Rotation curve of spiral galaxy Messier 33 (Triangulum)

The rotation curve of a

disc galaxy A disc galaxy (or disk galaxy) is a galaxy characterized by a galactic disc, a flattened circular volume of stars. These galaxies may or may not include a central non-disc-like region (a galactic bulge). Disc galaxy types include: * Spiral ga ...

(also called a velocity curve) is a plot of the orbital speeds of visible

star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...

s or gas in that galaxy versus their

radial distance In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the ...

from that galaxy's centre. It is typically rendered graphically as a

plot Plot or Plotting may refer to: Art, media and entertainment * Plot (narrative), the story of a piece of fiction Music * ''The Plot'' (album), a 1976 album by jazz trumpeter Enrico Rava * The Plot (band), a band formed in 2003 Other * ''Plot' ...

, and the data observed from each side of a spiral galaxy are generally asymmetric, so that data from each side are averaged to create the curve. A significant discrepancy exists between the experimental curves observed, and a curve derived by applying gravity theory to the matter observed in a galaxy. Theories involving

dark matter Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not ab ...

are the main postulated solutions to account for the variance. The rotational/orbital speeds of galaxies/stars do not follow the rules found in other orbital systems such as stars/planets and planets/moons that have most of their mass at the centre. Stars revolve around their galaxy's centre at equal or increasing speed over a large range of distances. In contrast, the orbital velocities of planets in planetary systems and moons orbiting planets decline with distance according to Kepler’s third law. This reflects the mass distributions within those systems. The mass estimations for galaxies based on the light they emit are far too low to explain the velocity observations. The galaxy rotation problem is the discrepancy between observed galaxy rotation curves and the theoretical prediction, assuming a centrally dominated mass associated with the observed luminous material. When mass profiles of galaxies are calculated from the distribution of stars in spirals and

mass-to-light ratio In astrophysics and physical cosmology the mass-to-light ratio, normally designated with the Greek letter upsilon, , is the quotient between the total mass of a spatial volume (typically on the scales of a galaxy or a cluster) and its luminosity. T ...

s in the stellar disks, they do not match with the masses derived from the observed rotation curves and the

law of gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...

. A solution to this conundrum is to hypothesize the existence of

and to assume its distribution from the galaxy's center out to its

halo Halo, halos or haloes usually refer to: * Halo (optical phenomenon) * Halo (religious iconography), a ring of light around the image of a head HALO, halo, halos or haloes may also refer to: Arts and entertainment Video games * ''Halo'' (franch ...

. Though dark matter is by far the most accepted explanation of the rotation problem, other proposals have been offered with varying degrees of success. Of the possible alternatives, one of the most notable is

modified Newtonian dynamics Modified Newtonian dynamics (MOND) is a hypothesis that proposes a modification of Newton's law of universal gravitation to account for observed properties of galaxies. It is an alternative to the hypothesis of dark matter in terms of explaining ...

(MOND), which involves modifying the laws of gravity.For an extensive discussion of the data and its fit to MOND see

History

In 1932,

Jan Hendrik Oort Jan Hendrik Oort ( or ; 28 April 1900 – 5 November 1992) was a Dutch astronomer who made significant contributions to the understanding of the Milky Way and who was a pioneer in the field of radio astronomy. His ''New York Times'' obituary ...

became the first to report that measurements of the stars in the

solar neighborhood The Local Interstellar Cloud (LIC), also known as the Local Fluff, is an interstellar cloud roughly across, through which the Solar System is moving. This feature overlaps a region around the Sun referred to as the solar neighborhood. It is unk ...

indicated that they moved faster than expected when a mass distribution based upon visible matter was assumed, but these measurements were later determined to be essentially erroneous. In 1939,

Horace Babcock Horace Welcome Babcock (September 13, 1912 – August 29, 2003) was an American astronomer. He was the son of Harold D. Babcock. Career Babcock invented and built a number of astronomical instruments, and in 1953 was the first to propose th ...

reported in his PhD thesis measurements of the rotation curve for Andromeda which suggested that the mass-to-luminosity ratio increases radially. He attributed that to either the absorption of light within the galaxy or to modified dynamics in the outer portions of the spiral and not to any form of missing matter. Babcock's measurements turned out to disagree substantially with those found later, and the first measurement of an extended rotation curve in good agreement with modern data was published in 1957 by Henk van de Hulst and collaborators, who studied M31 with the newly commissioned Dwingeloo 25 meter telescope. A companion paper by Maarten Schmidt showed that this rotation curve could be fit by a flattened mass distribution more extensive than the light. In 1959, Louise Volders used the same telescope to demonstrate that the spiral galaxy M33 also does not spin as expected according to Keplerian dynamics. Reporting on

NGC 3115 NGC 3115 (also called the Spindle Galaxy or Caldwell 53) is a field lenticular (S0) galaxy in the constellation Sextans. The galaxy was discovered by William Herschel on February 22, 1787. At about 32 million light-years away from Earth, it i ...

Jan Oort Jan Hendrik Oort ( or ; 28 April 1900 – 5 November 1992) was a Dutch astronomer who made significant contributions to the understanding of the Milky Way and who was a pioneer in the field of radio astronomy. His ''New York Times'' obituary ...

wrote that "the distribution of mass in the system appears to bear almost no relation to that of light... one finds the ratio of mass to light in the outer parts of NGC 3115 to be about 250". On page 302–303 of his journal article, he wrote that "The strongly condensed luminous system appears imbedded in a large and more or less homogeneous mass of great density" and although he went on to speculate that this mass may be either extremely faint dwarf stars or interstellar gas and dust, he had clearly detected the dark matter halo of this galaxy. The

Carnegie telescope The Carnegie telescope (i.e. Carnegie double astrograph) is a twin refractor telescope located at Lick Observatory in California, United States. The double telescope's construction began in the 1930s with a grant from the Carnegie institution, a ...

(Carnegie Double Astrograph) was intended to study this problem of Galactic rotation. In the late 1960s and early 1970s,

Vera Rubin Vera Florence Cooper Rubin (; July 23, 1928 – December 25, 2016) was an American astronomer who pioneered work on galaxy rotation rates. She uncovered the discrepancy between the predicted and observed angular motion of galaxies by studyi ...

, an astronomer at the Department of Terrestrial Magnetism at the

Carnegie Institution of Washington The Carnegie Institution of Washington (the organization's legal name), known also for public purposes as the Carnegie Institution for Science (CIS), is an organization in the United States established to fund and perform scientific research. Th ...

, worked with a new sensitive

spectrograph An optical spectrometer (spectrophotometer, spectrograph or spectroscope) is an instrument used to measure properties of light over a specific portion of the electromagnetic spectrum, typically used in spectroscopic analysis to identify mate ...

that could measure the velocity curve of edge-on

spiral galaxies Spiral galaxies form a class of galaxy originally described by Edwin Hubble in his 1936 work ''The Realm of the Nebulae'' Together with fellow staff-member Kent Ford, Rubin announced at a 1975 meeting of the

American Astronomical Society The American Astronomical Society (AAS, sometimes spoken as "double-A-S") is an American society of professional astronomers and other interested individuals, headquartered in Washington, DC. The primary objective of the AAS is to promote the adv ...

the discovery that most stars in spiral galaxies orbit at roughly the same speed, and that this implied that galaxy masses grow approximately linearly with radius well beyond the location of most of the stars (the

galactic bulge In astronomy, a galactic bulge (or simply bulge) is a tightly packed group of stars within a larger star formation. The term almost exclusively refers to the central group of stars found in most spiral galaxies (see galactic spheroid). Bulges ...

). Rubin presented her results in an influential paper in 1980. These results suggested either that

Newtonian gravity Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...

does not apply universally or that, conservatively, upwards of 50% of the mass of galaxies was contained in the relatively dark galactic halo. Although initially met with skepticism, Rubin's results have been confirmed over the subsequent decades. If

Newtonian mechanics Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...

is assumed to be correct, it would follow that most of the mass of the galaxy had to be in the galactic bulge near the center and that the stars and gas in the disk portion should orbit the center at decreasing velocities with radial distance from the galactic center (the dashed line in Fig. 1). Observations of the rotation curve of spirals, however, do not bear this out. Rather, the curves do not decrease in the expected inverse square root relationship but are "flat", i.e. outside of the central bulge the speed is nearly a constant (the solid line in Fig. 1). It is also observed that galaxies with a uniform distribution of luminous matter have a rotation curve that rises from the center to the edge, and most low-surface-brightness galaxies (LSB galaxies) have the same anomalous rotation curve. The rotation curves might be explained by hypothesizing the existence of a substantial amount of matter permeating the galaxy outside of the central bulge that is not emitting light in the mass-to-light ratio of the central bulge. The material responsible for the extra mass was dubbed

, the existence of which was first posited in the 1930s by Jan Oort in his measurements of the

Oort constants The Oort constants (discovered by Jan Oort) A and B are empirically derived parameters that characterize the local rotational properties of our galaxy, the Milky Way, in the following manner: : \begin & A=\frac\left(\frac-\frac\Bigg\vert_\right) \ ...

and

Fritz Zwicky Fritz Zwicky (; ; February 14, 1898 – February 8, 1974) was a Swiss astronomer. He worked most of his life at the California Institute of Technology in the United States of America, where he made many important contributions in theoretical and ...

in his studies of the masses of

galaxy cluster A galaxy cluster, or a cluster of galaxies, is a structure that consists of anywhere from hundreds to thousands of galaxies that are bound together by gravity, with typical masses ranging from 1014 to 1015 solar masses. They are the second-l ...

s. The existence of non-baryonic cold dark matter (CDM) is today a major feature of the

Lambda-CDM model The ΛCDM (Lambda cold dark matter) or Lambda-CDM model is a parameterization of the Big Bang cosmological model in which the universe contains three major components: first, a cosmological constant denoted by Lambda (Greek Λ) associated with d ...

that describes the

cosmology Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount (lexicographer), Thomas Blount's ''Glossographia'', and in 1731 taken up in ...

of the

universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...

Halo density profiles

In order to accommodate a flat rotation curve, a density profile for a galaxy and its environs must be different than one that is centrally concentrated. Newton's version of

Kepler's Third Law In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbi ...

implies that the spherically symmetric, radial density profile is: :

\rho(r) = \frac\left(1+2~ \frac\right)

where is the radial orbital velocity profile and is the

gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...

. This profile closely matches the expectations of a singular isothermal sphere profile where if is approximately constant then the density to some inner "core radius" where the density is then assumed constant. Observations do not comport with such a simple profile, as reported by Navarro, Frenk, and White in a seminal 1996 paper. The authors then remarked that a "gently changing logarithmic slope" for a density profile function could also accommodate approximately flat rotation curves over large scales. They found the famous Navarro–Frenk–White profile, which is consistent both with

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 ''n''-body problem for other applications). ''N''-body simulations ar ...

and observations given by :

\rho (r)=\frac

where the central density, , and the scale radius, , are parameters that vary from halo to halo. Because the slope of the density profile diverges at the center, other alternative profiles have been proposed, for example the Einasto profile, which has exhibited better agreement with certain dark matter halo simulations. Observations of orbit velocities in spiral galaxies suggest a mass structure according to: :

v(r)= (r \, d\Phi/dr)^

with the galaxy

gravitational potential In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...

. Since observations of galaxy rotation do not match the distribution expected from application of Kepler's laws, they do not match the distribution of luminous matter. This implies that spiral galaxies contain large amounts of dark matter or, alternatively, the existence of exotic physics in action on galactic scales. The additional invisible component becomes progressively more conspicuous in each galaxy at outer radii and among galaxies in the less luminous ones. A popular interpretation of these observations is that about 26% of the mass of the Universe is composed of dark matter, a

hypothetical A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous obser ...

type of matter which does not emit or interact with

electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...

. Dark matter is believed to dominate the gravitational potential of galaxies and clusters of galaxies. Under this theory, galaxies are baryonic condensations of stars and gas (namely hydrogen and helium) that lie at the centers of much larger haloes of dark matter, affected by a gravitational instability caused by primordial density fluctuations. Many cosmologists strive to understand the nature and the history of these ubiquitous dark haloes by investigating the properties of the galaxies they contain (i.e. their luminosities, kinematics, sizes, and morphologies). The measurement of the kinematics (their positions, velocities and accelerations) of the observable stars and gas has become a tool to investigate the nature of dark matter, as to its content and distribution relative to that of the various baryonic components of those galaxies.

Further investigations

Comparison of rotating disc galaxies in the distant Universe and the present day

The rotational dynamics of galaxies are well characterized by their position on the

Tully–Fisher relation In astronomy, the Tully–Fisher relation (TFR) is an empirical relationship between the mass or intrinsic luminosity of a spiral galaxy and its asymptotic rotation velocity or emission line width. It was first published in 1977 by astronomer ...

, which shows that for spiral galaxies the rotational velocity is uniquely related to their total luminosity. A consistent way to predict the rotational velocity of a spiral galaxy is to measure its

bolometric luminosity Luminosity is an absolute measure of radiated electromagnetic power (light), the radiant power emitted by a light-emitting object over time. In astronomy, luminosity is the total amount of electromagnetic energy emitted per unit of time by a s ...

and then read its rotation rate from its location on the Tully–Fisher diagram. Conversely, knowing the rotational velocity of a spiral galaxy gives its luminosity. Thus the magnitude of the galaxy rotation is related to the galaxy's visible mass. While precise fitting of the bulge, disk, and halo density profiles is a rather complicated process, it is straightforward to model the observables of rotating galaxies through this relationship. So, while state-of-the-art cosmological and

galaxy formation The study of galaxy formation and evolution is concerned with the processes that formed a heterogeneous universe from a homogeneous beginning, the formation of the first galaxies, the way galaxies change over time, and the processes that have ge ...

simulations of dark matter with normal

baryonic matter In particle physics, a baryon is a type of composite subatomic particle which contains an odd number of valence quarks (at least 3). Baryons belong to the hadron family of particles; hadrons are composed of quarks. Baryons are also classifie ...

included can be matched to galaxy observations, there is not yet any straightforward explanation as to why the observed scaling relationship exists. Additionally, detailed investigations of the rotation curves of low-surface-brightness galaxies (LSB galaxies) in the 1990s and of their position on the Tully–Fisher relation showed that LSB galaxies had to have dark matter haloes that are more extended and less dense than those of galaxies with high surface brightness, and thus surface brightness is related to the halo properties. Such dark-matter-dominated

dwarf galaxies A dwarf galaxy is a small galaxy composed of about 1000 up to several billion stars, as compared to the Milky Way's 200–400 billion stars. The Large Magellanic Cloud, which closely orbits the Milky Way and contains over 30 billion stars, is s ...

may hold the key to solving the

dwarf galaxy problem The dwarf galaxy problem, also known as the missing satellites problem, arises from a mismatch between observed dwarf galaxy numbers and collisionless numerical cosmological simulations that predict the evolution of the distribution of matter in ...

structure formation In physical cosmology, structure formation is the formation of galaxies, galaxy clusters and larger structures from small early density fluctuations. The universe, as is now known from observations of the cosmic microwave background radiation, beg ...

. Very importantly, the analysis of the inner parts of low and high surface brightness galaxies showed that the shape of the rotation curves in the centre of dark-matter dominated systems indicates a profile different from the NFW spatial mass distribution profile. This so-called

cuspy halo problem The cuspy halo problem (also known as the core-cusp problem) refers to a discrepancy between the inferred dark matter density profiles of low-mass galaxies and the density profiles predicted by cosmological N-body simulations. Nearly all simulation ...

is a persistent problem for the standard cold dark matter theory. Simulations involving the feedback of stellar energy into the interstellar medium in order to alter the predicted dark matter distribution in the innermost regions of galaxies are frequently invoked in this context.

Alternatives to dark matter

There have been a number of attempts to solve the problem of galaxy rotation by modifying gravity without invoking dark matter. One of the most discussed is

(MOND), originally proposed by

Mordehai Milgrom Mordehai "Moti" Milgrom is an Israeli physicist and professor in the department of Particle Physics and Astrophysics at the Weizmann Institute in Rehovot, Israel. Biography He received his B.Sc. degree from the Hebrew University of Jerusalem i ...

in 1983, which modifies the Newtonian force law at low accelerations to enhance the effective gravitational attraction. MOND has had a considerable amount of success in predicting the rotation curves of low-surface-brightness galaxies, matching the

baryonic Tully–Fisher relation In particle physics, a baryon is a type of composite subatomic particle which contains an odd number of valence quarks (at least 3). Baryons belong to the hadron family of particles; hadrons are composed of quarks. Baryons are also classifie ...

, and the velocity dispersions of the small satellite galaxies of the

Local Group The Local Group is the galaxy group that includes the Milky Way. It has a total diameter of roughly , and a total mass of the order of . It consists of two collections of galaxies in a "dumbbell" shape: the Milky Way and its satellites form ...

. Using data from the Spitzer Photometry and Accurate Rotation Curves (SPARC) database, a group has found that the radial acceleration traced by rotation curves could be predicted just from the observed baryon distribution (that is, including stars and gas but not dark matter). The same relation provided a good fit for 2693 samples in 153 rotating galaxies, with diverse shapes, masses, sizes, and gas fractions. Brightness in the near infrared, where the more stable light from red giants dominates, was used to estimate the density contribution due to stars more consistently. The results are consistent with MOND, and place limits on alternative explanations involving dark matter alone. However, cosmological simulations within a Lambda-CDM framework that include baryonic feedback effects reproduce the same relation, without the need to invoke new dynamics (such as MOND). Thus, a contribution due to dark matter itself can be fully predictable from that of the baryons, once the feedback effects due to the dissipative collapse of baryons are taken into account. MOND is not a relativistic theory, although relativistic theories which reduce to MOND have been proposed, such as

tensor–vector–scalar gravity Tensor–vector–scalar gravity (TeVeS), developed by Jacob Bekenstein in 2004, is a relativistic generalization of Mordehai Milgrom's Modified Newtonian dynamics (MOND) paradigm. The main features of TeVeS can be summarized as follows: * As i ...

(TeVeS), scalar–tensor–vector gravity (STVG), the f(R) theory of Capozziello and De Laurentis, not to mention a version of Superfluid Vacuum theory based on the

Logarithmic Schrödinger equation In theoretical physics, the logarithmic Schrödinger equation (sometimes abbreviated as LNSE or LogSE) is one of the nonlinear modifications of Schrödinger's equation. It is a classical wave equation with applications to extensions of quantum mech ...

. A model of galaxy based on a

general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

metric was also proposed, showing that the rotation curves for the

Milky Way The Milky Way is the galaxy that includes our Solar System, with the name describing the galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars that cannot be individually distinguished by the naked eye ...

, NGC 3031, NGC 3198 and NGC 7331 are consistent with the mass density distributions of the visible matter, avoiding the need for a massive halo of exotic dark matter. According to a 2020 analysis of the data produced by the

Gaia spacecraft ''Gaia'' is a space observatory of the European Space Agency (ESA), launched in 2013 and expected to operate until 2025. The spacecraft is designed for astrometry: measuring the positions, distances and motions of stars with unprecedented preci ...

, it would seem possible to explain at least the

's rotation curve without requiring any dark matter if instead of a

Newtonian approximation In physics, the Newtonian limit is a mathematical approximation applicable to physical systems exhibiting (1) weak gravitation, (2) objects moving slowly compared to the speed of light, and (3) slowly changing (or completely static) gravitational ...

the entire set of equations of

is adopted. In March 2021, Gerson Otto Ludwig published a model based on

that explains galaxy rotation curves with

gravitoelectromagnetism Gravitoelectromagnetism, abbreviated GEM, refers to a set of formal analogies between the equations for electromagnetism and relativistic gravitation; specifically: between Maxwell's field equations and an approximation, valid under certain ...

Footnotes

Bibliography

* This was the first detailed study of orbital rotation in galaxies. * Observations of a set of spiral galaxies gave evidence that orbital velocities of stars in galaxies were unexpectedly high at large distances from the nucleus. This paper was influential in convincing astronomers that most of the matter in the universe is dark, and much of it is clumped about galaxies. * ''Galactic Astronomy'', Dmitri Mihalas and Paul McRae.

W. H. Freeman W. H. Freeman and Company is an imprint of Macmillan Higher Education, a division of Macmillan Publishers. Macmillan publishes monographs and textbooks for the sciences under the imprint. History The company was founded in 1946 by William H. ...

1968.

External links

*
The Case Against Dark Matter
About

Erik Verlinde Erik Peter Verlinde (; born 21 January 1962) is a Dutch theoretical physicist and string theorist. He is the identical twin brother of physicist Herman Verlinde. The Verlinde formula, which is important in conformal field theory and topolog ...

's approach to the problem. (November 2016) {{Portal bar, Stars, Spaceflight, Outer space, Solar System Astrophysics

Rotation curve The rotation curve of a disc galaxy (also called a velocity curve) is a plot of the orbital speeds of visible stars or gas in that galaxy versus their radial distance from that galaxy's centre. It is typically rendered graphically as a plot, and ...

Articles containing video clips Rotation Physics beyond the Standard Model