Effects of Multiple Spiral Arm Patterns on the Abundance Gradients of Heavy Elements

Abstract

Understanding how spiral structures influence the chemical evolution of the Galactic disc remains a key issue in Galactic archaeology. Recent advances in two-dimensional chemical evolution modeling allow us to account for the impact of multiple spiral arm patterns, each characterized by different pattern speeds, on the redistribution of elements throughout the Galaxy. In this work, we explore the influence of multi-pattern spiral arms on the radial abundance gradients of heavy elements in the Galactic disc. We focus on a scenario in which, during the most recent stage of evolution, corotation spans the entire disc. Our results indicate that the observed dispersion in the abundance gradients of O, Fe, Eu, and Ba, as traced by Cepheids, can be successfully reproduced if all galactocentric radii have effectively acted as corotation regions over the past 1–3 Gyr. We also note that such an extended phase has previously been identified as necessary to explain the azimuthal abundance variations reported in Gaia DR3 and Gaia-ESO survey data along local and inner spiral arms.

1. Introduction

In the field of Galactic Archaeology using chemical evolution models [1,2,3], understanding the role of spiral arm perturbations is becoming increasingly important. Several investigations have interpreted the Milky Way’s spiral structure as arising from multiple overlapping spiral perturbations, each with a distinct pattern speed (e.g., [4,5]). Building on this perspective, ref. [6] employed the two-dimensional Galactic chemical evolution model originally introduced by [7], in which the spiral pattern is represented as three separate segments, each rotating at its own angular velocity. Their findings indicate that the passage of spiral arms locally boosts star formation, producing spatial variations in elemental abundances. The magnitude of these variations is closely linked to the element production timescales: elements formed by short-lived stars show rapid, significant fluctuations, whereas those released over longer periods exhibit smoother and less pronounced variations. This methodology was later applied by [8] to study the two-dimensional distribution of the short-lived radionuclide ²⁶Al. Some hydrodynamic simulations and gravity-only N-body models support a picture of spiral structure in which spiral arms are transient features that continuously form, dissipate, and re-form rather than remaining steady over time (e.g., [9,10,11,12]). In this framework, individual arms dissipate on dynamical timescales, while new ones continuously form, maintaining a long-lived overall spiral pattern. For example, studies such as [9,13] show that spiral arms in idealised N-body simulations behave as transient features that co-rotate with the stellar component at all radii, often referred to as dynamic or co-rotating spirals. These structures are thought to arise through a process similar to swing amplification [14,15,16], where the combined effects of differential rotation, stellar epicyclic motion, and self-gravity amplify local density perturbations. Hence, ref. [6] considered scenarios in which spiral arms act as transient features corotating with the Galactic disc across a broad radial range, consistent with predictions from the above-mentioned numerical simulations. In the present study, we investigate the combined influence of multiple spiral patterns under the assumption that, in the latest stages of Galactic evolution, corotation occurs across all radii. We particularly focus on how this setup affects the radial abundance gradients of heavy elements in the Galactic disc and assess whether the observed scatter in Cepheid measurements can be accounted for by these models.

In particular, the original contribution of this work lies in exploring a scenario in which corotation extends across the entire Galactic disc during the last 3 Gyr of Galactic evolution (Model C3; Section 2.2). The results for the last 1 Gyr and 0.3 Gyr are adapted from [6]. In addition, the comparison with Cepheid observations using a running-median analysis is presented here for the first time in the context of scenarios in which corotation spans the entire Galactic disc over the last 0.3, 1, and 3 Gyr, respectively.

2. Chemical Evolution Model in the Presence of Multiple Spiral Arm Patterns

2.1. Gas Accretion and Inside-Out Formation of the Thin Disc

The Galactic thin disc, as in [6], is assumed to form through the gradual accretion of primordial gas, following the classical chemical evolution framework of [17]. We adopt the inside-out formation scenario originally proposed by [17] and further developed by [18], in which the accretion timescale increases with Galactocentric radius as $\tau \left(R\right)=1.033R\left[\mathrm{kpc}\right]-1.27\mathrm{Gyr}$. This assumption is consistent with cosmological zoom-in simulations [19,20,21,22,23] and is essential to reproduce observed abundance gradients. The current total surface density in the solar neighborhood is taken as ${\mathsf{\Sigma }}_{\odot }=54$ M_⊙ pc⁻² [18,24,25,26], with a radial profile described by ${\mathsf{\Sigma }}_D(R,t_G)={\mathsf{\Sigma }}_{\odot }{e}^{-(R-R_{\odot })/R_D}$, where $t_G$ is the present Galactic age and $R_D=3.5$ kpc is the disc scale length. Finally, we assume a spatially and temporally invariant stellar initial mass function (IMF), adopting the formulation of [27].

2.2. Modeling Multiple Spiral Patterns

Expanding on the approach introduced by [7], ref. [6] incorporated multiple spiral patterns into a multi-zone two-dimensional Galactic chemical evolution model, examining their effects on elements synthesised on different timescales, such as oxygen, iron, europium, and barium. We partitioned the disc into concentric radial shells, each 1 kpc wide. Every shell was further subdivided into 36 angular sectors, each spanning 10^o, resulting in 36 distinct zones at any given Galactocentric radius. Following [28], the spiral structure is represented as a combination of localized features, each characterized by its own pattern speed.

The spiral-induced, time-dependent surface density perturbation adopted from [6] is defined as ${\mathsf{\Sigma }}_S(R,\varphi ,t)=\chi (R,t_G){\sum }_{j=1}^N{M}_j\left({\gamma }_j\right)$, where the present-day spiral amplitude is $\chi (R,t_G)={\mathsf{\Sigma }}_{S,0}exp[-(R-R_0)/R_S]$. The term $M_j\left({\gamma }_j\right)$ is modulation function of the j-th spiral component [29]:

$$M_j\left({\gamma }_j\right)=\left({\displaystyle \frac{8}{3\pi }}cos{\gamma }_j+{\displaystyle \frac{1}{2}}cos\left(2{\gamma }_j\right)+{\displaystyle \frac{8}{15\pi }}cos\left(3{\gamma }_j\right)\right)·\mathbb{1}[Rj,\min ,R_{j,\max }],$$

(1)

where the indicator function limits each pattern to the radial interval $[R_{j,\min },R_{j,\max }]$. The phase angle is given by:

$${\gamma }_j(R,\varphi ,t)=m_j\left[\varphi +{\mathsf{\Omega }}_{s,j}t-{\varphi }_p\left(R_0\right)-{\displaystyle \frac{ln(R/R_0)}{\tan \alpha }}\right],$$

(2)

with $m_j$ being the arm multiplicity, $\alpha ={15}^{\circ }$ the pitch angle, and ${\mathsf{\Omega }}_{s,j}$ the pattern speed. A more detailed description of the model can be found in [6]. Figure 1 (left panel) illustrates the adopted spiral configuration (Model A in

Table 1. Summary of the main characteristics of the models considered in this study. For each model, we list the pattern speed (${\mathsf{\Omega }}_{s,j}$), the radial extent of each spiral component ($R_{\min }$ to $R_{\max }$), and the associated corotation radius $R_{j,\mathrm{cor}.}$. All spiral structures follow the evolution of Model A (see [6]) for the majority of the Galactic disk’s lifetime. In the final stages, we explore three scenarios in which corotation extends across all radii, i.e., ${\mathsf{\Omega }}_{s,j}\left(R\right)={\mathsf{\Omega }}_d\left(R\right)$: for the last 0.3 Gyr (Model C0_3), 1 Gyr (Model C1), and 3 Gyr (Model C3) (see [6,31,32]). All models share the same global parameters: a radial scale length for the decrease in the spiral arm density amplitude of $R_S=7$ kpc, a disk scale length of $R_D=3.5$ kpc, and a reference spiral arm surface density at $R_0=8$ kpc of ${\mathsf{\Sigma }}_{S,0}$ = 20 M_⊙ pc⁻². In addition, a star formation efficiency of $\nu =1.1$ Gyr⁻¹ and a multiplicity parameter $m=2$ are adopted throughout.

Models	${\mathit{M}}_{\mathit{MS},\mathbf{1}}\mathbf{\left(}{\mathit{\gamma }}_{\mathbf{1}}\mathbf{\right)}$				${\mathit{M}}_{\mathit{MS},\mathbf{2}}\mathbf{\left(}{\mathit{\gamma }}_{\mathbf{2}}\mathbf{\right)}$				${\mathit{M}}_{\mathit{MS},\mathbf{3}}\mathbf{\left(}{\mathit{\gamma }}_{\mathbf{3}}\mathbf{\right)}$
	${\mathsf{\Omega }}_{\mathbf{s},\mathbf{1}}$	${\mathit{R}}_{\mathbf{1},\mathbf{min}}$	${\mathit{R}}_{\mathbf{1},\mathbf{max}}$	${\mathit{R}}_{\mathbf{1},\mathbf{cor}.}$	${\mathsf{\Omega }}_{\mathbf{s},\mathbf{2}}$	${\mathit{R}}_{\mathbf{2},\mathbf{min}}$	${\mathit{R}}_{\mathbf{2},\mathbf{max}}$	${\mathit{R}}_{\mathbf{2},\mathbf{cor}.}$	${\mathsf{\Omega }}_{\mathbf{s},\mathbf{3}}$	${\mathit{R}}_{\mathbf{3},\mathbf{min}}$	${\mathit{R}}_{\mathbf{3},\mathbf{max}}$	${\mathit{R}}_{\mathbf{3},\mathbf{cor}.}$
	[ $\frac{\mathbf{km}}{\mathbf{s}\mathbf{kpc}}$]	[kpc]	[kpc]	[kpc]	[ $\frac{\mathbf{km}}{\mathbf{s}\mathbf{kpc}}$]	[kpc]	[kpc]	[kpc]	[ $\frac{\mathbf{km}}{\mathbf{s}\mathbf{kpc}}$]	[kpc]	[kpc]	[kpc]
A	30.0	3.0	7.0	6.0	20.0	6.0	12.0	8.7	15.0	9.0	18.0	12.0
C0_3	${\mathsf{\Omega }}_d$	3.0	7.0	[3–7]	${\mathsf{\Omega }}_d$	6.0	12.0	[6–12]	${\mathsf{\Omega }}_d$	9.0	18.0	[9–18]
(last 0.3 Gyr)
C1	${\mathsf{\Omega }}_d$	3.0	7.0	[3–7]	${\mathsf{\Omega }}_d$	6.0	12.0	[6–12]	${\mathsf{\Omega }}_d$	9.0	18.0	[9–18]
(last 1 Gyr)
C3	${\mathsf{\Omega }}_d$	3.0	7.0	[3–7]	${\mathsf{\Omega }}_d$	6.0	12.0	[6–12]	${\mathsf{\Omega }}_d$	9.0	18.0	[9–18]
(last 3 Gyr)

), consisting of three segments with multiplicity $m=2$, each characterized by a distinct ${\mathsf{\Omega }}_{s,j}$ and radial extent. The disc angular velocity ${\mathsf{\Omega }}_d\left(R\right)$ follows [30]. In addition, N-body simulations of barred galaxies [9] suggest that spiral structures are transient, recurrent, and exhibit pattern speeds decreasing with radius toward corotation with the stellar disc. Following [6], we therefore assume that at recent times ${\mathsf{\Omega }}_{s,j}\left(R\right)={\mathsf{\Omega }}_d\left(R\right)$ for all spiral components (Models C0_3, C1, and C3 in Table 1). This prescription is necessary to reproduce the azimuthal abundance variations observed in Gaia DR3 [31] and Gaia-ESO data [32].

Figure 1 (left panel) shows the spiral structure adopted in this work: with multiplicity m = 2, comprising three spiral segments, each rotating at a different pattern speed ${\mathsf{\Omega }}_{s,j}$ and confined to a specific radial range (Model A in Table 1). The disc angular velocity ${\mathsf{\Omega }}_d\left(R\right)$ is taken from [30], with the central pattern speed set to ${\mathsf{\Omega }}_{s,2}=20$ km s⁻¹ kpc⁻¹. Spiral density waves are generally limited by their principal resonances, the ILR and OLR. Moreover, N-body simulations of barred galaxies [9] indicate that spiral arms are transient and recurrent, with pattern speeds that decrease with radius and tend to corotate with the stellar disc. To approximate this behavior, [6] assumed that, at recent times, the spiral pattern speeds follow the disc rotation curve. Consistently, in this work we adopt at most recent evolutionary times the condition ${\mathsf{\Omega }}_{s,j}\left(R\right)={\mathsf{\Omega }}_d\left(R\right)$ for all radii and spiral components (Models C0_3, C1, and C3 listed in Table 1), an assumption that has proven essential to reproducing the azimuthal abundance variations observed in Gaia DR3 [31] and Gaia-ESO data [32].

3. Nucleosynthesis Prescriptions

3.1. Oxygen and Iron

Following several previous chemical evolution studies (e.g., [1,33,34,35,36,37,38]), we adopt the nucleosynthesis prescriptions of [39] for oxygen and iron. In their work, the authors identified the sets of stellar yields that provide the best match to observational data (we refer the reader to [39] for full details on the datasets used). For Type II supernovae (SNe II), ref. [39] determined that the yields from [40] offer the best agreement with observations. Specifically, iron yields computed for solar metallicity require no adjustments, while oxygen yields are better reproduced when calculated as functions of metallicity. For Type Ia supernovae (SNe Ia), we adopt the theoretical yields from [41], and for single low- and intermediate-mass stars, we follow the prescriptions of [42].

3.2. Europium and Barium

Neutron star mergers (NSMs) are considered a primary site for europium production in our analysis. Following the approaches of [43,44], we define the realization probability of double neutron star systems (i.e., the fraction of massive stars that ultimately merge) as ${\alpha }_{\mathrm{NSM}}$. For each event, we adopt an Eu yield of 2 × 10⁻⁶ M_⊙, consistent with the range suggested by [45], which estimates that NSMs can produce between 10⁻⁷ and 10⁻⁵ M_⊙ of Eu per merger. It is further assumed that a fixed fraction of massive stars in the 10–30 M_⊙ range serve as NSM progenitors. To reproduce the observed Galactic NSM rate of $R_{\mathrm{NSM}}={83}_{-66}^{+209}$ Myr⁻¹ [46], the parameter ${\alpha }_{\mathrm{NSM}}$ is set to 0.05. Observations of GW170817 provide supporting evidence for this rate [2,47].

We assume a constant coalescence timescale of 1 Myr for all NSM events, following [43,44]. While this simplification neglects the distribution of merger timescales that is likely present in nature, similar to the delay-time distribution for Type Ia supernovae [48,49], the short, fixed delay is compatible with other potential r-process sources, such as magneto-rotationally driven (MRD) SNe [50,51] and collapsars [52]. For the s-process contribution, we adopt the yields from [53,54] for low-mass AGB stars (1.3–3 M_⊙), which primarily impact barium production in our model. We use the non-rotating stellar yields, which tend to overestimate s-process element production at solar metallicity; yields from rotating AGB stars, in contrast, produce insufficient neutron-capture elements. Following the approach of [55], we reduce the non-rotating yields by a factor of 2 to better match observational constraints at solar metallicity. Additionally, s-process production from rotating massive stars is included, using the updated nucleosynthesis prescriptions of [56], as detailed in Table 3 of [55], building on earlier work by [44,57,58,59].

4. Results

In this section, we explore how the presence of multiple spiral arm patterns influences the abundance gradients of different chemical elements. All spiral structures follow the evolution of Model A (see Table 1) for the majority of the Galactic disk’s lifetime. In the final stages, we explore three scenarios in which corotation extends across all radii during the last 0.3, 1, and 3 Gyr of Galactic evolution (models C0_3, C1, and C3, respectively). Previous studies indicate that maintaining corotation across all Galactic radii for an extended period of 1–3 Gyr is required to reproduce the azimuthal abundance variations observed in Gaia DR3 and Gaia-ESO data along local and inner spiral arms [31,32]. However, these timescales have not yet been directly tested against radial abundance gradients measured from Cepheids, especially for neutron-capture elements. Notably, the abundance gradients for model C3 (period of 3 Gyr where all radii are corotations) were not considered in [6] nor have its predictions for heavy elements been presented before.

In Figure 2, Figure 3 and Figure 4, we present the abundance gradients predicted by our models at six different azimuthal angles for oxygen, europium, iron, and barium. In this study, we focus in particular on how the spread varies with Galactocentric distance, as generated by the six different abundance gradients predicted at these azimuthal angles. These predictions are obtained under the assumption that corotation extends across all Galactic radii during the last 0.3, 1, and 3 Gyr of Galactic evolution. The model results are compared with observed abundance gradients derived from Cepheids: oxygen and europium from [60,61], iron from [62], and barium from [63]. To enable a quantitative comparison between model predictions and observations, we computed the running median of the data in radial bins of 0.5 kpc, adopting a 40% overlap between consecutive bins. We required a minimum of two stars per bin for O, Fe, and Eu, while for Ba, given the more limited dataset, we allowed bins with at least one star. The corresponding standard deviations were also calculated and included in the analysis to quantify the data dispersion.

Figure 2 shows that the observed dispersion in the data cannot be reproduced if corotation across all radii is assumed to have occurred only during the last 0.3 Gyr. Nevertheless, elements synthesized on relatively short timescales, such as oxygen and europium, display a larger spread, especially near the radii where corotation was previously confined before extending throughout the disk, in qualitative agreement with the observations. In contrast, Figure 3 and Figure 4 show a significantly improved agreement with the data. In particular, in the 3 Gyr scenario, the model is able to account for nearly the entire observed dispersion in the abundance gradients, indicating that spiral arm effects can largely explain the observed spread. However, we highlight that the observed dispersion appears to be more uniform and nearly constant across different radii, whereas the model predicts a more variable radial behaviour. Another caveat is that the model predictions for oxygen and europium appear to underestimate the observed abundances.

Finally, in

Table 2. Comparison of the abundance gradient slopes computed between 4 and 15 kpc (derived from linear fits), between observations and model predictions for oxygen, iron, and europium. In the models, the slopes are obtained from the median elemental abundances [X/H], averaged over all azimuthal coordinates as a function of the Galactocentric distance. For the observational data, the slopes are computed from median abundance values evaluated in radial bins of 1 kpc.

Gradient Slope Δ[X/H]/ΔR [dex/kpc]	DATA	Model C0_3	Model C1	Model C3
X = O	−0.057	−0.051	−0.051	−0.055
X = Fe	−0.072	−0.079	−0.080	−0.082
X = Eu	−0.038	−0.045	−0.045	−0.049

we compare the abundance gradient slopes, derived from linear fits, between observations and model predictions for oxygen, iron, and europium (barium is excluded due to the limited availability of observational data). In the models, the slopes are obtained from the median elemental abundances [X/H], averaged over all azimuthal coordinates as a function of Galactocentric distance. We find that models C0_3 and C1 predict very similar slopes, closely matching the observations. Model C3 improves the agreement for oxygen, but slightly overestimates the steepness of the gradients for Fe and Eu compared to the data.

5. Discussion

Our findings indicate that the presence of multiple spiral arm patterns has a significant impact on the abundance gradients of several elements, including neutron-capture species. We investigated scenarios in which corotation extends across the entire Galactic disc over the last 0.3, 1, and 3 Gyr. The shortest timescale (0.3 Gyr; Figure 2) is insufficient to reproduce the observed dispersion in the data. In contrast, longer durations of extended corotation (1–3 Gyr; Figure 3 and Figure 4) progressively improve the agreement with observations, with the 3 Gyr case accounting for nearly the full extent of the observed spread. This suggests that a long-lived corotation pattern spanning all Galactic radii plays a key role in generating azimuthal variations in chemical abundances. However, a fraction of the present-day dispersion is likely attributable to stellar radial migration, which is not included in our model. This process redistributes stars throughout the Galactic disk, thereby contributing to the observed spread in combination with spiral arm effects [64,65,66]. Overall, while corotation driven by spiral structure can explain a large part of the observed abundance patterns, a comprehensive interpretation requires accounting for both spiral dynamics and stellar migration within chemical evolution models. On the other hand, if future studies rule out the presence of co-rotation at all radii for such an extended evolutionary time as a viable scenario, then the observed spread in the abundance gradients cannot be attributed to the effects of spiral arms.