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Anton A. Smirnov

Publications and source records attributed to Anton A. Smirnov.

At least 19 recordsLinked to original sources
arXiv2026

Analysis of Galactic cirrus filaments in HSC-SSP high-resolution deep images using artificial neural networks

The existence of Galactic optical cirrus poses a challenge for observing faint objects within our Galaxy and dim extragalactic structures. To investigate individual cirrus filaments in the Hyper Suprime-Cam Subaru Strategic Program public data release 3 (HSC-SSP DR3) we use a technique based on convolutional neural networks and ensemble learning. This approach allows us to distinguish cirrus filaments from foreground and background objects across the entire HSC-SSP, using optical images in the $g$, $r$, and $i$ wavebands. A comparison with previous work using deep Sloan Digital Sky Survey Stripe~82 (SDSS Stripe~82) data reveals that the cirrus clouds identified in this study are highly consistent in location within the overlapping survey region. However, in the deeper HSC-SSP dataset, we were able to detect $4.5$ times more cirrus clouds. Our study indicates that the sky background in HSC-SSP coadd images is over-subtracted, as evidenced by the surface brightness distribution in cirrus filaments and surrounding regions. Objects with surface brightness of $m = 29~\mbox{mag~arcsec}^{-2}$ near large filaments can be dimmed by over-subtraction of $0.5$ magnitude in the $r$ band. This suggests that cirrus clouds should be taken into account in algorithms for estimating the sky background. For practical use, we provide a catalog of filaments and a framework that allows one to train neural network models for segmenting cirri in HSC-SSP coadd images.

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arXiv2026

Isochrone Fitting of Galactic Globular Clusters -- VII. NGC1904 (M79), NGC4372, and revision of NGC288, NGC362, NGC5904 (M5), NGC6205 (M13), and NGC6218 (M12)

We estimate key parameters for the Galactic globular clusters NGC1904 and NGC4372 and update the parameters for NGC288, NGC362, NGC5904, NGC6205, and NGC6218, which were analysed in our previous papers. We fit various colour-magnitude diagrams (CMDs) of the clusters using isochrones from the DSED and BaSTI. The CMDs are constructed from data sets provided by the HST, Gaia, SMSS, a large compilation of Stetson, and other sources, using multiple filters for each cluster. Our cross-identification of almost all the data sets with those from Gaia or HST allows us to use their astrometry to precisely select cluster members in all the data sets. We obtain the following estimates for NGC288, NGC362, NGC1904, NGC4372, NGC5904, NGC6205 and NGC6218, respectively: metallicities [Fe/H]$=-1.28$, $-1.26$, $-1.64$, $-2.28$, $-1.33$, $-1.56$, and $-1.27$ dex; ages $12.94$, $10.33$, $13.16$, $12.81$, $11.53$, $12.75$, and $13.03$ Gyr; distances $8.83$, $9.00$, $12.66$, $5.17$, $7.24$, $7.39$, and $4.92$ kpc; reddenings $E(B-V)=0.022$, $0.029$, $0.031$, $0.545$, $0.045$, $0.024$, and $0.210$ mag; extinctions $Av=0.09$, $0.09$, $0.11$, $1.58$, $0.13$, $0.09$, and $0.67$ mag; and extinction-to-reddening ratio $Rv=3.9$, $3.0$, $3.8$, $2.9$, $2.9$, $3.6$, and $3.2$. We confirm that the differences in horizontal branch morphology among the 16 Galactic globular clusters analysed in our studies can be explained by variations in their metallicity, age, mass-loss efficiency, and the loss of low-mass members during cluster evolution. Accordingly, most clusters indicate a relatively high mass-loss efficiency, consistent with the Reimers mass-loss law with $η>0.3$.

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arXiv2026

The Spiral Pattern Speed in the Milky Way Galaxy

For a sample of masers, the basic kinematic equations were solved by including the Galactic rotation parameters and the peculiar velocity of the Sun as the unknown variables. Based on spectral analysis, the following estimates were obtained: $|f|_{R,θ}=(7.0,5.1)\pm(1.2,1.4)$ km s$^{-1}$ and the corresponding wavelengths $λ_{R,θ}=(1.9,1.7)\pm(0.4,0.7)$ kpc, as well as $χ_\odot=-140^\circ\pm15^\circ$. The presence of periodic perturbations in the vertical velocities of masers with an amplitude of $|f|_W=3.1\pm1.4$ km s$^{-1}$ and a wavelength of $λ=1.9\pm0.8$ kpc was confirmed. It is shown that the velocities $f_R$ and $f_θ$ can have both the same and different signs. Therefore, we obtained a large scatter of estimates. Thus, if $f_R$ and $f_θ$ have the same signs, then $Ω_p=25.8\pm2.0$ km s$^{-1}$ kpc$^{-1}$ and $R_{cor}=9.1\pm0.8$ kpc. And when $f_R$ and $f_θ$ have different signs, then $Ω_p=35.4\pm2.0$ km s$^{-1}$ kpc$^{-1}$ and $R_{cor}=6.8\pm0.8$ kpc.

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arXiv2025

Phase-space distortion as a key to unraveling galactic bar buckling

For the first time, we investigate the resonant structure of $N$-body galactic bar at the stage of buckling using action-angle variables. We studied the evolution of vertical actions ($J_z$) and angles associated with vertical resonance ($θ_\mathrm{res}=θ_z - θ_R$) for all orbits in the bar. For this purpose, we divide the orbits into types according to the behavior (libration or circulation) of their resonant angle with respect to fixed points $θ_\mathrm{res}=0$ and $π$ (vertical resonance). We show that during buckling, flat bar orbits circulating with increasing $θ_\mathrm{res}$ transformed into banana-shaped librating orbits (resonant capture) or circulating orbits with decreasing $θ_\mathrm{res}$ (resonant heating). The orbital transformation is accompanied by an increase in $J_z$ and the formation of a boxy/peanut-shaped (B/PS) bulge. During buckling, the phase space $J_z - θ_\mathrm{res}$ undergoes a distortion creating an asymmetry in the position of the fixed points $θ_\mathrm{res}=0$ and $π$ and in banana-shaped orbits near these points. The fixed point $θ_\mathrm{res}=0$ may disappear completely. This also breaks the symmetry between the orbits, which are captured into resonance or go into circulation with decreasing $θ_\mathrm{res}$ near $θ_\mathrm{res}=0$ and $π$. At the same time, near $θ_\mathrm{res}=0$, banana-shaped orbits with low vertical action $J_z$ appear. This reopens the path of orbital transformation through the fixed point $θ_\mathrm{res}=0$. The phase space transformation and orbit transformation occur in a coordinated manner and lead to smoothing of phase space perturbations and restoration of symmetry between orbits.

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arXiv2024

Boxy/peanut shaping of a mature galactic bar in action-angle space

We study vertical resonant trapping and resonant heating of orbits. These two processes both lead to the growth of a boxy/peanut-shaped bulge in a typical $N$-body model. For the first time, we study this by means of the action variables and resonant angles of the actual orbits that compose the model itself. We used the resonant angle instead of the frequency ratio, which allowed us to clearly distinguish between these two processes in numerical simulations. We show that trapping and heating occur simultaneously, at least at the stage of a mature bar, that is, some orbits quickly pass through vertical resonance while at the same time, a substantial number of orbits remains trapped into this stage for a long time. Half of all bar orbits spend more than 2.5 Gyr in vertical resonance over an interval of 4 Gyr. Half of the orbits trapped into the bar over the last 3 Gyr of simulation remain captured in vertical resonance for more than 2 Gyr. We conclude that in the later stages of the bar evolution, the process of vertical trapping dominates in the ongoing process that causes the boxy/peanut shape of a bar in a typical $N$-body model. This contradicts the results of several recent works.

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arXiv2024

Positive Lynden-Bell derivative as a ticket to the bar trap?

We have translated the results of $N$-body simulations of one barred model into the language of action variables and frequencies. Using this language, we analysed the behaviour of all orbits in the model on a large time scale at the stage of a mature bar. We show that the orbits join the bar while preserving their adiabatic invariant, which takes into account the 3D structure of the orbits. This allows us to apply the concept of the Lynden-Bell derivative for each of these orbits and trace how the sign of the derivative changes, i.e. how asynchronous changes in angular momentum $L_z$ and orbital precession rate $Ω_\mathrm{pr}$ (normal orbital mode) change to synchronous (abnormal mode). The transition to the abnormal mode occurs when $Ω_\mathrm{pr}$ reaches the angular velocity of the pattern $Ω_\mathrm{p}$, after which the orbit becomes stuck in the bar trap. All this happens against the background of secular changes in actions ($L_z$ decreases, $J_\mathrm{R}$ and $J_z$ increase). At the same time, corotation particles near two stable Lagrange points are also subject to secular changes in their actions. They increase $L_z$ and drift to the periphery, shifting corotation outwards. We also show that a change in the orbital mode from normal to abnormal and the trapping of orbits in a bar is possible only when the bar speed decreases with time, regardless of what is causing the bar to slow down. Our findings clarify and expand the picture of bar formation and evolution in numerical models.

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arXiv2024

Galaxies decomposition with spiral arms -- II: A multiwavelength case study of M 51

Spiral structure can contribute significantly to a galaxy's luminosity. However, only rarely are proper photometric models of spiral arms used in decompositions. As we show in the previous work, including the spirals as a separate component in a photometric model of a galaxy would both allow to obtain their structural parameters, and reduce the systematic errors in estimating the parameters of other components. Doing so in different wavebands, one can explore how their properties vary with the wavelength. In this paper, second in this series, we perform decomposition of M 51 in 17 bands, from the far UV to far IR, using imaging from the DustPedia project. We use the same 2D photometric model of spiral structure where each arm is modelled independently. The complex and asymmetric spiral structure in M 51 is reproduced relatively well with our model. We analyze the differences between models with and without spiral arms, and investigate how the fit parameters change with wavelength. In particular, we find that the spiral arms demonstrate the largest width in the optical, whereas their contribution to the galaxy luminosity is most significant in the UV. The disk central intensity drops by a factor of 1.25--3 and its exponential scale changes by 5--10\% when spiral arms are included, depending on wavelength. Taking into account the full light distribution across the arms, we do not observe the signs of a long-lived density wave in the spiral pattern of M 51 as a whole.

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arXiv2024

Isochrone fitting of Galactic globular clusters -- V. NGC6397 and NGC6809 (M55)

We fit various colour-magnitude diagrams (CMDs) of the Galactic globular clusters NGC\,6397 and NGC\,6809 (M55) by isochrones from the Dartmouth Stellar Evolution Database (DSED) and Bag of Stellar Tracks and Isochrones (BaSTI) for $α$-enhanced [$α$/Fe]$=+0.4$. For the CMDs, we use data sets from {\it HST}, {\it Gaia}, VISTA, and other sources utilizing 32 and 23 photometric filters for NGC\,6397 and NGC\,6809, respectively, from the ultraviolet to mid-infrared. We obtain the following characteristics for NGC\,6397 and NGC\,6809, respectively: metallicities [Fe/H]$=-1.84\pm0.02\pm0.1$ and $-1.78\pm0.02\pm0.1$ (statistic and systematic uncertainties); distances $2.45\pm0.02\pm0.06$ and $5.24\pm0.02\pm0.18$ kpc; ages $12.9\pm0.1\pm0.8$ and $13.0\pm0.1\pm0.8$ Gyr; reddenings $E(B-V)=0.178\pm0.006\pm0.01$ and $0.118\pm0.004\pm0.01$ mag; extinctions $A_\mathrm{V}=0.59\pm0.01\pm0.02$ and $0.37\pm0.01\pm0.04$ mag; extinction-to-reddening ratio $R_\mathrm{V}=3.32^{+0.32}_{-0.28}$ and $3.16^{+0.66}_{-0.56}$. Our estimates agree with most estimates from the literature. BaSTI gives systematically higher [Fe/H] and lower reddenings than DSED. Despite nearly the same metallicity, age, and helium enrichment, these clusters show a considerable horizontal branch (HB) morphology difference, which must therefore be described by another parameter. This parameter must predominantly explain why the least massive HB stars (0.58-0.63 solar masses) are only found within NGC 6809. Probably they have been lost by the core-collapse cluster NGC\,6397 during its dynamical evolution and mass segregation. In contrast, NGC\,6809 has a very low central concentration and, hence, did not undergo this process.

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arXiv2024

Globular clusters and bar: captured or not captured?

Studies of the dynamics of globular clusters assume different values of bar parameters (mass, velocity, size) and analyse the results of orbit classifications over the range of the chosen values. It is also a usual thing that a spherical bulge component is converted into the bar to obtain a non-axisymmetric potential from an axisymmetric one. The choice of bar parameters and the way the bar is converted from the bulge introduce systematics into the orbit classifications that we explore in the present study. We integrate orbits of 30 bulge globular clusters residing in the inner area of the Galaxy ($R \lesssim 5$ kpc) backwards in time for three different potentials, two of which are obtained by fitting the rotation curve, and one is taken from the surrogate $N$-body model representing our Galaxy. We analyse each orbit in terms of dominant frequencies obtained from its coordinate spectra. We find that the bar pattern speed is a key factor in orbital classification. With an increase of it, frequencies deviate more and more from the ``bar'' frequency ratio 2:1. The bar-to-bulge mass ratio (assuming the total mass of the bar plus the bulge is fixed) and size of the bar play a smaller role. We also find that, in the $N$-body potential, the fraction of orbits that follow the bar is higher than in those obtained from fitting the rotation curve.

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arXiv2023

Galaxies decomposition with spiral arms -- II: 29 galaxies from S$^4$G

Spiral structure can occupy a significant part of the galaxy, but properly accounting for it in photometric decomposition is rarely done. This may lead to significant errors in the parameters determined. To estimate how exactly neglecting the presence of spiral arms affects the estimation of galaxy decomposition parameters, we perform fitting of 29 galaxies considering spiral arms as a separate component. In this study, we utilize 3.6$μ$m-band images from the S$^4$G survey and use a new 2D photometric model where each spiral arm is modeled independently. In our model, the light distribution both along and across the arm can be varied significantly, as well as its overall shape. We analyze the differences between models with and without spiral arms, and show that neglecting spiral arms in decomposition causes errors in estimating the parameters of the disk, the bulge, and the bar. We retrieve different parameters of the spiral arms themselves, including their pitch angles, widths, and spiral-to-total luminosity ratio, and examine various relations between them and other galaxy parameters. In particular, we find that the spiral-to-total ratio is higher for galaxies with more luminous discs and with higher bulge-to-total ratios. We report that the pitch angle of spiral arms decreases with increasing bulge or bar fraction. We measure the width of the spiral arms to be 53\% of the disc scale length, on average. We examine the contribution of the spiral arms to the azimuthally-averaged brightness profile and find that spiral arms produce a ``bump'' on this profile with a typical height of 0.3--0.7 mag.

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arXiv2023

B/PS bulges and barlenses from a kinematic viewpoint. I

A significant part of barred disc galaxies exhibits boxy/peanut-shaped structures (B/PS bulges) at high inclinations. Another structure also associated with the bar is a barlens, often observed in galaxies in a position close to face-on. At this viewing angle, special kinematic tests are required to detect a 3D extension of the bars in the vertical direction (B/PS bulges). We use four pure $N$-body models of galaxies with B/PS bulges, which have different bar morphology from bars with barlenses to the so-called face-on peanut bars. We analyse the kinematics of our models to establish how the structural features of B/PS bulges manifest themselves in the kinematics for galaxies at intermediate inclinations and whether these features are related to the barlenses. We apply the dissection of the bar into different orbital groups to determine which of them are responsible for the features of the LOSVD (line-of-sight velocity distribution), i.e., for the deep minima of the $h_4$ parameter along the major axis of the bar. As a result, we claim that for our models at the face-on position, the kinematic signatures of a `peanut' indeed track the vertical density distribution features. We conclude that orbits responsible for such kinematic signatures differ from model to model. We pay special attention to the barlens model. We show that orbits assembled into barlens are not responsible for the kinematic signatures of B/PS bulges. The results presented in this work are applicable to the interpretation of IFU observations of real galaxies.

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arXiv2023

Barless flocculent galaxies: a dynamic puzzle

We draw attention to the bright galaxies that do not show a bar in their structure but have a flocculent spiral structure. Using the THINGS' and HERACLES' kinematic data for four barless galaxies (NGC~2841, NGC~3512, NGC~5055, NGC~7331) we built their mass models including dark halos. We concluded that the fraction of the dark matter does not exceed 50\% within the optical radii of the galaxies. This is too little to explain the lack of a bar in these galaxies. In an attempt to understand the featureless structure of these galaxies we constructed several $N$-body models with an initially reduced content of dark matter. We concluded that, in addition to the low mass of the dark halo, the decisive factor that leads to a barless disc is the start from an initially unstable state. An isolated dynamically cold disc (with the Toomre parameter $Q < 0.5$) settled into rotational equilibrium passes trough the short stage of violent instability with fragmentation and formation of stellar clumps. After that, it evolves passively and ends up with a featureless structure. We assume that the barless flocculent galaxies studied in the present work may be descendants of galaxies at high redshifts with rotation curves which are consistent with the high mass fraction of baryons relative to the total dark matter halo.

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arXiv2023

Prospects for future studies using deep imaging: Analysis of individual Galactic cirrus filaments

The presence of Galactic cirrus is an obstacle for studying both faint objects in our Galaxy and low surface brightness extragalactic structures. With the aim of studying individual cirrus filaments in SDSS Stripe 82 data, we develop techniques based on machine learning and neural networks that allow one to isolate filaments from foreground and background sources in the entirety of Stripe 82 with a precision similar to that of the human expert. Our photometric study of individual filaments indicates that only those brighter than 26 mag arcsec$^{-2}$ in the SDSS $r$ band are likely to be identified in SDSS Stripe~82 data by their distinctive colours in the optical bands. We also show a significant impact of data processing (e.g. flat-fielding, masking of bright stars, and sky subtraction) on colour estimation. Analysing the distribution of filaments' colours with the help of mock simulations, we conclude that most filaments have colours in the following ranges: $0.55 \leq g-r \leq 0.73$ and $0.01 \leq r-i \leq 0.33$. Our work provides a useful framework for an analysis of all types of low surface brightness features (cirri, tidal tails, stellar streams, etc.) in existing and future deep optical surveys. For practical purposes, we provide the catalogue of dust filaments.

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arXiv2022

Isochrone fitting of Galactic globular clusters -- IV. NGC6362 and NGC6723

We present new isochrone fits to the colour-magnitude diagrams of the Galactic globular clusters NGC\,6362 and NGC\,6723. We utilize 22 and 26 photometric filters for NGC\,6362 and NGC\,6723, respectively, from the ultraviolet to mid-infrared using data sets from {\it HST}, {\it Gaia}, unWISE, and other photometric sources. We use models and isochrones from the Dartmouth Stellar Evolution Database (DSED) and Bag of Stellar Tracks and Isochrones (BaSTI) for $α$-enhanced [$α$/Fe]$=+0.4$ and different helium abundances. The metallicities [Fe/H]$=-1.04\pm0.07$ and $-1.09\pm0.06$ are derived from the red giant branch slopes in our fitting for NGC\,6362 and NGC\,6723, respectively. They agree with spectroscopic estimates from the literature. We find a differential reddening up to $ΔE(B-V)=0.13$ mag in the NGC\,6723 field due to the adjacent Corona Australis cloud complex. We derive the following for NGC\,6362 and NGC\,6723, respectively: distances $7.75\pm0.03\pm0.15$ (statistic and systematic error) and $8.15\pm0.04\pm0.15$ kpc; ages $12.0\pm0.1\pm0.8$ and $12.4\pm0.1\pm0.8$ Gyr; extinctions $A_\mathrm{V}=0.19\pm0.04\pm0.06$ and $0.24\pm0.03\pm0.06$ mag; reddenings $E(B-V)=0.056\pm0.01\pm0.02$ and $0.068\pm0.01\pm0.02$ mag. DSED provides systematically lower [Fe/H] and higher reddenings than BaSTI. However, the models agree in their relative estimates: NGC\,6723 is $0.44\pm0.04$ kpc further, $0.5\pm0.1$ Gyr older, $ΔE(B-V)=0.007\pm0.002$ more reddened, and with $0.05\pm0.01$ dex lower [Fe/H] than NGC\,6362. The lower metallicity and greater age of NGC\,6723 with respect to NGC\,6362 explain their horizontal branch morphology difference. This confirms age as the second parameter for these clusters. We provide lists of the cluster members from the {\it Gaia} Data Release 3.

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arXiv2022

The distribution of dust in edge-on galaxies: I. The global structure

In this first paper in a series we present a study of the global dust emission distribution in nearby edge-on spiral galaxies. Our sample consists of 16 angularly large and 13 less spatially resolved galaxies selected from the DustPedia sample. To explore the dust emission distribution, we exploit the Herschel photometry in the range 100-500 $μ$m. We employ Sérsic and three-dimensional disc models to fit the observed two-dimensional profiles of the galaxies. Both approaches give similar results. Our analysis unequivocally states the case for the presence of extraplanar dust in between 6 to 10 large galaxies. The results reveal that both the disc scale length and height increase as a function of wavelength between 100 and 500 $μ$m. The dust disc scale height positively correlates with the dust disc scale length, similar to what is observed for the stellar discs. We also find correlations between the scale lengths and scale heights in the near- and far-infrared which suggest that the stellar discs and their dust counterparts are tightly connected. Furthermore, the intrinsic flattening of the dust disc is inversely proportional to the maximum rotation velocity and the dust mass of the galaxy: more massive spiral galaxies host, on average, relatively thinner dust discs. Also, there is a tendency for the dust-to-stellar scale height ratio to decrease with the dust mass and rotation velocity. We conclude that low-mass spiral galaxies host a diffuse, puffed-up dust disc with a thickness similar to that of the stellar disc.

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arXiv2022

B/PS bulges in DESI Legacy edge-on galaxies I: Sample building

We present the biggest up-to-date sample of edge-on galaxies with B/PS bulges and X-structures. The sample was prepared using images from the DESI Legacy catalogue and contains about 2000 galaxies. To find suitable candidates in catalogue, we made the assumption that the residues (original images minus model) of galaxies with B/PS bulges should exhibit a characteristic X-shape. Galaxies with such features were selected by eye and then used as input data for a neural network training, which was applied to a bigger sample of edge-on galaxies. Using the available data and the photometric models from the literature, we investigated the observational and statistical properties of the sample created. Comparing the $B/D$ ratios for galaxies with and without B/PS bulges, we found that the $B/D$ ratio for galaxies from our sample is statistically higher, with typical values in the range $\approx 0.2-0.5$ depending on the decomposition procedure. We studied how the opening angles $φ$ of the X-structure and the length of its rays are distributed in the formed sample and found them to be consistent with previous measurements and predictions from $N$-body models, e.g. $φ\gtrsim 25~°$, but measured here for a much larger number of galaxies. We found a sharp increase in the B/PS bulge fraction for stellar masses $\log M_{\star} \gtrsim 10.4$, but for edge-on galaxies, which complements the results of previous works. The sample can be used in future work to test various bar models and their relationship with B/PS bulges, as well as to study their stability and evolution.

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arXiv2021

Fractal dimension of optical cirrus in Stripe82

The geometric characteristics of dust clouds provide important information on the physical processes that structure such clouds. One of such characteristics is the $2D$ fractal dimension $D$ of a cloud projected onto the sky plane. In previous studies, which were mostly based on infrared (IR) data, the fractal dimension of individual clouds was found to be in a range from 1.1 to 1.7 with a preferred value of 1.2--1.4. In the present work, we use data from Stripe82 of the Sloan Digital Sky Survey to measure the fractal dimension of the cirrus clouds. This is done here for the first time for optical data with significantly better resolution as compared to IR data. To determine the fractal dimension, the perimeter-area method is employed. We also consider IR (IRAS and Herschel) counterparts of the corresponding optical fields to compare the results between the optical and IR. We find that the averaged fractal dimension across all clouds in the optical is $\langle D \rangle =1.69^{+0.05}_{-0.05}$ which is significantly larger then the fractal dimension of its IR counterparts $\langle D\rangle=1.38^{+0.07}_{-0.06}$. We examine several reasons for this discrepancy (choice of masking and minimal contour level, image and angular resolution, etc.) and find that for approximately half of our fields the different angular resolution (point spread function) of the optical and IR data can explain the difference between the corresponding fractal dimensions. For the other half of the fields, the fractal dimensions of the IR and visual data remain inconsistent, which can be associated with physical properties of the clouds, but further physical simulations are required to prove it.

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arXiv2021

Isochrone fitting of Galactic globular clusters -- III. NGC\,288, NGC\,362, and NGC\,6218 (M12)

We present new isochrone fits to colour-magnitude diagrams of the Galactic globular clusters NGC\,288, NGC\,362, and NGC\,6218 (M12). We utilize a lot of photometric bands from the ultraviolet to mid-infrared by use of data from the {\it HST}, {\it Gaia}, unWISE, Pan-STARRS, and other photometric sources. In our isochrone fitting we use theoretical models and isochrones from the Dartmouth Stellar Evolution Program and Bag of Stellar Tracks and Isochrones for $α$-enhanced abundance [$α$/Fe]$=+0.40$, different helium abundances, and a metallicity of about [Fe/H]$=-1.3$ adopted from the literature. We derive the most probable distances $8.96\pm0.05$, $8.98\pm0.06$, and $5.04\pm0.05$ kpc, ages $13.5\pm1.1$, $11.0\pm0.6$, and $13.8\pm1.1$ Gyr, extinctions $A_\mathrm{V}=0.08\pm0.03$, $0.11\pm0.04$, and $0.63\pm0.03$ mag, and reddenings $E(B-V)=0.014\pm0.010$, $0.028\pm0.011$, and $0.189\pm0.010$ mag for NGC\,288, NGC\,362, and NGC\,6218, respectively. The distance estimates from the different models are consistent, while those of age, extinction, and reddening are not. The uncertainties of age, extinction, and reddening are dominated by some intrinsic systematic differences between the models. However, the models agree in their relative age estimates: NGC\,362 is $2.6\pm0.5$ Gyr younger than NGC\,288 and $2.8\pm0.5$ Gyr younger than NGC\,6218, confirming age as the second parameter for these clusters. We provide reliable lists of the cluster members and precise cluster proper motions from the {\it Gaia} Early Data Release 3.

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