# 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 DR2 Data Release2 pc Parsec mas Milli arc second myr Million Year bp Blue Photometry rp Red Photometry gp Green Photometry p Parallax d Distance σ Standered Deviation

## 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.

## 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
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 ( $m$) 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 ( $M$) 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 $\left({{{m}_{b}}_{o}}_{l}\right)$we observe the star in many wavelength in order to assign an exact magnitude. If a star is at a distance $d$ from earth, then​

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

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

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 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, ( $A_λ$) 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 $E(B-V)$. We have to correct the data for its extinction when we fit it with the theoretical Isochrone.

$A_λ= E(B-V) * (A_λ/(E(B-V)))$

where $A_λ/(E(B-V))$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.

$p=1/d$

where $p$ is the parallax in arcsecond and $d$ is the distance in parsecond.

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( $m$) of a star is given in terms of its intensity $I$ by

$m=-2.5logI$

Suppose we have two stars of equal mass binaries each with apparent magnitude $m_1$, intensity $I_1$. 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;

$I=2I_1$

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

​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 $m_1$ as the magnitude of brighter star;

$I<2I_1$

$m_1>m>m_1-.75$

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.

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 $L_2$position of the Sun- Earth system. It can measure upto 10 micro arcsecond.

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.

## 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.
• $M_1,M_2,M_3,M$are identified in the main sequence with reference to isochrone. Where $M_2$is the mass at which fragmentation of binary stars happen. $M_1and M_3$denote 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 $M3$ of that case.

## 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
Praesepe
Alpha Per
Trump10
Stock2
Call140
BGC6475
IC2391
IC2602
IC4665
Blanco1
NGC2232
NGC2437
NGC6633
NGC6774
NGC6793
NGC188
NGC2360
NGC2447
NGC2516
NGC7092
NGC884
IC4651
NGC2451
a,  Isochrone fitted CMD of different Clusters
NGC2682
NGC2323
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
 Cluster D.M. d(pc) D.M.1 d1(pc) D.M.2 d2(pc) Δ d1(pc) Δ d2(pc) log age Age(myr) Turn off mass(solar mass) Comment Hyades -3.5 50.12 -3.4 47.86 -3.6 52.48 -2.26 2.36 9 1000 1.9 at turnoff Comesber -4.6 83.18 -4.5 79.43 -4.7 87.1 -3.74 3.92 8.8 630.96 2.2 passed Pleiades -5.5 125.89 -5.3 114.82 -5.7 138.04 -11.08 12.15 8.6 398.11 2.7 at turnoff IC2391 -6.3 181.97 -6.2 173.78 -6.4 190.55 -8.19 8.58 7.6 39.81 6.8 approaching turn off IC2602 -6 158.49 -6.2 173.78 -5.8 144.54 -13.95 15.29 7.6 39.81 6.8 at turn off Alphaper -6.4 190.55 -6.3 181.97 -6.6 208.93 -8.58 18.38 7.8 63.1 6.2 Almost reached Preasepe -6.1 165.96 -6 158.49 -6.2 173.78 -7.47 7.82 9 1000 1.9 passed NGC 2451 -6.6 208.93 -6.5 199.53 -6.7 218.78 -9.40 9.85 7.6 39.81 6.8 approaching turn off Blanco 1 -7.2 275.42 -7 251.19 -7.4 302.0 -24.23 26.57 7.8 63.1 5.4 approaching turn off BGC 6475 -7.6 331.13 -7.5 316.23 -7.7 346.74 -14.90 15.61 8.4 251.19 3 Just passed turn off NGC6774 -8.1 416.87 -8 398.11 -8.2 436.52 -18.76 19.65 8.8 630.96 2.2 passed NGC2232 -8.1 416.87 -7.9 380.17 -8.3 457.09 -36.68 40.22 7.4 25.12 8.7 Just near turnoff IC4665 -8.1 416.87 -8 398.11 -8.2 436.52 -18.76 19.65 7.6 39.81 6.8 Just near turnoff NGC7092 -7.9 380.17 -7.8 363.08 -8 398.11 -17.11 17.92 8.2 158.49 3.6 At turn off Stock2 -7.9 380.17 -7.7 346.74 -8.1 416.87 -33.45 36.68 8.2 158.49 3.6 Haven’t reached NGC6633 -8.8 575.44 -8.7 549.54 -8.9 602.56 -25.9 27.12 8.6 398.11 2.7 At turn off CALL140 -8 398.11 -7.9 380.19 -8.3 457.09 -17.92 58.98 7.6 39.81 6.8 At turn off NGC2516 -8.4 478.63 -8.3 457.09 -8.6 524.81 -21.54 46.18 8.2 158.49 3.7 At turn off Trump10 -7.9 380.19 -7.7 346.74 -8.1 416.87 -33.45 36.68 8.2 158.49 3.7 At turnoff NGC6793 -9.7 870.96 -9.6 831.76 -9.8 912.01 -39.2 41.05 8.8 630.96 2.2 at turnoff NGC2682 -9.7 870.96 -9.6 831.76 -9.8 912.01 -39.2 41.05 9.6 3981.07 1.2 passed IC4651 -10.4 1202.26 -10.3 1148.15 -10.5 1258.93 -54.11 56.66 9.2 1584.89 1.7 reached turn off NGC2323 -10.3 1148.15 -10.2 1202.26 -10.4 1096.48 -51.68 54.11 7 10 12 havent reached NGC2447 -10 1000 -9.9 954.99 -10.2 1096.48 -45.00 96.48 8.8 630.96 2.2 at turnoff NGC2360 -10 1000 -9.9 954.99 -10 1000 -45.00 0 9.2 1584.89 1.6 passed NGC2437 -11.1 1659.59 -11 1584.89 -11.4 1905.46 -74.69 245.87 8.6 398.11 2.7 passed NGC188 -11.4 1905.46 -11.3 1819.70 -11.4 1905.46 -85.76 0 9.8 6309.57 1.1 passed NGC869 -13.2 4365.16 -13 3981.07 -13.3 4570.88 -384.09 205.72 7.2 15.85 9.7 at turnoff NGC884 -13.1 4168.69 -12.9 3801.89 -13.3 4570.88 -366.8 402.19 7.2 15.85 9.7 at turnoff NGC2158 -14.2 6918.31 -14 6309.57 -14.5 7943.28 -608.74 1024.97 9.2 1584.89 1.6 passed

*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.

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) Hyades 5 18.29 1.07 17.23 19.36 0.05 0.06 0.058 0.00 -0.00 10 18.28 0.94 17.35 19.22 0.05 0.06 0.05 0.00 -0.00 20 18.32 0.91 17.41 19.23 0.05 0.06 0.05 0.00 -0.00 Comesber 5 11.62 0.39 11.23 12.01 0.08 0.09 0.08 0.00 -0.00 Pleiades 5 7.37 0.26 7.11 7.63 0.14 0.14 0.13 0.01 -0.00 10 7.37 0.28 7.09 7.64 0.14 0.14 0.13 0.01 -0.00 20 7.36 0.29 7.08 7.65 0.14 0.14 0.13 0.01 -0.01 IC2391 5 6.6 0.26 6.39 6.80 0.15 0.16 0.15 0.00 -0.00 10 6.6 0.22 6.37 6.81 0.15 0.16 0.15 0.01 -0.00 IC2602 5 6.57 0.22 6.35 6.79 0.15 0.16 0.15 0.01 -0.00 Alpha Per 5 5.72 0.19 5.53 5.92 0.17 0.18 0.17 0.01 -0.01 10 5.72 0.2 5.52 5.92 0.17 0.18 0.17 0.01 -0.01 20 5.72 0.2 5.52 5.92 0.17 0.18 0.17 0.01 -0.01 Preasepe 5 5.37 0.16 5.22 5.52 0.18 0.19 0.18 0.01 -0.01 10 5.37 0.17 5.2 5.54 0.19 0.19 0.18 0.01 -0.01 15 5.37 0.17 5.19 5.54 0.19 0.19 0.18 0.01 -0.01 NGC2451 5 5.16 0.15 5.01 5.31 0.19 0.2 0.19 0.01 -0.01 10 5.16 0.15 5.01 5.31 0.19 0.2 0.19 0.01 -0.01 20 5.16 0.15 5.01 5.31 0.19 0.2 0.19 0.01 -0.01 Blanco1 5 4.22 0.12 4.10 4.33 0.24 0.24 0.23 0.01 -0.01 10 4.22 0.12 4.1 4.34 0.24 0.24 0.23 0.01 -0.01 20 4.22 0.12 4.1 4.34 0.24 0.24 0.23 0.01 -0.01 BGC6475 5 3.57 0.13 3.44 3.7 0.28 0.29 0.27 0.01 -0.01 10 3.57 0.15 3.42 3.71 0.28 0.29 0.27 0.01 -0.01 20 3.57 0.23 3.34 3.80 0.28 0.3 0.26 0.02 -0.02 NGC6774 5 3.25 0.11 3.14 3.37 0.31 0.32 0.3 0.01 -0.01 10 3.26 0.14 3.11 3.4 0.31 0.32 0.29 0.01 -0.01 20 3.23 0.26 2.97 3.49 0.31 0.34 0.29 0.03 -0.02 NGC2232 5 3.08 0.14 2.94 3.22 0.32 0.34 0.31 0.02 -0.01 10 3.06 0.18 2.88 3.24 0.33 0.35 0.31 0.02 -0.02 20 3.06 0.19 2.87 3.25 0.33 0.35 0.31 0.02 -0.02 IC4665 5 2.88 0.13 2.75 3.02 0.35 0.36 0.33 0.02 -0.02 10 2.86 0.18 2.68 3.04 0.35 0.37 0.33 0.02 -0.02 20 2.86 0.18 2.68 3.04 0.35 0.34 0.33 0.02 -0.02 NGC7092 5 2.64 0.1 2.55 2.74 0.38 0.39 0.37 0.01 -0.01 10 2.67 0.17 2.5 2.84 0.37 0.40 0.35 0.03 -0.02 20 2.66 0.24 2.41 2.9 0.38 0.41 0.34 0.04 -0.03 Stock2 5 2.64 0.07 2.57 2.71 0.38 0.39 0.37 0.01 -0.01 10 2.66 0.14 2.52 2.8 0.38 0.4 0.36 0.02 -0.02 20 2.67 0.22 2.45 2.89 0.37 0.41 0.35 0.03 -0.03 NGC6633 5 2.53 0.08 2.45 2.61 0.4 0.41 0.38 0.01 -0.01 10 2.52 0.12 2.41 2.64 0.4 0.42 0.38 0.02 -0.02 20 2.52 0.13 2.39 2.65 0.4 0.42 0.38 0.02 -0.02 CALL140 5 2.56 0.13 2.44 2.69 0.39 0.41 0.37 0.02 -0.02 10 2.56 0.17 2.39 2.73 0.39 0.42 0.37 0.03 -0.0 20 2.55 0.22 2.33 2.76 0.39 0.43 0.36 0.4 -0.03 NGC2516 5 2.41 0.07 2.34 2.48 0.41 0.43 0.40 0.01 -0.01 10 2.42 0.13 2.29 2.55 0.41 0.44 0.39 0.02 -0.02 20 2.41 0.17 2.24 2.57 0.42 0.45 0.39 0.03 -0.03 Trump10 5 2.28 0.09 2.18 2.35 0.44 0.46 0.42 0.02 -0.02 10 2.27 0.15 2.12 2.42 0.44 0.47 0.41 0.03 -0.03 20 2.28 0.24 2.04 2.53 0.44 0.49 0.4 0.05 -0.04 NGC6793 10 1.68 0.1 1.58 1.77 0.6 0.63 0.56 0.04 -0.03 20 1.70 0.20 1.50 1.90 0.59 0.67 0.53 0.08 -0.06 NGC2682 5 1.13 0.05 1.09 1.18 0.88 0.92 0.85 0.04 -0.03 10 1.14 0.06 1.08 1.20 0.88 0.93 0.83 0.05 -0.05 20 1.13 0.05 1.09 1.18 0.88 0.92 0.85 0.04 -0.03 IC4651 5 1.06 0.05 1.01 1.11 0.94 0.99 0.90 0.04 -0.04 10 1.06 0.06 1.00 1.13 0.94 1 0.89 0.06 -0.05 20 1.06 0.09 0.97 1.15 0.94 1.03 0.87 0.08 -0.07 NGC2323 5 1.00 0.04 0.96 1.04 1 1.04 0.96 0.04 -0.04 10 1.00 0.05 0.95 1.05 1.00 1.05 0.96 0.05 -0.05 20 1.00 0.06 0.94 1.05 1.00 1.06 0.95 0.06 -0.05 NGC2447 10 0.97 0.06 0.91 1.03 1.03 1.1 0.97 0.07 -0.06 20 0.98 0.09 0.89 1.06 1.02 1.12 0.94 0.1 -0.08 NGC2360 10 0.90 0.06 0.85 0.96 1.11 1.18 1.04 0.07 -0.06 20 0.92 0.1 0.82 1.01 1.09 1.22 0.99 0.13 -0.10 NGC2437 10 0.61 0.05 0.56 0.66 1.65 1.8 1.5 0.15 -0.13 20 0.61 0.07 0.54 0.69 1.63 1.85 1.45 0.22 -0.17 NGC188 5 0.51 0.03 0.48 0.53 1.98 2.09 1.88 0.11 -0.10 10 0.51 0.04 0.47 0.54 1.97 2.12 1.84 0.15 -0.13 20 0.51 0.05 0.46 0.55 1.97 2.17 1.81 0.2 -0.16 NGC869 10 0.41 0.04 0.37 0.44 2.47 2.71 2.27 0.24 -0.20 20 0.4 0.05 0.35 0.45 2.50 2.87 2.22 0.37 -0.28 NGC884 10 0.41 0.04 0.37 0.44 2.46 2.70 2.26 0.24 -0.20 20 0.41 0.05 0.36 0.46 2.47 2.82 2.19 0.35 -0.27 NGC2158 20 0.18 0.04 0.13 0.22 5.67 7.56 4.54 1.89 -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 Name fraction of Equal mass Binary Stars fraction of Unequal mass Binary Stars Total Binary Star Fraction Upper limit Mass=M1 (in solar mass) Lower limit Mass=M3 (in solar mass) Preasepe 0.02 0.11 0.13 2.2 0.35 Pleiades 0.03 0.15 0.18 2.60 0.16 NGC2682 0.02 0.2 0.22 1.21 0.6 NGC2360 0.06 0.24 0.30 1.62 0.4 NGC2447 0.06 0.27 0.33 1.99 0.35 NGC2516 0.04 0.14 0.18 4.2 0.54 BGC6475 0.03 0.2 0.23 2.3 0.3 NGC3532 0.05 0.20 0.25 3.12 0.55 NGC1039 0.06 0.18 0.24 1.57 0.45 NGC2422 0.05 0.25 0.30 4.38 0.58

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

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 $M_2$, 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 $M_2$. The results are tabulated below.

M1, M2, M3 marked in the CMD of NGC2422
Analysis for  $M_2$
 Cluster Name In the upper portion from M2 In the lower portion from M2 Upper limit Mass=M1(in solar mass) Lower limit Mass=M2(in solar mass) Equal mass binary fraction Unequal mass binary fraction Upper limit Mass=M2(in solar mass) Lower limit Mass=M3(in solar mass) Equal mass binary fraction Unequal mass binary fraction Pleiades 2.2 .9 0.01 0.19 .9 0.35 0.03 0.13 Preasepe 2.60 1.15 0.01 0.19 1.15 0.16 0.06 0.19 NGC3532 3.12 1.47 0.01 0.2 1.47 0.55 0.03 0.12 NGC2360 1.62 1.35 0 0.33 1.35 0.4 0.06 0.21 NGC2422 4.38 1.40 0.01 0.38 1.40 0.58 0.05 0.2 NGC1039 1.57 1.40 0.02 0.22 1.40 0.45 0.06 0.16 NGC2447 1.99 1.8 0 0.33 1.8 0.35 0.04 0.16 NGC2682 1.21 1.15 0 0.17 1.15 0.6 0.04 0.14 BGC6475 2.3 1 0.04 0.19 1 0.3 0.05 0.17

From the analysis, In the main sequence of a cluster, I could find a mass $M_2$ 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

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

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

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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