A direct test of density wave theory in a grand-design spiral galaxy

The exact nature of the arms of spiral galaxies is still an open question1. It has been widely assumed that spiral arms in galaxies with two distinct symmetrical arms are the products of density waves that propagate around the disk, with the spiral arms being visibly enhanced by the star formation that is triggered as the passing wave compresses gas in the galaxy disk1–3. Such a persistent wave would propagate with an approximately constant angular speed, its pattern speed ΩP. The quasi-stationary density wave theory can be tested by measuring this quantity and showing that it does not vary with radius in the galaxy. Unfortunately, this measurement is difficult because ΩP is only indirectly connected to observables such as the stellar rotation speed4–6. Here, we use the detailed information on stellar populations of the grand-design spiral galaxy UGC 3825, extracted from spectral mapping, to measure the offset between young stars of a known age and the spiral arm in which they formed, allowing a direct measurement of ΩP at a range of radii. The offset in this galaxy is found to be as expected for a pattern speed that varies little with radius, indicating consistency with a quasi-stationary density wave, and lending credence to this new method. Information on stellar populations of the grand-design spiral galaxy UGC 3825 is exploited to measure the offset between young stars of a known age and the spiral arm in which they formed. The measured offset is consistent with a quasi-stationary density wave.

The price paid for deriving this extra parameter was that Ω P had to be assumed to be constant with radius, making it less of a test of the quasi-stationary nature of the conventional density wave picture. However, we have now reached a point where the quality of optical spectroscopy and the associated modelling techniques allow for the extraction of a stellar population of a specified age from spectral data, so that Ω P can be measured as a function of the radius to see how constant it really is.
As a test case, we selected the galaxy UGC 3825. This isolated system 20 has a symmetrical grand-design structure, which makes it a prime candidate for being the product of a global density wave of internal (rather than tidal) origin 3,21 . According to the Galaxy Zoo citizen science project 22 , it does not contain a bar, which might complicate the interpretation of its spiral structure, and it is at an ideal intermediate angle to the line of sight, allowing us to identify its spiral structure and measure the rotational motion of material via the Doppler shift. It is also one of the targets of the Sloan Digital Sky Survey IV (SDSS-IV) Mapping Nearby Galaxies at the Apache Point Observatory (MaNGA) integral field spectroscopic survey 23 , which means that a high-quality spectrum has been obtained for every point across the face of the galaxy.
At each location across the face of the galaxy, we decompose the MaNGA spectrum into contributions from stars of differing ages 24 and current star formation (see Methods), allowing us to map the distribution of all of these various components. Figure 1 shows the resulting distribution of Hα emission and young stars. The Hα emission is a frequently used indicator for ongoing star formation 25 while the young stars are defined as those with ages up to 60 million years after formation. Since the youngest stars dominate the light, the map picks out the location of the stars at a time δτ ~ 2 × 10 7 years after they formed (see Methods), providing us with maps of different timescales since the onset of star formation. As a fiducial, the figure also shows the location of the spiral-arm regions determined as part of the ongoing Galaxy Zoo:3D (GZ:3D) citizen science project (see Methods), although this information on the locations of the arms is not required or used in the analysis here. Even in these raw maps, it is discernible that over most of the galaxy the young stellar population is found on the leading edge of the spiral arm. This is what one would expect from the spiral density wave picture, as the material in the inner parts of a galaxy is predicted to circulate at a higher angular velocity than the spiral pattern, so gas clouds overtake the arms and collapse to form stars in the density wave. These young stars continue to overtake the spiral arm to emerge out of the leading edge after a time interval determined by the difference between the pattern and material speeds.

NaTUre aSTroNomy
We can render this description more quantitative by measuring the small angular offset in azimuth between these two spiral-arm tracers as a function of radius, δθ(r), by cross-correlating data from the maps of current star formation and the young stellar population. We can also measure the circular angular speed of the galaxy as a function of radius, Ω(r), using the Doppler shift in the emission lines in the spectra (appropriately corrected for the galaxy's inclination; see Methods for details). By considering the rate at which material travelling around the galaxy at this angular speed will overtake the spiral pattern, it is straightforward to show 15 that the pattern speed is given by the formula: The results of this analysis for UGC 3825, as presented in Fig. 2, are entirely consistent with the predictions of the density wave theory. At small radii, as expected from the qualitative analysis of offsets, matter is rotating faster than the derived pattern speed, but eventually the measured angular speed of material drops to where it is rotating at the same speed as the spiral pattern-a point known as the co-rotation resonance. The location of this resonance, at a radius of ~6 kpc (0.6R E ), is consistent with estimates for other galaxies using less direct techniques 14 and with the predictions for a modal picture of quasi-stationary density waves 3,11 . The derived form for Ω P (r) is consistent with a constant value of 33 km s −1 kpc −1 . Such constancy was in no way imposed by the analysis, but rather again confirms the existence of a quasi-stationary density wave for this galaxy.
Thus, at least for the case of UGC 3825, a coherent story emerges in which the observed spiral structure is consistent with a quasistationary, internal-origin density wave. However, it has been suggested (and evidence is making it increasingly clear) that such a model can only explain the spiral structure found in a fraction of all galaxies 26,27 . Since different mechanisms for producing spiral arms should result in significantly different radial profiles in pattern speed 3,28 , we can distinguish between such physical processes using this new technique; large spectroscopic surveys of galaxies such as MaNGA will ultimately allow us to fully determine the circumstances under which galaxy spiral arms are produced by longlived density waves.

NaTUre aSTroNomy
This work makes use of the integral field spectroscopy mapping of UGC 3825 (MaNGA plate-integral field unit (IFU) 8132-12702), which is publicly available as part of SDSS data release 14 38 . As part of this work, we make use of the MaNGA data analysis pipeline (DAP) outputs as part of the internal data release MaNGA Product Launch 6 (MPL-6). These DAP outputs will be publicly available as part of SDSS Data Release 15, which is planned for December 2018. GZ:3D. Figure 1 makes use of spiral-arm masks generated by GZ:3D as a fiducial. This is an ongoing citizen science project at https://www.zooniverse.org/projects/ klmasters/galaxy-zoo-3d inspired by the Galaxy Zoo 39,40 galaxy classification project. In GZ:3D, volunteer users are shown images of galaxies in the MaNGA target catalogue and asked to draw boundaries marking the edges of the spiral arms and bars, as well as identifying the position of the galactic centre. The end result is an image with defined spiral and bar weights at each position, determined by the number of users who defined that position as part of a spiral arm or a bar. Further details will be published in a separate paper (K.L.M. and the MaNGA collaboration, manuscript in preparation). For UGC 3825, eight users drew spiral arms, so the fiducial in Fig. 1 is a contour denoting the contiguous region that at least two users defined as being part of the spiral arms.
As stated previously, the GZ:3D spiral-arm masks are only used for display purposes in Fig. 1, and were not included in the analysis.
Spectral fitting. MaNGA data provide a cube of information comprising a spectrum from 3,600 to 10,000 Å at each spatial location on the sky. Each such spectrum of UGC 3825 (MaNGA plate-IFU 8132-12702) was fitted using a set of 270 single stellar population (SSP) template spectra from the Medium-resolution Isaac Newton Telescope Library of Empirical Spectra (MILES) project 41 . These templates have a wavelength range similar to MaNGA (3,540 to 7,410 Å), with SSPs available for a large number of different ages and metallicities. We use templates covering a wide range of ages (27 values between 3 × 10 7 (~10 7.5 ) years and 13 × 10 9 (~10 10.1 ) years) and metallicities (10 values of [M/H] between − 1.79 and + 0.40). The finer sampling of the age parameter space was required to achieve the temporal resolution needed to separate out the young stellar components sought in this analysis; the coarser sampling in metallicity is entirely adequate for this work while keeping the total number of templates within the maximum that the software can process. We assume a Kroupa revised stellar initial mass function 42 with Padova isochrones 43 . We fit each spaxel's spectrum individually to ensure that we retain all of the spatial information possible. This will result in a decreased signal-to-noise ratio (SNR) at the edges of the galaxy, but within the region indicated in Fig. 2, no spaxel has a SNR of less than 5. Noise at a pixel-by-pixel basis is smoothed out by the cross-correlation techniques in any case. No regularization was imposed on any of the fitting processes.
As a first step to extract and remove emission-line contributions from the spectra, pPXF 44,45 was used to simultaneously fit the shape and kinematics of both the stellar spectra and a full set of emission lines, whose profiles were assumed to be Gaussian. The resulting Hα emission-line flux measurements provide the tracer of ongoing star formation, since Hα luminosity, L Hα , is directly proportional to the local star-formation rate 25 .
The spectra were initially logarithmically binned to allow the kinematics to be derived. After the emission lines had been subtracted out, the remaining stellar spectra were re-binned to a linear scale and fitted using the Starlight 46,47 code that is optimized for modelling stellar populations. Although Starlight is explicitly designed for this task, a cross-check of the results from the Starlight and pPXF codes confirmed very close agreement between the stellar population results obtained. This will be fully described in a separate paper (T.G.P., A.A.-S., M.R.M and the MaNGA collaboration, manuscript in preparation). To reproduce the observed spectra in the fitting process, we also allowed for dust obscuration using the variable-strength Cardelli-Clayton-Mathis reddening law 48 .
Measuring the time offset. The resulting output from Starlight provides the contribution of each SSP of specific age and metallicity to the spectrum at each location across the face of the galaxy. Thus, we are now in a position to create maps showing how stars with differing properties contribute to the total light of the galaxy. In this case, we are interested in mapping out the young stars, so we extract the contribution from all of the SSPs with ages of less than 6 × 10 7 years. The youngest SSP template age is 3 × 10 7 years, but the fitting process will attribute all younger stars to this population as well. We therefore assume that this template's 'true' age is approximately 1.5 × 10 7 years, and all other templates' weights account for stars of their listed age. By weighting by Starlight's fit's template weights, we find the mean age of the young population over the entire MaNGA field of view to be 1.9 × 10 7 years, with a conservative estimate of the uncertainty as 1.0 × 10 7 years. The peak in Hα emission occurs very soon after the onset of star formation; hence, the temporal offset between the young population and Hα emission in equation (1) is δτ = 19 ± 10 Myr.
Measuring the angular offset. Although the angular offset δθ(r) between the maps of young stars and Hα emission is visually apparent in the data (see Fig. 1), it is quite a subtle effect, so some care must be taken to optimize the signal when extracting it. As a first step, we de-project the maps to face-on using kinematic centre, inclination and position angle measurements, determined from the best-fit parameters to the gas disk kinematics, and convert the Cartesian images to polar ones, binned in radius with a step size of Δ r ~ 0.16 kpc. When the National Aeronautics and Space Administration Sloan Atlas 49 measurements are used instead for the centre, inclination and position angles (assuming the galaxy disk is intrinsically round), the results are unchanged.
For each such radius, we determine the offset between the spiral features in the Hα and young stellar map by cross-correlating the signal in the polar maps, as displayed in Supplementary Fig. 1. The location of the cross-correlation maximum is then refined to a sub-pixel value by fitting a second-order polynomial around the peak. The region of r < 3.2 kpc (approximately 0.32R E ) in Fig. 2 is ignored when calculating Ω P (r) since the azimuthal signal of Hα variations here is found to be too small to reliably measure δθ(r).
A conservative estimate for the uncertainty δ Δθ (r) in this offset can be obtained from the ratio of the full width at half maximum (FWHM) of the peak in the crosscorrelation signal to the SNR of the signal. The SNR is in turn estimated as the ratio of the peak height H to the s.d. of the cross-correlation signal σ δθ(r) ; that is: The radially varying FWHM allows the value of δ Δθ (r) to account for the radial variation in the MaNGA beam size effects in the polar-coordinate plots. At low r, the beam covers a large range in θ. The cross-correlation signal's peak will therefore be proportionally wider, increasing δ Δθ (r). At large radius, the beam will cover a small range in θ, allowing us to obtain a tighter constraint on the value of δθ(r).
Measuring the angular velocity of circular orbits. The other ingredient needed to determine the pattern speed is the angular speed of material following a circular orbit in the galaxy. We use gas velocity measurements to determine the angular velocity of material Ω(r) since the very young stars this material traces will not yet have been dynamically heated from their purely circular trajectories 50 . The MaNGA DAP (details to be described; K.B.W. and the MaNGA collaboration, manuscript in preparation) provides measurements of the line-of-sight gas velocity v los,gas for each pixel. This analysis uses the MaNGA Product Launch 6 (MPL-6) version of the DAP outputs.
Using the same process as described above, the v gas map can be remapped into polar coordinates. At each radius, the observed line-of-sight velocity will vary sinusoidally with azimuthal angle, and a simple least-squares fit yields the amplitude of this variation at each radius, V gas (r). The angular speed can then be simply calculated as Ω = , where i is the inclination angle of the galaxy to the line of sight (i = 0 for a face-on galaxy) derived from the galaxy's kinematics. The error in Ω is dominated by the contribution from the uncertainty in the sinusoidal fit, so this value is adopted.
Testing with an older stellar population. If the picture established here is correct, it should be possible to repeat the analysis using a somewhat older stellar population that will have had time to travel further from the peak of the spiral density wave. In practice, it appears that the residual spiral feature fades very rapidly into the noise from the more general disk population (T.G.P., A.A.-S., M.R.M. and the MaNGA collaboration, manuscript in preparation), but we were able to extract a consistent signal from a portion of the southern spiral arm for an intermediate-age population combining the templates with ages between 0.2 and 1.3 Gyr. This population mainly comprises B-and A-type stars; adopting a luminosity-weighted age of ~2 × 10 8 years, the results from the angular offset of these somewhat older stars, which we have shown in Supplementary Fig. 2, are entirely consistent with the pattern speed derived in the main text, giving some further confidence in both the method and the density wave theory.

Data availability
Integral field spectroscopy data of UGC 3825 are available as part of data release 14 of the SDSS 38 . The specific data that support the plots within this paper and other findings of this study are an updated version of these data, and will be made publicly available as part of SDSS data release 15, which will be described in a separate paper by the MaNGA collaboration in early 2019 (Aguado et al., manuscript in preparation). In the meantime, the data used here are available from the corresponding author upon reasonable request.