Testing the Tremaine–Weinberg Method Applied to Integral-field Spectroscopic Data Using a Simulated Barred Galaxy

As mentioned above, the TW method is the only model-independent means to infer Ω_p from observational data. Its original and simplest implementation uses long-slit spectroscopic observations, under the assumptions that the disk is flat and the bar (actually the whole disk) has a single well-defined pattern speed. Another key requirement is that the (surface brightness of the) tracer population obeys the continuity equation. From these assumptions, Ω_p can be derived as

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{p}}}\sin i=\displaystyle \frac{{\int }_{-\infty }^{+\infty }h(Y){dY}{\int }_{-\infty }^{+\infty }\,[{V}_{| | }(X,Y)-{V}_{\mathrm{sys}}]\,{\rm{\Sigma }}(X,Y){dX}}{{\int }_{-\infty }^{+\infty }h(Y){dY}{\int }_{-\infty }^{+\infty }\,[X-{X}_{{\rm{c}}}]\,{\rm{\Sigma }}(X,Y){dX}}\end{eqnarray} \tag{ 1a }$$

$$\begin{eqnarray}&&=\,\displaystyle \frac{\langle {V}_{| | }\rangle -{V}_{\mathrm{sys}}}{\langle X\rangle -{X}_{{\rm{c}}}},\end{eqnarray} \tag{ 1b }$$

where (X, Y) are sky plane coordinates with origin at the galaxy center, ${V}_{| | }(X,Y)$ is the mean line-of-sight (LOS) velocity, and Σ(X, Y) the surface brightness (or surface mass density) of the tracer adopted, V_sys is the galaxy systemic velocity, X_c is the position of the galaxy minor axis, and h(Y) is an arbitrary weighting function. The X-axis and the Y-axis must be aligned with the major and the minor axis of the galaxy disk, respectively, and the slits (one for each Y) should thus be aligned parallel to but offset from the disk major axis (so that the integrals along X correspond to summations along slits parallel to but offset from the major axis).

Merrifield & Kuijken (1995) refined the TW method, suggesting to simply collapse the long-slit spectra to yield higher signal-to-noise ratio (S/N) measurements. They thus rewrote Equation 1(a) into the equivalent Equation 1(b), where

$$\begin{eqnarray}&&\langle {V}_{| | }\rangle \,\equiv \,\displaystyle \frac{{\int }_{-\infty }^{+\infty }\,{V}_{| | }(X,Y){\rm{\Sigma }}(X,Y){dX}}{{\int }_{-\infty }^{+\infty }\,{\rm{\Sigma }}(X,Y){dX}}\end{eqnarray} \tag{ 2 }$$

and

$$\begin{eqnarray}&&\langle X\rangle \,\equiv \,\displaystyle \frac{{\int }_{-\infty }^{+\infty }\,X\,{\rm{\Sigma }}(X,Y){dX}}{{\int }_{-\infty }^{+\infty }\,{\rm{\Sigma }}(X,Y){dX}}\end{eqnarray} \tag{ 3 }$$

are, respectively, the luminosity-weighted mean velocity and the mean position obtained from collapsing one long-slit spectrum along, respectively, its spatial and its dispersion (wavelength) direction. Then, ${{\rm{\Omega }}}_{{\rm{p}}}\sin i$ can easily be obtained from the slope of a straight line fit to $\langle {V}_{| | }\rangle$ versus $\langle X\rangle$ for multiple slits. In addition to a single higher-S/N measurement for each of the numerator and denominator in Equation 1(b), another advantage of this procedure is that the uncertainties on V_sys and X_c are unimportant (while they can lead to systematic biases when using Equation 1(a) directly). We will refer to this method as the MK method.

The TW method infers the bar pattern speed by quantifying the asymmetries introduced by the bar in the galaxy disk. Indeed, for an axisymmetric disk with no bar, Σ(X, Y) is an even function of X (i.e., along the slit), while ${V}_{| | }(X,Y)$ and X are odd functions of X. Thus, the integrals in the numerator and denominator of Equation (1) both sum to zero, and Ω_p is undefined. The existence of a bar (or a spiral pattern) introduces an oddness into Σ(X, Y) and an evenness into ${V}_{| | }(X,Y)$, resulting in nonzero integrals and the desired measurement of Ω_p. This is the main power of the TW method but also its greatest weakness, in that the integrals in the numerator and denominator of Equation (1) effectively take the difference of two large numbers, making TW measurements extremely sensitive to imperfections in the data and prone to systematic biases. In particular, asymmetries due to (sub)structures other than the bar can lead to false signals. It is thus essential to ensure that the contributions to the integrals in Equation (1) are indeed due to the bar and not other artifacts.

There are various ways to measure bar pattern speeds using the TW method. According to Equation 1(a), Ω_p can be measured from a single measurement using a single slit or from the average of multiple measurements using multiple slits (e.g., Kent 1987; Bureau et al. 1999). We will refer to the latter method as the TW averaging method. As suggested by Equation 1(b), another way to measure Ω_p is to measure the slope of $\langle {V}_{| | }\rangle$ versus $\langle X\rangle$. However, instead of collapsing the spectra to measure $\langle {V}_{| | }\rangle$ and $\langle X\rangle$, as suggested by Merrifield & Kuijken (1995), $\langle {V}_{| | }\rangle$ and $\langle X\rangle$ can equally be derived from measurements of Σ(X, Y) and ${V}_{| | }$ at each position along the slits, following Equations (2) and (3) (see, e.g., Aguerri et al. 2015). We will refer to this as the TW fitting method. Overall, there are thus at least four variants of the TW method: the original TW method, the TW averaging method, the TW fitting method, and the MK method. The first two are sensitive to uncertainties in V_sys and X_c, while the last two are not.

The most important assumption of the TW method is arguably that the tracer population satisfies the continuity equation. Clearly, the old stellar populations of S0 galaxies are good targets for the TW method, as star formation (SF) and dust extinction are usually unimportant. As a result, S0 galaxies have been frequent targets of TW studies (e.g., Kent 1987; Merrifield & Kuijken 1995; Gerssen et al. 1999, 2003; Debattista et al. 2002; Aguerri et al. 2003; Corsini et al. 2003, 2007; Debattista & Williams 2004). However, the TW method has also been applied to late-type galaxies, despite risks that SF and dust will affect the measurements (e.g., Gerssen et al. 2003; Treuthardt et al. 2007; Aguerri et al. 2015; Guo et al. 2019). The application of the TW method to gas tracers rather than stars is also more recent and requires more caution. In galaxies, gas will normally cycle through multiple phases (molecular, atomic, and ionized), and any given phase may not obey the continuity equation. However, if the interstellar medium (ISM) is dominated by a single phase, the gas in that dominant phase may then approximately satisfy the continuity equation. TW bar pattern speed measurements have been carried out using observations of H i (e.g., Bureau et al. 1999; Banerjee et al. 2013) and H₂ (e.g., Rand & Wallin 2004; Zimmer et al. 2004). The ionized gas rarely dominates the ISM by mass, and any line flux does not reliably trace its mass. However, using N-body/SPH simulations, Hernandez et al. (2005) found that the TW method could be applied to the ionized gas component to get a rough estimate of Ω_p, on the condition that the shock regions are avoided and the measurements are confined to the gaseous bar region. Hα has indeed been used numerous times to measure bar pattern speeds with the TW method (e.g., Hernandez et al. 2005; Emsellem et al. 2006; Fathi et al. 2007, 2009; Chemin & Hernandez 2009; Gabbasov et al. 2009). Most of the above measurements, irrespective of the tracer, suggest that bars rotate fast.

Observations with IFSs, which yield 3D data, are becoming increasingly popular, and sizeable samples of barred galaxies with a wide range of morphologies can now be selected from large, homogeneous, and often publicly available IFS surveys—e.g., the Calar Alto Legacy Integral-field Spectroscopy Area Survey (CALIFA; Sánchez et al. 2012), Sydney-Australian-Astronomical-Observatory Multi-object Integral-field Spectrograph (SAMI; Croom et al. 2012) survey, and Mapping Nearby Galaxies at Apache Point Observatory (MaNGA; Bundy et al. 2015) survey. IFS observations have unique advantages over long-slit observations for TW measurements. In particular, multiple pseudo-slits can be constructed from a single IFS observation. Most importantly, Debattista (2003, hereafter D03) investigated slit misalignments and found that errors in the position angle of the disk (PA_disk; e.g., from intrinsic disk ellipticities) and thus of the slits significantly and systematically affect TW Ω_p measurements. IFSs naturally allow us to choose the orientation of the pseudo-slits after the observations are obtained, thus allowing us to test for systematic uncertainties associated with the choice of PA_disk.

The use of the TW method with IFS data also differs from that with long-slit observations in several ways.

1. Extensive spatial coverage is very important to the TW method, as the integrals in Equation 1(a) formally range from $-\infty$ to $+\infty$ (see, e.g., Debattista & Williams 2004; Rand & Wallin 2004; Zimmer et al. 2004; Chemin & Hernandez 2009; Fathi et al. 2009; Aguerri et al. 2015; Guo et al. 2019). In practice, the integrals only need to reach the (projected) radius where the disk becomes (nearly) axisymmetric, as the integrands will cancel out at larger distances. There is no specific prescription to identify this radius, and it is likely to be different from galaxy to galaxy. Most long slits extend to the disk outskirts, where Ω_p measurements should asymptotically converge before the slits end. However, many IFSs have a small FOV, and the disks observed may not reach axisymmetry within that FOV. Before using the TW method with IFS data, it is thus essential to establish the minimum spatial coverage required for reliable and accurate TW measurements. This is done in Section 4.1.

2. Similarly, IFS data allow us to create (i.e., overlay on the observations) an arbitrary number of pseudo-slits, at arbitrary positions (i.e., at arbitrary offsets from the galaxy major axis). However, it is unclear whether all the resulting measurements will be equally reliable, as the photometric and kinematic signatures of bars necessarily decrease with increasing radius. In Section 4.2, we will thus explore how the quality of a measurement varies as a function of the slit position.

3. Spatial binning is commonly used with IFSs to increase the S/N of the data within a given spatial region (i.e., a bin made up of several spaxels), especially in the outer parts of galaxies, where convergence of the TW integrals is necessary. The flux and velocity of each bin are thus altered and effectively replaced by their (luminosity-weighted) spatial averages. As these are the quantities that underlie the TW method, the positions and sizes of the bins are likely to impact TW Ω_p measurements. In Section 4.3, we will therefore quantify the effects of spatial binning on these measurements.

4. As the position angle of the disk (PA_disk) is generally not aligned with the grid/axes of the IFS data (X_grid, Y_grid), the footprints of pseudo-slits are irregular in shape (at least in terms of the integer spaxels contained within them), an effect absent with long-slit observations. This introduces extra asymmetries and can thus artificially contribute to the integrals in Equation 1(a), potentially affecting the resulting Ω_p measurements. A possible way to eliminate this effect is to calculate the weights/contributions of fractional spaxels along the slits. However, a more straightforward and practical solution is often to simply recast the data grid. We must then test to what extent this grid recasting can affect the measurements, and this is done in Section 4.4.

There are other issues that may affect TW bar pattern speed measurements using either IFS or long-slit observations.

1. A clear advantage of using IFS over long-slit data is that the slits' orientation (implicitly the disk position angle PA_disk) can readily be changed during the TW calculations. Although PA_disk does not appear explicitly in Equation (1), and consequently PA_disk uncertainties are not normally propagated through Ω_p uncertainties, PA_disk directly affects Ω_p measurements using both IFS and long-slit spectra. According to D03, we expect errors on PA_disk to be the largest (and systematic) source of error in TW Ω_p measurements, as the measured Ω_p can be under- or overestimated by ≈50% for a misalignment angle δPA_disk of only ≈5°. In particular, it might be that some of the so-called ultrafast bars recently reported in the literature (Guo et al. 2019) are due to such misalignments. We therefore carry out tests analogous to those of D03 in Section 4.5.1, exploring a larger range of misalignment angles and different disk view angles. We also explore potential empirical disk position angle diagnostics (Section 4.5.2).

2. The orientation of a barred disk with respect to the observer is very important. In particular, the bar angle relative to the galaxy major axis (ϕ_bar) is crucial. Indeed, if the bar is exactly parallel or perpendicular to the disk major axis, the surface brightness distribution and velocity field will remain symmetric with respect to the disk minor axis. The integrands in Equation 1(a) will thus remain odd, again yielding an undefined Ω_p. As TW measurements utilize ${V}_{| | }$, nearly face-on (i ≈ 0°) galaxies will also yield unreliable measurements, while bars are difficult to identify in nearly edge-on (i ≈ 90°) systems (the intrinsic thickness of the disks is then also apparent and may affect the results). We will thus identify the optimal ranges of the relative bar angle and inclination in Section 4.6, to derive a prescription to exclude a priori galaxies with inappropriate orientations.

3. There is also no specific rule to choose the optimal slit width to use. Generally, a slit width roughly equal to the PSF has been adopted. Guo et al. (2019) tested two different pseudo-slit widths and reported that they both yielded consistent measurements. We will explicitly test the influence of the slit width on TW Ω_p measurements in Section 4.7.

4. Lastly, the limited spatial resolution of the data may affect measurements. Indeed, observations do not yield the intrinsic surface brightness Σ(X, Y) and mean velocity ${V}_{| | }(X,Y)$ of the chosen tracer, but rather those quantities convolved by the PSF of the observations W(X, Y), such that the observed surface brightness Σ_o(X, Y) and velocity field V_o(X, Y) are given by

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{{\rm{o}}}(X,Y)=\int W(X-{X}^{{\prime} },Y-{Y}^{{\prime} }){\rm{\Sigma }}({X}^{{\prime} },{Y}^{{\prime} }){{dX}}^{{\prime} }{{dY}}^{{\prime} }\end{eqnarray} \tag{ 4 }$$

and

$$\begin{eqnarray}&&{V}_{{\rm{o}}}(X,Y)=\displaystyle \frac{\int W(X-{X}^{{\prime} },Y-{Y}^{{\prime} }){\rm{\Sigma }}({X}^{{\prime} },{Y}^{{\prime} }){V}_{| | }({X}^{{\prime} },{Y}^{{\prime} }){{dX}}^{{\prime} }{{dY}}^{{\prime} }}{{{\rm{\Sigma }}}_{{\rm{o}}}(X,Y)}.\end{eqnarray} \tag{ 5 }$$

Tremaine & Weinberg (1984) argued that the limited spatial resolution of observations is unimportant if the PSF W(X, Y) is an even function of X, which is usually the case. To verify this, however, we will directly test in Section 4.8 how the PSF affects TW Ω_p measurements.

The N-body simulation we adopt is shown in Figure 1. It is a simple simulation of a disk galaxy with 10⁶ live particles of total mass M_d = 4.25 × 10¹⁰ M_⊙, evolving in a rigid dark matter halo spherical potential ${\rm{\Phi }}(r)=\tfrac{1}{2}{V}_{{\rm{c}}}^{2}\mathrm{ln}(1+{r}^{2}/{r}_{{\rm{c}}}^{2})$, where r is the spherical radius, V_c ≈ 250 km s⁻¹ is the asymptotic circular velocity at infinity, and r_c = 15 kpc is the core radius. The particles are initially distributed in a dynamically cold (Toomre's Q ≈ 1.5) axisymmetric exponential disk with a scale length of ≈1.9 kpc and a scale height of ≈0.2 kpc. A bar forms spontaneously and quickly buckles in the vertical direction. To create our mock data sets, we selected a snapshot at 2.4 Gyr (shown in Figure 1), when the bar is quasi-steady.

Zoom In Zoom Out Reset image size

We measure a maximum bar ellipticity ${\epsilon }_{\mathrm{bar},\max }\approx 0.47$ using the Image Reduction and Analysis Facility (IRAF; Tody 1986, 1993) Ellipse task, from the first maximum of the radial profile of ellipticity of the isophotes. We estimate the bar (half-)length to be a_bar ≈ 4.6 kpc, from the average of the radius of the first minimum in the radial ellipticity profile (4.7 kpc) and the radius suggested by the bar–interbar contrast method (4.5 kpc; see Aguerri et al. 2000).⁷ This maximum ellipticity and the (half-)length of the bar compare favorably to those of typical bars (see, e.g., Marinova & Jogee 2007), although we could of course rescale our simulation to any desired physical size. The simulated galaxy effective (half-mass) radius is R_e = 2.4 kpc. Our simulated barred galaxy is thus well suited to realistic tests of TW measurements.

As a function of radius, Figure 2 shows the m = 2 Fourier amplitudes of features in the density distribution that have well-defined frequencies, thus revealing coherent patterns and their pattern speeds (measured using several snapshots closely spaced temporarily around our selected time of 2.4 Gyr; see Sparke & Sellwood 1987 for more details). The figure clearly shows that the bar has a constant intrinsic pattern speed Ω_p,int = 36.3 km s⁻¹ kpc⁻¹ throughout its extent, while a pair of transient spiral arms with a lower pattern speed (Ω_p,spiral = 20.7 km s⁻¹ kpc⁻¹) dominate the disk beyond the bar region.

Zoom In Zoom Out Reset image size

At first, we align the simulated bar with the data X_grid-axis, as shown in Figure 1. To create multiple mock data sets, the simulation is then rotated counterclockwise around the Z_grid-axis by ϕ_bar,⁸ and inclined with respect to the X_grid-axis (i.e., the major axis of the galaxy) by i, where ϕ_bar ranges from 0° to 180° in steps of 5° and i ranges from 0° to 90° also in steps of 5°. We generally assume that the disk major axis is aligned with the X-axis of the grid (PA_disk = 90°), although we discuss the case where it is not in Section 4.4.

After rotation, the simulated disk is placed at a distance of 100 Mpc, typical of the MaNGA survey sample galaxies (Bundy et al. 2015). Assuming a stellar mass-to-light ratio M/L = 2 ${M}_{\odot }/{L}_{\odot ,r}$ at r band, we then convert the projected mass to an r-band flux (or equivalently an r-band surface brightness Σ) map, with an original fine sampling (i.e., spaxel size) of ≈008 or 40 pc.

These finely binned flux maps are then convolved by a Gaussian PSF of variable width to mimic different seeings (i.e., different observing conditions). Our adopted PSF FWHMs range from 0'' to 5'' in steps of 05, corresponding respectively to extremely good (e.g., adaptive optics) and bad seeings. After convolution, to generate realistic mock images, we further bin all the flux maps to match the Sloan Digitized Sky Survey (SDSS) imaging survey sampling size and add the SDSS mean sky background and noise, resulting in a final coarse sampling (i.e., spaxel size) of 04 or 200 pc at D = 100 Mpc (Gunn et al. 2006).

The creation of the required velocity fields is analogous to that of the flux maps, except for the convolution step. Indeed, unlike mass (i.e., flux), mean velocity is not a conserved quantity under convolution, but momentum is. Using the original fine grid, we therefore first calculate the mean LOS velocity ${V}_{| | }$ in each spaxel and then multiply the resulting finely binned velocity map by the associated finely binned flux map, resulting in a finely binned map of the quantity ${V}_{| | }{\rm{\Sigma }}$. It is this map that we then convolve by the PSF, rather than the finely binned velocity map. The convolved finely binned ${V}_{| | }{\rm{\Sigma }}$ map is then more coarsely (re)binned as before and divided by the associated coarsely binned (and convolved) Σ map to obtain the desired properly convolved coarsely binned ${V}_{| | }$ map (see Equations (4) and (5)).

In the end, we thus have mock flux (or surface brightness) and mean velocity maps with a range of relative bar angles ϕ_bar, inclinations i, and spatial resolutions (PSFs), all of which have a spaxel size of 200 pc (04 at a distance of 100 Mpc). Figure 3 shows examples of mock surface brightness and mean velocity maps with ϕ_bar = 45°, i = 45°, and two different seeings.

Zoom In Zoom Out Reset image size

Pseudo-slits are positioned according to the projected disk orientation and bar length in the mock images. Most pseudo-slits are located within the bar region (in terms of their offset Y from the galaxy major axis), but as the influence of a bar extends beyond its extent, we also locate a few extra pseudo-slits beyond the bar, to test whether these slits still yield accurate measurements of Ω_p. Moreover, the slit widths are varied from 04 to 2'' in steps of 04, to investigate the slit width influence on the measurements. To obtain an independent measurement for each slit, adjacent pseudo-slits touch, but are not overlapped with, each other.

Figure 4 schematically shows how pseudo-slits are overlaid on mock images, for both the configuration used in most of the tests shown (ϕ_bar = 45°, i = 45°, and a slit width of 12) and a configuration with narrower 04 slits.

Zoom In Zoom Out Reset image size

In our study, as by construction the galaxy major axis is always aligned with the X_grid-axis, we generally use easily constructed, perfectly rectangular horizontal slits, which are symmetric with respect to the minor axis (see the left panel of Figure 5). However, as in practice the galaxy major axis (and thus the pseudo-slits) may be misaligned with respect to the axes of the data/grid, we also create pseudo-slits with irregular (zigzagged) asymmetric shapes, comprising only those spaxels whose center falls within the desired (perfectly rectangular) slit (see the right panel of Figure 5). We then use those irregularly shaped pseudo-slits to test their effects (and the potential effects of recasting the grid) on TW Ω_p measurements in Section 4.4.

Zoom In Zoom Out Reset image size

Equation 1(a) suggests that an exact determination of the minor-axis position (X_c) and the galaxy systemic velocity (V_sys) is very important for individual slit measurements and the TW averaging method. The center and mean velocity of our simulated galaxy are initially well determined (and by construction set to 0), but thereafter the galaxy can shift slightly during its evolution. It is thus preferable to determine the value of X_c and V_sys from the chosen snapshot of the simulation itself. In fact, by measuring the mean position and velocity of our simulated galaxy in increasingly large volumes (centered on 0), we see that they are indeed nearly 0 up to a radius of ≈10 kpc, but that significant offsets (like those discussed below in Figure 6) exist when considering particles farther out, presumably because of "rogue" particles and asymmetries in the galaxy outer parts. Since most of the slits considered in our analyses do reach those distances, it appears essential to measure X_c and V_sys empirically from the simulation data.

Zoom In Zoom Out Reset image size

Of course, an analogous determination of X_c and V_sys is likely to be essential for observational data, as they are unlikely to be known a priori and will need to be determined from the observational data themselves (i.e., a cube in the case of IFS data) or from ancillary imaging and spectroscopic data (see, e.g., Bureau et al. 1999).

Equation 1(a) suggests that if X_c and/or V_sys are wrong, slits with the same offset but on opposite sides of the galaxy major axis will yield Ω_p measurements that are systematically (and increasingly) biased in opposite ways (i.e., respectively larger and smaller than the truth). To measure X_c and V_sys (for each bar orientation ϕ_bar, disk inclination i, and spatial resolution), we thus first measure Ω_p for multiple slits (inner slits only; see Section 4.2) and for a wide range of possible X_c and V_sys values and then select the X_c and V_sys (and associated Ω_p) that minimize the sum of the squares of the differences of opposite slit pairs (thus ensuring that the two slits in each pair yield Ω_p measurements that are as similar to each other as possible).

We first let X_c and V_sys vary over a very wide range of values for every (ϕ_bar, i) pair. Most X_c and V_sys thus selected are small, as expected, but for extreme values of ϕ_bar and i the X_c and V_sys selected are nearly always at the extreme of the ranges allowed (no matter how large these are), and they often rapidly switch in an opposite manner from positive to negative values (suggesting that X_c and V_sys are somewhat degenerate, as expected from Equation 1(a)). Since we will later discard those extreme (ϕ_bar, i) pairs as inappropriate for Ω_p measurements (see Section 4.6), in practice we restrict X_c and V_sys to smaller ranges appropriate for most (ϕ_bar, i) pairs (−400 pc ≤ X_c ≤ 400 pc and −10 km s⁻¹ ≤ V_sys ≤ 10 km s⁻¹).

Figure 6 shows the resulting X_c and V_sys offsets (with respect to 0) as a function of ϕ_bar and i for a seeing of 2'', no binning, and no position angle misalignment. Except for extreme values of ϕ_bar and i, the systemic velocity offsets are nearly always small (generally ≲8 km s⁻¹) and show no obvious trend. The results are similar for the minor-axis position offsets, which can be large for extreme (ϕ_bar, i) pairs but are otherwise small (generally ≲1 pixel and always ≲2 pixels) and again show no clear trend. The increased minor-axis position offsets at small i are simply due to the deprojection (amplifying small offsets).

Importantly, Figure 7 shows that changes in Ω_p when applying X_c and V_sys offsets are small (compared to measurements with no offset), ≲3% excluding extreme (ϕ_bar, i) pairs, as expected from the fact that the simulation was centered to start with. Of course, the changes in the case of observational data are likely to be more significant, as a result of the poorer initial guesses for X_c and V_sys.

Zoom In Zoom Out Reset image size

Ideally, the integrals in Equation 1(a) should range from $-\infty$ to $+\infty$, but this range is necessarily limited in real observations. As the integration limits (i.e., the length of each pseudo-slit) increase, measurements of Ω_p based on Equation 1(a) should asymptotically converge and reach the true Ω_p value at the (projected) radius where the disk becomes axisymmetric. It is thus very important to establish the minimum pseudo-slit length (i.e., the minimum spatial coverage) required for accurate TW measurements.

To establish this, we plot in Figure 8 distance (i.e., projected radius) profiles of the quantities $\int {V}_{| | }\,{\rm{\Sigma }}\,{dX}$ (the numerator of Equation 1(a)), $\int X\,{\rm{\Sigma }}\,{dX}$ (the denominator of Equation 1(a)), and Ω_p (the ratio of the numerator and denominator of Equation 1(a) divided by the known $\sin i$), constructed by gradually increasing the (half-)slit length (i.e., the integration limits). Multiple pseudo-slits are shown, with different offsets from the disk major axis, for a mock data set with ϕ_bar = 45°, i = 45°, 2'' seeing, 12 slit width, no binning, and δPA_disk = 0°.

Zoom In Zoom Out Reset image size

As alluded to above, for ease of interpretation and comparison to the known intrinsic bar pattern speed Ω_p,int = 36.3 km s⁻¹ kpc⁻¹, in Figure 8 and all other figures and measurements in this section we divide each Ω_p sin i measurement by sin i using the known inclination (i.e., the inclination with which the simulation was projected). For real observations, the accuracy with which an ${{\rm{\Omega }}}_{{\rm{p}}}\sin i$ measurement can be deprojected clearly depends on the accuracy of the measured inclination, adding an additional uncertainty to the Ω_p measurement.

Figure 8 shows that $\int {V}_{| | }\,{\rm{\Sigma }}\,{dX}$ and $\int X\,{\rm{\Sigma }}\,{dX}$ have converged by a distance of ≈8 kpc (≈3.3R_e for our simulated galaxy, much larger than the bar half-length), where convergence is defined here as being within 10% of the known true value. However, somewhat surprisingly, Ω_p converges much earlier, by a distance of ≈3 kpc (≈1.3R_e, slightly shorter than the projected bar half-length). The convergence distance for $\int {V}_{| | }\,{\rm{\Sigma }}\,{dX}$ and $\int X\,{\rm{\Sigma }}\,{dX}$ is thus a rather conservative estimate of the required slit (half-)length, and a much smaller field coverage is sufficient for reliable TW Ω_p measurements.

While Figure 8 shows the convergence test for a single $({\phi }_{\mathrm{bar}},i)$ pair, convergence tests for the entire range of ϕ_bar and i yield similar results, i.e., while the convergence distances of the two integrals taken separately (numerator and denominator of Equation 1(a)) are always rather large (and slightly larger for smaller ϕ_bar), the convergence distance of their ratio (i.e., Ω_p) is always much smaller (and remains essentially unchanged). Guo et al. (2019) also tested the influence of the slit (half-)length on TW measurements and found that Ω_p distance profiles converge by ≈1.2 times the bar half-length. The convergence distances are similar for different inclinations and bar orientations.

Having said that, the convergence distance is likely to differ from galaxy to galaxy, and Ω_p distance profiles may never converge in the presence of large-scale asymmetries such as spiral arms and lopsidedness. We therefore recommend checking whether the Ω_p distance profiles converge on a slit-by-slit and galaxy-by-galaxy basis. Only those slits where Ω_p has converged yield reliable bar pattern speed measurements.

Here, for simplicity, we adopt as the pattern speed measured for one slit the value at a distance of 10 kpc (i.e., the absolute value of the integration limits and thus the half-slit length), well beyond the convergence distance of any slit.

The different colors in Figure 8 show pseudo-slits with different offsets from the simulated galaxy major axis (or, equivalently, different Y_grid positions), while open and filled circles show slits with the same offset but on opposite sides of the major axis. Except for the three outermost pseudo-slits, all other pseudo-slits (which we will refer to as inner slits) are located within the bar region. The three panels of Figure 8 show that the distance profiles of the outer slits have significant fluctuations, much greater than those of the inner slits. In particular, the (absolute) values of the profiles can decrease with radius. Most importantly, the bottom panel of Figure 8 shows that, as the offset from the major axis increases, all slits yield a systematically and increasingly biased measurement of Ω_p (toward lower values), with the three outer slits significantly worse. This is likely due to the two transient spiral arms with a lower pattern speed in the outer parts of the simulated disk (see Figure 2). Indeed, beyond the bar ends, the contribution of the bar to the pattern becomes less significant and the spiral arms begin to dominate. It is thus important to quantify this effect to ascertain the reliability of TW Ω_p measurements using increasingly offset pseudo-slits.

To investigate the effects of the spiral arms on the bar pattern speed measurements, we compute a simple weighted measure of Ω_p (${{\rm{\Omega }}}_{{\rm{p}},\mathrm{weighted}}$), by weighting the bar and spiral arm pattern speeds by their likely contributions to the integrals in Equation 1(a). We thus calculate the luminosity within and beyond the (projected) bar region along each pseudo-slit and then weight the two pattern speeds accordingly, i.e.,

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{p}},\mathrm{weighted}}\equiv \displaystyle \frac{{{\rm{\Sigma }}}_{\mathrm{bar}}\,{{\rm{\Omega }}}_{{\rm{p}},\mathrm{bar}}+{{\rm{\Sigma }}}_{\mathrm{spiral}}\,{{\rm{\Omega }}}_{{\rm{p}},\mathrm{spiral}}}{{{\rm{\Sigma }}}_{\mathrm{bar}}+{{\rm{\Sigma }}}_{\mathrm{spiral}}},\end{eqnarray} \tag{ 6 }$$

where ${{\rm{\Omega }}}_{{\rm{p}},\mathrm{bar}}={{\rm{\Omega }}}_{{\rm{p}},\mathrm{int}}=36.3$ km s⁻¹ kpc⁻¹, ${{\rm{\Omega }}}_{{\rm{p}},\mathrm{spiral}}\,=20.7$ km s⁻¹ kpc⁻¹ (see Figure 2), and Σ_bar and Σ_spiral are the fluxes within and beyond the bar region, respectively. The bar region is defined here as the elliptical region with an ellipticity equal to the maximum bar ellipticity (${\epsilon }_{\mathrm{bar},\max }=0.47$) and a semimajor-axis radius equal not to the bar half-length (a_bar = 4.6 kpc) but rather to the radial extent of the main feature with a pattern speed equal to that of the bar in Figure 2 (8 kpc).

The resulting weighted pattern speeds are shown as horizontal lines in the bottom panel of Figure 8, appropriately colored for each pseudo-slit (dotted and solid lines denote pseudo-slits with, respectively, positive and negative offsets from the major axis; see Figure 4). As expected, the figure shows that ${{\rm{\Omega }}}_{{\rm{p}},\mathrm{weighted}}$ systematically decreases with increasing major-axis offset. Most importantly, ${{\rm{\Omega }}}_{\mathrm{bar},\mathrm{weighted}}$ roughly agrees with the TW measurement for all the slits, and it slowly tends to ${{\rm{\Omega }}}_{{\rm{p}},\mathrm{spiral}}\approx 21$ km s⁻¹ kpc⁻¹ for the three outer slits. This indicates that pseudo-slits located outside the bar region will not yield accurate TW Ω_p measurements (as outer spiral arms increasingly bias the measurements), in turn suggesting that only pseudo-slits located within the bar region should be used.

Figure 9 shows a comparison of the TW Ω_p measurements for the different slits and for appropriate slit averages. In the left panel, the TW averaging method is used, whereby each measurement shown is the average of that for multiple pseudo-slits. The gray solid line shows the known intrinsic pattern speed of the bar in the simulated galaxy (${{\rm{\Omega }}}_{{\rm{p}},\mathrm{int}}=36.3$ km s⁻¹ kpc⁻¹), while each colored solid line shows the average pattern speed of all the slits within a given offset from the galaxy major axis (i.e., all slits within a given minor-axis position $| {Y}_{\mathrm{grid}}|$), identified by the large circles of the same color. Again, we see that the average Ω_p decreases as slits with increasingly large offsets from the major axis are used. In the test shown, the adopted TW Ω_p measurement would be >5% lower than the real value when all pseudo-slits are used, and ≈5% too low when only the inner pseudo-slits are used.

Zoom In Zoom Out Reset image size

The right panel of Figure 9 shows an analogous comparison of the TW Ω_p measurements when the TW fitting method is used instead. Here $\langle V\rangle$ and $\langle X\rangle$ are calculated for each pseudo-slit, and each adopted Ω_p is obtained from the fitted slope to the ($\langle V\rangle$ vs. $\langle X\rangle$) data points within a given offset from the galaxy major axis (again color-coded). The results are consistent with those of the TW averaging method, the fitted slope decreasing by a similar fraction as slits increasingly offset from the major axis are incorporated into the fit. The right panel of Figure 9 also shows that the (absolute) values of $\langle V\rangle$ and $\langle X\rangle$ decrease for the outer slits (after peaking for the pseudo-slits closest to the bar ends).

Compared to the inner slits, we thus see from Figures 8 and 9 that the distance profiles, measurement errors, and $\langle V\rangle$—$\langle X\rangle$ diagrams of the outer slits behave differently than those of the inner slits. In addition to an actual bar (half-)length measurement, these behaviors can help us to distinguish the inner and outer slits, and thus to select the appropriate pseudo-slits for a TW Ω_p measurement. Whether using the TW averaging or the TW fitting method, we therefore recommend using only pseudo-slits located within the bar region.

To test the effect of spatial binning, we bin the mock images over multiple spaxels using the commonly used Voronoi binning algorithm of Cappellari & Copin (2003). Differently from real observations, the noise is defined here as the Poisson noise of the total flux within each bin. The spaxels within each bin are labeled, and the surface brightness and velocity of the bin are then replaced by their (luminosity-weighted) spatial averages. Figure 10 shows binned flux and mean velocity maps with different S/N thresholds. Given the necessarily different S/N definitions, it is not possible to directly compare the S/N estimates of real observations with ours, so we use instead the total number of bins within the field of view to characterize the severity of the binning applied (more severe binning, or equivalently higher S/N thresholds, leading to fewer and larger bins). We tested the sensitivity of both the TW averaging method and the TW fitting method to binning, and both yield consistent results.

Zoom In Zoom Out Reset image size

Figure 11 shows the relative error of the bar pattern speed measurement (${\rm{\Delta }}{{\rm{\Omega }}}_{{\rm{p}}}/{{\rm{\Omega }}}_{{\rm{p}},\mathrm{int}}$) as a function of the number of bins for the TW averaging method, for a range of viewing angles. The FOV of the mock images is 20 × 20 kpc² in all cases. As expected, when the number of bins is large (here ≳200, i.e., little binning or low S/N threshold), spatial binning has no effect on the Ω_p measurements (independently of the viewing angle). Interestingly, the relative Ω_p error does not increase monotonically with the number of bins when the number of bins is decreased; it changes abruptly when the number of bins is sufficiently small (here ≲200). This is probably a direct consequence of the importance of the bins' positions and shapes to the TW integrals. The results of the TW integrals will be very different for pseudo-slits intersecting, e.g., a few small bin centers versus the boundaries of several large bins. Overall, Figure 11 shows a general trend of increasing (absolute value of the) relative error ${\rm{\Delta }}{{\rm{\Omega }}}_{{\rm{p}}}/{{\rm{\Omega }}}_{{\rm{p}},\mathrm{int}}$ with decreasing number of bins (i.e., more aggressive binning or higher S/N thresholds) when the number of bins is sufficiently small. The bar pattern speed Ω_p is then almost always underestimated, by as much as 15% for the most severe binning considered.

Zoom In Zoom Out Reset image size

From Figure 11, we can draw the conclusion that binned IFS data with a number of bins smaller than ≈200 in an FOV of 20 × 20 kpc² should be discarded for TW Ω_p measurements.

As shown in the right panel of Figure 5, when the major axis of the disk is misaligned from the data/grid axes, the integrations along the (pseudo-)slits required for TW measurements (see Equation 1(a)) are nontrivial. Naively, one can simply integrate (i.e., sum) along an irregularly shaped slit comprising only those spaxels whose center falls within the desired (perfectly rectangular) slit (see the red polygon in the right panel of Figure 5). Alternatively, one can recast (i.e., regrid) the grid to align the galaxy major axis with the X_grid-axis (see the blue rectangle in the left panel of Figure 5), or equivalently carry out a more complex summation along the slit (see the blue rectangle in the right panel of Figure 5).

To quantify the potential biases associated with summing along irregularly shaped slits and/or the recasting process, we create new mock data sets where the disk major axis is misaligned from the X_grid-axis. Specifically, after projection, we further rotate the model counterclockwise, so that the angle between the disk major axis and the data/grid axes (PA_disk) ranges from 0° to 90° in steps of 5°, this for all ϕ_bar and i.

To simulate the effects of recasting, we subsequently also recast (i.e., rotate and regrid) all the above mock data sets to align the disk major axis with the X_grid-axis. To achieve this, we apply a bilinear interpolation to the surface brightness (Σ) maps directly. Analogously to the convolutions described in Section 3.2, however, we first multiply the velocity fields ${V}_{| | }$ by their associated surface brightness maps Σ to obtain ${V}_{| | }{\rm{\Sigma }}$ maps. We then apply the bilinear interpolation to these ${V}_{| | }{\rm{\Sigma }}$ maps and divide the recast ${V}_{| | }{\rm{\Sigma }}$ maps by their associated recast Σ maps, to finally obtain the desired recast velocity field ${V}_{| | }$. All the recast mock surface brightness maps and velocity fields then have the disk major axis aligned with the X_grid-axis, and the pseudo-slits constructed are perfectly rectangular, symmetric with respect to the minor axis, and parallel to the X_grid-axis (see the left panel of Figure 5).

TW Ω_p measurements are then carried out on the two sets of simulated data, those recast and those with irregularly shaped slits. These Ω_p measurements are then compared to each other and to the known intrinsic bar pattern speed of the simulated galaxy (${{\rm{\Omega }}}_{{\rm{p}},\mathrm{int}}=36.3$ km s⁻¹ kpc⁻¹) in Figure 12, for mock data sets with i = 45°, a range of ϕ_bar, 2'' seeing, 0.4 and 12 slit widths, no binning, δPA_disk = 0°, and for both the TW averaging and TW fitting methods. Measurements carried out without recasting (i.e., with irregularly shaped slits) and with recasting (i.e., perfectly rectangular slits) are shown side by side for comparison.

Zoom In Zoom Out Reset image size

Comparing the naive integrations along irregularly shaped slits and recast measurements (i.e., each pair of panels in Figure 12), Figure 12 shows that the two sets of measurements are consistent. However, the recast measurements always show less scatter. Indeed, the relative difference between the measurements ${\rm{\Delta }}\equiv | {{\rm{\Omega }}}_{{\rm{p}},\max }-{{\rm{\Omega }}}_{{\rm{p}},\min }| /{{\rm{\Omega }}}_{{\rm{p}},\mathrm{int}}$ is then ≲2% for both narrow and wide slits, where ${{\rm{\Omega }}}_{{\rm{p}},\max }$ and ${{\rm{\Omega }}}_{{\rm{p}},\min }$ are, respectively, the maximum and minimum derived Ω_p at a given ϕ_bar. The relative difference Δ can be as large as 7% for narrow and wide slits before the recasting process. Simulated galaxies at different inclinations show the same trends. This suggests that naive integrations with irregularly shaped slits do affect TW Ω_p measurements, presumably due to the asymmetries introduced (and removed by recasting the grid, so that the only asymmetries left are those due to the bar itself, as required by TW). Recasting the grid (or, equivalently, carrying out a more complex summation along the slits) should thus help to reduce measurement uncertainties.

Interestingly, Figure 12 shows that, for any given ϕ_bar, the TW Ω_p measurements do not systematically depend on PA_disk, which simply introduces a little bit of scatter in the measurements. This is reassuring and confirms the suggestion above that a misalignment of the galaxy major axis from the data/grid axes (and potential associated recasting effects) does not significantly impact TW Ω_p measurements with IFSs. However, grid recasting does help to decrease the scatter of TW measurements with PA_disk.

Figure 12 also reveals that the TW fitting and TW averaging methods generally share the same trends. However, the TW fitting method systematically yields slightly lower TW Ω_p measurements than the TW averaging method, the difference increasing with increasing ϕ_bar. This is somewhat surprising, as one would naively expect the TW fitting method to be superior to the TW averaging method. We do not fully understand this tendency, but it seems to be related to the aforementioned fact that the Ω_p measurements decrease with increasing offset from the major axis. Indeed, the inner slits near the ends of the bar (i.e., the green data points in the right panel of Figure 9) yield both the lowest Ω_p estimates and the greatest (absolute) values of $\langle V\rangle$ and $\langle X\rangle$, thus effecting the greatest leverage (down) on Ω_p when fitting the straight line. This in turn yields lowered estimates compared to ones with effectively equal weight for all data points. This effect becomes more acute for larger ϕ_bar as the intersection of the bar and the slit decreases, and it can be minimized by neglecting the slits near the ends of the bar.

In any case, most importantly here, even after recasting Figure 12 shows that the scatter across the measurements is most significant at small ϕ_bar (i.e., ϕ_bar ≲ 15°; both ϕ_bar ≲ 15° and 75° ≲ ϕ_bar ≲ 105° before recasting). The relative difference Δ ≈ 2% when ϕ_bar ≳ 15°, but it can be as large as 7% for ϕ_bar ≲ 15° and 11% for 75° ≲ ϕ_bar ≲ 105° before recasting. This suggests that bar pattern speed measurements of galaxies with disadvantageous ϕ_bar will have large uncertainties, and that these galaxies should be excluded from observational samples. In Section 4.6, we shall therefore explore the optimal range of both ϕ_bar and i for sample selection.

4.5.1. Pattern Speed Uncertainties

To quantify the effects of misalignments (δPA_disk) of the (pseudo-)slits with respect to the true position angle of the disk (PA_disk), we must create new mock images. As described in Section 3.2, the simulated bar is normally first aligned with the data X_grid-axis, and the disk is then rotated about the Z_grid-axis counterclockwise by ϕ_bar and about the X_grid-axis by i. After these two rotations, the disk major axis lies along the X_grid-axis by construction. To introduce a misalignment of the pseudo-slits, we further rotate the disk about the Z_grid-axis by an angle δPA_disk (but continue to assume that the disk major axis is along the X_grid-axis for the purpose of the TW calculations). For these tests, i and ϕ_bar range from 15° to 165° in steps of 15° and δPA_disk ranges from −10° to 10° in steps of 1°. By convention (and as in D03), δPA_disk > 0 when the true disk major axis is rotated anticlockwise. For ϕ_bar ≲ 90°, the additional rotation increases the angle between the bar and the assumed major axis (i.e., the X_grid-axis), while for ϕ_bar ≳ 90° the additional rotation decreases that angle. We carry out this test using both the TW averaging method and the TW fitting method, but the results are very similar. The relative Ω_p errors introduced by position angle misalignments are thus shown in Figure 13 for the TW averaging method only.

Zoom In Zoom Out Reset image size

Figure 13 confirms the result of D03 that not only are position angle misalignments the most important factor potentially affecting TW Ω_p measurements, but their effect is also systematic. Indeed, Figure 13 suggests that the (absolute value of the) relative error ΔΩ_p/Ω_p,int is ≈35% for all inclinations i and most bar angles ϕ_bar when the misalignment angle is as small as 5°. When 75 ≲ ϕ_bar ≲ 105°, the relative Ω_p errors can be very large for even smaller δPA_disk. In fact, when δPA_disk ≈ 10°, ${\rm{\Delta }}{{\rm{\Omega }}}_{{\rm{p}}}/{{\rm{\Omega }}}_{{\rm{p}},\mathrm{int}}$ can reach 100%. This test thus suggests that the measured Ω_p can be severely overestimated when the angle between the bar and the assumed disk major axis is wrongly increased, while it can be severely underestimated when that angle is decreased.

We note that D03 also examined the effects of position angle misalignment on TW Ω_p measurements. He used a simulated barred galaxy with i = 45° and ϕ_bar = 30°, 45°, and 60° (his Figure 9), which can be directly compared with our tests. The results are analogous, except that our relative errors are slightly smaller, probably due to differences in the simulated galaxies.

4.5.2. Empirical Diagnostic

Given the importance of identifying the true disk position angle PA_disk to avoid the substantial systematic biases just described, it would be very useful to have an empirical diagnostic of it, i.e., a method to identify or measure the true disk position angle from the data themselves. In this subsection, we therefore explore such empirical diagnostics, unfortunately only with partial success.

We test three different parameters (scatters) to quantify the "goodness" of the position angle used: (a) the sum of the squares of the differences of opposite slit pair measurements, (b) the sum of the squares of the differences of all slit measurements with respect to the mean (i.e., the scatter across all pseudo-slit measurements), and (c) the sum of the previous two parameters. We define the goodness of a given position angle as $\left[1-(\sigma -{\sigma }_{\min })/({\sigma }_{\max }-{\sigma }_{\min })\right]$, where σ is the given measure of scatter evaluated at that position angle and σ_min and σ_max are, respectively, the minimum and maximum scatter measured within the entire position angle ranged probed. This goodness ranges from 0 to 1 by definition, with the maximum value (1) defining what naively would appear to be the most appropriate disk position angle to use. We compare the goodness of a range of δPA_disk for a number of (ϕ_bar, i) pairs (and for each of the three scatters defined) in Figure 14.

We naively expected the goodness to be maximum (1) for δPA_disk = 0°, thus allowing us to empirically identify the true position angle when it is not accurately known a priori. However, Figure 14 reveals the situation to be more complex. First, the maximum of the first parameter tested (a, red curves) is often not very well defined. Second, even for the second (b, green curves) and third (c, blue curves) parameters tested, although a clear maximum is often present near δPA_disk = 0°, it is often away from exactly δPA_disk = 0° (nevertheless suggesting that the second parameter, the scatter across all inner pseudo-slits, is the most promising diagnostic tested). Indeed, the peaks in the goodness profiles of the second and third parameters often deviate from δPA_disk = 0° by a few degrees (≲5°) when i ≲ 60° (the goodness profiles for i = 75° are uninformative). When ϕ_bar = 45°, the true disk position angle is accurately recovered (i.e., the goodness is maximum for δPA_disk ≈ 0°). However, when the bar is close to the disk major axis (ϕ_bar < 45°), the suggested disk position angle is always slightly larger than the truth (i.e., the goodness is maximum for δPA_disk > 0°). Conversely, when the bar is close to the disk minor axis (ϕ_bar > 45°), the suggested disk position angle is always smaller than the truth (i.e., the goodness is maximum for δPA_disk < 0°). Given the sensitivity of Ω_p to a disk position angle error of even a few degrees (see the previous Section 4.5.1), these diagnostics are simply not good enough.

Zoom In Zoom Out Reset image size

While showing some promise, the potential empirical diagnostics of the true disk position angle presented in Figure 14 thus fall short of what is required to ensure bar pattern measurements devoid of significant PA_disk-related systematic errors. We also do not fully understand the systematic dependence of the goodness adopted on the bar angle ϕ_bar. Further investigations are thus required to identify a more reliable empirical diagnostic of the true disk position angle.

As discussed in Section 2.1 (and according to Equation 1(a)), the TW integrals will sum to ≈0 if a bar is nearly parallel (ϕ_bar = 0°) or perpendicular (ϕ_bar = 90°) to the major axis of the disk. In addition, the LOS velocities will be very small (and undistinguishable within the uncertainties) if a galaxy is close to face-on (i = 0°), while a bar will be difficult to identify if a galaxy is close to edge-on (i = 90°). It is thus necessary to exclude those galaxies with disadvantageous ϕ_bar and i for reliable TW Ω_p measurements.

There is no specific prescription for the optimal Ω_p ranges, so we use our simulated galaxy with different projections to identify those optimal ranges here. To do this, we will consider two quantities. First, the relative difference of each average Ω_p measurement with respect to the known intrinsic bar pattern speed (${{\rm{\Omega }}}_{{\rm{p}},\mathrm{int}}=36.3$ km s⁻¹ kpc⁻¹), where the TW average is taken over all individual inner slit measurements of a given (ϕ_bar, i) pair. Second, the relative uncertainty of each average Ω_p measurement with respect to the known intrinsic bar pattern speed, where the uncertainty is defined as the rms scatter across all individual inner slit measurements of a given (ϕ_bar, i) pair.

We note that there are very few inner pseudo-slits when ϕ_bar ≲ 10° (or ϕ_bar ≳ 170°) and/or the disk is nearly edge-on, as the bar is then nearly parallel to the major axis of the disk and/or the projected bar length is too short. By the time i ≈ 90° is reached, measurements are essentially impossible for any ϕ_bar. In particular, the uncertainty measure defined above requires at least two slits.

Figure 15 shows the (absolute values of the) Ω_p relative differences and uncertainties derived from mock data sets with a 2'' seeing, 12 slit width, no binning, δPA_disk = 0°, and a range of ϕ_bar and i. From the figure, we see that the relative differences are ≳10% when ϕ_bar ≲ 10° and 75° ≲ ϕ_bar ≲ 105°, or i ≲ 15° and i ≳ 70°. Similarly, the relative uncertainties are ≳10% for 75° ≲ ϕ_bar ≲ 105° or i ≲ 15°. In any case, it is difficult to accurately measure the inclination of a nearly face-on galaxy, as the disk appears nearly (but may not be intrinsically perfectly) round, and the deprojection correction to the velocities (or equivalently the Ω_p sin i TW measurement) is very large. The projection of the bar on the minor axis is also too short to construct multiple pseudo-slits when ϕ_bar is small.

Zoom In Zoom Out Reset image size

As already mentioned, according to Equation 1(a), the TW integrals will sum to ≈0 if a bar is nearly parallel (ϕ_bar = 0°) or perpendicular (ϕ_bar = 90°) to the major axis of the disk. The Ω_p inferred is then expected to have large uncertainties. Figure 15 indeed suggests that the relative differences are very large when ϕ_bar ≈ 90°, but surprisingly the relative differences remain small for ϕ_bar ≈ 0° (or, equivalently, ϕ_bar ≈ 180°). To investigate this, we show in Figure 16 the TW Ω_p measurements for individual pseudo-slits (and their averages), for a range of bar angles ϕ_bar and inclinations i. Figure 16 reveals a clear systematic dependence of the measurements on the bar orientation. The relatively good quality of the measurements when ϕ_bar ≈ 0° is confirmed. However, unexpectedly, not only are the measurements poor for ϕ_bar ≈ 90°, but the curves appear antisymmetric about ϕ_bar ≈ 90° as well. We do not fully understand this, but we surmise that it is due to the decreasing contribution of the bar to the TW integrals when the angle between the bar and the major axis increases (while the influence of the spiral arms remains). In any case, TW Ω_p measurements should be treated with particular caution when the bar is close to the disk minor axis. It might also be that erroneous measurements at ϕ_bar ≈ 90° could explain so-called ultrafast bars (Guo et al. 2019).

Zoom In Zoom Out Reset image size

Our tests therefore provide useful guidelines for the selection of observational samples for TW Ω_p measurements. Indeed, using Figure 15, one can exclude inappropriate ϕ_bar and i that would result in significant measurement biases and/or uncertainties. For example, according to our simulation, and to keep both of these quantities to ≲10%, one should select barred disk galaxies with 10° ≲ ϕ_bar ≲ 75° and 105° ≲ ϕ_bar ≲ 170° and 15° ≲ i ≲ 70°. Measurements are particularly inaccurate when the bar is close to the disk minor axis (see also Figure 16). It is worth noting that the optimal ϕ_bar range is quoted in the intrinsic face-on values. In Figure 15, we also show the optimal observed ϕ_bar,obs range with a black box when projection effects are taken into account (see Footnote 8). The optimal observed ϕ_bar,obs range (black box) should be used when selecting samples for TW measurements.

Having said that, our tests are based on a single simulated barred galaxy, and the relative differences and uncertainties shown in Figure 15 would likely change slightly for other simulations/galaxies. In any case, as noted before, even favorable (ϕ_bar, i) pair TW Ω_p measurements can still suffer from large biases because of other large-scale asymmetric structures such as spiral arms and lopsidedness. Our recommendation thus remains to always look at the Ω_p distance profile for each slit and apply the convergence test discussed in Section 4.1 and Figure 8.

Our mock data sets span pseudo-slit widths of 04–2'' in steps of 04, with no overlap or gap between adjacent pseudo-slits. The number of slits that can be used thus decreases with increasing slit width (as the absolute size of the simulated galaxy is fixed). The bar pattern speeds measured from those individual pseudo-slits are shown in Figure 17, along with the averages using the TW averaging and TW fitting methods applied to the inner slits only.

Zoom In Zoom Out Reset image size

Overall, all the average measurements shown in Figure 17 are in agreement with each other to within ≈3%, well within any reasonable uncertainties (defined by, e.g., the scatter across individual measurements for each slit width). The slit width therefore has no significant effect on the measured average Ω_p.

As was already pointed out from Figure 12 (see also Figure 9), we again note that the TW fitting method yields slightly and systematically lower measurements than the TW averaging method, but again this is only by a few percent and within reasonable uncertainties.

We also note that the scatter between individual measurements (i.e., the Ω_p measured from individual slits) increases drastically for very narrow slits (i.e., slits significantly narrower than the seeing). As the measured Ω_p varies systematically with the offset from the galaxy major axis (see Section 4.2), this is probably simply due to the wider slits encompassing (and thus averaging) wider ranges of Ω_p. In any case, overly narrow pseudo-slits should be avoided for single slit measurements. Importantly, however, the average measurements remain reliable, so that again we conclude that the pseudo-slit width has no significant impact on the measurements.

Angular resolution is usually considered very important for observations, as it determines the ability to see intricate spatial details in the objects studied. In optical/infrared observations, this angular resolution is usually quantified by the FWHM of the Gaussian PSF (i.e., the seeing). In this study, we use spatial and angular resolution interchangeably, as the distance to our simulated galaxy is fixed, and we use our simulation to test the influence of spatial resolution on TW measurements.

Figure 18 shows TW Ω_p measurements using mock data sets with seeings ranging from 0'' to 5'' in steps of 05, and using both the TW averaging and the TW fitting method. The average measurements using the TW fitting method are again slightly lower than those using the TW averaging method. Of more interest here, however, there is a weak trend of decreasing Ω_p with increasing seeing for the innermost slits (i.e., the magenta, red, and orange data points in Figure 18). The same trend is seen with other mock data sets with different ϕ_bar and i, and it is easily understood from the trend of decreasing Ω_p with increasing offset from the galaxy major axis described in Section 4.2. Indeed, with increasingly bad seeing, particles at increasingly large Y_grid (or, equivalently, increasingly large major axis offsets) influence the observed surface brightnesses and velocities within any given pseudo-slit (see Equations (4) and (5)). The measured Ω_p thus decreases slightly but systematically with seeing. The opposite is true for the last inner slits near the ends of the bar (i.e., the green data points in Figure 18), presumably for the same reason, i.e., with increasingly bad seeing particles at increasingly small Y_grid contribute to the measurements, and in consequence Ω_p systematically increases with increasing seeing.

Zoom In Zoom Out Reset image size

Most important here, however, is that the average measurements are essentially independent of seeing. Indeed, even for an unrealistically large seeing of $5^{\prime\prime}$, the relative error (${{\rm{\Omega }}}_{{\rm{p}},{5}^{{\prime\prime} }}-{{\rm{\Omega }}}_{{\rm{p}},{0}^{{\prime\prime} }})/{{\rm{\Omega }}}_{{\rm{p}},\mathrm{int}}$ of the average measurements is <1%, where ${{\rm{\Omega }}}_{{\rm{p}},{5}^{{\prime\prime} }}$ and ${{\rm{\Omega }}}_{{\rm{p}},{0}^{{\prime\prime} }}$ are the measured bar pattern speeds for 5'' and 0'' seeing, respectively, and Ω_p,int = 36.3 km s⁻¹ kpc⁻¹ is the known intrinsic bar pattern speed of the simulation. For a typical MaNGA seeing of ≈2'', this relative error is also <1%. Considering realistic uncertainties (again, e.g., the scatter across individual measurements for each seeing), Figure 18 suggests that spatial resolution has no significant effect on TW measurements. The TW method is thus reliable even at low spatial resolutions, as originally suggested by Tremaine & Weinberg (1984). Interestingly, it is thus well suited for poor-seeing backup observing programs.

Combining the advantages of imaging and spectroscopy, IFSs are now available on most telescopes. These instruments, however, have different characteristics (wavelength coverage, FOV, spectral resolution, angular sampling, etc.), that make them most appropriate for different observational strategies and thus scientific goals. More IFSs will also come online in the near future.

For TW Ω_p measurements, only those IFSs with a relatively large FOV are appropriate (see Section 4.1). We therefore select some IFSs with a large FOV and summarize their characteristics in Table 1. All operate in the optical domain, with FOV diameters larger than 30'' and spectral resolutions and angular samplings appropriate for dynamical work on nearby galaxies. Two Fabry–Perot instruments (GHaFaS and FaNTOmM) specifically target Hα emission, thus providing high-quality spectra for application of the TW method to ionized gas. Their FOVs are very large, and spectral resolutions very high. A new generation of imaging Fourier transform spectrographs has similar strengths (SpIOMM and SITELLE). Instruments mounted on large telescopes and/or exploiting adaptive optics have particularly high angular samplings (MUSE, VIMOS, Kyoto3DII). Although a high spatial resolution is not crucial for TW measurements (see Section 4.8), the data from these instruments also generally have higher S/N and can be binned more flexibly, which should lead to higher-quality TW Ω_p measurements. The SparsePak, DensePak, and PPack family of fiber-fed IFSs have unusually high spectral resolutions, but this is unlikely to be a main driver for TW measurements.

Table 1.  Characteristics of IFS Instruments Appropriate for TW Pattern Speed Measurements

Instrument	Wavelength Range	Spectral Res.	Spatial Sampling	Field of View	Telescope
	(Å)
RSS	4300–8600	300–9000	013	8'	SALT 10 m
Kyoto3DII	4000–7000	400, 7000	0056	19 × 19	Subaru 8.2 m
MUSE	4650–9300	≈3000	02	1' × 1'	VLT 8 m
VIMOS	3600–10,000	200–2500	067	54'' × 54''	VLT 8 m
SAURON	4500–7000	≈1500	094	41'' × 33''	WHT 4.2 m
GHaFaS	4500–8500	≈20,000	04	34 × 34	WHT 4.2 m
WEAVEa	3700–9600	5000, 20,000	13, 26	11'' × 12'', 78'' × 90''	WHT 4.2 m
SAMI	3700–9500	1700–13,000	16	15''	AAT 3.9 m
DensePak	3700–11,000	5000–20,000	30	30'' × 45''	WIYN 3.8 m
SparsePak	5000–9000	5000–20,000	47	72'' × 713	WIYN 3.8 m
SITELLE	3500–9000	1–10,000	032	11' × 11'	CFHT 3.6 m
PPak	4000–9000	≈8000	27	74'' × 64''	Calar Alto 3.5 m
VIRUS-P	3500–6800	≈850	43	17 × 17	McDonald 2.7 m
VIRUS-W	4340–6040	2500, 6800	32	105'' × 75''	McDonald 2.7 m
MaNGA	3600–10,400	≈2000	20	125–325	APO 2.5 m
CHILIa	3600–7200	900, 1900	56	71'' × 65''	Lijiang 2.4 m
FaNTOmM	4000–9000	10,000–60,000	16	17'	OMM 1.6 m
SpIOMM	3500–9000	1–25,000	05	12' × 12'	OMM 1.6 m

Note.

^aFuture IFS.

Download table as:  ASCIITypeset image

Over the past two decades, the development of IFSs has motivated many IFS surveys of nearby galaxies, the most important of which are summarized in Table 2. The Spectral Areal Unit for Research on Optical Nebulae (SAURON) and subsequent Atlas^3D projects using the SAURON IFS pioneered large-scale surveys, observing a mass- (K-band luminosity) and volume-limited sample of 260 early-type galaxies (Cappellari et al. 2011). The Calar Alto Legacy Integral Field Area (CALIFA) survey used the PPak IFS to study a sample of ≈670 galaxies selected for their optical size (Sánchez et al. 2012). The VIRUS-P Exploration of Nearby Galaxies (VENGA) survey observed a sample of 30 nearby spiral galaxies chosen for their extensive ancillary multiwavelength data (Blanc et al. 2013). The Sydney-Australian-Astronomical-Observatory Multi-object Integral-field Spectrograph (SAMI) survey is currently targeting 3400 galaxies using a multiplexed fiber-fed IFS with two wavelength channels (Croom et al. 2012). The Mapping Nearby Galaxies at APO (MaNGA) project aims to observe 10,000 galaxies over a wide range of stellar masses and environments (Bundy et al. 2015).

Table 2.  Characteristics of Relevant IFS Surveys

Survey	Sample Size	Field of View	Spatial Elements	Wavelength Range	Velocity Res.	Angular Res.
				(Å)	(σ; km s−1)	(Reconstructed FWHM)
MaNGA	10,000	>1.5, >2.5 Re	19–127	3600–10,300	50–80	≈25
WEAVE	5000	11'' × 12'', 78'' × 90''	37, 540	3660–9840	15, 60	⋯
SAMI	3400	>1 Re	61	3700–5700	70	≈22
				6300–7400	30	≈22
CALIFA	667	>2.5 Re	331	3700–7000	85, 150	≈24
Atlas3D	260	≈1 Re	1431	4800–5380	105	⋯
SIGNALS	38	11' × 11'	≈4,500,000	3630–6850	60	≈08
VENGA	30	≈0.7 R25a	246	3600–6800	≈120	≈56

Note.

^aR₂₅ is the radius of the 25 mag arcsec⁻² isophote in the R band.

Download table as:  ASCIITypeset image

Table 2 shows that Atlas^3D has the best spatial sampling but a relatively small spatial coverage and poor velocity resolution. CALIFA has the largest FOV, helping to ensure that the optical extent of most galaxies is fully covered (and thus that the TW integrals converge). The FOV of VENGA is also large, but its velocity resolution is rather poor for TW measurements. SAMI and MaNGA have good compromises between FOV and spatial sampling, as well as between wavelength coverage and velocity resolution. The advantage of MaNGA is its large sample size, but SAMI covers a wider range of environments. Both should enable comprehensive studies of the correlations between bar pattern speeds and other properties of galaxies. The FOV of MaNGA reaches at least 1.5R_e for two-thirds of the sample and at least 2.5R_e for one-third, the latter being particularly promising given the convergence test discussed in Section 4.1 (showing that TW Ω_p measurements typically converge by ≈1.3R_e). As a second-generation IFS, the Multi Unit Spectroscopic Explorer (MUSE; Bacon et al. 2010) provides a unique combination of large FOV, high spatial sampling, and generally high S/N, offering new opportunities for accurate Ω_p measurements using the TW method.