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

While the defining features of spiral galaxies are the beautiful arms that they display, the exact nature of such spiral structures is still an open question. Since the 1960s, it has been widely assumed that spiral arms in galaxies with two distinct symmetrical arms (known as"grand design"systems) are the products of density waves that propagate around the disk as a persistent pattern, with the spiral arms being visibly enhanced by the star formation that is triggered as the passing wave compresses gas in the galaxy disk. Such a persistent wave would propagate with an approximately constant angular speed called its pattern speed, $\Omega_{\text{P}}$, and so a simple test of the density-wave theory is to measure this quantity and show that it does not vary with radius in the galaxy. Unfortunately, this measurement is difficult because $\Omega_{\text{P}}$ only has an indirect connection to readily-measurable quantities such as the stellar rotation speed. Here, we make use of the detailed information on stellar populations that can now be extracted from spectral mapping of a grand-design spiral galaxy (UGC 3825) to measure the offset between young stars of a known age and the spiral arm in which they formed, allowing the first direct measure of $\Omega_{\text{P}}$ 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, vindicating the global spiral density wave theory and establishing the reliability of this new method.

of density waves that propagate around the disk as a persistent pattern, with the spiral arms being visibly enhanced by the star formation that is triggered as the passing wave compresses gas in the galaxy disk 1, 3 . Such a persistent wave would propagate with an approximately constant angular speed called its pattern speed, Ω P , and so a simple test of the density-wave theory is to measure this quantity and show that it does not vary with radius in the galaxy. Unfortunately, this measurement is difficult because Ω P only has an indirect connection to readilymeasurable quantities such as the stellar rotation speed. 4-8 Here, we make use of the detailed information on stellar populations that can now be extracted from spectral mapping of a grand-design spiral galaxy (UGC 3825) to measure the offset between young stars of a known age and the spiral arm in which they formed, allowing the first direct measure of Ω P 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, vindicating the global spiral density wave theory and establishing the reliability of this new method.
The idea of using offsets between features of differing ages to determine pattern speeds is not new; it has been employed to good effect using the observed offset between the dense molecular gas that is currently being compressed by the spiral wave and young hot stars that formed previously in the spiral arm and have now moved from the peak of the wave. [9][10][11][12] However, in these previous analyses the offset in time between the two phases -essentially the timescale for star formation -was also unknown, so had to be solved for simultaneously. 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 persistent density wave picture. However, we have now reached a point where the quality of optical spectroscopy and the associated modelling techniques allow one to extract a stellar population of a specified age from spectral data, so that Ω P can be measured as a function of radius to see how constant it really is.
As a test case, we have selected the galaxy UGC 3825. This isolated system 13 has a symmetric grand-design structure, which makes it a prime candidate for being the product of a global density wave. According to the Galaxy Zoo citizen science project 14 , 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 both to identify its spiral structure and to measure the rotational motion of material via the Doppler shift. It is also one of the targets of the SDSS-IV MaNGA (Mapping Nearby Galaxies at APO) integralfield spectroscopic survey, 15 which means that for every point across the face of the galaxy a high-quality spectrum has been obtained. 16 At each location across the face of the galaxy, we decompose the MaNGA spectrum into the contributions from stars of differing ages 17, 18 and that from current star formation 19,20 (see Methods), allowing us to map the distribution of all these various components. Figure 1 shows the resulting distribution of young stars and star formation. The young stars were chosen to be those with ages between 20 and 60 million years, but 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. 1 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 2 (see Methods for details). Even in these raw maps, it is discernable 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.
We can render this description more quantitative by measuring the small angular offset 1 There is some uncertainty in the value adopted here, but it is notable that if it is left as a free parameter in the analysis then this is also the value that returns the most consistent estimate for Ω p at all radii.
2 https://www.zooniverse.org/projects/klmasters/galaxy-zoo-3d 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 10 that the pattern speed is given by the formula The results of this analysis for UGC 3825, presented in Figure 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 corotation resonance. The location of this resonance, at a radius of ∼ 6 kpc, is consistent with estimates for other galaxies using less direct techniques 11 , and approximately corresponds to where the arm-interarm flux contrast is greatest, as predicted by simulations 22 . The derived form for Ω P (r) is consistent with a constant value of 31 km s −1 kpc −1 . Such constancy was in no way imposed by the analysis, but rather again confirms the predictions of density wave theory 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 the predictions of simple density wave theory. However, it has been suggested 23 (and evidence is making it increasingly clear 24,25 ) that such a model can only explain the spiral structure found in a fraction of all galaxies. Since different mechanisms for producing spiral arms should result in significantly different radial profiles in pattern speed, 26 we can distinguish between such physical processes using this new technique; large spectroscopic surveys of galaxies like MaNGA will ultimately allow us to fully determine the circumstances under which galaxy spiral arms are produced by long-lived density waves.   . 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 (IMF) 45 with Padova isochrones 46 . We fit each spaxel spectra individually to ensure that we retain all of the spatial information possible.
This will result in decreased signal-to-noise (S/N) at the edges of the galaxy, but within the region indicated in Figure 2, no spaxel has a S/N less than 4.2. 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 20, 47 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 19 .
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 rebinned to a linear scale and fitted using the Starlight 18, 48 code that is optimised for modelling stellar populations. 9 To reproduce the observed spectra in the fitting process, we also allowed for dust obscuration using a variable-strength Calzetti, Clayton & Mathis reddening law. 50 .
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 can extract a map of the contribution of stars with differing properties to the total light from the galaxy. In this case, we are interested in mapping out the young stars, so we extract the contribution to the total light from all the SSPs with ages of less than 6 × 10 7 years. These will be mainly O-and B-type stars, with an average luminosity-weighted age of ≈ 2 × 10 7

years.
Measuring the offset Although the angular offset δθ(r) between the maps of young stars and Hα emission is visually apparent in the data (See Figure 1), it is quite a subtle effect, so some care must be taken to optimise the signal when extracting it. As a first step, we deproject 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 ∆r ≈ 0.16 kpc. When the NASA-Sloan Atlas 51 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, displayed in Figure   3. The location of the cross-correlation maximum was then refined to a sub-pixel value by fitting a 2 nd -order polynomial around the peak. The region of r < 3.2 kpc (approximately The galactic centre is at the bottom of the map, and the black outlines indicate the location of the same GZ:3D spiral arm mask used in Figure 1. The cross-correlation signal between these maps is also shown (right), with the cross-correlation angle δθ(r) shown as a red line. Figures 2 and 3 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).

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A conservative estimate for the uncertainty δ ∆θ (r) in this offset can be obtained from the ratio of the full-width-at-half-maximum (FWHM) of the peak in the cross-correlation signal to the signal-to-noise ratio (SNR) of the signal. The SNR is in turn estimated as the ratio of the peak height H to the standard deviation of the cross-correlation signal σ δθ(r) ; i.e. δ ∆θ (r) = FWHM(r) × σ δθ(r) H(r) . ( The radially-varying FWHM allows the value of δ ∆θ (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 ∆θ (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 52 . The MaNGA data analysis pipeline (DAP) 53 provides measurements of the line-of-sight gas velocity v los,gas for each pixel. This analysis uses the 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(r) = Vgas(r) r×sin(i) , where i is the inclination angle of the galaxy to the line of sight (i = 0 for a face-on galaxy) derived from the observed ellipticity in the NASA-Sloan Atlas. The error in Ω is dominated by the contribution from the uncertainty in the sinusoidal fit, and so this value is adopted.
Testing with an older stellar population If the picture established here is correct, then 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, but we were able to extract a consistent signal from a portion of the