The Relation of the Alpha Persei Star Cluster with the Nearby Stellar Stream

The first mention of the Alpha Persei star cluster (Melotte 20, Collinder 39) is in Eddington (1910), who first noticed a group of common motion bright stars in the vicinity of the star. Artyukhina (1972) identified candidate kinematic members brighter than B = 12 mag within 45 from the cluster center. The analysis of the vector point diagram allowed her to isolate 163 stars within $\sigma \sqrt{2}=10$ mas yr⁻¹ from the mean proper motion, 287 stars within 2σ = 14 mas yr⁻¹, and 503 stars within 3σ = 20 mas yr⁻¹. Artyukhina (1972) also investigated the cluster projected density profile and its luminosity function. Of the 503 3σ candidates, she isolated 83 bona fide members. The cluster corona was computed as 7.3 pc (25) and the core was computed as 3.7 pc (125).

Shatsova (1981) studied the Alpha Persei supercorona and extracted from the Smithsonian Astrophysical Observatory Star Catalog (R.A. = 2–4 hr, decl. = +40° to +60°) stars with spectroscopic information (B stars down to V = 8, and A0–A3 stars down to V = 9 mag). The circle 25 wide around the cluster center was therefore excluded. This supercorona was found to be composed of 93 B-A3 comoving stars.

Makarov (2006) investigated Alpha Persei with Tycho-2 and the Second USNO CCD Astrographic Catalog (UCAC2). He found 139 candidate members and estimated an age of 52 Myr. Makarov (2006) also noticed that the cluster is surrounded by a sparse halo of comoving dwarfs identified from 2MASS photometry.

The first suggestion that the Alpha Persei cluster might be associated with this extended group of common motion stars is in Mermilliod et al. (2008), who first introduced the term "stream."

van Leeuwen (2009) selected 50 stars from the Hipparcos catalog 8° around the cluster center and derived μ_α = 22.73 ± 0.17 mas yr⁻¹, μ_δ = −26.51 ± 0.17 mas yr⁻¹, π = 5.80 ± 0.09 mas, (m–M) = 6.18 ± 0.03 mag (hence 172.4 ± 2.7 pc of distance), and log(age) = 7.55. He concluded that the overall stars' properties characterize Alpha Persei more as a remnant of an OB association than a bound cluster.

By following Perryman et al. (1998) and Luri et al. (2018), Lodieu et al. (2019) investigated Alpha Persei with Gaia DR2 data. He inferred a tidal radius of 9.5 pc and found 554 stars within it. Extending the searching area to three tidal radii (28.5 pc) allowed him to find 2041 sources. Lodieu et al. (2019) derived a distance of 177.68 ± 0.84 pc, a tangential velocity of 28.7 ± 0.5 km s⁻¹, and a core radius of 2.3 ± 0.3 pc containing 21 stars.

The Gaia DR2 data release (Gaia Collaboration et al. 2016, 2018b) allows two different approaches to study star clusters.

The first approach consists of selecting the stars with the most reliable parameters (parallaxes—Plx, or π—, proper motion components—μ_α and μ_δ—, and radial velocities). Cluster members would then crowd in a five-dimensional or six-dimensional parameter space. This approach permits a detailed investigation of star clusters' three-dimensional structure and internal kinematics (Jeffries et al. 2014; Damiani 2018; Cantat-Gaudin et al. 2019; Beccari et al. 2020). The limitation of this approach is that only cluster stars for which parameters are precisely measured can be used.

The second approach consists of a statistical investigation and makes use of the virtue of Gaia as an all-sky survey complete down to a specified limiting magnitude. Within this statistical approach, the ultimate goal of the stars' selection is increasing the contrast of the cluster against the field. We set the limits of the parameters (parallaxes and the proper motions) in such a way that probable cluster members are not missing, despite the possibly large errors of the parameters. At the same time, we have the advantage that the number of field stars—and hence field stars density fluctuations—is kept low.

Statistical methods are then used to evaluate the distribution functions characterizing the cluster under study, such as surface or spatial densities, luminosity, and mass function (see, for example, Seleznev 1998, 2016a, 2016b; Seleznev et al. 2000, 2017). The ultimate goal is to obtain complete realizations of the distribution functions in a statistical sense.

An illustration of the application of both approaches is in our recent study of the old nearby star cluster Ruprecht 147 (Yeh et al. 2019). The first approach revealed the tidal tails of the cluster, while the second one was used to derive the radial density profile (with the cluster corona outside the cluster tidal radius), and the luminosity and mass functions. The first approach used 69 candidate members only given the very strict selection. From the application of the second approach the number of the cluster stars was 280 ± 67 and the cluster mass was 234 ± 52 M_⊙.

The goal of this paper is to study the area around Alpha Persei with the purpose of separating stars belonging to the cluster and to the stream from the general field and investigating their relationship.

This paper is organized as follows. Section 2 is a brief description of the Gaia DR2 sample. In Section 3 we discuss the properties of the cluster and the stream and their mutual relationship. Section 4 is an analysis of the spatial distribution of white dwarf stars in the area. Section 5 summarizes our results.

We first selected stars from Gaia DR2 in the five-dimensional space defined as: l ∈ [90; 200] deg, b ∈ [−55; 45] deg, π ∈ [3; 8] mas, μ_α ∈ [15; 30] mas yr⁻¹, μ_δ ∈ [−32; −18] mas yr⁻¹. This selection returned 60,603 stars. We will refer to this selection as Sample 1a.

The text of the ADQL query was as follows:

SELECT l, b, ra, ra_error, dec, dec_error, source_id, parallax, parallax_error, pmra, pmra_error, pmdec, pmdec_error, phot_g_mean_mag, phot_bp_mean_mag, phot_rp_mean_mag, bp_rp, radial_velocity, radial_velocity_error, phot_bp_mean_flux, phot_bp_mean_flux_error, phot_rp_mean_flux, phot_rp_mean_flux_error, phot_g_mean_flux, phot_g_mean_flux_error

FROM gaiadr2.gaia_source

WHERE b BETWEEN −55.0 and 45.0 AND l BETWEEN 90.0 and 200.0

AND pmra BETWEEN 15.0 and 30.0 AND pmdec BETWEEN −32.0 and −18.0

AND parallax BETWEEN 3.0 AND 8.0.

The corresponding area on sky is about 100° on a side. This wide area allowed us to detect structures around the cluster possibly associated with the stream. Since our goal is to study the cluster in a statistical manner, in order to preserve the largest possible number of cluster members, we did not use any quality flag. When we add the limitation on stellar magnitude, G < 18 mag, the sample diminishes to 32,442 stars. We will refer to this selection as Sample 1. We adopted the fundamental parameters of Alpha Persei from Loktin & Popova (2017); see Table 1.

Table 1.  Fundamental Parameters of Alpha Persei

Parameter	Value
R.A.	03h27m
Decl.	$+49^\circ 07^{\prime}$
Galactic longitude	1475
Galactic latitude	−065
Log (Age [yr])	7.9
Heliocentric distance	176 pc
Color excess $E(B-V)$	0.1 mag

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Figure 1 shows the surface density map for Sample 1 obtained using a kernel density estimator (KDE; Seleznev 2016b). This density map was plotted with the quartic kernel for two dimensions (Seleznev 2016b; just as all density maps in this paper) and the kernel half-width h = 120'. In this map one can see two populations—the cluster in the center and the stream (the structure is elongated about 70° in approximately the diagonal direction). The conclusion that the stream could be a tidal feature associated with the cluster is not plausible, since the stream direction is significantly different from the cluster motion.

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We then attempted to find a cluster sample by gradually squeezing the parameters' limits π, μ_l, and μ_b around their mean values ($\overline{\pi }=5.7$ mas, $\overline{{\mu }_{l}}=33$ mas yr⁻¹, $\overline{{\mu }_{b}}=-8.5$ mas yr⁻¹; these mean values were taken approximately from the diagrams "parameter-G"). We controlled this iterative process by monitoring visually the color–magnitude diagram (CMD) of the residual sample at every step. We terminated the process when the cluster sequence emerged clearly in the CMD against a minimum field star contamination. The final limits on the parallaxes and the proper motions are: π ∈ [5.05; 6.35] mas, μ_l ∈ [26; 40] mas yr⁻¹, μ_b ∈ [−11; −6] mas yr⁻¹.

The left panel of Figure 2 shows the CMD at the end of the process. Some field stars are still present close to the cluster main sequence between G = 15 mag and G = 18 mag. This selection returned 1413 stars, and we will refer to it as Sample 2. Adopting the Padova suite of stellar models (Bressan et al. 2012) and using cluster parameters as in Table 1, we computed Sample 2 total mass as 924 M_⊙. We do not accompany this value with an uncertainty since a large part of it resides in the mass–luminosity relation used by Bressan et al. (2012), which remains unpublished.

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For this mass, the tidal radius is estimated via the King (1962) expression, and the Oort constants for the solar vicinity are taken from Bobylev & Bajkova (2014). It amounts to R_t = 12.8 ± 0.4 pc. This is clearly a lower limit, because we applied a luminosity cut at G = 18 mag. The estimate of uncertainty is also lower because we took into account only the uncertainties on the Oort constants. The tidal-radius circle is indicated with a red circle in Figures 1 and 2 (right panel). The tangential velocity vector for the stars inside the tidal radius is shown as a black arrow.

Figure 2 (right panel) shows the surface density distribution for Sample 2 stars plotted by KDE with the quartic kernel half-width h = 100'. We would like to highlight two other pieces of evidence. First, the stream stars did not disappear completely. This means that the cluster and the stream lie very close in three-dimensional space of parallaxes and proper motions, and one needs more sophisticated methods to separate them. Second, we notice some hints of tidal features close to Alpha Persei's tidal radius that are not perfectly aligned with its velocity vector.

In order to separate the cluster, the stream, and the field, we used DBSCAN (Xu et al. 1997). Jerabkova et al. (2019) and Beccari et al. (2020) successfully used this procedure for selection of the filament-like structures in regions of active or recent star formation. We run DBSCAN for three different configurations of the parameter space. The first one (C1) is a five-dimensional (l, b, π, μ_α, μ_δ) space. The second one (C2) is a three-dimensional (π, μ_l, μ_b) space, where μ_l and μ_b are proper motions in Galactic coordinates. We refer to the third configuration (C3) as the reduced three-dimensional volume (π, μ_lr, μ_br), where μ_lr and μ_br are the residual proper motions with respect to the mean projected motion of cluster stars (see above). Table 2 lists the adopted DBSCAN parameters and the results of the selection of stars in the cluster, in the stream, and in the field (the field being leftover after cluster and stream removal). ε is the selection radius and minPts is the minimum points number.

Table 2.  Parameters' Values and Results for the Selection with DBSCAN

Space	$(l,b,{Plx},{\mu }_{\alpha },{\mu }_{\delta })$	$({Plx},{\mu }_{l},{\mu }_{b})$	$({Plx},{\mu }_{{lr}},{\mu }_{{br}})$
Configuration	C1	C2	C3
ε	0.2	0.05	0.05
min(Pts)	400	450	600
Cluster (stars)	2147	1884	1370
Stream (stars)	3881	9197	2977
Field (stars)	26414	21361	28095

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We explored DBSCAN parameters in a very wide limits' range ($\varepsilon \in [0.001;0.8]$, $\mathrm{minPts}\in [10;1000]$). However, the separation of the cluster and the stream is achieved in a very narrow interval of ε and minPts only, close to the values listed in Table 2. Even a small shift tended to situation when the cluster and the stream were not separated. The values of DBSCAN parameters were found by going over the ε with some step and looking for an appropriate minPts.

The outcome of the three different selection criteria (C1, C2, and C3) can be analyzed using stars' surface density maps (see Figure 3). All these maps were plotted by KDE with the kernel half-width of h = 120'. The effectiveness of the selection in particular can be tested against the general field stars' distribution, which ideally should be uniform. The selection C2 clearly provides the best result from this point of view. The cluster corona is elongated along the Galactic longitude. However, in the inner part of the cluster, the density contours are aligned with the cluster velocity vector (see Figure 1 for a comparison).

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In Figure 4 (left panel) we show the CMDs for the cluster (color coded in red) and the stream (color coded in black) for the best performing configuration C2. The right panel, on the other hand, shows the CMD for the stars in common in all three DBSCAN configurations.

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We then focused on the stream itself, and attempt at determining its fundamental parameters as a stellar population. In Figure 5 we compare the star distribution in the CMD with isochrones from the Padova suite of models (Bressan et al. 2012). Given the proximity of this structure to the Sun, we explored solar metallicity models. The fitting has been performed eyeballing the isochrone with respect to the star distribution in the vicinity of the turnoff point and in the red clump region. We found a plausible age of 5 ± 1 Gyr.

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To obtain this age estimate, we shifted the isochrones horizontally by E(B–V) ∼ 0.1 (see Table 1), which gives E(BP–RP) = 0.13 ± 0.03 according to Cardelli et al. (1989) and O'Donnell's (1994) extinction curve. This value of the reddening E(B–V) is confirmed by the analysis of the 2MASS (Skrutskie et al. 2006) color–color diagram J–K versus H–K. The reddening for the candidate stream stars (in accordance with the C2 DBSCAN configuration) as derived from the aforementioned diagram turns out to be E(B–V) = 0.09 ± 0.06 mag. The vertical shift yields for the stream a distance modulus (m–M) = 7.4 ± 0.2 mag and (m–M)₀ = 7.1 ± 0.2 mag with A_G = 0.27 mag (corresponding to the same extinction curve). It corresponds to a mean heliocentric distance of ${263}_{-23}^{+25}$ pc.

Then, from the inspection of these CMDs, we can conclude that the stream consists of a stellar population ∼5 Gyr old, significantly older than the cluster. This rules out a common origin scenario for the cluster and the stream. The most plausible explanation for the stream could be that it is the leftover of an old disrupted star cluster moving in the vicinity of the Alpha Persei cluster.

Additionally, we also note that the stream appears to lie generally behind the cluster. The difference between the adopted heliocentric distance of the cluster and the mean heliocentric distance of the stream is ≈90 pc. Finally, by looking more closely at the left panel of Figure 4 one can notice from the MS width that the stream looks extended along the line of sight by about 180 pc, from 190 to 370 pc from the Sun, approximately. These estimates have been obtained from the main-sequence band boundaries at, e.g., (BP–RP) = 2 mag, assuming the same extinction for the cluster and for the stream. A more detailed investigation of the spatial structure of the stream shows that this structure is more complicated.

The existence of the stream as a spatially confined structure is confirmed by the distribution of stellar heliocentric distances for a series of regions across the stream area that we outlined by isodensity contours (see Figure 6). Figure 7 shows such distributions plotted with KDE (quartic kernel for one dimension, with a kernel half-width of h = 25 pc for regions C, E3 (this region includes the cluster), G, and J (this region represents the general field)).

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In order to derive these distributions we downloaded a new sample from Gaia DR2 (Sample 3) that is the same as Sample 1 except for the parallax limit Plx ∈ [2; 8] mas, and therefore includes more distant stars. This Sample 3 contains 59,803 stars having G < 18. When analyzing the distance distributions one needs to remember that within the same solid angle and with the uniform spatial distribution of stars the number of stars will increase with the distance. So an increase of the number of stars at increasing distance for region J is due to this geometric effect.

We calculated the distances by inverting parallaxes. We are aware that the direct inversion of the parallax is only a crude approximation of the distance (Luri et al. 2018), but we expect that for the small distance range we are working with the implied errors are not very important. Parallaxes in Sample 3 have small relative errors: 90% of stars have a relative parallax error less than 0.06% and 97.7% of stars have an error less than 0.1.

These distance distributions show several interesting features. First, the stream is standing against the background as a net overdensity. Second, the lower left part of the stream is closer to us than the upper right part of the stream (the density maximum in region G is at 210 pc and in region C at 270 pc; the density maximum for the cluster in region E3 is at 176 pc, which is in excellent agreement with the cluster data). Third, the stream is clumpy. This is well seen on the density maps (Figures 1 and 3) and on distance distributions, which often have multiple maxima. Fourth, the stream is partially overlapping with the cluster. The width of the stream along the line of sight is in good agreement with that inferred from the CMD (see above).

With the goal of providing a sharper view of the structure of the stream we calculated coordinates X, Y, and Z for the stars of Sample 3. The center of this coordinate system coincides with the cluster center, while the XY-plane is parallel to Galactic plane. The X-axis is directed toward the Galactic anticenter, the Y-axis follows the tangent of the cluster Galactic circular orbit, and the Z-axis is perpendicular to the Galactic plane. Figure 8 shows the surface density distributions in a projection onto the XY-plane for 20 pc thick "slices" along the Z-axis (KDE with a quartic kernel half-width of h = 20 pc). The density levels in units of pc⁻² were selected to provide the best contrast and highlight the structure of the stream effectively. It is clear that the stream has a complicated structure consisting of several clumps, and it overlaps with the cluster. Our previous conclusions on the overall geometry of the stream are confirmed.

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An upper limit of the stream mass is ∼6000M_⊙. This has been obtained by merely counting the number of stream stars (C2) and it is an upper limits, since we cannot exclude some amount of field star contamination. Therefore, the stream seems to identify the leftover of quite a massive cluster that is now on the verge of dissolving into the Galactic field.

To lend further support to this hypothesis we checked the distribution of various subsystems in the proper motion plane (vector point diagram). Figure 9(a) shows three subsystems selected with the C2 DBSCAN configuration: the violet dots denote the field, the gray dots denote the stream, and red dots denote the cluster. In order to make the difference between the field and the stream more evident we plotted the density maps in a vector point diagram for these subsystems (Figure 9(b) and Figure 9(c), respectively). It is readily seen that the stream has a well-marked central concentration, unlike the field. The dispersion of proper motions for the stream is larger than that for the cluster. This picture resembles a disrupting star cluster. The kernel half-width for these maps was taken to be h = 3 mas yr⁻¹.

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Figure 9(d) shows the distribution of white dwarf (WD) candidates (see below) in the proper motion plane. This distribution shows two maxima. The upper one is close to the upper maximum of the field distribution (Figure 9(b)), and the lower one is close to the maximum of the stream distribution (Figure 9(c)). We stress here that the WD sample consists of stars with G > 18 mag (Sample 1a, see below). These stars usually have large errors of proper motions. Consequently, it is difficult to draw firm conclusions only considering the proper motion distribution of these stars.

We could not find a set of DBSCAN parameters suitable to isolate the tidal tails structure we have seen in Figure 2 as belonging to the cluster. In order to clarify the nature of this structure, we compiled a sample of stars (hereafter called TTS) in the following way. We took the stars of Sample 2 that formed the tidal tails structure in the projection to a tangent plane, namely, the stars inside red circles at Figure 10(a). Nearly all stars of the TTS sample lie outside the cluster sample selected by the DBSCAN procedure.

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Figure 10(b) shows the CMD, where the gray points are from the right panel of Figure 4 (the cluster and the stream stars in common for all the three DBSCAN selection criteria) and the red points correspond to stars that lie inside the encircled regions in Figure 10(a) (TTS). The vast majority of the red points lie very close to the cluster sequence, with a few exceptions for stars scattered away from the MS. This fact supports the cluster as the origin of tidal tails stars.

Additional support for the hypothesis that the TTS stars originated from the cluster is in the proper motion distribution. Figure 10(c) shows the proper motion diagram for the cluster stars (C2 DBSCAN configuration, blue dots) and stars of the TTS sample (red dots). The stars of the TTS sample lie inside the region occupied by the cluster stars. Thereby, we can confidently claim that the tidal tail-like structures from Figure 2 are real and probably connected to the cluster.

A possible explanation for why the DBSCAN did not add the TTS stars to the cluster sample is that these structures are poorly populated with respect to other features.

What is the origin of these tidal tails? A gravitational interaction with the stream seems unlikely due to the low mass of the stream. Danilov (1994) and Danilov & Seleznev (1995a, 1995b) showed that a massive gas-stellar complex (with the mass of 10⁵–10⁸ solar masses) can exert prominent gravitational influence on an open star cluster. The Gould Belt, with a mass of about 10⁶ solar masses, could play such role, but the Alpha Persei open cluster is very close to its center (see a review in Bobylev 2014). Our mind is that these tidal tails, although not perfectly aligned with Alpha Persei's velocity vector, are probably of Galactic origin, analogous to other cases reported in the literature (Yeh et al. 2019).

In Table 3 we list the mean velocities of the cluster stars, the mean residual velocities of the stream stars, and the mean velocities of the tidal tails stars (residual velocity with respect to the cluster). The mean velocity of the tidal tails is very close to the mean velocity of the cluster stars. The dispersion, though, is high.

Table 3 also illustrates that all three DBSCAN selections yield similar results concerning the cluster but different results concerning the stream (this is also clearly seen in Figure 3).

To lend further support to our findings we looked at the space distribution of white dwarf (WD) stars in the area. In fact, the stream is old and populous enough to harbor a noticeable population of WDs. Since we do not expect WDs from Alpha Persei, these stars should be genuine tracers of the stream. Their distribution should therefore confirm both the structure and the age of the stream.

Unfortunately, Sample 1 contains very few WDs. Most WDs have in fact G > 18 mag. Because of this, to isolate the WD population of the stream, we turned to Sample 1a, relaxing the magnitude constraint. Recall that this sample contains 60,603 stars with the same limits on parallaxes and proper motions as Sample 1, but with G magnitudes up to the limit of the Gaia DR2 catalog. The left panel of Figure 11 shows the CMD of this sample with WD candidates highlighted in red. We also compared the expected position of WDs with the work of Gaia Collaboration et al. (2018a). We relaxed the limit on the stellar magnitude because WD candidates with G ≤ 18 mag were very few and did not show any spatial concentration.

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The right panel of Figure 11 shows the density map for the CMD-selected WDs (KDE with a quartic kernel half-width of h = 230'). The concentration of stars close to the stream region is readily seen. The depression (low density region) at the cluster position is due to the screening effect of the cluster, which lies in front of the stream along the line of sight.

In order to confirm the spatial association of WDs with the stream, we selected two regions with the maximum surface density of the WD stars (the red rectangles in the right panel of Figure 11) and plotted the distance distributions of stars from these regions. We present these distributions in Figure 12. These distributions were plotted in the same way as Figure 7. The distance distributions of WDs show distinct maxima. We can compare these distributions with Figure 7. The maximum in the left panel of Figure 12 (for a lower rectangle of Figure 11, right panel) coincides well with the maximum for field G of Figure 7. The maxima in the right panel of Figure 12 (for an upper rectangle of Figure 11, right panel) coincides well with the maxima for field C of Figure 7. Note that the lower rectangle corresponds roughly to regions F, G, H of Figure 6, and the upper rectangle corresponds roughly to region C of Figure 6. Thus, the distance distributions of WDs correspond well to the distance distributions of the stream stars.

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Finally, in Figure 9(d) we showed the density distribution of the WDs (selected as described above) in the proper motion plane. Unfortunately, the errors of the proper motions for stars with G > 18 mag are large, and it is difficult to make any firm conclusions about the similarity of distributions in Figures 9(b)–(d). Nevertheless, a visual comparison of these figures suggests that the WD proper motion distribution shares features of both the field and the stream (see the discussion above).

The spatial concentration is clear evidence that most of the selected WDs are members of the stream. This concentration is also a confirmation of the age estimate of the stream, ∼5 Gyr.

In this work we investigated in detail the projected spatial distribution of stars surrounding the Alpha Persei star cluster with the aim of isolating the stellar stream with which the cluster is mixed. The evidence and existence of this stream have been repeatedly discussed in recent literature.

We succeeded in separating the cluster stars and the stream stars from the general Galactic field using DBSCAN. The stream exhibits a large radial extent (about 180 pc). The distance between the cluster center and the stream central line is about 90 pc; the stream lies generally in the background of the cluster and has a clumpy structure. The stream is significantly older than the cluster, with an age of ∼5 Gyr. The presence of a conspicuous WD population that follows the stream spatial distribution lends further support to the age estimate of this structure. We estimated the stream mass to be ∼6000M_⊙.

The most likely interpretation of our results is that the stream is the relict of an initially massive star cluster that the Galactic tidal forces are bringing to the brink of dissolution. This point of view does not contradict the distribution of the proper motions of the stream stars.

The density map of Sample 2 (taken with very narrow limits on parallaxes) shows the presence of structures resembling tidal tails. An analysis of the proper motions and photometric properties of stars composing these tidal tails confirmed that these stars probably originated from the cluster. It is conceivable that had the stream exerted significant tidal action on the cluster, if it would be much more massive (Danilov 1994; Danilov & Seleznev 1995a, 1995b). However, by considering similar cases in the solar vicinity, we favor a scenario in which Alpha Persei tidal tails have a Galactic origin.

This work was supported by the Ministry of Science and Higher Education of the Russian Federation, FEUZ-2020-0030, and by Act No. 211 of the Government of the Russian Federation, agreement No. 02.A03.21.0006. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. The input of the anonymous referee has been greatly appreciated.