Hierarchical Absorption in LAMOST low Resolution Normalized Spectra

According to the hierarchical clustering scenario, galaxies are assembled by merging and accretion of many clusters of different sizes and masses(Klypin et al., 1999). Cold dark matter model also predicts that the galaxies like the Milky Way form hierarchically(Bullock & Johnston, 2005; Komatsu et al., 2011; Bullock & Boylan-Kolchin, 2017). However, the disk stars, which account for the vast majority of the Galaxy, appear to be continuous. The interaction between stars and the Galaxy could be the explanation of the contradiction. A star cluster in or near the Galaxy initially had clustering properties; under the interaction with the Galaxy, it gradually disintegrates and forms a stellar stream(Ibata et al., 2002; Johnston et al., 2002; Carlberg, 2012; Ibata et al., 2021); then it will lose spatial distribution characteristics and only retain kinematic features, such as Gaia SausageBelokurov et al. (2018); finally, the kinematic features of the stars will be lost, leaving only chemical information. The above process has been verified in the galactic halo. As for the disk, some works suggest that a significant fraction of the Galactic Disk also formed hierarchically(Gilmore et al., 2002; Abadi et al., 2003), but the evidence is limited. If the Galactic disk is indeed formed hierarchically, it is believed that most of the clusters in the disk are only recognizable by their element abundances.

The elemental abundances obtained by spectral line fitting are difficult to use for distinguishing potential substructures within the Galactic disk. On the one hand, the elemental abundances derived in this way inherently contain errors, leading to a broadening of the abundance distribution. For low-resolution spectra, the unclear spectral line profiles introduce errors. For medium-to-high-resolution spectra, errors can be introduced by factors such as the difficulty in determining the continuum spectrum, uncertainties in atmosphere parameters and atomic parameters, and non-local thermodynamic equilibrium (NLTE) effects in metal-poor stars(e.g. Amarsi et al., 2016; Li et al., 2022; Lombardo et al., 2022; Shen et al., 2023). However, even if there were ways to significantly improve the accuracy of abundances derived from spectral line fitting, it may still not be possible to identify clusters within the Galactic disk if we only focus on Fe and light elements, due to the high degeneracy of these elements(e.g. Horta et al., 2023). Abundant sources of elements lead to higher degeneracy. Almost all the stars can generate iron and light elements. In contrast, the sources of heavy elements, especially neutron capture elements, are much fewer. The neutron capture process contains rapid neutron capture process (r-process) and slow neutron capture process (s-process). For elements heavier than Fe (or Zn), based on current understanding, they can only produced by neutron capture process. The astrophysical sites of the r-process have been debated, such as type II supernovae, binary neutron star mergers, cataclysmic variable stars, and other white dwarf systems(Lattimer & Schramm, 1974; Rosswog et al., 1999; Arcones et al., 2007; Hoffman et al., 2008; Fischer et al., 2010; Wanajo et al., 2011; Arcones & Montes, 2011; Goriely et al., 2011; Wanajo et al., 2013; Wanajo, 2013; Wanajo et al., 2014). Among them, only the binary neutron star merger is confirmed till now(Chornock et al., 2017; Drout et al., 2017; Kilpatrick et al., 2017; Pian et al., 2017; Abbott, 2017a; Smartt, 2017; Valenti et al., 2017; Shappee et al., 2017; Abbott, 2017b; Watson et al., 2019). The s-process occurs in the late evolution stage of small and medium-mass stars, but due to the uncertainty of the reaction rate of neutron source reactions, theoretical models cannot well constrain element yields(Busso et al., 1999; Herwig, 2005; Karakas & Lattanzio, 2014; Pignatari et al., 2010; Frischknecht et al., 2015; Choplin, Arthur et al., 2018; Limongi & Chieffi, 2018). The abundance of the neutron capture process elements in stars is necessary for in-depth understanding of the r-process and s-process, as well as the analysis of the substructures within the Galactic disk.

The measurement of the abundance of neutron capture process elements is difficult because fitting their spectral lines requires mid or high spectral resolution. So, the sample size of stars with s-process abundances(Jorissen & Mayor, 1988; McClure & Woodsworth, 1990; Van Eck & Jorissen, 1999; Shetye et al., 2018, 2020) and r-process abundances(Frebel, 2018; Hansen et al., 2018; Roederer et al., 2018; Sakari et al., 2018a, b, 2019; Ezzeddine et al., 2020; Holmbeck et al., 2020) is small. Accurate abundance measurements must fit the spectral lines, but rough abundance estimate sometimes works in photometric data or low-resolution spectra (R $\sim$2000). There have been successful methods to estimate [Fe/H] using photometry(e.g. Huang et al., 2023) or low-resolution spectra(e.g. Wu et al., 2011). As for neutron capture process elements, if the abundance of them is obviously higher than average (so called s-type stars or r-type stars), photometric data and low-resolution spectra can help to identify them(Jorissen et al., 1993; Chen et al., 1998; Wang, X.-H. & Chen, P.-S., 2002; Yang et al., 2006; Chen et al., 2019, 2023), but hard to provide the abundance. Besides, the low-resolution spectra still have potential. Even though there are only a few spectral lines that can be fitted in low-resolution spectra, it still retains a lot of information about elements(Ting et al., 2017). If the overall effect of all spectral lines of a certain element can be identified on low-resolution spectra, abundance measurement can be achieved. However, even with this method, only few heavy elements can be measured(Xiang et al., 2019), because the abundances of many heavy elements are too low, or the distribution of the absorption lines of several heavy elements are too close.

It is difficult to estimate the abundance of a single heavy element, but it should be easier to estimate the overall abundance of several heavy elements. The s- and r-processes always form a series of elements at the same time, so it is reasonable to regard all neutron capture elements as a whole. Most of the absorption lines of s-process elements and r-process elements are more concentrated in the violet end of the optical spectrum, so the effects of them in low-resolution spectra are close, which will be shown in this paper.

This paper is organized as follows. Section 2 introduces the synthetic and observed data used in this work. Section 3 provides the detailed process for obtaining evidence of the Hierarchical absorption in LAMOST low resolution normalized spectra. Section 4 mainly provides a simulation, and some other discussions.

It is found that as the frequency of the spectrum increases, the spectral absorption depth becomes more sensitive to changes in the abundances of the neutron capture elements. Therefore, Eq. 1 is selected to fit the difference spectrum between the actual spectrum and the baseline spectrum.

Where the unit of x is Å, and the wavelength range of spectral is 4000Åto 8000Å. The fitting parameters must contain information about neutron capture elements, but they must also include other information. As shown in Fig. 2, for solar-like stars, the fitting parameters are sensitive to Ca, Sc, Ti, Cr, Mn, Fe, Co, Ni, Sr, Zr, La, Ce, Pr, Nd, Sm, Gd, and Dy. The change in the other elements will not affect the fitting parameters too much. So, $p_{1}$ and $p_{2}$ can be regarded as the projections of atmospheric parameters, abundances of neutron capture elements, and several other elements in this specific two-dimensional parameter space. Among atmospheric parameters, $T_{\mathrm{eff}}$, $\log g$, and microturbulence have no direct relationship with the substructures in the Galactic disk, and should be decoupled from $p_{1}$ and $p_{2}$ before analysis. The specific decoupling process is as follows:

First, the median spectrum of one thousandth of the stars randomly selected from the LAMOST LRS Stellar Parameter Catalog of A, F, G and K Stars is calculated. One thousandth of the spectra are chosen randomly instead of the entire catalog to reduce computational costs. The median spectrum is chosen, instead of the average spectrum which has a lower computational cost, to exclude the impact of outlier spectra. The $p_{1}$ and $p_{2}$ distribution of the entire catalog is shown in Fig. 3. It can be seen that $p_{1}$ and $p_{2}$ are coupled with $T_{\mathrm{eff}}$, which means that the same cluster is split into multiple clusters due to different $T_{\mathrm{eff}}$. Decoupling of the $T_{\mathrm{eff}}$ must be performed. Selecting a narrow temperature range can solve the problem, as shown in Fig. 4. Fig. 4 is a deformation of Fig. 3, which shows that $p_{1}$ and $T_{\mathrm{eff}}$ are not coupled, $p_{2}$ and $T_{\mathrm{eff}}$ are strongly coupled, however, in small intervals, $p_{2}$ and $T_{\mathrm{eff}}$ are almost decoupled. Besides, $\log g$ causes both $p_{1}$ and $p_{2}$ to stratify. As a result, we separate the LAMOST sample in to 16 $T_{\mathrm{eff}}$ ranges: 3900K, 4120K, 4340K, 4520K, 4710K, 4900K, 5090K, 5290K, 5460K, 5670K, 5860K, 6050K, 6240K, 6420K, 6680K, 6860K, 7000K, and recalculate the fitting parameters within each range. Only the main sequence star is retained, there are two reasons: On the one hand, remove the giants and sub-giants can decouple the fitting parameters and $\log g$. On the other hand, some mixed giants can transport the elements produced in their core onto the surface. Fig. 5 shows the distribution of the fitting parameters versus $T_{\mathrm{eff}}$ and surface gravity of the sample between 5670K and 5860K, demonstrating that decoupling has indeed been achieved. The decoupled fitting parameters are written as $\tilde{p}_{1}$ and $\tilde{p}_{2}$.

Among the known factors, microturbulence has not yet been decoupled from the fitting parameters so far. However, Fuhrmann (2004) found that microturbulence has a strong correlation with $T_{\mathrm{eff}}$. Therefore, to a certain extent, $\tilde{p}_{1}$ and $\tilde{p}_{2}$ have been decoupled from the microturbulence. Even if the decoupling is not complete, the residual effect of microturbulence is unlikely to cause clustering in the distribution of $\tilde{p}_{1}$ and $\tilde{p}_{2}$.

To enhance our understanding of the interrelationships among data clusters, we performed a clustering analysis using the HDBSCAN algorithm from sklearn.cluster package. The clustering procedure was carried out in a two-dimensional parameter space, encompassing $\tilde{p}_{1}$, $\tilde{p}_{2}$. During this process, we employed the natural logarithm of pp1, and imposed constraints on $\ln\tilde{p}_{1}$ to reside between -4 and 70, and on $\tilde{p}_{2}$ to fall within the interval of -0.5 to 0.5, ensuring an absolute value greater than 10^-5 to preclude the inclusion of outliers. Stars with $\mathrm{SNR}_{u}<10$ and the error of radial velocity larger than 10 (possible binaries) are also removed. Prior to the clustering analysis, the input parameters were normalized utilizing StandardScaler from the sklearn.preprocessing package, thereby facilitating a more robust evaluation of the data clusters.

Our investigation was particularly focused on two hyperparameters, namely min_cluster_size and min_samples. For HDBSCAN method, the hyperparameters significantly influence the outcomes of clustering, evaluating the clustering results under different hyperparameters can be achieved visually. To improve the classification performance, in each $T_{\mathrm{eff}}$ range, the data are manually separated into two parts, low density and high-density areas. A grid search was executed to ascertain the optimal pairing of these hyperparameters, considering the values for min_cluster_size within the range of [400,300,200,100,50,40,30] for low density areas; [600,500,400,300,200,100] for high density areas. For min_samples, both low and high density areas are set within [9, 7, 5, 3, 2, 1]. Upon applying the best hyperparameters in each $T_{\mathrm{eff}}$ interval, 16, 23, 18, 24, 28, 26, 48, 45, 50, 50, 61, 40, 58, 51, 38, 28 clusters are found. For colder stars, their selection effect is strong; For hotter stars, their sample size is small, both factors can lead to cluster loss. An example of clusters is shown in Fig. 6. Fig. 7 depicts that the clusters do not have obvious kinematic features.

Among the 604 clusters we found, 538 of them have average [Fe/H], [Fe/H] scatter, average [$\alpha$/M], and [$\alpha$/M] scatter. K-means algorithm is performed in this four-dimensional parameter space. This time we focused on two hyperparameters, the first is the number of clusters required for K-means algorithm, which is set from 20 to 60; The second is the cluster which contains more than $n$ $T_{\mathrm{eff}}$ ranges, which is set from 3 to 8. The result is shown in Fig. 8. When $n=6$, the absolute slope of the number of clusters span more than n $T_{\mathrm{eff}}$ intervals and the number of K-mean clusters is the smallest, and there are about 10-12 clusters span more than 6 $T_{\mathrm{eff}}$ intervals.

To gain a clearer understanding of the meanings of $\tilde{p}_{1}$, $\tilde{p}_{2}$, we conducted tests using more synthetic spectra. In these tests, all synthetic spectra had $T_{\mathrm{eff}}$=5500K, $\log g$=4.5, and [Fe/H] ranging from [-1.5,-1,-0.75,-0.5,-0.25,0,0.25,0.5]. The abundances of the elements were changed to [0.1,0.5,1,5,10] times the solar abundance, with the changed elements falling into three categories:

Heavy

Ga, Ge, As, Se, Br, Kr, Rb, Sr, Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, Cd, In, Sn, Sb, Te, I, Xe, Cs, Ba, La, Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, Hf, Ta, W, Re, Os, Ir, Pt, Au, Hg, Tl, Pb, Bi, Po, At, Rn, Fr, Ra, Ac, Th, Pa, U, Np, Pu, Am, Cm, Bk, Cf, Es

S-only

Mo, Tc, Ru, Rh, Pd, Ag, Cd, In, Sn, Sb, Te, I, Xe, Cs, Ba, La, Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, Hf, Ta, W, Re, Os, Ir, Pt, Au, Hg, Tl, Pb

Alpha

Mg, Si, Ca, Ti

Therefore, we obtained $8\cdot 5\cdot 3=120$ synthetic spectra. Subsequently, we added random noise to each synthetic spectrum, with the noise following a Gaussian distribution and the variance being [0.1,0.01,0.001] of the normalized flux at each wavelength, corresponding to SNRs of [10,100,1000]. For each parameter combination, we generated 1000 spectra with noise. To ensure that the median spectrum’s [Fe/H] approximates 0, reflecting the actual abundance distribution of stars in our sample, we generated three times more samples with [Fe/H] = 0.25 and two times more samples with [Fe/H] = 0.5.

It is worth noting that these SNRs do not correspond to the SNR of the original spectra, as all spectra in this work were processed to a resolution of 10 nm. Considering that the noise at each point in the original spectra should not be independent, the SNR should increase significantly after reducing the resolution. This work limits the $\mathrm{SNR}_{u}>=10$, and after reducing the resolution, the SNR of our sample should be significantly higher than 10.

As shown in Fig. 9, under a resolution of 10 nm, no clusters can be distinguished when SNR = 10; obvious clusters can be identified when the SNR = 100, but they overlap significantly; and when SNR = 1000, not only can the clusters be clearly discerned, but also the three abundance variation patterns of “heavy”, “s-only”, and “alpha” can be distinguished within a certain range of parameters. However, this does not imply that $\tilde{p}_{1}$, $\tilde{p}_{2}$ can identify all three abundance patterns when the SNR is sufficiently high. There are two reasons for this: 1) The parameters of the synthetic spectra are discrete, while those of the actual spectra are continuous. 2) $\tilde{p}_{1}$, $\tilde{p}_{2}$ contain insufficient information since they are merely projections of a high-dimensional distribution onto this plane. Therefore, when using $\tilde{p}_{1}$ and $\tilde{p}_{2}$, only the lower limit of the number of clusters can be provided.

Fig. 10 compares the synthetic results of SNR = 100 with actual data ($5460\mathrm{K}<T_{\mathrm{eff}}<5670\mathrm{K}$). It can be seen that compared with the synthetic distribution, the actual distribution has a smaller $\tilde{p}_{1}$. This is because the continuous spectrum of the synthetic data is a perfect black-body spectrum, which is impossible for actual data. LAMOST lacks absolute flux calibration and has a lower resolution, resulting in a poorer quality of the continuous spectrum compared to the synthetic spectrum, making the normalized spectrum flatter. This leads to deviations in $\tilde{p}_{2}$. However, the impact of this effect on $\tilde{p}_{1}$ is limited. As can be seen from Eq. 1, $\tilde{p}_{2}$ is much more sensitive to the data than $\tilde{p}_{1}$.