The Stellar"Snake"-- II: The Mass Function

We present a comprehensive investigation on the mass function (MF) of a snake-like stellar structure in the solar neighbourhood, building on our previous discovery. To ensure the reliability of the data, we reselect the member stars of the Stellar ``Snake'' in the latest {\it Gaia} Data Release 3 using the same approach as the initial series of articles. We also precisely measure the physical parameters of the clusters within the Stellar Snake. In light of the high completeness of the member stars in the cluster regions, we develop a simulated model color-magnitude diagram-based inference method to derive the mass function, binary fraction, and mass-ratio distribution of the clusters in the Stellar Snake. Notably, despite their similar ages and metallicity, we discover systematic variations in the MFs along the elongation direction of the Snake in the mass range of 0.5 to 2.0 M$_\odot$. The ``head'' of the Snake conforms to a canonical initial mass function with a power-law slope of $\alpha\sim-2.3$. Extending towards the ``tail,'' the MF becomes more top-light, indicating a deficiency of massive stars within these clusters. This result provides evidence for the delayed formation of massive stars in the clusters. Such clues give support to the hypothesis that the Stellar Snake constitutes as a hierarchically primordial structure.


INTRODUCTION
The stellar initial mass function (IMF) describes the distribution of the number of stars formed as a function of their mass.Originally proposed by Salpeter (1955) and further studied by seminal works like Miller & Scalo (1979) and Kroupa (2001), the IMF is typically described as a power-law distribution.At high masses, the IMF follows a power-law slope of around −2.3, while slopes of around −1.3 and −0.3 are seen at intermediate and low masses, respectively.
The IMF is fundamentally important because it determines the initial properties of stellar populations and galaxies as they evolve over time.In particular, the shape of the IMF at its low-mass end, encompassing brown dwarfs, very-low-mass stars and low-mass stars, has crucial implications for understanding the unseen, or "dark", mass budget of galaxies.Furthermore, a well-determined IMF is an essential ingredient to many astrophysical models, including galaxy dynamics and stellar population synthesis; such models rely on assumptions and observations related to the IMF (Romano et al. 2006;Lamers et al. 2013).However, observational limitations pose challenges to measuring the IMF.Even within a few kpc of the Sun, we ★ E-mail: hjtian@hdu.edu.cn,sarahbird@ctgu.edu.cncan only completely survey the most massive stars, and low-mass stars remain elusive beyond such volume.Moreover, direct mass measurements are only feasible for binary systems, yet binary orbits are unavailable for the vast majority of stars.As a result, masses must typically be inferred through luminosities, colors, and spectra compared to theoretical stellar models.
Earlier works assumed a universal IMF (canonical IMF, Kroupa 2001) based on studies finding its robustness across many environments (e.g., Bastian et al. 2010).However, highly improved observations now probing a diversity of astrophysical environments and physical scales are revealing the IMF may not be constant, as recent evidence of IMF variations demonstrates.Bastian et al. (2010) and Zhang et al. (2018) find top-heavy IMFs measured in high star formation rate densities like galactic centers and starburst galaxies.Van Dokkum & Conroy (2010) and Conroy & Van Dokkum (2012) find bottom-heavy IMFs detected in metal-rich environments.Geha et al. (2013) measure bottom-light IMFs observed in metal-poor Milky Way dwarf satellites.
Further observational constraints on the IMF under a more full range of astrophysical conditions are needed to elucidate what drives its diversity and properly account for IMF variations in galaxy models.Specifically, observational studies have shown that stellar pop-ulations forming at early cosmic times contained a deficit of lowmass stars compared to predictions of the canonical IMF, and the present-day IMF exhibits an enhanced proportion of low-mass stars at higher stellar metallicities, as demonstrated through large-scale spectroscopic surveys (Li et al. 2023).Understanding IMF variations remains an important outstanding problem with wide-ranging implications.
Achieving a robust characterization of the IMF requires deeper insight into stellar birth and evolution.Theoretical models predict the IMF arises from fragmentation and competitive accretion within clustered environments, implying dependencies on the initial composition and structure of natal clouds (Jappsen et al. 2005;Chabrier et al. 2014).Apart from direct measurements of the present-day IMF from the prestellar and star forming natal clouds, studying the present-day MF (PDMF) of young stellar systems is a close avenue for gaining insight on the present-day IMF.
Open clusters and associations are excellent sites to measure the PDMF.This function is defined as the relative proportion of coeval stars with different masses.This quantity reflects the initial physical conditions (density and metal content) affecting the fragmentation efficiency of the primordial cloud from which these stars form (Hennebelle & Chabrier 2008).Open clusters are composed of coeval and chemically homogeneous stars located at the same distance, thus eliminating the uncertainties linked to relative distance and differences in the age and chemical composition of individual stars (Ebrahimi et al. 2022).Consequently, this allow us to avoid potential influences from different stages of evolution and metallicity trends.
In recent years, numerous studies have focused on determining the PDMF of open clusters (e.g., Massey et al. 1995;Slesnick et al. 2002;Moraux et al. 2004;Pandey et al. 2007;Dib 2014;Kalari et al. 2018).Due to various factors such as dynamical evolution, differences in metallicity, variations in star-forming environments, and the relatively short time needed for massive stars to leave the main sequence, the MFs of these open clusters do not always perfectly match the distribution expected for the canonical IMF.
In the case of some older open clusters, dynamical evolution has led to mass segregation, where more massive stars are concentrated toward the center of the cluster, while less massive stars have gradually moved outward beyond the cluster's radius (Schneider & Elmegreen 1979;Portegies Zwart et al. 2010).This results in the PDMF of some older star clusters becoming flatter, such that the proportion of high-mass stars relative to the canonical IMF is higher (Kraus & Hillenbrand 2007;Goldman et al. 2013;Bisht et al. 2020) For relatively young open clusters and associations, the PDMF is more akin to the IMF because the majority of member stars are located in the main-sequence or pre-main sequence stages; they have yet to evolve long enough for the massive stars to leave the main sequence.Additionally, the impact of dynamical evolution is not as pronounced as in older clusters.However, the number of cluster members will decrease with the dynamic evolution of the cluster, even within a few tens of millions of years from its formation (Dinnbier et al. 2022).Consequently, for these relatively young clusters, while they are less affected by dynamical evolution and the departure of high-mass stars from the main sequence, the PDMF is slightly different from the IMF.In spite of this, the PDMF remains crucial for uncovering the patterns of star formation and temporal variations in dynamic processes within young clusters, and exposing the physical characteristics of star-forming regions.Such research not only contributes to a comprehensive understanding of the evolutionary processes of star clusters but also provides essential clues for further exploration of the characteristics and formation mechanisms of the IMF.
The Stellar Snake (first reported by Tian 2020) is a more recently discovered filamentary structure composed of at least nine open clusters (Wang et al. 2022).This formation mechanism of the Stellar Snake was elucidated by the identification of coeval stars that bridge the open clusters, as demonstrated by Beccari et al. (2020).The Stellar Snake consists of numerous member stars with approximately the same age (30−40 Myr) and metallicity (solar metallicity), and the structure is located at a relatively close distance (300 pc from the Sun).Under the conditions of relatively fixed metallicity and age (ranging only from 30 − 40 Myr), the Snake is likely not significantly dynamically evolved and most member stars are likely retained.The Snake is expected to exhibit a relatively uniform MF that closely resembles the IMF.Therefore, the Snake provides an ideal laboratory for analyzing stellar formation and evolution in cluster environments as well as tracking the IMF.
On top of the Snake constituting as an ideal laboratory, Gaia DR3 (Vallenari et al. 2023) presents unprecedentedly accurate and precise information for over 1.59 billion sources.These data include positions (, ), proper motions (  ,   ), and parallaxes () with average systematic uncertainties ranging from 0.05−0.1 mas (for  < 20 mag).Additionally, high-precision measurements are available for ,  BP , and  RP magnitudes, with typical errors ranging from 0.3−6.0mmag (for  < 20 mag).These advancements enable us to make more accurate determinations of the members of the star clusters within the Stellar Snake, consequently, obtain more precise parameters and more reliable MFs for each cluster.
In this paper, we investigate the variations in the PDMF of the open clusters in the Stellar Snake.To do this we utilize data from Gaia DR3 and construct a simulated CMD model to calculate the PDMF for nine member clusters of the Snake.In Section 2, we utilize data from Gaia DR3 to identify member stars of the open clusters within the Stellar Snake.In Section 3, we construct a robust simulated CMD model to calculate the cluster's MF (), binary fraction (   ), and binary mass ratio distribution ().In Section 4, we present the results characterizing the Stellar Snake open clusters using our simulated CMD model; we also present intriguing trends in the spatial distribution along the Snake of the cluster MFs and give a potential physical explanation of our results.In Section 5, we evaluate the selection of the open cluster member stars and the performance of our model, as well as compare the MF derived from the Stellar Snake field stars to those of the open clusters.Finally, we conclude in Section 6.

DATA
In this section, we carefully select the member stars of the Stellar Snake (Section 2.1) and its open clusters as well as derive essential cluster parameters (Section 2.2), such as stellar age, metallicity, cluster radius ( cl ) and spatial distribution.By leveraging off of the comprehensive Gaia DR3 dataset and implementing a systematic approach, we ensure high quality and quantity samples that contribute to a comprehensive understanding of the Snake's characteristics.Con-sidering the high demand for a complete sample from which to build the MF, the overall MF of the Snake (field stars and cluster members) may be affected by incompleteness.Therefore, we primarily focus on studying the open clusters within the Snake to enhance the reliability of our results.Based on the member stars of the Stellar Snake data and using the parameters specific to the star clusters, we identify a reliable sample of member stars for each cluster (Section 2.3).Finally, we validate the completeness of the sample to ensure a more accurate reflection of the true properties of the clusters (Section 2.4).
In order to select member stars of the Stellar Snake, we first need to impose certain restrictions on the search range and filtering criteria to obtain preliminary samples.Based on the previously discovered adjacent serpent-like (Tian 2020) and filamentary (Beccari et al. 2020) structures, two search regions, Part I and Part II, are defined to study the interconnection between these structures.We follow the empirical criteria mentioned by Wang et al. (2022) and re-screen the samples from Parts I and II using Gaia DR3.
After data selection, we adopt the FoF algorithm using the software ROCKSTAR (Behroozi et al. 2012) to search for members of the Snake from both Parts I and II.FoF is an algorithm used to search for members of a group; this algorithm has been well implemented in ROCKSTAR (Behroozi et al. 2012), which employs a technique of adaptive hierarchical refinement in 6D phase space to divide all stars into several FoF groups by tracking the high number density clusters and excluding those stars that could not be grouped in star aggregates.Taking into account the completeness of the Gaia data, the member stars might not be complete at fainter magnitudes, potentially impacting the results of the MFs.We conservatively impose a limit on the -band magnitude to be less than 18 mag for our subsequent research in order to ensure the completeness of the sample.Finally we retain 2217 and 6991 candidate Snake member stars of Parts I and II, respectively.In Figure 1, the cyan markers illustrate the distribution of the Snake members with  < 18 mag in coordinates (RA, DEC).For ease of description, we spatially categorized the Snake based on its distribution (Wang et al. 2022).The region with relatively older ages, specifically Trumpler 10 (high  ), is defined as the "head" of the Snake, while the region with relatively younger ages, Tian 2 (low  ), is defined as the "tail" of the Snake.

Parameter Determination of Open Clusters in the Stellar
"Snake" In order to study the mass function of each cluster in the Snake, it is essential to determine the fundamental cluster parameters accurately.This includes utilizing precise isochrones characterized by metallicity and age to establish reliable evolutionary tracks for the member stars of the clusters in the CMD.Additionally, considering the distance of the clusters from us, we need to factor in the distance modulus to convert apparent magnitudes to absolute magnitudes.On the other hand, extinction makes stars appear redder and dimmer, requiring corrections to restore their true luminosity.Therefore, accounting for the effects of the distance modulus and extinction is crucial for accurately describing the distribution of stars of different masses in the CMD.These considerations enable us to determine the masses of member stars and gain insights into the evolutionary processes of the clusters.We obtain PARSEC 1 (Bressan et al. 2012) isochrones ranging from 0.0005 to 0.0305 in metallicity (Z) and from 6 to 9 in logarithmic age (log Age), incrementing by 0.0005 and 0.005, respectively.We utilize the package Automated Stellar Cluster Analysis (ASteCA, Perren et al. 2015) to determine the cluster parameters.The process of determining the cluster center and radius using ASteCA involves several steps.Below we summarize the steps taken by ASteCA to select the initial cluster members: (i) Input Nearby Samples: ASteCA takes as input a sample of stars that are near the suspected star cluster.This sample typically includes both cluster members and field stars.Taking into consideration that ASteCA can only analyze one star cluster at a time, for each star cluster, we select a preliminary sample from the data obtained from Section 2.1.This preliminary sample includes stars about the approximate central region within a radius of 3 to 5 degrees and with  < 18 mag as input.
(ii) Fitting the Radial Density Profile (RDP): ASteCA calculates the RDP of the input sample.The RDP represents the spatial distribution of stars as a function of distance from the cluster center, providing information about the cluster's extent.
(iii) Cluster Radius ( cl ) Determination: Using the RDP, ASteCA determines the  cl by identifying the point where the density of stars transitions from the higher density of the cluster region to the lower density of the field region.This transition point corresponds to the estimated radius of the star cluster, which marks the estimated cluster boundary.
(iv) Initial Sample Mixing: ASteCA creates an initial sample by mixing the input sample of suspected cluster members with the nearby field stars.This mixed sample helps establish a more robust statistical analysis.
(v) Fitting PARSEC Isochrones: ASteCA employs the Markov Chain Monte Carlo (MCMC) method to fit the mixed sample with the corresponding PARSEC isochrones.The MCMC analysis infer the parameters of age (log Age), metallicity (), extinction (  , where   =   × 0.836,  BP_RP =   × 0.449), and distance modulus (DM = ( − ) 0 ) to find the best-fitting values that reproduce the observed properties of the stars in the mixed sample.
(vi) Determining Cluster Parameters: Through the MCMC posteriors, ASteCA determines the best-fitting cluster parameters, including age, metallicity, extinction, and distance modulus.These parameters provide insights into the characteristics and evolutionary stage of the star cluster.
By following these steps, ASteCA facilitates the determination of the cluster center,  cl , and other important parameters, enabling a comprehensive analysis of the star clusters based on their observed properties and theoretical isochrones.
We employ ASteCA to obtain the best-fit parameters for the open clusters in the Stellar Snake.Here we show the results for the cluster NGC 2232 as an working example.Figure 2 shows the respective radii and associated uncertainties of the star cluster's RDP, and Figure A1 shows the MCMC results for the optimal age, metallicity, distance modulus, and extinction of the star cluster using the bestfitting PARSEC isochrones.
Finally, we obtain the fundamental parameters for the nine open clusters in the Stellar Snake through ASteCA and summarize the results in Table 1.

Member Stars of Open Clusters
Based on the star cluster radius ( cl ) and cluster centers obtained from ASteCA, we select the corresponding RA and DEC coordinates from Table 1 as the centers of each cluster.We select all Stellar Snake member stars within  cl as the final sample of member stars of each open cluster.
Subsequently, we calculate the median parallax value and its associated error for all candidate stars within the selected radius range.Gaia data provides precise parallax measurements, allowing us to obtain reliable samples of member stars for all open clusters in the Stellar Snake by restricting the parallax values.Applying excessively strict selection criteria may lead to the exclusion of a significant number of member stars, resulting in potential biases in the results.Therefore, we apply the parallax restriction only to the data within the cluster's radius to strike a balance between sample reliability and inclusivity.We calculate the median parallax and its standard deviation for all candidate stars within  cl , and retain candidate stars whose parallax is within the range of the median value ±3 times the corresponding error (within a 3 range).The outliers are either spurious members, or true Snake members but not cluster members.
These selected stars are considered as the member stars of the open clusters.The final number of member stars  for each star cluster is shown in Table 1. Figure 1 shows the distribution of our final Snake cluster member star samples in coordinates (RA, DEC).

Completeness Verification
In calculating the MF, balance is essential between the quality of the data and the completeness of the sample.The MF is primarily influenced by two factors: on one hand, the completeness of Gaia itself, which unavoidably results in varying completeness under different conditions and in different regions due to the observational limitations of Gaia; on the other hand, our quality control measures for the initial sample selection.We apply criteria such as RUWE < 1.4 (RUWE is a normalized chi-square value provided in the data releases that is obtained by fitting Gaia sources to the point-spread function) and /  > 10.0 to choose a higher quality sample.These conditions, to some extent, have implications for data completeness.
To investigate the first aforementioned factor that influences the MF, we employ the empirically built analytical model2 proposed by Cantat-Gaudin et al. (2023) to test the completeness of all the Gaia data within the ASteCA-derived open cluster  cl centered on the cluster centers (Table 1).The model is calibrated using the deep imaging of the Dark Energy Camera Plane Survey DECaPS as a complete sample and predicts Gaia's completeness values with a precision of up to a few per cent across the sky.To develop their model, Cantat-Gaudin et al. ( 2023) explore several quantities and find that the best indicator of completeness is the  magnitude of the sources with the smallest number of observations.They make use of the number of observations used to compute the astrometric solution of a given source which is labeled in the Gaia DR3 catalogue as astrometric_matched_transits.Defining  10 as the median magnitude of Gaia sources within a patch of sky with astrometric_matched_transits ≤ 10, Cantat-Gaudin et al. (2023) build their model to depend on  10 as a proxy for completeness and find the model accounts for both the effects of crowding and the Gaia scanning law.We use their online package to assess the completeness of our data at various  magnitudes within the region of each open cluster.
We conduct the completeness check by selecting all Gaia sources within  cl of each Snake cluster and applying the Cantat-Gaudin  Considering that our study focuses only on stars with  < 18 mag, we can reasonably conclude that the initial Gaia data within the scope of our research is complete.Therefore, we conclude that the samples within these regions are not significantly affected by selection biases originating from Gaia itself.
After verifying the completeness of the Gaia data within the regions of our Stellar Snake clusters, we next move on to investigate the second aforementioned factor influencing the MF, this being due to the criteria we applied to select high quality Stellar Snake members.
We consider the selection criteria that we follow as presented in Wang et al. (2022) to assess their potential impact on data completeness.For the following two tests, we explore the completeness changes by comparing all Gaia data within each  cl with and without applying the two different selection criteria.Specifically, to assess the impact of RUWE and parallax_over_error on data completeness, we divide all Gaia sources within  cl for each open cluster into two groups.The first group includes all Gaia data within the radius without any restrictions, while the second group consists of sources with restrictions only on RUWE < 1.4 and those with limitations only on /  > 10.0.We compare the ratio of the number of sources in different  magnitude intervals between the two groups to evaluate Specifically, as shown in the lower panel of Figure 4, we investigate our selection criteria on the relative precision criterion parallax_over_error (/  > 10.0).This condition aims to ensure more accurate parallax measurements, thereby obtaining more reliable member stars.Sources with higher parallax_over_error generally correspond to more distant sources and fainter sources.Considering that the distance to the Snake is approximately 300 pc and  less than 18 mag, we expect that by applying the criterion of /  > 10.0, we are able to improve the overall data quality without compromising the completeness of the sample, thus aiding in the identification of reliable member stars.
We now test our expectation that the completeness remains uncompromised by this criterion.For all Gaia sources within  cl of each cluster, we search for changes in the completeness under different  magnitudes when restricting /  > 10.0.We observe, as shown in Figure 4, that as  decreases, the differences between the two data sets also decrease and eventually approach unity.This indicates that even under the worst-case scenario, the completeness remains above 0.8.Therefore, we conclude that /  > 10.0 does not significantly impact our results.
Our Stellar Snake members are also limited to sources with RUWE < 1.4.We now consider this additional limiting condition and its impact on data completeness.The RUWE is a metric defined in Lindegren et al. (2018) that quantifies the goodness-of-fit of the astrometric solution provided by Gaia.Sources with high RUWE values may be indicative of either of two scenarios: 1) the source is an unresolved binary system that Gaia did not resolve, or 2) the quality of the source's observation is poor and has higher overall errors.
In the upper panel of Figure 4, we find that in most cases, the ratio between the two groups (with and without the RUWE restriction) remains above 0.7, indicating good data completeness.However, for the BBJ 1 open cluster, we find an anomaly where the completeness ratio drops to around 0.3 at  magnitudes around 11.5 mag.This discrepancy might be attributed to a higher number of unresolved binary stars and poor observational quality in  magnitudes around 11.5 mag, resulting in significant data incompleteness.
To obtain more reliable results, we further divide the member star data into two groups of preliminary samples using the ROCKSTAR algorithm: one with RUWE < 1.4 (Sec.2.3), and the other group with the same initial conditions (Sec.2.1) and  cl and parallax selections (Sec.2.3) but without the RUWE restriction.In the following sections, we analyze the data by dividing it into two groups to discuss the impact of RUWE restrictions on data completeness and the potential errors introduced when RUWE is not considered, which can lead to unreliable data.
Finally, we obtain data for two groups of member stars of the open clusters.The distribution of the two groups of member stars of the open clusters, along with the best-fitting isochrones obtained from ASteCA, are displayed in the CMD as shown in Figure 5.The blue inverted triangles represent the stars that are additionally selected when we do not impose the RUWE restriction compared to the case with the RUWE restriction (gray markers).For the BBJ 1 open cluster, we observe a significant difference between the two groups around  = 11.5 mag.Specifically, the RUWE restriction did not identify any member stars in this region, while the sample without RUWE restriction included 10 additional member stars.This discrepancy highlights the issue of data completeness, as the completeness ratio between the two groups is only 0.3 in this region.Despite the completeness discrepancy of BBJ 1, we find that BBJ 1 and all the other eight Stellar Snake clusters that we analyze have best-fitting ages and metallicities that are consistent with our expectations, such that the ages range from 30 − 40 Myr and the metallicities are slightly higher than solar metallicity.

SIMULATED COLOR-MAGNITUDE MODEL
One of the commonly used methods to determine the MF is by obtaining the mass of coeval stars through their positions in the colormagnitude diagram (CMD, Sollima 2019; Hallakoun & Maoz 2021;Pang et al. 2021).Taking into account the relatively small number of member stars in open clusters, directly calculating the mass of each star and fitting the corresponding histograms using polynomials to compute the MFs faces certain difficulties.The process of binning data when fitting a distribution often introduces unnecessary ambiguity in selecting the bins, which can result in biases in the inferred distribution.This is especially true when the number of samples in each bin is unequal, as demonstrated by studies such as Apellániz & Úbeda (2005) and Maschberger & Kroupa (2009).
On the other hand, only as an approximation can we directly treat all cluster member stars as single stars and calculate their MFs.More precisely, we need to consider the impact of binary stars, as binary proportions are generally higher in open clusters.If we treat multiple lower-mass stars in binary systems as single stars with higher mass, this results in a flattening of the MFs.Therefore, for accurate MF measurements, we need to consider the influence of binaries.Fortunately, for binary systems, due to their overall luminosity being brighter and redder compared to single stars, we can identify these binaries in the CMD, usually distributed at the red side of the main sequence (e.g., Romani & Weinberg 1991; Bolte 1992; Rubenstein Taking into account the aforementioned factors, we build a simulated CMD model using the MCMC algorithm to determine the MF, binary fraction (   ), and binary mass ratio distribution () within each cluster.This approach helps avoid biases introduced by binning the data and also considers the impact of binaries on the MFs.
We now proceed to model the color-magnitude diagram (CMD) based on prior knowledge such as the initial mass function (IMF), binary fraction, and mass-ratio distribution.This CMD model, synthesized from theoretical recipes (Section 3.1), will then be meticulously compared with the observed CMD to determine the model parameters (Section 3.2).

Synthetic stars
In order to study the properties of our open clusters, we generate synthetic stars that resemble the observed stellar populations following the proceeding steps.
(i) We generate a population of synthetic single stars and binary systems within a specific mass range by utilizing the mass function formula, where  represents the stellar mass and / is the number of stars within the mass range of  to  + .The MF power-law index is represented as .
We generate 100, 000 × (1 −  b ) synthetic single stars with masses   and 100, 000 ×  b binary systems with primary stellar masses  p  within the specified mass range.
(ii) We obtain the best-fit PARSEC isochrones for the star clusters by using the optimal metallicity and age values provided in Table 1.These isochrones provide the corresponding range of colors and magnitudes in the CMD for stars within the specific mass range of the cluster.
(iii) Using the mass   of each synthetic star, we perform interpolation to assign appropriate intrinsic colors Col 0, and absolute magnitudes   0 , based on the best-fitting isochrone.Similarly, for the binary systems, we assign intrinsic colors Col  0, and absolute magnitudes    0 , to the primary stars using their masses    .(iv) For the single stars, we add the corresponding distance modulus DM and extinction   to each single star, as well as the extinction in the  BP_RP color index  BP_RP .Here,   is calculated as   × 0.836, and  BP_RP is calculated as 0.449 ×   as provided by Cardelli et al. (1989).These extinction coefficients are consistent with those used in ASteCA.By incorporating the DM,   , and  bp_rp values, we obtain the mock observed colors Col 1, and apparent magnitudes  1, for each single synthetic star, accounting for the effects of distance and extinction.
(v) For the errors in Col 1, and  1, , we divide them into multiple small bins based on the errors on the real data,  Col (color error) and   (magnitude error).We use the median value of the error within each bin as the representative error for the synthetic stars within that bin.Considering the systematic errors introduced during the processing of Gaia data, uneven distribution of extinction among cluster member stars, various physical factors inherent to stars, etc., the actual distribution of member stars may be more scattered than the simulated star distribution obtained through observational errors.To simulate more realistic observational results, we follow the method proposed by Li & Shao (2022) and add an additional scatter of 0.01 mag to the errors to mimic the observations.Then we add random errors with standard deviations  Col and   to each synthetic star within the corresponding bins, to obtain values Col *  and  *  , representing the final mock observed colors and magnitudes of the synthetic single stars.
(vi) For binary stars, we only consider binary systems with mass ratios ( = 1/2) greater than 0.3.This is because unresolved binary systems with  < 0.3 are difficult to distinguish from single stars.To assign masses to the secondary stars (   ) for each primary star, we use a given value of , which is the power-law slope in the equation, which represents the distribution of binary mass ratios.We generate the masses of the secondary stars based on this distribution.Subsequently, we perform interpolation using the mass of the secondary star    to determine the corresponding color Col  0, and absolute magnitudes    0 , of the secondary star for the binary systems.(vii) For both the primary and secondary stars in the binary systems, we synthesize their absolute magnitudes  tot  0 , and colors Col tot by using the formula,  tot  0 , = −2.5 log (10 After synthesizing the magnitudes and colors, we add the corresponding errors, DM, and   to obtain the synthetic magnitudes  * tot, and colors Col * tot, of the binary stars.The errors, DM, and   account for uncertainties in the observations and the effects of distance and extinction. With these steps, we obtain the colors (Col *  , Col * tot, ) and magnitudes ( *  ,  * tot, ) for the synthetic single stars and binary stars.These values are used to characterize the synthetic stellar samples generated based on the corresponding ,   , and  parameters.Figure 6 shows a comparison between the synthetic stars and the observed Stellar Snake open cluster member stars for NGC 2232.

Bayesian Statistical Inference
We treat   (Col  ,   |Θ) as the probability density distribution for each member star, where Θ represents the set of parameter describing the distribution of the star cluster, which includes ,   , and .The prior probability distributions for these parameters are listed in Table 2. To obtain the probability density distribution, we use the kernel density estimation (KDE) method to analyze the colors (Col *  , Col * tot, ) and magnitudes ( *  ,  * tot, ) of 100,000 synthetic stars generated within the specific mass intervals in the CMD.For the selection of bandwidth and kernel function, we perform a 10-fold crossvalidation using the GridSearchCV module from scikit-learn3 .We consider different bandwidth and kernel function combinations, calculating the scores for each combination to determine the optimal bandwidth and kernel function.
Subsequently, we apply the true data to the KDE model to obtain the probability density   for each star.The final likelihood is then computed as follows: According to Bayes' theorem, we obtain the posterior probability density distribution of Θ given the observed data (Col  ,   ): where (Θ | Col  ,   ) is the posterior probability density distribution of the parameter set Θ, L (Col  ,   | ) is the likelihood function representing the probability density distribution of the observed data given the parameter set Θ, and Π() is the prior probability density distribution of the parameter set Θ.
We utilize the MCMC algorithm within the emcee4 (Foreman-Mackey et al. 2013) package, employing 120 walkers and 3000 steps to sample the fitting parameters and obtain the posterior probability density function of the parameters.Finally, we use the median (50th percentile) as the best-fit result and the 16th and 84th percentiles as the 1 uncertainty interval.

RESULTS
In order to make use of the model built in the previous section, we utilize the fundamental parameters from Table 1 for each individual open cluster calculated by ASteCA.We download the appropriate PARSEC isochrones corresponding to the specified age and metallicity values for each cluster.By inputting the calculated extinction and distance modulus, we use our comprehensive model described in Section 3 to determine the masses corresponding to the brightest and faintest stars within each cluster.Subsequently, we calculate the respective mass functions and binary ratios for the Stellar Snake star clusters and present the intriguing trends in Section 4.1, and provide explanation for these results in Section 4.2.

Variations in the Mass Function
In order to facilitate meaningful comparisons between different open clusters within the Stellar Snake, we ensure that the fitting range for each star cluster falls within the same mass interval.Taking into account that at  = 18 mag, the minimum mass for these clusters is approximately 0.45 M ⊙ (Trumpler 10), and the lower limit for the maximum mass for all clusters is around 2.0 M ⊙ (UBC 7).We thus calculate the mass functions for each cluster in the same mass range of [0.5 − 2.0 M ⊙ ], separately for the two datasets, one with RUWE restricted and the other without the RUWE restriction.By selecting this mass range, we eliminate members of the clusters; but this approach using the same mass range eliminates the dim cluster members with higher uncertainties and allows us to better compare the mass function relationships of different open clusters within the Snake region.The final results are presented in Table 3.
From the results obtained within the mass range of 0.5 M ⊙ to 2.0 M ⊙ , we observe that the binary fraction ( > 0.3) for most of the clusters is concentrated between 30% to 50%, and  is predominantly negative, indicating that the majority of members in these open clusters have lower-mass companions in binary systems.Regarding their mass functions, the majority of clusters follow a mass function close to the Kroupa initial mass function with a value of  ∼ −2.3.However, there are also four clusters (Collinder 140, Tian 2, NGC 2232, BBJ 1) with a value of  < −2.3, indicating that these clusters have a higher proportion of low-mass stars and relatively fewer high-mass stars.
To further validate the reliability of our results, we employ XGBoost (Chen & Guestrin 2016) to compare the member stars'  BP −  RP and  with the corresponding PARSEC isochrones, taking into account the extinction and distance modulus, to directly compute the mass of the member stars.Subsequently, we calculate the histogram for the computed masses within the specified mass range and compare it to the best-fit slope obtained from our simulated CMD model (Figure 7).Specifically, for BBJ 1, NGC 2232, Collinder 140, and Tian 2, where the results deviate from the canonical Kroupa IMF, as shown in Figure 7, we clearly observe that the fitting results (red curve) better match the histogram distribution compared to the Kroupa IMF (blue curve).In contrast, the other clusters exhibit results similar to the Kroupa IMF.Thus, we validate the reliability of our simulated CMD model and the fitting results by comparing them with the histograms.
Comparing the results show in Table 3 for the two sample groups, we can see that, except for BBJ 1, the results of the other eight open clusters are very similar.Taking into account the corresponding errors, the results of the two groups are consistent.However, for BBJ 1, as previously indicated in the upper panel of Figure 4, the impact of RUWE on its completeness is significant, leading to substantial differences in its results.The difference in  obtained from the two sample groups is also beyond their error ranges, indicating two distinct results.Therefore, we can conclude that whether RUWE is restricted has a significant impact on the results for BBJ 1, while for the other clusters, it only leads to minor differences.
Intriguingly, we note a decreasing trend in the parameter  from the head-to-tail of the Snake.The coordinates RA, DEC, and  effectively trace the length of the Snake from head-to-tail, therefore, the parameter  exhibits a significant correlation with RA, DEC, and  as show in Figure 8. Specifically, for the circle markers (RUWE restrictions), as RA decreases,  becomes more negative.However, it is worth noting that BBJ 1 does not entirely follow this pattern in the direction of DEC and .Except for BBJ 1, there seems to be a trend in DEC and , where  becomes more negative as DEC increases and  decreases.As for the triangle markers (without RUWE restrictions), apart from BBJ 1, the triangle markers are similar to the circle markers.Due to the variability in  for BBJ 1, these triangle markers exhibit more pronounced trends in DEC and , where  becomes more negative as the coordinate values increase and decrease, respectively.However, in RA, this trend is more pronounced only when BBJ 1 is excluded.The distribution of the Snake's length along coordinate  exhibits a U-shaped pattern, indicating that  does not effectively capture the head-to-tail distribution of the Snake; therefore, no clear trend in  is evident along .

Proposed Explanations for the Results
In Section 4.1, we calculate the results for two sets of samples: one with the restriction RUWE < 1.4 and the other without this restriction.We find that the results for BBJ 1 show significant differences between these two samples, consistent with what we find in Section 2.4, where BBJ 1 exhibits a pronounced completeness issue at the brighter end.The results for the other clusters remain largely consistent between these two sets of samples.The noticeable completeness issue for BBJ 1 when limiting RUWE < 1.4, particularly around  = 11.5 mag, may be attributed to the overall Gaia data quality in the observed region or potentially the presence of a higher number of unresolved binary stars in the vicinity of  = 11.5 mag.
We also find that whether RUWE is restricted or not, the MFs of the member clusters in the Stellar Snake appear to exhibit variations along the length of the Snake.Additionally, the Kroupa IMF ( = −2.35for mass > 0.5 M ⊙ ) lies outside the error range for four of the Snake open clusters that have smaller  values, suggesting they might contain more low-mass stars within the mass range [0.5 − 2.0 M ⊙ ].
A plausible explanation for this is that the clusters near the "tail" are relatively younger compared to those near the "head" (Wang et al. 2022).Additionally, the delayed formation of massive stars (Krause et al. 2020) leads to a lack of high-mass stars in the MF within the mass range [0.5−2.0M ⊙ ].As star-forming regions evolve, they form more massive stars, thus the most massive stars tend to be younger, because they only form when the star formation rate of the whole region has increased sufficiently to sample the massive regions of the IMF, according to the simulations of Vázquez-Semadeni et al. (2017, hereafter, VS17).The formation of massive stars necessitates environments with higher gas density, and reaching such conditions takes time.Hence, this leads to a delayed formation of massive stars.This is exemplified in figure 7 of VS17, which shows the distribution of the cumulative masses of early-stage cluster stars over time as obtained through simulations.They show that the formation of massive stars commences only when the merger of group 1 and group 2 generates a new cluster with a higher star formation rate, which they denote as group 12. Therefore, massive stars form only when the star formation   rate is sufficiently high, predominantly in high-density regions, and thus massive stars are concentrated near the center of the cluster.
In Figure 9, we compare the masses of the Stellar Snake member stars in the nine open clusters and their distances from the cluster centers.The results closely resemble the model of VS17, with massive stars predominantly clustered near the center of the clusters, while low-mass stars exhibit a more dispersed distribution.Some young clusters also appear "mass segregated," where the more-massive stars are more concentrated in the centers of clusters (Bonnell & Davies 1998;De Grĳs et al. 2002;Gouliermis et al. 2004;Chen et al. 2007).Considering that the Snake clusters have relatively young ages and are less likely to be influenced by significant dynamical evolution, we expect that the concentration of massive stars toward the center of the clusters is likely to be primordial.This delayed formation of massive stars also implies that the age range of the massive stars is slightly smaller (and they are younger) than that of the low-mass stars, which begin to form since the onset of the star formation activity in the region.Equivalently, the mass range of the younger stars is larger and extends to higher masses than that of the older stars (Krause et al. 2020).Therefore, the delayed formation of massive stars suggests that, compared to the expected number produced by the overall stellar population, there is a scarcity of massive stars in the early phases of evolution in the ensembles of cores.This finding aligns with observations reported by Povich et al. (2016).
As mentioned by Wang et al. (2022), the Snake is a hierarchically primordial structure and probably formed from a filamentary giant molecular cloud.Gravitationally bound stellar clusters in this structure arise naturally at localized high-density regions, while unbound associations form in situ at low density regions.Therefore,   The Stellar Snake region exhibits a layered formation along the extent of its filamentary structure, with a significant spread in the properties of its open clusters along RA, DEC, and .Interestingly, the "head" of the Stellar Snake, corresponding to larger values of RA and  (smaller DEC), appears to house relatively older open clusters.Conversely, the "tail" of the Stellar Snake, characterized by smaller values of RA and  (larger DEC), coincides with the four open clusters that do not conform to the Kroupa IMF.This observation seems plausible after taking into account the delay in the formation of high-mass stars, these relatively young clusters lack high-mass stars due to stellar feedback.This provides a reasonable explanation for the anomalous  observed in these open clusters.
The best-fit ages obtained from ASteCA provide a rough estimate of the cluster's age; the error range is approximately 5 Myr.Such errors give an explanation to the lack of strong correlation between  and the best-fit age.The correlation between  and the spatial distribution of the clusters shows a much stronger trend.

Determination of Open Cluster Member Stars
The selection of cluster members has always been a challenging aspect in stellar cluster studies (e.g., Lada & Lada 2003;Sung et al. 2004;Bouy et al. 2013;Kuhn et al. 2014).Different clustering methods can yield different results (e.g., Hunt & Reffert 2021), and the optimal method may vary depending on the research objectives.Determining the MF is particularly challenging, as obtaining the most precise MF results requires not only ensuring a large sample of cluster members but also avoiding the influence of field stars that might be introduced by relaxing selection criteria.
The comparison of different star clusters requires that the initial sample selection criteria and restrictions applied are as similar as possible for each cluster being compared.This ensures that the results obtained are most reliable.
Taking these factors into consideration, we propose that the Stellar Snake, with its approximately uniform metallicity and relatively young age, provides an excellent environment for studying the MF.We use a combination of the ROCKSTAR FoF method and ASteCA to obtain the corresponding cluster members.Using FoF to filter the overall structure of the Stellar Snake helps ensure that cluster members are selected under uniform criteria, making the results more reliable.However, FoF alone cannot provide the cluster's fundamental parameters.Therefore, we use ASteCA to obtain the cluster's fundamental parameters and, by applying the corresponding parallax restrictions, remove spuriously selected stars that do not actually belong to the Stellar Snake clusters.These steps allow us to distinguish the member stars from field stars for each cluster within the Stellar Snake.
We only select member stars in the mass range of 0.5 to 2.0 M ⊙ for calculating the MF to ensure that we compare results within the same selection criteria, making the comparison between clusters more reliable.We also observe that member stars at  = 18 mag exhibit a broader spread in the CMD.Considering that their errors could introduce contamination from field stars that are challenging to distinguish, our restriction on the mass range of 0.5 to 2.0 M ⊙ helps select stars with smaller errors.However, this also results in fewer member stars per cluster which increases the uncertainties in the obtained results.

Stellar Snake Field Stars
We observe changing trends among the Snake clusters To understand whether these trends are specific to the clusters or are instead characteristic of the entire filamentary Stellar Snake structure, we select several non-cluster regions (field regions) within the Snake for further investigation.
We initially compute the kernel density estimate (KDE) distribution map for the spatial structure of the Snake member stars, as shown in Figure 11.Next, we select three regions with a radius of 1.5 degrees from the "head," "middle," and "tail" parts of the Snake as non-cluster samples (field samples Field 1, Field 2, and Field 3) with the aim to calculate their MF.Considering the significant differences in distances and extinction among the member stars in non-cluster regions, we directly convert the apparent magnitudes to absolute magnitudes using For extinction, we utilize the Galactic extinction catalog (Leike et al. 2020) to assign extinction values to each member star.We use the extinction corrected   and  BP_RP magnitudes for all stars within a specified radius in the Snake field, with DM and   set to 0 as input parameters.We perform the fitting by using ASteCA to obtain the best isochrone.Finally, we employ our simulated CMD model to fit these three regions, and present the results in Table 4. Interestingly, we observe that the MF in these three regions also seem to follow a decreasing trend from head-to-tail, similar to the trend observed in the Stellar Snake open clusters.Therefore, we can infer that this trend appears to be present throughout the entire Snake structure, rather than being specific to cluster regions.

CONCLUSIONS
We characterize the Stellar Snake's open clusters, measuring their age, metallicity, MF, binary star fraction, and binary star mass ratio distribution, by utilizing the high quality and quantity of data from Gaia DR3.We employ the FoF algorithm ROCKSTAR to identify a preliminary sample of candidate member stars for each Snake cluster.Subsequently, we employ ASteCA to measure parameters such as the cluster radius  cl , age, and center.We then select the final sample of member stars of each Stellar Snake open cluster as Snake members within the ASteCA-determined  cl .Through ASteCA's parameter determination, we find that the age distribution of the Snake clusters is consistent with previous expectations, falling in the range of 30 − 40 Myr.Additionally, these clusters exhibit metallicity similar to that of the Sun.
Given the high demand for completeness in determining the MF of open clusters, we conduct an assessment of data completeness within  cl and the impact of our selection criteria on completeness.We observe that Gaia's data is generally complete for  < 18 mag.However, we note that limiting RUWE has a significant impact on the completeness of data within the Stellar Snake open cluster BBJ 1.This situation may be attributed to issues intrinsic to the Gaia observations or to the presence of numerous unresolved binary stars within BBJ 1's  cl .Therefore, we divide the data into two groups: one with a restriction of RUWE < 1.4 and another without RUWE restrictions.This division is made to compare the influence of RUWE restrictions on the results.
Subsequently, to calculate the MF (), binary star ratio (   ), and binary mass ratio distribution () for the Snake clusters, we construct a comprehensive simulated CMD model to obtain the corresponding results.After validating the reliability of the model, we compute the best-fitting results for ,   , and  for each cluster within the [0.5 − 2.0 M ⊙ ] mass range.We find that the Snake cluster binary star ratios are primarily concentrated between 30 − 50%.Concerning the MF, the comparison between the two data samples shows that, except for BBJ 1, restricting RUWE did not lead to significant differences in the MF.
Additionally, we observe similar trends in both samples such that tracing the Snake from head-to-tail reveals a trend with the power law index  of the MF becoming more negative.We find that the MF of the star clusters near the Snake's head are well described by a Kroupa IMF within their respective mass ranges, whereas the four relatively young star clusters near the tail (BBJ 1, NGC 2232, Tian 2, Collinder 140) are not as well represented by a Kroupa IMF.These clusters with more negative  imply that within the mass range of [0.5−2.0M ⊙ ], clusters with more negative  have a relatively higher proportion of low-mass stars compared to the typical Kroupa IMF ( = −2.3).Similarly, we also calculate the MF for the field stars in the "head," "middle," and "tail" regions of the Snake.We find the same trend is also present in the field star sample.One plausible explanation for this trend is that the Stellar Snake is a hierarchically primordial structure.The head of the structure (e.g., Trumpler 10, the physical pair of Collinder 135 and UBC 7, NGC 2547, and NGC 2451B) formed initially, and around a few Myrs later, the formation of the Snake's tail (e.g., Tian 2) commenced.Due to the delayed formation of massive stars, when massive stars formed in clusters at the head of Snake, their feedback in the form of stellar winds blew away the surrounding gas, gradually halting star formation.The four relatively young clusters in the tail of Snake (BBJ 1, NGC 2232, Tian 2, Collinder 140), influenced by the stellar winds, did not form as massive stars and were consequently forced to cease star formation before doing so, thus resulting in the lack of massive stars.Consequently, for stars that deviate significantly from the isochrones, i.e., those in regions not covered by the synthetic stars, the returned probability values may be lower.
Our simulated CMD model is similar to the approach used in Ebrahimi et al. (2022), where 10 5 stars are generated based on the best-fit isochrones to model the true distribution of stars in a star cluster.However, Ebrahimi et al. (2022) uses an iterative method to obtain the final values of ,   , and , while our simulated CMD model uses MCMC, which may be considered more reliable.Additionally, our simulated CMD model provides an estimate for  as well, although this may come at the cost of increased computation time.
The cluster modelling of Li & Shao (2022), MIxture Model for Open clusters (MiMO), is able to skip the member star selection process and directly estimate parameters like member star fraction, best-fit isochrones, ,   , , etc., when dealing with mixed populations of member and field stars.This makes MiMO a more rigorous and comprehensive approach.In contrast, our simulated CMD model can seamlessly integrate with different clustering methods, allowing the member stars obtained through clustering to be directly applied to our simulated CMD model.
Ebrahimi et al. (2022) also uncover a similar phenomenon.They find that open clusters with ages less than half of the half-mass relaxation time exhibit an MF resembling the canonical IMF.On the other hand, for clusters with ages equal to or greater than half of the half-mass relaxation time, the MF tends to become flatter.However, in a study of 773 open clusters, Almeida et al. (2023) find no clear trend in age-related mass segregation.In general, the majority of open clusters exhibit MFs that are similar to the canonical IMF (Bonatto et al. 2005; Niedzielski et al. 2007; Angelo et al. 2019; Cordoni et al. 2023).

Figure 1 .
Figure1.Spatial distribution of candidate member stars in the Stellar Snake in terms of (RA, DEC).The cyan background points represent the member field stars of the filamentary Stellar Snake structure (Section 2.1), our curated samples of member stars for each selected open clusters, detailed in Section 2.3, is highlighted with distinct colors corresponding to each cluster as indicated in the legend.The "head" to the "tail" of the structure is annotated for orientation.

Figure 2 .
Figure 2. Radial density profile (RDP) of the open cluster NGC 2232 derived using ASteCA.Blue points represent the stars/arcmin 2 RDP with the cluster center as the origin; the corresponding error bars are the standard deviations.The horizontal black dashed line is the field density value and the black dotted lines represent its upper and lower limits.The red vertical line is the derived cluster radius  cl with the uncertainty region marked as the gray shaded area.A 2P King profile fit is indicated with the green dashed curve and its associated uncertainties are depicted by the green shaded region with the core radius  c shown as a green vertical dotted line.The inset plot in the upper right corner displays the results in logarithmic coordinates.

Figure 3 .
Figure 3. Completeness analysis of all Gaia data sources within  cl for the nine Stellar Snake open clusters.The -axis represents the -band magnitude, the -axis depicts the completeness of the data at different  magnitudes, and the color scale indicated by the colorbar represents the corresponding  10 (median magnitude of sources with astrometric_matched_transits ≤ 10) values.The blue vertical dashed line indicates  = 18 mag, which is the maximum magnitude we investigate in our study.Based on the model results, we can consider Gaia data with  < 18 mag within the cluster radius to be complete.

Figure 4 .
Figure 4. Variation of data completeness along  for all Gaia sources within the  cl of the different Snake open clusters.The upper panel compares the effect of the RUWE restriction (RUWE < 1.4) on data completeness and the lower panel compares the effect of the parallax_over_error restriction (/   > 10.0) on data completeness.The -axis represents different  intervals.The -axis in the upper panel represents the ratio of the number of all Gaia data sources with RUWE < 1.4 to the number of all Gaia data sources without the RUWE restriction.The -axis in the lower panel represents the ratio of the number of all Gaia data sources with /   > 10.0 to the number of all Gaia data sources without parallax_over_error restrictions.Each open cluster is represented by a different colored data point.Limiting RUWE < 1.4, except for BBJ 1, generally maintains completeness around 0.8 for  > 9 mag data, while completeness is consistently at or above 0.9 for limiting /   > 10.0.

Figure 5 .
Figure 5. CMDs for the 9 open clusters of the Stellar Snake's Parts I and II.The gray points represent the member stars selected when restricting RUWE < 1.4, the blue inverted triangles represent the additional member stars that are selected when we do not impose the RUWE < 1.4 restriction (Section 2.3).The gray curves represent the best-fitting isochrones.The blue horizontal dashed lines demark the apparent magnitude range for single stars within the mass range [0.5 − 2.0 M ⊙ ].The parameters related to the isochrone fitting are shown in the upper-right corner of each panel.DM, Age,   and  represents the best-fitting distance modulus, age, extinction and metallicity.These clusters have ages ranging from 30 − 40 Myr and metallicities slightly higher than solar metallicity.

Figure 6 .
Figure 6.Comparison between the synthetic and observed member stars for the Stellar Snake open cluster NGC 2232.The gray curve represents the best-fitting isochrone.The blue dots represent the 100,000 synthetic stars generated with specific ,   , and  within a given mass range [ 1 ,  2 ] (where  2 >  1 ).The blue horizontal dashed lines demark the apparent magnitude range for single stars within the mass range [ 1 ,  2 ].The red dots represent the Gaia DR3 observed member stars of NGC 2232.
The values of ,   , and  represent the mass function, binary fraction ( > 0.3), and binary mass ratio distribution, respectively, obtained for data restricted to RUWE < 1.4 within the mass range of 0.5 to 2.0 M ⊙ .On the other hand,  * ,   * , and  * correspond to the same except for the data when RUWE is unrestricted.

Figure 7 .
Figure 7. Verification of results for Snake clusters with RUWE < 1.4 in the mass range of 0.5 M ⊙ to 2.0 M ⊙ .The -axis represents the number of stars ( ), and the -axis represents the mass.The histogram presents the distribution of member stars within various mass intervals calculated directly from the CMD and the best-fit PARSEC isochrone found by ASteCA.The red line corresponds to the best-fit result from our model while the blue line represents the Kroupa IMF value for masses greater than 0.5 M ⊙ ( = −2.3).

Figure 8 .
Figure 8. Variation of the MF slopes  for member stars of each cluster in the range of 0.5 to 2.0 solar masses, along coordinates RA, DEC and .Each open cluster is represented with a different color (as indicated in the legend).Circles represent the results obtained from our data sample with RUWE < 1.4, and inverted triangles represent the results from the control group where we do not apply the RUWE restrictions.The horizontal dashed line represents the Kroupa initial mass function slope ( = −2.3)for masses greater than 0.5 solar masses.Except for BBJ 1, the results agree between our data sample and the control group.Intriguingly, we find a trend in  along RA, DEC, and  from the head to the tail of the Stellar Snake (the head/tail direction is indicated in the upper right of each panel).

Figure 9 .
Figure 9. Mass distribution of stars in the nine open clusters of the Stellar Snake.The distance  of member stars from the cluster center is displayed on the -axis and the mass of the member stars on the -axis.The horizontal blue dashed line represents the boundary at a mass of 1 M ⊙ .The red and black vertical lines indicate the median distances from the cluster center  for all member stars with mass > 1 M ⊙ and mass < 1 M ⊙ .
Figure 10.Proportion of massive and low-mass stars within each star cluster as a function of the spatial position (RA, DEC) of each cluster along the Stellar Snake.The blue dashed line and circles represent the trend of the proportion of cluster member stars with masses less than 1 M ⊙ relative to all member stars, while the red dashed line and circles depict the trend of the proportion of cluster member stars with masses greater than 1 M ⊙ relative to all member stars.The dotted line and inverted triangles indicate the variation trend for the control group without applying RUWE restrictions.
Field centers. Parameters obtained using ASteCA (Section 2.2).  Fitting results from our simulated CMD model.

Figure 11 .
Figure 11.KDE distribution map of the Stellar Snake in the spatial coordinates  and .Different colors, as indicated by the colorbar, represent varying stellar number densities.The regions enclosed by the red circles represent our selected field star samples belonging to the Snake.

Figure B1 .
Figure B1.Kernel density estimation plot of the central region of the open clusters.The green dashed lines intersect at the ASteCA-determined center of the cluster, while the red circle denotes the ASteCA-determined cluster radius  cl .The blue curves represent density contour lines.The different colors on the colorbars represent the varying star number density within different regions.BBJ3 and BBJ3 lack red circles because ASteCA is unable to determine their centers.

Figure C1 .
Figure C1.Simulated CMD model evaluation using the emcee 1D and 2D posterior samples.The red vertical lines represent the true values of the parameters, while the black dashed vertical lines indicate the 18th, 50th, and 84th percentiles (also shown above each 1D posterior panel).The parameters being fit include  (MF power-law index),   (binary fraction for  > 0.3) and  (power-law index of the distribution of binary mass ratios).

Figure C2 .
Figure C2.Residuals (fitting values − true values) for ,   , and .The horizontal axis represents the number of stars used in the fitting process, while the vertical axis represents the residuals.The error bars indicate the 1 range of errors on the fitting values.The different colors on the colorbar represent the various real values used for the synthetic stars.

Table 1 .
Derived Stellar Snake open cluster parameters.

Table 2 .
Bayesian priors used to generate synthetic star clusters.

Table 3 .
Fitting results from our simulated CMD model for each Stellar Snake open cluster.

Table 4 .
Properties for each Stellar Snake non-cluster region.