GEOMETRIC OFFSETS ACROSS SPIRAL ARMS IN M51: NATURE OF GAS AND STAR FORMATION TRACERS

A measurement method for the geometrical offset was developed by Egusa et al. (2004) and updated in E09. The offset is measured by eye as the azimuthal angular offset between the emission peaks of gas and star-forming region tracers. This Peak Tracing method starts by binning the phase diagrams in the radial direction with an appropriate bin size to match the spatial resolution of all tracer images (here we use a bin size of 5''). The emission peaks are visually determined in an azimuthal intensity profile of each bin. The offsets are defined as the azimuthal shifts of the peaks between the two tracers. We find that this visual identification tends to pick up only peaks with a high S/N. Obviously, some radii are not included in the analysis if no emission peaks could be identified in their azimuthal profiles. Figure 2 shows examples of identified peaks in all the four tracers in a part of Arm 1: (a) H i, (b) CO, (c) 24 μm, and (d) Hα. Gas is flowing in a counterclockwise direction.

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T08 cross-correlate the azimuthal profiles of the gas and SF tracers in each radial bin. They define a normalized cross-correlated function,

$$\begin{equation} cc_{x,y}(l)=\frac{\Sigma _{k}[(x_{k}-\bar{x})(y_{k-l}-\bar{y})]}{\sqrt{\Sigma _{k}(x_{k}-\bar{x})^{2}\Sigma _{k}(y_{k}-\bar{y})^{2}}}, \end{equation} \tag{ 2 }$$

where x_k and y_k are the fluxes in gas and SF tracers (in T08 and F11, H i and 24 μm, respectively), $\bar{x}$ and $\bar{y}$ are the mean values of the fluxes, and l is the lag (or the shift) in pixel units between the two profiles. The denominator normalizes the cross-correlation function based on the variance of each data set; this normalized cross-correlation function is useful when adopting a cutoff value to remove low signal-to-noise data at different wavelengths. In principle, the cross-correlation function shows a maximum at the value of l = l_max that best aligns the two profiles—finding the peak of the cross-correlation function should be enough to obtain l_max. In practice, we fit a polynomial to the profile of the cross-correlation function around the peak to reduce the effect of noise. T08 use a fourth-degree polynomial, and we follow their method. Examples of the cross-correlation function for a radial bin (at the galactic radius of 2.51 kpc) are shown in Figure 3 for each set of tracer combinations. The offset is simply ΔΘ(r) = l_max(r). T08 adopted a lower threshold (i.e., 0.2) for the normalized cross-correlation function to reject measurements with low significance and F11 used a threshold of 0.3. We set the threshold for an acceptable cross-correlation value to 0.3 to avoid low signal-to-noise measurements.

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We apply both the Peak Tracing method and the Cross-Correlation method to two gas tracers (CO and H i) and two SF tracers (Hα and 24 μm). The phase diagrams are binned radially in 5'' increments for all data. This bin size is chosen to match the resolutions of the H i and 24 μm data. We denote the spatial offset between the gas (tracer A) and star-forming regions (B) as ΔΘ_{A → B} and measure ΔΘ(r)_{H I→24 μm}, ΔΘ(r)_H I→Hα, ΔΘ(r)_{CO → 24 μm}, and ΔΘ(r)_{CO → Hα}. If multiple peaks exist in the cross-correlation function, the largest is chosen, since by definition this is the location where the peaks of the spiral arms are best aligned. For the Peak Tracing method, the locations of any multiple peaks are checked against maps of M51 to make sure that they correspond to the spiral arms instead of emission in the interarm regions (Koda et al. 2009).

Two types of error could be introduced in the analyses: (1) due to uncertainties in the determination of peak positions in each tracer, and (2) due to the misassociation of the peaks in gas and SF (e.g., in the case of multiple peaks). The first type of error is easy to estimate and is simply related to the resolution of the images in both methods. This relation may not be so obvious in the case of the Cross-Correlation method, but, for example, if there are point sources convolved with point-spread functions (PSF; i.e., resolutions) the cross-correlation function is simply a multiplication of the two PSFs with a variable lag. Its width (or the associated error) is therefore determined by the sizes of the resolutions. In all the data, we measure only the positions of bright peaks and the positional accuracy should be a small fraction of the resolutions. We here conservatively adopt half of the resolution as the error. This choice provides an error similar to that estimated by E09, but much larger than that of T08. The second type of error is difficult to estimate, and we neglect it for now. This error, however, should appear as scatter in our measurements.

Figure 5 compares the offsets measured using the same tracers and methods as the previous studies. The left column shows the results from the Peak Tracing method applied to one spiral arm (i.e., Arm 1; as in E09). The right column shows the same offsets but from the Cross-Correlation method applied to both arms (as done in T08 and F11). The offsets between H i and 24 μm are shown in the top panels, and the offsets between CO and Hα are shown in the bottom panels.

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The measured offsets in Figure 5 are consistent with the previous measurements of T08, E09, and F11 in terms of the ranges of their amplitudes. E09 compared CO and Hα data using the Peak Tracing method and found offsets up to 25°–30° in Arm 1. Figure 5 (bottom left) shows our corresponding measurement. The amplitude range of our offsets between CO and Hα is similar to that in E09 (see their Figure 5, Arm 1). Our range in Arm 2 is also consistent with that of E09 (see Figure 6). E09 found irregular offsets between CO and Hα along Arm 2, and our measurements for Arm 2 see similar irregularities, which are discussed later in this section. T08 analyzed H i and 24 μm using the Cross-Correlation method (see their Figure 4) and found smaller offsets (mostly less than 5°–10°), except for the innermost radii (some large offsets of 15°–20° in ≲  2 kpc). In our comparison of H i and 24 μm (Figure 5, top right), we find similar small offsets, most of which are under 10° in ≳ 2 kpc, consistent with T08 results. We were unable to measure offsets at the innermost radii (discussed above). F11 also measured offsets between H i and 24 μm using the Cross-Correlation method and found that they are mostly under 10° (their Figure 10), which is comparable to our amplitude range. The amplitude of the offsets found by T08, E09, and F11 agrees with our measured offsets when we use the same set of tracers and measurement methods. Our measurements reproduce the amplitude discrepancy in the previous studies.

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Figure 6 shows plots of all of the azimuthal offsets measured for each set of tracers as a function of galactic radius for the Peak Tracing method (left two columns) and the Cross-Correlation method (middle two columns). We measure the offsets in the two arms separately in these four columns. T08 and F11 analyzed the two arms together to measure one offset at each radius. For comparison, we also analyze the two arms together and show the plots in Figure 6 (right column). E09 found a strange trend in Arm 2, where the offset increases with radius (as opposed to the prediction in Figure 1). Our results for Arm 2 from the Peak Tracing method show similar behavior. Along Arm 2, the emission from the SF tracers appear to be scattered over a large area (though mostly at the leading side of the arm). Therefore, we discuss only Arm 1 in the following analysis when we analyze the two arms separately.

T08 also analyzed CO and 24 μm data (their Figure 7) and found that the offsets are even smaller than those seen between H i and 24 μm. We found the contrary. The offsets between CO and 24 μm in Figure 6 (third row) are larger compared to those between the H i and 24 μm (top row). F11 also compared a lower-resolution CO image with a 24 μm image and noted possible evidence for ordered offsets in M51, though they did not quantify the CO → 24 μm offsets further. Indeed, compared to their Figure 11, we find even larger offsets at the small radii (Figure 5 (top right)). Even if we smooth our CO data to a 13'' resolution to match F11's, we still find larger offsets than F11. F11 also analyzed the two arms separately, but report no difference. Again, this is inconsistent with our results. Part of the reason for this difference may be that they used higher transition CO (J = 2 − 1) data while we used CO (J = 1 − 0). The higher transition emission might be excited due to heating by young stars (Koda et al. 2012). If that is the case, CO (2 − 1) should appear closer to SF tracers (i.e., smaller offsets).

The two measurement methods provide, in general, consistent results and do not appear to be the cause of the discrepancy. Figure 7 directly compares the measured offsets in Arm 1 from the Peak Tracing method and the Cross-Correlation method (i.e., the points in Figure 6) and shows good agreement, with some outliers. The dotted lines have a slope of one. Two solid lines in each panel show ±5° from the dotted line. We find that 70% of the data points are within ±5°. Except for some outliers, most of the measured offsets are comparable, and neither method appears to bias the offsets in any systematic way in any of the panels. The two methods give roughly consistent offsets. The top left panel only has 3 out of 19 radii where the two measurement methods do not show a correlation. For the top right panel 6 out of 19 points do not agree. The bottom left panel has 5 points out of 19 and the bottom right panel has 6 out of 17 points that do not show a correlation.

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E09 discussed that the two arms show quantitatively different offsets and analyzed them separately with the Peak Tracing method. The difference between the two arms is clear in Figure 6 (first and second columns). Since the two methods generally provide consistent results when applied to single arms, it is natural to expect differences in the measured offsets when the two arms are analyzed simultaneously using the Cross-Correlation method. We see this in our analyses of both arms simultaneously (Figure 6 (fifth column) and separately (Figure 6 (third and fourth columns)) with the Cross-Correlation method. We note that both T08 and F11 analyzed the arms simultaneously, while F11 also isolated individual arms; the difference between the two arms was not mentioned in either case. From our work, regardless of the measurement method (the Peak Tracing or Cross-Correlation methods) and the analysis (the two arms separately or simultaneously), the offsets are mostly positive, but do not appear ordered.

The choice of tracers appears to be the dominant cause of the discrepancy. Large differences are seen among the panels in Figure 6 (left and middle columns), where the two arms are analyzed separately. Most remarkably, H i and 24 μm—the combination employed by T08 and F11—show the smallest, if non-zero, offsets for M51. Similarly, very small and zero offsets are seen between H i → Hα. If we use CO as a tracer of the dense, compressed gas (e.g., CO → Hα used by Egusa et al. 2004 and E09), the offsets are much larger. Clearly, the difference in gas tracer is the dominant cause.

Figure 8 shows a direct comparison of the offsets between gas tracers and SF tracers in a given radial bin. The left column shows the offsets from the Cross-Correlation method, while the right column shows those from the Peak Tracing method. The top panels compare the offsets found using H i with Hα and CO with Hα. The bottom panels show the same, but for 24 μm. The offsets between H i and the SF tracers are mainly zero; however, at the same radii the offsets between CO and the SF tracers are larger. In addition, when non-zero offsets are measured for both gas tracers, the offsets between CO and the SF tracers appear larger than the offsets between H i and the SF tracers. This is evident in Figure 8 as most data points appear in the top left parts of the panels. Occasional outliers exist, but there are no more than two in each panel.

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There are multiple components that could be contributing to the H i emission, which would give different offsets (and possibly non-zero offsets) depending on which component is dominant at the radius in question. For example, H i may trace some of the gas compressed upon entry into a spiral arm, resulting in positive offsets, but would also trace the gas photodissociated by newly formed young stars, leading to zero offsets. There is also an extended background component in the H i emission that is virtually everywhere across the galactic disk. We can also find some spatial displacements between H i and CO peaks in Figure 4—H i is mostly downstream of CO, indicating that the H i emission is not simply from the dense gas, but from multiple sources. This trend is also seen in Figure 9, where ΔΘ_CO→H I is plotted against radius. The positive offsets suggest that H i is downstream of the CO. We will discuss this more in Section 5.1.

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As for SF tracers, 24 μm and Hα peaks are mostly consistent; therefore, the offsets with both tracers are more or less the same in Figure 6. However, there are some noticeable differences at some radii. For example, in Figure 4 there are radii at which Hα peaks appear downstream of 24 μm peaks. This is perhaps due to the fact that the distribution of 24 μm emission is affected by the distributions of both dense gas/dust and illuminating stars. If newly formed stars escape from the dense parental gas (as we see in Figure 4), the 24 μm emission may not trace the locations of the young stars any more because the peak of dust density is left behind. The relevance of the tracers to the offset analysis is discussed in Section 5.1.

4.4.1. Resolution

4.4.2. Bin Size

We found that the most significant cause of the discrepancy between the previous studies is the difference in the gas and SF tracers used. We also found that CO emission is the best tracer of the compressed gas in spiral arms and that Hα is the best tracer of the associated star-forming regions. Thus, they are the most useful for the offset measurement among currently available archival data. The previous studies adopted different tracers for tenable and practical reasons. Here we discuss the advantages and disadvantages of each tracer.

Hα emission is from H ii regions around young massive stars whose lifetimes are short (≲ 10 Myr). There is no doubt that discrete Hα peaks pinpoint the locations of very recent star-forming regions. In addition, many sets of Hα data are readily available in archives, and typically have a higher resolution than the Spitzer 24 μm data, another widely used tracer of SF. On the other hand, Hα emission could be easily obscured, especially in dense star-forming gas in spiral arms. We might be preferentially finding H ii regions far away from the parental gas, biasing the measured offsets toward the large side (T08). This, however, has proved not to be the case. The ability of Hα emission to trace the locations of star-forming regions has been shown recently by a comparison of Hα and Paschen α images of M51 (Egusa et al. 2011). Even in the densest regions in a spiral arm, the majority of H ii regions, if not all, can be seen in Hα emission, though their fluxes could be significantly attenuated. Hα emission is the best locater of H ii regions for offset measurement since their positions are the only parameter required.

The 24 μm emission is less sensitive to extinction and might be a better locator of star-forming regions that are embedded deeply in a dense star-forming spiral arm. However, it may not be as straightforward an SF tracer as previously thought, since the emission could be from dust heated by longer-lived, older stellar populations (Liu et al. 2011). In addition, the dust could be heated by the continuum emission longward of the Lyman limit from young stars. The shorter-wavelength emission is mostly absorbed in the vicinity of the young stars and forms H ii regions, but the longer-wavelength photons can potentially travel farther and heat up dust not immediately associated with the young stars. If this is the case, the 24 μm may not pinpoint sites of recent SF. The 24 μm image is also sensitive to the distributions of both heating sources and the gas/dust. If the gas spiral arm and recently formed stellar spiral arm are spatially offset (e.g., the stellar arm should be downstream within a corotation radius; Roberts 1969), the young stars may illuminate the gas/dust spiral arm from the front side and shift the peak of the 24 μm emission. There is little doubt that young stars are the dominant source of dust heating around star-forming regions, and their discrete appearance is very clear in the 24 μm image. However, contamination by the escaped photons from H ii regions and the older stellar population could be possibly biasing the offset measurements using 24 μm emission to the small side.

A concentration of CO (J = 1 − 0) emission is the clearest tracer of dense molecular spiral arms and is aligned perfectly with narrow dust lanes in optical images (Koda et al. 2009). The critical density for collisional excitation of the low-J CO line is high (e.g., ∼a few × 100 cm⁻³; Scoville & Sanders 1987), and all young star-forming regions in the Milky Way are associated with GMCs, which are bright in CO. There is no doubt that high-resolution CO data show the locations of the gas spiral arms that are being compressed by the spiral density wave.

It also had seemed reasonable to assume that the H i emission is adequate to locate the enhancements of gas density in a spiral arm (T08 and F11). However, the small, and often zero, offsets of SF tracers from H i, as opposed to the large offsets from CO, are a clear sign that the H i emission peaks are not tracing the compressed gas in the spiral arm. In fact, the H i peaks are almost always at the downstream side of the CO peaks (Figures 2 and 9). The zero offsets to SF tracers most likely indicate that the H i emission is tracing gas that is photodissociated by recent SF (Allen 2002). Blitz et al. (2007) showed that GMCs are often found at the peaks of H i emission in the H i-dominated galaxy M33, but there is also more extended H i emission across the galaxy in regions without GMCs. H i peaks may trace locations where the gas will not form into stars. It is remarkable that the fraction of molecular gas does not change azimuthally across spiral arms in M51 (Koda et al. 2009), and therefore the effect of photodissociation is small in terms of the global evolution of the gas phase in the spiral galaxy. However, this small change makes a huge impact in the identification of the H i arm—since overall the abundance of atomic gas is low—and likely caused the discrepancy in the previous studies. Perhaps there is some enhancement of H i emission along the gas spiral arm, but in most cases it is washed out by the stronger emission of the photodissociated H i gas, as indicated by the zero/small offsets between H i and SF (T08 and F11). These observational results require a reconsideration of the notion of a short GMC life cycle, an argument largely based on the offset measurements with H i and 24 μm emissions (T08). The offset measurement does not indicate the lifetime of GMCs, since the gas remains mostly molecular even after spiral arm passages and SF.

Therefore, we conclude that CO and Hα are the best tracers of the compressed gas in spiral arms and the associated star-forming regions, respectively. The H i emission does not necessarily trace the compressed, star-forming gas in the spiral arm; instead, it traces predominantly the gas photodissociated by recent SF. The 24 μm emission could be tracing an older stellar population in addition to areas of recent SF and may not pinpoint the site of SF due to the offsets between the dust and young star distributions.

In this work, our discussion of the offset is limited to M51, and we cannot make any general conclusion. Nevertheless, our finding that H i traces the photodissociated gas, rather than star-forming dense gas, offers a natural explanation for the general discrepancies in the previous studies. The best combination of tracers (CO and Hα) shows clear spatial offsets and indicates larger, mostly positive offsets than previously suggested using the H i data (T08), although the positive offsets are not ordered as predicted by the standard density-wave theory.

The offset method is based on only a few assumptions, i.e., a constant Ω_p and Δt and pure circular rotation. Its simplicity assures robustness, but has some weaknesses as well. We note and summarize several limitations here which are likely related to the large scatter found in the plots presented above. Some of the limitations come from the model (Equation (1)), and the others are from intrinsic location-to-location variations of gas conditions and SF in spiral arms.

An intrinsic spatial variation of CO and Hα distributions along spiral arms is a source of a systematic error. As discussed in Section 4, the spiral arms are not simple, continuous structures; at small scales they appear as ensembles of more discrete clumps in both CO and Hα images. The spiral arms go back and forth between the downstream and upstream sides, and this wiggling directly affects the offset measurement. The typical amplitude of the wiggle is about 200 pc and contributes to the scatter in Figure 6. We should note that this type of error tends to cancel out and does not bias offset values systematically. In addition, the molecular gas shows filamentary structures (or spurs) in the interarm regions at the downstream side of spiral arms (Koda et al. 2009). These interarm structures can potentially limit the offset measurement, especially using the Cross-Correlation method where such individual structures are neither identified nor rejected in the analysis.

One of the other limitations is the finite lifetime of H ii regions. The SF timescale Δt derived by the previous studies ranges between ≲ 3 Myr (T08) and ≳ 10 Myr (E09). Some of these timescales are as short as (or less than) the typical lifetime of H ii regions (∼10 Myr). The derived timescales are meaningful only when they are comparable with or longer than the lifetime of H ii regions. If one obtains a shorter timescale, more careful consideration, such as taking into account the age distribution of H ii regions, is necessary before any physical interpretation of the results.

The assumption of a constant pattern speed is commonly adopted, but it is possible that this assumption is not valid. Indeed, it has been suggested that Ω_p changes radially in M51 (Meidt et al. 2008). In particular, the grand-design spiral arms in M51 could be driven by the companion galaxy (NGC 5195). Oh et al. (2008) discussed, based on theoretical modeling, that Ω_p in tidally driven spiral arms may vary with radius. The mostly positive offsets that we found in M51 are an indication that the material is flowing through the spiral pattern. Therefore, the spiral needs to be a wave, not a concentration, of material. This, however, does not mean that the spiral pattern speed is constant with radius (i.e., the traditional density wave). If the stellar spiral density enhancement lives longer than the gas crossing time, the observed results can be explained without a stationary spiral pattern. Note that the assumption of constant Ω_p has an impact neither on the offset measurement itself nor on our conclusion that the cause of the discrepancy in the previous studies is the choice of tracers.

The circular rotation of galaxies is another assumption that most studies adopt. T08 calculated the directional changes in gas orbits due to the spiral arm potential and concluded that its impact on Δt is very small (i.e., substantially less than a factor of two). Another important factor is the change of the amplitude, not only the direction, of gas velocity due to a symmetric galactic potential. Gas orbits in spiral galaxies are often described as ovals (Wada 1994; Onodera et al. 2004; Koda & Sofue 2006). In an axisymmetric galactic potential the gas moves inward and outward, the speed being a maximum at the points on the semi-minor axis of the orbit, and a minimum on the semi-major axis. Spiral arms form primarily due to this slowdown on the semi-major axis; the gas stays longer on the spiral arms, which leads to the density enhancement. More precisely, the spiral density enhancement occurs even if we do not account for the shock (which produces even more enhancement), and is proportional to the passage times through the spiral structure t_cross∝1/(Ω(r) − Ω_p). Ω(r), and therefore t_cross, change substantially during the orbital motion and can change the surface density in spiral arms by an order of magnitude (Onodera et al. 2004, their Figure 11).

To estimate the effect of this slow down to Δt, we estimate a difference in rotational velocities between circular orbits assumed in Equation (1) and elliptical orbits that form spiral arms. Assuming that the gas is traveling in an elliptical orbit with the eccentricity $e=\sqrt{1-b^{2}/a^{2}}$ (maximum and minimum distances a and b, respectively) and that the galactic potential is isothermal (i.e., flat rotation curve, Φ∝ln (r)), conservation of energy and angular momentum gives

$$\begin{equation} \frac{v_{a}}{v_{c}}= \frac{\Omega _{a}}{\Omega _{c}}=\sqrt{\ln (1-e^{2})\frac{(e^{2}-1)}{e^2}}, \end{equation} \tag{ 3 }$$

where v_a is the rotation velocity at the maximum distance a for an elliptical orbit, v_c is the circular rotation velocity at radius a. Ω_a and Ω_c are the angular speeds corresponding to v_a and v_c. For example, assuming an elongated orbit (e = 0.6–0.8), derived pattern speed (Ω_p = 30 km s⁻¹ kpc⁻¹), and rotation velocity (∼200 km s⁻¹), Δt∝1/(Ω − Ω_p) changes from its value under circular rotation by a factor of 2–3 (but up to 10–30) at radii of 4–6 kpc. If the non-circular motion, the very essence of spiral arms, is taken into account, the derived Δt under the assumption of circular rotation is an underestimate by a factor of 2–3. It is important to note that the non-circular motions do not explain the discrepancy in the measured offsets. They affect the interpretation of the azimuthal offsets, but not their measurements. (Note again that the main focus of this study is the measurements.) In order to confirm our simple analytic calculations, we compare the simplistic circular rotation model with the numerical simulations by Dobbs & Pringle (2010) and F11 that include non-circular motion. We use the offsets measured by F11 between the gas and 100 Myr old clusters in the simulations. The pattern speed of their stationary spiral model is 20 km s⁻¹ kpc⁻¹ and the rotation curve is roughly flat with a peak velocity of 220 km s⁻¹. We adopt these parameters for our circular rotation model. Figure 10 shows the expected offsets for the circular rotation model, Equation (1), compared to the offset from the numerical simulations. It is clear that the circular rotation model overpredicts the offsets by a factor of two in this radius range. Therefore, the gas motion across the spiral arm slows down in the more realistic numerical simulations. The difference is expected to be larger around the corotation radius (∼11 kpc in this parameter set). These results are consistent with our simple calculation of non-circular orbits. We conclude that the circular rotation model adopted in the previous studies tends to underestimate the SF timescale by a factor of 2–3. Development of an orbit model that fits M51 is beyond the scope of this paper, but this discussion provides a caveat that the Δt derived from the offset method is likely a lower limit of the SF timescale, which could be about an order of magnitude greater when the non-circular motion is taken into account. This could also be evidence against a short GMC lifetime.

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