2.1. 2MASS

In this paper, we have used The Two Micros Allsky Survey (2MASS, ^{Skrutskie et al. 2006}) data for the cluster NGC 5288. This data set uses two highly automated 1.3m telescopes, one at Mt. Hopkins, Arizona (AZ), USA and the other at CTIO, Chile, with a 3-channel camera (256×256 array of HgCdTe detectors in each channel). The 2MASS catalog provides J (1.25 µm), H (1.65 µm) and K _s (2.17 µm) band photometry for millions of galaxies and nearly a half-billion stars (Carpenter, 2001). The sensitivity of this catalog is 15.8 mag for J, 15.1 mag for H and 14.3 mag for K _s band at S/N=10.

2.2. GAIA DR3

The Gaia DR3 data gives a complete astrometric solution for more than 1.46 billion sources. It consists of their positions on the sky (α,δ), parallaxes, and proper motions within 3 to 21 mag limit in the G band (^{Gaia Collaboration et al. 2016}; ^{Babusiaux et al. 2022}). The Gaia photometric bands: G, G _BP , and G _RP gather measurements over the wavelength ranges of 330−1050,330−680, and 640−1050 nm, respectively.

We collected the data within a radius of 10 arcmin, taking into account the estimated cluster radius of 6.3 arcmin from an earlier study (^{Piatti et al. 2006}) of the cluster. The identification map of the cluster under study is shown in Figure 1 and is taken from the Digitized Sky Survey (DSS). The proper motions and their respective errors for all stars in the Gaia data base are plotted against G mag in Figure 2.

Fig. 1 Identification map of NGC 5288 taken from DSS. 

Fig. 2 Plot of proper motions and their errors against G mag 

3.1. Vector Point Diagrams

As mentioned above, the cluster stars share similar kinematic properties and the same mean motions. Therefore, proper motions of stars are a crucial parameter for identifying cluster members and field stars (^{Yadav et al. 2013}; ^{Bisht et al. 2021b}). A graph is plotted between the proper motion components of the stars, µ _α cosδ (mas yr⁻¹) and µ _δ (mas yr⁻¹) to obtain a vector point diagram (VPD), as shown in the top panel of Figure 3. A CMD corresponding to each of these VPDs is plotted in the bottom panel of Figure 3. Proper motion is the change in the apparent positions of a celestial object when observed from the center of mass of the Solar System, and each point in a VPD gives the proper motion of one star. The dense region in a VPD depicts the stars that share similar motions in space compared with the background field stars. This is one of the primary conditions of being cluster members. Hence, we have used these stars for a preliminary cluster member selection. We define an eye estimated radius around dense region in the VPD as shown with red color in Figure 3. Finally we calculate the membership probability for the selected stars. The CMD in the middle of the bottom panel of Figure 3 appears much cleaner than the other two, confirming the effectiveness of this primary selection method.

Fig. 3 (Top panel) VPDs for cluster NGC 5288. (Bottom panel) G vs (GBP − GRP ) CMDs of the stars shown in the VPD above them. (Left panel) includes all stars. (Middle panel) includes only the stars within the region inside the red circle; which we classify as the probable cluster members. (Right panel) includes the probable background/foreground field stars). The color figure can be viewed online. 

3.2. Membership Probabilities

In this section, we aim to estimate the membership probability of stars using precise measurements of PMs and parallax from Gaia-DR3. ^{Vasilevskis et al. (1958)} described the mathematical setup for determining the membership probability of stars. For NGC 5288, we have used the method devised by ^{Balaguer-Núñez et al. (1998)} to calculate the membership probability of individual stars. This method has been used previously for both open and globular star clusters (^{Yadav et al. 2013}, ^{Sariya & Yadav 2015}; ^{Sariya et al. 2021a}, ^2021b), and a detailed description of this method can be found in ^{Bisht et al. (2020}).

To derive the two distribution functions defined in this method, ϕcν (cluster star distribution) and ϕfν (field star distribution), we considered only those stars which have PM errors better than 1 mas yr⁻¹. A group of the cluster’s preliminary members shown in the VPD is found to be centered at µ _xc =−3.840mas yr⁻¹, µ _yc =−1.934mas yr⁻¹. We have estimated the PM dispersion for the cluster population as (σ _c )=0.08mas yr⁻¹. For the field region, we have estimated (µ _xf , µ _yf ) = (−6.18, −1.95)mas yr⁻¹ and (σ _xf , σ _yf ) = (0.84, 0.70)mas yr⁻¹.

The membership probabilities (P _µ ) are thus determined, and they are shown as a function of Gaia’s G magnitude in Figure 4. Finally, we identified 304 stars with P _µ > 50%.

Fig. 4 Plot of cluster membership probabilities against the G mag. The red open circles represent the cluster members having membership probability greater than 50%. The color figure can be viewed online. 

To estimate the mean PM of NGC 5288, we considered the most probable cluster members and constructed the histograms for µ _α cosδ and µ _δ as shown in the right panel of Figure 6. The fitting of a Gaussian function to the histograms provides the mean PM in both directions. We obtained the mean PM of NGC 5288 as −3.840±0.230 and −1.934 ± 0.162mas yr⁻¹ for µ _α cosδ and µ _δ respectively. The estimated values of mean PM for this object is in very good agreement with the mean PM value given by ^{Cantat-Gaudin et al. (2018)} (µ _α cosδ =−3.850mas yr⁻¹, µ _δ =−1.935mas yr⁻¹).

Fig. 5 Top, middle and bottom panels represent plots of parallax distribution PM in declination and PM in right ascension vs. G mag, respectively. The red dots represent the cluster members, while the black ones are the field stars. The color figure can be viewed online. 

Fig. 6 Histograms of star count in RA(top left), DEC(bottom left), pmRA(top right), and pmDEC(bottom left) fitted with a Gaussian. The vertical dashed-line at the Gaussian center represents the respective mean values. The color figure can be viewed online. 

3.3. Cluster Membership Validation

We have categorized the stars with membership probability greater than 50% as cluster members, and only these stars have been used for evaluating the properties of the cluster in the current study. To validate the accuracy of our identified cluster members, we have plotted proper motions, µ _α cosδ (mas yr⁻¹) and µ _δ (mas yr⁻¹), and parallax as a function of G mag in Figure 5. The graph shows a narrow distance range of parallax and proper motion coordinates for the stars categorized as cluster members, but this range is significant for field stars. Hence it is fair to say that we have successfully separated the cluster members.

4.1. Cluster Center

Precise estimations of the central coordinates of a cluster play a vital role in evaluating its fundamental properties. We use a star count approach that states that the cluster center is located at the point with the highest stellar density in the cluster region to determine central coordinates (^{Bisht et al.2021a}). For this, a histogram of the star count in RA and DEC is plotted and fitted with Gaussian profiles. The mean values of RA and DEC are estimated to be 207.188 ± 0.156 deg (13 ^h 48 ^m 45 ^s ), and −64.679 ± 0.066 deg (−64^◦40^′44^′′), respectively (refer to Figure 6). The center of the cluster is approximated to be at these coordinates. These values agree well with the values estimated in other studies, as shown in Table 2.

A CMD, an identification chart and the proper motion distribution of cluster members (with the cluster center highlighted) are plotted in Figure 7.

Fig. 7 Plots of CMD((GBP − GRP vs. G), identification chart and proper motion distribution of cluster members represented as the red dots. The plus sign denotes the cluster center. The color figure can be viewed online. 

4.2. Radial Density Profile

The radius of a cluster gives us a direct indication of its physical dimensions. We plotted a radial density profile (RDP) of NGC 5288 as shown in Figure 8. We divided the cluster region into several concentric circular bins and calculated the number density, ρi=NiAi of each bin using cluster members. ρ _i is the number of cluster members lying in the i ^th radial bin divided by the area of the i ^th bin, A _i . The i ^th bin corresponds to the region between the i ^th and (i−1) ^th concentric circles. The observed radial profile is fitted by the ^{King (1962)} model, represented as the smooth continuous line in Figure 8. The middle one of the three horizontal lines plotted in Figure 8 corresponds to the background density, and the two lines on either side of it indicate the error in it. A mathematical expression of the model used for fitting is given as :

where f _bg ,f ₀ ,r _c are background density, central density and core radius, respectively. Using the ^{King (1962)} fit, we estimate the structural parameters of NGC 5288, which are given in Table 1. We estimate the cluster radius as the point after which the cluster density merges with the field density. Following this criterion, we derived the cluster radius as ≈ 5.5 arcmin. The previous study by ^{Piatti et al. (2006)} estimates the radius of NGC 5288 as 6.3 arcmin using CCD photometry. We also calculate the density contrast parameter, δ _c , for the cluster using the formula, δc=1+f0fbg and obtain its value as 17±2. Our estimated value of δ _c lies within the limit (7 ≤ δ _c ≤ 23) given by ^{Bica & Bonatto (2005)}, suggesting that the cluster is compact.

Fig. 8 Radial density profile of NGC 5288 using cluster members. The smooth continuous line represents the fitted ^{King (1962)} model. The middle dotted horizontal line depicts the background density and the long and short dashed lines represent the error in the background density. 

5.2. Interstellar Reddening from JHK Colors

The cluster reddening in the near-IR region has been estimated using the (J − H) vs (J − K) colorcolor diagrams as shown in Figure 10. The black dots are the cluster members, the solid red line represents the cluster’s zero-age main sequence (ZAMS) taken from ^{Caldwell et al. (1993)}, while the red dashed line is the ZAMS displaced by the value of E(J − H) and E(J − K), which are estimated as 0.15 ± 0.04 and 0.28±0.07 respectively. The reddening is calculated using the following equations (^{Fiorucci & Munari 2003}) :

Using the above equations, we derived the interstellar reddening of the cluster, E(B − V ) as 0.45.

Fig. 10 (J − H) vs (J − K) two color diagram. The red solid and the dashed lines are the ZAMS taken from ^{Caldwell et al. (1993)}. The red dashed line is the ZAMS shifted by the value given in the text. The color figure can be viewed online. 

5.3. Age and Distance from Isochrone Fitting

We estimate important cluster parameters such as age and metallicity by fitting the theoretical isochrones of ^{Marigo et al. (2017)} to the CMDs, (G, G _BP − G _RP ) and (J, J − H) as shown in Figure 11. Color-magnitude diagrams (CMDs) represent the relationship between the absolute magnitudes of the stars and their surface temperatures, usually identified by their color. CMDs are extensively used to study star clusters and are beneficial for estimating their properties (^{Kalirai & Tosi 2004}; ^{Sariya et al. 2021b}). We tried to fit many isochrones with different metallicity and ages and found the best fit at Z=0.03, which shows a good agreement with the literature value of 0.04 given by ^{Piatti et al. (2006)}. In Figure 11, we have superimposed theoretical isochrones of different ages (log age = 8.5, 8.7 and 8.9) and Z=0.03 over the observed (G, G _BP − G _RP ) and (J, J − H) CMDs. We estimated the cluster’s age to be 510±190 Myr. Our estimated age lies between the age given by ^{Dias et al. (2014)} and ^{Kharchenko et al. (2016)}. We found the color excess value, E(G _BP − G _RP ) of the cluster as 0.60. The isochrone fitting is performed so that it will result in a cluster distance close to what we estimated from the parallax. From this fitting, we calculated the distance modulus (m − M) as 13.50 ± 0.25 mag, corresponding to a heliocentric distance of 2.64±0.11 kpc. This value is comparable with the distance we approximated using the parallax method in the above section. The A _G value for this cluster is 0.99±0.62, which is calculated by the weighted mean method using the A _G values of cluster members from the Gaia DR3 data. The value of the total to selective extinction ratio using values determined in the present analysis is compatible with the value given by ^{Wang & Chen (2019)} within errors. A comparative analysis of the fundamental properties of cluster members derived in this study with the values in the literature is given in Table 2.

Fig. 11 The (G, G _BP −G _RP ) and (J, J −H) color-magnitude diagrams of the open star cluster NGC 5288. The black dots represent cluster members and the curves represent the theoretical isochrones of three different ages (8.5, 8.7 and 8.9) with Z = 0.03. The color figure can be viewed online. 

6.1. Dynamical Relaxation Time

In the lifetime of star clusters, encounters between their member stars gradually lead to increased energy equipartition throughout the clusters. In this process, massive stars are concentrated towards the cluster core and transfer their kinetic energy to fainter stars. The relaxation time T _R is defined as the time in which the stellar velocity distribution becomes Maxwellian and is expressed by the following formula:

where N represents the number of stars in the clusters (in our case, the ones with membership probability higher than 50%), R _h is the cluster half mass radius expressed in parsec and m¯ is the average mass of the cluster members (^{Spitzer & Hart 1971}) in solar units. The value of R _h is assumed as 2.56 pc, which is equal to half of the cluster’s extent. Using the above formula, the value of the dynamical relaxation time T _R is determined as 24.6 Myr, significantly less than the cluster’s age. Hence, we conclude that NGC 5288 is a dynamically relaxed cluster.