What is a Star Cluster? (Part 1)

Updated: May 16


Star clusters are large groups of stars and there are two types of main star clusters. They are

  • Globular clusters: These types of stars are tight groups of hundreds to millions of old stars which are gravitationally bound

  • Open Clusters: These types of stars are more loosely clustered groups of stars. These stars are very young and they generally contain only a few hundred stars only. Open clusters become disrupted over time by the gravitational influence of giant molecular clouds as they move through the galaxy. The cluster members will continue to move in broadly the same direction through space even though they are no longer gravitationally bound. These types of stars are known as stellar associations or moving groups.

Globular cluster


Globular clusters are roughly spherical groupings of from 10,000 to several million stars packed into regions from 10 to 30 light-years across. Globular clusters are very tightly bound by gravity, which gives them their spherical shapes and high stellar densities toward their centers. The name of this category of star cluster is derived from the Latin, globulus, a small sphere. Occasionally, a globular cluster is known simply as a globular. They commonly consist of very old Population II stars and just a few hundred million years younger than the universe itself. They are mostly yellow and red with masses less than two solar masses. These types of stars predominate within clusters because hotter and more massive stars have exploded as supernovae, or evolved through planetary nebula phases to end as white dwarfs. But there are a few rare blue stars that exist in the globular cluster but it is believed that a number of stellar mergers in their dense inner regions. These stars are known as blue stragglers.


Milky Way globular clusters are distributed roughly spherically in the galactic halo around the Galactic Centre and orbit the center in highly elliptical orbits. Globular clusters tend to be older, more massive, denser, and contain fewer heavy elements than open clusters, which are generally found in the disks of spiral galaxies. But the Andromeda Galaxy compares in size to the Milky Way may have as many stars existing. Some giant elliptical galaxies particularly those at the centers of galaxy clusters, such as M87, have as many as 13,000 globular clusters.


Every galaxy of sufficient mass in the Local Group has an associated group of globular clusters, and almost every large galaxy surveyed has been found to possess a system of globular clusters. Some globular clusters likely originally formed in dwarf galaxies and have since been removed from their birth galaxies by tidal forces and joined the Milky Way. Although it appears that globular clusters contain some of the first stars to be produced in the galaxy, their origins and their role in galactic evolution are still unclear. However, it does appear clear that globular clusters are significantly different from dwarf elliptical galaxies and were formed as part of the star formation of the parent galaxy, rather than as a separate galaxy


In 1917, the astronomer Harlow Shapley made the first reliable estimate of the Sun's distance from the galactic center based on the distribution of globular clusters.


Until the mid-1990s, globular clusters were the cause of a great mystery in astronomy, as theories of stellar evolution gave ages for the oldest members of globular clusters that were greater than the estimated age of the universe. Our improved distance measurements of the Hipparcos satellite used in the globular clusters and increasingly accurate measurements of the Hubble constant resolved the paradox. This helps to give an age for the universe of about 13 billion years and an age for the oldest stars of a few hundred million years less.

Milky Way has about 150 globular clusters and some of which may have been captured from small galaxies disrupted by the Milky Way. This seems to be the case for the globular cluster M79. Some galaxies are much richer in the globular clusters but the giant elliptical galaxy M87 contains over a thousand.


A few of the brightest globular clusters are visible to the naked eye including the brightest, Omega Centauri. This has been known since antiquity and cataloged as a star before the telescopic age. The brightest globular cluster in the northern hemisphere is Messier 13 in the constellation of Hercules.


Observation history


The first known globular cluster is now called M 22. It was discovered in 1665 by Abraham Ihle and he was a German amateur astronomer. Even the small aperture of early telescopes and individual stars within a globular cluster was not resolved until Charles Messier observed M 4 in 1764. Abbé Lacaille would list NGC 104, NGC 4833, M 55, M 69, and NGC 6397 in his 1751–1752 catalog.


When William Herschel began his comprehensive survey of the sky using large telescopes in 1782, there were 34 known globular clusters. Herschel discovered another 36 himself and was the first to resolve virtually all of them into stars. He coined the term globular cluster in his Catalogue of a Second Thousand New Nebulae and Clusters of Stars published in 1789.


The number of globular clusters discovered continued to increase, reaching 83 in 1915, 93 in 1930, 97 by 1947, and 157 in 2010. Astronomers are believing that undiscovered globular clusters are hidden behind the gas and dust of the Milky Way.


Beginning in 1914, Harlow Shapley began a series of studies of globular clusters published in about 40 scientific papers. He examined the RR Lyrae variables in the clusters (which he assumed were Cepheid variables) and used their period-luminosity relationship for distance estimates. Later, it was found that RR Lyrae variables are fainter than Cepheid variables, which caused Shapley to overestimate the distances of the clusters.

Of the globular clusters within the Milky Way, the majority are found in a halo around the galactic core, and the large majority are located in the celestial sky centered on the core. In 1918, this strongly asymmetrical distribution was used by Shapley to make a determination of the overall dimensions of the galaxy. By assuming a roughly spherical distribution of globular clusters around the galaxy’s center, he used the positions of the clusters to estimate the position of the Sun relative to the galactic center. While his distance estimate was an insignificant error (although within the same order of magnitude as the currently accepted value), it did demonstrate that the dimensions of the galaxy were much greater than had been previously thought.


Shapley's measurements indicated that the Sun is relatively far from the center of the galaxy, also contrary to what had previously been inferred from the apparently nearly even distribution of ordinary stars. In reality, most ordinary stars lie within the galaxy's disk, and those stars that lie in the direction of the galactic center and beyond are thus obscured by gas and dust, whereas globular clusters lie outside the disk and can be seen at much further distances.


Classification


Shapley was subsequently assisted in his studies of clusters by Henrietta Swope and Helen Battles Sawyer (later Hogg). From 1927to 929, Shapley and Sawyer categorized clusters according to the degree of concentration each system has toward its core. The most highly concentrated clusters such as M75 are classified as Class I, with successively diminishing concentrations ranging to Class XII, such as Palomar 12. The class is sometimes given with numbers from Class 1–12 shown with Roman numerals. This became known as the Shapley–Sawyer Concentration Class. The Shapley–Sawyer Concentration Class is a classification system on a scale of one to twelve using Roman numerals for globular clusters according to their concentration.


History


From 1927–1929, Harlow Shapley and Helen Sawyer Hogg began categorizing clusters according to the degree of concentration the system has toward the core using this scale. This became known as the Shapley–Sawyer Concentration Class.


Classes


  1. Class I - High concentration toward the center, for example, Messier 75

  2. Class II - Dense central concentration, for example, Messier 2

  3. Class III - Strong inner core of stars, for example, Messier 54

  4. Class IV - Intermediate rich concentrations, for example, Messier 15

  5. Class V - Intermediate concentrations, for example, Messier 30

  6. Class VI - Intermediate mild concentration, for example, Messier 3

  7. Class VII - Intermediate loose concentration, for example, Messier 22

  8. Class VIII - Rather loosely concentrated towards the center, for example, Messier 14

  9. Class IX - Loose towards the center, for example, Messier 12

  10. Class X - Loose, for example, Messier 68

  11. Class XI - Very loose towards the center, for example, Messier 55

  12. Class XII - Almost no concentration on the center, for example, Palomar 12


Formation


The formation of globular clusters is poorly understood because the traditionally, globular clusters have been described as a simple stellar population, in which all of the stars formed from the same giant molecular cloud. Thus have roughly the same age and metallicity. But modern observations show that nearly all globular clusters contain multiple populations. An example of this is the globular clusters in the Large Magellanic Cloud (LMC) that exhibit a bimodal population. During their youth, these LMC clusters may have encountered giant molecular clouds that triggered the second round of star formation. This star-forming period is relatively brief, compared with the age of many globular clusters. It has been proposed that the reason for this multiplicity in stellar populations could have a dynamical origin. In the Antennae galaxy, for example, the Hubble Space Telescope has observed clusters of clusters, regions in the galaxy that span hundreds of parsecs, where many of the clusters will eventually collide and merge. Many of them present a significant range in ages, hence possibly metallicities, and their merger could plausibly lead to clusters with bimodal or even multiple distributions of populations.


Observations of globular clusters show that these stellar formations arise primarily in regions of efficient star formation, and where the interstellar medium is at a higher density than in normal star-forming regions. Globular cluster formation is prevalent in starburst regions and in interacting galaxies. Research indicates a correlation between the mass of central supermassive black holes (SMBH) and the extent of the globular cluster systems of elliptical and lenticular galaxies. The mass of the SMBH in such a galaxy is often close to the combined mass of the galaxy's globular clusters.


No known globular clusters display active star formation, which is consistent with the view that globular clusters are typically the oldest objects in the Galaxy and were among the first collections of stars to form. Very large regions of star formation known as superstar clusters, such as Westerlund 1 in the Milky Way, maybe the precursors of globular clusters. There are many globular clusters with a retrograde orbit around the Milky Way Galaxy including the most massive known Milky Way globular cluster, Omega Centauri. The retrograde orbit may suggest that ω Cen is the remnant of a dwarf galaxy that was captured by the Milky Way.


Composition


Globular clusters are generally composed of hundreds of thousands of low-metal, old stars. The type of stars found in a globular cluster are similar to those in the bulge of a spiral galaxy but confined to a spheroid in which half the light is emitted within a radius of only a few to a few tens of parsecs. They are free of gas and dust and it is presumed that all of the gas and dust were long ago either turned into stars or blown out of the cluster by the massive first-generation stars.


Globular clusters can contain a high density of stars on average about 0.4 stars per cubic parsec, increasing to 100 or 1000 stars per cubic parsec in the core of the cluster. The typical distance between stars in a globular cluster is about 1 light-year, but at its core, the separation between stars averages about a third of a light-year, 13 times closer than Proxima Centauri.


Globular clusters are not thought to be favorable locations for the survival of planetary systems. Planetary orbits are dynamically unstable within the cores of dense clusters because of the perturbations of passing stars. A planet orbiting at 1 astronomical unit around a star that is within the core of a dense cluster such as 47 Tucanae would only survive on the order of 108 years. There is a planetary system orbiting a pulsar (PSR B1620−26) that belongs to the globular cluster M4, but these planets likely formed after the event that created the pulsar.


Some globular clusters, like Omega Centauri in the Milky Way and Mayall II in the Andromeda Galaxy, are extraordinarily massive, with several million solar masses (M☉) and multiple stellar populations. Both can be regarded as evidence that supermassive globular clusters are in fact the cores of dwarf galaxies that are consumed by the larger galaxies.[50] About a quarter of the globular cluster population in the Milky Way may have been accreted along with their host dwarf galaxy. More than 60% of the globular clusters in the outer halo (more than 25 kiloparsecs (82,000 ly) from the center) of the Andromeda Galaxy were likely accreted in a similar fashion.


Heavy element content


Globular clusters normally consist of Population II stars, which have a low proportion of elements other than hydrogen and helium when compared with Population I stars such as the Sun. Astronomers refer to these heavier elements as metals and to the proportions of these elements as metallicity. These elements are produced by stellar nucleosynthesis and then are recycled into the interstellar medium, where they enter the next generation of stars. Hence the proportion of metals can be an indication of the age of a star, with older stars typically having a lower metallicity. The Dutch astronomer Pieter Oosterhoff noticed that there appear to be two populations of globular clusters, which became known as Oosterhoff groups. The second group has a slightly longer period of RR Lyrae variable stars. Both groups have weak spectral lines of metallic elements. But the lines in the stars of the Oosterhoff type I (Oo I) cluster are not quite as weak as those in type II (Oo II).[56] Hence type I is referred to as metal-rich (for example, Terzan 7), while type II is metal-poor (for example, ESO 280-SC06).


These two populations have been observed in many galaxies, especially massive elliptical galaxies. Both groups are nearly as old as the universe itself and are of similar ages, but differ in their metal abundances. Many scenarios have been suggested to explain these subpopulations, including violent gas-rich galaxy mergers, the accretion of dwarf galaxies, and multiple phases of star formation in a single galaxy. In the Milky Way, the metal-poor clusters are associated with the halo and the metal-rich clusters with the bulge. In the Milky Way, it has been discovered that the large majority of the low metallicity clusters are aligned along a plane in the outer part of the galaxy's halo. This result argues in favor of the view that type II clusters in the galaxy were captured from a satellite galaxy, rather than being the oldest members of the Milky Way's globular cluster system as had been previously thought. The difference between the two cluster types would then be explained by a time delay between when the two galaxies formed their cluster systems.


Exotic components


Globular clusters have a very high star density, and therefore close interactions and near-collisions of stars occur relatively often. Due to these chance encounters, some exotic classes of stars, such as blue stragglers, millisecond pulsars, and low-mass X-ray binaries, are much more common in globular clusters. A blue straggler is thought to form from the merger of two stars, possibly as a result of an encounter with a binary system. The resulting star has a higher temperature than comparable stars in the cluster with the same luminosity and thus differs from the main sequence stars formed at the beginning of the cluster.


Astronomers have searched for black holes within globular clusters since the 1970s. The resolution requirements for this task are exacting, and it is only with the Hubble Space Telescope that the first claimed discoveries were made in 2002 and 2003. In independent programs, a 4,000 M☉ intermediate-mass black hole was suggested to exist based on HST observations in the globular cluster M15 and a 20,000 M☉ black hole in the Mayall II cluster in the Andromeda Galaxy. Both x-ray and radio emissions from Mayall II appear to be consistent with an intermediate-mass black hole. However, these claimed detections are controversial. The heaviest objects in globular clusters are expected to migrate to the cluster center due to mass segregation. As pointed out in two papers by Holger Baumgardt and collaborators, the mass-to-light ratio should rise sharply towards the center of the cluster, even without a black hole, in both M15 and Mayall II. Observations from 2018 find no evidence for an intermediate-mass black hole in any globular cluster, including M15, but cannot definitively rule out a 500–1000 M☉ one.

Confirmed intermediate-mass black holes in globular clusters would be significant for theories of galactic development and as possible sources for the supermassive black holes at their centers. The mass of these intermediate-mass black holes is proportional to the mass of the clusters, following a pattern previously discovered between supermassive black holes and their surrounding galaxies.


Color-magnitude diagram

Color-magnitude diagram for the globular cluster M3. There is a characteristic "knee" in the curve at magnitude 19 where stars begin entering the giant stage of their evolutionary path.
Color-magnitude diagram for the globular cluster M3. There is a characteristic knee in the curve at magnitude 19 where stars begin entering the giant stage of their evolutionary path.

The Hertzsprung-Russell diagram (HR-diagram) is a graph of a large sample of stars that plots their visual absolute magnitude against their color index. The color index, B−V, is the difference between the magnitude of the star in blue light, or B, and the magnitude in visual light (green-yellow), or V. Large positive values indicate a red star with a cool surface temperature, while negative values imply a blue star with a hotter surface. When the stars near the Sun are plotted on an HR diagram, it displays a distribution of stars of various masses, ages, and compositions. Many of the stars lie relatively close to a sloping curve with increasing absolute magnitude as the stars are hotter, known as main-sequence stars. The diagram also typically includes stars that are in later stages of their evolution and have wandered away from this main-sequence curve.


As all the stars of a globular cluster are at approximately the same distance from the Earth, their absolute magnitudes differ from their visual magnitude by about the same amount. The main-sequence stars in the globular cluster will fall along a line that is believed to be comparable to similar stars in the solar neighborhood. The accuracy of this assumption is confirmed by comparable results obtained by comparing the magnitudes of nearby short-period variables, such as RR Lyrae stars and cepheid variables, with those in the cluster. By matching up these curves on the HR diagram the absolute magnitude of main-sequence stars in the cluster can be determined. This in turn provides a distance estimate to the cluster, based on the visual magnitude of the stars. The difference between the relative and absolute magnitude, the distance modulus, yields this estimate of the distance.


When the stars of a particular globular cluster are plotted on an HR diagram, in many cases nearly all of the stars fall upon a relatively well-defined curve. This differs from the HR diagram of stars near the Sun, which lumps together stars of differing ages and origins. The shape of the curve for a globular cluster is characteristic of a grouping of stars that were formed at approximately the same time and from the same materials, differing only in their initial mass. As the position of each star in the HR diagram varies with age, the shape of the curve for a globular cluster can be used to measure the overall age of the star population.


However, the above-mentioned historic process of determining the age and distance to globular clusters is not as robust as first thought, since the morphology and luminosity of globular cluster stars in color-magnitude diagrams are influenced by numerous parameters, many of which are still being actively researched. Certain clusters even display populations that are absent from other globular clusters (e.g., blue hook stars), or feature multiple populations. The historical paradigm that all globular clusters consist of stars born at exactly the same time, or sharing exactly the same chemical abundance, has likewise been overturned (e.g., NGC 2808).


Further, the morphology of the cluster stars in a color-magnitude diagram, which includes the brightnesses of distance indicators such as RR Lyrae variable members, can be influenced by observational biases. One such effect is called blending, and it arises because the cores of globular clusters are so dense that in low-resolution observations multiple (unresolved) stars may appear as a single target. Thus the brightness measured for that seemingly single star (e.g., an RR Lyrae variable) is erroneously too bright, given those unresolved stars contributed to the brightness determined. Consequently, the computed distance is wrong, so the blending effect can introduce a systematic uncertainty into the cosmic distance ladder and may bias the estimated age of the Universe and the Hubble constant.


The most massive main-sequence stars will have the highest absolute magnitude, and these will be the first to evolve into the giant star stage. As the cluster ages, stars of successively lower masses will also enter the giant star stage. Thus the age of a single population cluster can be measured by looking for the stars that are just beginning to enter the giant star stage. This forms a knee in the HR diagram, bending to the upper right from the main-sequence line. The absolute magnitude at this bend is directly a function of the age of the globular cluster, so an age scale can be plotted on an axis parallel to the magnitude.


Globular clusters can be dated by looking at the temperatures of the coolest white dwarfs. Typical results for globular clusters are that they may be as old as 12.7 billion years. This is in contrast to open clusters which are rarely older than about 500 million years. The ages of globular clusters place a bound on the age limit of the entire universe. This lower limit has been a significant constraint in cosmology. Historically, astronomers were faced with age estimates of globular clusters that appeared older than cosmological models would allow.[80] However, better measurements of cosmological parameters through deep sky surveys and satellites such as the Hubble Space Telescope appear to have resolved this issue.


Evolutionary studies of globular clusters can be used to determine changes due to the starting composition of the gas and dust that formed the cluster. That is, the evolutionary tracks change with changes in the abundance of heavy elements. The data obtained from studies of globular clusters are then used to study the evolution of the Milky Way as a whole. In globular clusters a few stars known as blue stragglers are observed, apparently continuing the main sequence in the direction of brighter, bluer stars. In a few clusters, two sequences of blue stragglers can be detected, one more blue than the other. How blue stragglers form is still unclear, but most models suggest that these stars are the result of interactions in multiple star systems, either by stellar mergers or by the transfer of material from one star to another.

Morphology


In contrast to open clusters, most globular clusters remain gravitationally bound for time periods comparable to the life spans of the majority of their stars. However, strong tidal interactions with other large masses result in the dispersal of the stars, leaving behind tidal tails of stars removed from the cluster. After they are formed, the stars in the globular cluster begin to interact gravitationally with each other. As a result, the velocity vectors of the stars are steadily modified, and the stars lose any history of their original velocity. The characteristic interval for this to occur is the relaxation time. This is related to the characteristic length of time a star needs to cross the cluster as well as the number of stellar masses in the system. The value of the relaxation time varies by cluster, but a typical value is on the order of 109 years.


Although globular clusters generally appear spherical in form, ellipticities can occur due to tidal interactions. Clusters within the Milky Way and the Andromeda Galaxy are typically oblate spheroids in shape, while those in the Large Magellanic Cloud are more elliptical.


Radii


Astronomers characterize the morphology of a globular cluster by means of standard radii. These are the core radius (rc), the half-light radius (rh), and the tidal (or Jacobi) radius (rt). The overall luminosity of the cluster steadily decreases with distance from the core, and the core radius is the distance at which the apparent surface luminosity has dropped by half. A comparable quantity is the half-light radius or the distance from the core within which half the total luminosity from the cluster is received. This is typically larger than the core radius. The half-light radius includes stars in the outer part of the cluster that happens to lie along the line of sight, so theorists will also use the half-mass radius (rm), the radius from the core that contains half the total mass of the cluster. When the half-mass radius of a cluster is small relative to the overall size, it has a dense core. An example of this is Messier 3 (M3), which has an overall visible dimension of about 18 arc minutes, but a half-mass radius of only 1.12 arc minutes.


Almost all globular clusters have a half-light radius of less than 10 pc, although there are well-established globular clusters with very large radii (i.e. NGC 2419 (Rh = 18 pc) and Palomar 14 (Rh = 25 pc)). Finally, the tidal radius, or Hill sphere, is the distance from the center of the globular cluster at which the external gravitation of the galaxy has more influence over the stars in the cluster than does the cluster itself. This is the distance at which the individual stars belonging to a cluster can be separated away by the galaxy. The tidal radius of M3 is about 40 arc minutes or about 113 pc[98] at a distance of 10.4 kpc.


Mass segregation, luminosity, and core-collapse


In measuring the luminosity curve of a given globular cluster as a function of distance from the core, most clusters in the Milky Way increase steadily in luminosity as this distance decreases, up to a certain distance from the core, then the luminosity levels off. Typically this distance is about 1–2 parsecs from the core. About 20% of the globular clusters have undergone a process termed core collapse. In this type of cluster, the luminosity continues to increase steadily all the way to the core region. Examples of core-collapsed globular clusters include M15 and M30.


Core-collapse is thought to occur when the more massive stars in a globular cluster encounter their less massive companions. Over time, dynamic processes cause individual stars to migrate from the center of the cluster to the outside. This results in a net loss of kinetic energy from the core region, leading the remaining stars grouped in the core region to occupy a more compact volume. When this gravothermal instability occurs, the central region of the cluster becomes densely crowded with stars and the surface brightness of the cluster forms a power-law cusp. A core-collapse is not the only mechanism that can cause such a luminosity distribution; a massive black hole at the core can also result in a luminosity cusp. Over a lengthy period of time, this leads to a concentration of massive stars near the core, a phenomenon called mass segregation.

The dynamical heating effect of binary star systems works to prevent an initial core collapse of the cluster. When a star passes near a binary system, the orbit of the latter pair tends to contract, releasing energy. Only after the primordial supply of binaries is exhausted due to interactions can a deeper core-collapse proceed. In contrast, the effect of tidal shocks as a globular cluster repeatedly passes through the plane of a spiral galaxy tends to significantly accelerate the core collapse. The different stages of core collapse may be divided into three phases. During a globular cluster's adolescence, the process of core collapse begins with stars near the core. The interactions between binary star systems prevent further collapse as the cluster approaches middle age. Finally, the central binaries are either disrupted or ejected, resulting in a tighter concentration at the core.


The interaction of stars in the collapsed core region causes tight binary systems to form. As other stars interact with these tight binaries, they increase the energy at the core, which causes the cluster to re-expand. In the meantime for a core-collapse is typically less than the age of the galaxy, many of a galaxy's globular clusters may have passed through a core-collapse stage, then re-expanded.


The Hubble Space Telescope has been used to provide convincing observational evidence of this stellar mass-sorting process in globular clusters. Heavier stars slow down and crowd at the cluster's core, while lighter stars pick up speed and tend to spend more time at the cluster's periphery. The globular star cluster 47 Tucanae, which is made up of about 1 million stars, is one of the densest globular clusters in the Southern Hemisphere. This cluster was subjected to an intensive photographic survey, which allowed astronomers to track the motion of its stars. Precise velocities were obtained for nearly 15,000 stars in this cluster. The overall luminosities of the globular clusters within the Milky Way and the Andromeda Galaxy can be modeled by means of a Gaussian curve. This gaussian can be represented by means of an average magnitude Mv and a variance σ2. This distribution of globular cluster luminosities is called the Globular Cluster Luminosity Function (GCLF). For the Milky Way, Mv = −7.20 ± 0.13, σ = 1.1 ± 0.1 magnitudes. The GCLF has been used as a standard candle for measuring the distance to other galaxies, under the assumption that the globular clusters in remote galaxies follow the same principles as they do in the Milky Way.


N-body simulations


In physics and astronomy, an N-body simulation is a simulation of a dynamical system of particles, usually under the influence of physical forces, such as gravity (see the n-body problem). N-body simulations are widely used tools in astrophysics, from investigating the dynamics of few-body systems like the Earth-Moon-Sun system to understanding the evolution of the large-scale structure of the universe. In physical cosmology, N-body simulations are used to study processes of non-linear structure formation such as galaxy filaments and galaxy halos from the influence of dark matter. Direct N-body simulations are used to study the dynamical evolution of star clusters. An N-body simulation of the cosmological formation of a cluster of galaxies in an expanding universe.


Computing the interactions between the stars within a globular cluster requires solving what is termed the N-body problem. That is, each of the stars within the cluster continually interacts with the other N−1 stars, where N is the total number of stars in the cluster. The naive CPU computational cost for a dynamic simulation increases in proportion to N 2 (each of N objects must interact pairwise with each of the other N objects), so the potential computing requirements to accurately simulate such a cluster can be enormous. An efficient method of mathematically simulating the N-body dynamics of a globular cluster is done by subdividing into small volumes and velocity ranges and using probabilities to describe the locations of the stars. The motions are then described by means of a formula called the Fokker–Planck equation, often using a model describing the mass density as a function of radius such as a Plummer model. This can be solved by a simplified form of the equation, or by running Monte Carlo simulations and using random values. The simulation becomes more difficult when the effects of binaries and the interaction with external gravitation forces (such as from the Milky Way galaxy) must also be included.


The results of N-body simulations have shown that the stars can follow unusual paths through the cluster, often forming loops and often falling more directly toward the core than would a single star orbiting a central mass. In addition, due to interactions with other stars that result in an increase in velocity, some of the stars gain sufficient energy to escape the cluster. Over long periods of time, this will result in a dissipation of the cluster, a process termed evaporation. The typical time scale for the evaporation of a globular cluster is 1010 years. In 2010 it became possible to directly compute, star by star, N-body simulations of a low-density globular cluster over the course of its lifetime.


Binary stars form a significant portion of the total population of stellar systems, with up to half of all field stars and open cluster stars occurring in binary systems. The present-day binary fraction in globular clusters is difficult to measure, and any information about their initial binary fraction is lost by subsequent dynamical evolution. Numerical simulations of globular clusters have demonstrated that binaries can hinder and even reverse the process of core-collapse in globular clusters. When a star in a cluster has a gravitational encounter with a binary system, a possible result is that the binary becomes more tightly bound and kinetic energy is added to the solitary star. When the massive stars in the cluster are sped up by this process, it reduces the contraction at the core and limits core collapse. The ultimate fate of a globular cluster must be either to accrete stars at its core, causing its steady contraction, or gradual shedding of stars from its outer layers.

  • Nature of the particles

The particles treated by the simulation may or may not correspond to physical objects which are particulate in nature. For example, an N-body simulation of a star cluster might have a particle per star, so each particle has some physical significance. On the other hand, a simulation of a gas cloud cannot afford to have a particle for each atom or molecule of gas as this would require on the order of 1023 particles for each mole of material (see Avogadro constant), so a single particle would represent some much larger quantity of gas (often implemented using Smoothed Particle Hydrodynamics). This quantity need not have any physical significance but must be chosen as a compromise between accuracy and manageable computer requirements.

  • Direct gravitational N-body simulations

N-body simulation of 400 objects with parameters close to those of Solar System planets.


In direct gravitational N-body simulations, the equations of motion of a system of N particles under the influence of their mutual gravitational forces are integrated numerically without any simplifying approximations. These calculations are used in situations where interactions between individual objects, such as stars or planets, are important to the evolution of the system. The first direct N-body simulations were carried out by Erik Holmberg at the Lund Observatory in 1941, determining the forces between stars in encountering galaxies via the mathematical equivalence between light propagation and gravitational interaction: putting light bulbs at the positions of the stars and measuring the directional light fluxes at the positions of the stars by a photocell, the equations of motion can be integrated with O(N) effort.


The first purely calculational simulations were then done by Sebastian von Hoerner at the Astronomisches Rechen-Institut in Heidelberg, Germany. Sverre Aarseth at the University of Cambridge (UK) has dedicated his entire scientific life to the development of a series of highly efficient N-body codes for astrophysical applications which use adaptive (hierarchical) time steps, an Ahmad-Cohen neighbor scheme, and regularization of close encounters. Regularization is a mathematical trick to remove the singularity in the Newtonian law of gravitation for two particles that approach each other arbitrarily close. Sverre Aarseth's codes are used to study the dynamics of star clusters, planetary systems, and galactic nuclei.

  • General relativity simulations

Many simulations are large enough that the effects of general relativity in establishing a Friedmann-Lemaitre-Robertson-Walker cosmology are significant. This is incorporated in the simulation as an evolving measure of distance (or scale factor) in a comoving coordinate system, which causes the particles to slow in comoving coordinates (as well as due to the redshifting of their physical energy). However, the contributions of general relativity and the finite speed of gravity can otherwise be ignored, as typical dynamical timescales are long compared to the light crossing time for the simulation, and the space-time curvature induced by the particles and the particle velocities are small. The boundary conditions of these cosmological simulations are usually periodic (or toroidal), so that one edge of the simulation volume matches up with the opposite edge.

  • Calculation optimizations

N-body simulations are simple in principle because they involve merely integrating the 6N ordinary differential equations defining the particle motions in Newtonian gravity. In practice, the number N of particles involved is usually very large (typical simulations include many millions, the Millennium simulation included ten billion) and the number of particle-particle interactions needing to be computed increases on the order of N^2, and so direct integration of the differential equations can be prohibitively computationally expensive. Therefore, a number of refinements are commonly used.


Numerical integration is usually performed over small timesteps using a method such as a leapfrog integration. However, all numerical integration leads to errors. Smaller steps give lower errors but run more slowly. Leapfrog integration is roughly 2nd order on the timestep, other integrators such as Runge–Kutta methods can have 4th order accuracy or much higher. One of the simplest refinements is that each particle carries with it its own timestep variable so that particles with widely different dynamical times don't all have to be evolved forward at the rate that with the shortest time. There are two basic approximation schemes to decrease the computational time for such simulations. These can reduce the computational complexity to O(N log N) or better, at the loss of accuracy.

  • Tree methods

In tree methods, such as a Barnes–Hut simulation, an octree is usually used to divide the volume into cubic cells and only interactions between particles from nearby cells need to be treated individually; particles in distant cells can be treated collectively as a single large particle centered at the distant cell's center of mass (or as a low-order multipole expansion). This can dramatically reduce the number of particle pair interactions that must be computed. To prevent the simulation from becoming swamped by computing particle-particle interactions, the cells must be refined to smaller cells in denser parts of the simulation which contain many particles per cell. For simulations where particles are not evenly distributed, the well-separated pair decomposition methods of Callahan and Kosaraju yield optimal O(n log n) time per iteration with a fixed dimension.

  • Particle mesh method

Another possibility is the particle mesh method in which space is discretized on a mesh and, for the purposes of computing the gravitational potential, particles are assumed to be divided between the nearby vertices of the mesh. Finding the potential energy Φ is easy because the Poisson equation


where G is Newton's constant and p is the density (number of particles at the mesh points), is trivial to solve by using the fast Fourier transform to go to the frequency domain where the Poisson equation has the simple form



where K vector is the comoving wavenumber and the hats denote Fourier transforms. Since the gravitational field can now be found by multiplying by -ik vector

and computing the inverse Fourier transform (or computing the inverse transform and then using some other method). Since this method is limited by the mesh size, in practice a smaller mesh or some other technique (such as combining with a tree or a simple particle-particle algorithm) is used to compute the small-scale forces. Sometimes an adaptive mesh is used, in which the mesh cells are much smaller in the denser regions of the simulation.

  • Special-case optimizations

Several different gravitational perturbation algorithms are used to get fairly accurate estimates of the path of objects in the solar system. People often decide to put a satellite in a frozen orbit. The path of a satellite closely orbiting the Earth can be accurately modeled starting from the 2-body elliptical orbit around the center of the Earth, and adding small corrections due to the oblateness of the Earth, gravitational attraction of the Sun and Moon, atmospheric drag, etc. It is possible to find a frozen orbit without calculating the actual path of the satellite.


The path of a small planet, comet, or long-range spacecraft can often be accurately modeled starting from the 2-body elliptical orbit around the sun, and adding small corrections from the gravitational attraction of the larger planets in their known orbits. Some characteristics of the long-term paths of a system of particles can be calculated directly. The actual path of any particular particle does not need to be calculated as an intermediate step. Such characteristics include Lyapunov stability, Lyapunov time, various measurements from ergodic theory, etc.

  • Two-particle systems

Although there are millions or billions of particles in typical simulations, they typically correspond to a real particle with a very large mass, typically 109 solar masses. This can introduce problems with short-range interactions between the particles such as the formation of two-particle binary systems. As the particles are meant to represent large numbers of dark matter particles or groups of stars, these binaries are unphysical. To prevent this, a softened Newtonian force law is used, which does not diverge as the inverse-square radius at short distances. Most simulations implement this quite naturally by running the simulations on cells of finite size. It is important to implement the discretization procedure in such a way that particles always exert a vanishing force on themselves.

  • Softening

Softening is a numerical trick used in N-body techniques to prevent numerical divergences when a particle comes too close to another (and the force goes to infinity). This is obtained by modifying the regularized gravitational potential of each particle as

,(rather than 1/r) where epsilon is the softening parameter. The value of the softening parameter should be set small enough to keep simulations realistic.

  • Incorporating baryons, leptons, and photons into simulations

Many simulations simulate only cold dark matter and thus include only the gravitational force. Incorporating baryons, leptons, and photons into the simulations dramatically increases their complexity, and often radical simplifications of the underlying physics must be made. However, this is an extremely important area and many modern simulations are now trying to understand processes that occur during galaxy formation which could account for galaxy bias.

  • Computational complexity

Reif and Tate[3] prove that if the n-body reachability problem is defined as follows given n bodies satisfying a fixed electrostatic potential law, determining if a body reaches a destination ball in a given time bound where we require a poly(n) bits of accuracy and the target time is poly(n) is in PSPACE. On the other hand, if the question is whether the body eventually reaches the destination ball, the problem is PSPACE-hard. These bounds are based on similar complexity bounds obtained for ray tracing.


Intermediate forms


The distinction between cluster types is not always clear-cut, and objects have been found that blur the lines between the categories. For example, BH 176 in the southern part of the Milky Way has properties of both an open and a globular cluster. In 2005, astronomers discovered a completely new type of star cluster in the Andromeda Galaxy (M31), which is, in several ways, very similar to globular clusters. The new-found clusters contain hundreds of thousands of stars, a similar number to that found in globular clusters. The clusters share other characteristics with globular clusters such as stellar populations and metallicity. What distinguishes them from the globular clusters is that they are much larger several hundred light-years across and hundreds of times less dense. The distances between the stars are, therefore, much greater within the newly discovered extended clusters. Parametrically, these clusters lie somewhere between a globular cluster and a dwarf spheroidal galaxy. The formation of these Andromeda Galaxy halo clusters is likely related in some way to accretion. Why Andromeda has such clusters, while the Milky Way does not, is not yet known. It is also unknown if any other galaxy contains these types of clusters, but it would be very unlikely that Andromeda is the sole galaxy with extended clusters.


Tidal encounters


When a globular cluster has a close encounter with a large mass, such as the core region of a galaxy, it undergoes a tidal interaction. The difference in the pull of gravity between the part of the cluster nearest the mass and the pull on the furthest part of the cluster results in a tidal force. A tidal shock occurs whenever the orbit of a cluster takes it through the plane of a galaxy.


As a result of a tidal shock, streams of stars can be pulled away from the cluster halo, leaving only the core part of the cluster. These tidal interaction effects create tails of stars that can extend up to several degrees of arc away from the cluster. These tails typically both precede and follow the cluster along its orbit. The tails can accumulate significant portions of the original mass of the cluster and can form clumplike features.


The globular cluster Palomar 5, for example, is near the apogalactic point of its orbit after passing through the Milky Way. Streams of stars extend outward toward the front and rear of the orbital path of this cluster, stretching out to distances of 13,000 light-years. Tidal interactions have stripped away much of the mass from Palomar 5, and further interactions as it passes through the galactic core are expected to transform it into a long stream of stars orbiting the Milky Way halo.


The Milky Way is in the process of tidally stripping the Sagittarius Dwarf Spheroidal Galaxy of stars and globular clusters through the Sagittarius Stream. As many as 20% of the globular clusters in the Milky Way's outer halo may have originated in that galaxy. Palomar 12, for example, most likely originated in the Sagittarius Dwarf Spheroidal but is now associated with the stream of the Milky Way. Tidal interactions like these add kinetic energy to a globular cluster, dramatically increasing the evaporation rate and shrinking the size of the cluster.[88] Not only does tidal shock strip off the outer stars from a globular cluster, but the increased evaporation accelerates the process of core-collapse.


Planets


Astronomers are searching for exoplanets of stars in globular star clusters. In 2000, the results of a search for giant planets in the globular cluster 47 Tucanae were announced. The lack of any successful discoveries suggests that the abundance of elements (other than hydrogen or helium) necessary to build these planets may need to be at least 40% of the abundance in the Sun. Terrestrial planets are built from heavier elements such as silicon, iron, and magnesium. The very low abundance of these elements in globular clusters means that the member stars have a far lower likelihood of hosting Earth-mass planets when compared with stars in the neighborhood of the Sun. Hence the halo region of the Milky Way galaxy, including globular cluster members, are unlikely to host habitable terrestrial planets.


In spite of the lower likelihood of giant planet formation, just such an object has been found in the globular cluster Messier 4. This planet was detected orbiting a pulsar in the binary star system PSR B1620-26. The eccentric and highly inclined orbit of the planet suggests it may have been formed around another star in the cluster, then was later exchanged into its current arrangement. The likelihood of close encounters between stars in a globular cluster can disrupt planetary systems, some of which break free to become rogue planets. Even close orbiting planets can become disrupted, potentially leading to orbital decay and an increase in orbital eccentricity and tidal effects.


Dark globular cluster


A dark globular cluster is a proposed type of globular star cluster that has an unusually high mass for the number of stars within it. Proposed in 2015 on the basis of observational data, dark globular clusters are believed to be populated by objects with significant dark matter components, such as central massive black holes.

Observations with ESO's Very Large Telescope in Chile have discovered a new class of dark globular star clusters around this galaxy. In the image, the dark globulars are circled in red. Normal globulars are circled in blue, and globulars showing similar properties to dwarf galaxies are in green.
Observations with ESO's Very Large Telescope in Chile have discovered a new class of dark globular star clusters around this galaxy. In the image, the dark globulars are circled in red. Normal globulars are circled in blue, and globulars showing similar properties to dwarf galaxies are in green.

The observational data for dark globular clusters comes from the Very Large Telescope (VLT) in Chile which observed the vicinity of the galaxy Centaurus A. Many of the globular clusters inside that galaxy are brighter and more massive than those orbiting the Milky Way and a sample of 125 globular clusters around Centaurus A was studied using the VLT's FLAMES instrument. While globular clusters are normally considered to be almost devoid of dark matter. The study of the dynamical properties of sampled clusters suggested the presence of exotically concentrated dark matter. The existence of dark globular clusters would suggest that their formation and evolution are markedly different from other globular clusters in Centaurus A and the Local Group.

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