9 March 2022 Doctor of Philosophy Joint Astronomy Programme
Department of Physics Faculty of Science

# Panchromatic study of star clusters: binaries, blue lurkers, blue stragglers
and membership

Vikrant Vinayak Jadhav

###### Abstract

Binary systems can evolve into immensely different exotic systems such as blue
straggler stars (BSSs), yellow straggler stars, cataclysmic variables, type Ia
supernovae depending on their initial mass, the orbital parameters and how the
two components evolve. The formation and evolution scenarios for some of these
exotic objects are still ambiguous, as they differ significantly from the
standard single stellar evolution theory. The UV to infrared panchromatic
study of binary stars can characterise them, determine evolutionary history
and forecast their evolution.

The aim of this thesis is to study the demographics of post-mass-transfer
systems (BSSs, white dwarfs (WDs) and blue lurkers) in the open clusters and
their formation pathways. There are multiple formation pathways theorised for
the BSSs, but which pathways are favoured and where is not entirely known. We
also need a homogeneous catalogue of such systems to analyse the pathways. The
Gaia DR2 release in 2018 has provided deep and precise all-sky data required
to determine cluster membership and find post-mass-transfer systems.
Similarly, UV imaging from UVIT/AstroSat telescope is crucial to study the hot
components in the interacting systems such as WDs and BSSs. In this thesis, I
have utilised UVIT, Gaia and other archival data to do a comprehensive
panchromatic study of open cluster BSSs and post-mass-transfer systems.

This thesis consists of 7 chapters. Chapter 1 contains the basics of star
formation, star clusters and evolution of low mass stars. The chapter also
contains the overview of binary evolution and the resulting stellar exotica:
BSSs are the most massive stars in a cluster formed via binary or higher-order
stellar interactions; blue lurkers are lower mass stars formed through such
interactions; extremely low mass (ELM) WDs and hot subdwarfs are results of
mass loss by the progenitor. Chapter 2 contains the details about
observational facilities and telescopes used in this thesis. The chapter also
contains the details of the research methods and models: photometry, spectral
energy distributions of single and binary sources, the isochrone models and
evolutionary tracks.

In chapter 3, I present the study of six open clusters (Berkeley 67, King 2,
NGC 2420, NGC 2477, NGC 2682 and NGC 6940) using the UVIT and Gaia EDR3. We
used combinations of a Gaussian mixture model and a supervised machine-
learning algorithm to determine cluster membership. This technique is robust,
reproducible, and versatile in various cluster environments. We could detect
200–2500 additional members per cluster using our method with respect to
previous studies, which helped estimate mean space velocities, distances,
number of members and core radii. The UVIT photometric catalogues, including
BSSs, main-sequence and red giants, are also provided. The UV–optical
catalogues are used to briefly analyse the six clusters using colour-magnitude
diagrams.

Chapter 4 describes the detailed study of open cluster M67. The UV–optical
colour-magnitude diagrams suggested the presence of excess UV flux in many
members, which could be extrinsic or intrinsic to them. We constructed multi-
wavelength spectral energy distributions (SEDs) using photometric data from
the UVIT, Gaia DR2, 2MASS and WISE surveys along with optical photometry. We
fitted model SEDs to 7 WDs and found 4 of them have mass $>$ 0.5 M⊙ and
cooling age of less than 200 Myr, thus demanding BSS progenitors. SED fits to
23 stars detected ELM WD companions to WOCS2007, WOCS6006 and WOCS2002, and a
low mass WD to WOCS3001, which suggest these to be post-mass-transfer systems.
12 sources with possible WD companions need further confirmation. 9 sources
have X-ray and excess UV flux, possibly arising from stellar activity. The
increasing detection of post-mass-transfer systems among BSSs and main-
sequence stars suggest a strong mass-transfer pathway and stellar interactions
in M67.

Chapter 5 presents the study of BSSs in open cluster King 2 with an age of
$\sim$ 6 Gyr and a distance of $\sim 5700$ pc. The Gaia EDR3 membership showed
the presence of 39 BSS candidates in the cluster. We created multi-wavelength
SEDs of all the BSSs. Out of 10 UV detected BSSs, 6 bright ones fitted with
double component SEDs and were found to have hotter companions with properties
similar to extreme horizontal branch/subdwarf B (sdB) stars, with a range in
luminosity and temperature, suggesting a diversity among the hot companions.
We suggest that at least 15% of BSSs in this cluster are formed via the mass
transfer pathway.

Chapter 6 presents the census and a statistical study of BSSs in the Galactic
open clusters. We first created a catalogue of BSSs using Gaia DR2 data. Among
the 670 clusters older than 300 Myr, we identified 868 BSSs in 228 clusters
and 500 BSS candidates in 208 clusters. In general, all clusters older than 1
Gyr and having mass greater than 1000 M⊙ have BSSs. The average number of BSSs
increases with the age and mass of the cluster, and there is a power-law
relation between the cluster mass and the maximum number of BSSs in the
cluster. We introduced the term fractional mass excess ( $\mathcal{M}_{e}$)
for the BSSs. We find that at least 54% of BSSs have $\mathcal{M}_{e}$ $<$ 0.5
(likely to have gained mass through a binary mass transfer), 30% in the $1.0<$
$\mathcal{M}_{e}$ $<0.5$ range (likely to have gained mass through a merger)
and up to 16% with $\mathcal{M}_{e}$ $>$ 1.0 (likely from multiple
mergers/mass transfer). We also find that the percentage of low
$\mathcal{M}_{e}$ BSSs increases with age, beyond 1–2 Gyr, suggesting an
increase in formation through mass transfer in older clusters. The BSSs are
radially segregated, and the extent of segregation depends on the dynamical
relaxation of the cluster.

Finally, in chapter 7, we present the conclusions and summary of the work. The
chapter also contains ideas for future studies, including panchromatic and
spectroscopic analysis of noteworthy clusters and simulations to compare the
observational properties to the expected properties.

© Vikrant Vinayak Jadhav

All rights reserved

I hereby declare that the work reported in this doctoral thesis titled
“Panchromatic study of star clusters: binaries, blue lurkers, blue stragglers
and membership” is entirely original and is the result of investigations
carried out by me in the Department of Physics, Indian Institute of Science,
Bangalore, under the supervision of Prof. Annapurni Subramaniam at the Indian
Institute of Astrophysics, Bangalore and Dr. Rajeev Kumar Jain at the Indian
Institute of Science, Bangalore.

I further declare that this work has not formed the basis for the award of any
degree, diploma, fellowship, associateship or similar title of any University
or Institution.

To,
My Family and Friends

###### Acknowledgements.

Here, I want to express my gratitude to my supervisor, Prof. Annapurni
Subramaniam, for constant motivation and guidance. I am thankful for the
friendly environment, informative discussions, constructive criticism and the
long and exhaustive reviews of the manuscripts. I would also like to thank
Prof. Ram Sagar, Prof. Kaushar Vaidya, Prof. Smitha Subramanian and Prof.
Sudhanshu Barway for exciting and thought-provoking discussions. I also want
to thank Prof. Rajeev Kumar Jain, Prof. Nirupam Roy, Prof. Maheswar
Gopinathan, Prof. Aruna Goswami, Prof. Prateek Sharma, Prof. Banibrata
Mukhopadhyay, Prof. Sivarani Thirupathi, Prof. Piyali Chatterjee, Prof. Bacham
Eswar Reddy, Prof. Gajendra Pandey and Prof. Tarun Deep Saini for their
support. I also thank all the JAP instructors who taught us in our coursework.
I would also like to thank all the staff members at IIA and office staff of
the Department of Physics, IISc, for their help in administrative works. I
thank my friends, contemporaries, and seniors in IISc, Bhaskara and various
institutes - Sahel, Prerna, Jyoti, Anirban, Snehalata, Prasanta, Chayan,
Deepthi, Dhanush, Ankit, Manika, Sharmila, Anju, Khushboo, Raghu, Sipra,
Samyaday, Gaurav, Ranjan, Suchira, Atanu, Shubham, Sudeb, Surajit, Rashid,
Samriddhi for making my life memorable. Special thanks to Sindhu for the
discussions and help at the beginning of my PhD. Finally, I want to thank my
family members, without whom I would not have been here.

Publications in Refereed Journals

  1. 1.

UVIT Open Cluster Study. I. Detection of a White Dwarf Companion to a Blue
Straggler in M67: Evidence of Formation through Mass Transfer,
Sindhu, N., Subramaniam, A., Jadhav, V. V., Chatterjee, S., Geller, A. M.,
Knigge, C., Leigh, N., Puzia, T. H., Shara, M., & Simunovic, M. (2019),
The Astrophysical Journal, 882, 43

  2. 2.

UVIT Open Cluster Study. II. Detection of Extremely Low Mass White Dwarfs and
Post-Mass Transfer Binaries in M67,
Jadhav, V. V., Sindhu, N., & Subramaniam, A. (2019),
The Astrophysical Journal, 886, 13111presented in Chapter 4

  3. 3.

UVIT/ASTROSAT studies of Blue Straggler stars and post-mass transfer systems
in star clusters: Detection of one more blue lurker in M67,
Subramaniam, A., Pandey Sindhu, Jadhav V. and Sahu Snehalata (2020),
Journal of Astrophysics and Astronomy, 41,45

  4. 4.

UOCS. III. UVIT catalogue of open clusters with machine learning based
membership using Gaia DR2 astrometry,
Jadhav, V. V., Pennock, C. M., Subramaniam, A., Sagar R., & Nayak P. K.
(2020),
Monthly Notices of the Royal Astronomical Society, 503, 236222presented in
Chapter 3

  5. 5.

UOCS. IV. Characterising blue straggler stars in old open cluster King 2 with
ASTROSAT,
Vikrant V. Jadhav, Sindhu Pandey, Annapurni Subramaniam & Ram Sagar (2021),
Journal of Astrophysics and Astronomy, 42, 89 333presented in Chapter 5

  6. 6.

Blue Straggler Stars in Open Clusters using Gaia: Dependence to Cluster
Parameters and Possible Formation Pathways,
Vikrant V. Jadhav & Annapurni Subramaniam (2021),
Monthly Notices of the Royal Astronomical Society, 507, 1699 444presented in
Chapter 6

  7. 7.

High Mass-ratio Binary Population of Open Clusters and their radial
segregation,
Vikrant V. Jadhav, Kaustubh Roy, Naman Joshi & Annapurni Subramaniam (2021),
The Astronomical Journal, 162, 264

  8. 8.

UOCS.VI. UVIT/AstroSat detection of low-mass white dwarf companions to 4 more
blue stragglers in M67,
Sindhu Pandey, Annapurni Subramaniam & Vikrant V. Jadhav (2021),
Monthly Notices of the Royal Astronomical Society, 507, 2373

  9. 9.

UOCS - VII. Blue Straggler Populations of Open Cluster NGC 7789 with
UVIT/AstroSat,
Kaushar Vaidya, Anju Panthi, Manan Agarwal, Sindhu Pandey, Khushboo K. Rao,
Vikrant Jadhav & Annapurni Subramaniam (2022),
Monthly Notices of the Royal Astronomical Society, 511, 2274

  10. 10.

Characterization of hot populations of Melotte 66 open cluster using
Swift/UVOT,
Khushboo K. Rao, Kaushar Vaidya, Manan Agarwal, Anju Panthi, Vikrant Jadhav &
Annapurni Subramaniam (2022, submitted)
Monthly Notices of the Royal Astronomical Society

  11. 11.

UOCS–VIII. UV Study of the open cluster NGC 2506 using ASTROSAT,
Anju Panthi, Kaushar Vaidya, Vikrant Jadhav, Khushboo K. Rao, Annapurni
Subramaniam, Manan Agarwal & Sindhu Pandey (2022 submitted),
Monthly Notices of the Royal Astronomical Society

Proceedings

  1. 1.

Detection of White Dwarf Companions to Blue Straggler Stars from UVIT
Observations of M67,
N, Sindhu., Subramaniam, A., Geller, A. M., Jadhav, V., Knigge, C., Simunovic,
M., Leigh, N., Shara, M., & Puzia, T. H. (2019),
IAUS, 351, 482

###### Contents

  1. 1 Introduction
    1. 1.1 Star formation and star clusters
      1. 1.1.1 Cluster membership
    2. 1.2 Evolution of low mass stars
    3. 1.3 Stellar multiplicity
      1. 1.3.1 Evolution of binary systems
    4. 1.4 Stellar exotica
      1. 1.4.1 Blue straggler stars
        1. 1.4.1.1 Evolution of blue stragglers
      2. 1.4.2 Extremely low-mass white dwarfs
      3. 1.4.3 Hot subdwarf stars
      4. 1.4.4 Blue lurkers
      5. 1.4.5 Yellow stragglers/yellow giants
    5. 1.5 Dynamical evolution of star clusters
    6. 1.6 Thesis aim and structure
  2. 2 Data and Methods
    1. 2.1 Telescopes
      1. 2.1.1 Ultra-Violet Imaging Telescope
      2. 2.1.2 Gaia
      3. 2.1.3 GALEX
      4. 2.1.4 0.9 m Kitt Peak National Observatory
      5. 2.1.5 3.5 m Calar Alto Observatory
      6. 2.1.6 1.8 m Panoramic Survey Telescope and Rapid Response System
      7. 2.1.7 6.5 m MMT
      8. 2.1.8 2.2 m MPG/ESO telescope
      9. 2.1.9 Two Micron All-Sky Survey
      10. 2.1.10 Wide-field Infrared Survey Explorer
    2. 2.2 Research methods and models
      1. 2.2.1 Photometry
      2. 2.2.2 Spectral energy distribution
      3. 2.2.3 Binary SED fitting
        1. 2.2.3.1 Error estimation
      4. 2.2.4 Isochrones and evolutionary tracks
  3. 3 Cluster Membership and UV Catalogues
    1. 3.1 Introduction
      1. 3.1.1 Literature information of the OCs
    2. 3.2 Data and models
      1. 3.2.1 UVIT data
      2. 3.2.2 Gaia data
      3. 3.2.3 Isochrones and evolutionary tracks
    3. 3.3 Membership determination
      1. 3.3.1 Gaussian Mixture Model
      2. 3.3.2 Probabilistic Random Forest
      3. 3.3.3 Selection of features and membership criteria
      4. 3.3.4 Comparison with literature
    4. 3.4 Results
      1. 3.4.1 The catalogues
      2. 3.4.2 Cluster properties
      3. 3.4.3 Individual clusters
    5. 3.5 Discussion
      1. 3.5.1 Membership determination
      2. 3.5.2 Individual clusters
      3. 3.5.3 General discussion for all clusters
    6. 3.6 Conclusions and summary
  4. 4 Extremely Low Mass White Dwarfs and Post–Mass Transfer Binaries in NGC 2682
    1. 4.1 Introduction
    2. 4.2 Observations and analysis
      1. 4.2.1 UVIT data
      2. 4.2.2 Archival data
    3. 4.3 Analysis
      1. 4.3.1 Membership
      2. 4.3.2 Colour-magnitude diagrams
      3. 4.3.3 Spectral energy distribution
      4. 4.3.4 Mass and age estimation
    4. 4.4 Results
      1. 4.4.1 Method to interpret the UV properties
      2. 4.4.2 White dwarfs
      3. 4.4.3 WOCS2002/S1040
      4. 4.4.4 WOCS2007/S984
      5. 4.4.5 WOCS3001/S1031
      6. 4.4.6 WOCS6006/S1271
      7. 4.4.7 Other members
        1. 4.4.7.1 WOCS1001/S1024
        2. 4.4.7.2 WOCS11005/S995
        3. 4.4.7.3 WOCS11011/S757
        4. 4.4.7.4 WOCS2003/S1045
        5. 4.4.7.5 WOCS2008/S1072
        6. 4.4.7.6 WOCS2012/S756
        7. 4.4.7.7 WOCS2015/S792
        8. 4.4.7.8 WOCS3009/S1273
        9. 4.4.7.9 WOCS4003/S1036
        10. 4.4.7.10 WOCS4015/S1456
        11. 4.4.7.11 WOCS5007/S1071
        12. 4.4.7.12 WOCS5013/S1230
        13. 4.4.7.13 WOCS7005/S1274
        14. 4.4.7.14 WOCS7010/S1083
        15. 4.4.7.15 WOCS8005/MMJ5951
        16. 4.4.7.16 WOCS8006/S2204
        17. 4.4.7.17 WOCS9005/S1005
      8. 4.4.8 Triple systems
        1. 4.4.8.1 WOCS2009/S1082
        2. 4.4.8.2 WOCS7009/S1282
    5. 4.5 Discussion
    6. 4.6 Summary and conclusions
      1. 4.6.1 Conclusions
  5. 5 Blue Stragglers in King 2
    1. 5.1 Introduction
    2. 5.2 UVIT and archival data
    3. 5.3 SED fitting and colour-magnitude diagrams
    4. 5.4 Results and discussion
    5. 5.5 Conclusions and summary
  6. 6 Blue Stragglers in Galactic Open Clusters
    1. 6.1 Introduction
    2. 6.2 Data and classification
    3. 6.3 Calculations of stellar and cluster parameters
      1. 6.3.1 Absolute magnitudes of the stars
      2. 6.3.2 Mass of the stars
      3. 6.3.3 Mass and LF of the cluster
      4. 6.3.4 Effective radius
      5. 6.3.5 Dynamical relaxation time of the cluster
    4. 6.4 Results
      1. 6.4.1 Relation between BSSs and cluster properties
      2. 6.4.2 Fractional mass excess
      3. 6.4.3 BSSs properties in age and mass binned clusters
      4. 6.4.4 BSS segregation
      5. 6.4.5 Probable BSSs and data limitations
      6. 6.4.6 Contamination in the blue straggler regime
    5. 6.5 Discussion
      1. 6.5.1 Reference mass for calculating $\mathcal{M}_{e}$
      2. 6.5.2 BSS formation pathways
      3. 6.5.3 Comparison with literature
    6. 6.6 Summary
  7. 7 Conclusions and Future Work
    1. 7.1 Summary and conclusions
    2. 7.2 Ongoing and future work

###### List of Figures

  1. 1.1 Composite images of open and globular star clusters (Credits: NASA APOD). Top left: a pair of OCs, M35 and NGC 2158 (Credits: Dieter Willasch), top right: globular cluster Omega Centauri (Credits: Ignacio Diaz Bobillo), bottom left: OC NGC 2682 (Credits: Noel Carboni and Greg Parker) and bottom right: OC NGC 7142 and nearby reflection nebula NGC 7129. (Credits: Steve Cannistra)
  2. 1.2 Schematic plots that help in membership determination. Plots show the spatial distribution, CMD, VPD and parallax-proper motion (PM) plots for members (as red) and field stars (as grey) of NGC 2682.
  3. 1.3 Evolutionary tracks for single stars in mass range 0.5–60 M⊙ indicating the various phases each star passes through. (Credits: Wikipedia Commons)
  4. 1.4 Lifecycle of a low mass star (0.8–8M⊙). (Credits: Wikimedia Commons)
  5. 1.5 Example of the Roche lobe of an accreting binary system. (Credits: Pearson Prentice Hall, Inc, 2005)
  6. 1.6 Primary formation pathways of BSSs: collisions/mergers, case A/B MT and case B/C MT.
  7. 1.7 The relation between the WD mass and the progenitor mass in isolated and binary systems. (Credits: Iben1991ApJS...76...55I)
  8. 1.8 HRD showing position of hot subdwarfs with respect to MS, giants, WDs and cool subdwarf stars. (Credits: Heber2009ARA&A..47..211H).
  9. 1.9 Evolution of binary components in BSSs and BLs in the HRD. The left HRD shows the evolution of a binary system, which goes through MT to form a BSS. While the right HRD shows an example of a system which forms a BL. Both panels show a donor (A) and accretor (B) moving through the HRD according to their evolution, assuming that there was an MT event which transferred mass from A to B.
  10. 2.1 Schematic of the UVIT showing the twin FUV and NUV/VIS telescopes. (Credits: AstroSat support cell).
  11. 2.2 A UVIT NUV image before and after drift correction.
  12. 2.3 A schematic of the Gaia telescope. (Credits: EADS Astrium)
  13. 2.4 Focal plane of the Gaia telescope. (Credits: European Space Agency)
  14. 2.5 Instruments schematic of GALEX telescope.
  15. 2.6 Schematic example SED of a model star with a temperature of 6250 K. The Kurucz spectrum is shown in red. Selective filters profiles from GALEX, Gaia, 2MASS and WISE are shown in the top panel. The effective flux sampled by each filter is shown by the coloured regions in the bottom panel. The black square markers show the integrated flux detected by each filter. The integrated flux plotted as a function of effective wavelengths forms the SED.
  16. 2.7 Schematic of a composite spectrum of MS-WD binary star (green spectrum). The red and blue spectra correspond to a typical MS (1 M⊙) and WD (0.5 M⊙) respectively. The top panel shows scaled transmission curves of various missions used for NGC 2682.
  17. 2.8 Interpolated Bergeron WD cooling curves. Left panels show the publicly provided cooling curves for 0.2–1.3 M⊙ WDs. The right panels show the interpolated models. The WD cooling curves are coloured according to their mass and age in the top and bottom panels. Solar metallicity isochrones of log(age) 8.0, 9.0 and 10.0 are shown in grey for reference.
  18. 3.1 UVIT images of the six clusters: Berkeley 67 in N245W, King 2 in N219M, NGC 2420 in F148W, NGC 2477 in N263M, NGC 2682 in F148W and NGC 6940 in F169M.
  19. 3.2 The photometric error in the magnitudes. Each subplot shows the magnitude-error plots for a cluster in all available filters.
  20. 3.3 Example of a decision tree.
  21. 3.4 Comparison of different MPs from prf feature-combinations for NGC 2682. The numbers in brackets represent the number of stars in that particular category. (a) Comparison of F6 and F10, where black dots are good quality sources (qf = 1) and red circles are poor quality sources (qf = 0). The green dashed box represents sources with $P\\_F10<0.7\leq P\\_F6$. (b) CMD of NGC 2682 according to membership criteria in Eq. 3.5, the grey dots are members ($P\\_F10>0.7$), while the green triangles are the candidates ($P\\_F10<0.7\leq P\\_F6$). (c) Comparison of F6 and F8 to demonstrate the MP dependence on CMD location. The colour is according to (P_F6 $-$ P_F8), as shown in the right most panel. (d) CMD of stars coloured according to (P_F6 $-$ P_F8). The stars with peculiar CMD position are bluer.
  22. 3.5 Grouped histogram for common candidates, common members, added members and rejected stars. The totals of four classes are normalised to 100 for easy visualisation and the numbers are tabulated in Table 3.5. The likely members by prf and other techniques are represented by Common Memb (dark green) and Common Cand (lime). Red bars are stars added as members by prf (classified as field by other techniques or missing from literature catalogues). Blue bars are stars rejected (classified as field) by prf, but these are members in other techniques. Most of the rejected stars are faint and with larger pmR0. The dotted lines separate the comparisons with different methods and papers viz. gmm, Cantat2018A&A...618A..93C, Gao2018b, Gao2018c and Geller2015. All comparisons are done for the same FOV.
  23. 3.6 Comparison of MPs from prf and gmm. Common members (P_F10.3 $>$ cutoff and P_GMM) $>$ 0.6) are shown as grey dots, common candidates (P_F10.3 $<$ cutoff and P_F6.4 $>$ cutoff and P_GMM) $>$ 0.6) are shown as green triangles, common field stars (P_F10.3 $<$ cutoff and P_F6.4 $<$ cutoff and P_GMM) $<$ 0.6) are shown as black dots, added members (P_F10.3 $>$ cutoff and P_GMM) $<$ 0.6) are shown as red dots and rejected stars (P_F10.3 $<$ cutoff and P_GMM) $>$ 0.6).
  24. 3.7 (continued…)
  25. 3.8 Comparison of MPs from prf and Cantat2018A&A...618A..93C. The markers for different types of stars and the individual panels are similar to Fig. 3.6.
  26. 3.9 (continued…)
  27. 3.10 Analysis of membership from prf and Gao2018b; Gao2018c and Geller2015. The markers for different types of stars and the individual panels are similar to Fig. 3.6.
  28. 3.11 VPDs and CMDs of the six clusters. First and third rows are VPDs of members, candidates and field for respective clusters. Second and fourth rows are CMDs of members, candidates and field for respective clusters. Isochrones are plotted as a blue line for reference.
  29. 3.12 (a) Spatial distribution for Berkeley 67. All Gaia stars are denoted by grey dots and cluster members are denoted by black points. (b)–(d) Distribution of g magnitude with P_GMM, P_F10.3 and parallax. (e) Calculation of core radius by fitting radial density with King’s surface density profile. (f) Histogram of P_10.3. (g)–(j) and (k)–(n) show the VPD and CMD of all stars, members, candidates and field respectively. (k)–(n) show the number of stars plotted in the brackets along with the Padova isochrone as a blue curve.
  30. 3.13 Spatial distribution, VPD and CMDs of King 2. The subplot descriptions are the same as Fig. 3.12.
  31. 3.14 Spatial distribution, VPD and CMDs of NGC 2420. The subplot descriptions are the same as Fig. 3.12.
  32. 3.15 Spatial distribution, VPD and CMDs of NGC 2477. The subplot descriptions are the same as Fig. 3.12.
  33. 3.16 Spatial distribution, VPD and CMDs of NGC 2682. The subplot descriptions are the same as Fig. 3.12.
  34. 3.17 Spatial distribution, VPD and CMDs of NGC 6940. The subplot descriptions are the same as Fig. 3.12.
  35. 3.18 The CMDs of NGC 2682 with UVIT and Gaia photometry. Gray dots are Gaia EDR3 members according to Eq. 3.5. Red squares, green triangles and black crosses are members, candidates and field stars detected in particular filters. The blue line is an isochrone with $Log(Age)=9.6,DistanceModulus=9.64,E(B-V)=0.05\ mag,[M/H]=0\ dex$; the dashed blue line is a WD cooling curve for a 0.5 M⊙ WD; the light blue arrows represent the reddening vectors with direction and magnitude in each CMD. (a), (b) and (c) The optical CMDs with stars detected in F148W, F154W and F169M filters respectively. (d) The UV–optical CMDs with F148W and g filters. (e) The UV CMD with F148W $-$ F169M colour. (f) UV–optical CMD of sources cross-matched with Williams2018 catalogue of possible WDs. The CMD also shows three quasars (blue stars) detected by F148W filter.
  36. 3.19 (a), (b) and (c) The optical and optical–UV CMDs of Berkeley 67. The markers are the same as Fig. 3.18. The isochrone for Berkeley 67 has $Log(Age)=9.2,DistanceModulus=11.53,E(B-V)=0.8\ mag,[M/H]=0\ dex$. (d) Optical CMD of King 2. with F148W detected members denoted by red squares and N219M detected members denoted by blue filled triangles. The isochrone for King 2 has $Log(Age)=9.7,DistanceModulus=13.8,E(B-V)=0.45\ mag,[M/H]=-0.4\ dex$. (e) and (f) The optical and UV–optical CMDs of NGC 2420. The markers are the same as Fig. 3.18. The isochrone for NGC 2420 has $Log(Age)=9.3,DistanceModulus=12.0,E(B-V)=0.15\ mag,[M/H]=-0.4\ dex$; WD cooling curve is for 0.6 M⊙ WD.
  37. 3.20 (a) and (b) The optical CMDs of NGC 2477. (c), (d) and (e) The UV–optical and UV CMDs of NGC 2477. The markers are the same as Fig. 3.18. The isochrone is for $Log(Age)=8.9,DistanceModulus=10.9,E(B-V)=0.4\ mag,[M/H]=0\ dex$; WD cooling curve is for a 0.7 M⊙ WD.
  38. 3.21 (a) and (b) The optical and UV–optical CMDs of NGC 6940. The markers are the same as Fig. 3.18. The isochrone is for $Log(Age)=9.0,DistanceModulus=10.0,E(B-V)=0.2\ mag,[M/H]=0.0\ dex$; WD cooling curve is for a 0.6 M⊙ WD.
  39. 3.22 Normalised feature-importance for F10 for all six OCs.
  40. 3.23 Spatial location of RGs in Berkeley 67 over-plotted on the reddening map. The plot contains all Gaia members/candidates (as black dots), N242W members/candidates (as red circles) and the RG detected in N242W filter (as red square).
  41. 4.1 UVIT F148W image of NGC 2682.
  42. 4.2 Photometric errors in magnitudes for all three filters. Filled points are NGC 2682 members, while hollow points are other detected stars.
  43. 4.3 Spatial distribution of NGC 2682 members as observed by UVIT along with the members from the Geller2015 catalogue. The NGC 2682 members according to Geller2015 are shown in grey, while the members detected by UVIT are stylised according to their known classification.
  44. 4.4 (a) Optical CMD of NGC 2682. The members listed by Geller2015 are shown in grey dots, while the unclassified members detected by UVIT are shown as black dots. The isochrone (age = 3.98 Gyr) is generated with the fsps code using the BaSTI model. The MS/subgiant/RG phase is shown in red; expected locations of BSSs and WDs are shown in blue and grey, respectively. The inset shows the expanded view of the turn-off region. (b) UV–optical CMD (V, F148W$-$V) of NGC 2682. The inset shows the expanded view of the fainter end of the BS sequence. (c) UV–optical CMD (F148W, F148$-$V) of NGC 2682. The inset shows the expanded view of the turn-off region. (d) UV CMD (F148W, F148W$-$F169M) of NGC 2682.
  45. 4.5 Example of the method used to fit SEDs using WOCS3009. (a) The top panel shows the least $\chi^{2}$ fit for a single SED over UV–IR data points. The legend notes the $T_{eff}$ of the fit, the model used (Kr: Kurucz; Koe: Koester WD models), and the log $g$. The fitted points are shown with error bars, while not fitted points are shown as circles. The middle panel shows the residual for the fit. The bottom panel shows the individual $\chi^{2}$ values calculated at each point. (b) Same as panel (a) but fit is done over the optical–IR region. (c) The top bar shows the transmission curves (not to scale) for the filters of respective telescopes. The second panel shows the composite double fit SED. The figure shows the ‘A component’ (Kurucz model; red dot–dashed), ‘B component’ (Koester WD model; blue dashed), ‘Model’ i.e., total flux of 2 components (green solid) and observed flux (black, as only error bars to simplify the graph) and unfitted points (hollow circle). The light-coloured solid lines in blue, red, and green show the higher-resolution spectra corresponding to Kurucz, Koester (WD), and composite models, respectively. The third and fourth panels are similar to the residual and $\chi^{2}$ panels in panel (a).
  46. 4.6 HRD of WDs and hotter components plotted over interpolated DA type (Tremblay2009ApJ...696.1755T) (as solid band) and He-core (Panei2007MNRAS.382..779P) (as dashed lines) WD cooling curves. The errors are plotted as ellipses. Legends for all figures are same as in (b). (a) The gradient corresponds to the mass of the DA model WDs. The single WDs and the hotter components’ parameters from ’log $g$ = 7’ fits are over-plotted to estimate their mass. (b) Same as ’a’ for the SED fits with ’log $g$ = 9’. (c) The gradient corresponds to the cooling age of the DA model WDs. The single WDs and the hotter components’ parameters from ’log $g$ = 7’ fits are over-plotted to estimate their age. (d) Same as ’c’ for the SED fits with ’log $g$ = 9’.
  47. 4.7 SED fits of all isolated WDs with Koester WD model SEDs of log $g$ = 7.
  48. 4.8 Double SED fits of WOCS2002, WOCS2007, WOCS3001 and WOCS6006, same as 4.5c
  49. 4.9 Double SED fits of WOCS1001, WOCS11005, WOCS11011 and WOCS2003
  50. 4.10 Double SED fits of WOCS2008, WOCS2012, WOCS2015 and WOCS4003
  51. 4.11 Double SED fits of WOCS4015, WOCS5007, WOCS5013 and WOCS7005
  52. 4.12 Double SED fits of WOCS7010, WOCS8005, WOCS8006 and WOCS9005
  53. 4.13 SEDs of triple systems, WOCS2009: Model SED using two known components (Aa and Ab) with third component B fitted only to the optical to IR residual. WOCS7009: System with 2 known components (Aa and Ab) and a possible third component fitted to the UV residual.
  54. 4.14 Comparison of UV and X-ray luminosity for all X-ray detected stars with X-ray luminosity taken from Belloni1998, Van2004 and Mooley2015
  55. 5.1 CMD of King 2 cluster candidates using Gaia EDR3 data. All BSS members are shown as blue circles. The UV bright BSS (see § 5.3) are shown as red squares, and other UVIT detected BSS are shown as red X’s. The Gaia members are shown as black dots along with the PARSEC isochrone of log $age$ = 9.7, [M/H] = -0.4, DM = 13.8 and E(B$-$V) = 0.45.
  56. 5.2 Two-component SED of BSS1. (a) The composite SED (green curve) is shown along with the observed flux (as black error bars). The unfitted point (in this case: CAHA.R) is shown as an orange dot. The cooler (BSS, orange dot-dashed curve) and hotter (blue dashed curve) components are also shown with their Teff and log $g$. The model, B component and residuals of noisy iterations are also shown as light coloured lines. (b) The fractional residual is shown for single component fit (orange dot-dashed curve) and composite fit (solid green curve). The fractional observational errors are also indicated on X-axis. (c) The $\chi_{i}^{2}$ of each data point. (d) HRD of the two components along with the isochrone for reference. The density distribution of the noisy B component fits is plotted in blue. (e) Teff–$\chi^{2}$ distribution for the noisy B component fits coloured according to their radii. The dashed lines are the quoted limits of the temperature.
  57. 5.3 Double component fits of BSS2, BSS3 and BSS4. The descriptions of double component fits are the same as Fig. 5.2. (Continued…)
  58. 5.4 (Double component fits of BSS2, BSS3, BSS4, BSS5 and BSS7. And the single component fits of BSS10 and BSS15 are shown as an example with model fit (blue curve), fitted data points (red points) with 1 $\sigma$ and 3 $\sigma$ errors as solid and dashed lines. The theoretical spectra (in grey) is added for reference. The observed (reddening affected) SED is shown in grey below the corrected data-points. The title mentions the Teff, log $g$, metallicity and AV of the model fit.
  59. 5.5 HRD of locations of components of binaries in King 2, NGC 2682 (Sindhu2019ApJ...882...43S; Jadhav2019ApJ...886...13J; Pandey2021MNRAS.507.2373P), NGC 188 (Subramaniam2016ApJ...833L..27S) and EHB stars in NGC 1851 (Singh2020ApJ...905...44S). In King 2, the UV faint BSS (blue dots), UV bright BSS (blue diamonds with their ID) and hotter components in BSSs (coloured filled circles with numbers and error bars) are shown. Hotter components in NGC 2682 and NGC 188 are shown as orange triangles and crosses. EHB stars in NGC 1851 are shown as stars. The PARSEC isochrone (black curve), PARSEC zero-age MS (ZAMS; dashed grey line), WD cooling curves (thick grey curves) and BaSTI ZAHB (dotted grey curve) are shown for reference.
  60. 6.1 Schematic of classifying BSSs and pBSs in the colour-magnitude plane for OC NGC 2682, 2243, 2420 and 6791. Stars in blue and green boxes are classified as BSS and pBS respectively. The isochrone and ZAMS are shows for comparison. The manually identified A and B points are shown as red stars.
  61. 6.2 (a) Age and (b) mass distribution of the BSSs as blue histograms (and BSS+pBS as dashed green histograms). CMDs of (c) BSSs and (d) pBSs coloured according to the cluster age. All stars are corrected for distance and reddening using cluster parameters in Cantat2020. An isochrone of 3 Gyr and ZAMS are shown for reference.
  62. 6.3 Upper panels show the histograms of cluster (a) age, (b) mass and (c) density for all clusters (grey), clusters with BSSs (blue) and clusters with BSS/pBS (dashed green). Bottom panels show the dependence of number of BSS (as blue circles) and BSS+pBS (as green dots) on the cluster (d) age, (e) mass and (f) density. The panel (e) shows a linear fit to clusters with at least 10 BSS (black filled circles) as a solid red line. The dashed red line represents the empirical upper limit on the number of BSSs in a cluster based on cluster mass.
  63. 6.4 Variation of BSS fraction with number of BSSs, cluster age, relaxation periods passed and mass. The increase in fraction after including pBS is indicated by arrows. The clusters are coloured according to the number of BSSs.
  64. 6.5 (a) 2-D density plots for visualising the variation of mass of the BSSs with respect to the cluster MSTO mass, where the points are coloured according to the crowding (yellow means crowding). The dashed, and solid red lines represent the BSSs with 1.5$M_{TO}$ and 2$M_{TO}$ mass, respectively. (b) Variation of $\mathcal{M}_{e}$ with cluster MSTO mass. (c) $\mathcal{M}_{e}$ distribution of the BSSs and pBSs. Bottom panels: Variation of $\mathcal{M}_{e}$ with (d) cluster age, (e) cluster mass and (f) cluster density and (g) dynamical relaxation.
  65. 6.6 Top panels: CMDs of BSSs across various age bins. Middle panels: CMDs of BSSs across various cluster mass bins. The ZAMS are shown in the CMDs for reference. The BSSs are coloured according to $\mathcal{M}_{e}$. Bottom panels: The variation of percentage of stars in the three BSS classes for the above age and cluster mass bins. The shaded regions indicate the possible percentages if pBSs were included in the tally.
  66. 6.7 Box plots of the relationship of normalised effective radii of BSSs with the cluster age and dynamic relaxation time. The upper panels show the relationship for all clusters, while the lower panels only show for clusters with 5 or more BSSs.

###### List of Tables

  1. 2.1 Details of UVIT filters (Tandon2017AJ....154..128T; Tandon2020AJ....159..158T).
  2. 3.1 Cluster coordinates, ages, distances, mean PMs and radii are taken from Cantat2018A&A...618A..93C; Cantat2020. The metallicity of Berkeley 67 is from Lata2004 and other metallicities are from Dias2002A&A...389..871D.
  3. 3.2 The log of UVIT observations in different filters are given along with exposure time and FWHM. The number of detected stars in each filter as well as the number of cluster members/candidates determined according to the Eq. 3.5 are also listed.
  4. 3.3 Definition and formulae for Gaia parameters and derived parameters used in this study.
  5. 3.4 Feature-combinations used in prf algorithm to calculate MP.
  6. 3.5 Comparison of membership classification by prf with gmm, Cantat2018A&A...618A..93C, Gao2018b, Gao2018c and Geller2015.
  7. 3.6 Example of Gaia EDR3 membership catalogue with MP using gmm and machine learning. The spatial coordinates, g and g_rp along with MP obtained by gmm, F6, F8 and F10 feature-combinations are included. The ‘class’ column shows the classification according to Eq. 3.5 (M: member, C: candidate and F: Field).
  8. 3.7 Example of photometric catalogue of all the detected stars in the UVIT images of NGC 6940. The catalogue includes UV magnitudes and errors along with the membership classification. Similar tables for each cluster are available online. The magnitudes of saturated stars are listed as ‘F169M_sat’. The last column shows the classification according to Eq. 3.5 (M: member, C: candidate and F: Field).
  9. 3.8 The cluster parameters as derived from the members (see Eq. 3.5) of the six clusters.
  10. 4.1 The measured magnitudes (not corrected for extinction) of WD and other stellar sources of NGC 2682 from three UVIT filters. Probability of PM membership (PPM) and probability of RV membership (PRV) are obtained from a Yadav2008, b Zhao1993, c Girard1989, d Geller2015, e Williams2018, f Photometric member (Williams2018).
  11. 4.2 The $\chi^{2}_{red}$ comparison between single fits and double fits. Single fits are done by fitting single Kurucz model SED to all available points ($\chi^{2}_{S1}$) or excluding UV points ($\chi^{2}_{S2}$). Double fits are the combination of one Kurucz SED ($T_{A}$) and one WD SED ($T_{B}$ ) at log $g=$ 7 ($\chi^{2}_{Dob}$).
  12. 4.3 The best-fit parameters of all sources estimated using $\chi^{2}$ fits. First column has identification from  a WOCS: Geller2015, S: Sanders1977, Y: Yadav2008, WD: Williams2018.
  13. 4.4 Continued…
  14. 5.1 Age, distance, reddening (E(B$-$V)) and metallicity of King 2 estimated by various investigators are listed. [1] Dias2002A&A...389..871D, [2] Aparicio1990A&A...240..262A, [3] Tadross2001NewA....6..293T, [4] Kaluzny1989AcA....39...13K.
  15. 5.2 Fitting parameters of the best fit of the double and single-component fits of BSSs with the hotter component. The scaling factor is the value by which the model has to be multiplied to fit the data, Nfit is the number of data points fitted, and $\chi_{r}^{2}$ is the reduced $\chi^{2}$ for the composite fit. The $\chi_{r}^{2}$ values of single fits of the cooler components are given in brackets.  Note: the log $g$ values are imprecise due to the insensitivity of the SED to log $g$.
  16. 6.1 Example list of all clusters with log($age$) $>$ 8.5. Full table is available online.
  17. 6.2 Example list of all BSSs in our sample. A complete list of all BSSs and pBSs is available online.
  18. 6.3 The distribution of BSSs in different $\mathcal{M}_{e}$ classes for clusters binned by age and mass. The first column shows the limits of cluster age and mass for each bin. Second and third columns have the number of clusters and BSSs in the respective bins. The last three columns show the number of BSSs divided into low- $\mathcal{M}_{e}$, high- $\mathcal{M}_{e}$ and extreme- $\mathcal{M}_{e}$ BSSs. The last row shows the $\mathcal{M}_{e}$ class distribution of all BSSs.

Asymptotic Giant Branch | AGB | Near Ultra-Violet | NUV
---|---|---|---
Blue Straggler Star | BSS | Open Cluster | OC
Charge-Coupled Device | CCD | Proper Motion | PM
Colour-Magnitude Diagram | CMD | Radial Velocity | RV
Extreme Horizontal Branch | EHB | Red Giant | RG
Extremely low-mass White Dwarf | ELM WD | Red Giant Branch | RGB
Far Ultra-Violet | FUV | Spectral Energy Distribution | SED
Field Of View | FOV | UltraViolet | UV
Full Width Half Maximum | FWHM | Ultra-Violet Imaging Telescope | UVIT
Giant Molecular Cloud | GMC | UVIT Open Cluster Study | UOCS
Hertzsprung–Russell Diagram | HRD | Vector Point Diagram | VPD
Horizontal Branch | HB | White Dwarf | WD
InfraRed | IR | WIYN Open Cluster Study | WOCS
Main Sequence | MS | Yellow Straggler Star | YSS
Main Sequence Turn-Off | MSTO | Zero-Age Horizontal Branch | ZAHB
Mass Transfer | MT | Zero-Age Main Sequence | ZAMS
Membership Probability | MP |  |

[100mm] An outlier is an observation that differs so much from other
observations as to arouse suspicion that it was generated by a different
mechanism D. M. Hawkins

##  Chapter 1 Introduction

> The night sky is filled with uncountable stars with assorted brightness and
> colours. All these stars are born through the gravitational collapse of
> gaseous nebulae known as giant molecular clouds (GMCs). The GMCs are massive
> ($\sim 10^{7}$ M⊙) cold dense gas clouds which break into smaller fragments
> due to self-gravity and inherent turbulence of the diffuse interstellar
> medium (McKee2007ARA&A..45..565M). The temperature and pressure at the cores
> of these gravitationally collapsing fragments increase with time, and
> eventually, they become hot and dense enough to start hydrogen fusion. And a
> star is born.

###  1.1 Star formation and star clusters

A star is rarely born in isolation. The GMCs break into multiple fragments of
various sizes. The hierarchical fragmentation leads to the observed initial
mass function (Salpeter1955ApJ...121..161S; Kroupa2001MNRAS.322..231K). The
protostars accrete the gas from the surroundings and simultaneously ionise the
surrounding material with radiation and winds. The formation of an individual
star depends on the complex interplay between the GMC properties (temperature,
density, pressure, magnetic field) and stellar feedback (winds, supernovae).
Over time, most of the gas in the GMC is driven out by the stellar feedback or
accreted by individual stars resulting in star formation efficiency of
0.2%–20% (Shu1987ARA&A..25...23S; Murray2011ApJ...729..133M). Once the stars
are formed, their future is defined by the internal and external dynamical
forces. If the stars stay gravitationally bound to each other, they remain as
a cluster; otherwise, they will be dispersed in the galaxy
(Adamo2020SSRv..216...69A).

Krause2020SSRv..216...64K defined a cluster as a gravitationally bound group
of at least 12 stars not dominated by dark matter. However, kinematic
information is needed to verify the bound-ness of a group of stars, which is
not possible in every scenario, especially extragalactic.
Gieles2011MNRAS.410L...6G provided a kinematic definition for differentiating
between clusters and associations.

$T_{cr}\equiv 10\left(\frac{R_{eff}^{3}}{GM}\right)^{1/2}\qquad\Pi=Age/T_{cr}$
(1.1)

Where, $T_{cr}$ is stellar crossing time of in the cluster, $R_{eff}$ is the
projected half-light radius, $M$ is cluster mass and $Age$ is cluster age. For
stellar agglomerates older than 10 Myr, $\Pi>1$ would indicate clusters while
$\Pi<1$ would indicate stellar associations.

There are two main classes of clusters in the Milky Way: i) open clusters
(OCs) and ii) globular clusters.

Fig. 1.1: Composite images of open and globular star clusters (Credits: NASA
APOD). Top left: a pair of OCs, M35 and NGC 2158 (Credits: Dieter Willasch),
top right: globular cluster Omega Centauri (Credits: Ignacio Diaz Bobillo),
bottom left: OC NGC 2682 (Credits: Noel Carboni and Greg Parker) and bottom
right: OC NGC 7142 and nearby reflection nebula NGC 7129. (Credits: Steve
Cannistra)

OCs are smaller clusters found throughout the Milky Way disc. OCs are
typically

  * •

young (a few Myr to a few Gyr)

  * •

less massive ($\lesssim 10^{5}$ M⊙)

  * •

up to 10000 stars

  * •

near the Galactic plane

The globular clusters are aptly named as such due to their obvious spherical
shape. The Galactic globular clusters are typically

  * •

old ($>10$ Gyr)

  * •

massive ($>10^{4}$ M⊙)

  * •

more than 10000 stars

  * •

present in the Galactic halo

Fig. 1.1 shows images of a few OCs and a globular cluster. However, this
distinction in classification is not absolute. The Magellanic Clouds are known
to harbour massive young clusters, and the Galactic bulge contains ancient and
massive clusters (Palma2019MNRAS.487.3140P; Ferraro2021NatAs...5..311F). There
are also loosely bound collections of stars present in the spiral arms known
as stellar associations. Overall, there are thousands of OCs, more than 150
globular clusters and thousands of stellar associations in the Milky Way.

The star clusters are test-beds for the study of stellar evolution in diverse
physical environments because all the stars are formed from the same GMC
(Krause2020SSRv..216...64K). Consequently, their chemical makeup is similar to
each other, and their evolution only differs due to the stellar mass and
possible interactions. The coeval nature of all members is then used to
observe the different phases of stellar evolution across the stellar mass
range (Krumholz2019ARA&A..57..227K). The same can be done across clusters of
different ages to get the complete picture of stellar evolution phases across
various ages and masses. Additionally, as all cluster members are situated at
the same location, the improved statistics can be used to accurately calculate
their distance, project their extinction and estimate their age, and
metallicity.

####  1.1.1 Cluster membership

Fig. 1.2: Schematic plots that help in membership determination. Plots show
the spatial distribution, CMD, VPD and parallax-proper motion (PM) plots for
members (as red) and field stars (as grey) of NGC 2682.

Before commencing the study of individual stars in a cluster, it is vital to
establish their cluster membership. In the early twentieth century, all stars
in the vicinity of the cluster were used to study the cluster. van1942 made
significant improvements in the membership determination using the PM of stars
in the vicinity of the clusters. Vasilevskis1958 and Sanders1971 pioneered the
techniques of membership probability (MP) determination using vector point
diagrams (VPDs). As the accuracy of PM measurements improved, membership
determination using VPDs were also enhanced (Sagar1987; Zhao1990;
Balaguer1998; Bellini2009 and references therein). Assuming that the field and
cluster stars produce overlapping Gaussian distributions in the VPD, new
techniques were developed to separate the two populations (Bovy2011;
Vasiliev2019).

The arrival of Gaia was instrumental in study of star clusters
(Gaia2016A&A...595A...1G). Trigonometric parallaxes and accurate PMs from Gaia
DR2/EDR3 have led to accurate identification of cluster members: using
Hertzsprung–Russell diagram (HRD) of star clusters to improve the models of
stellar evolution. (Gaia2018a); Cantat2018A&A...618A..93C identified cluster
members in 1229 OCs and discovered 60 new clusters using
($\mu_{\alpha},\mu_{\delta},\varpi$) clustering. Similarly, Liu2019; Sim2019;
He2020; Castro2020 discovered new OCs by applying visual or machine learning
techniques to Gaia DR2 and identified cluster members.

Fig. 1.2 shows a sample of schematic plots that help in membership
determination. The first two panels from the left show that the spatial
distribution and CMD location of the field and member stars have significant
overlap. Comparatively, VPD shown in the third panel suggests that the cluster
members are well separated from the field stars in the velocity plane. The
last panel shows the segregation of cluster members in the parallax versus
modified PM (see chapter 3) plane. The figure, therefore, demonstrates that a
combination of spatial location, CMD, VPD and parallax would be the ideal
method to select members in a cluster. Chapter 3 gives more details about
cluster membership of OCs using the latest Gaia EDR3 data
(Gaia2021A&A...649A...1G).

###  1.2 Evolution of low mass stars

Fig. 1.3: Evolutionary tracks for single stars in mass range 0.5–60 M⊙
indicating the various phases each star passes through. (Credits: Wikipedia
Commons)

Fig. 1.3 shows how the stars of different masses move through the HRD. The low
mass stars go through the sub-giant, red giant branch (RGB) and asymptotic
giant branch (AGB) phases. In comparison, high mass stars go through yellow,
blue, and red supergiant phases. This thesis work pertains to less than 8 M⊙
stars.

Fig. 1.4: Lifecycle of a low mass star (0.8–8M⊙). (Credits: Wikimedia Commons)

The evolution of single stars primarily depends on its mass. Fig. 1.4 shows
the lifecycle of Sun-like stars. The star starts its life as a T-Tauri star
and reaches the main sequence (MS) where it spends most of its time
(Kippenhahn1990sse..book.....K). As the hydrogen in the stellar core depletes,
the core begins gravitational collapse. The heat released from this collapse
begins H-shell burning, and the H-shell burning gets stronger and stronger as
the mass of the core increases due to the He-ash generated from the shell
burning. The radiation pressure from the H-shell burning causes the star to
expand. The surface temperature drops due to the expansion, and the star
becomes a sub-giant. At lower temperatures, the radiation can more freely
travel outwards. Hence, the star gets brighter while staying at the same
temperature. The continued H-shell forces the star to expand more and become a
red giant (RG) (Iben1991ApJS...76...55I; Boehm1992itsa.book.....B).

Depending on the mass of the He core, it may or may not ignite. Stars with an
initial mass of $<0.5$ M⊙ will never start He-burning and become He-core white
dwarfs (WDs). The cores of 0.5–3 M⊙ stars contract so much that the electrons
become degenerate before the start of He-burning. The degeneracy allows the
temperature to increase ($\sim 10^{8}$ K) without any increase in pressure,
and without increasing pressure, the core cannot expand and cool. This leads
to a runaway start to the He-burning, called He-flash
(Boehm1992itsa.book.....B). The stellar cores of stars massive than 3 M⊙ do
not become degenerate, and they start the He-burning through the triple-alpha
process ($3\ ^{4}He\longrightarrow^{12}C$) in a quasi-equilibrium manner. Once
the He-burning starts in the core, the energy generation in the He-core and
H-shell burning leads to a slight increase in temperature but reduces the
luminosity due to the reduction in the radius. Stars in the 0.5–3 M⊙ range,
after He-flash, will have similar core mass, and they form a clump in the HRD
called the red clump. In older clusters, in the He-core-burning phase, their
luminosity is almost constant, but their temperature is determined by the
amount of H envelop present around the core. Overall, these stars create a
constant luminosity locus near $10^{1.5}$ L⊙ collectively called as the
horizontal branch (HB; deBoer2008sse..book.....D).

The core He burns for roughly 100 Myr, and then the core starts gravitational
collapse again. The resulting heat and pressure start the He-shell burning and
H-shell burning at the edges of the core. Low mass stars cannot ascend the AGB
due to the reduced mass in the envelope, and their cores contract to become a
WD. More massive stars ascend the AGB and eventually become a CO core WD
(Vassiliadis1994ApJS...92..125V; deBoer2008sse..book.....D).

###  1.3 Stellar multiplicity

The evolution scenarios given above are for single stars evolving in
isolation. However, the majority of stars in the universe are not single
stars. The binary fraction of FGK, B and O type stars is 34%, 56–58% and
42–69% respectively (Luo2021arXiv210811120L and references therein). Most of
these stars are formed as bound binary stars during the GMC fragmentation. A
minority of binary pairs, typically in dense clusters, can also form by
capturing one star by another.

The binary stars can be classified and identified depending on their
properties as follows.

  * •

Astrometric binaries: These binaries can be resolved through a sufficiently
large telescope. These are only identifiable in the solar neighbourhood.
However, we need multi-epoch imaging to confirm that it is not a coincidence
and that the stars are bound to each other. Recently, resolved binaries are
being identified using Gaia astrometric data (ElBadry2021MNRAS.506.2269E). In
addition, Belokurov2020MNRAS.496.1922B showed that it is possible to identify
nearly unresolved binary stars using the wobble in the Gaia astrometry.

  * •

Spectroscopic binaries: Unresolved binary stars have their own predefined
spectra. The orbital motion of the stars shifts the lines in the stellar
spectra in a cyclic fashion, which can be used to identify and characterise
the binary components. The binary is called double-lined spectroscopic binary
(SB2) if lines from both components are seen oscillating. This happens when
both components have similar flux. If one of the component is too faint, only
one set of lines oscillates and the binary is called single-lined
spectroscopic binary (SB1). We can calculate orbital periods, eccentricity,
mass function and primary temperature from SB1 binaries. SB2s can provide
further information about the mass ratio, minimum masses, and temperature of
both stars. However, time-consuming multi-epoch spectroscopic monitoring and
significant post-processing are required to reduce the data and determine
binary parameters (e.g., Mathieu2000ASPC..198..517M;
Pourbaix2004A&A...424..727P).

  * •

Photometric binaries: Unresolved binaries with different temperatures can be
identified using their position in the colour-magnitude diagram (CMD) and
their spectral energy distribution (SED), assuming multiwavelength
observations spanning ultraviolet (UV) to infrared (IR) are available.
Thompson2021AJ....161..160T demonstrated that OC binaries could be identified
using optical–IR SEDs. Binaries with different temperatures (especially WD+MS)
can be identified using UV–IR SEDs (Jadhav2019ApJ...886...13J;
Rebassa2021MNRAS.506.5201R).

  * •

Eclipsing binaries: If the binary orbital plane lies along the line of sight,
the binary components can eclipse one another. Eclipsing binaries provide the
most information such as distance, inclination, radii, and masses of
components in addition to the typical information from spectroscopic binaries.
Detecting such eclipses requires continuous monitoring of the sky with high
precision photometric instruments. Recent monitoring by Kepler and TESS
missions has greatly increased the sample of known eclipsing binaries
(Kirk2016AJ....151...68K).

####  1.3.1 Evolution of binary systems

If the binary stars are far apart, both stars evolve independently without
affecting the end products. However, components of an interacting binary
evolve quite differently from single stars. Close tidal interactions can lead
to non-spheroidal shapes and also induce/reduce rotational periods affecting
the evolution of one or both components. However, the most significant factor
in determining the evolution of a binary system is the mass transfer (MT)
between the two components. There are two main ways for MT:

Fig. 1.5: Example of the Roche lobe of an accreting binary system. (Credits:
Pearson Prentice Hall, Inc, 2005)

  * •

Roche lobe overflow: Roche lobe is the gravitational equipotential surface of
a binary system. The gravitational force on the stellar content within one’s
own Roche lobe is dominated by the same star. In contrast, the material
outside the Roche lobe is pulled away from the star by its companion. Fig. 1.5
shows an example of a Roche lobe overflow in a binary system. Here, the left
star has evolved to the point where its surface touches the Roche lobe. Hence,
MT is happening through the L1 Lagrange point towards its companion.

  * •

Wind accretion: As a star evolves and increases in radius, the radiation
pressure pushes material outside the star in the form of stellar wind. The
companion can accrete this gas. The gas typically forms an accretion disc due
to the angular momentum in the material. Although most of the stellar wind is
lost, simulations have shown that wind accretion can reach an efficiency of up
to 45% for binaries with 2000–10000 day orbits (Abate2013A&A...552A..26A).

The method of MT primarily depends on the separation of the binaries. Wide
binaries do not allow Roche lobe overflow, while wind accretion occurs in both
close or wide binaries. Short period binaries can become common envelop
binaries due to the small Roche lobe. Both forms of MT lead to angular
momentum transfer and changes in orbital period and eccentricity. In general,
the MT process leads to loss of angular momentum and circularisation of the
orbit. If the binaries are close, they can even become tidally locked. Binary
systems can even undergo multiple instances of MT depending on which component
is undergoing Roche lobe overflow. Overall, the ultimate fate of the binary
depends on the duration of MT, initial masses of the components, initial/final
separation of the binary and instances of MT. This dependency on so many
parameters is the reason for the diversity seen in the binary systems and
their evolution.

###  1.4 Stellar exotica

####  1.4.1 Blue straggler stars

Blue straggler stars (BSSs) are the most massive stars in a cluster. They
stand out from other cluster members due to their bluer and brighter position
in the CMD. Since their first reporting (Sandage_1953AJ.....58...61S),
multiple mechanisms have been proposed for their formation. All the stars in a
cluster are formed almost simultaneously, so there should not be stars
brighter than the MS turn-off (MSTO). This apparently longer life is justified
by some type of mass accretion by the progenitor of the BSS. The primary
formation pathways are as follows:

  1. 1.

McCrea_1964MNRAS.128..147M proposed that MT from a binary companion can lead
to rejuvenation of the acceptor and formation of a BSS. The MT efficiency
depends on the orbital periods: wider orbits have non-conservative MT and
leave a remnant behind, while close binaries can have conservative MT and lead
to mergers. Depending on the type of MT, the binary becomes a BSS and a WD
with lower (case A/B MT) or normal (case B/C MT) mass. Signatures of
hotter/compact companion can be used to detect such systems, using methods
such as deconvolving SEDs (e.g., Sindhu2019ApJ...882...43S) and variability in
radial velocity (RV). MT systems are typically expected to have circular
orbits due to the past MT event, but MT in elliptical orbits has also been
detected (e.g., Boffin2014A&A...564A...1B).

  2. 2.

Collisions of individual stars or mergers from collisions of binary pairs
(2+2) are also linked to the BSS formation (Hills_1976ApL....17...87H;
Leonard_1989AJ.....98..217L). Collisions typically happen in dense
environments (such as globular clusters) and do not show chemical
peculiarities of an MT event (Sills_1999ApJ...513..428S). However,
Leigh_2013MNRAS.428..897L found no correlation between collision rate and
number of BSSs, meaning collisions are not the dominant pathway, even in dense
globular clusters.

  3. 3.

Naoz_2014ApJ...793..137N showed that the eccentric Kozai–Lidov mechanism could
tighten the inner binary in a hierarchical triple system.
Perets_2009ApJ...697.1048P suggested that such merger of inner binary has a
significant role in BSSs formation in OCs. Presently, such systems will likely
be MS+BSS binaries with long eccentric orbits.

However, different mechanisms are said to dominate in different cluster
environments: (i) less dense clusters favour binary MT pathway while high-
density clusters favour collisional pathway (Davies_2004MNRAS.349..129D), (ii)
binary pathway is dominant in globular clusters of all masses
(Leigh_2007ApJ...661..210L; Knigge_2009Natur.457..288K), (iii) core collapse
of a globular cluster can trigger a burst of BSS formation
(Ferraro_2009Natur.462.1028F), (iv) old, less dense and relaxed clusters
favour binary pathway (Mathieu_2015ASSL..413...29M).

#####  1.4.1.1 Evolution of blue stragglers

Fig. 1.6: Primary formation pathways of BSSs: collisions/mergers, case A/B MT
and case B/C MT.

Fig. 1.6 shows a schematic of formation and evolution of BSSs. The
collisional/merger products follow a relatively straightforward path of two
stars forming a BSS and evolving to a WD. Comparatively, the MT pathway is
quite complicated. The MT efficiency depends on the separation of two
components, the mass of the two stars and influence from their neighbours. The
evolutionary phase of stars at the start of the MT is of utmost importance in
terms of the final products. There are three main MT types:

  * •

Case A: MT when the donor is in MS

  * •

Case B: MT when the donor is in RG phase of evolving towards it

  * •

Case C: MT in supergiant phase

In the case B and C MT scenario, the donor’s core is already formed. Thus, the
resultant mass loss does not affect the final evolutionary product of the
donor. The top-right panel in Fig. 1.6 shows such a scenario where the donor
becomes a typical WD while the accretor becomes a BSS. The resultant BSS+WD
system can again go through MT if the BSS can fill its Roche lobe. However,
the system is unlikely to merge as it has already survived one MT phase. Thus,
the final product of such a system will be WD+WD. Here, the donor WD mass will
correlate with donor mass, but the accretor WD will have mass corresponding to
the BSS mass, which is larger than the original accretor star.

If the binary stars are close enough for case A or early case B MT and still
far enough not to merge, then they can form a BSS+WD system. In this case, the
donor’s core is not fully formed before the MT begins. Hence, the mass loss
will also lead to a lower mass core. Depending on the severity of mass loss,
the donor can evolve into a low mass WD. Furthermore, identifying such low
mass WD can confirm the case A/B MT pathway as the formation scenario for the
BSS.

####  1.4.2 Extremely low-mass white dwarfs

Fig. 1.7: The relation between the WD mass and the progenitor mass in isolated
and binary systems. (Credits: Iben1991ApJS...76...55I)

90% of all stars evolve into a WD. However, the mass of the WD depends on the
mass of the progenitor. More massive stars lead to massive WDs and vice versa
(Iben1991ApJS...76...55I; Cummings2018). The exact relation depends on the
star’s metallicity, mass, rotation, and environment. However, Fig. 1.7 shows
the rough relation between the mass of a progenitor and the resultant WD. The
most massive stars massive become ONe-core WDs, while lower mass stars become
CO-core WDs. Determining the final WD mass relation and the exact dependencies
is an ongoing problem that can only be solved with observations of WDs in
clusters, WD binaries and their complete characterisation.

The lower end of the WD mass also has one other limiting factor: the age of
the universe. As the low mass stars evolve slower, such stars will not evolve
within the $\sim$13 Gyr passed since the big bang. Hence, there is a lower
limit of $\sim$0.4 M⊙ on the mass of a WD formed through single stellar
evolution (Kilic2007ApJ...671..761K). The oldest stars exhaust their H-fuel
and begin the rise in the RGB. At the tip of the RGB, the He-core ignites and
begins the conversion of He to C. Hence, the WDs of low mass stars have CO
cores. However, there are observations of lower mass WDs with He-cores. WDs of
mass of 0.1–0.4 $M_{\odot}$ were found to be part of compact binaries
(Marsh1995MNRAS.275..828M; Benvenuto2005; Brown2010). Such extremely low-mass
WDs (ELM WDs) are generally found in binary systems where the companions are
neutron stars/pulsars (Driebe1998; Lorimer2008), WDs (Brown2016), or A/F MS
stars in EL CVn-type systems (Maxted2014; Wang2018). Recently two R CMa-type
eclipsing binaries are suggested to have precursors of low mass He WDs
(Wang2019).

The mass loss in the early stage of the evolution can explain the lower mass
of the ELM WD. Fig. 1.7 shows that WDs from close binaries have lower mass
than WD from isolated stars. Case-A/B MT can lead to early loss of envelope
and result in the He-core WD (Iben1991ApJS...76...55I;
Marsh1995MNRAS.275..828M). Thus, MT in close binary systems is a must for the
formation of ELM WDs, where companion strips of the envelope of the ELM WD
progenitor and the low mass core fails to ignite the core He. The MT scenario
can also be linked to the acceptor star. While the ELM WD progenitor lost
mass, the companion star gained some portion. The companion thus becomes a BSS
or a blue lurker (BL; see §1.4.4). The similar formation scenarios mean that
detection of an ELM WD alongside a BSS confirms the post-MT nature of both
components.

####  1.4.3 Hot subdwarf stars

Fig. 1.8: HRD showing position of hot subdwarfs with respect to MS, giants,
WDs and cool subdwarf stars. (Credits: Heber2009ARA&A..47..211H).

Hot subdwarfs are blue and compact stars that are more luminous than WDs.
Depending on their spectral type, they are classified as subdwarf B (sdB) and
subdwarf O (sdO). Fig. 1.8 shows the position of hot subdwarfs in the HRD. Hot
subdwarfs have $\sim 40$% binary fraction with a period of fewer than 10 days.
The review by Heber2009ARA&A..47..211H provides detailed observational
findings and evolutionary scenarios of hot subdwarfs. There are two primary
scenarios proposed for the formation of hot subdwarfs:

  * •

Close binary scenarios: Common envelop ejection(s) in close binaries can lead
to formation of sdB+MS star (Han2003MNRAS.341..669H). A merger of two He-core
WDs can also form a rapidly rotating hot subdwarf (Saio2000MNRAS.313..671S).

  * •

Single star evolution: Hot-flash in the He-core of an RGB star can lead to the
star leaving the RGB towards the blue end of the zero-age HB (ZAHB;
Dcruz1996ApJ...466..359D). If such a hot flash occurs while the star is on the
WD cooling curve, it can become a hot subdwarf and enrich it in He, C, and N.

Extreme HB (EHB) stars in globular clusters also occupy similar space as sdB
stars in the HRD, and they are proposed to have similar formation mechanisms.
However, field hot subdwarfs have more binary fraction than globular cluster
EHB stars, likely due to higher merger fractions in globular clusters. Most
field sdB stars have WDs or low mass MS stars as companions. However, the
exact formalisms for the sdB/sdO origin are still being debated due to the
complicated nature of hot flashes and common envelope ejection.

####  1.4.4 Blue lurkers

Fig. 1.9: Evolution of binary components in BSSs and BLs in the HRD. The left
HRD shows the evolution of a binary system, which goes through MT to form a
BSS. While the right HRD shows an example of a system which forms a BL. Both
panels show a donor (A) and accretor (B) moving through the HRD according to
their evolution, assuming that there was an MT event which transferred mass
from A to B.

BSSs are the post-MT systems that are brighter and bluer than the MSTO.
However, this only happens because the accreted mass is enough to make the
progenitor brighter than MSTO. If the amount of accreted gas was less or the
accretor star was too small, the jump in the CMD would not be enough to make
the accretor brighter than the MSTO. Fig. 1.9 shows the two scenarios. The
left panel gives an example of BSS formation, while the right panel gives an
example of an accretor that failed to cross the MSTO. The faint accretor is
essentially the same as a BSS, but it cannot be classified as a BSSs simply
because it is not bluer and brighter than the MSTO. Hence, these stars were
dubbed as blue lurkers (Leiner2019ApJ...881...47L).

Although the figure shows formation through MT, BLs are products of binary (or
multiple) evolution which became brighter due to mass accretion similar to the
BSSs. Detection of BLs is difficult because they appear as typical stars on
the MS. There are a handful of techniques to identify BLs:

  * •

Observation of higher than average rotation ($v$ sin$i$) which is an
indication of recent MT

  * •

Identification of a companion which can only form through mass donation (e.g.,
ELM WDs, hot sub-dwarfs)

  * •

Presence of chemical peculiarities (r-process, s-process) which are only
possible from mass accretion

Unfortunately, the abundance studies alone are not convincing evidence of MT
without further research into the explicit effects of mergers and MT on
stellar abundances. Rotational signatures and hot companions are easily
detectable with high-resolution spectra and multiwavelength photometry.
However, the uncharacteristic nature of cool compact companions, slowing down
of BL after MT, transient nature of atmospheric signatures make the
identification of BLs challenging. Similarly, the classification of merger
products as BLs is difficult due to the absence of companions and our novice
understanding of chemical signatures of mergers. Chapter 4 presents the
detection of BLs in NGC 2682 using the characterisation of the hotter
companions to MS stars.

####  1.4.5 Yellow stragglers/yellow giants

Yellow straggler stars (YSSs) or yellow giants are brighter than the subgiant
branch but hotter than the RGB. They are thought to be evolving BSSs moving
through the subgiant phase. Their cluster membership and stellar mass
estimates are required before theorising about the formation mechanisms.
Landsman1997 characterised a YSS+WD (WOCS 2002) pair in NGC 2682 using GHRS
and FOS spectra. The WD was found to be an ELM (0.22 M⊙), and thus the most
likely formation pathway is the Roche lobe overflow from the WD progenitor
leading to the formation of BSS+ELM WD pair which evolved into the YSS+ELM WD.
Leiner2016 used Kepler K2 observations to study a YSS (WOCS 1015) in NGC 2682
and found that it has a mass twice that of MSTO. The possible formation
scenario proposed was one or more binary encounters. However, the formation
mechanisms of all YSSs are still not confirmed. Recently,
Rain_2021arXiv210306004R found 77 YSS candidates among 408 OCs. A detailed
multi-epoch study to determine orbital parameters and
spectroscopic/multiwavelength study to characterise the evolutionary states of
YSSs and possible companions is necessary to understand the different
formation pathways and their frequency clearly.

###  1.5 Dynamical evolution of star clusters

A star cluster, with its members in motion with respect to the common centre
of mass, is located in the gravitational potential of the parent galaxy. The
internal movement of the members with a given velocity dispersion in the
presence of the galactic potential drives the dynamical evolution of the
cluster. Over the life of a cluster, it loses mass due to stellar evolution
and dynamical evolution. And the star clusters eventually dissolve into the
field of the galaxy.

It takes up to a few million years for the cluster to form and get rid of the
residual gas in the parent molecular cloud. The most massive stars evolve
through the supernova phase in the next 3–40 Myr. Their stellar remnants
(black holes and neutron stars) are also kicked out of the cluster
(Faucher2006ApJ...643..332F) leading to a 20% reduction in mass. In the next
40–100 Myr period, the stellar winds from AGB stars become the dominant mass
loss phenomenon. The stellar evolution ceases to be the dominant phenomenon
for timescales longer than 100 Myr and the dynamical relaxation becomes the
most critical factor in the mass loss of a cluster.

The stars (and multiple systems) in a cluster exchange kinetic energy through
random interactions, increasing the velocity of low mass stars and reducing
for low mass stars. The stars with high velocity escape the cluster’s
gravitational potential, while the massive stars sink to the centre. The
timescale over which these interactions substantially affect the motion of
each star is known as dynamical relaxation time ($T_{relax}$). In a
collisional system like a cluster, $T_{relax}$ depends on the number density
and mass of a cluster. We calculated the two-body dynamical relaxation time
for the cluster as follows (Spitzer1971ApJ...164..399S;
Subramaniam1993A&A...273..100S):

$\begin{split}T_{relax}&=\frac{8.9\times 10^{5}(N_{cluster}\
r^{3})^{0.5}}{\langle m\rangle^{0.5}\textrm{log}(0.4N_{cluster})}[yr]\\\
N_{relax}&=Age/T_{relax}\end{split}$ (1.2)

where $r$ is the half-mass radius and $\langle m\rangle$ is the average mass
of the stars in the cluster.

The typical $T_{relax}$ is $\sim 10^{7}$ yr for OCs, $\sim 10^{8}$ yr for
globular clusters, and $>10^{14}$ yr for galaxies (Binney1987gady.book.....B).
Comparing this to the typical ages of OCs (0.1–10 Gyr), globular clusters
($>10$ Gyr), and galaxies ($>12$ Gyr) indicates that most clusters are relaxed
while galaxies are not. Thus, we expect to find mass segregation in OCs. In
addition to the two body relaxation, interaction with Galactic disc also
affects the dynamical evolution of clusters. (Piatti2019MNRAS.489.4367P)
showed that the radii of globular cluster decrease with increasing Galactic
potential. Similarly, (Piatti2020RNAAS...4..248P) found that the outer
globular clusters have smaller BSS segregation compared to the inner globular
clusters which have experienced stronger tidal forces by Milky Way. However,
most OCs lie within the disc of the Milky Way, hence the tidal forces would
strongly depend on the local neighbourhood and their orbit, and the detailed
analysis of its effect is beyond the scope of this work. Within a cluster,
BSSs are the most massive stars, hence they are good indicators of mass
segregation and dynamical evolution of the cluster
(Ferraro2012Natur.492..393F; Alessandrini2016ApJ...833..252A). Binary stars
are also more massive than single stars; hence their segregation can also be
used to study the dynamical evolution of clusters (Jadhav2021AJ....162..264J).

###  1.6 Thesis aim and structure

Close binary systems can evolve into immensely different exotic systems such
as BSSs, YSSs, cataclysmic variables, type Ia supernovae depending on the
orbital parameters and how the two components evolve. The formation and
evolution scenario for some of these exotic objects are still ambiguous, as
they differ significantly from standard single star evolution theory. This
thesis is focused on understanding the binary stars and their evolutionary
products in OCs.

The structure of this thesis is as follows:

  * •

Chapter 1 contains the introduction of star formation, stellar evolution and
stellar exotica such as BSS and BLs.

  * •

Chapter 2 contains the information about the various telescopes used in this
thesis. The chapter also details the methods used for data analysis, such as
photometry and SED.

  * •

Chapter 3 shows the method to determine cluster membership using Gaia data. We
also provide the UV–optical catalogues of the six OCs and their brief
analysis.

  * •

Chapter 4 presents the study of BSSs and BLs in OC NGC 2682.

  * •

Chapter 5 presents the study of BSSs in OC King 2.

  * •

Chapter 6 provides the BSS catalogue based on Gaia DR2 data of all known OCs.

  * •

Chapter 7 summaries the studies done in this thesis and presents the possible
future directions we can take.

[100mm] Light brings the news of the universe William Bragg

##  Chapter 2 Data and Methods

> Stellar emissions in all wavelengths of the electromagnetic spectrum are
> essential to understand stellar properties. Optical data are the most
> prevalent due to the possibility of observations from the ground and our
> advanced understanding of the data. In addition, UV data are vital to study
> hot and young stellar populations, while IR data is important for colder
> stars. Thus, multiwavelength observations can allow us to study unresolved
> binary stars with different temperatures, such as MS+WD systems. In this
> work, I have used UV data (UVIT and GALEX), optical data (Gaia, 0.9 m KPNO,
> 3.5 m CAO, Pan-STARRS, 6.5 m MMT and 2.2 m MPG/ESO) and IR data (2MASS and
> WISE). The details of the telescopes and used data are given in § 2.1. The
> details about methods developed for analysing the data are provided in § 2.2

###  2.1 Telescopes

####  2.1.1 Ultra-Violet Imaging Telescope

Fig. 2.1: Schematic of the UVIT showing the twin FUV and NUV/VIS telescopes. (Credits: AstroSat support cell). Table 2.1: Details of UVIT filters (Tandon2017AJ....154..128T; Tandon2020AJ....159..158T). Name | Filter | Mean $\lambda$ [Å] |  $\Delta\lambda$ [Å] | Zero Point
---|---|---|---|---
F148W | CaF2-1 | 1481 | 500 | 18.097$\pm$0.010
F154W | BaF2 | 1541 | 380 | 17.771$\pm$0.010
F169M | Sapphire | 1608 | 290 | 17.410$\pm$0.010
F172M | Silica | 1717 | 125 | 16.274$\pm$0.020
N242W | Silica-1 | 2418 | 758 | 19.763$\pm$0.002
N219M | NUVB15 | 2196 | 270 | 16.654$\pm$0.020
N245M | NUVB13 | 2447 | 280 | 18.452$\pm$0.005
N263M | NUVB4 | 2632 | 275 | 18.146$\pm$0.010
N279N | NUVN2 | 2792 | 90 | 16.416$\pm$0.010

Ultra-Violet Imaging Telescope (UVIT) onboard AstroSat is the first Indian
space observatory launched on 2015 September 28. The Ritchey–Chretien UV
telescope consists of two 37 cm telescopes. One observing in far UV (FUV;
130–180 nm) and another in near UV (NUV; 200–300 nm) and visible (VIS; 350–550
nm). The VIS channel is only used to correct the drift of the spacecraft. Each
channel has multiple narrower filters along with a grating for slit-less
spectroscopy. Fig. 2.1 shows the configuration of the UVIT telescope. The
details of effective area curves, UVIT calibrations and instrumentation can be
found in Tandon2017a; Tandon2020AJ....159..158T and Kumar2012.

The NUV and FUV channels work on photon counting mode with CMOS detectors.
Each incoming photon is focused on a photocathode. The electron shower from
the photocathode is amplified by a factor of $\sim 10^{7}$ using a
microchannel plate (Hutchings2007PASP..119.1152H). This event causes the
illumination of several pixels on the CMOS. $3\times 3$ pixel centroiding is
performed onboard for each event and is typically read-out at a rate of 29 Hz.
The final resolution of the UVIT images depends on the spread of
photoelectrons on the detector ($\sim$1′′) and tracking jitter
($\sim$0$.\\!\\!^{\prime\prime}$5). The VIS channel is used to calculate the
telescope’s drift due to the large abundance of optically bright sources
compared to UV sources. It works in integration mode with exposures of 1/16 s.
16 such exposures are stacked onboard before transmitting the VIS data
resulting in effective exposure of 1 s.

Fig. 2.2: A UVIT NUV image before and after drift correction.

The spacecraft transmits Level 0 (L0) data to ISSDC, ISRO. The L0 data is
combined with spacecraft metadata and reformatted to Level 1 (L1) data. I used
ccdlab (Postma2017PASP..129k5002P; Postma2021JApA...42...30P) for processing
the L1 UVIT data. The left panel in Fig. 2.2 shows an example of an integrated
NUV image for one orbit without any corrections. ccdlab performs a number of
functions including removal of duplicated data from L1, performing field
distortion correction, correcting for centroiding bias and flat fielding. A
critical functionality performed by ccdlab is drift correction. As mentioned
earlier, the VIS channel is used for correcting the drift. In case of
corruption of VIS data, UV data can also be drift corrected by integrating
short exposures (2–5 s) and then stacking these exposures. The drift corrected
images for each orbit are then aligned and merged to create the final image.
The merged images are still imperfect due to the 1 Hz sampling of VIS data
compared to 29 Hz for the UV data. Furthermore, there is a slight difference
found in the pointing of the VIS and FUV telescope due to thermal stick-slip
(Postma2021JApA...42...30P). To correct this, ccdlab has an optimising the PSF
(point spread function) functionality which stacks the images of bright
sources with $\sim$20 s exposures to obtain a minor drift correction and the
best possible PSF. The right panel of Fig. 2.2 shows an example of a UVIT
image after correcting for the satellite drift. The circular field of view
(FOV) of the UVIT has a radius of 14′, whereas the typical radius of the
usable UVIT images is slightly less (Tandon2017a). For the images used in this
thesis, the range for full width half maximum (FWHM) of the PSF was 1′′–2′′
for various targets and filters.

After creating science ready images, we used Gaia and GALEX point source
catalogues for doing astrometry. We used astrometric coordinates from GALEX
images and ccmap task of iraf to create astrometric solution of NGC 2682 data
from April 2017. For all other UVIT images, we have used the coordinate
matching algorithm within the updated ccdlab (Postma2020PASP..132e4503P). It
first extracts point sources within UV images. Then selects bright UV sources
($\sim 50$) and bright Gaia GBP sources as reference (this reference catalogue
can be changed if needed). A comparison of similar triangles within two
catalogues is made using a least-square solution to obtain corresponding
triangles within UV and reference catalogues. Further refining is done by
adding more bright sources, and the final WCS solution is appended in the fits
file header. An error of $\sim 0\mbox{$.\\!\\!^{\prime\prime}$}2$ can be
expected in the WCS solution.

####  2.1.2 Gaia

Fig. 2.3: A schematic of the Gaia telescope. (Credits: EADS Astrium)

Gaia is the revolutionary space telescope launched by European Space Agency
(ESA) in 2013 (Gaia2016A&A...595A...1G). Gaia consists of two 1-m class
optical telescopes sharing their focal plane, which contains 106 charge-
coupled devices (CCDs). Fig. 2.3 shows the schematic of Gaia telescope
indicating the position of primary mirrors, focal plane assembly and spin
axis. Fig. 2.4 shows the CCD arrangement in the focal plane. The primary
components of the focal plane are as follows:

Fig. 2.4: Focal plane of the Gaia telescope. (Credits: European Space Agency)

  * •

Sky mapper CCDs: These 14 CCDs are used to detect sources up to $\sim$20 mag
and convey the position of each source to the astrometric field CCDs for
tracking purposes.

  * •

Astrometric field CCDs: 62 CCDs for astrometric measurements (position, PM,
and parallax) and broadband photometric measurements (G-band) up to 21 mag.

  * •

Blue and red photometers: 14 CCDs dedicated to spectrophotometric observations
of stars in GBP (330–680 nm) and GRP (640–1050 nm) bands.

  * •

Radial velocity spectrometer (RVS): 12 CCDs collecting medium resolution
spectra ($\lambda/\Delta\lambda\sim 11500$) of the Calcium triplet are used to
estimate RV, metallicity, log $g$ and temperature of objects brighter than
$\sim 15$ mag.

The twin telescopes in Gaia continuously scan the sky defined by the Gaia
scanning law (spin rate of 60′′ s-1). The sky mapper CCDs detect the source,
and then time-delayed integration is used in the rest of the detectors to
measure position, flux (G, GBP and GRP) and RVS spectra in a collectively
$\sim$80 sec exposure. Around 70–80 transits are expected for each object
brighter than 20 mag in the 5-year operation.

The first data release (Gaia DR1) was done with 14 months of data; however, PM
data were not available for all sources, and spectrophotometric data were
omitted entirely. The major data release was Gaia DR2
(Gaia2018A&A...616A...1G), which used 22 months of data. It contained
position, PM and 3 band photometric data of more than a billion sources. The
most recent data release is Gaia EDR3 (Gaia2021A&A...649A...1G), which
contains a similar number of stars as DR2, but has more precise parameters due
to observations spanning 34 months. The typical performance of Gaia EDR3 is as
follows:

  * •

Completeness: Essentially complete for 12–17 G-mag, but actual completeness
depends on the crowding and the scanning law pattern.

  * •

Astrometry: Positional accuracy of better than 1 mas, PM accuracy of better
than 1 mas yr-1, parallax accuracy of better than 1.3 mas at 21 G-mag.

  * •

Photometry: Accuracy of 6 mmag in G-band, 108 mmag in GBP and 52 mmag in GRP
at 20 G-mag.

In this thesis, I have used both Gaia DR2 (chapter 4, 6) and EDR3 (chapter 3,
5) data.

####  2.1.3 GALEX

Fig. 2.5: Instruments schematic of GALEX telescope.

The Galaxy Evolution Explorer, GALEX, is a 50 cm Ritchey–Chretien space
telescope launched in 2003 by the National Aeronautics and Space
Administration (NASA). It simultaneously observes in two filters, FUV
(1344–1786 Å) and NUV (1771–2831 Å). It has a large FOV of 1$.\\!\\!^{\circ}$2
with an effective area of 36.8 and 67.7 cm2 in FUV and NUV, respectively. The
plate scale of GALEX detectors is 1$.\\!\\!^{\prime\prime}$5 pix-1 with FWHM
of 4$.\\!\\!^{\prime\prime}$2 and 5$.\\!\\!^{\prime\prime}$3 in FUV and NUV
channels, respectively. Fig. 2.5 shows the cross-section of GALEX along with
the light path and focal plane instruments. GALEX has done two main sky
surveys:

  * •

All-sky imaging survey (AIS): Typical exposure of 100 s with FUV and NUV depth
of 20 and 21 ABmag, respectively.

  * •

Medium-depth imaging survey (MIS): Typical exposure of 1500 s with FUV and NUV
depth of 22.7 ABmag in both filters.

In this thesis, I have used the photometric data from the revised catalogue of
GALEX UV sources (Bianchi2017ApJS..230...24B).

####  2.1.4 0.9 m Kitt Peak National Observatory

We have used archival photometric data of NGC 2682 (Montgomery1993) from the
0.9 m telescope at Kitt Peak National Observatory (KPNO). It is a Cassegrain
telescope with a FOV of 6$.\mkern-4.0mu^{\prime}$6\. Montgomery1993 observed
NGC 2682 in UBVRI
filters111https://www.noao.edu/kpno/mosaic/filters/filters.html using 25
overlapping field to cover the cluster. Typical exposures were 4$\times$60 s
(VI), 4$\times$120 s (B) and 900 s (U). More details about the observations
can be found in Montgomery1993.

####  2.1.5 3.5 m Calar Alto Observatory

We have used archival data from the 3.5 m telescope at Calar Alto, Spain for
OC King 2. Aparicio1990A&A...240..262A observed King 2 in 1988 with UBVR
filters and exposure time of 2500, 590, 230 and 110 s respectively. The FWHMs
of the images were $\leq$ 1′′.

####  2.1.6 1.8 m Panoramic Survey Telescope and Rapid Response System

Panoramic Survey Telescope and Rapid Response System (Pan-STARRS1) is the
first data release that used the 1.8 m Ritchey–Chretien telescope in
Haleakala, Hawaii (Chambers2016arXiv161205560C). It has 3∘ FOV with a 1.4
Gigapixel camera. The survey was carried out in grizyP1 filters with
sensitivity of 23.3, 23.2, 23.1, 22.3 and 21.4 respectively (S/N = 5). Typical
exposure times were 30–60 s in each filter.

####  2.1.7 6.5 m MMT

MMT is a classical Cassegrain telescope with a 6.5 m primary mirror with a
large FOV of up to 1∘ depending on the secondary mirror. Williams2018 used the
wide field CCD imager Megacam with FOV of $25\mbox{${}^{\prime}$}\times
25\mbox{${}^{\prime}$}$ to observe OC NGC 2682 in three filters: u
(35$\times$600 s), g (10$\times$240 s) and r (10$\times$180 s). The FWHM in
the ugr filters were 0$.\\!\\!^{\prime\prime}$6, 0$.\\!\\!^{\prime\prime}$8
and 1$.\\!\\!^{\prime\prime}$3 respectively.

####  2.1.8 2.2 m MPG/ESO telescope

MPG/ESO 2.2-metre telescope is a Ritchey–Chretien reflector based at La Silla,
Chile. Yadav2008 used the NGC 2682 images taken by Wide Field Imager (WFI)
with FOV of $34\mbox{${}^{\prime}$}\times 33\mbox{${}^{\prime}$}$ in 2000 and
2004 to calculate PMs. The WFI camera has a pixel scale of 238 mas, which
results in positional precision of $\sim$7 mas for a bright star. The PM
accuracy from these observations was 3 mas yr-1 at 18 V-mag and 6 mas yr-1 at
20 V-mag.

####  2.1.9 Two Micron All-Sky Survey

Two Micron All-Sky Survey (2MASS) is an all IR sky survey carried out at two
1.3 m telescopes at Mount Hopkins, Arizona, and Cerro Tololo, Chile between
June 1997 and February 2001 (Skrutskie2006AJ....131.1163S). 2MASS covered
99.998% of the sky simultaneously in three near-IR bands (JHK). 2MASS used
6$\times$1.3 s exposures at each sky location with FOV of
8$.\mkern-4.0mu^{\prime}$5$\times$8$.\mkern-4.0mu^{\prime}$5\. The sensitivity
of the final catalogue is 15.8, 15.1 and 14.3 mag in JHK respectively (S/N =
10). The typical FWHM of 2MASS images is 2.5–3′′with photometric uncertainties
of 0.02–0.03 mag till 13 mag.

####  2.1.10 Wide-field Infrared Survey Explorer

Wide-field Infrared Survey Explorer (WISE) is an IR space telescope launched
by NASA in 2009 (Wright2010AJ....140.1868W). The telescope has a 40 cm mirror
with a FOV of 47′. The FOV was split into 4 detectors using dichroic beam
splitters. WISE operated in continuous scanning mode using a secondary mirror
to freeze the sky onto the detector for the exposure time of 8.8 s. Each
source was observed at least 8 times with sensitivity of 17.11, 15.66, 11.40
and 7.97 mag for W1, W2 W3 and W4 filters, respectively (S/N = 5). Typical
FWHM for the WISE filters are 6$.\\!\\!^{\prime\prime}$1,
6$.\\!\\!^{\prime\prime}$4, 6$.\\!\\!^{\prime\prime}$5 and
12$.\\!\\!^{\prime\prime}$0 respectively.

###  2.2 Research methods and models

####  2.2.1 Photometry

In this thesis, I have used the daophot package of Image Reduction and
Analysis Facility (iraf; Tody1993) to do photometry on the UVIT images. I have
followed the photometry manual by W. E.
Harris222https://physics.mcmaster.ca/~harris/daophot_irafmanual.txt for doing
the photometry. The summary of the tasks used is given below.

  * •

imexam: Estimation of background counts and FWHM of the image.

  * •

daofind: Identification of sources above a certain threshold (typically 3–10
times background).

  * •

phot: Doing aperture photometry on the stars selected by daofind.

  * •

pstsel and psf: Selection of isolated stars to create a PSF model.

  * •

allstar: Estimating PSF magnitudes of all stars simultaneously through
multiple iterations of PSF fitting to groups of stars.

  * •

wcsctran: Applying world coordinate solution (WCS) using manually cross-
correlated sources between UVIT and GALEX/Gaia reference sources.

The PSF magnitudes from allstar are then updated using (i) aperture
correction: effect of the curve of growth (ii) PSF correction: systematic
difference between allstar magnitudes and phot magnitudes. As the UVIT FUV &
NUV detectors work in photon counting mode, further saturation correction was
performed using the steps given in Tandon2017AJ....154..128T.

####  2.2.2 Spectral energy distribution

Fig. 2.6: Schematic example SED of a model star with a temperature of 6250 K.
The Kurucz spectrum is shown in red. Selective filters profiles from GALEX,
Gaia, 2MASS and WISE are shown in the top panel. The effective flux sampled by
each filter is shown by the coloured regions in the bottom panel. The black
square markers show the integrated flux detected by each filter. The
integrated flux plotted as a function of effective wavelengths forms the SED.

Ideally, the spectrum of a star is used to estimate the temperature and log
$g$ of a star. However, obtaining spectra of a large number of stars is time-
consuming and prohibitively expensive. The cheaper alternative (observation
time wise, instrument complexity wise and monetarily) is an SED. An SED can be
created by combining imaging done across the electromagnetic spectrum. Fig.
2.6 shows an example SED of a model star assuming it was observed with GALEX,
Gaia, 2MASS and WISE. The SED (shown with black points) contains information
about the luminosity and temperature of the star. Additionally, log $g$ and/or
metallicity can also be derived from SEDs depending on the sensitivity of the
models. The archival photometry of stars is available from multiple wide-field
surveys and through dedicated observations. Hence, SEDs are an effective tool
to characterise a large number of stars.

We used the virtual observatory tool, VO SED Analyzer
(vosa333http://svo2.cab.inta-csic.es/theory/vosa/index.php; Bayo2008), for SED
analysis. vosa includes spectral libraries and virtual observatory facilities.
The conjoined SVO Filter Profile Service444http://svo2.cab.inta-
csic.es/theory/fps/ provides the filter profiles and characteristics (such as
effective wavelengths and zero points). One can create an SED from spatial
location alone using the virtual observatory services included in vosa.

For the work done in this thesis, I uploaded the UVIT photometry to vosa and
used the integrated virtual observatory service to get archival photometry
from GALEX, Gaia, Pan-STARRS1, 2MASS and WISE. Depending on the availability
of targeted observations, I also uploaded photometry from MMT, MPG/ESO, Calar
Alto and KPNO to increase the data points in the SEDs. The SEDs are then
corrected for extinction before fitting the models. The corrections for
distance are applied later to estimate the luminosity and radius.

After getting the flux of the source in all filters, vosa calculates synthetic
photometry, for a selected theoretical model, using filter transmission
curves. It performs a $\chi^{2}$ minimisation test by comparing the synthetic
photometry with observed data to get the best-fit parameters of the SED. The
reduced $\chi^{2}$, $\chi^{2}_{r}$, value is given by

$\small\chi^{2}_{r}=\frac{1}{N-N_{f}}\chi^{2}=\frac{1}{N-N_{f}}\sum_{i=1}^{N}\Big{\\{}\frac{(F_{o,i}-M_{d}F_{m,i})^{2}}{\sigma_{o,i}^{2}}\Big{\\}}$
(2.1)

where N is the number of photometric data points, Nf is the number of free
parameters in the model, $F_{o,i}$ is the observed flux, $M_{d}F_{m,i}$ is the
model flux of the star, $\displaystyle{M_{d}=\bigg{(}\frac{R}{D}\bigg{)}^{2}}$
is the scaling factor corresponding to the star (where R is the radius of the
star and D is the distance to the star) and $\sigma_{o,i}$ is the error in the
observed flux. In this thesis, the value of $N$ for stellar sources changes
from 10–16 depending on the number of detections in all available filters. The
$N_{f}$ is 2 for single fits (temperature and scaling factor).

The radius and luminosity of the source can be calculated from the scaling
factor and temperature as follows:

$\begin{split}R&=\sqrt{D^{2}M_{d}}\\\ L&=4\pi\sigma_{SB}R^{2}T^{4}\end{split}$
(2.2)

Typically, $\chi^{2}_{r}\sim 1$ fits are considered as good fits while
$\chi^{2}_{r}<1$ is considered overfitting and $\chi^{2}_{r}>1$ is considered
bad fit. However, this is only true assuming the model is correct and is
linear (Andrae2010arXiv1009.2755A; Andrae2010arXiv1012.3754A). The models in
SEDs are not linear, and the $\chi^{2}_{r}$ value is highly dependent on
errors in the photometry. It can be as high as 100 for data with very small
errors. Hence, keeping a lower limit to the fractional errors is recommended
so that each data point has enough weightage during the fitting process. For
example, Lodieu2019A&A...628A..61L limited the smallest fractional errors to
half of the averaged fractional error by increasing the smaller errors. We
have also used a similar technique to increase errors in Chapter 5.

####  2.2.3 Binary SED fitting

Upon investigation of SED fits obtained via vosa, we found some stars show
poor fits, especially in the UV region. A UV excess flux can be caused by
lower than expected metallicity, chromospheric activity or hot companion. The
hot companion can be characterised by using double component SEDs. SEDs have
previously been used to distinguish between single and binary stars. Recently,
Thompson2021AJ....161..160T used optical–IR (griJH$K_{s}$[3.6][4.5]) SEDs to
study binary population in OCs: NGC 1960, NGC 2099, NGC 2420, and NGC2682.
They presumed that the members should follow the mass-luminosity-radius
relations given by modified PARSEC isochrones. However, this study was
designed to identify MS+MS type binary stars with mass ratio $>0.3$. This
approach does not work for MS+hot companion systems as hot companions are wide
variety, and they do not follow a simple mass-luminosity-radius.

Fig 2.7 shows an example spectrum of an MS+WD system. The optical–IR emission
is dominated by the MS star, while FUV emission is dominated by the WD with a
mix of two in the NUV. The exact transition region depends on the relative
temperature difference between the two stars, but usually, the previous
statement applies to most MS+WD systems.

Fig. 2.7: Schematic of a composite spectrum of MS-WD binary star (green
spectrum). The red and blue spectra correspond to a typical MS (1 M⊙) and WD
(0.5 M⊙) respectively. The top panel shows scaled transmission curves of
various missions used for NGC 2682.

I have developed a python
code555https://github.com/jikrant3/Binary_SED_Fitting to fit two-component
SEDs and give the component parameters. In this thesis, I have searched for
optically sub-luminous hotter companions. Hence, I will refer to the cooler MS
type star as primary and the hotter compact star as secondary. The steps for
double fitting are as follows:

  * •

Selection of sources with no neighbours within 5′′. This step makes sure that
all the photometric points (e.g., GALEX, 2MASS, WISE) have accurate flux.

  * •

Fitting of single Kurucz model (Castelli1997A&A...318..841C) SEDs to all these
isolated stars using UV–IR data points using vosa.

  * •

Identification of sources with unsatisfactory SED fits (e.g., unusually large
$\chi^{2}$, significant fractional residual).

  * •

Refitting of these sources using only optical–IR data ($>3000$Å) in vosa. The
fits should have minimal residual in the optical–IR region.

  * •

Selection of models for primary and secondary components. In this work, all
cooler components were fitted with Kurucz models
(Castelli1997A&A...318..841C), while the hotter components were fitted using
either Koester models (Koester2010) for WDs (in chapter 4) or Kurucz models
for hot sub-dwarfs and MS type stars(in chapter 4 and 5).

  * •

The residual flux after primary fitting is then fitted with a hotter SED. This
step gives the temperature and radius of the secondary source.

  * •

The errors in temperature and radii are estimated from a combination of the
steps sizes in the models, errors in distance and errors in photometry.

After fitting a satisfactory double fits, we checked whether these
temperatures and radii are physically possible and what type of objects these
could be. Chromospherically active sources, X-ray sources and binaries with
ongoing MT can give off excess UV flux. Hence, such sources were not confirmed
to have a hotter component even after successful double component fitting.

#####  2.2.3.1 Error estimation

Estimating the errors in $\chi^{2}$ fitting is not a straightforward process
due to the non-linear nature of SED fitting. One way to estimate errors is
using the grid size in the temperature to estimate the errors. In this case,
the errors in temperature, scaling factor, radius, and luminosity are
estimated as follows:

$\begin{split}{\Delta T}&=\frac{Temperature\ step\ size}{2}\\\ \frac{\Delta
R}{R}&=\frac{\Delta D}{D}\\\ \frac{\Delta L}{L}&=2\frac{\Delta
R}{R}+4\frac{\Delta T}{T}\end{split}$ (2.3)

Alternatively, we can use bootstrapping to get statistical errors in
temperature and other parameters. We first add a Gaussian noise proportional
to the error in the observed flux. We fit this noisy observed flux with a
model SED and get a new set of $T$, $R$ and $L$. We use 100 random noisy SED
fits to get a distribution of fitting parameters. The 32nd, 50th and 68th
percentiles (e.g., $T_{32},\ T_{50}$ and $T_{68}$) of distributions are used
to get the asymmetric 1-$\sigma$ errors in $T$, $R$ and $L$. For example,

$\begin{split}T&=T_{50}\\\ left\\_error&=T_{50}-T_{32}\\\
right\\_error&=T_{68}-T_{50}\end{split}$ (2.4)

To get the total errors in $R$ and $L$, we combine the statistical errors and
distance errors as follows:

$\begin{split}R&=R_{50}\\\
left\\_error&=\sqrt{(R_{50}-R_{32})^{2}+\left(\frac{R\Delta
D}{D}\right)^{2}}\\\
right\\_error&=\sqrt{(R_{68}-R_{50})^{2}+\left(\frac{R\Delta
D}{D}\right)^{2}}\end{split}$ (2.5)

$\begin{split}L&=L_{50}\\\
left\\_error&=\sqrt{(L_{50}-L_{32})^{2}+\left(\frac{2L\Delta
D}{D}\right)^{2}}\\\
right\\_error&=\sqrt{(L_{68}-L_{50})^{2}+\left(\frac{2L\Delta
D}{D}\right)^{2}}\end{split}$ (2.6)

The methods and techniques mentioned above evolving since 2018 and are under
active development. Hence, there will be differences between the exact method
and codes used in Chapter 4, Chapter 5, Vaidya2022arXiv220108773V and the
latest version available in GitHub. Chapter 4 used eq. 2.3 to estimate errors
while Chapter 5 used eq. 2.4–2.6 for estimating errors.

####  2.2.4 Isochrones and evolutionary tracks

An isochrone is the locus of coeval stars with different masses in the
HRD/CMD. The stars lie in various locations according to their evolutionary
phase. As stars in a cluster are formed at approximately the same time, a CMD
of a cluster looks like an isochrone except for a few deviations: (i) varying
number density according to the initial mass function and (ii) small shifts in
the position due to unresolved stars (binaries, multiple systems and
coincidental spatial overlap).

I have primarily used parsec isochrones 666http://stev.oapd.inaf.it/cgi-
bin/cmd (Bressan2012MNRAS.427..127B) generated with known cluster metallicity
and age. The isochrones used Kroupa2001MNRAS.322..231K initial mass function
and the extinction correction was done using Cardelli1989 and Odonnell1994.
Examples of isochrones are given as grey curves in Fig. 2.8.

Fig. 2.8: Interpolated Bergeron WD cooling curves. Left panels show the
publicly provided cooling curves for 0.2–1.3 M⊙ WDs. The right panels show the
interpolated models. The WD cooling curves are coloured according to their
mass and age in the top and bottom panels. Solar metallicity isochrones of
log(age) 8.0, 9.0 and 10.0 are shown in grey for reference.

The WD cooling curves give the luminosity, temperature, and log $g$ of WDs
with different masses and ages. We have used the Bergeron WD
models777http://www.astro.umontreal.ca/~bergeron/CoolingModels/ (Fontaine2001;
Tremblay2011) for hydrogen atmosphere (DA type). The models span cooling
curves of masses 0.2–1.3 M⊙. Fig. 2.8 (a) shows the HRD of the WD models. The
models are convolved with commonly used filters to obtain the model magnitudes
of WDs in different telescopes. The convolutions with UVIT filters were
obtained via personal communication with P. Bergeron. To estimate the mass and
age of detected WDs, we needed finer data points than provided. Hence, we
interpolated the models into a finer regular grid of 500 points spanning the
log(L) and log(T) plane. Fig. 2.8 (b) and (d) show the interpolated models in
the HRD. In Chapter 4, we use these models to estimate the mass and age of the
WDs. The WD temperature and luminosity errors are compared with the
interpolated models, and corresponding errors in age and mass are derived.

For lower mass WDs, I have used He-core WD models 888
http://fcaglp.fcaglp.unlp.edu.ar/~panei/models.html by
Panei2007MNRAS.382..779P. These models span a mass range of 0.1869–0.4481 M⊙.
Due to the unavailability of convolved magnitudes, I have only used these
models in the luminosity–temperature plane. [100mm] The skies are painted with
unnumber’d sparks William Shakespeare

##  Chapter 3 Cluster Membership and UV Catalogues
Jadhav et al., 2021, MNRAS, 503, 236

> ###  3.1 Introduction

Multi-wavelength studies of stars in clusters help to reveal the possible
formation mechanism of non-standard stellar populations (Thomson2012;
Jadhav2019ApJ...886...13J). OCs in the Milky Way span a wide range in ages,
distances and chemical compositions (Dias2002A&A...389..871D; Kharchenko2013;
Netopil2016; Cantat2020). The relatively low stellar density in the OCs is
also an essential factor that helps in understanding the properties of binary
systems in a tidally non-disruptive environment.

Table 3.1: Cluster coordinates, ages, distances, mean PMs and radii are taken
from Cantat2018A&A...618A..93C; Cantat2020. The metallicity of Berkeley 67 is
from Lata2004 and other metallicities are from Dias2002A&A...389..871D.

Name |  $\alpha_{c}$ (J2015.5) |  $\delta_{c}$ (J2015.5) | l | b | $D$ | Age | [M/H] | $\mu_{\alpha,c}cos\delta$ | $\mu_{\delta,c}$ | r50
---|---|---|---|---|---|---|---|---|---|---
| (∘) | (∘) | (∘) | (∘) | (pc) | (Gyr) |  | (mas yr-1) | (mas yr-1) | (’)
Berkeley 67 | 69.472 | 50.755 | 154.85 | 2.48 | 2216 | 1.3 | +0.02 | 2.3 | -1.4 | 4.9
King 2 | 12.741 | 58.188 | 122.87 | -4.68 | 6760 | 4.1 | -0.41 | -1.4 | -0.8 | 3.1
NGC 2420 | 114.602 | 21.575 | 198.11 | 19.64 | 2587 | 1.7 | -0.38 | -1.2 | -2.1 | 3.2
NGC 2477 | 118.046 | -38.537 | 253.57 | -5.84 | 1442 | 1.1 | +0.07 | -2.4 | 0.9 | 9.0
NGC 2682 | 132.846 | 11.814 | 215.69 | 31.92 | 889 | 4.3 | +0.03 | -11.0 | -3.0 | 10.0
NGC 6940 | 308.626 | 28.278 | 69.87 | -7.16 | 1101 | 1.3 | +0.01 | -2.0 | -9.4 | 15.0

The OCs of our Galaxy are located at various distances from us. Thus, stars
detected in any observation will be a mixture of cluster members as well as
both foreground and background field stars. The identification of cluster
members using a reliable method is therefore extremely important. Earlier,
this was accomplished using the spatial location of stars in the cluster
region, as well as their location on different phases of single stellar
evolution, i.e., the MS, sub-giant branch and RGB in the CMDs of star clusters
(Shapley1916). However, many intriguing and astrophysically significant stars
such as BSSs, sub-sub-giants were not considered members due to their peculiar
locations in the OC CMDs. In this chapter, we refer to locations other than
MS, sub-giant branch and RGB, which are part of the single star evolution, as
peculiar.

To study exotic stellar populations in OCs, we selected clusters that are safe
to be observed using UVIT (those at the high galactic latitude and without
bright stars in the UVIT FOV) and have a high probability of detecting UV
stars. The OCs were selected such that enough bright members will be detected
with specific focus on UV bright population such as BSSs and WDs. Some
clusters are (and will be) looked into individually. This work focuses on OCs
Berkeley 67, King 2, NGC 2420, NGC 2477, NGC 2682 and NGC 6940. Fig. 3.1 shows
the UVIT images of the six clusters. They span a range of age (0.7–6 Gyr) and
distance (0.8–5.8 kpc). Table 3.1 lists the parameters such as location in the
sky, distance, age, mean PM and radius of the OCs under study. The relevant
literature surveys are included in §3.1.1.

Fig. 3.1: UVIT images of the six clusters: Berkeley 67 in N245W, King 2 in
N219M, NGC 2420 in F148W, NGC 2477 in N263M, NGC 2682 in F148W and NGC 6940 in
F169M.

UVIT study of NGC 2682 is presented in Sindhu2019ApJ...882...43S;
Subramaniam2020JApA...41...45S; Pandey2021MNRAS.507.2373P, and chapter 4
(Jadhav2019ApJ...886...13J). However, this chapter includes more recent and
deeper photometry for NGC 2682 compared to chapter 4. It is also one of the
most studied OCs with well-established CMD; hence we compare the behaviour of
other OCs with NGC 2682 to interpret the optical and UV CMDs in further
sections. We also use it to validate our membership determination method
against previous efforts. This chapter aims to analyse the UV–optical CMDs and
the overall UV characteristics of these clusters.

Here, we have used multi-modal astrometric and photometric data from the
latest Gaia EDR3 (Gaia2021A&A...649A...1G) for cluster membership. The
membership determination of OC stars, in particular the UV bright population
of BSSs, binaries and WDs, requires careful incorporation of data quality
indicators from Gaia EDR3. PMs of field and cluster stars can be approximated
by Gaussian distributions (Sanders1971) which can be separated analytically,
and individual MP can be estimated from the distance of a star from field and
cluster centre in the VPD. However, this method does not distinguish between
field stars with the same PM as cluster members. Therefore, parallax and CMD
position could be used to remove such field stars. Also, parallax, colour, and
magnitudes have non-Gaussian distributions. To optimally use all the Gaia
parameters, we chose supervised machine learning to segregate the cluster
members. The use of machine learning techniques is increasing in astronomy to
automate classification tasks, including cluster membership (Gao2018a;
Gao2018b; Gao2018c; Gao2018d; Zhang2020; Castro2020). However, as most machine
learning techniques do not include errors in the data, we used probabilistic
random forest (prf, Reis2019), which incorporates errors in the data. To train
the prf, we first selected the cluster members by deconvolving the PM Gaussian
distributions using a Gaussian Mixture Model (gmm, Vasiliev2019). The overall
method also provides the much-needed MPs necessary for stars with non-standard
evolution.

This chapter is arranged as follows: § 3.1.1 contains literature surveys of
the six clusters. § 3.2 has the details of UVIT observations, Gaia data and
isochrone models. The membership determination technique is explained in §
3.3. The membership results and UV–optical photometry are presented in § 3.4
and discussed in § 3.5. The full versions of Gaia EDR3 membership catalogue
(Table 3.6) and UV photometric catalogues of the six OCs (Table 3.7) are
available online.

####  3.1.1 Literature information of the OCs

Berkeley 67 is a $\sim$1 Gyr old OC located at a distance of $\sim$2.45 kpc.
It is a low-density cluster with an angular diameter of $\sim$14′. Lata2004
carried out deep Johnson UBV and Cousins RI CCD photometry of this cluster
while Maciejewski2007 obtained BV CCD data as part of a survey of 42 open star
clusters. Both studies are based on the optical CMD of the cluster.

King 2 is a $\sim$5 Gyr old OC located at a distance of $\sim$6 kpc towards
the Galactic anti-centre direction. It is a faint but rich cluster situated in
a dense stellar field. It lags behind the local disc population by 60–100 km
s-1 and could be part of the Monoceros tidal stream
(Warren2009MNRAS.393..272W). Kaluzny1989AcA....39...13K obtained BV CCD
photometric data for the cluster. A deep Johnson–Cousins UBVR CCD photometric
study of the cluster was carried out by Aparicio1990A&A...240..262A. They
estimated E(B $-$ V) = 0.31 mag in the cluster’s direction and indicated the
presence of $>25\%$ of binary stars, based on the observed scatter in the CMD
of the cluster.

The OC NGC 2420 is $\sim$1 Gyr old and located at a distance of $\sim$3 kpc.
Cannon1970 obtained relative PMs and also determined BV photographic
magnitudes. The broadband optical CCD photometric study was carried out by
Sharma2006. The ubyCaH intermediate-band CCD photometry of this star cluster
was performed by Anthony2006. All these studies indicate that NGC 2420 is
older than 1 Gyr.

The intermediate-age ($\sim$0.9 Gyr) southern rich OC NGC 2477 is located at a
distance of $\sim$1.4 kpc (Hartwick1974; Smith1983; Kassis1997; Eigenbrod2004;
Jeffery2011). This cluster has a metallicity near Solar ([Fe/H] $\sim$
$-$0.17–0.07 dex; Friel2002AJ....124.2693F; Bragaglia2008) and a high binary
frequency ($\sim$36%) for the RGs (Eigenbrod2004). Presence of significant
differential reddening (E(B $-$ V) = 0.2–0.4 mag) across the cluster was
indicated (Hartwick1972; Smith1983; Eigenbrod2004). Using Gaia DR2 data down
to $\sim$21 mag, Gao2018b identified more than 2000 cluster members. A deep
HST photometric study of the NGC 2477 was carried out by Jeffery2011 to
identify WD candidates and estimate their age.

NGC 2682 (M67) is a nearby OC with an age of $\sim$3–4 Gyr (Montgomery1993;
Bonatto2015) and located at a distance of $\sim$800–900 pc (Stello2016). It is
a well-studied cluster from X-rays to IR (Mathieu1986; Belloni1998;
Bertelli2018; Sindhu2018). There are various studies on the membership
determination of NGC 2682 (Sanders1977; Yadav2008; Geller2015; Gao2018c). It
contains stars in various stellar evolutionary phases such as MS, RGs, BSSs,
and WDs. NGC 2682 contains 38% photometric binaries (Montgomery1993) and 23%
spectroscopic binaries (Geller2015). Recently Sindhu2019ApJ...882...43S and
Jadhav2019ApJ...886...13J detected massive and ELM WDs with UVIT observations.
The presence of 24 BSSs, four YSSs, two sub-subgiants, massive WDs and ELM WDs
indicates that constant stellar interactions occur in NGC 2682.

NGC 6940 is a well-known intermediate-age ($\sim$1 Gyr) OC located at a
distance of about 0.8 kpc. The membership of the cluster was investigated by
Vasilevskis1957 and Sanders1972; while photometric studies were carried out by
Walker1958, Johnson1961, Larsson1964 and Jennens1975. Baratella2018 presented
medium resolution (R $\sim$13000), high signal-to-noise (S/N $\sim$100),
spectroscopic observations of seven RG members.

###  3.2 Data and models

####  3.2.1 UVIT data

Table 3.2: The log of UVIT observations in different filters are given along
with exposure time and FWHM. The number of detected stars in each filter as
well as the number of cluster members/candidates determined according to the
Eq. 3.5 are also listed.

Cluster | Filter | Observation Date | Exposure Time | Detected Stars | Members | Candidates | FWHM
---|---|---|---|---|---|---|---
|  | (yyyy-mm-dd) | (s) |  |  |  | (′′)
Berkeley 67 | N242W | 2016-12-21 | 2700 | 469 | 64 | 5 | 1.15
| N245M | 2016-12-21 | 2722 | 258 | 19 | 3 | 1.09
King 2 | F148W | 2016-12-17 | 2666 | 150 | 5 | 1 | 1.33
| N219M | 2016-12-17 | 2714 | 303 | 3 | 1 | 1.35
NGC 2420 | F148W | 2018-04-30 | 2136 | 177 | 57 | 2 | 1.70
NGC 2477 | F148W | 2017-12-18 | 2278 | 301 | 92 | 16 | 1.56
| N263M | 2017-12-18 | 1881 | 1637 | 576 | 53 | 1.34
NGC 2682 | F148W | 2018-12-19 | 6575 | 918 | 84 | 18 | 1.76
| F154W | 2017-04-23 | 2428 | 267 | 31 | 7 | 1.47
| F169M | 2018-12-19 | 6596 | 259 | 58 | 15 | 2.01
NGC 6940 | F169M | 2018-06-13 | 1875 | 151 | 43 | 12 | 1.73

Fig. 3.2: The photometric error in the magnitudes. Each subplot shows the
magnitude-error plots for a cluster in all available filters.

The observations were carried out from December 2016 to December 2018 using
different UV filters of UVIT (AstroSat proposal IDs: A02_170, A04_075, G07_007
and A05_068). The log of UVIT observations is presented in Table 3.2, along
with total exposure time in each filter. We planned observations in at least
one FUV and one NUV broadband filter to get wavelength coverage across the UV
regime for detailed study. Due to payload related issues, NUV observations
were done only for early observations such as Berkeley 67, King 2 and NGC
2477. Unfortunately, no cluster members could be detected in FUV observations
of Berkeley 67 despite observing it in two FUV filters due to the lack of FUV
bright stars. The remaining three OCs (NGC 2420, NGC 2682 and NGC 6940) were
observed in FUV filters alone. Exposure time for different filters ranges from
1875–6596 sec with a typical value of $\sim$2000 sec.

We performed PSF photometry on all UVIT images using daophot package of iraf.
We used 5-$\sigma$ detection and limited the catalogue to the detections with
magnitude errors $<$0.4 mag. The magnitude vs PSF error plots for all the
images are shown in Fig. 3.2. The magnitudes were corrected for saturation
following Tandon2017a. We removed artefacts arising from saturated/bright
stars and false detection at the edge to create the final list of UVIT
detected sources for each observed filter. We included the saturated stars in
the catalogue, however, their magnitudes represent the upper limit (they are
brighter than these values), and their astrometry may be incorrect by a few
arcseconds. In this way, over 100 stars were detected in each image and the
details of this are listed in Table 3.2.

####  3.2.2 Gaia data

The Gaia EDR3 data for all clusters were compiled by constraining spatial and
parallax measurements. We used the r50 (radius containing half the members)
mentioned in Cantat2020 to get the majority of the members with minimal
contamination in the VPD. This region was used to calculate MPs in the gmm
model. We tripled the radius for running the prf algorithm to detect more
members in the outer region. The definition and formulae of
independent/derived Gaia parameters used in this work are shown in Table 3.3.
The errors of ra, dec, pmra, pmdec and parallax are taken from Gaia EDR3.
Upper limits of the photometric errors in g are calculated using the
‘mag_error’ formula in Table 3.3 (similar for errors in gbp, grp, bp_rp bp_g
and g_rp). Errors in ruwe and aen are assumed to be zero.

The parallax_cluster, ra_cluster, dec_cluster and radius_cluster (as mentioned
in Table 3.1) are used to select sources near the cluster using the following
ADQL query:

select *
from gaiaedr3.gaia_source
where
pmra is not null and parallax is not null and
ABS(parallax-cluster_parallax)$<$3* parallax_error and
contains(point(’icrs’, gaiaedr3.gaia_source.ra, gaiaedr3.gaia_source.dec),
circle(’icrs’, cluster_ra, cluster_dec, cluster_radius)) = 1

Table 3.3: Definition and formulae for Gaia parameters and derived parameters
used in this study.

Gaia EDR3 Parameters †
---
ra, dec, pmra, pmdec, parallax, ra_error, dec_error, pmra_error, pmdec_error,
parallax_error,
phot_g_mean_mag (AS g), phot_bp_mean_mag (AS gbp), phot_rp_mean_mag (AS grp),
bp_rp, bp_g, g_rp, phot_g_mean_flux, phot_bp_mean_flux, phot_rp_mean_flux,
phot_g_mean_flux_error, phot_bp_mean_flux_error, phot_rp_mean_flux_error,
g_zero_point_error, bp_zero_point_error, rp_zero_point_error,
astrometric_excess_noise (AS aen), astrometric_excess_noise_sig (AS aen_sig),
ruwe
Cluster parameters (taken from Table 3.1)
ra_cluster, dec_cluster, parallax_cluster, radius_cluster
Derived parameters
mag_error |  $\sqrt{\left(1.086\times\frac{flux\\_error}{flux}\right)^{2}+zero\\_point\\_error^{2}}$ |  [mag]
---|---|---
pmR0 |  $\sqrt{(\textsc{pmra}-cluster\\_pmra)^{2}+(\textsc{pmdec}-cluster\\_pmdec)^{2}}$ |  [mas yr-1]
$\textsc{qf}\ \text{ (Quality filter)}\quad\bigg{\\{}$ |  0, if (ruwe > 1.4) or (aen > 1 and aen_sig > 2) |
|  $1,\qquad\text{otherwise}$ |

†
gea.esac.esa.int/archive/documentation/GEDR3/Gaia_archive/chap_datamodel/sec_dm_main_tables/ssec_dm_gaia_source.html

####  3.2.3 Isochrones and evolutionary tracks

We used parsec isochrones 111http://stev.oapd.inaf.it/cgi-bin/cmd
(Bressan2012MNRAS.427..127B) generated for cluster metallicity and age,
adopted from Dias2002A&A...389..871D and
WEBDA222https://webda.physics.muni.cz/. As the UV images would detect WDs, we
included WD (hydrogen-rich atmosphere, type DA) cooling curves in the CMDs. As
the turn-off masses of the OCs under study range from 1.1–2.3 M⊙, we included
WD cooling curves of mass 0.5–0.7 M⊙ (Fontaine2001; Tremblay2011;
Cummings2018). We used reddened isochrones and WD cooling curves in this
chapter.

###  3.3 Membership determination

####  3.3.1 Gaussian Mixture Model

The distribution of stars in the PM space is assumed to be an overlap of two
Gaussian distributions. The sum of which can be written as,

$f(\mu|\overline{\mu_{j}},\Sigma_{j})=\sum_{j=1}^{2}w_{j}\frac{exp\left[-1/2(\mu-\overline{\mu_{j}})^{T}\Sigma^{-1}_{j}(\mu-\overline{\mu_{j}})\right]}{2\pi\sqrt{det\Sigma_{j}}}$
(3.1) $w_{j}>0,\qquad\sum_{j=1}^{2}w_{j}=1$ (3.2)

where $\mu$ is individual PM vector, $\overline{\mu_{j}}$ are field and
cluster mean PMs, $\Sigma$ is the symmetric covariance matrix and $w_{j}$ are
weights for the two Gaussian distributions. The generalised formalism for the
n-D case and details of fitting the Gaussian distributions to Gaia data are
available in the appendix of Vasiliev2019.

We selected stars within r50 of cluster centre and removed sources with
following quality filters (Lindegren2018; Riello2020) to keep stars with good
astrometric solutions:

$\begin{split}\textsc{ruwe}>1.4\\\ \textsc{aen >
1.0}\quad\textsc{and}\quad\textsc{aen\\_sig > 2.0}\\\
|\textsc{parallax}-parallax\\_cluster|>3\times\textsc{parallax\\_error}\end{split}$
(3.3)

For such sources, a gmm is created using pmra and pmdec, as only these
parameters have distinct Gaussian distribution for the cluster members. Two
isotropic Gaussian distributions are assumed for the field and member stars.
These were initialised with previously known values of cluster PM and internal
velocity dispersion. We used GaiaTools333https://github.com/GalacticDynamics-
Oxford/GaiaTools to maximise the likelihood of the gmm and get the mean and
standard deviation of the two Gaussian distributions. Simultaneously, the MPs
of all stars in the field are calculated.

gmm cannot use the other parameters provided by Gaia EDR3 catalogue (parallax,
ra, dec, g, bp_rp, etc.) due to their non-Gaussian distributions. gmm does not
organically account for systematic parameters leading to loss of interesting
stellar systems with variability, binarity and atypical spectra. However, gmm
can convincingly give the average CMD and VPD distribution of stars in a
cluster. This can be further enhanced with the inclusion of photometric and
systematic information. Hence, we used a supervised machine learning method to
improve membership determination and utilise the non-Gaussian parameters.

####  3.3.2 Probabilistic Random Forest

A random forest consists of multiple decision trees. Fig. 3.3 shows an example
of a decision tree for deciding whether to play golf. There are many factors
to be accounted for, such as outlook, humidity, and wind. These factors are
called as features. The data-set, S, contains the information about the
features. Then each node (root and internal node) uses one feature and one
threshold to make a decision. The next node uses one of the remaining features
to make its decision. The last node, leaf node, gives the final decision to
play or not to play. Overall, the layered nodes make a decision tree. The
random forest is constructed using multiple such trees. Each tree starts with
a randomised subset of the data-set to avoid overfitting and uses random order
for nodes with specific features. The average classification of each tree
(leaf nodes) gives the decision of the random forest.

Fig. 3.3: Example of a decision tree.

The random forest needs to be trained in case of a supervised process. We use
a labelled data-set with values of the features and the expected decision in
each case. The random forest takes the labelled data-set and modifies the
order of nodes, threshold values and the number of layers to replicate the
expected decisions closely. This optimising process is known as training. Now
the random forest can make decisions based on new input features.

The major drawback of traditional random forest is that no uncertainties
(errors) are assumed during these calculations. The probabilistic random
forest (prf) algorithm 444https://github.com/ireis/PRF, developed by Reis2019,
takes care of errors in the data, which is essential for any astronomical
data-set. It assumes all features and labels as probability distribution
functions and out-performs traditional random forest algorithms in the case of
noisy data sets.

In this study, we used the photometric, astrometric and systematic parameters
from Gaia as features. To create a training set for the prf algorithm, we
first calculated the MPs using the gmm method. We used stars within r50 radius
(Cantat2020) from the cluster centre to reduce field star contamination. The
training set was created for each cluster by labelling $P\\_GMM>0.5$ as
members and others as non-members. The training and testing sets were created
by randomly splitting the data set into 3:1 ratios. The prf requires the
features, their errors and the known class (gmm membership labels) as inputs
for training. The output contains fractional MP for each star and the feature-
importance. After training the prf on stars within radius r50, we applied the
algorithm on the stars within 3$\times$r50 of the cluster centre to increase
the sample size.

We assessed the performance of the following parameters as features that can
impact the membership determination: ra, dec, pmra, pmdec, parallax, g, g_rp,
ruwe, aen, pmR0 and many others. The meaning of the features used is mentioned
in Table 3.3. RVs are limited to stars with g $<15$ mag; hence they were not
used as a feature.

We tried more than 22 feature-combinations to optimise the membership
determination. We judged the different combinations by:

  1. 1.

their ability to recreate MP similar to gmm using accuracy score in testing
phase. The accuracy score is defined as:

$accuracy\ score=\frac{correctly\ predicted\ class}{total\ testing\
class}\times 100$ (3.4)

Although the accuracy score itself is not enough to select the final feature-
combination, one can weed out poorly performing combinations.

  2. 2.

the distribution of members in VPDs (Cluster should occupy a compact circular
region in the VPD e.g., Fig. 3.11).

  3. 3.

the distribution of members in CMDs (Minimal contamination to the CMD,
although it is a subjective judgement).

  4. 4.

the distribution of members in PM–parallax plot (the cluster should occupy
small range in both PM and distance).

Based on their individual merits, the notable feature-combinations are listed
in Table 3.4.

####  3.3.3 Selection of features and membership criteria

Table 3.4: Feature-combinations used in prf algorithm to calculate MP. Name |  Features |  Information
---|---|---
F6 |  ra, dec, pmra, pmdec, parallax, pmR0 |  Astrometry
F8 |  ra, dec, pmra, pmdec, parallax, pmR0, g, g_rp |  Astrometry+ Photometry
F10 |  ra, dec, pmra, pmdec, parallax, pmR0, g, g_rp, ruwe, aen |  Astrometry+ Photometry+ Systematics

We trained the prf using 1–1000 trees and saw a plateau in accuracy score
after 150–200 trees. As oshiro_how_2012 suggested, the optimum number of trees
lies between 64–128; hence we chose 200 trees for further analysis. Almost all
feature-combinations had an accuracy score of 92–98%, as all were designed to
select the cluster members. Hence, choosing the best combination was not
trivial.

As expected pmra, pmdec and parallax are important features for membership.
The cluster’s distribution in VPD is a 2D Gaussian, and the random forest does
not completely replicate this quadratic relation between pmra and pmdec.
Hence, we created a new parameter called pmR0, which is the separation of the
source from the cluster centre in the VPD. The PM cluster centre was obtained
from the gmm results. pmR0 helped constrain the cluster distribution to a
circular shape. To test the importance of individual features, we introduced a
column with random numbers as a feature. Among the Gaia features, ra and dec
showed very comparable feature-importance as the random column. However, upon
further inspection, we found that inclusion of ra/dec does not harm the prf
while improving the membership determination is some cases (King 2 is an
example, which is the farthest cluster in our set and has the smallest sky
footprint). Due to known overestimation of gbp flux in fainter and redder
stars (Riello2020), we have used g and g_rp as features.

Fig. 3.4: Comparison of different MPs from prf feature-combinations for NGC
2682. The numbers in brackets represent the number of stars in that particular
category. (a) Comparison of F6 and F10, where black dots are good quality
sources (qf = 1) and red circles are poor quality sources (qf = 0). The green
dashed box represents sources with $P\\_F10<0.7\leq P\\_F6$. (b) CMD of NGC
2682 according to membership criteria in Eq. 3.5, the grey dots are members
($P\\_F10>0.7$), while the green triangles are the candidates
($P\\_F10<0.7\leq P\\_F6$). (c) Comparison of F6 and F8 to demonstrate the MP
dependence on CMD location. The colour is according to (P_F6 $-$ P_F8), as
shown in the right most panel. (d) CMD of stars coloured according to (P_F6
$-$ P_F8). The stars with peculiar CMD position are bluer.