gas ratio is higher than the unperturbed value.
The ratio of the arm and inter-arm DGRs is
dgrarm
Σg1
≈ 1 − 2Cdg
(26)
dgri−arm
Σg0
0.4
a=0.25Μm
CR2
2
3
4
Radius HkpcL
fsg =1
fsg =8
-0.1
fsg =9
-0.2
CR1
-0.3
CR2
fsg =10
-0.4
0
1
2
3
4
Radius HkpcL
Fig. 2.— Sketch of the radial dependence of the perturbed gas
surface density (top panel) and the DGR (bottom two panels)
expected in the presence of two spiral perturbations, with two
corotation radii CR1 and CR2 , at the location of the spiral arms.
The ratio of the perturbed to unperturbed gas surface densities
Σg1 /Σg0 =η is modeled as ηmax cos 2 [(x − x0 )/2d], which goes to
zero at positions CR1 and CR2 separated by distance d from the
mid-point at x0 , where the ratio is at its peak ηmax . The direction
of radial gas motions, which also go to zero at corotation, are
depicted by arrows. The fractional change to the DGR from its
unperturbed value adopting the model Σg1 /Σg0 is shown for a
range of drift velocities, from low vdrif t /vg1 (light gray) to high
vdrif t /vg1 (black), depending on the dust grain size and stellarto-gas mass surface density ratio fsg following § 3.1 (and see
Figure 1). In the middle panel, a constant fsg = 9 is assumed
and the grain size is allowed to vary, while the grain size is fixed
to 0.2 µm in the bottom panel and fsg varies from 1-10. In all
cases we assume a constant Σg =20 M pc−2 , Qs /Q=2 and a
dust grain density appropriate for silicates, ρd =3.5 g cm−2 . At
corotation, there is no change to the DGR from its unperturbed
value.
7
themselves go to zero there. At corotation, then, the
DGR remains identical to its unperturbed value DGR0
(see Figure 2).
Elsewhere along the arm, the DGR becomes progressively lower as the distance from corotation increases.
At the corotation radius the sign of radial streaming
motions switches: inside (outside) corotation gas is
driven radially inwards (outwards). This will tend to
evacuate the zone around corotation of material, leading
also to a lower Σg1 /Σg0 .5 But as a result of the smaller
(radial) velocity of the dust compared to the gas, when
both move away from corotation the gas leaves the dust
behind. This leads to a surplus of dust around corotation compared to the rest of the arm (which may become less recognizable with azimuthal-averaging) and a
bump in the radial profile of the DGR/DGR0 .
In the presence of multiple patterns, and thus multiple corotation radii, the profile of DGR will show more
than one bump (as depicted in Figure 2). The radial extent of a given DGR bump will be dictated by the length
of the unique spiral pattern, i.e. the distance from the
inner to the outer Lindblad resonance (ILR and OLR),
which is typically several kpc’s: for flat (and/or rising)
rotation curves, Lspiral ∼1-2 Vc,max /Ωp .
4.3. Other Predictions
4.3.1. Radial Migration of the dust
Based on their simulations with a realistic dust-gas
coupling timescale set by neutral gas drag, Vorobyova
& Shchekino (2006) predict that the dust and gas decouple at corotation, allowing the dust to migrate radially
outward. However, those simulations do not demonstrate whether it is only the dust (and not also the gas)
that is preferentially deposited at larger radii. As a natural consequence of torquing by the spiral pattern, both
gas and dust will be simultaneously driven radially outwards outside corotation, thus explaining the existence
of dust beyond the corotation radius. (Both gas and
dust from inside corotation can be transported beyond
the corotation radius.) In this case, we would expect no
change in the average dust-to-gas ratio at radii outside,
as compared to inside, corotation.
Our expression for the relation between the gas and
dust velocities, on the other hand, explicitly predicts
that the largest grains can be preferentially transported
outward. According to eq. (13), dust motions are the
reverse of the gas motions for grains with sizes
1/2
3Σg
Σg0
√ .
a&
(27)
Σg1
4ρd fsg 4π
Adopting ρd =3.5 g cm−2 , Σg0 /Σg1 = 0.4 Σg =20 M
pc−2 and stellar-to-gas mass surface density ratio fsg
f =10, this corresponds to grains with a>0.4 µm.
Large grains can therefore move outward even inside
corotation, where gas and small grains otherwise move
5 The (local) reduction in star formation expected in the case
of the corotation gas deficit is one of the most commonly used
morphological characteristics used to locate corotation (REF;
Elmegreen;).
inward. Big grains from large radii will also be driven to
smaller radii (toward corotation). But in the presence
of multiple patterns, such inward dust motions may be
inhibited while outward dust motions can extend over a
large radial range. When multiple patterns exist, each
transitioning to the next at, or near, the former’s corotation radius, disk material is always inside the corotation of a pattern. Consequently, gas motions (and those
of the smallest grains that are well-coupled to the gas)
remain radially inward throughout the total spatial extent of the combined patterns.6 The motions of large
grains, in contrast, will be continually radially outward.
In this way, the outermost gas disk can host dust
that was formed at, and transported from, smaller radii.
These grains would be preferentially larger than those in
the inner disk. Since we expect only a small fraction of
the total dust mass to be in the form of the very largest
grains, this is consistent much lower values of the dustto-gas ratio at the largest radii than inside R25 .
On the other hand, Lorentz forces (see § 3.2) may
act to prevent the radial migration of large grains,
given that they acquire more charge than smaller grains
(Draine 2003) and their radial drift motions are thus expected to be most strongly suppressed. Whether large
grains are prevented from efficiently moving outwards
would then depend on the nature and orientation of the
field in the spiral arm and interarm.
4.3.2. Complexity of 2D DGR structure
According to eq. (11), the dust velocity includes an
additional term at a phase π/2 from the spiral arm
peak. While this will tend to be small, as the gas velocity is itself expected to be negligible at this azimuth, this
may introduce a noticeable phase offset between peaks
in the gas and dust distributions. The DGR distribution may therefore also exhibit structure beyond the
pure arm-interarm variations described in § 4.1. This
might be consistent with some degree of the complexity
in the DGR found in the numerical simulations of Theis
& Orlova (2006).
5. APPLICATION TO NEARBY SPIRAL
GALAXIES
5.1. Overview
In the previous section we demonstrated how nonaxisymmetric perturbations introduce a non-zero drift
velocity between gas and dust. In the presence of
this drift velocity, the dust-to-gas ratio varies both azimuthally and radially. The size of the deviation from
the unperturbed DGR is governed primarily by the location and strength of the spiral perturbation7 , the geometry of the spiral arms (multiplicity and pitch angle; set by the stellar-to-gas mass surface density ratio),
and the size of the dust grains. Stronger, more tightly
6 This scenario is expected to be necessary in order to efficiently move gas from larger radii to small radii, thereby making
it accessable for star formation and for fueling central activity.
7 The strength of the spiral perturbation dictates the magnitude of the streaming motions, and both can be inferred from the
ratio of the perturbed to unperturbed gas surface densities.
8
wound spirals in more massive stellar disks lead to more
variation in the DGR from its unperturbed value.
The location of maximum deviation along a given
spiral is governed by distance from its corotation radius; smaller radial motions nearer to corotation prevent significant deviation from the unperturbed value
of the DGR. Higher streaming motions elsewhere along
the arm reduce the DGR everywhere except in the
vicinity of CR, introducing a bump in the profile of
DGR/DGR0 . In the presence of multiple patterns, the
radial profile of the DGR will exhibit multiple bumps.
These may become progressively smaller with each successive pattern, according to the roughly inverse dependence of k on radius R (e.g. kvs ∝Ω).
The latest observations suggest tentative agreement
with these predictions. Radial profiles of the DGR
in SINGS galaxies measured by Munoz-Mateos et al.
(2009) are indeed characterized by non-monotonic radial variation. Moreover, the variation in these profiles
tends to become less prominent in later Hubble Types
(with weaker spirals). In the radial profile of the DGR
in M100 (Pohlen et al. 2010), three localized bumps
appear at radii remarkably near the corotation radii estimated for each of the three main structures (the bar,
spiral, and nuclear bar; Hernandez et al. 2005)
However, reliably asserting that such features are dynamical in origin is complicated by the fact that there
are several other factors that can influence the DGR, including observational artifacts. We note that the effect
of variations in the dust-gas coupling should be manifest
on a more local, rather than global level. The nature of
the smooth radial decline characteristic of DGR profiles
is likely less impacted by dynamics than large-scale gradients in the dust temperature or ISM metallicity (although dynamics may introduce a segregation of grain
sizes from small to large galactocentric radius).
5.2. Distinguishing from other influences on the DGR
Successfully relating variation in the DGR to the dynamics of gas and dust requires that the quantity Σd /Σg
measure as accurately as possible the true dust-to-gas
ratio. But both Σd , even when derived from state-of-the
art pixel-by-pixel SED modeling, and Σg , representing
the total ISM surface density, are subject to uncertainties.
5.2.1. Uncertainties in Σg
Over-or under-estimation in the CO-to-H2 conversion
factor XCO could conceivably introduce bumps in the
DGR such as predicted here. These could occur either
near the radius of the transition from primarily molecular to primarily atomic gas in the ISM, or as a result of
the dependence of XCO on metallicity, which has been
found to decrease with radius. However, there is little to
no indication that metallicity (and hence XCO ) varies
azimuthally (as such variation would be wiped away
on the order of an orbital period). This suggests that
metallicity is not likely responsible for azimuthal variation in the DGR. Even if opacity effects in the densest
(spiral arm) regions imply that CO intensity is not a
sufficient measure of the total molecular gas in these
zones (Bolatto et al. 2013), a low XCO there will lead
to overestimation of the DGR in the densest regions,
and thus a very different signature in the DGR than
predicted here. (This scenario would, however, lead to
the underestimation of the difference between arm and
inter-arm DGRs.)
Similarly, variations in the optical thickness of HI,
which we have assumed to be optically thin, are unlikely
to introduce the expected pattern of enhanced DGR
in the interarm relative to the arm: an increase in HI
opacity in the cold, densest spiral arm regions would
lead to underestimation of its mass and thus an increase
in the DGR in the arm compared to the interarm.
5.2.2. Uncertainties in Σd
Errors may also be introduced into the estimate of Σd
used here if the dust emissivity varies with ISM phase
(and gas surface density), as suggested by recent evidence (Martin et al. 2012). An increase in dust emissivity with increasing density would lead to an overestimation of the dust mass at the locations of the spiral
arms, in particular. But, if anything, the dust-to-gas
ratio in the spiral arms would be raised above its true
value, again leading to the opposite of the signature we
predict here (reduction in the DGR at the location of
the spiral arms).
On the other hand, because we are insensitive to cold
dust contained at the highest densities in giant molecular clouds at our current resolution (see Sandstrom et al.
2013), dust masses could be underestimated. If GMCs
in the spiral arm contain more high density material
than their counterparts in the inter-arm (or if there are
simply more GMCs per unit gas mass in the spiral arm),
then the DGR in the spiral arm would appear preferentially lowered in comparison. This scenario therefore
appears nearly indistinguishable from the dynamical effect considered here, except for the radial variation predicted for the DGR along the arm.
5.2.3. Additional factors
Several other factors can lead to genuine variation in
the DGR. Dust grains are predicted to be both created
and destroyed at locations of spiral arms. (The spiral
shock can lead to dust creation or destruction, while the
high UV radiation field associated with star formation
along the spiral arms can lead to dust destruction.) But
the timescales to add or remove significant quantities
of dust via these processes are expected to be on the
order of, or longer than, a dynamical time (∼100-200
Myr; Draine 2003; REFs). Localized rings of star
formation, and thus elevated UV radiation field, could
lead to localized dust destruction at radii very near
where we expect the largest reduction in the DGR; in
contrast to the corotation radius, the inner and outer
Lindblad resonances are often associated with the
pile-up of gas (and thus enhanced SF; see Elmegreen).
But we would expect this to introduce a much sharper
decrease in DGR/DGR0 at these locations, rather than
the gradual decrease away from corotation we predict.
9
Fig. 3.— From Aniano et al. (2012); to be replaced. Maps of the dust mass surface density derived at three different resolutions:
11” (from PACS 160 µm; left), 18” (SPIRE/250 µm; middle) and 40” (MIPS 160 µm; right) for NGC 628 (top) and NGC 6946 (bottom).
6. EVIDENCE FOR DYNAMICAL SIGNATURES
IN THE DGRs OF KINGFISH GALAXIES
In this section we compare the gas and dust distributions in 2 nearby galaxies from KINGFISH with our
predictions. According to eq. (24), to constrain the
impact of dynamics on the DGR=Σdust /Σgas requires
only indirect measures of the instantaneous gas/dust
kinematics, namely the ratio of the perturbed to unperturbed gas surface density Σg1 /Σg0 and the stellarto-gas mass surface density ratio.
6.1. the data
Our primary observables are the total (molecular plus
atomic) gas surface density Σgas =ΣHI +ΣH2 and the
dust mass surface density Σdust , measured and spatially
sampled in the manner employed by Sandstrom et al.
(2013). Maps of Σgas are constructed using the combination of CO (2-1) from HERACLES (Leroy et al.
2009a, Leroy et. al 2012) and HI from THINGS (Walter et al. 2008) adopting a Galactic CO-to-H2 conversion factor αCO =4.4 M pc−2 (K km s−1 )−1 and CO(21)/CO(1-0)=0.7 (see Leroy et al. 2008; Sandstrom et al.
2013). These maps are convolved (using the techniques
described by Aniano et al. 2011) to the same resolution
as the dust distribution Σdust measured by Aniano et
al. 2012, which is set by the resolution of the longest
wavelength data considered in the modelling of the dust
SED. For this work we adopt dust maps constructed
with observations out to SPIRE/250µm. These afford
the high spatial resolution suitable for discerning the
major non-axisymmetric structures present in the dust
and gas distributions, which are less prominent at the
low resolution of the 350 µm data. Although the dust
masses constructed with data out to 250 µm may be
biased high relative to those measured to 160 µm, they
are expected to be within 10% of those for which the
SPIRE 350 µm data is included (Aniano et al. 2012).
For a discussion of the role of uncertainty in Σdust and
Σgas on signatures of variation in the DGR see § 5.2.
6.2. the galaxies
We study two nearly face-on nearby star-forming
galaxies, NGC 628 (D=7.3 Mpc; SA(s)c) and NGC
6946 (D=5.9 Mpc; SAB(rs)cd). Both galaxies host
prominent two-armed spiral patterns and show evidence
for more complex (three-armed) structure (Elmegreen,
Elmegreen & Montenegro 1992; Korchagin et al. 2005).
Both also exhibit central non-axisymmetries in NIR images tracing the underlying stellar mass distribution
(and potential). The bar in the center of NGC 6946 extends out to 20”, followed by a weaker oval elongation
out to R∼100” (Elmegreen, Chromey & Santos 1998;
Regan & Vogel 1995). NGC 628 hosts a nuclear bar in
the inner 2” (Laine et al. 2002), as well as a central
oval distortion (weak bar) out to ∼43” (James & Seigar
1999).
10
Fig. 4.— From Aniano et al. (2012); to be replaced. Maps of the dust-to-gas ratio (DGR) calculated as the ratio of the dust
maps in Figure 3 and maps of the total gas surface density for NGC 628 (top) and NGC 6946 (bottom).
Estimates for the pattern speeds and corotation radii
of these non-axisymmetric structures, determined with
a variety of techniques, exist in the literature. These
are discussed in detail in Appendix C (and summarized
in Table 1) and later compared with features in the radial profiles of DGR/DGR0 in § 6.4. When only the
pattern speed estimate exists we locate the corresponding corotation radius through comparison with the angular frequency curves shown in Figure 6. These are
defined by the HI rotation curve fits made by Leroy et
al. (2008). Together with several morphology-based estimates for the corotation radii of the spiral structures,
we also add tentative estimates (and pattern speeds)
for the oval structures/weak bars, under the assumption that each must end at its corotation (Contopoulos
1980).
6.3. Notes on other assumed and measured quantities
We measure average values of Σgas , Σdust and
Σdust /Σgas in a series of radial bins spaced at the resolution of the data. For the unperturbed gas and dust
surface densities we assume the azimuthally-averaged
value (see the radial profiles in the top row of Figure
5). Perturbed quantities in each ring are measured as
the average value at the location where the gas surface density exceeds a (radially varying) threshold, assuming that the gas distribution traces the potential.
TABLE 1
Pattern speeds and corotation radii
NGC628
bar/oval
spiral
RCR
Ωp (km s−1 kpc−1 )
45”±7”
205”±24”
114+17
−8
30+4
−1
NGC6946
bar/oval
92”±17”
60±9
spiral
248”+10
26+4
−34
−1
Note. — All values have been adjusted to our
adopted distance. A detailed description can be found
in Appendix C.
(Although more sophisticated arm-based definitions are
possible, e.g., based on kinematics, this morphological
definition should be suitable at our resolution). The resulting maps of the non-axisymmetric structure in the
two galaxies are shown in Figure ??, adopting the azimuthal average as the surface density threshold at a
given radius.
To measure the size of the perturbation Σg1 , we subtract the unperturbed Σg0 from the perturbed Σg . The
radial profile of Σg1 /Σg0 along the arm is plotted in
middle row of Figure 5. According to eq. (17), Σg1 /Σg0
0.5
0.0
50
100
150
200 250
arcsecL
300
350
50
100
0.1
150
200 250
arcsecL
300
350
0.0
-0.1
-0.2
0
50
100
150 200 250
Radius HarcsecL
300
350
NGC 6946
2.0
1.5
1.0
0.5
0
0.6
Sg1 Sg0
0.60
0.5
0.4
0.3
0.2
0.1
0.0
0.20
log Sg0 , log Sd0 ‰500
NGC 628
1.0
DGRDGR0 -1
DGRDGR0 -1
Sg1 Sg0
log Sg0 , log Sd0 ‰500
11
100
200
300
arcsecL
400
100
200
300
arcsecL
400
100
200
300
Radius HarcsecL
400
0.4
0.2
0.0
0.20
0.1
0.0
-0.1
-0.2
-0.3
0
Fig. 5.— (Top) Radial profiles of the gas and dust surface densities in NGC 628 (left column) and NGC 6946 (right column).
Error bars represent the variance at each radius, which everywhere exceeds the random uncertainties in the data. (Middle) Ratio of
the perturbed to unperturbed gas surface density Σg1 /Σg0 . (Bottom) Fractional difference in the perturbed and unperturbed DGRs
(DGR/DGR0 -1). Vertical gray boxes in the bottom two panels indicate the locations of corotation radii. The dashed gray line in
the bottom panel represents a toy model for the expected pattern in the DGR relative to its unperturbed value adopting a realistic
dust grain size and locally representative values of fsg , Σg1 /Σg0 and Σg in each galaxy, calculated as the average of the value at the
minimum in the DGR with values in adjacent radial bins: for NGC 628 fsg =3.5, Σg = 14 M pc−2 , Σg1 /Σg0 =0.32 and a=0.24µm
(with Qs /Q=2) and for NGC 6946 fsg =8, Σg =17 M pc−2 , Σg1 /Σg0 =0.49 and a=0.15µm (with Qs /Q=2).
should go to zero at corotation, as it is proportional to
kvgr1 /m(Ω − Ωp ) and vgr1 goes to zero at corotation
(see eq.[7]). But because we a priori define the measurement region based on an excess in surface density,
our selection by definition consists of regions for which
Σg1 /Σg0 is non-zero.
We base our estimate of the stellar-to-gas mass ratio
on the total stellar, HI and H2 surface density profiles
estimated by Leroy et al. (2008). We expect a single, constant value of fsg =9±1 to be representative in
the case of NGC 6946 outside R∼50”. In NGC 628,
fsg =5.5 on average, but with a factor of ∼2 under(over-)estimation at radii R.100” (R&100”).
6.4. Measured vs. predicted dust-to-gas ratios
Two-dimensional maps of the DGR in NGC 628 and
NGC 6946 very clearly exhibit strong azimuthal variation. The DGR is lower in the spiral arms (at high
gas density) than in the inter-arm. This is more clear
in the bottom row of Figure 5, where we plot the fractional difference of the perturbed DGR (at high gas density, in the spiral arms) from the unperturbed value, i.e.
DGR/DGR0 -1. The difference is as large as 15% and
30% in NGC 628 and NGC 6946, respectively.
In agreement with our predictions, the maximum reduction occurs far from corotation. In both galaxies,
there are two corotation radii that each mark the location of little to no variation in the DGR. Only the corotation of the innermost (bar) structures (see Appendix
C) have no counterpart in the DGR profile, but these
are unresolved at the current resolution. Moreover, using average estimates of fsg , Σg and Σg1 /Σg0 appropriate for each galaxy, even our simple toy model can
reproduce the radial trend remarkably well. Both model
curves are consistent with the observations adopting realistic dust grain sizes a∼0.2 µm.
In principle, directly comparing the left and right
hand sides of eq. (20) at each radius would be the ideal
way to assess agreement with our predictions – and thus
test how well the overall radial behavior in the DGR
can be reproduced considering dynamical effects, alone.
However, this is complicated by uncertainty in several
observables, especially those related to the stellar disk.
The radial behavior of the stellar-to-gas mass surface
density ratio fsg is highly sensitive to the adopted stellar M/L (used to convert from observed stellar surface
12
brightness to Σs ) in addition to XCO (which specifies
Σgas ). The radial behavior of Qs (and Q) is also difficult
to constrain with certainty. Although globally we can
assert that both the gas, which is assumed to be star
forming, and the stars, which exhibit non-axisymmetric
structure, must be at or near the threshold for stability, i.e., Q∼1 and Qs ∼2, the stellar and gaseous velocity
dispersions (and Σs and Σgas ) must be securely known
in order to reliably estimate how Qs and Q vary with
radius. Dust grain sizes (and their possible radial variation) would also need to be well known (although it
should be possible to solve for the dust mass-weighted
grain size at each radius). A more detailed test is beyond the scope of this paper.
7. DISCUSSION: RELATION TO OTHER
OBSERVATIONS ??
Paradis et al. (2012; Hi-Gal): increased emissivity
in MW spiral arms and at large radius interpreted to
suggest changes in structure of grains in spiral arms.
8. SUMMARY AND CONCLUSIONS
APPENDIX
A. THE SPIRAL INTERACTION TIMESCALE
We can make a realistic estimate of the wavenumber k, and how it varies from galaxy to galaxy, using the fact
that the internal spiral perturbation hosted by any given galaxy disk is intimately linked to the properties of the
galaxy (i.e. galaxies are unstable to density waves with very specific properties).
Assuming that the pattern present in the stellar (and gaseous) density is responsible for the potential perturbation
(i.e. the pattern is a self-consistent density wave) then the two are related through Poisson’s equation
Φ1 =
2πGΣtot
1
|k|
stars
where Σtot
+ Σgas
1 = Σ1
1 . Now we can use eq. (17), with vgr1 replaced using eq. (7), to write
2κ Σtot
1
k 2 vs2 + kvs
+ κ2 − m2 (Ω − Ωp )2 = 0
Q Σgas
1
(A1)
(A2)
taking Q=vs κ/πGΣ to describe the stability of the gas disk and ignoring the (negligible) contribution of dust to
the mass budget. (Note that the equations in sections hold in the general case and describe the response of the
gas disk to any (internal or external) perturbation Φa . Here we impose the additional requirement that the density
perturbation is itself responsible for the potential perturbation.)
We now solve this quadratic equation for kvs , finding


s
gas 2 2 2
2
Σ
Q
m
(Ω
−
Ω
)
κ Σtot
p
1
1
1 ± 1 +
−1 
(A3)
kvs =
Q Σgas
Σtot
κ2
1
1
Considering the continuity equation obeyed by the stars in the tight-winding limit (see eq. [17]), the ratio of
the perturbations in the stellar and gaseous disks is roughly equivalent to the ratio of the unperturbed densities
stars
(Σstars
/Σgas
/Σgas
1
1 ≈Σ0
0 ; when the streaming velocities driven in response the potential are the same in both).
In nearby spiral galaxies the gas-to-stellar mass ratio8 , is typically low, ∼10-30% (see, e.g. Blanton; Leroy et al.
2008), i.e. fsg = Σstars
/Σgas
1
1 ≈3-10. In this case eq. (A3) reduces to
κ
(A4)
kvs = (1 + fsg ) ≈ fsg Ω
Q
taking Q≈ 1-2 and assuming a flat rotation curve for which κ=2Ω. The spiral arm interaction timescale (kvs )−1
will therefore be a fraction 1/fsg of the orbital period.
B. THE DUST-GAS COUPLING TIMESCALE
Here we evaluate the coupling timescale A−1 set by the dust-gas collision timescale when the dust and gas disks
have similar scale heights.
√
We assume a gaussian vertical distribution
√ for the gas, with mid-plane density ρg = Σg / 2πH and scale-height
H set by the stellar gravity, i.e. H = vs / 4πGρs with ρs the density of stars, in the limit of small gas-to-stellar
density. (Ignoring stellar gravity and instead considering only gas self-gravity, H ≈ vs Ω−1 , as assumed by Noh et
8 The ratio of the total mass in stars to the total mass in gas is roughly equivalent to the ratio of stellar and gas surface densities
when the gaseous and stellar disks have similar scalelengths.
140
120
100
80
60
40
20
0
0
50
100 150 200
Radius HarcsecL
250
300
angular frequency Hkm s-1 kpc-1 L
angular frequency Hkm s-1 kpc-1 L
13
100
80
60
40
20
0
0
50
100 150 200
Radius HarcsecL
250
300
Fig. 6.— Angular Frequency curves for NGC 628 and NGC 6924. In both panels, the solid black line shows the orbital angular
frequency Ω estimated from the rotation curve fit to HI velocities by Leroy et al. (2008), while the gray lines represent curves for
Ω±κ/2. The main corotation radii are indicated by gray bands. Red dashed lines show the average pattern speed associated with each
corotation.
al. 1991). Now writing the stability of the stellar disk
Qs =
σs κ
πGΣs
(B1)
then for an isothermal stellar disk with the surface density Σs = ρs 2Hs where Hs is the stellar scale height, the gas
scale height becomes
1/2
1 Qs
Qs 2Hs
(B2)
= vs √
H = vs
4σs κ
2 κ
assuming the stellar disk is in hydrostatic equilibrium, i.e. Hs = σs2 /πGΣs = σs Qs /κ.
In this case,
√
4 πρd aQs
A−1 =
3Σg κ
(B3)
with dust grain mass md =ρd (4π3)a3 (where ρd is the density and a is the radius of the dust grain) and dust
collisional cross section σ = πa2 .
C. COROTATION RADII AND PATTERN SPEEDS OF NON-AXISYMMETRIC STRUCTURES IN
NGC 628 AND NGC 6946
The non-axisymmetric (bar and spiral) structures in NGC 628 and NGC 6946 are some of the most well-studied
in the literature, and their pattern speeds and corotation radii have been estimated with a variety of techniques. In
the following sections we summarize the existing measurements and converge on the most likely corotation radius
for each of the independent structures in the two galaxies. We tend to prefer estimates derived from kinematics, as
the association of morphological features with resonances can be especially model-dependent along spiral arms in
particular (see below). In some cases, both morphologically- and kinematically-based techniques provide consistent
estimates, and as outlined below, we consider these to be more reliable than those with little additional supporting
evidence.
When only the pattern speed estimate exists (e.g., measured with the model-independent TW method) we locate
the corresponding corotation radius through comparison with the angular frequency curves shown in Figure 6. These
are defined by the HI rotation curve fits made by Leroy et al. (2008). Throughout, we adopt the galaxy distance
tabulated by Leroy et al. (2008).
NGC 628
The harmonic decomposition of projected non-circular motions in the HI velocity field (the line-of-sight residual
spiral
velocity field; see also Schoenmakers, Franx & de Zeeuw 1997) imply a corotation RCR
=206” (Sakhibov & Smirnov
2004) for the main spiral pattern. This value is consistent with the corotation at R=175” derived by Korchagin
14
et al. (2005) with a global modal approach, as well as the corotation at R=235” implied by the pattern speed
estimated by Tamburro et al. (2008) through the measurement of angular offsets in the gas and young stars along
spiral
the spiral arms. For the main spiral, then, we adopt RCR
=205”±24” (the average of three estimates), from which
−1
we estimate a pattern speed Ωspiral
=30+4
kpc−1 (see Figure 6).
p
−3 kms
For the nuclear bar and inner oval/weak bar structures we infer pattern speeds and corotation radii based on the
expectation that each must end at, or near, its corotation (according to stellar orbit theory; Contopoulos 1980).
Adopting the lengths estimated by Laine et al. (2002) and James & Seigar (1999), respectively, we estimate Ωbar1
∼
p
bar1
bar2
bar2
(with RCR =2”) and Ωp = (with RCR ∼45”±7”; conservatively assuming 15% uncertainty) from the angular
frequency curves in Figure 6. (Laine et al. 2000)
NGC 6946
Fathi et al. (2007) apply the model-independent method proposed by Tremaine & Weinberg (1984; hereafter
−1
theTW method) using ionized gas as their kinematic tracer. They find a dominant Ωspiral
=26+4
kpc−1 , for
p
−1 kms
spiral
which we estimate corotation RCR =248”+10
−34 with our adopted rotation curve. They also find a hint of a higher
−1
−1
pattern speed Ωp =50+4
kms
kpc
measured
from the central-most TW integrals, which they attribute to the
−2
inner bar structure. However, the higher pattern speed measured by Fathi et al. (2007) should be considered a lower
bound on the true pattern speed in this zone. Because the TW integrals correspond to an intensity-weighted average,
when they intersect with more than one pattern (such as a those that are centrally located and pass through a bar
and an outer spiral) the traditional TW method returns a weighted-average of all pattern speeds (Meidt et al. 2008).
A higher pattern speed Ωp =60 kms−1 kpc−1 has in fact been estimated by Peton (1982) from the analysis of central
bar
=92”±14” (again assuming 15% uncertainty), very near the
ionized gas kinematics. This places corotation at RCR
location where the inner oval structure is measured to end (∼100”±20”; Elmegreen, Chromey & Santos 1998; Regan
& Vogel 1995). At smaller radii, we estimate corotation from the end of inner bar, RCR =20” (Elmegreen et al.
1998) and thus a pattern speed Ωp ∼109 kms−1 kpc−1 .
The averaging of speeds measured with the TW method in the presence of more than one structure (and pattern
speed) is likely responsible for the intermediate spiral speed estimated by Rand & Wallin (2004; from lower resolution
data, and thus fewer independent TW integrals, than analyzed by Fathi et al. 2007). We therefore consider only
the former speeds for our work.
Other estimates
There is at least one morphology-based estimate for corotation in each galaxy supported by a coincidence with a
minimum (or truncation) in the global pattern of star formation: RCR =110” in NGC 628 (Cepa & Beckman 1990)
and RCR =154” in NGC 6946 (Elmegreen, Elmegreen & Montenegro 1992). This estimate for corotation is set apart
from those made with independent techniques that are generally in very good agreement with eachother (see Table
1). Rather than trace the gas-evacuated corotation circle, it therefore seems plausible that the reduced star formation
in these zones is the result of large streaming motions expected far from corotation, which can lead to a suppression
of star formation (Meidt et al. 2013). We therefore do not consider this type of estimate for corotation to be reliable.
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