Olbers' paradox

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Olbers' paradox in action

 

In astrophysics and physical cosmology, Olbers' paradox, named after the German astronomer Heinrich Wilhelm Olbers (1758–1840) and also called the "dark night sky paradox", is the argument that the darkness of the night sky conflicts with the assumption of an infinite and eternal static universe. The darkness of the night sky is one of the pieces of evidence for a non-static universe such as the Big Bang model. If the universe is static, homogeneous at a large scale, and populated by an infinite number of stars, any sight line from Earth must end at the (very bright) surface of a star, so the night sky should be completely bright. This contradicts the observed darkness of the night.

History

Edward Robert Harrison's Darkness at Night: A Riddle of the Universe (1987) gives an account of the dark night sky paradox, seen as a problem in the history of science. According to Harrison, the first to conceive of anything like the paradox was Thomas Digges, who was also the first to expound the Copernican system in English and also postulated an infinite universe with infinitely many stars.^[1] Kepler also posed the problem in 1610, and the paradox took its mature form in the 18th century work of Halley and Cheseaux.^[2] The paradox is commonly attributed to the German amateur astronomer Heinrich Wilhelm Olbers, who described it in 1823, but Harrison shows convincingly that Olbers was far from the first to pose the problem, nor was his thinking about it particularly valuable.
Harrison argues that the first to set out a satisfactory resolution of the paradox was Lord Kelvin, in a little known 1901 paper,^[3] and that Edgar Allan Poe's essay Eureka (1848) curiously anticipated some qualitative aspects of Kelvin's argument:

Were the succession of stars endless, then the background of the sky would present us a uniform luminosity, like that displayed by the Galaxy – since there could be absolutely no point, in all that background, at which would not exist a star. The only mode, therefore, in which, under such a state of affairs, we could comprehend the voids which our telescopes find in innumerable directions, would be by supposing the distance of the invisible background so immense that no ray from it has yet been able to reach us at all.^[4]

The paradox

What if every line of sight ended in a star? (Infinite universe assumption #2)

The paradox is that a static, infinitely old universe with an infinite number of stars distributed in an infinitely large space would be bright rather than dark.

To show this, we divide the universe into a series of concentric shells, 1 light year thick. Thus, a certain number of stars will be in the shell 1,000,000,000 to 1,000,000,001 light years away. If the universe is homogeneous at a large scale, then there would be four times as many stars in a second shell between 2,000,000,000 to 2,000,000,001 light years away. However, the second shell is twice as far away, so each star in it would appear four times dimmer than the first shell. Thus the total light received from the second shell is the same as the total light received from the first shell.

Thus each shell of a given thickness will produce the same net amount of light regardless of how far away it is. That is, the light of each shell adds to the total amount. Thus the more shells, the more light. And with infinitely many shells there would be a bright night sky.

Dark clouds could obstruct the light. But in that case the clouds would heat up, until they were as hot as stars, and then radiate the same amount of light.

Kepler saw this as an argument for a finite observable universe, or at least for a finite number of stars. In general relativity theory, it is still possible for the paradox to hold in a finite universe:^[5] though the sky would not be infinitely bright, every point in the sky would still be like the surface of a star.

In a universe of three dimensions with stars distributed evenly, the number of stars would be proportional to volume. If the surface of concentric sphere shells were considered, the number of stars on each shell would be proportional to the square of the radius of the shell. In the picture above, the shells are reduced to rings in two dimensions with all of the stars on them.

The mainstream explanation

Poet Edgar Allan Poe suggested that the finite size of the observable universe resolves the apparent paradox.^[6] More specifically, because the universe is finitely old and the speed of light is finite, only finitely many stars can be observed within a given volume of space visible from Earth (although the whole universe can be infinite in space).^[7] The density of stars within this finite volume is sufficiently low that any line of sight from Earth is unlikely to reach a star.
However, the Big Bang theory introduces a new paradox: it states that the sky was much brighter in the past, especially at the end of the recombination era, when it first became transparent. All points of the local sky at that era were comparable in brightness to the surface of the sun, due to the high temperature of the universe in that era; and most light rays will terminate not in a star but in the relic of the Big Bang.

This paradox is explained by the fact that the Big Bang theory also involves the expansion of space which can cause the energy of emitted light to be reduced via redshift. More specifically, the extreme levels of radiation from the Big Bang have been redshifted to microwave wavelengths (1100 times longer than its original wavelength) as a result of the cosmic expansion, and thus form the cosmic microwave background radiation. This explains the relatively low light densities present in most of our sky despite the assumed bright nature of the Big Bang. The redshift also affects light from distant stars and quasars, but the diminution is minor, since the most distant galaxies and quasars have redshifts of only around 5 to 8.6.

Alternative explanations

Steady state

The redshift hypothesised in the Big Bang model would by itself explain the darkness of the night sky, even if the universe were infinitely old. The steady state cosmological model assumed that the universe is infinitely old and uniform in time as well as space. There is no Big Bang in this model, but there are stars and quasars at arbitrarily great distances. The light from these distant stars and quasars will be redshifted accordingly (by the Doppler effect and thermalisation^[8]), so that the total light flux from the sky remains finite. Thus the observed radiation density (the sky brightness of extragalactic background light) can be independent of finiteness of the Universe. Mathematically, the total electromagnetic energy density (radiation energy density) in thermodynamic equilibrium from Planck's law is

e.g. for temperature 2.7 K it is 40 fJ/m³ ... 4.5×10⁻³¹ kg/m³ and for visible temperature 6000 K we get 1 J/m³ ... 1.1×10⁻¹⁷ kg/m³. But the total radiation emitted by a star (or other cosmic object) is at most equal to the total nuclear binding energy of isotopes in the star. For the density of the observable universe of about 4.6×10⁻²⁸ kg/m³ and given the known abundance of the chemical elements, the corresponding maximal radiation energy density of 9.2×10⁻³¹ kg/m³, i.e. temperature 3.2 K.^[9][10]
This is close to the summed energy density of the cosmic microwave background and the cosmic neutrino background. The Big Bang hypothesis, by contrast, predicts that the CBR should have the same energy density as the binding energy density of the primordial helium, which is much greater than the binding energy density of the non-primordial elements; so it gives almost the same result. But (neglecting quantum fluctuations in the early universe) the Big Bang would also predict a uniform distribution of CBR, while the steady-state model predicts nothing about its distribution^{[citation needed]}. Nevertheless the isotropy is very probable in steady state as in the kinetic theory.

Since the speed of light is a constant value, regardless of the shift towards infrared frequencies, the universe is still sharply constrained to finite sizes in space as well as in time. Some models of an infinite (Steady State theory or static universe) solution of the universe are still viable, and Olber's paradox cannot sharply distinguish between them from some variants of the Big Bang model.

Finite age of stars

Stars have a finite age and a finite power, thereby implying that each star has a finite impact on a sky's light field density. But if the universe were infinitely old, there would be infinitely many other stars in the same angular direction, with an infinite total impact.

Absorption

A commonly proposed alternative explanation is that the universe is not transparent, and the light from distant stars is blocked by intermediate dark stars or absorbed by dust or gas, so that there is a bound on the distance from which light can reach the observer.

This would not resolve the paradox given the following argument: according to the laws of thermodynamics, the intermediate matter must eventually heat up (or cool down, if it was initially hotter) until it is in thermal equilibrium with the surrounding stars. Once this happens, the matter would then radiate the energy it receives from the stars at the same (average) temperature, so the sky would still appear uniformly bright.

Brightness

Suppose that the universe were not expanding, and always had the same stellar density; then the temperature of the universe would continually increase as the stars put out more radiation. Eventually, it would reach 3000 K (corresponding to a typical photon energy of 0.3 eV and so a frequency of 7.5×10¹³ Hz), and the photons would begin to be absorbed by the hydrogen plasma filling most of the universe, rendering outer space opaque. This maximal radiation density corresponds to about 1.2×10¹⁷ eV/m³ = 2.1×10⁻¹⁹ kg/m³, which is much greater than the observed value of 4.7×10⁻³¹ kg/m³.^[2] So the sky is about fifty billion times darker than it would be if the universe were neither expanding nor too young to have reached equilibrium yet.

Fractal star distribution

A different resolution, which does not rely on the Big Bang theory, was first proposed by Carl Charlier in 1908 and later rediscovered by Benoît Mandelbrot in 1974. They both postulated that if the stars in the universe were distributed in a hierarchical fractal cosmology (e.g., similar to Cantor dust)—the average density of any region diminishes as the region considered increases—it would not be necessary to rely on the Big Bang theory to explain Olbers' paradox. This model would not rule out a Big Bang but would allow for a dark sky even if the Big Bang had not occurred.

Mathematically, the light received from stars as a function of star distance in a hypothetical fractal cosmos is:

where:
r₀ = the distance of the nearest star. r₀ > 0;
r = the variable measuring distance from the Earth;
L(r) = average luminosity per star at distance r;
N(r) = number of stars at distance r.
The function of luminosity from a given distance L(r)N(r) determines whether the light received is finite or infinite. For any luminosity from a given distance L(r)N(r) proportional to r^a, is infinite for a ≥ −1 but finite for a < −1. So if L(r) is proportional to r⁻², then for to be finite, N(r) must be proportional to r^b, where b < 1. For b = 1, the numbers of stars at a given radius is proportional to that radius. When integrated over the radius, this implies that for b = 1, the total number of stars is proportional to r². This would correspond to a fractal dimension of 2. Thus the fractal dimension of the universe would need to be less than 2 for this explanation to work.

This explanation is not widely accepted among cosmologists since the evidence suggests that the fractal dimension of the universe is at least 2.^[11][12][13] Moreover, the majority of cosmologists take the cosmological principle as a given, which assumes that matter at the scale of billions of light years is distributed isotropically. Contrasting this, fractal cosmology requires anisotropic matter distribution at the largest scales.

Globular cluster

From Wikipedia, the free encyclopedia

 

The Messier 80 globular cluster in the constellation Scorpius is located about 30,000 light-years from the Sun and contains hundreds of thousands of stars.^[1]

A globular cluster is a spherical collection of stars that orbits a galactic core as a satellite. Globular clusters are very tightly bound by gravity, which gives them their spherical shapes and relatively high stellar densities toward their centers. The name of this category of star cluster is derived from the Latin globulus—a small sphere. A globular cluster is sometimes known more simply as a globular.
Globular clusters, which are found in the halo of a galaxy, contain considerably more stars and are much older than the less dense galactic, or open clusters, which are found in the disk. Globular clusters are fairly common; there are about 150^[2] to 158^[3] currently known globular clusters in the Milky Way, with perhaps 10 to 20 more still undiscovered.^[4] Large galaxies can have more: Andromeda, for instance, may have as many as 500.^[5] Some giant elliptical galaxies, particularly those at the centers of galaxy clusters, such as M87,^[6] have as many as 13,000 globular clusters. These globular clusters orbit the galaxy out to large radii, 40 kiloparsecs (approximately 131,000 light-years) or more.^[7]

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.^[8] The Sagittarius Dwarf galaxy and the disputed Canis Major Dwarf galaxy appear to be in the process of donating their associated globular clusters (such as Palomar 12) to the Milky Way.^[9] This demonstrates how many of this galaxy's globular clusters might have been acquired in the past.

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. 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.^[10] However, recent conjectures by astronomers suggest that globular clusters and dwarf spheroidals may not be clearly separate and distinct types of objects.^[11]

Observation history

Early Globular Cluster Discoveries

Cluster name	Discovered by	Year
M22	Abraham Ihle	1665
ω Cen	Edmond Halley	1677
M5	Gottfried Kirch	1702
M13	Edmond Halley	1714
M71	Philippe Loys de Chéseaux	1745
M4	Philippe Loys de Chéseaux	1746
M15	Jean-Dominique Maraldi	1746
M2	Jean-Dominique Maraldi	1746

The first globular cluster discovered was M22 in 1665 by Abraham Ihle, a German amateur astronomer.^[12] However, given the small aperture of early telescopes, individual stars within a globular cluster were not resolved until Charles Messier observed M4.^[13] The first eight globular clusters discovered are shown in the table. Subsequently, Abbé Lacaille would list NGC 104, NGC 4833, M55, M69, and NGC 6397 in his 1751–52 catalogue. The M before a number refers to the catalogue of Charles Messier, while NGC is from the New General Catalogue by John Dreyer.

William Herschel began a survey program in 1782 using larger telescopes and was able to resolve the stars in all 33 of the known globular clusters. In addition he found 37 additional clusters. In Herschel's 1789 catalog of deep sky objects, his second such, he became the first to use the name globular cluster as their description.^[13]

The number of globular clusters discovered continued to increase, reaching 83 in 1915, 93 in 1930 and 97 by 1947. A total of 152 globular clusters have now been discovered in the Milky Way galaxy, out of an estimated total of 180 ± 20.^[4] These additional, undiscovered globular clusters are believed to be 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 would use 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 distance to the clusters.^[14]

NGC 7006 is a highly concentrated, Class I globular cluster.

Of the globular clusters within our Milky Way, the majority are found in the vicinity of the galactic core, and the large majority lie on the side of the celestial sky centered on the core. In 1918 this strongly asymmetrical distribution was used by Harlow 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.^[15] While his distance estimate was significantly in error, it did demonstrate that the dimensions of the galaxy were much greater than had been previously thought. His error was because dust in the Milky Way diminished the amount of light from a globular cluster that reached the earth, thus making it appear farther away. Shapley's estimate was, however, within the same order of magnitude as the currently accepted value.

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

Classification of globulars

Shapley was subsequently assisted in his studies of clusters by Henrietta Swope and Helen Battles Sawyer (later Hogg). In 1927–29, Harlow Shapley and Helen Sawyer began categorizing clusters according to the degree of concentration the system has toward the core. The most concentrated clusters were identified as Class I, with successively diminishing concentrations ranging to Class XII. This became known as the Shapley–Sawyer Concentration Class. (It is sometimes given with numbers [Class 1–12] rather than Roman numerals.)^[16]

Formation

NGC 2808 contains three distinct generations of stars.^[17] NASA image

At present, the formation of globular clusters remains a poorly understood phenomenon, and it remains uncertain whether the stars in a globular cluster form in a single generation, or are spawned across multiple generations over a period of several hundred million years. In many globular clusters, most of the stars are at approximately the same stage in stellar evolution, suggesting that they formed at about the same time.^[18] However, the star formation history varies from cluster to cluster, with some clusters showing distinct populations of stars. 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 a second round of star formation.^[19] This star-forming period is relatively brief, compared to the age of many globular clusters.^[20] It has also been proposed that the reason for this multiplicity in stellar populations could have a dynamical origin. In e.g. the Antennae galaxy we observe thanks to the Hubble Space Telescope clusters of clusters, regions in the galaxy that span hundreds of parsec, 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 a bimodal or even multiple distribution of populations.^[21]

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.^[22] Research indicates a correlation between the mass of a 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.^[23]

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 super star clusters, such as Westerlund 1 in the Milky Way, may be the precursors of globular clusters.^[24]

Composition

Djorgovski 1's stars contain hydrogen and helium, but not much else. In astronomical terms, they are described as "metal-poor".^[25]

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 volume of only a few million cubic parsecs. They are free of gas and dust and it is presumed that all of the gas and dust was long ago turned into 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.^[26] The typical distance between stars in a globular cluster is about 1 light year,^[27] but at its core, the separation is comparable to the size of the Solar System (100 to 1000 times closer than stars near the Solar System).^[28]

However, they 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.^[29] 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.^[30]

Some globular clusters, like Omega Centauri in our Milky Way and G1 in M31, are extraordinarily massive, with several million solar masses 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.^[31] About a quarter of the globular cluster population in the Milky Way may have been accreted along with their host dwarf galaxy.^[32]

Several globular clusters (like M15) have extremely massive cores which may harbor black holes,^[33] although simulations suggest that a less massive black hole or central concentration of neutron stars or massive white dwarfs explain observations equally well.

Metallic content

Messier 53 has surprised astronomers with its unusual number of a type of star called blue stragglers.^[34]

Globular clusters normally consist of Population II stars, which have a low proportion of elements other than hydrogen and helium when compared to Population I stars such as the Sun. Astronomers refer to these heavier elements as metals and to the proportions of these elements as the 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.^[35]

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.^[36] Both groups have weak lines of metallic elements. But the lines in the stars of Oosterhoff type I (OoI) cluster are not quite as weak as those in type II (OoII).^[36] Hence type I are referred to as "metal-rich" (e.g. Terzan 7^[37]) while type II are "metal-poor" (e.g. ESO 280-SC06^[38]).

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 our Milky Way, the metal-poor clusters are associated with the halo and the metal-rich clusters with the bulge.^[39]

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.^[40]

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 formed from the merger of two stars, possibly as a result of an encounter with a binary system.^[41] 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.^[42]

Globular cluster M15 may have an intermediate-mass black hole at its core. NASA image.

Astronomers have searched for black holes within globular clusters since the 1970s. The resolution requirements for this task, however, are exacting, and it is only with the Hubble space telescope that the first confirmed discoveries have been made. In independent programs, a 4,000 solar mass intermediate-mass black hole has been suggested to exist based on HST observations in the globular cluster M15 and a 20,000 solar mass black hole in the Mayall II cluster in the Andromeda Galaxy.^[43] Both x-ray and radio emissions from Mayall II appear to be consistent with an intermediate-mass black hole.^[44]

These are of particular interest because they are the first black holes discovered that were intermediate in mass between the conventional stellar-mass black hole and the supermassive black holes discovered at the cores of galaxies. 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.

Claims of intermediate mass black holes have been met with some skepticism. 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^[45] and Mayall II.^[46]

Color-magnitude diagram

Messier 5 is a globular cluster consisting of hundreds of thousands of stars bound together by their collective gravity.^[47]

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. However 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 us, 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.^[48]

By matching up these curves on the HR diagram the absolute magnitude of main-sequence stars in the cluster can also 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.^[49]

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.^[50]

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

The most massive main-sequence stars will also 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 globular cluster, so an age scale can be plotted on an axis parallel to the magnitude.

In addition, 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.^[51] This is in contrast to open clusters which are only tens of millions of years old.

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. During the early 1990s, astronomers were faced with age estimates of globular clusters that appeared older than cosmological models would allow. However, better measurements of cosmological parameters through deep sky surveys and satellites such as COBE have resolved this issue as have computer models of stellar evolution that have different models of mixing.^{[citation needed]}

Evolutionary studies of globular clusters can also 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.^[52]

In globular clusters a few stars known as blue stragglers are observed, apparently continuing the main sequence in the direction of brighter, bluer stars. The origins of these stars is still unclear, but most models suggest that these stars are the result of mass transfer in multiple star systems.^[53]

Morphology

NGC 411 is classified as an open cluster.^[54]

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, a possible exception is when strong tidal interactions with other large masses result in the dispersal of the stars.

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.^[55] The value of the relaxation time varies by cluster, but the mean value is on the order of 10⁹ years.

Ellipticity of Globulars

Galaxy	Ellipticity[56]
Milky Way	0.07±0.04
LMC	0.16±0.05
SMC	0.19±0.06
M31	0.09±0.04

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.^[57]

Radii

Astronomers characterize the morphology of a globular cluster by means of standard radii. These are the core radius (r_c), the half-light radius (r_h) and the tidal radius (r_t). 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.^[58] 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.

Note that the half-light radius includes stars in the outer part of the cluster that happen to lie along the line of sight, so theorists will also use the half-mass radius (r_m)—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.^[59]

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 (R_h = 18 pc) and Palomar 14 (R_h = 25 pc)).^[11]

Finally the tidal radius 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 38 arc minutes.

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. However 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.^[60] An example of a core-collapsed globular is M15.

47 Tucanae – the second most luminous globular cluster in the Milky Way, after Omega Centauri.

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.^[61] (Note that 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.)^[62] 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.^[63][64] In contrast, the effect of tidal shocks as a globular cluster repeatedly passes through the plane of a spiral galaxy tends to significantly accelerate core collapse.^[65]

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. However, the interactions between binary star systems prevents 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. As the mean time 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.^[66]

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.^[67]

A 2008 study by John Fregeau of 13 globular clusters in the Milky Way shows that three of them have an unusually large number of X-ray sources, or X-ray binaries, suggesting the clusters are middle-aged. Previously, these globular clusters had been classified as being in old age because they had very tight concentrations of stars in their centers, another test of age used by astronomers. The implication is that most globular clusters, including the other ten studied by Fregeau, are not in middle age as previously thought, but are actually in 'adolescence'.^[68]

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 M_v and a variance σ². This distribution of globular cluster luminosities is called the Globular Cluster Luminosity Function (GCLF). (For the Milky Way, M_v = −7.20 ± 0.13, σ = 1.1 ± 0.1 magnitudes.)^[69] The GCLF has also 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

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 N³,^[70][71] so the potential computing requirements to accurately simulate such a cluster can be enormous.^[72] 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. This can be solved by a simplified form of the equation, or by running Monte Carlo simulations and using random values. However 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.^[73]

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.^[74] The typical time scale for the evaporation of a globular cluster is 10¹⁰ years.^[55] In 2010 it became possible to directly compute, star by star, N-body simulations of a globular cluster over the course of its lifetime.^[75]

Binary stars form a significant portion of the total population of stellar systems, with up to half of all stars occurring in binary systems. 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.^[42]

The ultimate fate of a globular cluster must be either to accrete stars at its core, causing its steady contraction,^[76] or gradual shedding of stars from its outer layers.^[77]

Intermediate forms

Messier 10 is a ball of stars that lies about 15000 light-years from Earth, in the constellation of Ophiuchus.^[78]

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.^[79]

In 2005, astronomers discovered a completely new type of star cluster in the Andromeda Galaxy, 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.^[80]

How these clusters are formed is not yet known, but their formation might well be related to that of globular clusters. Why M31 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 M31 is the sole galaxy with extended clusters.^[80]

 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.^[81] 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.^[82]

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.^[83] 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.

Tidal interactions add kinetic energy into a globular cluster, dramatically increasing the evaporation rate and shrinking the size of the cluster.^[55] Not only does tidal shock strip off the outer stars from a globular cluster, but the increased evaporation accelerates the process of core collapse. The same physical mechanism may be at work in Dwarf spheroidal galaxies such as the Sagittarius Dwarf, which appears to be undergoing tidal disruption due to its proximity to the Milky Way.

Orbits

There are many globular clusters with a retrograde orbit round the Milky Way Galaxy.^[84] A hypervelocity globular cluster was discovered around Messier 87 in 2014, having a velocity in excess of the escape velocity of M87.^[85]

Planets

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 to 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.^[86]

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.^[87] The likelihood of close encounters between stars in a globular cluster can disrupt planetary systems, some of which break loose to become free floating planets. Even close orbiting planets can become disrupted, potentially leading to orbital decay and an increase in orbital eccentricity and tidal effects.^[88]