Study of open star clusters
Keywords: CMD, parallax, Turn-off mass, D.M., Isochrone
|H.R.D.||Hertzsprung Russel Diagram|
|C.M.D.||Colour Magnitude Diagram|
|mas||Milli arc second|
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
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.
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 ( ) 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 ( ) 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 we observe the star in many wavelength in order to assign an exact magnitude. If a star is at a distance from earth, then
where, is called the distance modulus, which is the difference between and .
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.
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, ( ) 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 . We have to correct the data for its extinction when we fit it with the theoretical Isochrone.
where is known as the extinction coefficient, which only depends on the wavelength of observation.
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.
where is the parallax in arcsecond and is the distance in parsecond.
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.
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( ) of a star is given in terms of its intensity by
Suppose we have two stars of equal mass binaries each with apparent magnitude , intensity . 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;
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 as the magnitude of brighter star;
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.
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 position of the Sun- Earth system. It can measure upto 10 micro arcsecond.
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 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.
- are identified in the main sequence with reference to isochrone. Where is the mass at which fragmentation of binary stars happen. 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 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.
*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.
|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|
|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|
|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|
|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|
|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|
|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|
*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.
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.
|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)|
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.
|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)|
I plotted the equal mass binary and unequal mass binary points that I gathered via python programming. The plot is given below.
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 , 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 . The results are tabulated below.
|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|
From the analysis, In the main sequence of a cluster, I could find a mass 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
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.
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.
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.
1. 'Introduction to Stellar Astrophysics' by Erika Bohm.
2. 'Astrophysics in a Nutshell' by Dan Maoz.
3. Astrophysics Data System by NASA,
5. YES:York Extinction Solver-http://www.cdac-ccda.hia.nrc-cnrc.gc.ca/community/YorkExtinctionSolver/output.cgi