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THE ASTROPHYSICAL JOURNAL SUPPLEMENT SERIES, 125:99È121, 1999 November
1999. The American Astronomical Society. All rights reserved. Printed in U.S.A.(
PRECISION VELOCITY FIELDS IN SPIRAL GALAXIES. I. NONCIRCULAR MOTIONS
AND RMS NOISE IN DISKS
CHARLES BEAUVAIS AND G. BOTHUN
Department of Physics, University of Oregon, Eugene, OR 97403; torch=jps.net, nuts=bigmoo.uoregon.edu
Received 1998 October 21; accepted 1999 June 3
ABSTRACT
Investigation of the symmetry of the major- and minor-axis rotation curves reveals strong evidence of
nonconcentric gas orbits with the maximum center shift of D300 pc. Comparisons between kinematic
and photometric structure (e.g., position angles, inclinations, centers) show considerable noise on small
scales. Although large-scale averages are in agreement, this noise is a matter of some concern in the
application of the Tully-Fisher method to disk galaxies. Moreover, cases of signiÐcant misalignment in
position angle between the inner and outer disks are seen in two of the sample galaxies and may indicate
the transition between luminous and dark-matterÈdominated regions (i.e., where the maximum-disk
hypothesis begins to fail).
The kinematic disk models are used to Ðnd the residual velocity Ðelds, and typical residuals are found
to be 10È15 km s~1 over regions 0.5È1.5 kpc in diameter. Correlations are shown to exist between the
residual velocity Ðelds and both the Haintensity and the velocity dispersion images. This suggests that
kinematic feedback to the gas from star formation is an important source of noncircular motion.
However, the relative quiescence of the large-scale velocity Ðeld indicates that the e†ect does not cause a
signiÐcant deviation from circular symmetry, kinematically indicating that star formation is not a hidden
parameter in the Tully-Fisher relation. Finally, the residual velocity Ðelds are examined for signs of non-
circular orbits by looking for azimuthal angular harmonics that would be present if disk galaxies are
embedded in a triaxial dark matter potential. For our sample we Ðnd the ellipticity of the gas orbits to
be which implies the potential is relatively round. This is consistent with disks being maximal.[0.08,
Subject headings: galaxies : kinematics and dynamics È techniques : interferometric
1.INTRODUCTION
The study of velocity Ðelds in spiral galaxies have been
traditionally done using 21 cm H Iemission as the tracer
feature. This allows the kinematics and dynamics of the gas
to be explored on large scales, typically at a resolution of a
few kiloparsecs. Early e†orts by Rots (1975), van der Hulst
(1979), Bosma (1981), and Gottesman (1982) showed the
power of these techniques in improving our physical under-
standing of disk galaxies and their implied mass distribu-
tions. The large-scale dynamics, as manifested by the
observed motion of the gas, also forms the basis for the
assertion that disk galaxies contain substantial amounts of
nonluminous material (e.g., Kent 1986). In addition to
probing the mass distribution, the large-scale behavior of
the gas is a highly useful diagnostic to compare the
observed radial and tangential streaming motions against
predictions of density wave theory (e.g., Yuan & Cheng
1989) and to determine critical components such as the
inner and outer Lindblad resonances as well as the corota-
tion radius (see Adler & Westphal 1996; Canzian & Allen
1997). On the other hand, the large-scale gaseous com-
ponent in some disk galaxies can show kinematic features
that suggest the complete absence of density waves
(Thornley & Mundy 1997).
Since the beam size associated with 21 cm interferometric
observations is usually at least as large as the disk scale
length of the target galaxy, very little small-scale structure
in the velocity Ðeld can be detected in this manner.
Improved spatial resolution can be achieved when using
CO as the tracer gas (e.g., Turner & Hurt 1992), but such
gas is usually strongly conÐned to the spiral arms and/or
nuclear regions and is not, therefore, a pervasive tracer of
disk galaxy kinematics. To achieve signiÐcantly better
spatial resolution requires a tracer gas that radiates at
optical wavelengths and instrumentation that allows this
tracer gas to be mapped in a two-dimensional fashion ana-
logous to 21 cm interferometry.
The best candidate for this is, of course, Haemission
associated with the general star formation rate in a galactic
disk. The two-dimensional kinematics of this gas can be
e†ectively mapped by using a Fabry-Perot (FP) detector
which can provide information on the kinematics of disk
galaxies on a spatial scale of a few arcseconds. Moreover,
since many late-type spirals have ubiquitous star formation,
Ha-emitting gas is a pervasive component of the optical
disk. FP observations at Haof external galaxies were Ðrst
performed photographically by Tully (1972) in his detailed
study of the kinematics and dynamics of M51 (see Tully
1974). Following the pioneering work of Tully, Deharveng
& Pellet (1975) performed a kinematical and dynamical
study of M31 using Haas the tracer gas, and Dubout et al.
(1976) made the Ðrst observations of M33. The Ðrst system-
atic studies of the velocity Ðelds of late-type spirals using an
FP detector are those of de Vaucouleurs & Pence (1980)
and Davoust & de Vaucouleurs (1980). Marcelin et al.
(1982) extended this technique to the more complex velocity
99
100 BEAUVAIS & BOTHUN Vol. 125
Ðelds that arise in interacting galaxies through their obser-
vations of NGC 5128. Williams, Caldwell, & Schommer
(1984) Ðrst used a CCD as the detector being the Fabry-
Perot in their study of the ionized hydrogen in M82.
To date, FP observations of spiral galaxies at Hahave
been used to construct high signal-to-noise ratio (S/N) rota-
tion curves for several late-type galaxies (e.g., Marcelin,
Boulesteix, & Georgelin 1985; Bonnarel et al. 1988 ; Pence,
Taylor, & Atherton 1990; Corradi et al. 1991 ; Cecil, Wilson,
& Tully 1992; Vogel et al. 1993 ; Veilleux, Cecil, & Bland-
Hawthorn 1995; Sicotte, Caignan, & Durand 1996 ; Ryder
et al. 1998; Jimenez-Vicente et al. 1999). Most of those
studies have concentrated only on speciÐc galaxies and not
samples. By far the largest single sample comes from
Schommer et al. (1993, hereafter SBWM), who observed
approximately 100 galaxies in the Hydra-Centaurus
regions. In addition to late-type spirals, FP observations
have been used to study the detailed velocity Ðeld in barred
spirals (e.g., Teuben et al. 1986; Pence et al. 1988 ; Schom-
mer et al. 1988; Duval et al. 1991 ; Regan et al. 1996).
Finally, strongly interacting galaxies are natural FP targets,
so that detailed comparisons can be done between obser-
vations and models of interacting galaxies (see Marcelin et
al. 1987; Unger et al. 1990 ; Caulet et al. 1992 ; Mihos &
Bothun 1997, 1998).
The resolution and overall data qualities of the pre-
viously cited observations show considerable variation. It is
the aim of the present study to improve upon this founda-
tion by performing FP observations with higher S/N and
velocity resolution than have been achieved before. Our
goal is to derive as precise a velocity Ðeld as possible. As
emphasized by Lyakhovich et al. (1997), recovery of the full
vector velocity Ðeld can result in a fairly precise determi-
nation of the mass distribution within that disk. Potentially,
this means that the small-scale mass distribution can be
compared with the light distribution to provide a more
e†ective constraint on the maximal disk hypothesis, cur-
rently a contentious issue (see Sackett et al. 1994; Sackett
1997; McGaugh & de Blok 1998 ; Courteau & Rix 1999).
Since the S/N of FP observations is limited only by Ha
surface brightness per pixel, in general there is no strong
radial degradation in the quality of the velocity Ðeld. That
is, the errors on the velocity measurements at less than 1
scale length are similar to those present at 3 scale lengths.
The major limiting factor is the Ðlling factor of Haemission
within the disk and the velocity resolution of the obser-
vations.
In this paper, we present the results of FP Haobser-
vations of seven spiral galaxies that have large disk volume
Ðlling factors of Haemission and velocities less than 4000
km s~1. For these galaxies we are able to determine preci-
sion velocity Ðelds (errors ¹15 km s~1) over most of the
optical extent of the disk on a subkiloparsec scale. We use
these data to study noncircular motions in spiral galaxies.
The presence of noncircular motions in spiral galaxies can
have a considerable impact on the shape of the rotation
curve and thus on the shape and amount of dark matter
inferred. In turn, this impacts one of the major assumptions
in the Tully-Fisher (TF) relation, namely, that disk galaxies
are circularly symmetric in both a photometric and a kine-
matic sense. To Ðrst order, the modes scatter observed in
the TF relation does imply that, on large scales, the velocity
Ðeld must be fairly quiet (see Tully & 1985). We areFouque
interested here in detecting possible systematic departures
from circular symmetry, as such departures could easily be
an important source of scatter in the TF relation (Franx &
de Zeeuw 1992; Bothun & McGaugh 1999). Currently,
there are very few available data that constrain the actual
small-scale rms noise in the velocity Ðeld, and the sample
discussed here will help to overcome this limitation.
There are many possible sources for the occurrence of
noncircular motions. The presence of a bar will cause local
deviations due to the elongated potential (e.g., Amram et al.
1992, 1996). Strong spiral structure can introduce shocks
and streaming motions in the gas (Marcelin et al. 1985).
Radiation pressure from young stars can drive gas motions,
and this kinematic feedback to the ISM can be directly
explored in these data, as the Hastrength is directly related
to the star formation rate. In addition, any perturbation in
the local mass density within the disk will produce a corre-
sponding perturbation in the orbits, so, in e†ect, the con-
struction of precision velocity Ðelds can reveal a lumpy
mass distribution if it exists. Finally, warping in the inner
disk would lead to noncoplanar gas orbits that would e†ec-
tively render the derived rotation curve meaningless.
The possibility of exploring all of the above-mentioned
e†ects via a single data set has motivated our FP study of
seven galaxies whose angular diameters roughly match the
Ðeld of view of the instrument. This paper reports the initial
results of that study. The bulk of this paper is related to
describing the data reduction and the use of the tilted ring
model to extract the velocity Ðeld from the FP data cube.
This is fully discussed in °2. In °3 we present the derived
velocity Ðelds and discuss evidence for noncircular motions
in these disks. We also show how the velocity Ðelds can be
reconstructed from modeling and how such models lead to
Ðrm constraints on the rms noise present in the overall disk
velocity Ðeld. Concluding remarks are presented in °4.
Future papers in this series will explore other aspects of this
rich data set, including the extraction of rotation curves and
a comparison between dynamical features and the local star
formation rate in the disk.
2.OBSERVATIONS AND DATA REDUCTION
Imaging Fabry-Perot data were taken for a number of
nearby late-type disk galaxies explicitly selected to have an
angular size well matched to the e†ective Ðeld of view of the
instrument (D5@). While we observed more galaxies that are
discussed here, our Ðnal sample of seven galaxies was selec-
ted on the basis of the overall quality of the data and is
listed in Table 1. The sample is certainly not complete but
hopefully is representative of late-type disk galaxies that
span a large range of disk mass. In Table 1, column (1) gives
the galaxy identiÐcation, column (2) gives the major and
minor axes, column (3) is column (4) is the heliocentricBT,
radial velocity, column (5) is the observed B[Rcolor (i.e.,
not corrected for foreground or internal reddening), while
column (6) gives the derived maximum circular velocity
(from Beauvais 1997). Finally, column (7) gives the linear
resolution of the data, in units of parsecs per arcsecond.
Distances to individual galaxies were obtained from either
SBWM or Bureau, Mould, & Staveley-Smith (1996). As the
seeing was approximately in the combined FP cube, the1A.5
last column shows that we are e†ectively sampling the
velocity Ðeld on the subkiloparsec scale.
The data were acquired with the 1.5 m telescope located
at CTIO on the nights of 1994 November 14È19. We used
the Rutgers Imaging Fabry-Perot in the wide-band mode
No. 1, 1999 PRECISION VELOCITY FIELDS. I. 101
TABLE 1
THE SAMPLE
a]bB
TVsys B[RV
max Scale
Galaxy Name (arcmin) (mag) (km s~1) (mag) (km s~1) (arcsec pc~1)
(1) (2) (3) (4) (5) (6) (7)
ESO 358ÈG63...... 4.7 ]1.3 12.60 1932 1.10 130 75
ESO 437ÈG30...... 3.1 ]0.6 13.65 3759 1.50 200 210
NGC 1292 .......... 3.0 ]1.3 12.80 1367 1.00 105 75
NGC 1406 .......... 3.8 ]0.8 12.40 1075 1.20 165 75
NGC 2417 .......... 2.8 ]1.9 12.80 3184 1.20 190 140
NGC 3463 .......... 1.5 ]0.7 13.75 3950 1.40 215 205
IC 2559 ............. 1.7 ]0.7 14.35 2974 1.35 140 130
(FWHM \2 ordering Ðlters from the standard CTIOA),
set, and their TEK1 1k CCD. The combination of telescope
and CCD gave a resolution of pixel~1. Between 15 and0A.65
30 frames were taken for each galaxy, depending on the
individual mass and inclination, to obtain full line-width
coverage. Each frame was a 300 s exposure and was visually
inspected in real time to determine when the end of the Ha
Ñux was reached in velocity space.
Calibration frames were taken at regular intervals to
correct for drift in both the zero point and the position of
the optical axis. Flat-Ðeld and bias images were taken at
each wavelength for processing. The individual frames were
debiased, trimmed, and Ñat-Ðelded in the usual way with
standard IRAF tasks, corrected for air mass [assuming an
extinction of 0.09 mag (air mass)~1], aligned to a common
center, and then assembled into data cubes. The reduction
of the data to cubes follows the procedure outlined in detail
by SBWM. For each pixel in a data cube, the z-column
represents the line proÐle of the gas imaged by that pixel
(once proper corrections are made for the position of the
optical axis in each frame). The line proÐles (Ñux as a func-
tion of wavelength or velocity) of each pixel were Ðtted to
recover four parameters: the continuum level (background),
the position of the peak (line-of-sight [LOS] velocity, vLOS),
the velocity dispersion (half-width), and the peak Hainten-
sity (above background). The parameters and their errors
were put into image maps for subsequent analysis.
2.1. T he Fit Algorithm
While a number of procedures exist for Ðtting the line
proÐles of a Fabry-Perot data cube, it remains somewhat
unclear what kind of proÐle shape should be used. The
procedure of SBWM Ðtted Gaussian line proÐles using a
least squares minimization procedure to identify the best Ðt.
The analysis employed here uses a more rigorous pro-
cedure. To begin with, a s2minimization routine was sub-
stituted for the least squares minimization routine. The new
routine was more stable, and the parameters were less sensi-
tive to the initial estimates. A good initial estimate of the
parameters was important for either Ðt algorithm to con-
verge, especially in areas of poor S/N, so a new method for
computing the estimates was devised. To remove spurious
Ðts, a series of cuts was introduced. A minimum Ñux level
was required, and pixels whose intensity fell below this were
discarded and a failure code output to the image maps. This
was adjusted for each galaxy, along with the smoothing
length (see below), to maximize the area of the galaxy with
reliable Ðts.
The only major modiÐcation to the raw data involved
data smoothing. In regions of low S/N it is desirable to
average the Ñux over some region in each frame of the cube.
The line proÐles obtained from the smoothed data are more
regular and have smaller errors, increasing the chance of a
successful Ðt to the proÐle. SBWM smooth on a 3 ]3 pixel
scale for pixels with low S/N, but use no smoothing other-
wise. Another standard code for Ðtting line proÐles
obtained from B. Weiner (1996, private communication)
uses several di†erent smoothing lengths, depending on the
value of S/N. Such a variable smoothing length allows the
best spatial resolution possible at each pixel while still guar-
anteeing that a Ðt is found if possible, but it also introduces
artiÐcial correlations between the various parameters.
To avoid these correlations, a constant smoothing length
was used for all pixels in each data cube. The smoothing
length was chosen to give the best spatial resolution for
each galaxy, while giving good coverage by successfully
Ðtting a large fraction of the low S/N pixels. For the galaxies
in the sample, the seeing (1A) was larger than the individual
pixel scale, which resulted in overlap between adjacent
pixels. The alignment process used in the assembly of the
data cubes also introduces a slight uncertainty in the posi-
tion of individual pixels in the data cube. For these reasons,
a minimum smoothing size of 3 ]3 pixels was chosen. This
is slightly larger than the averaging seeing in the Ðnal image
map. Smoothing every pixel allowed the use of rms devi-
ations rather than Poisson counting statistics for the error
estimates on the line proÐle. These were typically smaller
than the Poisson errors, although they exhibited the same
sort of dependence on the Ñux level, i.e., greatest in the
proÐle peaks and smaller along the sides of the peaks. The
smaller errors resulted in a better determination of the
parameters. It also e†ectively removed points contaminated
by cosmic-ray events from the Ðts by assigning them enor-
mous errors, obviating the need for a separate screening
procedure.
2.2. Optimizing the L ine ProÐles
The primary consideration in the reduction of Fabry-
Perot data lies in the choice of Ðtting function used to
model the line proÐle. Of interest here is the most robust
and accurate way of determining the total Ñux under the
line and its width. Issues of determining the line center are
generally immune to the choice of Ðtting function. In the
past, Gaussians have been the function of choice, since they
are easily calculated and provide a reasonable approx-
imation to the data. However, the Gaussian tends to under-
102 BEAUVAIS & BOTHUN Vol. 125
represent the data in regions of high intensity. This can
compromise the accuracy of the integrated Ñux under the
line (i.e., the intensity). Recently, Voigt functions have been
gaining in popularity (e.g., Mihos & Bothun 1997, 1998). As
the convolution of a Gaussian with a Lorentzian, the Voigt
function has two dispersion components allowing the width
to be adjusted separately (but not independently) in the
peak and the tails.
To compare the performance of the two functions, a large
number of proÐles (300]) were modeled, and for complete-
ness a Lorentzian was tried as well. Sample line proÐle Ðts
are shown in Figure 1. Listed for each of the Ðts are the s2,
Q(the s2probability function, indicating the goodness of
Ðt), and the LOS velocity. The s2values (and thus Q) were
found to be dominated by the continuum data points in the
line proÐle whose errors are small. Thus, even Ðts with large
FIG. 1.ÈSample line proÐles Ðtted with Gaussian (solid line), Lorentzian (dotted line), and Voigt (dashed line) functions. The value of s2,Q(goodness of Ðt),
and central (peak) velocity are indicated for each model.
No. 1, 1999 PRECISION VELOCITY FIELDS. I. 103
FIG. 1.ÈContinued
s2(and small Q) provide a good Ðt to the data. That said,
the Voigt proÐle gave the best Ðt. As noted earlier, the
Gaussian tended to be overly short both in the peak and at
its base. The Lorentzian was often too tall and too narrow,
allowing systematic shifts in caused by the asymmetryVLOS
in the size of the error bars across the peak. Although the
Voigt function gave the best Ðt to the data, the Gaussian
gave an identical estimate of in every case. Where theVLOS
Voigt function surpassed the Gaussian was in providing a
better estimate of the intensity and the integrated Ñux (area
under the line proÐle), thus providing more reliable Hapho-
tometry.
The added degree of freedom makes the Voigt function a
better Ðt to the data, but it also has a major drawback. The
interdependence of the dispersion components made it
impossible to unambiguously determine the two individ-
ually. This resulted in large errors on the dispersion, typi-
cally greater than 35% and often much higher, and the
errors on the intensity ran higher than 30% but often went
as high as 150%. Another difficulty with the Voigt function
104 BEAUVAIS & BOTHUN Vol. 125
was that for the majority of the proÐles the Ðt failed to
converge at all unless run interactively. One remedy for this,
employed by the standard codes that use Voigt functions to
model the line proÐles, is to use a four-parameter Voigt
function, holding the Lorentzian component of the disper-
sion Ðxed. Sample line proÐles Ðtted in this manner are
shown in Figure 2. This greatly increased the number of
proÐles successfully Ðtted, but it also removed most of the
advantage gained by using the Voigt function. Since the
Gaussian is a good model for low-intensity proÐles, the
Lorentzian component of the dispersion must be kept small,
making the two nearly identical for high-intensity proÐles.
Thus, the four-parameter Voigt function does not give any
better photometry than the simple Gaussian.
As a Ðnal check on the utility of the various functions, the
entire velocity Ðeld of E358 was Ðtted with Gaussian, Lor-
FIG. 2.ÈSample line proÐles Ðtted holding the Lorentzian component of the dispersion ÐxedÈGaussian (solid line), Lorentzian (dotted line), and
four-parameter Voigt (dashed line) functions. The value of s2,Q(goodness of Ðt), and central (peak) velocity are indicated for each model.
No. 1, 1999 PRECISION VELOCITY FIELDS. I. 105
FIG. 2.ÈContinued
entzian, and four- and Ðve-parameter Voigt functions. The
results are shown in Figure 3 and were as expected. The
Gaussian and four-parameter Voigt function gave essen-
tially identical velocity Ðelds. The full Ðve-parameter Voigt
function failed to converge or failed the (liberal) error cuts
for the majority of the line proÐles. And the Lorentzian
Ðtted a larger number of proÐles because of its tendency to
peak high and make the Ñux cut where the others fail. But
the small-scale structure is di†erent, as this is skewed by the
errors (as noted above). This foray into alternative Ðtting
functions then provides a cautionary note that one needs to
understand, in detail, the behavior of the Ðtting function in
order to avoid introducing spurious errors, on various size
scales, in the derived velocity Ðelds. Although a Gaussian
does underestimate the Ñux in the line, it seems to be the
most free of these kinds of errors and thus provides the most
robust measure of the velocity Ðeld, which is the principal
item of interest. Since the four-parameter Voigt function
does not provide a signiÐcantly better model of the line
proÐle shape and the full Ðve-parameter Voigt function
106 BEAUVAIS & BOTHUN Vol. 125
FIG. 3.ÈVelocity Ðelds for E358 derived from Gaussian (top left), Lorentzian (bottom left), and four- (top right) and Ðve-parameter Voigt (bottom right)
functions.This is a gray-scale image; the full color images are available in the online edition of the Astrophysical Journal.
must be run interactively, we conclude that Gaussians
should be used to Ðt Fabry-Perot line proÐles and that a
separate narrowband image should be taken for the most
precise measure of HaÑux.
2.3. Bimodal L ine ProÐles
In most cases, line proÐles were deÐned by a single peak.
However, the data from NGC 1406 exhibited bimodal line
proÐles. Secondary peaks are expected if the gasdynamics
becomes complicated when two separate dynamical struc-
tures are imaged by the same pixel. This can happen in
expanding shells or in tidal tails of interacting galaxies
(Mihos & Bothun 1997). Bimodal line proÐles can be well
modeled by summing two Gaussians as shown in Figure 4.
On closer examination, all line proÐles in NGC 1406 had a
second peak one-sixth as high as the primary peak and
separated by a distance of D600 km s~1. The regularity of
the separation and relative size of the two peaks over a large
range of intensities argues against gasdynamics being the
source of the second peak. Most likely the secondary peak is
the [N II]j6548. None of the other galaxies showed evi-
dence of bimodal line proÐles, perhaps indicating that [N II]
j6548 is unusually strong in this particular disk.
2.4. Photometric Imaging Data
To complement the FP data, we acquired, in photometric
weather, images of the three highest angular size galaxies
(E358, NGC 1292, and NGC 1406). V- and I-band images
were taken using the CTIO 0.9 m telescope and standard
Ðlter set. Again, the images were processed in the usual way
using standard IRAF tasks. The surface brightness proÐles
were obtained using the ellipse-Ðtting GASP software
package described by Cornell et al. (1987). The zero-point
calibrations were made using stars in Landolt standard
Ðelds (Landolt 1983). The surface brightness proÐles are
shown in Figure 5. These galaxies were a priori chosen to
have little or no bulge component (veriÐed by inspection of
Fig. 5), so that any possible deviations from circular orbits
should be driven entirely by the Ñattened dark matter halo
(if it exists) and not by any bulge (or bar).
2.5. Kinematic Modeling: T he T ilted-Ring Model
Since only line-of-sight velocities are observable, any
attempt at using kinematic data to probe galaxy structure
must deal with projection e†ects. The relevant coordinate
systems are the sky (detector) coordinates (x,y,D), with D
the distance from the galaxy to the observer, and a cylin-
No. 1, 1999 PRECISION VELOCITY FIELDS. I. 107
FIG. 4.ÈSample bimodal line proÐles Ðtted with a double Gaussian
drical coordinate system centered on the disk (R,/,z). They
are related by
(
t
:
t
t
x
y
D
)
t
;
t
t\
(
t
:
t
t
cos asin a0
[sin acos a0
001
)
t
;
t
t
(
t
:
t
t
10 0
0 cos isin i
0[sin icos i
)
t
;
t
t
]
(
t
:
t
t
Rcos /
Rsin /
z
)
t
;
t
t]
(
t
:
t
t
x0
y0
D0
)
t
;
t
t,
with ithe inclination of the disk to the LOS (for a face-on
disk i\0), athe position angle (P.A.) of the high velocity
end of the galaxy on the detector, and /measured from the
P.A. This gives an observed LOS velocity of
vLOS \vsys ]vc(R,/)cos/sin i
]vs(R,/) sin /sin i]vz(R,/) cos i,
where is the LOS systemic (recessional) velocity of thevsys
galaxy, is the circular velocity in the /-direction, is thevcvs
FIG. 5.ÈCalibrated surface brightness proÐles for E358, N1292, and N1406. Vband (black) and Iband (gray)
108 BEAUVAIS & BOTHUN
radial (streaming) velocity, and represents any infall/vz
outÑow motions.
The standard method for deprojecting galaxies from
kinematic data is to assume that the galaxy is made up of a
number of rotating solid body annuli, the tilted-ring model.
The velocity Ðeld is Ðtted with the equation vLOS \vsys
for Ðve free parameters: the center and]vccos /sin ix
0
the position angle (P.A.) aand inclination i, and they0,
rotational speed The systemic velocity, was heldvc.vsys,
Ðxed during the Ðt. The annular Ðts smooth both radially
and azimuthally, which minimizes the inÑuences of ran-
domly distributed noncircular motions. (Systematic devi-
ations from circular motions are more insidious and harder
to deal with and are discussed later.) In spite of the smooth-
ing, the radial distribution of recovered parameters showed
rapid Ñuctuations about more slowly changing values. This
was found to be an artifact of the Ðtting algorithm; chang-
ing the parameters changed which pixels were contained in
the annulus, causing a slight degeneracy in the parameters.
This was dealt with in two ways. First, rather than the
common least-squares minimization, the rms deviation was
minimized instead. This encouraged the Ðtted algorithm to
choose a better balance between a small least-squares value
and keeping a large number of pixels in each annulus, since
the least-squares value is minimal (zero) when there are no
pixels in the annulus. This improved the performance,
though there was still a degeneracy near the minimum rms
value. To remove the remaining Ñuctuations, the parameter
distributions were lightly smoothed (on a smaller scale than
the annular width) after the Ðt was complete. Near the edge
of the velocity Ðelds, the data become sparse and the Ðts
eventually failed, leading to a maximum radius to which the
tilted-ring model could be extended, so they were truncated
at that point. The Ðnal step of the procedure was to project
the annuli back onto the sky and obtain a predicted value
for each point, giving model images which could bevLOSat
compared directly to the velocity Ðelds of galaxies. The gen-
eration of these model images will later allow us to add
random noise to compare to real images in order to assess
the overall amplitude of rms disturbances in the disk veloc-
ity Ðeld.
2.6. T he Pre-Fit
Since the systemic velocity represents the bulk motion of
a galaxy, it should not vary between annuli, and there are
theoretical reasons to expect gas orbits to be concentric so
that the center should not be allowed to vary either. SBWM
allow both to vary in their kinematic disk Ðts, which would
be acceptable if the parameters only varied a little. For
some galaxies, however, the di†erence in systemic velocity
between annuli can be as high as 30 km s~1. This large a
shift in seems artiÐcial but can have a large impact onvsys
the other parameters, especially the position of the center
which is directly linked to On the other hand, we arevsys.
using emission lines as a diagnostic, and it is well known
that emission lines are a poor tracer of the systematic veloc-
ity. Hence, shifts between annulus could occur if say, kine-
matic feedback to the gas from star formation, has caused a
signiÐcant noncircular motion at some radius. At some level
it is difficult to assess the reality of these apparent shifts and
whether or not they are mostly due to the Ðtting procedure
or are a real asymmetry in the velocity distribution of the
emission line gas.
TABLE 2
PRE-FIT VALUES FROM TEST VF
xcycvsys
Level (arcsec) (arcsec) (km s~1)
(1) (2) (3) (4)
0 ....... 110.5 109.0 1369
5 ....... 109.9 108.9 1371
10...... 109.7 108.9 1372
15...... 109.4 108.5 1373
20...... 109.3 108.4 1374
To determine the systemic velocity and the position of the
center, the entire velocity Ðeld was Ðtted as a single solid-
body disk with the geometric parameters a,i)(x0,y0,
allowed to vary. This method was chosen because in many
galaxies the gas is distributed asymmetrically. Performing
an initial Ðt and taking the average was rejected becausevsys
certain pixels are in multiple annuli, making it difficult to
obtain a proper weighting. The single-disk method Ðnds the
point about which the cos /term is minimized for the
entire velocity Ðeld. This should give a good estimate. It
also insures the proper connection between the choice of
center and the systemic velocity.
The pre-Ðt procedure was tested on the same artiÐcial
velocity Ðelds used to test the tilted-ring model (see next
section). Small amounts (from 5 to 20 km s~1) of velocity
structure were added in the artiÐcial velocity Ðeld
(generated using NGC 1292 as a model) at various radial
locations to determine the extent to which this changed the
geometric center of the ring and/or the systemic velocity
(see Fig. 6). Table 2 shows that the added structure had little
e†ect on the center estimate and the systemic velocity.
Another check of the pre-Ðt procedure is to compare the
systemic velocities obtained to the accepted values of the
heliocentric velocities of the galaxies in the sample, taken
from the NASA Extragalactic Database. This comparison is
given in Table 3. These systemic velocities are generally
from 21 cm measurements but are sometimes absorption
(opt) or emission (Ha) line measurements. This is so noted
in column (3) of Table 3. We estimate that the systemic
velocities derived from the tilted-ring model method are
accurate to 5È15 km s~1, and hence the agreement with the
other data is reasonable. The only case which is cause for
some concern is that of NGC 3463. Giovanelli et al. (1997)
measure a 21 cm velocity of 3989 ^5 while the NED veloc-
ity, based on Hameasurements is 3950 ^10. Our value of
3917 ^15 km s~1 is signiÐcantly lower. In this case, its very
likely that the emission line velocity is not acting as a good
tracer of the systemic velocity. Overall, however, our results
TABLE 3
VELOCITY COMPARISON
Vhel vsys
Galaxy Name (km s~1) (km s~1)
(1) (2) (3)
ESO 358ÈG63...... 1932 1950
ESO 437ÈG30...... 3759 3732
NGC 1292 .......... 1367 1368
NGC 1406 .......... 1075 1066
NGC 2417 .......... 3184 3175
NGC 3463 .......... 3950 3917
IC 2559 ............. 2974 2955
FIG. 6.ÈDisk parameters recovered from test velocity Ðelds. (Top) Parameters before Ðnal smoothing (light gray), after Ðnal smoothing (dark gray), and
recovered from test velocity Ðeld with no structure (black). (Bottom) Parameters recovered from test velocity Ðeld with 10 (black), 15 (dark gray), and 20 km
s~1 (light gray) rms structure levels. Linear units measured in arcseconds and angles measured in degrees.
110 BEAUVAIS & BOTHUN Vol. 125
are consistent with those of SBWM, who also Ðnd similar
good agreement between 21 cm systemic velocities and
those obtained via the tilted-ring model.
3.TWO DIMENSIONAL VELOCITY FIELDS IN SPIRAL
GALAXIES
3.1. Testing the T ilted-Ring Model
The tilted-ring model can be used to construct a velocity
Ðeld from which a rotation curve can be extracted. In addi-
tion, the tilted-ring model provides the mechanism to detect
possible noncircular motions in these disks. But how well
can the method work? To determine the ability of the tilted-
ring model to accurately recover the structure of galactic
disks from real velocity Ðelds, an artiÐcial velocity Ðeld was
generated for a disk with known parameters. To create
another test velocity Ðeld with parameters that mimic those
found in real disks, N1292 was modeled, and the resulting
parameters used to build the test velocity Ðeld. The test
velocity Ðeld was Ðtted with a tilted-ring model, and the
parameters are shown in Figure 6. They are nearly identical
to those used in the construction of the test velocity Ðeld but
give an indication of the radial smoothing due to the Ðnite
annular widths.
Real velocity Ðelds contain noncircular motions which
were removed from the test velocity Ðeld by the smoothing
in the tilted-ring model procedure. To examine the e†ects
noncircular motions have on the parameter determinations,
they were simulated by producing an image of Gaussian
noise which was smoothed until its structure resembled that
seen in the real velocity Ðeld of N1292. This was then scaled
to rms levels of 5, 10, and 15 km s~1 and added to the test
velocity Ðeld. The resulting images were then modeled in
turn, and the recovered parameters are shown in Figure 7.
These generally followed those obtained from the test veloc-
ity Ðeld without the added small-scale structure, but the size
of the deviations was found to scale with the size of the
structure. The noncircular motions thus have a small but
noticeable inÑuence on the derived parameters, typically 2A
on the center and 3¡ on the orientation angles. This e†ect is
most evident at large radii (Rº50@@) where the outer edge
of the annulus passes the edge of the test velocity ÐeldÏs disk.
In real galaxies the data normally become increasingly
sparse with radius, so this truncation e†ect should not be
apparent. Visual comparison of the test velocity Ðeld images
shown in Figure 7 indicates that the noncircular motions
in N1292 are D10 km s~1, typical of the galaxies studied
(see °4).
FIG. 7.ÈTest velocity Ðelds. Shown are (left to right,top to bottom) the real velocity Ðeld of N1292, and the test velocity Ðelds with small-scale structure of
5, 10, and 15 km s~1. This is a gray-scale image; the full color images are available in the online edition of the Astrophysical Journal.
No. 1, 1999 PRECISION VELOCITY FIELDS. I. 111
3.2. Centering and Inclination Issues
The measurement of the disk galaxy inclination, however,
can be a tricky procedure. SBWM have called attention to
the rather large scatter that results in the comparison of
photometric versus kinematically determined inclinations
(see also Sakai et al. 1999). This is a matter of some concern
in the practical application of the TF relation. In some
cases, the scatter is caused by a slight di†erence between the
photometric and kinematic centers of the galaxies. This
e†ect could even be real if the light distribution does not
exactly trace the mass distribution in the inner parts of disk
galaxies. The submaximal disk hypothesis of Courteau and
Rix would suggest that such a center mismatch could occur.
For E358, N1292, and N1406 in our sample it is possible to
obtain a photometric estimate of the center. The photo-
metric center of a galaxy is taken as the average center at
large radii. While photometric data extend much past the
optical kinematic data, and so may be susceptible to warps,
etc., the kinematic and photometric estimates are nearly
identical. Only for E358 is the di†erence appreciableÈ5A.5
(\410 pc). This made a 12 km s~1 lowering of the systemic
velocity necessary when using the photometric center in the
tilted-ring model, which places it farther from the rather
accurately determined published heliocentric velocity.
While the center is expected to be constant, during the
testing of the tilted-ring model process it was allowed to
vary. This was done for two reasons. The Ðrst is that allow-
ing the center to vary resulted in more complicated param-
eter sets and thus provided a more rigorous test of the
tilted-ring model. The second was to compare the kinematic
parameters to those found during the ellipse Ðtting in the
photometric reductions. The kinematic and photometric
parameters are plotted in Figure 8. While the agreement is
not spectacular, the parameters share many of the same
trends and possess similar features. On large scales, this
indicates a relatively tight connection between the distribu-
tions of light and mass. The small-scale noise, however, may
represent features in the light distribution that are able to
generate local noncircular motions. This is one of the
primary results we are interested in obtaining, and it will be
more fully discussed in °4. For now, Figure 8 should be
viewed as an affirmation of the more global results obtained
by SBWM that the correlation between the photometric
and kinematic structure of disk galaxies is noisy. Inspection
of Figure 8 shows that the inclination proÐles are fairly
noisy and often have ““ features ÏÏ in them. These features
may be due to the inÑuences of noncircular motions at
certain radii (e.g., streaming motions). At the very least, the
inclination proÐles, coupled with the global comparison of
SWBM strongly suggests that inclination measures of disk
galaxies are not particularly robust unless they are very
edge-on.
In the case of NGC 1292, there seems to be poor align-
ment between the kinematic and photometric determi-
nations of P.A. in the inner regions of the galaxy. In
particular, there is a mean di†erence of approximately 10¡
in the inner 30A. The reason for this misalignment is not at
all clear but it again does raise the issue of what is the best
tracer of the P.A. (and inclination) of a disk. Certainly, the
emission line velocity Ðeld is measuring a much di†erent
component of the stellar population than I-band photo-
metry. The relatively good agreement between the V- and
I-band observations, compared to the kinematic data, to
Ðrst order suggest that the underlying stellar population
deÐnes a di†erent P.A. than the emission line gas that is
ionized by the very youngest population. The small mis-
alignment observed here could result from (a) inadequate
sampling of the velocity Ðeld in the inner regions (this seems
unlikely), (b) kinematic feedback from star formation which
has disturbed the emission line gas, or (c) an asymmetric
distribution of star formation (most likely). This suggests
that photometric tracers provide more reliable indications
of disk P.A. and inclination.
Choice of center, however, appears to be far less problem-
atical than the determination of the inclination. Recall that
the center of the velocity Ðeld can be chosen either by using
the center of symmetry of the tilted rings (the kinematic
center) or by using the optical isophotes (the photometric
center). In addition, we have also allowed the center to
become a Ðtting parameter of the tilted-ring model (the
variable center). All three center estimates (kinematic,
photometric, and variable) gave tilted-ring modelÏs that
reproduce the observed velocity Ðelds equally well. No
simple statistical test showed one model as better than the
others, since the true rotational structure is masked by the
noncircular motions. Thus, models were created using all
three and the radial proÐles of P.A., inclination, and Xand
Ypixel centroid are shown in Figure 9. Overall, the choice
of center had little e†ect on the P.A. or the inclination, so
the structure exhibited by them is likely to be real. In the
variable center model, the centers show shifts of D300 pc.
As discussed in Paper II, this shift has a signiÐcant impact
on the robustness of the extracted rotation curve. Many of
the galaxies show the largest center shift at large R, which
may indicate a physical misalignment of the inner and outer
disk rotation axes. Two galaxies, N1292 and E437, show
signs of signiÐcant misalignment between inner and outer
disk position angles.
3.3. T he Velocity Fields
The velocity Ðelds that have been derived for our sample
galaxies from Fabry-Perot data cube line proÐles Ðtted with
Gaussians are shown in Figure 10. As the velocity Ðelds
predicted by the models with the di†erent centers are all
similar in appearance, only the variable-center models are
shown here. Of course, velocity Ðelds presented in gray scale
have limited information and we cannot a†ord to publish
color images. We encourage the reader to see the color
images available in the online edition of the Astrophysical
Journal.
In Figure 10, all aspects of the data and the data
reduction are shown. The top row shows the continuum,
intensity and dispersion maps which we do not generally
make use of in this paper. The bottom row shows the veloc-
ity Ðeld, the model velocity Ðeld which is reconstructed from
the tilted-ring model Ðts, and the residual velocity map
obtained by comparing the model with the data. It is with
this residual map that we can estimate the rms noise in the
velocity Ðeld in a given galaxy. We discuss this in detail
below. Here we give some brief comments on the individual
galaxies in our sample. The reader may wish to conÐrm
these through an examination of the color images available
in the online edition of the Astrophysical Journal.
E358.ÈThis fairly inclined disk galaxy is quite well
sampled at Ha. The intensity map reveals signiÐcant star
formation in the inner regions (including the nucleus) and
more star formation on the leading side than the trailing
FIG. 8.ÈKinematic and photometric disk parameters for E358, N1292, and N1406. Kinematic parameters (black), and Vband (dark gray) and Iband
(light gray) photometric parameters. Linear units measured in arcseconds and angles in degrees.
PRECISION VELOCITY FIELDS. I. 113
FIG. 8.ÈContinued
side. The velocity Ðeld is very well sampled, and a detached
region of Haemission can be clearly seen in both the inten-
sity and velocity maps.
N1292.ÈThis galaxy is extraordinary in the amount of
di†use Hathat is present (analyzed in more detail in Paper
III). There are only a few prominent H II region complexes
and the nucleus is not particularly strong. As a result of the
distributed di†use Ha, the velocity Ðeld is very well
sampled.
N1406.ÈThis is also a fairly inclined disk. The star for-
mation activity is primarily limited to the inner disk and
would appear to deÐne two spiral arms. The trailing side of
the galaxy exhibits a discontinuity in the velocity Ðeld
coverage due to a break in Haemission from the disk.
E437.ÈThis fairly inclined disk has relatively spotty
coverage in the outer parts of both ends of the disk. The
bulk of the Haemission is conÐned to large complexes in
the inner disk and on the leading edge. The trailing edge has
no large Haemitting complexes except for one at the very
end of the disk.
N2417.ÈThis late-type spiral clearly shows Haemission
strongly conÐned to multiple spiral arms. Unlike the case of
NGC 1292, there is not a pervasive di†use component to
the Haemission. As a result, the velocity Ðeld is relatively
sparsely sampled and there is little Haemitting gas at the
velocity endpoints of the rotation curve. Comparison of the
velocity Ðeld image with the continuum image immediately
shows some of the di†erence that exist between photometric
and kinematic inclinations and position angles.
N3463.ÈThis is the only galaxy in the sample of seven
that shows evidence for a ringlike structure of star forma-
tion. A very bright Haemitting region can be seen to the
North of the nucleus. The nucleus itself has no Haemission.
This somewhat asymmetric emission line structure may
therefore account for the order 50 km s~1 di†erence
between the Haand 21 cm systemic velocities. Like E358, a
detached region of star formation is present in both the
intensity and velocity maps. The velocity Ðeld is very well
sampled in this case, except for a hole in the southern
(leading) end of the disk.
IC 2559.ÈThis relatively small galaxy has an unusual
distribution of star formation which is heavily conÐned to
the inner regions. The overall distribution of star formation
is fairly asymmetric but the velocity Ðeld appears quite
regular and well sampled.
3.4. Residual Velocity Fields
While rotation accounts for the bulk of the motions in
disk galaxies, noncircular motions are present as well. The
residual velocity is the di†erence between the observed LOS
velocity and that expected due to rotation. Thus, the
residual velocity is positive where the LOS velocity exceeds
the predicted value and negative where it falls short. Maps
of the residual velocities, often called residual velocity Ðelds,
were obtained for each galaxy by subtracting the model
velocity Ðeld from the true velocity Ðeld and are shown in
the bottom right panel for each galaxy in Figure 10.
Because the residual velocity Ðeld depends on the predicted
rotation, careful construction of the tilted-ring model is
crucial and we have again used the variable-center model to
produce the residual velocity Ðeld.
Earlier we displayed the test model velocity Ðeld, with
noise added, for NGC 1292 as a check on the tilted-ring
model (see Fig. 7). The same procedure was done for all the
FIG. 9.ÈModel disk parameters for E358, N1292, and N1406. Second page is for E437, N2417, and N3463. Parameters from variable center (black), from
kinematic center (dark gray), and photometric center (light gray). Linear units measured in arcseconds and angles measured in degrees.
114
FIG. 9.ÈContinued
115
FIG. 9.ÈContinued
PRECISION VELOCITY FIELDS. I. 117
FIG. 10.ÈColor images of the velocity Ðelds for the seven sample galaxies. In order, the galaxies shown are E358, N1292, N1406, E437, N2417, N3463,
and I2559. In each Ðgure the top row shows the continuum, intensity, and dispersion maps, and the bottom row shows the velocity Ðeld, the model velocity
Ðeld, and the residual velocity Ðled. This is a gray-scale image; the full color images are available in the online edition of the Astrophysical Journal.
other galaxies in the sample and, the results were the same
as seen for NGC 1292. In general, by comparing the models
with noise to the data we Ðnd that the average size of the
residual velocity is D10È15 km s~1. This agrees with well
with the amplitudes of the bumps and wiggles noticed in the
FP rotation curves obtained by SBWM and also by us in
Paper II. The residual velocity Ðelds show none of the sys-
tematic patterns that would indicate that the tilted-ring
model parameters are incorrect (Athanassoula 1984),
although there is some evidence of radial motion, particu-
larly in the case of NGC 1406. The residual velocity Ðelds,
in general, do not show large-scale features or much asym-
metry. This strongly suggests that the velocity Ðeld of a
typical disk galaxy is adequately described by the tilted-ring
model upon which 10È15 km s~1 of Gaussian noise, due to
noncircular motions, is sprinkled. At local places in the
disk, these noncircular motions could be of larger ampli-
tude, but globally averaged, our data suggest an rms value
of 10È15 km s~1. This is good for the TF relation as it
suggests the velocity Ðelds in most spiral galaxies are
quiescent, a point which is clearly consistent with the
observed scatter in most TF samples.
3.5. Possible Sources of Residual Velocities
3.5.1. Kinematic Feedback from Star Formation
One of the principal attributes of spiral arm density wave
theory (e.g., Yuan & Cheng 1989) is the production of
streaming motions by the spiral structure itself. There is
ample evidence, from 21 cm mapping of the Galaxy, that
this is occurring at the D10 km s~1 level. To the extent that
our data conÐrm this, we are not concerned with further
elucidation of this source of noncircular motions but
instead focus on other sources that hitherto have not been
strongly considered. Indeed, many processes that can
produce residual velocities also have other observable
e†ects. For example, regions of intense star formation are
easily seen in the Haintensity image and turbulent motions
in those regions may lead to an increase in the local velocity
dispersion. Thus, correlations may exist between the
residual velocity Ðeld and the features in the intensity and
dispersion maps. At some level, kinematic feedback to gas
from the process of star formation must be important.
Clearly, in the case of low mass irregular galaxies, this feed-
back may well be the major factor in determining evolution-
ary history as, in extreme cases, signiÐcant winds can be
generated which can blow the gas to large distances (e.g.,
Devost, Roy, & Drissen 1997; Martin & Kern 1999 ; see
also Martin 1997). The situation in normal spiral galaxies
has not been scrutinized yet, but our data allow for a pre-
liminary investigation.
To look for the signature of kinematic feedback to the gas
from star formation, contours of constant residual velocity
were plotted over gray-scale images of the intensity and
dispersion features, and vice versa. This may seem
redundant, but because the contour levels were chosen
somewhat arbitrarily, correlations are sometimes more
visible in one plot than the other. To create maps of just the
intensity and dispersion features, the images were boxcar
118 BEAUVAIS & BOTHUN Vol. 125
averaged to create background images which were then
subtracted from the originals. The relevant images are
shown in Figure 11. As is apparent, not all regions of large
residual velocity are associated with an intensity or disper-
sion feature, and neither are all intensity and dispersion
features related to large residual velocities, but there are
obviously some correlations. These can be seen as the
straightforward tracing of a gray-scale feature by a contour,
while in other areas a contour may avoid a feature. The
intensity and dispersion correlate equally well with both
positive or negative residual velocities and the correlation
between the residual velocity Ðeld and velocity dispersion is
slightly stronger than between the residual velocity Ðeld and
the Haintensity for most galaxies. There are few instances
where the same residual velocity correlates with both the
intensity and dispersion.
In general, the correlations with are found are on rela-
tively small spatial scales (e.g., 0.5È1 kpc), which is consis-
tent with the idea that many velocity residuals arise from to
local causes, such as star formation. Of course, in many
cases star formation is coincident with strong spiral struc-
ture and the kinematic feedback may be secondary to the
spiral density waves in producing the observed residual
velocities. On the other hand, galaxies with multiple but
weak spiral structure like NGC 2147 have residual veloci-
ties of the same amplitude as those observed in disks with
stronger spiral structure (e.g., NGC 1292). This would
suggest that the residual velocities can be driven by the local
star formation rate. Because the dispersion maps are gener-
ally noisy, it is difficult to determine if the overall velocity
dispersion of the Haemitting gas, on scales of a few kpc, is
a†ected by enhanced star formation. The use of a narrow
etalon would be required to make a better assessment, but
this would be an extremely time-consuming observation per
galaxy.
3.5.2. T riaxial Potentials
Another possible source of residual velocities, albeit oper-
ative over a larger spatial scale, would be related to the
failure of the tilted-ring model to adequately describe the
FIG. 11.ÈCorrelation of residual velocities with intensity and dispersion images. Shown are residual velocity contours over intensity feature gray-scale
(top left) and over velocity dispersion feature gray-scale (bottom left), and intensity (top right) and dispersion (bottom right) contours over gray-scale of residual
velocity Ðelds. The galaxies displayed here are E358, E437, N1292, N1406, N2417, and N3463. This is a gray-scale image ; the full color images are available in
the online edition of the Astrophysical Journal.
No. 1, 1999 PRECISION VELOCITY FIELDS. I. 119
FIG. 12.ÈPlots of for E358, N1292, N1406, E437, N2417, and N3463i\lsin 2/obs
TABLE 4
MEAN VALUES OF i
Galaxy Name i
ESO 358ÈG63...... 0.035
ESO 437ÈG30...... [0.034
NGC 1292 .......... 0.032
NGC 1406 .......... [0.079
NGC 2417 .......... 0.005
NGC 3463 .......... 0.019
orbital motions of the gas. As mentioned in the intro-
duction, there is the possibility that disk galaxies are
embedded in triaxial rather than axisymmetric potentials. A
triaxial potential has a preferred axis which drives elliptical
gas orbits whose eccentricity is determined by the aspect
ratio of the three principal axes. Potentially, these elliptical
orbits could be present at any radius.
As is well known from isophotal Ðtting, approximating
an elliptical distribution with one that is circular introduces
a regular pattern of angular harmonics (Jedrzejewski 1987).
120 BEAUVAIS & BOTHUN Vol. 125
The residual velocities at each R(i.e., in a given annulus of
the tilted-ring model) should satisfy
dv\vcsin iC;
k/1
3ckcos k/]sksin k/D.
The coefficients and are calculated for large inclina-cksk
tions iand to Ðrst order in the ellipticity of the orbits byeR
Franx, van Gorkom, & de Zeeuw (1994):
c1\[eRcos 2/obs ,s1\(1 [A[B)eRsin 2/obs ,
c2\0, s2\0,
c3\0, s3\([A[B)eRsin 2/obs .
The angle is measured from the long axis of the orbits/obs
(short axis of the potential) to the line of sight. The terms A
and Brelate to the slope of the rotation curve and the
inclination at a given R; if and with q4cos ithenvcDRa
A\1
2
a
1]a,B\(1 [q2)[3q2]1[2A(q2]1)]
(1 [q2)2](3q2]1)2.
Unfortunately, the above equations are only valid when the
true parameters (P.A., inclination, circular velocity, and
center) are used. These, of course, are returned by the tilted-
ring model as only approximate values precisely because it
assumes (zeroth-order Ðt). The second-order harmo-c1\0
nics vanish when the true center is used, and the remaining
coefficients are calculated assuming this. Franx et al. also
show that the tilted-ring model returns the true inclination
if the rotation curve is Ñat, causing the term to vanish. Allc3
information about is lost in the zeroth-order tilted ringc1
Ðt, so instead of information about both andeRcos 2/obs
which would allow the separate determinationeRsin 2/obs,
of and only information about is avail-eR/obs,eRsin 2/obs
able. The calculated coefficients are di†erent if the disk
parameters are derived from the photometry rather than
the kinematics, and allow the determination of both the eR
and However, the inclination found from the photo-/obs.
metry often di†ers by [5¡ from that found from the kine-
matics, making the overall determination uncertain.
The residual velocity Ðeld was Ðtted with
dv\vcsin iCs1sin /]s3sin 3/D
for and for each tilted-ring model annulus, providings1s3
three separate, but not independent, estimates of i\
These are shown in Figure 12. The Ðrst estimatelsin 2/obs.
is simply The remaining two come fromi\s1[s3.
solving for ifrom and separately under the assump-s1s3
tion of a Ñat rotation curve (A\a\0), and using the
known inclination to Ðnd B. A Ðt was performed rather
than a Fourier analysis because at certain radii there are
gaps in the residual velocity Ðeld which makes the Fourier
analysis impractical.
Although two of the three estimates depend on the rota-
tion curve being Ñat, they are actually in good agreement
over a surprisingly large range of R. At small R, the lack of
agreement between the estimates is likely due to the steep
slope of the rotation curve which is not well resolved at our
pixel size. At large R(and throughout N2417) large gaps
appear in the velocity Ðeld. Both of these e†ects result in s3
being ill-determined. For any galaxy, the value of iÑuctu-
ates greatly with R, though where the three estimates agree,
the values are in line with the estimate of found by[0.1, eR
Franx et al. The variation may be due to a changing ellip-
ticity or due to the changing disk geometry a†ecting /obs.
Density wave streaming possibly could account for the
change in disk geometry. Local sources of residual velocities
do not account for the variation of iwith radius. Corre-
lations between the residual velocities and the intensity and
dispersion images are with the small-scale structure and
should have little or no inÑuence on low-order harmonics.
The plots in Figure 12 can be used to estimate the size of
mean ellipticities of the galaxies. This was done by assuming
that lwas constant between galaxies and that viewing
angles, were randomly distributed. The average value/obs,
of ifor each galaxy was taken over the range where the
three estimates agreed, and these appear in Table 4. Since
the mean of sin 2/\0, the rms values were used for
This gives acomparisonÈirms \l[sin 2/obs]rms \l/J2.
mean ellipticity of so the ellipticities areeR\0.059 ^0.024,
most likely which is consistent with that found by[0.08,
Franx & de Zeeuw (1992). This implies that possible stream-
ing motions associated with Ñattened dark matter halos are
small and do not e†ectively contribute to the observed
scatter in the TF relation. These values are signiÐcantly
smaller than have been inferred based on the anisotropy of
the stellar distribution function in some ellipticals and
bulges (e.g., Gerhard et al. 1998). Our implied Ñattenings are
also much smaller (by several factors) than the strongly
triaxial halos in cosmological N-body simulations (e.g.,
Dubinski & Carlberg 1991). This may be an indication that
disks dominate the dynamics inside the optical radii of gal-
axies and that the presence of a strong disk serves to circu-
larize the potential. This is consistent with disks being
maximal.
4.SUMMARY
Two-dimensional velocity data provide detailed informa-
tion about the structure of spiral galaxies. In this paper we
have used relatively state-of-the art reduction schemes on
Fabry-Perot data, taken at Hato produce precision veloc-
ity Ðelds of seven late-type spiral galaxies. These velocity
Ðelds have a linear resolution of 0.5È1.0 kpc, which is sub-
stantially higher than 21 cm data can provide. This data set
allows us to probe many interesting areas of disk galaxy
kinematics. In this, the Ðrst paper of a series, we have
focussed attention on detecting small-scale noncircular
motions and assessing the larger scale rms noise that is
present in the velocity Ðelds. Part of the motivation for this
particular investigation lies in establishing the degree to
which disk galaxies are truly circularly symmetric. Interpre-
tation of the TF relation implicitly assumes this is the case.
If it is not, systematic errors can arise. Based on exami-
nation of these seven galaxies we reach the following con-
clusions:
1. The tilted-ring model method does a good job of
reproducing the observed velocity Ðelds of spiral galaxies
and are only modestly inÑuenced by the presence of non-
circular motions. Both the kinematic modeling and the
presence of small-scale structure in the velocity Ðelds of the
galaxies examined point to galaxies being more compli-
cated, on small scales, than simple di†erentially rotating
disks. The observed small-scale structure in the velocity
Ðeld is real and not an artifact of the reduction procedures,
No. 1, 1999 PRECISION VELOCITY FIELDS. I. 121
since the line-of-sight velocity is independent of whether the
line proÐles of the Fabry-Perot data are modeled as a Voigt
or Gaussian function. Averaged over the disk of the galaxy
these small-scale e†ects contribute 10È15 km s~1 of random
noise, indicating that velocity Ðelds on the large scale are
relatively quiescent.
2. A likely source of the small-scale noncircular motions
is kinematic feedback to the gas from regions of star forma-
tion. However, there is little evidence that suggests the
larger scale velocity dispersion of the gas is e†ected by
overall star formation rate. This is good news for the TF
relation as it suggests that the star formation rate is not a
secondary parameter. While this has been indirectly tested
in the past (there is no color dependence on residuals from
the TF relation), this study is perhaps the Ðrst to address
this issue on a strictly kinematical basis using kinematical
data.
3. We examined a subset of the velocity Ðelds for evi-
dence of triaxiality and detect only a small degree. This is at
odds with what has been observed for the case of some
spiral bulges and/or ellipticals and suggests that in our disk
dominated systems, the disk indeed provides a large frac-
tion of the mass inside the optical radius. In this case, the
disk would then serve to circularize the potential in the
azimuthal sense thus minimizing velocity residuals in the
plane of rotation. This supports KentÏs (1986, 1987) conclu-
sion that a maximum disk is consistent with the rotation
curves of spiral disks. It is certainly possible, though, for the
disk to somewhat less than maximal and for it still to
(nearly) circularize the potential inside the optical radius. If
this is the case, the P.A. misalignment between the inner and
outer optical disk could be evidence of a transition between
luminous and dark matter dominated regions. Interestingly,
two galaxies in our sample show evidence for this e†ect.
4. The noisy correlation between the photometric and
kinematic properties of our sample disk galaxies is cause for
some concern in the application of the TF relation. While
large-scale averages are in agreement, on small scales, there
are signiÐcant di†erences in position angle and orientation
of the photometric versus kinematic structure. As such dif-
ferences are not strongly correlated with where the residual
velocities appear, it seems that more than just spiral struc-
ture is the cause of these di†erences. This means that the
details of the rotation curve can be quite sensitive to the
presence of noncircular motions and/or a variable kine-
matic center with radius (cf. Persic & Salucci 1988, 1990).
Indeed, the variable center kinematic model provides a sig-
niÐcantly better match to the observed velocity Ðeld, espe-
cially in terms of the symmetry of the rotation curve. This is
explored in detail in Paper II of this series where it is argued
that the variable center is mostly likely the reÑection of a
clumpy mass distribution in the interior regions of disks. As
the light distribution is clumpy there as well, this may
provide another indication that maximal disks are present
in high surface brightness spiral galaxies.
We thank the referee, Brent Tully, for a careful reading of
this manuscript. We also thank conversations with Chris
Mihos and Ben Weiner regarding Ðtting line proÐles.
Support for this project was enabled by the Miller Trust
Fund for support of astronomical research at the University
of Oregon.
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