Variable stars in the Sh 2-170 H ii region

Optical photometric observations of the Sh 2-170 region were taken in the $I_{c}$ band on 17 nights and in the $V$ band on 5 nights starting from 30th September 2016 to 24th November 2017 with the 1.3m Devasthal Fast Optical Telescope (DFOT, India), 0.7m Thai Robotic Telescope (TRT-GAO, Gao Mei Gu Observatory, China) and 0.5m telescope of Thai National Observatory (TNO, Thailand). All the telescopes have a 2048$\times$2048 pixel square CCD for imaging. The 1.3m Devasthal telescope covers a field of view (FOV) of $18^{\prime}.4\times 18^{\prime}.4$, whereas the 0.7m and 0.5m telescopes have a FOV of $\sim 20^{\prime}.9\times 20^{\prime}.9$ and $\sim 23^{\prime}\times 23^{\prime}$, respectively. In total, 776 and 13 frames were taken in $I_{c}$ and $V$ filters, respectively, with exposure times of 180s on the 1.3m, 240s on the 0.7m and 300s on the 0.5m telescopes. Bias and flat frames were also taken in each night along with the target frames. The typical seeing size (estimated from the FWHM of stellar images) during the observations on 1.3m DFOT were about $1^{\prime\prime}.5$ - $2^{\prime\prime}$ and for the 0.5m and 0.7m telescopes, it was $3^{\prime\prime}$ - $4^{\prime\prime}$. Details of the observations are given in Table 1. In Fig. 1, we have shown the colour-composite image of the observed region by using the 12 $\mu$m (WISE, red colour), 2.17 $\mu$m ($K_{s}$, 2MASS, green colour) and 0.8 $\mu$m ($I_{c}$, present observation, blue colour) images.

The basic image processing, such as bias subtraction, flat fielding, cosmic ray rejection, were done by using the tasks available within IRAF¹¹1IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation.. The instrumental magnitude was obtained by using the DAOPHOT (Stetson, 1987) package. As the cluster region is very crowded, we carried out PSF photometry to get the magnitudes of the stars. We have used the DAOMATCH and DAOMASTER routine of DAOPHOT-II (Stetson, 1992) to estimate the shifts in individual frames with respect to a reference frame and to get the magnitudes of stars detected in different frames. The CCD pixel coordinates of the stars have been converted to celestial coordinates (RA and Dec) for the J2000 epoch by using the GAIA software²²2http://star-www.dur.ac.uk/ pdraper/gaia/gaia.html. The observations obtained with different telescopes were cross-matched by using their astrometry within a 1^′′ search radius.

We have calibrated the instrumental magnitudes into the standard system using the photometric data published by Bhatt et al. (2012) for the $\sim 13^{\prime}\times 13^{\prime}$ FOV (shown with the cyan square box in Fig. 1) of the cluster region. The transformation equations used for photometric calibration are as given below:

where $V$, $I_{c}$ and $v$, $i_{c}$ are standard magnitudes and instrumental magnitudes, respectively.

In addition to the data from the present observations, we have used the available photometric data from Bhatt et al. (2012) for our analysis. The catalog by Bhatt et al. (2012) provides data for 2261 stars in a FOV of $\sim 13^{\prime}\times 13^{\prime}$. We have detected extra 4103 stars in a total FOV of $\sim 18^{\prime}.4\times 18^{\prime}.4$ in the Sh 2-170 region.

Since NIR and MIR data are very useful to study the spectral energy distribution (SED) and disc properties of young stellar objects (YSOs), we have used NIR/MIR photometric data from available archives described below:

(i) NIR JHKs photometric data have been taken from the 2MASS All-Sky Point Source Catalog (Skrutskie et al., 2006; Cutri et al., 2003).

(ii) Spitzer-IRAC observations at 3.6 and 4.5 $\mu$m have been taken from the GLIMPSE360 Catalog and Archive (Werner et al., 2004).

(iii) MIR data at 3.4, 4.6, 12 and 22 $\mu$m have been taken from the Wide-field Infrared Survey Explorer (WISE) All-sky Survey Data release (Wright et al., 2010).

The NIR/MIR data having photometric error $\leq 0.2$ mag and a matching radius of $1^{\prime\prime}$ were used to identify their optical counterparts.

To study the stellar density distribution in the Sh 2-170 region, we have obtained a surface density map for a sample of stars taken from the 2MASS All-Sky Point Source Catalog (less affected by gas and dust distribution), covering $18^{\prime}.4\times 18^{\prime}.4$ FOV around this region. We have generated a surface density map using the nearest neighbour (NN) method as described by Gutermuth et al. (2005); Gutermuth et al. (2009). We took the radial distance necessary to encompass the $20^{th}$ nearest star and computed the local surface density in a grid size of 11^′′, which was then smoothened to a grid size of $3\times 3$ pixel². The density contours derived are plotted in Fig. 1 as white curves. The lowest contour is 1$\sigma$ above the mean stellar density (i.e., 31$\pm$9 stars/arcmin²) and the step size is equal to 1$\sigma$ (9 stars/arcmin²). The stellar density enhancement in the centre region of Sh 2-170, which is the Stock 18 cluster, can be easily seen from the contours. The core of the cluster is almost circular, whereas the outer contours are elongated. The density distribution shows a peak at $\alpha_{2000}$: $00^{h}01^{m}34^{s}.4$, $\delta_{J2000}$: $+64^{\circ}37^{\prime}48^{\prime\prime}$ with a core radius (defined as the point where density becomes half of the peak density) of $\sim 50^{\prime\prime}$. The extent of the cluster is shown with a circle having a radius of $\sim$2^′.72. On the basis of the radial density profile using 2MASS data, Bhatt et al. (2012) have reported the core and cluster radius as 18^′′ and 3^′.5, respectively.

The new and precise parallax measurements up to a very faint magnitude limits ($G$ (330-1050 nm) $\approx$ 21 mag) by $Gaia$ DR2 have opened a new dimension in the studies of membership determination in star clusters. The catalog can be queried on the $Gaia$ archive by using ADQL at http://gea.esac.esa.int/archive/. $Gaia$ PM data have been used to determine the membership probability of stars located within the Stock 18 cluster region (radius = 2^′.72). The PMs $\mu_{\alpha}$cos($\delta$) and $\mu_{\delta}$ are plotted as a vector-point diagram (VPD) in the top sub-panels of Fig. 2 (left panel). The bottom sub-panels show the corresponding $G$/($G_{BP}-G_{RP}$) CMDs, where $G_{BP}$ (330-680 nm) and $G_{RP}$ (630-1050 nm) covers the blue part and red part of the $G$ band. The left sub-panels show all stars, while the middle and right sub-panels show the probable cluster members and field stars, respectively. There seems to be an obvious clustering around $\mu_{\alpha}$cos($\delta$) = -2.62 mas yr^-1 and $\mu_{\delta}$ = -0.49 mas yr^-1. A circular area having a radius of 0.51 mas yr^-1 around the cluster centroid in the VPD seems to define the PMs of the cluster. The chosen radius is a compromise between the exclusion of cluster members with poor PMs and the inclusion of field stars sharing the cluster mean PM.

Assuming a distance of 2.8 kpc for Stock 18 (cf. Bhatt et al., 2012) and a radial velocity dispersion of 1 kms^-1 for open clusters (Girard et al., 1989), the expected dispersion, $\sigma_{c}$, in the PMs of the Stock 18 members would be 0.075 mas yr^-1 . For the remaining stars in the region (i.e., probable field stars), we have calculated $\mu_{xf}$ = -1.12 mas yr^-1, $\mu_{yf}$ = -0.67 mas yr^-1 , $\sigma_{xf}$ = 6.22 mas yr^-1 and $\sigma_{yf}$ = 2.68 mas yr^-1 (where $\mu_{xf}$ and $\mu_{yf}$ are the field PM centre and $\sigma_{xf}$ and $\sigma_{yf}$ are the field intrinsic PM dispersion). These values are further used to construct the frequency distributions of the cluster stars ($\phi^{\nu}_{c}$ ) and field stars ($\phi^{\nu}_{f}$ ). By using the procedure described by Yadav et al. (2013) the membership probability is estimated as per the following equation,

where $n_{c}$ (=0.19) and $n_{f}$ (=0.81) are the fractions of the cluster members and that of field stars, respectively. The membership probability of the sources within the cluster region (r < 2^′.72) is plotted as a function of $G$ magnitude in the top sub-panel of Fig. 2 (right panel). As can be seen a high membership probability (P${}_{\mu}\geqslant$ 80%) extends down to $G$ $\sim$ 20 mag. The bottom sub-panel of Fig. 2 (right panel) displays the parallax of the same stars as a function of $G$ magnitude. Except few outliers, most of the stars with high membership probability (P${}_{\mu}\geqslant$ 80%) follow a tight distribution. We estimated the membership probability for 463 stars in the cluster region and found 86 stars as cluster members (P${}_{\mu}\geqslant$ 80%). The details of these members are given in Table 2.

Bailer-Jones et al. (2018) have estimated distances of 1.33 billion stars using the data published in the $Gaia$ DR2 and those can be downloaded from $VizieR$³³3http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=I/347&-to=3. The individual distances of the 12 identified cluster members having parallax values with high accuracy (i.e. $error<0.05$ mas, shown as red triangles in the bottom sub-panel of Fig. 2 (right panel)), have been obtained from Bailer-Jones et al. (2018). The mean distance of these members comes out to be $3.1\pm 0.2$ kpc. This distance estimation is comparable, within the error, to that obtained by Bhatt et al. (2012, 2.8$\pm$0.2 kpc) based on the $V/(B-V)$ CMD.

Fig. 3 displays the $V/(V-I_{c})$ CMD of the member stars in the Stock 18 cluster (radius $<$ 2^′.72 and having P_μ $\geq 80\%$; 77 stars) shown by open square symbols. The post-MS isochrone for 2 Myr for the solar metallicity (black thick curve) by Marigo et al. (2008) along with the PMS isochrones of 0.1, 2, 7 Myr (red dashed curves) by Siess et al. (2000) are also shown. All the isochrones and evolutionary tracks are corrected for the distance of 3.1 kpc and for the reddening $E(V-I_{c})$ = 0.88 mag ($E(B-V)$= 0.7 mag, Bhatt et al., 2012). The $V/(V-I_{c})$ CMD indicates that a majority of the probable members are PMS stars. Few of the faint ‘members’ located near the MS isochrones at $V$ $\sim$ 20.5 mag are probably mis-identified stars due to large error in their parallax values near fainter magnitude limits (cf. Fig. 2). To investigate further about the age of the cluster, we have used NIR and MIR data from the 2MASS, WISE and $Spitzer$ surveys (cf. Section 2.2) to identify PMS stars showing excess IR emission, in the observed $18^{\prime}.4\times 18^{\prime}.4$ area of the Sh 2-170 region. We have used similar schemes as used in the recent literature (Sharma et al., 2016; Koenig & Leisawitz, 2014; Gutermuth et al., 2009). On the basis of NIR/MIR excess emission we have identified 66 Class ii YSOs in the region. A comparison with the optical data within 1^′′ matching radius yields optical counterparts for 46 YSOs. The optical counterparts of Class ii YSOs are also plotted as star symbols in Fig. 3. The locations of the YSOs in the CMD indicate that a majority of them are younger than 2 Myr with an upper age limit of 7 Myr, confirming the youth of this region. Thus, based on the distribution of the YSOs and most of the members in the optical CMD, we define an upper age limit of 7 Myr for stars associated with Sh 2-170. These are shown with red symbols in Fig. 3.

Differential photometry have been used to identify variables in $18^{\prime}.4\times 18^{\prime}.4$ FOV of the Sh 2-170 region. This procedure automatically cancels out the other interfering effects such as changes in photon counts due to sky variation, effects of airmass, instrumental signatures etc. Briefly, we divided all the stars in the frame in different magnitude bins (e.g., 10-11, 11-12, 12-13 mag and so on), and in each magnitude bin we identified a comparison star to generate differential LCs of our target sources. For this we created all the possible pairs of stars in a particular magnitude bin and calculated the difference between the magnitudes of the pair stars for all the observed frames. One of the star of the pair having lowest standard deviation for the difference in magnitude has been selected as the comparison star for that magnitude bin. For better accuracy, we have used only those stars which are having photometric error better than 0.1 mag in $I_{c}$ band. We have also avoided those stars which are located in the nebulosity, near the bright star or at the corners/edges of the CCDs.

The $I_{c}$ band differential magnitude ($\Delta m$) in the sense ‘$target-comparison$’ were plotted as a function of Julian date to generate LCs of the target stars and to identify variables. In the first step, we visually checked all the LCs for any variability. Any star which exhibited a systematic visual variation larger than the scatter in comparison star is considered as a variable star. There were some stars which do not show any intra-night variability but exhibit night-night variation. In the second step, we used the Period⁴⁴4http://www.starlink.rl.ac.uk/docs/sun167.htx/sun167.html software based on the Lomb-Scargle (LS) periodogram (Lomb, 1976; Scargle, 1982) to determine the periods of all the stars and phase-folded the LCs to identify the periodic variables visually. The LS method is effective even in the case of unevenly sampled data sets which are common in a majority of the astronomical observations. To check whether the data gaps could produce any false periodicity, we performed LS periodogram analysis on the periodic star after randomizing its amplitude by using linux command-line utility i.e., shuf⁵⁵5http://www.gnu.org/software/coreutils/shuf. None of the power spectra of randomized LC of identified periodic variables show any periodic signature. As an example, power spectra of the periodic variable V56 and its randomized LC are shown in Fig. 4. The maximum power is found at 0.173 day for this periodic variable whereas no signature of periodicity is visible in the randomized LC. We have also verified the periods using the NASA Exoplanet Archive Periodogram service⁶⁶6https://exoplanetarchive.ipac.caltech.edu/cgi-bin/Pgram/nph-pgram and PERIOD04⁷⁷7http://www.univie.ac.at/tops/Period04 (Lenz & Breger, 2005). The periods obtained using these programs generally matched well. Once the periodic variables were identified, the remaining LCs were once more checked visually to identify non-periodic variables.

The RMS dispersion of the magnitudes as a function of mean instrumental magnitude for all the target stars is shown in Fig. 5. As expected the dispersion increases towards the fainter end. Identified variables are shown with filled (periodic) and open (non-periodic) circles. Some of the stars with a very high rms in Fig 5 were not designated as variables due to unusual high photometric errors (possible reason may be bad pixels, bright background of the nearby star, residual from the cosmic corrections, etc) as compared to stars in the same magnitude bin or large deviation in magnitude estimation. These stars were further checked visually to ascertain the variability.

We identified 47 and 24 stars as periodic and non-periodic variables. The identification number, coordinates, period and other parameters of the identified variables are listed in Table LABEL:Variables. The periods and amplitude of the variables range between 4 hrs and 18 days and from 0.1 to 2.0 magnitude, respectively. The optical $V/(V-I_{c})$ CMD as discussed in Section 3.3 will further be used to classify the identified variables as PMS members or field stars (cf. next section). Therefore, the LCs of most of the variables (66) will be discussed in ensuing sections except for five having no $V$ band detection, i.e., V41, V42, V66, V68 and V71. The LCs of these variables are shown in Appendix A. For the two variables (V26 and V43; cf. Table LABEL:Variables) without $V$ band photometry, we transformed their available $SDSS$ magnitudes (g, r, i) to $V$ and $I_{c}$ magnitudes using the available transformation equations⁸⁸8https://www.sdss.org/dr12/algorithms/sdssubvritransform/.

The association of the 71 identified variables with Sh 2-170 has been checked on the basis of their membership probability (cf. Section 3.2), the position in the CMD (cf. Section 3.5.1), and the presence or absence of excess IR emission (cf. Section 3.3), and the details about their classification and nature of variability are given in Table LABEL:variability_nature. Periodic/non-periodic nature of the variables are also mentioned in the table. Five of the identified variables could not be classified due to non-availability of their $V$ band data. Out of the remaining 66 variables, 44 (66%) exhibit periodicity. Forty nine variables are found to be PMS sources, of which 29 (59%) variables are periodic. Most of the PMS variables have age and mass in the range of $\sim$ 0.1 - 2 Myr and $\sim$ 0.2 - 3 M_⊙, indicating that they are most probably T-Tauri stars. Seventeen of them exhibit excess IR emission and are classified as Class ii sources (cf. Section 3.3). The remaining 32 PMS variable stars having either insignificant or no IR-excess, may belong to the Class iii category. Seventeen stars are found to be MS/field stars and 15 (88%) of them have periodic LCs.

PMS stars are supposed to release their angular momentum through different mechanisms as they make the transition from the protostellar state down to MS. Disc-locking is the most commonly invoked mechanism to regulate the angular momentum (see e.g., Herbst et al., 2000). Herbst et al. (2000); Herbst et al. (2002) have found a bimodal period distribution around 1 and 8 days for PMS variables with M > 0.25 M_⊙ in Orion Nebula Cluster (ONC). Several studies (e.g., Herbst et al., 2000) explained this bimodal distribution through disc-locking mechanism. If a star is disc-locked further increase in its rotation speed due to contraction would be slowed down and a significant number of star will end up having similar rotation periods. After released from disc-locking, the stars will again increase their rotation speed.

Twenty nine of our PMS variable stars show periodicity in their LCs. In Fig. 12 and Fig. 13, we have plotted the phase folded LC of Class ii (10) and Class iii (19) periodic variables, respectively. In general the periods and amplitudes of Class ii variables are larger than Class iii variables. All the Class iii variables have period $<$ 6 days. As the variability signatures in WTTSs are dominated by the asymmetric distribution of cool spots on the stellar surface, 19 Class iii sources showing periodicity in the range of $\sim$4 hrs to 6 days with amplitudes of $\sim$0.15-0.68 mag (with an exception of V32 having an amplitude of 1.38 mag) are classified as WTTSs (see also, Bhardwaj et al., 2019). We show the period distribution for all 29 periodic PMS stars in the upper panel of Fig. 14. The distribution is bimodal with peaks near $\sim$1.5 and $\sim$4.5 days. We further subdivided this sample into Class ii and Class iii sources and their period distributions are also shown in the lower panel of Fig. 14. The bimodality is more clearly visible for Class iii sources, whereas the period distribution for the Class ii sources is more or less flat. Here we would like to mention that uneven sampling and lack of observations for longer periods may be the reason for the small number of relatively slow rotators. Lamm et al. (2005) have also found a similar bimodal period distribution for variables having $(R_{c}-I_{c})$ $>$ 1.3 (mass $\sim$0.25 M_⊙) with peaks at 1 and 4 days in NGC 2264. In the case of IC 348, Littlefair et al. (2005) have found a bimodal period distribution around 3 and 8 days similar to that seen in ONC for the stars having masses $>$ 0.25 M_⊙, but have reported a statistically significant lack of rapidly rotating stars with respect to the ONC. They also concluded that in spite of the similarity in the ages and disc fractions between NGC 2264 and IC 348, the marked difference in the period distributions between the two clusters presents a serious challenge to the disc-locking paradigm, as it provides no explanation for the difference. However, differences in the cluster environments and the physical conditions leading to the star formation can lead to these observed differences in the period distribution (see also, Littlefair et al., 2005).

Here, we would also like to mention the peculiar LC of V34 as shown in Fig. 15. In addition to the inter-night variability it also shows small intra-night variability. In the same night (within few hours) we see a small change in brightness in a periodic fashion. Multiple physical phenomena such as spot modulation, inner disc co-rotation and extinction events could be responsible for this.

An increase in accretion rate from the circumstellar disc onto the star can give rise to a significant burst in magnitude (few mags) that lasts over hours to days (Cody & Hillenbrand, 2018). Similarly, variable circumstellar extinction can cause brightening/fading events of a few tenths of magnitude. They are of short phenomena of up to a few hours typically (for e.g., AA Tau, Guo et al., 2018).

Twenty of our PMS variable stars do not show periodicity in their LCs. In Fig. 16 and Fig. 17, we have plotted the LCs of Class ii (7) and Class iii (13) non-periodic variables, respectively. Their amplitudes vary between 0.1 - 2 mag. The Class ii variables show significantly large magnitude variations as compared to the Class iii variables. Also, their variability features include either single or multiple fading/brightening events that last for different time-spans.These seven Class ii variables are therefore classified as CTTSs (see also, Bhardwaj et al., 2019). The photometric variations of these sources are found to be in between $I_{c}$$\sim$0.5 mag to $I_{c}$$\sim$2.0 mag. The source V35 shows a huge variation of 2 mag . On $11^{th}$ Nov 2016 it showed a decrease in magnitude from its semi-stable maximum. Again on $17^{th}$ Oct 2017 it showed a larger decrease, and with a few fluctuations reached to its stable maximum brightness around $24^{th}$ Nov 2017. The source V27 is almost stable in most part of its LC with little fluctuation. After $18^{th}$ Oct 2017 it started fluctuating with a maximum magnitude decrease of around 1 mag on $21^{th}$ Nov 2017. In the case of V14 and V36 we see significant intra-night fading and brightening events in few nights. We don’t see any intra-night variation in V19, but inter-night variations are present in night to night with a maximum variation of 0.87 mag. The characteristics of variability found in these sources are typical of CTTSs.

To check the dependence of the variation of the PMS variables on their evolutionary status, in Fig. 18 we plot the normalized cumulative amplitude distribution of Class ii and Class iii sources. This distribution clearly indicates that Class ii sources exhibit larger amplitude variations as compared to Class iii sources with a 96% confidence level as calculated by the Kolmogorov–Smirnov test. Similar results have been reported by Lata et al. (2011, 2012) and Bhardwaj et al. (2019). To further investigate how the period of variability evolves with the age and how the mass of PMS stars influence their periods, we plot the period of the PMS variables as a function of age and mass in the left and right panels of Fig. 19, respectively. The left panel of Fig. 19 indicates that stars with periods up to $\sim$6 days are uniformly distributed for the entire range of ages of the PMS sources, whereas all the PMS variables having periods $>$ 6 days are younger than 2 Myr. Lata et al. (2014); Lata et al. (2016) have also found a similar result that PMS stars with age $\geqslant$ 3 Myr seem to be relatively fast rotators. The right panel of Fig. 19 displays the dependence of period on the stellar mass. Although the sample is small, but we can see a trend in the distribution: out of 5 slow rotators (period $>$ 6 days), 4 have masses less than 1 M_⊙. and variables with periods $<$ 6 days have masses $\leqslant$ 2 M_⊙, whereas out of 3 high mass variables (M $>$ 3 M_⊙), 2 have periods $\sim$1 days. A similar trend has been reported by Lata et al. (2014); Lata et al. (2016). Littlefair et al. (2005) have also found a strong correlation between stellar mass and rotation rate in the case of IC 348. They concluded that the strong mass dependence of rotation rate seen in ONC (Herbst et al., 2000) may well be a common feature of young stellar populations.

Fig. 20 plots amplitude of variability as a function of age and mass of the PMS variables, which roughly indicates that their amplitudes decrease with the increase in mass and age. This is similar to the previous findings (Herbst et al., 2000; Herbst et al., 2002; Littlefair et al., 2005; Lata et al., 2014; Lata et al., 2016). The amplitude decrease with age could be due to the dispersal of the disc. The present result further supports that of our previous studies (Lata et al., 2011, 2012, 2016) that the disc dispersal mechanism is less efficient for relatively low mass stars and that a significant amount of discs is dispersed by $\sim$ 5 Myr. This is also in accordance with the result obtained by Haisch et al. (2001).

To understand the influence of circumstellar discs on the period and amplitude of young stars, we have to first identify a suitable disc indicator. Various disc indicators, such as equivalent widths of $H\alpha$ emission line and Ca II triplet lines, $\Delta(H-K_{s})$ and $\Delta(I_{c}-K_{s})$ indices, disc fraction, etc., have been used in the previous studies (e.g., Herbst et al., 2000; Littlefair et al., 2005; Cieza & Baliber, 2007). Since $I_{c}$ band fluxes originate solely from the photosphere unlike $K_{s}$ band flux which originates in circumstellar disc emission as well as photospheric emission, this provides $(I_{c}-K_{s})$ index with a longer wavelength base compared to other NIR indices (e.g. $(J-H)$ or $(H-K_{s})$). This index is expressed as (cf. Hillenbrand et al., 1998):

where $(I_{c}-K_{s})_{obs}$ is the observed colour of the star, $(I_{c}-K_{s})_{0}$ is the intrinsic colour and $A_{I_{c}}$ and $A_{K_{s}}$ are the interstellar extinction in the $I_{c}$ and $K_{s}$ bands, respectively. We have used the $A_{V}$ value ($2.6\pm 0.3$ mag) as determined by using $Q$ parameter (Johnson & Morgan, 1953) to correct the stars for their extinction. Intrinsic colour $(I-K)_{0}$ of the variables were taken from the PMS isochrones of Siess et al. (2000) according to the age and mass as derived by their CMD and then corrected for the distance and extinction.

Since $\Delta(I_{c}-K_{s})$ is sensitive to the inner part of the disc, we also used the available MIR data of these variables to compute an MIR index $[[3.6]-[4.5]]$, which is a better indicator of the presence of disc and its evolution (Lada et al. 2000; Rodríguez-Ledesma et al. 2010). Since the presence of disc also induces the accretion activity, we have also used the information such as the disc accretion rate and the disc mass obtained from the SED fitting tool (cf. Sec 3.5.2). The upper panels of Fig. 21 show the variation of amplitude as a function of $\Delta(I_{c}-K_{s})$ and $[[3.6]-[4.5]]$. A trend can be clearly seen in the sense that the larger values of the disc indicators, i.e., $\Delta(I_{c}-K_{s})$ or $[[3.6]-[4.5]]$ correspond to relatively larger amplitude variations. The $[[3.6]-[4.5]]$ versus amplitude diagram manifests that Class ii sources have active circumstellar discs as compared to Class iii sources, hence these sources display more active and dynamic variability due to the accretion activities. In the case of NGC 2282, Dutta et al. (2018) have also reported a positive correlation between RMS amplitudes and $\Delta(I_{c}-K_{s})$/$[[3.6]-[4.5]]$ disc indicators. The lower panels of Fig. 21 show amplitudes as a function of the disc accretion rate and disc mass, which indicates that higher disc accretion activity ($>10^{-8}$ M${}_{\odot}~{}yr^{-1}$) induces higher amplitude variation. The amplitude variation also depends on the mass of the stellar disc in the sense that discs with mass $\geqslant$ $10^{-3}$ M_⊙ show relatively larger amplitude variation.

The extensive study of rotation rates in ONC carried out by Herbst et al. (2000); Herbst et al. (2002) reveals a strong correlation between rotation period and infrared excess, suggesting that the observed rotation period distribution could be due to the disc-locking mechanism. In Fig. 22 (upper panels) we plot rotation period as a function of the $\Delta(I_{c}-K_{s})$ and $[[3.6]-[4.5]]$ disc indicators. The $[[3.6]-[4.5]]$ MIR disc indicator clearly suggests that all the five slow rotators (having periods $>$ 6 days) are Class ii sources having higher value of $[[3.6]-[4.5]]$ index ($\sim$0.4 mag), whereas Class iii sources with $[[3.6]-[4.5]]$ $\sim$0.0 mag have periods $\leqslant$ 6 days. This result indicates a reasonable likelihood that the presence of disc affects the rotation of the star. Similar results were reported by Edwards et al. (1993) and the physical interpretation proposed by the authors was that the discs slows the rotation of stars through magnetic interaction (Koenigl, 1991; Ostriker & Shu, 1995). Hence, stars having discs either locked or recently released tends to be slower rotators than stars which already dispersed their disc. However, Littlefair et al. (2005) did not find any correlation between the $H\alpha$ equivalent width or $(K_{s}-L)$ excess and the rotation period of the stars. They cautioned that the correlation between rotation period and infrared excess expressed by $(I_{c}-K_{s})$ index can also arise due to the strong dependence of $(I_{c}-K_{s})$ on stellar mass in the sense that high-mass stars have larger $(I_{c}-K_{s})$ values as compared to low-mass stars (Hillenbrand et al., 1998). To check the dependence of $\Delta(I_{c}-K_{s})$ and $[[3.6]-[4.5]]$ NIR/ MIR excess on stellar mass, we plot these parameters in the bottom panels of Fig. 22. The plots indicate a weak correlation between NIR/ MIR excess with stellar mass in the sense that higher values of NIR/MIR excess are associated with relatively low mass stars. Hence relatively low mass stars are preferably slow rotators.

Cieza & Baliber (2007) reported a clear increase of the disc fraction with period of stars in ONC and NGC 2264. They showed that the long-period peak (P $\sim$ 8 days) of the bimodal distribution observed in ONC is dominated by the population of stars possessing a disc, while the short-period peak (P $\sim$ 2 days) is dominated by the discless population and concluded that this is a strong evidence that the star-disc interaction regulates the angular momentum of young stars.

Fig. 23 shows the effect of disc accretion rate and stellar disc mass on the period. It is evident from this figure that both the disc accretion rate and stellar disc mass affects the stellar rotation in the sense that slow rotators are highly accreting stars (disc accretion rate $\gtrsim$ 5$\times 10^{-9}$ M${}_{\odot}~{}yr^{-1}$ ) and have higher stellar disc mass ($\gtrsim$ 3 $\times 10^{-4}$ M_⊙ ).