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Summer Research Fellowship Programme of India's Science Academies

Study of open star clusters

Anjana M

First Year MSc. Physics Student, School of Physics, University of Hyderabad, Gachibowli, Hyderabad 400056

Prof. Annapurni Subramaniam

Indian Institute of Astrophysics, 2nd Bloack, Koramangala, Bengaluru 560034

Abstract

My work here is the study of Open Star Clusters. Star clusters, as the name suggests are very large group of stars that are gravitationally bound. They are at the same distance from earth and are of the same age. By analysing them we can find information about the evolution of different type of stars and clusters. There can be two type of clusters; Open star clusters and Globular star clusters. On comparison with Globular star clusters, Open star clusters are gravitationally loosely bound and young, mainly consisting of blue stars. Open clusters will often disrupt due to the gravitational influence of nearby giant molecular cloud even before the stars in it die. It will be having a few hundreds of stars and they are confined to the galactic planes, in the spiral arms of the galaxies. Parallax estimating precise distance to the open clusters can be helpful as a reference in various distance measurements methods. They can be helpful to precisely calculate the distance to objects further away. In this project I am using the data from Gaia DR2, which comprises of 46 open star clusters. I find out the turn off mass and age of 30 clusters by fitting isochrone with their CMD in Topcat software. I find the distance to these clusters using two methods; Parallax method and D.M. methods. I also estimate the fraction of binary stars in a cluster, equal mass and unequal mass, in 10 clusters with the help of Python, Excel and Topcat. I also find the fragmentation mass of them.

Keywords: CMD, parallax, Turn-off mass, D.M., Isochrone

Abbreviations

Abbreviations
 H.R.D. Hertzsprung Russel Diagram
C.M.D. Colour Magnitude Diagram
 D.M. Distance Modulus
 DR2Data Release2 
 pcParsec 
 masMilli arc second 
 myrMillion Year 
bpBlue Photometry 
 rpRed Photometry 
 gpGreen Photometry 
 pParallax 
 dDistance 
 σStandered Deviation 

INTRODUCTION

Background

In astronomy, measuring distance, mass, temperature, age, etc are very important. What we observe is the brightness of different astronomical objects, and we have to find all other properties of that object from its magnitude. Finding out the properties of nearby object can serve as a reference to the measurements and understanding of far away objects.

Binary stars are different from single stars in their evolution and properties and can serve as a valuable tool for cosmology.

In this project I am analysing and finding different parameters of open star clusters; age of the cluster, distance to the cluster, Turn off mass of the cluster. I am trying to find the exact distance to the clusters by parallax method and D.M. method. I will analyse how the distance will affect the data. I will formulate a python code to find the binary star fraction in a cluster, equal and unequal mass binary, in different portions of the main sequence, and will find the mass at which equal mass and unequal mass binary proportion drastically changes.

Turn off mass of a star, its age and its present mass can give many valuable information about the composition of the star, and evolution of the cluster. The verified observations can give us clues about new systems; events that occurred in the past, and those to be expected in the future. Distance measurement is very important as it is a valuable tool in many astronomical analyses. Accurately measured distances can serve as a reference to estimate distance to far away objects using Cosmic Ladder method, Fixed Candle method, etc. among others. It is further useful to derive other parameters of that system from their relationship with distance. How distance affects the data is essential to understand how to process raw data. Binary stars have a different evolution than single stars. There are equal and unequal mass binaries, the study of which again gives us more information about the evolution of the cluster.

Objectives of the Research

  • Find the Turn off mass of a cluster.
  • Find the age of a cluster.
  • Find the distance to the cluster.
  • Find how distance affects the data.
  • Find the fraction of equal mass and unequal mass binaries in a cluster.
  • Find the mass at which equal mass binary to unequal mass binary proportion drastically changes.

Some Basic Principles & Formula Used

Hertzsprung-Russel diagram

​It is a scatter plot of stars showing the relationship between the stars absolute magnitude or Luminosity versus their stellar classification or absolute temperature. It is an aid in stellar evolution studies. The related Colour magnitude Diagram (CMD) plots the apparent magnitude of stars against their color, usually for a cluster so that the stars are all at the same distance. As the star evolve it changes its position in the HRD.

HRD OF M55.jpg
    HRD of M55
    stellar evolution_1.png
      Flowchart showing stellar evolution

      Magnitude of stars

      Magnitude refers to the brightness of a star. There are mainly 3 magnitude scales; Apparent Magnitude, Absolute Magnitude and Bolometric Magnitude. Apparent Magnitude ( mm) measures the magnitude of a star as seen from earth with respect to the brightness of star Vega, which is taken as the 0 mag point in this scale. Absolute Magnitude ( MM) is the magnitude of the star if it was at a distance of 10pc from Earth. Previously mentioned scales look at a star at any particular wavelength range depending on the magnitude systems they are following. It can give biased results when a particular star emits more radiation in a particular frequency alone, which can be due to the surface temperature of the star among other reasons. In Bolometric Magnitude (mbol)we observe the star in many wavelength in order to assign an exact magnitude. If a star is at a distance dd from earth, then​

      M=m5log(d/10pc)M=m-5log(d/10pc)

      where, 5log(d/10pc)-5log(d/10pc)​ is called the distance modulus, which is the difference between MMand m.​

      Colour of stars

      Colour of star can tell us about the surface temperature of the stars, abundant materials in the star, mass, evolutionary phase of the star,etc. Considering all these parameters the stars are classified into different spectral classes.

      spectral classification.jpg
        Spectral Classification of stars

        Reddening & Extinction

        As electromagnetic radiation travels from a distant emitter to an observer on earth, it can be absorbed and scattered by the interstellar dust and gas particles on its way. This causes extinction, ( AλA_λ) of the data. The size of these dust causes the blue side of the spectrum to be absorbed more, causing the data to have a greater proportion of red part of the spectrum. This phenomenon is called Reddening, denoted by E(BV)E(B-V). We have to correct the data for its extinction when we fit it with the theoretical Isochrone.

        Aλ=E(BV)(Aλ/(E(BV)))A_λ= E(B-V) * (A_λ/(E(B-V)))

        where Aλ/(E(BV))A_λ/(E(B-V))is known as the extinction coefficient, which only depends on the wavelength of observation.​

        Parallax

        Parallax is the difference in the apparent position of an object viewed along two different lines of sight with respect to some fixed point in space. In astronomy its a common practice to measure the large distances to stars using trigonometric parallax, with the help of distant fixed stars in the background. We observe the star 6 months apart when earth is at the two extremums of its orbit, then we will get the diameter of the earth's orbit around the sun as the baseline for this method.

        p=1/dp=1/d

        where pp is the parallax in arcsecond and dd is the distance in parsecond.

        1pc=3.26lightyears   =3.09x1016m

        star.jpg
          Stellar Parallx

          Turn off mass of the cluster

          Turn of mass of a cluster is the mass at which the stars in the cluster just deviates from main sequence and goes to the giant phase. The stars in the turn off point have just quit hydrogen burning and proton-proton cycle, and are going to helium burning and CNO cycle, which is an essential phase change in stellar evolution.

          Binary Stars

          The binary stars can be of two types; Equal mass binary - when two stars of the binary system are having the same mass, and Unequal mass binary- where two stars in a binary systems are not having equal mass. When equal mass binary system consists of high mass stars, the stars may fragment to smaller mass ones. This fragmentation has a higher probability to lead to unequal binaries. Which is one among the reasons of fragmentation, and following unequal mass binary formation.

          The magnitude( mm) of a star is given in terms of its intensity II by

          m=2.5logIm=-2.5logI

          Suppose we have two stars of equal mass binaries each with apparent magnitude m1m_1, intensity I1I_1. With the aid of observation we won't be able to differentiate between the two stars in a binary system, as they are close enough for us to differentiate. The system will appear as a single star of higher brightness. The new intensity and magnitude are respectively;

          I=2I1I=2I_1

          m=2.5log(I1).75m=-2.5log(I_1)-.75

          ​Irrespective of the magnitude of individual stars, equal mass binary stars will have a magnitude which is less by .75 than their individual magnitude.

          For Unequal mass binary, when we take m1m_1 as the magnitude of brighter star;

          I<2I1I<2I_1

          m1>m>m1.75m_1>m>m_1-.75

          In the CMD of a cluster, the equal binary stars will appear as a parallel curve over the main sequence at a distance of .75 along the magnitude axis. It will be less dense in the upper portion of the main sequence due to many reasons, fragmentation being one among them. Unequal mass binaries lie in the gap between single stars and equal mass binaries. As mentioned earlier, equal mass binary system with higher mass has the tendency to fragment and become unequal mass binary. This causes a non interrupted parallel curve in the upper/massive portion of the spectrum.

          Binary2447.png
            Ngc2447 with Equal mass binary star line marked. its seen that the line gets less denser in the upper part of the curve corresponding to higher mass part of the curve.

            GAIA

            Gaia is a space observatory of the European Space Agency, launched in 2013 and expected to operate until 2022. The spacecraft is designed for astrometry, measuring the positions, distances and motions of stars with unprecedented precision. The mission aims to construct by far the largest and most precise 3D space catalog ever made, totalling approximately 1 billion astronomical objects, mainly stars, but also planets, comets, asteroids and quasars among others. Its a successor of Hipparcos, and it orbits at the L2L_2position of the Sun- Earth system. It can measure upto 10 micro arcsecond.

            gaia.jpg
              Artist's impresssion of Gaia Spacecraft

              Stellar isochrone

              It is a theoretical curve on the HRD, representing a population of stars of the same age. Isochrones can be used to date open clusters because their members all have roughly the same age. If the initial mass function of the open cluster is known, isochrones can be calculated at any age by taking every star in the initial population, using numerical simulations to evolve it forward to the desired age, and plotting the star's luminosity and magnitude on the HR diagram.

              Topcat

              Topcat is an interactive graphical viewer and editor for tabular data. It aim to provide most of the facilities that astronomers need for analysis and manipulation of source catalogs and other tables, though it can be used for non-astronomical data as well. It understands a number of different astronomically important formats like FITS, VOTable & CDF and many more can be added. *reference 1,2,10 are used.

              METHODOLOGY

              Data Collection & Analysis

              • Data was taken from Gaia DR2. 22 month charting of the cosmos by Gaia has given the most vast stellar data till date. I analyse the data of 30 open star clusters. I also used the Isochrone data obtained from Gaia library. In the obtained Gaia data, magnitude was measured using green photometry. Extinction Coefficient was calculated using York Extinction Solver, for the relevant wavelength under study. Reddening value was taken from different journals, astrophysics catalogues and astrophysics data systems.
              • The vast data set was analysed with the help of softwares-Topcat, Excel, and by employing Python programming in Spyder. The details of which are summarised below.
              • Data is plotted in Topcat using plane plot option with bp-rp along X-axis and Gmag along inverted Y-axes. The correction for extinction and absolute magnitude are given in respective axes. The isochrone is also plotted in the same plot. By comparing the two plots, age, turn off mass and the phase of the cluster is found out.
              • D.M. method: The correction in Gmag is equated to D.M., and the corresponding distance is found out. Here I tried to give a range to the D.M. value that kept the CMD fitted with the isochrone. From it the range/error in distance is found out. The effect of distance on data points is also analysed.
              • Parallax Method: The distance can be found out from parallax. The positive and negative error in parallax won't affect the distance symmetrically as they have reciprocal relation. So in order to omit the largely deviated data, both positive and negative, I gave error limits in parallax error; like 5%, 10%, 20% . Using the bar plot option in Topcat, I found the mean parallax and standard deviation in parallax for each error limit. And from them I found the distance to the cluster with possible error/range and compared it with the distance obtained by D.M. method. The effect of distance on data is also analysed after repeating the process for 30 clusters..
              • Isochrone fitted CMD of clusters are analysed and the equal mass binary line is found to be present in some. Then I made a Python program to get the equal mass and unequal mass binary fraction of the cluster, with the help of Excel and Topcat softwares. The program divides the main sequence of the CMD, along the colour axis into a given no of subarrays with a given no. of data points in each subarray, say 2 points in a subarray. It then finds out the point that has Gmag value 0.75 less than the maximum Gmag in that subarray, with a small range given to account for the width of the main sequence. Adding these points of every subarray gives the total equal mass binary stars in the cluster. Parallely, the points between main sequence and equal mass binary is found out to give the total unequal mass binary stars in the cluster.
              • M1,M2,M3,MM_1,M_2,M_3,Mare identified in the main sequence with reference to isochrone. Where M2M_2is the mass at which fragmentation of binary stars happen. M1andM3M_1and M_3denote the masses at both ends of the main sequence top end and bottom end respectively. At times, due to the spreading of data points at the low end of the main sequence, which is also the lower mass end, the fraction analysis is only carried upto a certain mass, which is also denoted as M3M3 of that case.

              RESULTS AND DISCUSSION

              CMD Fitting & Analysis

              Isochrone fitting of different Clusters

              Using calculated extinction and D.M. the CMD of different clusters are fitted with the theoretical isochrone, images of which are given below.

              comesber dm.png
              Comesber
                hyades dm.png
                Hyades
                  pliades dm.png
                  Pleiades
                    preasepe dm.png
                    Praesepe
                      alphaper dm.png
                      Alpha Per
                        trump10 dm.png
                        Trump10
                          stock2 dm.png
                          Stock2
                            Call140 dm.png
                            Call140
                              BGC 6475 dm.png
                              BGC6475
                                IC 2391 dm.png
                                IC2391
                                  IC 2602 dm.png
                                  IC2602
                                    IC 4665 dm.png
                                    IC4665
                                      IC blanco1 dm.png
                                      Blanco1
                                        NGC 2232 dm.png
                                        NGC2232
                                          NGC 2437 8.6 dm.png
                                          NGC2437
                                            NGC 6633 dm.png
                                            NGC6633
                                              NGC 6774 dm 8.8.png
                                              NGC6774
                                                NGC 6793 8.8 dm.png
                                                NGC6793
                                                  ngc188 dm.png
                                                  NGC188
                                                    ngc2360 9.2dm.png
                                                    NGC2360
                                                      NGC2447 8.8dm.png
                                                      NGC2447
                                                        ngc2516 dm.png
                                                        NGC2516
                                                          NGC7092 dm.png
                                                          NGC7092
                                                            NGC 884 7.2 dm.png
                                                            NGC884
                                                              IC 4651 dm2.png
                                                              IC4651
                                                                NGC 2451 dm_1.png
                                                                NGC2451
                                                                  a,  Isochrone fitted CMD of different Clusters
                                                                  NGC 2682 9.6 dm2.png
                                                                  NGC2682
                                                                    NGC 2323 dm 7_1.png
                                                                    NGC2323
                                                                      NGC 869 7.2 dm.png
                                                                      NGC869
                                                                         Isochrone fitted CMD of different Clusters(cont)

                                                                        *reference 3-10 are used for data.

                                                                        Results of Analysing CMD for Distance, Turn-off mass & Age

                                                                        After fitting CMD with Isochrones for different clusters, using Topcat, I could find their log age and Turn off mass of it. Finding the possible range of D.M. that can fit the CMD to the Isochrone, I could find the mean distance to the cluster along with the possible error in it. The results are tabulated below.

                                                                        Analysis of Isochrone fitted CMD of different clusters for Age, Turn off mass & Distance
                                                                        ClusterD.M.d(pc)D.M.1d1(pc)D.M.2d2(pc)Δ d1(pc)Δ d2(pc)log ageAge(myr)Turn off mass(solar mass)Comment
                                                                        Hyades-3.550.12-3.447.86-3.652.48-2.262.36910001.9at turnoff
                                                                        Comesber-4.683.18-4.579.43-4.787.1-3.743.928.8630.962.2passed
                                                                        Pleiades-5.5125.89-5.3114.82-5.7138.04-11.0812.158.6398.112.7at turnoff
                                                                        IC2391-6.3181.97-6.2173.78-6.4190.55-8.198.587.639.816.8approaching turn off
                                                                        IC2602-6158.49-6.2173.78-5.8144.54-13.9515.297.639.816.8at turn off
                                                                        Alphaper-6.4190.55-6.3181.97-6.6208.93-8.5818.387.863.16.2Almost reached
                                                                        Preasepe-6.1165.96-6158.49-6.2173.78-7.477.82910001.9passed
                                                                        NGC 2451-6.6208.93-6.5199.53-6.7218.78-9.409.857.639.816.8approaching turn off
                                                                        Blanco 1-7.2275.42-7251.19-7.4302.0-24.2326.577.863.15.4approaching turn off
                                                                        BGC 6475-7.6331.13-7.5316.23-7.7346.74-14.9015.618.4251.193Just passed turn off
                                                                        NGC6774-8.1416.87-8398.11-8.2436.52-18.7619.658.8630.962.2passed
                                                                        NGC2232-8.1416.87-7.9380.17-8.3457.09-36.6840.227.425.128.7Just near turnoff
                                                                        IC4665-8.1416.87-8398.11-8.2436.52-18.7619.657.639.816.8Just near turnoff
                                                                        NGC7092-7.9380.17-7.8363.08-8398.11-17.1117.928.2158.493.6At turn off
                                                                        Stock2-7.9380.17-7.7346.74-8.1416.87-33.4536.688.2158.493.6Haven’t reached
                                                                        NGC6633-8.8575.44-8.7549.54-8.9602.56-25.927.128.6398.112.7At turn off
                                                                        CALL140-8398.11-7.9380.19-8.3457.09-17.9258.987.639.816.8At turn off
                                                                        NGC2516-8.4478.63-8.3457.09-8.6524.81-21.5446.188.2158.493.7At turn off
                                                                        Trump10-7.9380.19-7.7346.74-8.1416.87-33.4536.688.2158.493.7At turnoff
                                                                        NGC6793-9.7870.96-9.6831.76-9.8912.01-39.241.058.8630.962.2at turnoff
                                                                        NGC2682-9.7870.96-9.6831.76-9.8912.01-39.241.059.63981.071.2passed
                                                                        IC4651-10.41202.26-10.31148.15-10.51258.93-54.1156.669.21584.891.7reached turn off
                                                                        NGC2323-10.31148.15-10.21202.26-10.41096.48-51.6854.1171012havent reached
                                                                        NGC2447-101000-9.9954.99-10.21096.48-45.0096.488.8630.962.2at turnoff
                                                                        NGC2360-101000-9.9954.99-101000-45.0009.21584.891.6passed
                                                                        NGC2437-11.11659.59-111584.89-11.41905.46-74.69245.878.6398.112.7passed
                                                                        NGC188-11.41905.46-11.31819.70-11.41905.46-85.7609.86309.571.1passed
                                                                        NGC869-13.24365.16-133981.07-13.34570.88-384.09205.727.215.859.7at turnoff
                                                                        NGC884-13.14168.69-12.93801.89-13.34570.88-366.8402.197.215.859.7at turnoff
                                                                        NGC2158-14.26918.31-146309.57-14.57943.28-608.741024.979.21584.891.6passed

                                                                        *reference 3-10 are used for data.

                                                                        Parallax Method of Finding Distance

                                                                        Using plane plot option in Topcat I eliminated the data points with large errors. I gave error limits of 5%,10% and 20% for the calculation. Then using bar plot option in Topcat I plotted the histograms of the data, for each error limit.

                                                                        6793 new histo.png
                                                                          Histogram plotted for NGC6793

                                                                          Using Gaussian fit option in bar plot, I found the mean p and σ . From mean p, I found mean distance, d. To find the error in distance, I formulated p1 and p2, and their respective distances d1 and d2. The results are tabulated below.

                                                                          Distance Measurement from Parallax Method
                                                                          Cluster(p. error)/p(%)Mean p(mas)σ in p(mas)p1=p-σ (mas)p2=p+σ (mas)Mean d=1/p (kpc)d1=1/p1 (kpc)d2=1/p2  (kpc)Δd1=d1-d  (kpc)Δd2=d2-d  (kpc)
                                                                          Hyades518.291.0717.2319.360.050.060.0580.00-0.00
                                                                          1018.280.9417.3519.220.050.060.050.00-0.00
                                                                          2018.320.9117.4119.230.050.060.050.00-0.00
                                                                          Comesber511.620.3911.2312.010.080.090.080.00-0.00
                                                                          Pleiades57.370.267.117.630.140.140.130.01-0.00
                                                                          107.370.287.097.640.140.140.130.01-0.00
                                                                          207.360.297.087.650.140.140.130.01-0.01
                                                                          IC239156.60.266.396.800.150.160.150.00-0.00
                                                                          106.60.226.376.810.150.160.150.01-0.00
                                                                          IC260256.570.226.356.790.150.160.150.01-0.00
                                                                          Alpha Per55.720.195.535.920.170.180.170.01-0.01
                                                                          105.720.25.525.920.170.180.170.01-0.01
                                                                          205.720.25.525.920.170.180.170.01-0.01
                                                                          Preasepe55.370.165.225.520.180.190.180.01-0.01
                                                                          105.370.175.25.540.190.190.180.01-0.01
                                                                          155.370.175.195.540.190.190.180.01-0.01
                                                                          NGC245155.160.155.015.310.190.20.190.01-0.01
                                                                          105.160.155.015.310.190.20.190.01-0.01
                                                                          205.160.155.015.310.190.20.190.01-0.01
                                                                          Blanco154.220.124.104.330.240.240.230.01-0.01
                                                                          104.220.124.14.340.240.240.230.01-0.01
                                                                          204.220.124.14.340.240.240.230.01-0.01
                                                                          BGC647553.570.133.443.70.280.290.270.01-0.01
                                                                          103.570.153.423.710.280.290.270.01-0.01
                                                                          203.570.233.343.800.280.30.260.02-0.02
                                                                          NGC677453.250.113.143.370.310.320.30.01-0.01
                                                                          103.260.143.113.40.310.320.290.01-0.01
                                                                          203.230.262.973.490.310.340.290.03-0.02
                                                                          NGC223253.080.142.943.220.320.340.310.02-0.01
                                                                          103.060.182.883.240.330.350.310.02-0.02
                                                                          203.060.192.873.250.330.350.310.02-0.02
                                                                          IC466552.880.132.753.020.350.360.330.02-0.02
                                                                          102.860.182.683.040.350.370.330.02-0.02
                                                                          202.860.182.683.040.350.340.330.02-0.02
                                                                          NGC709252.640.12.552.740.380.390.370.01-0.01
                                                                          102.670.172.52.840.370.400.350.03-0.02
                                                                          202.660.242.412.90.380.410.340.04-0.03
                                                                          Stock252.640.072.572.710.380.390.370.01-0.01
                                                                          102.660.142.522.80.380.40.360.02-0.02
                                                                          202.670.222.452.890.370.410.350.03-0.03
                                                                          NGC663352.530.082.452.610.40.410.380.01-0.01
                                                                          102.520.122.412.640.40.420.380.02-0.02
                                                                          202.520.132.392.650.40.420.380.02-0.02
                                                                          CALL14052.560.132.442.690.390.410.370.02-0.02
                                                                          102.560.172.392.730.390.420.370.03-0.0
                                                                          202.550.222.332.760.390.430.360.4-0.03
                                                                          NGC251652.410.072.342.480.410.430.400.01-0.01
                                                                          102.420.132.292.550.410.440.390.02-0.02
                                                                          202.410.172.242.570.420.450.390.03-0.03
                                                                          Trump1052.280.092.182.350.440.460.420.02-0.02
                                                                          102.270.152.122.420.440.470.410.03-0.03
                                                                          202.280.242.042.530.440.490.40.05-0.04
                                                                          NGC6793101.680.11.581.770.60.630.560.04-0.03
                                                                          201.700.201.501.900.590.670.530.08-0.06
                                                                          NGC268251.130.051.091.180.880.920.850.04-0.03
                                                                          101.140.061.081.200.880.930.830.05-0.05
                                                                          201.130.051.091.180.880.920.850.04-0.03
                                                                          IC465151.060.051.011.110.940.990.900.04-0.04
                                                                          101.060.061.001.130.9410.890.06-0.05
                                                                          201.060.090.971.150.941.030.870.08-0.07
                                                                          NGC232351.000.040.961.0411.040.960.04-0.04
                                                                          101.000.050.951.051.001.050.960.05-0.05
                                                                          201.000.060.941.051.001.060.950.06-0.05
                                                                          NGC2447100.970.060.911.031.031.10.970.07-0.06
                                                                          200.980.090.891.061.021.120.940.1-0.08
                                                                          NGC2360100.900.060.850.961.111.181.040.07-0.06
                                                                          200.920.10.821.011.091.220.990.13-0.10
                                                                          NGC2437100.610.050.560.661.651.81.50.15-0.13
                                                                          200.610.070.540.691.631.851.450.22-0.17
                                                                          NGC18850.510.030.480.531.982.091.880.11-0.10
                                                                          100.510.040.470.541.972.121.840.15-0.13
                                                                          200.510.050.460.551.972.171.810.2-0.16
                                                                          NGC869100.410.040.370.442.472.712.270.24-0.20
                                                                          200.40.050.350.452.502.872.220.37-0.28
                                                                          NGC884100.410.040.370.442.462.702.260.24-0.20
                                                                          200.410.050.360.462.472.822.190.35-0.27
                                                                          NGC2158200.180.040.130.225.677.564.541.89-1.13

                                                                          Distance found by Parallax method agrees with the one found from D.M. method except for the cluster NGC2158.

                                                                          Binary star fraction

                                                                          I selected the main sequence points using Topcat, arranged it into a list of x and y coordinates in Excel and made a program in Python to calculate the fraction of equal mass binary and unequal mass binary in the cluster. The results are tabulated below.

                                                                          Fraction of Binary Stars
                                                                          Cluster Namefraction of Equal mass Binary Starsfraction of Unequal mass Binary StarsTotal Binary Star FractionUpper limit Mass=M1 (in solar mass)Lower limit Mass=M3 (in solar mass)
                                                                          Preasepe0.020.110.132.20.35
                                                                          Pleiades0.030.150.182.600.16
                                                                          NGC26820.020.20.221.210.6
                                                                          NGC23600.060.240.301.620.4
                                                                          NGC24470.060.270.331.990.35
                                                                          NGC25160.040.140.184.20.54
                                                                          BGC64750.030.20.232.30.3
                                                                          NGC35320.050.200.253.120.55
                                                                          NGC10390.060.180.241.570.45
                                                                          NGC24220.050.250.304.380.58

                                                                          I plotted the equal mass binary and unequal mass binary points that I gathered via python programming. The plot is given below.

                                                                          2422 paintpyplot.png
                                                                            Equal mass and unequal mass binaries of NGC2422 plotted in its CMD

                                                                            For the binary stars I analysed, the total binary fraction is found to be in the range ~15%-30%. And in the total binary fraction the equal mass binary contribution is found to be 9%-25% , and the rest is unequal mass binary.

                                                                            To find the fragmentating mass

                                                                            On checking the CMD, I could find that the equal mass binary lines are getting less dense as we go up the curve, i.e to the higher mass end. One of the possible reasons can be the fragmentation of one of the two stars of equal mass binary leading it into unequal mass binary. I could check it by cutting the curve into two parts at the mass M2M_2, we can name it as the fragmentating mass, point at which the equal mass binary to unequal mass binary proportion considerably changes. And, I tried finding separately, the equal mass and unequal mass binary fractions above and below M2M_2. The results are tabulated below.

                                                                            paint2_1.png
                                                                              M1, M2, M3 marked in the CMD of NGC2422
                                                                              Analysis for  M2M_2
                                                                              Cluster NameIn the upper portion from M2In the lower portion from M2

                                                                              Upper limit Mass=M1

                                                                              (in solar mass)

                                                                              Lower limit Mass=M2

                                                                              (in solar mass)

                                                                              Equal mass binary fractionUnequal mass binary fraction

                                                                              Upper limit Mass=M2

                                                                              (in solar mass)

                                                                              Lower limit Mass=M3

                                                                              (in solar mass)

                                                                              Equal mass binary fractionUnequal mass binary fraction
                                                                              Pleiades2.2.90.010.19.90.350.030.13
                                                                              Preasepe2.601.150.010.191.150.160.060.19
                                                                              NGC35323.121.470.010.21.470.550.030.12
                                                                              NGC23601.621.3500.331.350.40.060.21
                                                                              NGC24224.381.400.010.381.400.580.050.2
                                                                              NGC10391.571.400.020.221.400.450.060.16
                                                                              NGC24471.991.800.331.80.350.040.16
                                                                              NGC26821.211.1500.171.150.60.040.14
                                                                              BGC64752.310.040.1910.30.050.17

                                                                              From the analysis, In the main sequence of a cluster, I could find a mass M2M_2 such that, above this mass the equal mass binary fraction is very less than unequal mass binary compared to the respective fractions below it.

                                                                              CONCLUSION AND RECOMMENDATIONS

                                                                              Conclusion

                                                                              1. Distance, Age and Turn off mass of 30 clusters are calculated.

                                                                              2. Equal mass binary fraction and unequal mass binary fraction of 10 clusters are found out. For the clusters I analysed, Binary stars amount to ~15-30% of the stars in a cluster.

                                                                              3. Mass at which the equal mass binary to unequal mass binary proportion considerably changes is found out for 9 clusters.

                                                                              4. With large distance, especially above 2kpc, the distance calculated from parallax method found to have large error.

                                                                              Recommendations

                                                                              Future research can aim;

                                                                              1. To recognize any trend in the fragmentating mass of different clusters. We can also find how other factors influence the fragmentation of binary stars.

                                                                              2. To formulate a method to find the exact binary star fraction for the cluster, not only for the unspread portion of the main sequence of the cluster.

                                                                              3. We have assumed interstellar dust to be evenly distributed and of homogeneous in nature, which is not necessarily true. If we are to analyse the stars in different wavelengths, we can find the property of interstellar dust distributed in that direction. This knowledge can help us to better manipulate the observed data to find different astrometric parameters.

                                                                              ACKNOWLEDGEMENT

                                                                              I thank IASc-INSA-NASI Summer Research Fellowship 2019 for giving me this oppurtunity to be an IAS-intern. I express my sincere gratitude to Prof. Annapurni Subramaniam for selecting me as her intern, for her invaluable guidance and support. She corrected me whenever necessary and motivated me to go forward. I thank IIA and its administration for providing the needed facilities. I thank Prof. S V S Nageswara Rao (Department of Physics, University of Hyderabad) who recommended me for this internship program. I thank my family for their love and support. I would also like to mention and express my gratitude to the students at IIA who cleared my doubts and helped me all along doing this project. Thank you all.

                                                                              REFERENCES

                                                                              1. 'Introduction to Stellar Astrophysics' by Erika Bohm.

                                                                              2. 'Astrophysics in a Nutshell' by Dan Maoz.

                                                                              3. Astrophysics Data System by NASA, https://ui.adsabs.harvard.edu/

                                                                              4. Webda, https://webda.physics.muni.cz/

                                                                              5. YES:York Extinction Solver-http://www.cdac-ccda.hia.nrc-cnrc.gc.ca/community/YorkExtinctionSolver/output.cgi

                                                                              6. https://arxiv.org/pdf/1804.09378.pdf

                                                                              7. https://ui.adsabs.harvard.edu/abs/2013PASP..125..115D/abstract

                                                                              8. https://en.wikipedia.org/wiki

                                                                              9.http://www.star.bris.ac.uk/~mbt/topcat/

                                                                              10. http://sci.esa.int/gaia/

                                                                              SOURCE

                                                                              fig1:https://in.images.search.yahoo.com/search/images

                                                                              fig2:https://in.images.search.yahoo.com/search/images

                                                                              fig3:https://in.images.search.yahoo.com/search/images

                                                                              fig4:https://in.images.search.yahoo.com/search/images

                                                                              fig6:https://in.images.search.yahoo.com​/search/images

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