https://orcid.org/0000-0002-0529-7427
ABSTRACT
Spiral structure is ubiquitous in the Universe, and the pitch angle of arms in spiral galaxies provide an important observable in efforts to discriminate between different mechanisms of spiral arm formation and evolution. In this paper, we present a hierarchical Bayesian approach to galaxy pitch angle determination, using spiral arm data obtained through the Galaxy Builder citizen science project. We present a new approach to deal with the large variations in pitch angle between different arms in a single galaxy, which obtains full posterior distributions on parameters. We make use of our pitch angles to examine previously reported links between bulge and bar strength and pitch angle, finding no correlation in our data (with a caveat that we use observational proxies for both bulge size and bar strength which differ from other work). We test a recent model for spiral arm winding, which predicts uniformity of the cotangent of pitch angle between some unknown upper and lower limits, finding our observations are consistent with this model of transient and recurrent spiral pitch angle as long as the pitch angle at which most winding spirals dissipate or disappear is larger than 10°.
1 INTRODUCTION
Spiral structure is present in a majority of massive galaxies (e.g. Buta 1989, Lintott et al. 2008) yet the formation mechanisms through which spiral structure originates are still hotly debated (e.g. Dobbs & Baba 2014). Spirals are as diverse as the theories proposed to govern their evolution, from the quintessential pair of well-defined arcs of the grand design spiral, to the fragmented arm segments of the flocculent spiral, to the disjointed multi-armed spiral. (Elmegreen et al. 2011; examples of each type are shown in Fig. 1). The Hubble classification scheme (Hubble 1926) and its revisions and expansions (Sandage 1961; de Vaucouleurs et al. 1991) contain detailed variations of different types of spiral galaxy, divided by the presence of a bar and ordered by the openness of the spiral arms, the degree of resolution into condensations, and the prominence of a central bulge. Building on this, Elmegreen & Elmegreen (1982) found that flocculent spirals are more prevalent in unbarred, isolated galaxies. The presence of a bar, a binary companion or group membership result in a higher fraction of observed grand design spiral patterns.
Examples of the different types of spiral galaxy present in the sky. The left-hand column shows the grand design spiral NGC 5248. The middle shows the many-armed spiral NGC-3184 and the right shows the flocculent spiral NGC 2841. Images were taken with the Sloan Digital Sky Survey Telescope.
Whatever kind of spiral is present in a disc galaxy, there is plenty of evidence to suggest that they have a significant role on the overall evolution of that galaxy. For example, a majority of the population of young stars in a galaxy are located in its spiral arms (Elmegreen 2011), and there is evidence that spiral arms may trigger star formation (Cedrés et al. 2013) perhaps via their ability to promote the growth of Giant Molecular Clouds (Dobbs 2014). The rearrangement of disc gas and stars driven by spiral arms (e.g. Daniel & Wyse 2018) may lead to the formation of disc-like bulges (commonly called ‘pseudo-bulges’; e.g. Kormendy & Kennicutt 2004), which are prevalent in most spiral galaxies, including those without bars (Fisher & Drory 2010). Studies of spiral morphology have also found interesting correlations with other galactic properties, such as a correlation between the tightness of spiral arms and central mass concentration (Yu & Ho 2019, though neither Hart et al. 2017 nor Masters et al. 2019 found such a relation in large samples). Spiral tightness is also observed to correlate with rotation curve shape (Seigar et al. 2005), with galaxies with rising rotation curves having more open spiral structure. These predictions and observations provide compelling reasons for continued investigation of the underlying rules and dynamics of spiral structure, as doing so is essential for understanding the secular evolution of disc galaxies.
Our current understanding of the mechanisms which drive spiral growth and evolution suggests that different forms of spiral arms in a galaxy may be triggered by different processes. Grand design spirals are thought to have undergone a tidal interaction (Dobbs et al. 2010; Semczuk, Łokas & del Pino 2017), be driven by a bar (as seen in gas simulations, Sanders & Huntley 1976; Rodriguez-Fernandez & Combes 2008, and suggested for stars by Manifold theory, Romero-Gómez et al. 2006; Athanassoula, Romero-Gómez & Masdemont 2009a; Athanassoula et al. 2009b), or be obeying (quasi-stationary) density wave theory (QSDW theory), in which spiral arms are slowly evolving, ever-present structures in the disc (as first proposed by Lin & Shu 1964). Flocculent spirals are thought to be formed through swing amplification (shearing of small gravitational instabilities in the disc), and be transient and recurrent in nature (Julian & Toomre 1966).
One of the fundamental assumptions of early work on spiral formation mechanisms (primarily QSDW) was that the disc of a galaxy, if unstable to spiral perturbations, would create a stable, static wave which would exist unchanged for many rotational periods (Lin & Shu 1964). The motivation for static waves with small numbers of arms was primarily observational: most disc galaxies observed at the time showed spiral structure with low spiral arm numbers, suggesting that spirals exist for a long time or are continually rebuilt. This, in combination with theoretical arguments about the ‘winding problem’, motivated the original static density waves of Lin & Shu (1964), to which swing amplification was added by Toomre (1981) to provide a way to counteract the short lifetime of stellar density waves.
More recently, simulations demonstrate that spirals do not maintain a constant tightness (often quantified by pitch angle, the angle between the spiral and the tangent to a circle centred on the galaxy, Binney & Tremaine 1987, illustrated in Fig. 2), and instead wind-up over time due to the differential rotation of the disc (Baba, Saitoh & Wada 2013). Recent research suggests that spirals arms are transient, and continually dissipate and re-form (Dobbs & Baba 2014). These spirals can be maintained through the same mechanisms that drive QSDW spirals (i.e. ‘wave amplification by stimulated emission of radiation’, Mark 1976; swing amplification, Goldreich & Lynden-Bell 1965), but do not require the idealistic disc conditions required for the formation and maintenance of a stationary wave. The pitch angles of these transient spiral arms will decrease due to the differential rotation of the disc, with the density of the arm peaking at some critical pitch angle, before dissipating to be reformed.
Illustration of the definition of pitch angle. It is given as |$\phi = \tan ^{-1}\left(\frac{\mathrm{d}r}{\mathrm{d}\theta }\ /\ r\right)$|, or the angle between the spiral (red) and the tangent to a circle centred on the galaxy (blue).
In this dynamic picture of spiral arms, pitch angle monotonically decreases from a spiral arm’s formation to its dissipation. As a particular example of this, Pringle & Dobbs (2019) propose a simple test of the winding of spiral arms, predicting that the cotangent of the pitch angle of a spiral arm (|$\cot \phi$|) evolves linearly with time. They found that the distribution of pitch angles of their sample of 86 galaxies was consistent with this prediction, which they present as evidence against QSDW theory in favour of the dynamic spirals produced in many simulations.
We aim to test this idea of spiral winding using data from the Galaxy Builder citizen science project for the spiral galaxies present in Lingard et al. (2020). We make use of Bayesian hierarchical modelling to measure galaxy pitch angle from the spiral arms produced by Galaxy Builder. This methodology allows us to quantify the differences in pitch angles between arms in a single galaxy, as well as investigate the distribution of pitch angles in the galaxy population and investigate relationships between pitch angle and galaxy morphology.
Using Galaxy Zoo 2 data (Willett et al. 2013) we further separate the galaxies by the presence and strength of a stellar bar. This will allow us to test simulations of gas in barred galaxies, which often demonstrate that bars can drive long-term spiral evolution (Rodriguez-Fernandez & Combes 2008), or boost transient spiral structure (Grand, Kawata & Cropper 2012). Manifold theory (Romero-Gómez et al. 2006; Athanassoula et al. 2009a,b) is one attempt to determine the orbits of stars in bar-driven spiral arms: it proposes that stars in the vicinity of the unstable Lagrangian points at either end of the bar tend to escape along predictable orbits, governed by invariant manifolds. One of the primary factors influencing the shape of this invariant manifold is the relative strength of the non-axisymmetric forcing caused by the bar, with stronger bars resulting in spirals with larger pitch angles.
Many other galactic components may correlate with spiral morphology, including bulge fraction (Yoshizawa & Wakamatsu 1975; Savchenko & Reshetnikov 2013; Masters et al. 2019) and black hole mass (Seigar et al. 2008; Davis, Graham & Seigar 2017; Al-Baidhany et al. 2019). Larger bulges and more massive central black holes have both been observed to correlate with more tightly wound spiral arms. We can also test this with the data presented in this paper.
This paper is structured as follows: In Section 2, we introduce methods to measure galaxy pitch angle, and present our sample, and our Bayesian hierarchical modelling method making use of Galaxy Builder to estimate galaxy and population pitch angles. Section 3.1 presents our general constraints on pitch angles in our section, while Section 3.2 examines the correlation between pitch angle and bulge size implied by the Hubble sequence, and pitch angle and bar strength implied by Manifold theory. Section 3.3 investigates spiral arm winding using the test derived by Pringle & Dobbs (2019) (uniformity of galaxy pitch angle in |$\cot \,\,\phi$|). We provide a summary and conclusions in Section 4. Where necessary, we make use of H0 = 70 km s−1 Mpc−1.
2 METHOD
2.1 Measuring galaxy pitch angle
Many methodologies have been proposed and implemented to measure spiral arm properties, including visual inspection (Herrera-Endoqui et al. 2015), Fourier analysis (i.e. 2dfft; Davis et al. 2012), texture analysis (i.e. sparcfire; Davis & Hayes 2014), and combinations of automated methods and human classifiers (Hart et al. 2017; Hewitt & Treuthardt 2020). One potentially underused method of obtaining measurements of spirals is through photometric fitting of spiral structure, as possible using tools such as galfit (Peng et al. 2010) and Galaxy Builder (Lingard et al. 2020). These methods attempt to separate light from an image of a galaxy into distinct subcomponents, such as a galaxy disc, bulge, bar, and spiral arms, generally finding the optimum solution using computational optimization. This optimization process, however, is often not robust for complex, many-component models and requires significant supervision to converge to a physically meaningful result (Gao & Ho 2017). Lingard et al. (2020) proposed a solution to this problem through the use of citizen science to provide priors on parameters used in computational fitting.
The most commonly used measurement of uncertainty of length-weighted pitch angles is the unweighted sample variance between the arm segments which were identified.
A notable drawback of length-weighted pitch angle is sensitivity to the number and quality of the spiral arm segments; Hart et al. (2017) found that only 15 per cent of the arm segments which were identified using sparcfire (Davis & Hayes 2014) were identified as ‘good’ matches to real spiral arms by citizen science classifiers.
Fourier analysis in 1D and 2D (as performed by Díaz-García et al. 2019, Davis et al. 2012, Mutlu-Pakdil et al. 2018, and dating back to the seminal work of Considere & Athanassoula 1988) is another widely used method of computationally obtaining galaxy pitch angles. 2D Fourier methods generally decompose a deprojected image of a galaxy into a superposition of logarithmic spirals between inner and outer annuli (Davis et al. 2012) and report the pitch angle with the highest amplitude as the galaxy’s pitch angle. Hewitt & Treuthardt (2020) combined Fourier analysis of spiral galaxies with a visual tracing of spiral arms, successfully eliminating observed bias in a sample of toy images of galaxies. It is unclear how the variation between pitch angles of individual arms impacts this measurement. We note that while this method is able to model non-logarithmic spirals – as a sum of logarithmic spirals with differing pitch angles, most applications use models which assume that the pitch angle is constant with radius, in some cases picking regions of a galaxy in which this is true – e.g. see section 4.3.2 of Davis et al. (2012).
2.2 The galaxy sample
The galaxies analysed in this paper are those for which photometric models were obtained in Lingard et al. (2020). These are a subset of the stellar mass-complete sample in Hart et al. (2017), a sample of low-redshift (0.02 < z < 0.055) face-on spiral galaxies selected using data from the NASA–Sloan Atlas (Blanton et al. 2011) and Galaxy Zoo 2 (Willett et al. 2013). The stellar mass-complete sample ranged in stellar mass from 9.45 < log (M*/M⊙) < 11.05, with most of the sample between 9.5 < log (M*/M⊙) < 10.0. A histogram of stellar masses for our subset can be seen in Section 3.1, where variation with stellar mass, to check the impact of this limited mass range, is also investigated. For the reader’s convenience we also reproduce fig. 4 from Lingard et al. (2020) here (see Fig. 3) which shows the redshift and stellar mass distribution of our analysis sample compared to the full stellar mass-complete sample from Hart et al. (2017). Our choice (see Lingard et al. (2020) for details) to prefer lower redshift galaxies for Galaxy Builder analysis is clear in the mass distribution which results in a sample favouring galaxies 9.5 < log (M*/M⊙) < 10.0, and includes a smaller number of spirals with masses up to log (M*/M⊙) = 11.05.
A plot of redshift against stellar mass for the stellar mass-complete sample from Hart et al. (2017); the subset we use for analysis in Galaxy Builder are shown in red. At right we show a histogram of the stellar masses. This figure is identical to fig. 4 from Lingard et al. (2020) who use the same sample.
Some galaxies in Lingard et al. (2020) were shown to volunteers a second time in a repeat validation subset to create a second aggregate model used to test internal consistency. Section 3.2 of Lingard et al. (2020) presents a comparison of these classifications to investigate volunteer consistency. We can see that arm number is highly reliable within ±1 (93 per cent galaxies have arm number counts which agree in this range; 53 per cent have exactly identical arm number counts). In this work, we combine the 30 classifications of galaxies in this validation subset with the 30 original classifications. Clustering of drawn spiral arms and cleaning of points was then performed as detailed in Lingard et al. (2020). We remove any galaxies for which no spiral arms were identified, resulting in a hierarchical data structure of 129 galaxies, 247 spiral arms, and 238 433 points. This breaks down further to 68 galaxies (53 per cent of the sample) having two arms identified (meaning that they were marked by enough users to cluster into an arm), 19 (15 per cent) with three such arms, 4 (3 per cent) with four, and the remainder (38 or 29 per cent) with a single identified arm. We believe that our reported number of arms per galaxy is in many cases an underestimate, and since the most common number of arms is two, this most often results in a single arm being measured when two are present. The most common way we miss an arm is if the clustering did not converge when users have appeared to identify a second arm. For example, we find by visual inspection that just three galaxies in the sample are truly one-armed spirals, while in the remaining 35 galaxies for which only one arm was identified following clustering, the other arms were simply not recovered due to noise in the data set. Given this, we do not recommend using these statistics to make general conclusions about the number of arms per galaxy. However, as our hierarchical model incorporates the uncertainty involved with missing spiral arms, the results related to pitch angle should not be significantly affected.
Spiral arm points are deprojected to a face-on orientation using the disc inclination and position angle obtained through photometric model fitting performed in Lingard et al. (2020). Arms are individually corrected to all have the same chirality (a pitch angle greater than or equal to zero) using the logarithmic spiral fit in Lingard et al. (2020). This was achieved by multiplying the polar coordinate θ by −1 for arms identified as winding counterclockwise.
2.3 Bayesian modelling of spiral arms in galaxy builder
In this section, we lay out our Bayesian hierarchical model for galaxy pitch angle. We fit directly to clustered, cleaned points from polylines drawn in Galaxy Builder, deprojected and unwrapped to polar coordinates. We fit a logarithmic spiral to each clustered arm (examples are shown in Fig. 4), with the pitch angles of multiple arms in a single galaxy being drawn from a single parent distribution.
Examples of spiral profiles fit using the hierarchical model described in Section 2.3. Deprojected points from Galaxy Builder clustered, cleaned spiral arms are shown in black; fit logarithmic spiral arms are shown in red, with the width of the line corresponding to the 2σ interval on predicted values of |$\widetilde{r_\mathrm{arm}}$|. The two one-armed spirals in the top left-hand panels are instances where the spiral clustering algorithm failed to identify all spiral arms present in the galaxy.
Logarithmic spirals have the desirable properties of a constant pitch angle and a small number of free parameters. For this first analysis of the Galaxy Builder models we choose to make use of it here without an explicit comparison to other models. A simple visual inspection of the fitted logarithmic spirals suggests that it is an appropriate model, however, a comparison of a logarithmic spiral profile to other spiral forms (i.e. Archimedean or polynomial) is another important piece of work, outside of the scope of this research, as it has been reported that galaxy arms do not have constant pitch angles (Kennicutt 1981; Ringermacher & Mead 2009).
The choice to assume all galaxies show the same interarm variation in pitch angle (represented by a common value of σgal across all galaxies) was motivated by our small sample size and the low number of arms measured per galaxy. With this sample size we do not find, nor expect to be sensitive to variations in this parameter. It is possible that it does vary between galaxies, and that this variation is physically interesting. Several authors have previously made attempts to measure this parameter. In a seminal work, Kennicutt (1981) fit logarithmic spirals to 113 nearby (NGC) galaxies, and note the dominant error in average galaxy pitch angle comes from interarm variation, which they measure to have an average value of 5°. Davis & Hayes (2014) is primarily a machine learning method paper, and while details on the galaxies investigated are not clear, their table 1 presents the median difference in pitch angle between pairs of arms with different lengths, which varies from 14.5° in very short arms, to 2.6° in the longest traced arms. It is unclear how much this encodes error in their method versus real variation in the galaxy population. Within our own Milky Way, Vallée (2015) did a meta analysis and comparison of several technique to conclude a range of 12–14° (i.e. 2°) was reasonable for all Milky Way spiral arms. In a very detailed study of four very nearby spirals, Honig & Reid (2015) conclude there are large variations of pitch angles between spirals, and among arms in a given spiral, but made no comments as to if the variation was consistent with being constant. Further investigation of this issue in a larger Galaxy Builder sample would be an interesting follow-up project.
where |$\widetilde{r_\mathrm{arm}}$| is the model’s prediction for the radii of the deprojected points in a Galaxy Builder arm (|$\overrightarrow{r_\mathrm{arm}}$|), carm is the amplitude parameter (equivalent to A in equation 1), and |$\overrightarrow{\theta _\mathrm{arm}}$| is the polar angles of the points.
We assume that the radial error is Gaussian for simplicity of analysis, however, Shapiro–Wilk tests on the residuals of the logarithmic spirals fit in Lingard et al. (2020) suggest that this is not a good assumption, and a more robust likelihood (such as the Student’s t-distribution) would possibly more appropriate.
To perform inference, we make use of the No-U-Turn-Sampler (NUTS; Hoffman & Gelman 2011), implemented in pymc3,1 an open-source probabilistic programming framework written in Python (Salvatier, Wiecki & Fonnesbeck 2016). To aid the convergence of MC chains, we scale the radii of deprojected points to have unit variance.
3 RESULTS
3.1 Constraints on galaxy pitch angle
![Scatter plot showing how arm pitch angle compares to galaxy pitch angle for galaxies with different pitch angles and number of arms. The top panel shows a Gaussian KDE for E[ϕgal], and the right-hand panel shows a Gaussian KDE for E[ϕarm − ϕgal]. The galaxy pitch angle is consistent with the mean of its arms, with large scatter and a slight bias against values near the lower bound of 0 due to the lower limit applied.](https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/mnras/504/3/10.1093_mnras_stab1072/1/m_stab1072fig5.jpeg?Expires=1749137277&Signature=o0N9zZbmSCkOtqd6kANMZNBAJBftnspCj32ghssgmk~DZCz-4-hNFXE4uO3D93AVUAg1iTU0ILlzhxiZXe1IKlnkQbuMsN2BkNZIgPKoQ~TnpOpOZpxYVEaeyxQMhpbfOB30mmn~NRJM0BKjNU3SR9cIUVeBKlxcNFIn~kelvgEOOtse-CkFvpqi1MnUPBy7OBS7HMAxcD4puLOZNPZLQA-QNrm-D9-tClodkuuqB6KstXLSvNIwoz9dKggd-ljqsREdWtE2-kvKiccvAw8cc5~Y~1ib5gm0CfWGBLroQRM2ok~aJthJFITmkqshNlT2PR1Xf4~W18akxNsrnqZaTQ__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
Scatter plot showing how arm pitch angle compares to galaxy pitch angle for galaxies with different pitch angles and number of arms. The top panel shows a Gaussian KDE for E[ϕgal], and the right-hand panel shows a Gaussian KDE for E[ϕarm − ϕgal]. The galaxy pitch angle is consistent with the mean of its arms, with large scatter and a slight bias against values near the lower bound of 0 due to the lower limit applied.
In Fig. 6 we present the stellar mass distribution of the sample, and investigate how the global galaxy pitch angle depends on this parameter. The majority of our sample has stellar masses 9.5 < log (M*/M⊙) < 10.0, and galaxy average pitch angles 10°–20°. This should be remembered in our physical interpretation of other results. We observe no significant trend of the global pitch angle with mass, although there is a hint that more massive spirals may on average have more tightly wound arms.
The stellar mass distribution of the sample and the galaxy average pitch angle distribution shown as a 2D histogram (centre) and also projected along each axis.
Density plot showing bulge strength (Bavg; left-hand panel, orange) and bar strength (pbar; right-hand panel, green) against galaxy pitch angle (ϕgal). Split points for the marginalized Anderson–Darling tests are labelled. There is no statistically significant relationship for either bulge or bar strength.
3.2 Dependence of pitch angle on galaxy morphology
To test the possible progenitor distribution of our estimated arm pitch angles, we repeatedly perform an Anderson–Darling test (Stephens 1974, implemented in scipy, Jones et al. 2001) over each draw present in the MC trace, resulting in a distribution of Anderson–Darling statistics. We will refer to this test as the marginalized Anderson–Darling test. We also make use of the two-sample Anderson–Darling (Scholz & Stephens 1987) test in a similar manner.
We make use of Galaxy Zoo 2 data for morphological comparison. Two of the galaxies in our sample could not be matched to Galaxy Zoo 2 data, and as such have been dropped from this comparison (leaving 127 galaxies).
3.2.1 Pitch angle versus bulge size
We see no correlation between galaxy pitch angle derived from the hierarchical model and Bavg. The Pearson correlation coefficient between the expectation value of galaxy pitch angle (E[ϕgal]) and Bavg is 0.00 (with a p-value of 0.95). The relationship between galaxy pitch angle Bavg and is shown in the left-hand panel of Fig. 7.
We separate our sample into galaxies with weaker bulges (Bavg < 0.28, 79 galaxies) and those with stronger bulges (Bavg ≥ 0.28, 48 galaxies), to test whether their pitch angles could be drawn from significantly different distributions. A marginalized two-sample Anderson–Darling test comparing the distributions of ϕgal for the samples does not find evidence that galaxy pitch angles were drawn from different distributions: we reject the null hypothesis at the 1 per cent level for only 1 per cent of the samples. Similarly comparing arm pitch angles for galaxies in the different samples results in not rejecting the null hypothesis at the 1 per cent level for any of the samples. The distributions of the Anderson–Darling test statistic for ϕgal and ϕarm are shown in the upper panel of Fig. 8 in blue and orange, respectively.
The results of marginalized two-sample Anderson–Darling tests examining whether pitch angles (ϕgal in blue and ϕgal in orange) for galaxies with Bavg < 0.28 and Bavg ≥ 0.28 are drawn from the same distribution (top panel), and the results of marginalized three-sample Anderson–Darling tests for galaxies with no bar (pbar < 0.2), a weak bar (0.2 ≤ pbar ≤ 0.5) and a strong bar (pbar > 0.5) (bottom panel). Confidence intervals are shown, with moving rightwards indicating more confidence in rejecting the null hypothesis that the compared values were drawn from the same parent distribution. We cannot reject the null hypothesis at the 1 per cent level for any of the tests conducted, meaning there is no evidence in this sample that bulge size or bar strength impacts pitch angle.
One limitation of this result is that our sample does not contain many galaxies with dominant bulges: Bavg only varied from 0.09 to 0.75 (the allowed maximum being 1.0), with only four galaxies having Bavg > 0.5. The split point of 0.28 was also chosen to produce evenly sized comparison samples rather than from some physical motivation. However, the lack of any form of correlation implies that there is no evidence in our data for the link between bulge size and pitch angle predicted by the Hubble sequence and observed in other studies.
3.2.2 Pitch angle versus bar strength
One of the predictions of Manifold theory is that pitch angle increases with bar strength as evaluated by the Quadrupole moment, Q at the Lagrangian L1 point (a value that differs from typical ‘bar strength’, which is this value averaged over all radii in the bar; Athanassoula et al. 2009b). In this work, we do not have any similar measurement of bar strength and we note that Athanassoula et al. (2009b) caution that other measures of bar strength may not show this relation; we also do not have significant numbers of strongly barred galaxies in our sample. However in an attempt to investigate this relationship in our data, we make use of Galaxy Zoo 2’s bar fraction (pbar), which has been demonstrated to be a good measure of bar length (Willett et al. 2013) and bar strength (Masters et al. 2012; Skibba et al. 2012; Kruk et al. 2018) and therefore a good measure of the torque applied on the disc gas.
We do not observe a correlation between pbar and E[ϕgal] (Pearson correlation coefficient of −0.05, with a p-value of 0.54, with the relationship between pbar and ϕgal shown in the left-hand panel of Fig. 7). Following Masters et al. (2012) and Skibba et al. (2012), we separate the sample into galaxies without a bar (pbar < 0.2, 50 galaxies), with a weak bar (0.2 ≤ pbar ≤ 0.5, 44 galaxies) and with a strong bar (pbar > 0.5, 33 galaxies). Performing marginalized three-sample Anderson–Darling tests does not find that pitch angles (ϕgal or ϕarm) of galaxies with different bar strengths were drawn from different distributions; we do not reject the null hypothesis at the 1 per cent level for any samples for the test of ϕgal, and at the 10 per cent level for the test of ϕarm. The distributions of Anderson–Darling test statistic is shown in the lower panel of Fig. 8.
The fact that we do not find any link between our measure of bar strength (based on the prominence of the bar in Galaxy Zoo 2) and pitch angle is suggestive that there is actually no link between bar strength and pitch angle, which would exclude Manifold theory as the primary mechanism driving the evolution of the spirals in our sample. However, since we use only a proxy for bar strength, which has not been well tested, this is not conclusive.
3.3 Spiral winding
In QSDW theory, the pattern speed Ωp is a constant in R, as spiral arms obey rigid-body rotation. If Ωp instead varies with radius we would expect |$\cot {\phi }$| to be uniformly distributed between |$\cot {\phi _\mathrm{max}}$| and |$\cot {\phi _\mathrm{min}}$|. The model presented in Pringle & Dobbs (2019) does not give any physical justification for what |$\cot {\phi _\mathrm{max}}$| and |$\cot {\phi _\mathrm{min}}$| should be.
To test this theory, Pringle & Dobbs (2019) used a Kolmogorov–Smirnov test to examine whether a sample of 113 galaxies with measured pitch angles was likely to have been drawn from a distribution uniform in its cotangent. Pitch angles were measured by Yu & Ho (2019) using discrete Fourier transformations in 1D and 2D, and as such do not account for interarm variations. They conclude the model works within limits of |$\cot {\phi } \in [1.00, 4.75]$| (roughly 11.9° < ϕ < 45.0°), motivated by examination of the data.
We perform a similar test in this work, using our sample and methods. We will make use of the marginalized Anderson–Darling test described above, and examine winding on a per-arm basis, as well as a per-galaxy basis. Observation of the distribution of arm pitch angles in our sample (Fig. 9) suggests they are close to uniform in cotangent within limits of 15° < ϕ < 50.0°.
The distributions of pitch angles (orange and blue) relative to one uniform in |$\cot \phi$| (black). Histograms have been normalized by the area between the limits such that they are comparable. The histogram was recalculated with identical bins for each posterior sample of ϕgal and ϕarm, we plot the mean value of each bin, with the sample standard deviation shown as error bars. It is evident that the distributions are very similar between the chosen limits.
3.3.1 Galaxy pitch angle
Testing the uniformity of |$\cot \phi _\mathrm{gal}$| between 15° and 50° using a marginalized Anderson–Darling test results in rejecting the null hypothesis at the 1 per cent level for just 5 per cent of samples, with a large spread in observed test values. The full distribution of Anderson–Darling statistics can be seen in Fig. 10. The large spread in results is caused by the large uncertainties in ϕgal.
The results of a marginalized Anderson–Darling test for uniformity in |$\cot$| for ϕgal (blue) and ϕarm (orange), with values corresponding to various confidence intervals shown. Moving rightwards on the x-axis implies greater confidence in rejecting the null hypothesis that the sample was drawn from a distribution uniform in |$\cot$| between 15° < ϕ < 50.0°. In this instance, we would not be able to reject the null hypothesis at the 1 per cent level for either ϕgal or ϕarm, meaning our sample is consistent with a cot uniform distribution. The larger error in ϕgal means that this result is more significant for ϕarm, which is also physically motivated, as arms can wind independently.
This result suggests that our data are consistent with a cot-uniform source distribution for galaxy pitch angle, but the large uncertainty in ϕgal makes it difficult to make any conclusive statements. This result is also highly sensitive to the lower limit of ϕ: decreasing it to 10° results in us rejecting the cot-uniform model at greater than the 0.1 per cent level for 96 per cent of the posterior samples. We can conclude from this that the Pringle & Dobbs (2019) cot-uniform model is an adequate fit to the data, as long as the minimum pitch angle, |$\cot {\phi _\mathrm{min}}$|, at which the majority of winding dissipate or disappear is ϕmin > 10°, and more confidently ϕmin = 15°. We reiterate that there is no prediction in Pringle & Dobbs (2019) as to what this minimum pitch angle should be, so our observation constrains the allowed range.
3.3.2 Arm pitch angle
The inconclusive result for ϕgal is perhaps unsurprising: were we to assume that spiral arms are transient and recurrent instabilities, there is little reason for all of the arms to be at precisely the same evolutionary stage at the same time. This is supported by the large observed spread in interarm pitch angles (Section 3.1).
If we assume instead that spirals form and wind independently inside a galaxy, and that their evolution over time can be described by equation (12), the distribution of the cotangent of pitch angles of individual arms should be uniform between our limits, rather than that of the galaxy’s pitch angle as a whole.
Using the marginalized Anderson–Darling test we cannot reject the null hypothesis at even the 5 per cent level for any of the possible realizations of arm pitch angle. The resulting distribution of Anderson–Darling statistics is shown in the lower panel of Fig. 10. This result is highly consistent with the model for spiral winding proposed by Pringle & Dobbs (2019) with |$\cot {\phi _\mathrm{min}} = {15^\circ }$| and can be interpreted as evidence that spirals are formed through local disc perturbations, and are primarily governed by local forces.
4 SUMMARY AND CONCLUSIONS
This paper presents a new Bayesian approach to estimate galaxy pitch angle, making use of citizen science results to measure spiral arms through photometric modelling. We introduce an adaptation of the Anderson–Darling test, which we name the marginalized Anderson–Darling test, to incorporate full Bayesian posterior probabilities and use this test to investigate theories governing spiral formation and evolution.
The hierarchical Bayesian approach implemented in this paper allows a more thorough examination of pitch angle than length-weighted pitch angle calculation obtaining posterior distributions of measured parameters. It better accounts for the large variations observed in interarm pitch angle than Fourier analysis, which assumes all arms in a given symmetric mode have the same pitch angle. In this work, we find that the mean interarm difference in pitch angle is 11.0° ± 0.9°.
There is no evidence in our data for the link between bulge size and pitch angle predicted by the Hubble sequence and observed in other studies (see Section 3.2.1).
We do not find any link between our measure of bar strength and pitch angle in our sample. However, rather than a direct measure of bar strength, we make use of an available parameter which correlates with bar strength, so at best this observation is suggestive that the primary mechanism driving the evolution of the spirals in our sample may not be Manifold theory (see Section 3.2.2). Since this is not the measure of bar strength predicted to correlate with pitch angle by Athanassoula et al. (2009b), and those authors caution that the details of the bar strength measure can wash out the predicted correlation, this is not strong evidence against Manifold theory models.
Our results are consistent with spiral winding of the form described by Pringle & Dobbs (2019), in which spiral arms are transient and recurrent, evolve through mechanisms such as swing amplification (Goldreich & Lynden-Bell 1965) and which wind up over time. This model predict a distribution of pitch angles that is uniform in cotangent space across some range. No prediction is provided as to what that range should be. Our data are consistent with this model, if the minimum pitch angle is ϕmin = 15°, but rule it out if the minimum pitch angle is ϕmin = 10°. The assumptions of this model of spiral winding are highly simplistic, and it leaves many unanswered questions: what determines the limits on ϕ? Is the spiral arm equally apparent at all pitch angles, or is a selection effect present? Our observations suggest that any further development of this model needs to predict that the minimum pitch angle, ϕmin > 10°. This result is also not evidence against QSDW, as our distribution of pitch angles may be dictated by other factors such as disc shear.
In this work, we assume that spiral arms are equally likely to be identified and recovered at all pitch angles, which suggests the absence of galaxies at low pitch angles is not due to an inability of us to measure such arms. This is not an unfair assumption given the amount of human effort that went into obtaining spiral arm measurements (more so than any other pitch angle measurement method, with each galaxy receiving at least 30 human classifications). The galaxy sample used is a random subset of a volume limited sample (see Fig. 3), and is comparable in size to those used in other similar studies (Savchenko & Reshetnikov 2013; Pringle & Dobbs 2019; Yu & Ho 2019). The sample covers a range of masses (9.45 < log (M*/M⊙) < 11.05) and spiral types, however it is possible that tightly wound spirals are preferentially missed by the pre-selection from Galaxy Zoo, if they are less obviously identified as having spirals at these distances (0.02 < z < 0.055). The Kennicutt (1981) sample of 113 much more nearby spirals (all at z < 0.019 and most at z < 0.009), includes several with arms at much lower pitch angles, but very few arms which are loosely wound (none with |$\cot \phi \gt 31^\circ$|), meaning it does not match the same |$\cot \phi$| constant model well, although as at the time the sample was set by data availability, it is unclear how conclusive this is, and Kennicutt (1981) note the incompleteness of their sample for open armed spirals. In a future version of Galaxy Builder, we intend to include this sample as a comparison set.
We have presented evidence that the methodology proposed here is a robust solution to the problems facing investigation of spiral morphology, namely that of reliably identifying spiral arms, and properly accounting for the spread in pitch angles of arms within a galaxy. This is one of the largest samples for which this test has been done and is scaleable to larger samples; such a sample would make possible further comparisons, such as splitting galaxies into spiral type (grand design / many-armed / flocculent), examining the differences between populations, investigating if the interarm spread depends on other galaxy properties.
The processes governing the formation and evolution of spiral arms are complicated, but the prevalence of spiral galaxies in the Universe, their impact for understanding star formation, and the spiral nature of our own Milky Way, makes investigating their dynamics of fundamental importance to the scientific aims of understanding, predicting, and explaining the nature of the cosmos.
DATA AVAILABILITY
The data and software analysis underlying this paper are available on GitHub at https://github.com/tingard/gzbuilder_results and https://github.com/tingard/hierarchical-modelling-of-spiral-pitch-angle.
ACKNOWLEDGEMENTS
This publication made use of Sloan Digital Sky Survey-I/II data. Funding for the Sloan Digital Sky Survey and Sloan Digital Sky Survey-II was provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The Sloan Digital Sky Survey Web Site is http://www.sdss.org/.
This publication uses data generated via the Zooniverse.org platform, development of which is funded by generous support, including a Global Impact Award from Google, and by a grant from the Alfred P. Sloan Foundation. We would also like to thank the 2340 volunteers who have submitted classifications to the Galaxy Builder project, especially user EliabethB, whose presence on the Galaxy Builder forum on top of a large number of galaxies modelled has been a huge help.
This project was partially funded by a Google Faculty Research Award to Karen Masters (https://ai.google/research/outreach/faculty-research-awards/), and Timothy Lingard acknowledges studentship funding from the Science and Technology Facilities Council (ST/N504245/1).
Footnotes
https://docs.pymc.io/ (accessed April 4, 2021)
REFERENCES
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