Tremaine-Weinberg Method Research Articles

The Effects of Bar Strength and Kinematics on Galaxy Evolution: Slow Strong Bars Affect Their Hosts the Most

We study how bar strength and bar kinematics affect star formation in different regions of the bar by creating radial profiles of EW[Hα] and Dn4000 using data from Sloan Digital Sky Survey-IV Mapping Nearby Galaxies at Apache Point Observatory (MaNGA). Bars in galaxies are classified as strong or weak using Galaxy Zoo DESI, and they are classified as fast and slow bars using the Tremaine–Weinberg method on stellar kinematic data from the MaNGA survey. In agreement with previous studies, we find that strong bars in star-forming (SF) galaxies have enhanced star formation in their center and beyond the bar-end region, while star formation is suppressed in the arms of the bar. This is not found for weakly barred galaxies, which have very similar radial profiles to unbarred galaxies. In addition, we find that slow bars in SF galaxies have significantly higher star formation along the bar than fast bars. However, the global star formation rate is not significantly different between galaxies with fast and slow bars. This suggests that the kinematics of the bar do not affect star formation globally, but changes where star formation occurs in the galaxy. Thus, we find that a bar will influence its host the most if it is both strong and slow.

The bar pattern speed of the Large Magellanic Cloud

Context. The internal kinematics of the Large Magellanic Cloud (LMC) have been studied in unprecedented depth thanks to the excellent quality of the Gaia mission data, thus revealing the disc’s non-axisymmetric structure. Aims. We seek to constrain the LMC bar pattern speed using the astrometric and spectroscopic data from the Gaia mission. Methods. We applied three methods to evaluate the bar pattern speed by measuring it via: the Tremaine-Weinberg (TW) method, the Dehnen method, and a bisymmetric velocity (BV) model. These methods provide additional information on the bar properties, such as the corotation radius as well as the bar length and strength. We tested the validity of the methods with numerical simulations. Results. A wide range of pattern speeds are inferred by the TW method, owing to a strong dependency on the orientation of the galaxy frame and the viewing angle of the bar perturbation. The simulated bar pattern speeds (corotation radii, respectively) are well recovered by the Dehnen method (BV model). Applied to the LMC data, the Dehnen method finds a pattern speed of Ωp = −1.0 ± 0.5 km s−1 kpc−1, thus corresponding to a bar that barely rotates and is only slightly counter-rotating with respect to the LMC disc. The BV method gives a LMC bar corotation radius of Rc = 4.20 ± 0.25 kpc, corresponding to a pattern speed of Ωp = 18.5−1.1+1.2 km s−1 kpc−1. Conclusions. It is not possible to determine which global value best represents an LMC bar pattern speed with the TW method, due to the strong variation with the orientation of the reference frame. The non-rotating bar from the Dehnen method would be at odds with the structure and kinematics of the LMC disc. The BV method result is consistent with previous estimates and gives a bar corotation-to-length ratio of 1.8 ± 0.1, suggesting that the LMC is hosting a slow bar.

On the Tremaine–Weinberg method: how much can we trust gas tracers to measure pattern speeds?

ABSTRACT Pattern speeds are a fundamental parameter of the dynamical features (e.g. bars, spiral arms) of a galaxy, setting resonance locations. Pattern speeds are not directly observable, so the Tremaine–Weinberg (TW) method has become the most common method used to measure them in galaxies. However, it has not been tested properly whether this method can straightforwardly be applied to gas tracers, despite this being widely done in the literature. When applied to observations, the TW method may return invalid results, which are difficult to diagnose due to a lack of ground truth for comparison. Although some works applying the TW method to simulated galaxies exist, only stellar populations have been tested. Therefore, here we explore the applicability of the TW method for gas tracers, by applying it to hydrodynamical simulations of galaxies, where we know the true value of the bar pattern speed. We perform some simple tests to see if the TW method has a physically reasonable output. We add different kinds of uncertainties (e.g. in position angle or flux) to the data to mock observational errors based on the magnitude of uncertainty present in the observations. Secondly, we test the method on 3D simulations with chemical networks. We show that in general, applying TW to observations of gas will not recover the true pattern speed. These results have implications for many ‘pattern speeds’ reported in the literature, and based on these tests we also give some best practices for measuring pattern speeds using gas tracers going forwards.

Five methods for determining pattern speeds in galaxies

Context. After matter distribution and kinematics, the bar or spiral pattern speeds are the next fundamental parameters determining the dynamics of a galaxy. Aims. New or refined methods for determining instantaneous scalar and vector pattern speeds from a restricted domain are developed for applications in N-body simulations or in galaxies such as the Milky Way, for which the stellar coordinates become increasingly better known. Methods. The general feature used throughout follows from the fact that the time derivative of a function of the coordinates is linearly proportional to its rotation rate and its particle velocities. Knowing these therefore allows retrieving the instantaneous pattern speed vector by linear optimization. Similarly, if an invariant function depends on the position and velocities, then its instantaneous rotation vectors in space can be retrieved. Knowing the accelerations also allows determining the pattern rotation of velocity space. Results. The first three methods are based on the assumed rotational invariance of functions at each point in space or velocity space: (1) the 6D invariant function method, measuring the pattern speed vectors in space and velocity space, (2) the differential/regional 3D Tremaine-Weinberg method, evaluated over regions with a high signal-to-noise ratio, (3) the 3D Jacobi integral method, yielding the potential pattern speed. Extensions to derive the rotation center position, speed, and acceleration are introduced in the first and third methods. The last two methods are based on the assumed invariance of average functions of the particle coordinates: (4) the 2D and 3D moment of inertia methods by using the derivative of the singular value decomposition, (5) the 2D Fourier method (3D for m = 2 mode), giving the mode rotation speeds. Pattern speed accelerations are also derived in the fourth and fifth methods. Conclusions. Depending on the available data in specific problems, the different methods provide a choice of approaches.

Galaxy Zoo: kinematics of strongly and weakly barred galaxies

ABSTRACTWe study the bar pattern speeds and corotation radii of 225 barred galaxies, using integral field unit data from MaNGA and the Tremaine–Weinberg method. Our sample, which is divided between strongly and weakly barred galaxies identified via Galaxy Zoo, is the largest that this method has been applied to. We find lower pattern speeds for strongly barred galaxies than for weakly barred galaxies. As simulations show that the pattern speed decreases as the bar exchanges angular momentum with its host, these results suggest that strong bars are more evolved than weak bars. Interestingly, the corotation radius is not different between weakly and strongly barred galaxies, despite being proportional to bar length. We also find that the corotation radius is significantly different between quenching and star-forming galaxies. Additionally, we find that strongly barred galaxies have significantly lower values for $\mathcal {R}$, the ratio between the corotation radius and the bar radius, than weakly barred galaxies, despite a big overlap in both distributions. This ratio classifies bars into ultrafast bars ($\mathcal {R} \lt $ 1.0; 11 per cent of our sample), fast bars (1.0 $\lt \mathcal {R} \lt $ 1.4; 27 per cent), and slow bars ($\mathcal {R} \gt $ 1.4; 62 per cent). Simulations show that $\mathcal {R}$ is correlated with the bar formation mechanism, so our results suggest that strong bars are more likely to be formed by different mechanisms than weak bars. Finally, we find a lower fraction of ultrafast bars than most other studies, which decreases the recently claimed tension with Lambda cold dark matter. However, the median value of $\mathcal {R}$ is still lower than what is predicted by simulations.

Bar pattern speed at z∼1–2 to explore challenges of the Standard Cosmology

AbstractStellar bars drive the galaxy secular evolution. While rotating around the galaxy centre with a given angular frequency, the bar pattern speed, they sweep material and modify the galaxy structure. In the LCDM model, bars are expected to slow down by exchanging angular momentum with the other omponents and/or through dynamical friction exerted by the dark matter halo. The only direct method to derive the bar pattern speed, the Tremaine-Weinberg method, revealed that real bars rotate fast, stressing a tension between the observations, conducted to date in the local universe, and the LCDM model. Measuring the bar pattern speed to bars up to z∼1-2 will reveal if the expected bar evolutionary path is actually taking place and/or to confirm if the dark matter is able to exert friction. Using high resolution N-body simulations we tested the applicability of the Tremaine-Weinberg method to deep spectroscopy of the NIRSpec@JWST for a sample of bars at z∼1-2. Our analysis can be used to prepare an observational proposal to get dedicated data.

SDSS IV MaNGA: bar pattern speed in Milky Way analogue galaxies

ABSTRACT Most secular effects produced by stellar bars strongly depend on the pattern speed. Unfortunately, it is also the most difficult observational parameter to estimate. In this work, we measured the bar pattern speed of 97 Milky-Way analogue galaxies from the MaNGA survey using the Tremaine–Weinberg method. The sample was selected by constraining the stellar mass and morphological type. We improve our measurements by weighting three independent estimates of the disc position angle. To recover the disc rotation curve, we fit a kinematic model to the H α velocity maps correcting for the non-circular motions produced by the bar. The complete sample has a smooth distribution of the bar pattern speed ($\Omega _{\rm Bar}=28.14^{+12.30}_{-9.55}$ km s−1 kpc −1), corotation radius ($R_{CR} = 7.82^{+3.99}_{-2.96}$ kpc), and the rotation rate ($\mathcal {R} = 1.35^{+0.60}_{-0.40}$). We found two sets of correlations: (i) between the bar pattern speed, the bar length and the logarithmic stellar mass (ii) between the bar pattern speed, the disc circular velocity and the bar rotation rate. If we constrain our sample by inclination within 30° < i < 60° and relative orientation 20° < |PAdisc − PAbar| < 70°, the correlations become stronger and the fraction of ultra-fast bars is reduced from 20 to 10 per cent of the sample. This suggests that a significant fraction of ultra-fast bars in our sample could be associated with the geometric limitations of the TW method. By further constraining the bar size and disc circular velocity, we obtain a subsample of 25 Milky-Way analogues galaxies with distributions $\Omega _{\rm Bar}=30.48^{+10.94}_{-6.57}$ km s−1 kpc−1, $R_{\rm CR} = 6.77^{+2.32}_{-1.91}$ kpc, and $\mathcal {R} = 1.45^{+0.57}_{-0.43}$, in good agreement with the current estimations for our Galaxy.

A slow bar in the lenticular barred galaxy NGC 4277

Aims.We characterised the properties of the bar hosted in lenticular galaxy NGC 4277, which is located behind the Virgo cluster.Methods.We measured the bar length and strength from the surface photometry obtained from the broad-band imaging of the Sloan Digital Sky Survey and we derived the bar pattern speed from the stellar kinematics obtained from the integral-field spectroscopy performed with the Multi Unit Spectroscopic Explorer at the Very Large Telescope. We also estimated the co-rotation radius from the circular velocity, which we constrained by correcting the stellar streaming motions for asymmetric drift, and we finally derived the bar rotation rate.Results.We found that NGC 4277 hosts a short (Rbar = 3.2−0.6+0.9kpc), weak (Sbar = 0.21 ± 0.02), and slow (ℛ = 1.8−0.3+0.5) bar and its pattern speed (Ωbar = 24.7 ± 3.4 km s−1kpc−1) is amongst the best-constrained ones ever obtained with the Tremaine–Weinberg (TW) method with relative statistical errors of ∼0.2.Conclusions.NGC 4277 is the first clear-cut case of a galaxy hosting a slow stellar bar (ℛ > 1.4 at more than a 1σconfidence level) measured with the model-independent TW method. A possible interaction with the neighbour galaxy NGC 4273 could have triggered the formation of such a slow bar and/or the bar could be slowed down due to the dynamical friction with a significant amount of dark matter within the bar region.

Properties of Fast and Slow Bars Classified by Epicyclic Frequency Curves from Photometry of Barred Galaxies

Abstract We test the idea that bar pattern speeds decrease with time owing to angular momentum exchange with a dark matter halo. If this process actually occurs, then the radii of the corotation resonance and other resonances should generally increase with time. We therefore derive the angular velocity Ω and epicyclic frequency κ as functions of galactocentric radius for 85 barred galaxies using photometric data. Mass maps are constructed by assuming a dynamical mass-to-light ratio and then solving the Poisson equation for the gravitational potential. The locations of Lindblad resonances and the corotation resonance radius are then derived using the standard precession frequency curves in conjunction with bar pattern speeds recently estimated from the Tremaine–Weinberg method as applied to integral field spectroscopy data. Correlations between physical properties of bars and their host galaxies indicate that bar length and the corotation radius depend on the disk circular velocity while bar strength and pattern speed do not. As the bar pattern speed decreases, bar strength, length, and corotation radius increase, but when bars are subclassified into fast, medium, and slow domains, no significant change in bar length is found. Only a hint of an increase in bar strength from fast to slow bars is found. These results suggest that bar length in a galaxy undergoes little evolution, and is determined instead mainly by the size of the host galaxy.

Bar pattern speeds in CALIFA galaxies

Context. More than 10% of barred galaxies with a direct measurement of the bar pattern speed host an ultrafast bar. These bars extend well beyond the corotation radius and challenge our understanding of the orbital structure of barred galaxies. Most of the bars are found in spiral galaxies, rather than in lenticular galaxies. Aims. We analyse the properties of the ultrafast bars detected in the Calar Alto Legacy Integral Field Spectroscopy Area Survey to investigate whether they are an artefact resulting from an overestimation of the bar radius and/or an underestimation of the corotation radius or a new class of bars, whose orbital structure has not been understood yet. Methods. We revised the available measurements of the bar radius based on ellipse fitting and Fourier analysis and of the bar pattern speed from the Tremaine-Weinberg method. In addition, we measured the bar radius from the analysis of the maps tracing the transverse-to-radial force ratio, which we obtained from the deprojected i-band images of the galaxies retrieved from the Sloan Digital Sky Survey. Results. We found that nearly all the sample galaxies are spirals with an inner ring or pseudo-ring circling the bar and/or with strong spiral arms, which hamper the measurement of the bar radius from the ellipse fitting and Fourier analysis. According to these methods, the bar ends overlap with the ring or the spiral arms, thereby making the adopted bar radius unreliable. On the contrary, the bar radius from the ratio maps are shorter than the corotation radius. This agrees with the theoretical predictions and findings of numerical simulations regarding the extension and stability of the stellar orbits supporting the bars. Conclusions. We conclude that ultrafast bars are no longer observed when the correct measurement of the bar radius is adopted. Deriving the bar radius in galaxies with rings and strong spiral arms is not straightforward and a solid measurement method based on both photometric and kinematic data is still missing.

Relations among structural parameters in barred galaxies with a direct measurement of bar pattern speed

We investigate the relations between the properties of bars and their host galaxies in a sample of 77 nearby barred galaxies, spanning a wide range of morphological types and luminosities, with 34 SB0-SBa and 43 SBab-SBc galaxies. The sample includes all the galaxies with reliable direct measurement of their bar pattern speed based on long-slit or integral-field stellar spectroscopy using the Tremaine-Weinberg method. We limited our analysis to the galaxies with a relatively small relative error on the bar pattern speed (≤50%) and that do not host an ultrafast bar. For each galaxy, we collected the radius, strength, pattern speed, corotation radius, and rotation rate for the bar and we also collected the Hubble type and absolute SDSSr-band magnitude. We also used literature bulge-to-total luminosity ratios for a subsample of 53 galaxies with an available photometric decomposition. We confirmed earlier observational findings that longer bars rotate at lower bar pattern speeds, shorter bars are weaker, and bars with a low rate of bar rotation rotate at faster bar pattern speeds and have smaller corotation radii. In addition, we found that stronger bars rotate at lower bar pattern speeds, as predicted from the interchange of angular momentum during bar evolution, which in turn may depend on different galaxy properties. Moreover, we report that brighter galaxies host longer bars, which rotate at lower bar pattern speeds and have larger corotation radii. This result is in agreement with a scenario of downsizing in bar formation, if more massive galaxies formed earlier and had sufficient time to slow down, grow in length, and push corotation outwards.

SDSS-IV MaNGA: Bar pattern speed estimates with the Tremaine-Weinberg method and their error sources

Abstract Estimating the bar pattern speed (Ωbar) is one of the main challenges faced in understanding the role of stellar bars in galaxy dynamical evolution. This work aims to characterise different uncertainty sources affecting the Tremaine Weinberg (TW)-method to study the correlation between bar and galaxies physical parameters. We use a sample of 15 MaNGA SDSS-IV galaxies and 3 CALIFA galaxies from Aguerri et al. (2015). We studied the errors related with (i) galaxy centre determination, (ii) disc position angle (PA) emphasising the difficulties triggered by outer non-axisymmetric structures besides the bar, (iii) the slits length and (iv) the spatial resolution. In average, the PA uncertainties range $\sim 15 \%$, the slit length $\sim 9 \%$ and the centring error $\sim 5 \%$. Reducing the spatial resolution increases the sensitivity to the PA error. Through Monte Carlo simulations, we estimate the probability distribution of the $\mathcal {R}$ bar speed parameter. The present sample is composed of 7 slow, 4 fast and 7 ultrafast bars, with no trend with morphological types. Although uncertainties and low sample numbers may mask potential correlations between physical properties, we present a discussion of them: We observe an anti-correlation of Ωbar with the bar length and the stellar mass, suggesting that massive galaxies tend to host longer and slower bars. We also observe a correlation of the molecular gas fraction with $\mathcal {R}$, and a weak anti-correlation with Ωbar, suggesting that bars rotate slower in gaseous discs. Confirmation of such trends awaits future studies.

Evidence of a fast bar in the weakly-interacting galaxy NGC 4264 with MUSE

ABSTRACT We present surface photometry and stellar kinematics of NGC 4264, a barred lenticular galaxy in the region of the Virgo Cluster undergoing a tidal interaction with one of its neighbours, NGC 4261. We measured the bar radius (abar = 3.2 ± 0.5 kpc) and strength (Sbar = 0.31 ± 0.04) of NGC 4264 from Sloan Digital Sky Survey imaging and its bar pattern speed (Ωbar = 71 ± 4 km s−1 kpc−1) using the Tremaine–Weinberg method with stellar-absorption integral-field spectroscopy performed with the Multi Unit Spectroscopic Explorer at the Very Large Telescope. We derived the circular velocity (Vcirc = 189 ± 10 km s−1) by correcting the stellar streaming velocity for asymmetric drift and calculated the corotation radius (Rcor = 2.8 ± 0.2 kpc) from the bar pattern speed. Finally, we estimated the bar rotation rate (Rcor/abar = 0.88 ± 0.23). We find that NGC 4264 hosts a strong and large bar extending out to the corotation radius. This means that the bar is rotating as fast as it can like nearly all the other bars measured so far even when the systematic error due to the uncertainty on the disc position angle is taken into account. The accurate measurement of the bar rotation rate allows us to infer that the formation of the bar of NGC 4264 was due to self-generated internal processes and not triggered by the ongoing interaction.

THE DYNAMICAL RELATIONSHIP BETWEEN THE BAR AND SPIRAL PATTERNS OF NGC 1365

ABSTRACT Theories that attempt to explain the dynamical relationship between bar and spiral patterns in galactic disks make different predictions about the radial profile of the pattern speed. These are tested for the H-alpha bar and spiral patterns of NGC 1365. The radial profile of the pattern speed is measured by fitting mathematical models that are based on the Tremaine–Weinberg method. The results show convincing evidence for the bar rotating at a faster rate than the spiral pattern, inconsistent with a global wave mode or a manifold. There is evidence for mode coupling of the bar and spiral patterns at the overlap of corotation and inner Lindblad resonances (ILRs), but the evidence is unreliable and inconsistent. The results are the most consistent with the bar and spiral patterns being dynamically distinct features. The pattern speed of the bar begins near an ILR and ends near the corotation resonance (CR). The radial profile of the pattern speed beyond the bar most closely resembles what is expected for coupled spiral modes and tidal interactions.

A slow bar in the dwarf irregular galaxy NGC 3741

Using the Tremaine-Weinberg method, we measure the speed of the HI bar seen in the disk of NGC 3741. NGC 3741 is an extremely gas rich galaxy with an {H\,{\sc i}} disk which extends to about 8.3 times its Holmberg radius. It is also highly dark matter-dominated. Our calculated value of the pattern speed $\Omega_p$ is 17.1 $\pm$ 3.4 km $\textrm{s}^{-1}\textrm{kpc}^{-1}$. We also find the ratio of the co-rotation radius to the bar semi-major axis to be (1.6 $\pm$ 0.3), indicating a slow bar. This is consistent with bar models in which dynamical friction results in a slow bar in dark matter dominated galaxies.

PATTERN SPEEDS OF BARS AND SPIRAL ARMS FROM Hα VELOCITY FIELDS

We have applied the Tremaine-Weinberg method to 10 late-type barred spiral galaxies using data cubes, in H-alpha emission, from the GHAFAS and FANTOMM Fabry-Perot spectrometers. We have combined the derived bar (and/or spiral) pattern speeds with angular frequency plots to measure the corotation radii for the bars in these galaxies. We base our results on a combination of this method with a morphological analysis designed to estimate the corotation radius to bar-length ratio using two independent techniques on archival near infrared images, and although we are aware of the limitation of the application of the Tremaine-Weinberg method using ionised gas observations, we find consistently excellent agreement between bar and spiral arm parameters derived using different methods. In general, the corotation radius, measured using the Tremaine-Weinberg method, is closely related to the bar length, measured independently from photometry and consistent with previous studies. Our corotation/bar-length ratios and pattern speed values are in good agreement with general results from numerical simulations of bars. In systems with identified secondary bars, we measure higher H-alpha velocity dispersion in the circumnuclear regions, whereas in all the other galaxies, we detect flat velocity dispersion profiles. The excellent agreement between the Tremaine-Weinberg method results and the morphological analysis and bar parameters in numerical simulations, suggests that although the H-alpha emitting gas does not obey the continuity equation, it can be used to derive the bar pattern speed.

ON THE BAR PATTERN SPEED DETERMINATION OF NGC 3367

An important dynamic parameter of barred galaxies is the bar pattern speed. Among several methods that are used for the determination of the pattern speed the Tremaine-Weinberg method has the advantage of model independency and accuracy. In this work we apply the method to a simulated bar including gas dynamics and study the effect of 2D spectroscopy data quality on robustness of the method. We added a white noise and a Gaussian random field to the data and measured the corresponding errors in the pattern speed. We found that a signal to noise ratio in surface density ~5 introduces errors of ~20% for the Gaussian noise, while for the white noise the corresponding errors reach ~50%. At the same time the velocity field is less sensitive to contamination. On the basis of the performed study we applied the method to the NGC 3367 spiral galaxy using H{\alpha} Fabry-Perot interferometry data. We found for the pattern speed 43 \pm 6 km/s/kpc for this galaxy.

A slow bar in a dark matter dominated galaxy

Aims. We report on an estimate of the bar pattern speed Ω p for the low surface brightness spiral galaxy UGC 628. Methods. We applied the Tremaine-Weinberg method to high resolution Hα velocity and integrated emission maps of this dark matter dominated galaxy. Observations were made at the CFHT using the optical Fabry-Perot interferometer, F A NTO M M. Results. The Tremaine-Weinberg method estimates a bar pattern speed of (11.3 ± 2.0) km s ―1 kpc ―1 for UGC 628, which is among the lowest values found for a spiral galaxy. The corotation radius R c of the bar and the gaseous disc is R c = 9.8 +2.9 ―2.0 kpc, implying a ratio R = R c /a B of 2.0 +0.5 ―0.3 , where a B is the bar radius. The ratio is well beyond the usual range of values, 1.0 ≤ R ≤ 1.4, found for fast bars of high surface brightness barred galaxies. This implies that the bar in UGC 628 is slow. Conclusions. As shown through the use of numerical simulations, fast bars survive when the inner mass distribution of galaxies is dominated by the baryons over the dark matter. Our result suggests that the presence of slow bars in galaxies is likely related to the dominance of dark matter over the mass distribution.

OBSERVABLE PROPERTIES OF DOUBLE-BARRED GALAXIES INN-BODY SIMULATIONS

Although at least one quarter of early-type barred galaxies host secondary stellar bars embedded in their large-scale primary counterparts, the dynamics of such double-barred galaxies are still not well understood. Recently we reported success at simulating such systems in a repeatable way in collisionless systems. In order to further our understanding of double-barred galaxies, here we characterize the density and kinematics of the N-body simulations of these galaxies. This will facilitate comparison with observations and lead to a better understanding of the observed double-barred galaxies. We find the shape and size of our simulated secondary bars are quite reasonable compared to the observed ones. We demonstrate that an authentic decoupled secondary bar may produce only a weak twist of the kinematic minor axis in the stellar velocity field, due to the relatively large random motion of stars in the central region. We also find that the edge-on nuclear bars are probably not related to boxy peanut-shaped bulges which are most likely to be edge-on primary large-scale bars. Another kinematic feature often present in our double-barred models is a ring-like feature in the fourth-order Gauss-Hermite moment h4 maps. Finally, we demonstrate that the non-rigid rotation of the secondary bar causes its pattern speed to not be derived with great accuracy using the Tremaine-Weinberg method. We also compare with observations of NGC 2950, a prototypical double-barred early-type galaxy, which suggest that the nuclear bar may be rotating in the opposite sense as the primary.

Radial Dependence of the Pattern Speed of M51

The grand-design spiral galaxy M51 has long been a crucial target for theories of spiral structure. Studies of this iconic spiral can address the question of whether strong spiral structure is transient (e.g. interaction-driven) or long-lasting. As a clue to the origin of the structure in M51, we investigate evidence for radial variation in the spiral pattern speed using the radial Tremaine-Weinberg (TWR) method. We implement the method on CO observations tracing the ISM-dominant molecular component. Results from the method's numerical implementation--combined with regularization, which smooths intrinsically noisy solutions--indicate two distinct patterns speeds inside 4 kpc at our derived major axis PA=170 deg., both ending at corotation and both significantly higher than the conventionally adopted global value. Inspection of the rotation curve suggests that the pattern speed interior to 2 kpc lacks an ILR, consistent with the leading structure seen in HST near-IR observations. We also find tentative evidence for a lower pattern speed between 4 and 5.3 kpc measured by extending the regularized zone. As with the original TW method, uncertainty in major axis position angle (PA) is the largest source of error in the calculation; in this study, where \delta PA=+/-5 deg. a ~20% error is introduced to the parameters of the speeds at PA=170 deg. Accessory to this standard uncertainty, solutions with PA=175 deg. (also admitted by the data) exhibit only one pattern speed inside 4 kpc, and we consider this circumstance under the semblance of a radially varying PA.