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CHAPTER 1 INTRODUCTION The clear night sight sky is full of stars and is one of the most fascinating sights to our eyes

CHAPTER 1
INTRODUCTION
The clear night sight sky is full of stars and is one of the most fascinating sights to our eyes. It is very difficult to count the number of stars in the Universe. They are not distributed randomly. They are grouped into galaxies which are vast conglomerates of stars held together by the self-gravity of the stars as well as what is called the dark matter content of each. Sun, the center of solar system belongs to the Milky Way galaxy. Stars are the chief light producing ingredients of galaxies. We see the universe through the electromagnetic waves generated mostly by stars – all the way from radio waves to gamma rays. Other than electromagnetic waves neutrino detectors as well as material collected by space missions in the solar system are material messengers from the Universe. The recent detection of gravitational waves by the LIGO collaboration has opened a new non-material window to the Universe. The detections have been of signals from the spiralling-in, merger and ring down of binaries comprised of stellar mass black holes / neutron stars.
The hosts of these stellar mass black hole binaries are still not known. Their formation poses an interesting problem in star and stellar cluster formation and evolution. To determine the individual masses of the black hole binary members from the GW signal statistical methods are used. In using Bayesian methods it is noticed that the final masses inferred has a dependence on the assumed prior and through the prior on the stellar Initial Mass Function (IMF) (***********). In this context we explore in this dissertation stellar multiplicity as a function of primary mass in stellar clusters with flatter / steeper IMFs than Salpeter.
Stellar mass black holes are expected to arise as the end stage in the normal course of evolution of massive stars. In the rest of this introductory chapter brief reviews of a) the life of a star b) stellar IMF and c) stellar multiplicity are included. In the second Chapter ************. In the third chapter we present our results of the investigation on how multiplicity as a function of primary mass changes, with the steepness / flatness of the IMF at the high mass end.
1.1 A brief introduction to star life
Stars starts its life in a huge clouds of gas and dust known as molecular clouds. The energy produced during the nuclear reactions, which takes place at the core of a star, control its life stages. In the beginning, due to the effect of self-gravity, the molecular cloud collapses and fragments into dense cores which are the birth places of even more condensed proto-stars. In proto-stars sufficient gas and dust combined together to form a massive contracting ball. When the temperature at the center of the ball rises to the order of a million Kelvin, thermonuclear nuclear fusion reactions will start. Then the ball begins to shine and a new star is born in the Universe. The mass of gas will become a gas giant, a giant planet if the mass is insufficient to raise the temperature at the core for nuclear reaction to takes place 1.

Fig: 1.1 Image showing the life cycle of a sun like star (credit: NASA and the Night Sky Network)
When a star burns, it begins its life on the Main Sequence. The lifespan of a star on the main sequence depends practically solely on its mass. When the star is on the main sequence, most of its hydrogen in the core gets converted into helium by nuclear fusion. After numerous years hydrogen in the core depletes and this leads to the end of nuclear reactions since the temperature is not high enough for Helium burning reactions to start. With energy generation stopping the hot core slowly contracts and with temperatures rising to 108 Kelvin the star moves into the Helium burning phase. The outer layers of the star expand tremendously and now the star becomes a red giant. The next stage of star life is determined by its mass 2.
For a low mass star, as years pass the star will discard its outer layers, begin to cool down and shrink as well. It has become a white dwarf. White dwarfs cool down over billions of years to become a black dwarf. This is the death of a low mass star.

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For a high mass star, due to the gravity, the core of the red giant gets shrinks resulting in a series of nuclear reactions. Elements of higher and higher atomic number are successively produced and that star gets an onion shell structure with heavier and heavier elements inside. When the iron peak elements are finally produced all energy generation will cease since further reactions are endothermic. The sudden collapse of the core will lead to neutronization of the core and the released blast of neutrinos will throw out the collapsing and fast burning envelope at speeds close to ten thousand kilometre per second. This is the famous supernova explosion. The explosion leaves behind a neutron star. As time elapses the neutron star, which is the densest object known slowly cools as well as slows down its rapid spin losing energy all the while. This is the end of a high mass star 3.

The upper mass limit for a neutron star is around 2.77 to 3 solar masses. Beyond this the degeneracy pressure of neutrons will be insufficient to combat gravity and the star collapse further and ends up as a black hole. Such high masses for the end neutron star are supposed to arise from the evolution of stars with masses greater than 15 – 20 solar masses.
1.2 Stellar Clusters
Groups of gravitationally bound stars are called stellar clusters or star clusters. Clusters can be bright or faint, large or small, young or old and so on 4. Stellar clusters areclassified into two main types. Globular clusters and open clusters.

Open clusters are those which consist of loosely bound stars. They are found in the galactic disk and contain stars younger than those in the globular clusters. The stars in these clusters are metal rich and are supposed to form from gas in the galaxy that has been enriched by multiple, successive supernova explosions. Since the number of stars in these clusters is roughly ranging from a dozen to a few hundred members, their own self gravity is not very high.So they can be easily disrupted by collisions with other clusters / the tidal field of the galaxy 5.

Globular clusters are tightly bound stellar systems. Their distribution is not just in the galactic disk but extends in to the galactic halo. The diameter of this halo is almost two times greater than that of the galactic disk. Stars in globular clusters are very old, which implies that the cluster itself is older than open clusters. They have very low metallicity. These clusters consist of groups of tens of thousands to hundreds of thousands of stars. Since the cluster contains population II stars formed mostly during the early stages of the formation of the galaxy, the cluster is poor in metals 4.

Fig: 1.2.2 Schematic representation of open and globular clusters in Milky Way (credit: profjohn.com)

Fig: 1.2.1 Open and Globular clusters in Milky Way (credit:profjohn.com)
1.3 Binary systems
Stars in galaxies are predominantly found in binary or multiple systems 6. Binary stars are the most common star systems among the multiple star systems. Sir William Herschel was the one, who put forward the term binary star in his paper “On the Construction of the Universe” 7. A binary star system consists of a pair of stars orbiting about their common center of mass. The brightest star of the pair is usually called as primary star and the other is the secondary star.

Fig: 1.3Schematic representations of binary stars (credit: AstronomyOnline.org)
Binary stars are mainly classified according to their way of detection. Visual binary is a pair of stars, in which both of them can be viewed individually through a telescope.

— In spectroscopic binaries, stars are orbiting too closely to be resolved by the most efficient telescope. But by analysing thespectrum their respective spectra may be separated and their orbital motions can be studied 8.

— In eclipsing binaries, line of sight of the observer and the orbital plane of the binary systemare almost aligned. Hence a part of light is blocked, when one star moves in front of the other star relative to the observer. As a result one or both stars will be partially or completely hidden by other,during their revolution about the center of mass 9.

— If only one star is visible or both of them are extremely close to each other or one is too much faint compared to the other, in such cases stars cannot be distinguished appropriately8. Using astrometric methods position and movements of the system can be explained. Such binaries are called as astrometric binaries9.

According to the separation between pairing stars, a binary system can be classified into wide binaries or close binaries.
In close binaries stars are very much close to each other and sometimes able to destroy each other, i.e. evolution of one star can affect the other.

Most of the time low mass stars do not expand when they leave the main sequence and directly degrade into dwarfs. Stars belong to this category can be situated very close to each other without affecting the evolution of each other. These types of systems also come under the label of 6+45wide binaries 10.

1.4 Stellar Multiplicity
Gravity is a long range force. So concept of isolated star or binary systems or multiple systems are somewhat complicated. In many cases the closest neighbour of a star is situated at a distance much closer than the average separation between stars in a typical neighbourhood. Life time of such combination of stars is comparatively very long. Perhaps there may be seen the existence of some clusters containing even thousands to million stars or a small number of stars. They also live long but less than a binary system. Hence multiple systems are actually the midway between these clusters and binarysystems 11.
Stellar multiplicity comes to be an unquestionable feature of star formation process. The total frequency of binary and multiple systems, distribution of mass ratio and orbital periods are the most important factors of stellar evolutionin binary systems 12.

In the case of spectroscopic binaries, orbital period of a binary system may be easily calculated.The period distribution may be parametrized using a power law as f (P) ? P?, where P represents the orbital period and in most cases ? is taken as -1. A log- normal representation with P and a width for the distribution ?logPis also used by many authors.
The mass ratio q=MsecMprim?1 can be obtained from the flux ratio. The mass ratio distribution is flat for most of the range of primary masses and has an even weaker dependence on binary separation. The lowest mass ratio can be seen on the cores, in which most of the angular momentum is thrown away. As we go further into the lowest mass systems, steepness of the mass ratio distribution appears to be increasing. It acts as a restriction to the formation of the lowest mass systems 13.
Since most configurations of three body systems have inherent instabilities, there is greater probability for their break up into a binary and a third body. The theory which describes the three body break up process states that the probability of a final state is proportional to the volume of the phase space that allows this particular final state. This results in the expression, f (e) = 2e, where e is the eccentricity and f(e) is the distribution of eccentricity of the remnant binaries 12. Hence eccentricity is another important parameter in stellar evolution.

Multiplicity frequency of main sequence stars is a steep, monotonic function of stellar mass.Multiplicity frequency is seen to be increasing with increase in primary mass implying that high mass cores produce more fragments on average, because initially they carry more jeans masses, where multiplicity arises due to breakup 13.
The statement that “most stellar systems formed in the Galaxy are likely single and not binary” (Lada 2006) is not yet confirmed. But due to the greater number of low mass stars, most field stars are treated as single stars. Hence these fall in a region of less dense low mass star distribution 14.

In going from brown dwarfs to solar type stars multiplicity seems to be a smooth function of primary mass. By avoiding the case of short period spectroscopic binaries the above sentence is applicable to intermediate and high mass starstoo 13.
1.5 Initial Mass Function
The most important parameter describing the structure and evolution of a star is its mass. Distribution of stellar masses during birth is known as the Initial Mass Function (IMF) 15. IMF can be defined as the distribution of stellar mass formed during a star formation process in a given volume of space. Since masses of stars differs from each other, it is important to calculate the number of stars in a particular mass range to study the variation of IMF. Unfortunately mass of a star is very difficult to calculate 16. So it is calculated through an indirect way. Using mass – luminosity relation, conversion is possible from a single star luminosity function to a mass function. The number of stars formed in a certain range of mass and amount of mass accumulated in a star can be inferred from the conversion. The distribution of stellar systems and the importance of binary stars in star formation events can be gained from the comparison of single star luminosity function with system luminosity function 17.

IMF can be defined as the number of stars per unit volume per unit logarithmic mass,
nmdm=cm-?From simple power law,
?mdm=Cm-?EdwinSalpeter was the one who put forward the concept of IMF in 1955.
In terms of logarithmic mass,
?logmdlogm ~ m-?According to Salpeter,
dN=?logmdlogm?m-??logm=dNd(logm)?m-?Where ?=?-1m: mass of a star
N: number of stars in the logarithmic mass range log (m) and log (m) + d (log (m))
Hence,
?logm=dNd(logm)?m-?(1.1)
Salpeter derived the mass function of Galactic field stars of masses 0.4M?<M<10M?as a power law with?~1.35.

By integrating equation 1.1 and providing appropriate normalization the number of stars within a logarithmic mass interval can be estimated. Further studies on IMF revealed that IMF is not a single power law function. Later Kroupa established a multi segment power law for the IMF 18.

1.6 Stellar IMF- Variations in high stellar density
Variation in IMF is observed from region to region. Salpeter’s function gives a steep fall in the case of massive stars, i.e stars with masses greater than 5M?. Many authors have explored the IF in many situations – in the field versus in clusters, in galaxies with different morphologies etc. In 1979 Miller and Scalo derive a mass function of stars in the solar neighbourhood. They fitted it with a lognormal function and obtained a slope steeper than the Salpeter slope in the high mass region19.They made studies in the low mass region and came to the conclusion that the number of stars in the solar and sub-solar region in the solar neighbourhood is less than the extrapolated Salpeter mass function. In 1987 Rana proposed a mass function for the solar neighbourhood within a range of 0.08 to 100M?. As said earlier mass of a star is calculated from its luminosity, there should be a lower limit for the mass of a star to be detected. 0.1 M? is usually used as the lower mass limit of a star 20.
Results of previous works on the IMF are the following. In 1955 Salpeter found a power law function with slope ? ~ -1.35. He introduced the function based on the population of stars that could be observed in those days. At present the stellar population that is accessible to observations is somewhat different. Investigations of Scalo resulted in a function with slope ? ~ -1.7, for intermediate to high mass stars. Rana obtained a slope ? ~ -1.8 for M ; 1.6M?. Further studies leads to the conclusion, which says that slope of low density regions are steep and slope of high density regions, are shallow 21.
1.7 IMF in stellar clusters and ETGsTo be included
References
1 K D Abhyankar, Astrophysics Stars and Galaxies, 2001
2 Derek Ward-Thompson, Antony P Whitworth, An Introduction to Star Formation
3https://www.schoolsobservatory.org/learn/astro/stars
4 Mike Inglis, Observer’s Guide to Star Clusters, 2013
5 Charles A Cardona III, Star Clusters, 2010
6 Frank H. Shu, Star Formation in Molecular Clouds: Observation and Theory, 1987
7 Robert G Aitken, The Binary Stars, Douglas C. McMurtrie, 1918
8 S. Pattnaik, Binary Star System- A Spectral Analysis, Latin American Journal of Physics Education, 5 (2011), 438-442
9 Da?saRozmus, Eclipsing binary stars, 2010
10 Alan H. Batten, Binary and Multiple Systems of Stars, Pergamon Press Ltd, 1973
11 Peter Eggleton, Evolutionary Processes in Binary and Multiple Stars, 2006
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78578813 Gaspard Duchˆene1, Stellar Multiplicity, Annual Review of Astronomy and Astrophysics, 51(2013), 269-310
14 F.C. L´azaro, and M.J. Ar´evalo, Binary Stars, Selected Topics on Observations and Physical Processes, 1999
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16 Pavel Kroupa, The Initial Mass Function of Stars: Evidence for Uniformity in Variable Systems, Science, 295 (2002), 82–91
17 Pavel Kroupa, The distribution of low-mass stars in the Galactic disc, Astronomical Society of India, 262 (1993), 545-587
18Nate Bastian, A Universal Stellar Initial Mass Function? A Critical Look at Variations, Annual Review of Astronomy and Astrophysics, 48 (2010), 339-389
19Sami Dib, Testing the universality of the IMF with Bayesian statistics: young clusters, Astronomical Society of India, 2014, 1-26
20S. Ninkovi´c and V.Trajkovska, On the Mass Distribution of Stars in the Solar Neighbourhood, Serbian Astronomical Journal, 172 (2016), 17-26
21Bruce G. Elmegreen, Cluster Density and the IMF, 2004, 1-14

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