Galaxy rotation curve

The rotation curve of a disc galaxy (also called a velocity curve) is the rotational velocity of visible stars or gas in that galaxy as a function of their radial distance from that galaxy's center, typically rendered graphically as a scatter plot of the orbital speed (in km/s) of the stars or gas in the galaxy on the ordinate against the distance from the center of the galaxy on the abscissa.

A general feature of the galaxy rotation curves that have been measured is that rotational velocity of stars and gas is constant as far from the galactic centre as can be measured (line B in the illustration): stars are observed to revolve around the centre of these galaxies at a constant speed over a large range of distances from the centre of any galaxy. If disc galaxies had mass distributions similar to the observed distribution of stars and gas, the rotation curves velocities should decline at large distances (dotted line A in illustration) in the same way as do other systems with most of their mass in the centre, such as the Solar System or the moons of Jupiter, following the prediction of Kepler's Laws.

The galaxy rotation problem is the discrepancy between observed galaxy rotation curves and the Newtonian-Keplerian prediction, assuming a centrally-dominated mass associated with the observed luminous material. When masses of galaxies are calculated solely from the luminosities and mass-to-light ratios in the disk, and if core portions of spiral galaxies are assumed to approximate to those of stars, the masses derived from the kinematics of the observed rotation and the law of gravity do not match. This discrepancy can be accounted for by a large amount of dark matter that permeates the galaxy and extends into the galaxy's halo.

Though dark matter is by far the most accepted explanation for the resolution to the galaxy rotation problem, other proposals have been offered with varying degrees of success. Of the possible alternatives, the most notable is Modified Newtonian Dynamics (MOND), which involves modifying the laws of gravity.^[2]

In 1932 Jan Hendrik Oort became the first to report measurements that the stars in the Solar neighborhood moved faster than expected when a mass distribution based upon visible matter was assumed, but this measurement was later determined to be essentially erroneous.^[3] In 1933, Fritz Zwicky postulated "missing mass" to account for the orbital velocities of galaxies in clusters. In 1939, Horace Babcock reported in his PhD thesis measurements of the rotation curve for Andromeda which suggested that the mass-to-luminosity ratio increases radially.^[4] He, however, attributed it to either absorption of light within the galaxy or modified dynamics in the outer portions of the spiral and not to any form of missing matter. In 1959, Louise Volders demonstrated that spiral galaxy M33 does not spin as expected according to Keplerian dynamics.^[5] Following this, in the late 1960s and early 1970s, Vera Rubin, a young astronomer at the Department of Terrestrial Magnetism at the Carnegie Institution of Washington worked with a new sensitive spectrograph that could measure the velocity curve of edge-on spiral galaxies to a greater degree of accuracy than had ever before been achieved.^[6] Together with fellow staff-member Kent Ford, Rubin announced at a 1975 meeting of the American Astronomical Society the discovery that most stars in spiral galaxies orbit at roughly the same speed,^{[citation needed]} which implied that their mass densities were uniform well beyond the location with most of the stars (the galactic bulge), a result independently found in 1978.^[7] Rubin presented her results in an influential paper in 1980.^[8] These results suggest that either Newtonian gravity does not apply universally or that, conservatively, upwards of 50% of the mass of galaxies was contained in the relatively dark galactic halo. Met with skepticism, Rubin insisted that the observations were correct.

Based on Newtonian mechanics and assuming, as was originally thought, that most of the mass of the galaxy had to be in the galactic bulge near the center, matter (such as stars and gas) in the disk portion of a spiral should orbit the center of the galaxy similar to the way in which planets in the solar system orbit the sun, i.e. where the average orbital speed of an object at a specified distance away from the majority of the mass distribution would decrease inversely with the square root of the radius of the orbit (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 slopes up from the center to the edge, and most low-surface-brightness galaxies (LSB galaxies) rotate with a rotation curve that slopes up from the center, indicating little core bulge.

The rotation curves can be explained if there is a substantial amount of matter permeating the galaxy that is not emitting light in the mass-to-light ratio of the central bulge. The material responsible for the extra mass was dubbed, "dark matter", the existence of which was first posited in the 1930s by Jan Oort in his measurements of the Oort constants and Fritz Zwicky in his studies of the masses of galaxy clusters, though these proposals were left unexplored until after Rubin's work was accepted as correct. The existence of non-baryonic cold dark matter (CDM) is today a major feature of the Lambda-CDM model that describes the cosmology of the universe.

In order to accommodate a flat rotation curve, a density profile for galactic environs must be different than one that is centrally concentrated. Newton's version of Kepler's Third Law states that the radial density profile ρ(r) equals

${\displaystyle \rho (r)={\frac {3[v(r)]^{2}}{4\pi Gr^{2}}}}$

where v(r) is the radial orbital velocity profile and G is the gravitational constant. This profile closely matches the expectations of a singular isothermal sphere profile where if v(r) is approximately constant then the density ${\displaystyle \rho \sim r^{-2}}$ to some inner "core radius" where the density leveled off to a constant. Observations did not comport with such a simple profile as reported by Navarro, Frenk, and White in a seminal 1996 paper:

If more massive halos were indeed associated with faster rotating disks and so with brighter galaxies, a correlation would be expected between the luminosity of binary galaxies and the relative velocity of their components. Similarly, there should be a correlation between the velocity of a satellite galaxy relative to its primary and the rotation velocity of the primary's disk. No such correlations are apparent in existing data.^[9]

The authors then remarked, as did a few others before them, that a "gently changing logarithmic slope" for a density profile could also accommodate approximately flat rotation curves over large scales. They wrote down the famous Navarro–Frenk–White profile which is consistent both with N-body simulations and observations given by

${\displaystyle \rho (r)={\frac {\rho _{0}}{{\frac {r}{R_{s}}}\left(1~+~{\frac {r}{R_{s}}}\right)^{2}}}}$

where the central density, ρ₀, and the scale radius, R_s, are parameters that vary from halo to halo. In part 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 as good or better agreement with certain dark matter halo simulations.^[10][11]

The rotational dynamics of galaxies are, in fact, extremely well characterized by their position on the Tully-Fisher relation which shows that for spiral galaxies that rotational velocity is uniquely related to its total luminosity with essentially no scatter. A consistent way to predict the rotational velocity of a spiral galaxy is to measure its bolometric luminosity and then extrapolate its rotation curve from its location on the Tully-Fisher diagram. Likewise, knowing the rotational velocity of a spiral galaxy is an excellent indication of its luminosity. Thus the amplitude of the galaxy rotation curve is related to the galaxy's visible mass.

While precise fitting bulge, disk, and halo density profiles is a rather complicated process, it is straightforward to model the observables of rotating galaxies through this relationship.^[12] So, while state-of-the-art cosmological and galaxy formation simulations of dark matter with normal baryonic matter included can be matched to galaxy observations, there is not yet any straightforward explanation as to why the scaling relationship exists as observed.^[13][14] Additionally, detailed investigations of the rotation curves of low-surface-brightness galaxies (LSB galaxies) in the 1990s^[15] and of their position on the Tully-Fisher relation^[16] showed that LSB galaxies had to have dark matter haloes that are more extended and less dense than those of HSB galaxies and thus surface brightness is related to the halo properties. Such dark matter-dominated dwarf galaxies may hold the key to solving the dwarf galaxy problem of structure formation.

Additionally, analysis of the centres of low surface brightness galaxies showed that the shape of the rotation curves in the centre of dark-matter dominated systems, indicated a profile that differed from the NFW spatial mass distribution profile.^[17] This so-called cuspy halo problem of cold dark matter is requires detail modeling and understanding of the feedback mechanisms in the innermost regions of galaxies.^[18]

That dark matter theory continues to be supported as an explanation for galaxy rotation curves is because the evidence for dark matter is not solely derived from these curves. It has been uniquely successful in simulating the formation of the large scale structure seen in the distribution of galaxies and in explaining the dynamics of groups and clusters of galaxies.^[19] Dark matter also correctly predicts the results of gravitational lensing observations, see especially the Bullet Cluster.

There are a number of attempts to solve the problem of galaxy rotation curves without invoking dark matter.

One of the most discussed alternatives is MOND (Modified Newtonian Dynamics), originally proposed by Mordehai Milgrom as a phenomenological explanation back in 1983 but which has been seen to have predictive power in accounting for galaxy rotation curves. This posits that the physics of gravity changes at large scale but, until recently, was not a relativistic theory. However, this changed with the development by Jacob Bekenstein of the tensor–vector–scalar gravity (TeVeS) theory,^[2][20] enabling gravitational lensing to be covered by the theory.

Another, similar, alternative is the relativistic modified gravity (MOG) theory, also called scalar–tensor–vector gravity (STVG), of John Moffat.^[21] Brownstein and Moffat ^[22] applied MOG and MOND to the question of galaxy rotation curves, and demonstrated excellent fits to a large sample of over 100 low-surface-brightness (LSB), high surface brightness (HSB) and dwarf galaxies.^[23] Each galaxy rotation curve fit was made without dark matter, using only the available photometric data (stellar matter and visible gas) and a two-parameter mass distribution model which made no assumption regarding the mass to light ratio. The MOG results were compared to MOND and were nearly indistinguishable right out to the edge of the rotation curve data, where MOND predicts a forever flat rotation curve, but MOG predicts an eventual return to the familiar inverse-square gravitational force law.

Although these alternatives are not considered by the astronomical community to be as convincing as the dark matter model,^[24][25] gravitational lensing studies have been proposed as the means to separate the predictions of the different theories. Indeed, gravitational lensing by the Bullet Cluster was reported to provide the best current evidence for the nature of dark matter^[26][27] and to provide "evidence against some of the more popular versions of Modified Newtonian Dynamics (MOND)" as applied to large galactic clusters.^[28] In retort, Milgrom, the original proposer of MOND, posted a rejoinder online^[29] that claims MOND correctly accounts for the dynamics of galaxies outside of galaxy clusters, and removes the need for most dark matter in clusters, leaving twice as much matter as is visible, which Milgrom expects to be simply unseen ordinary matter rather than exotic cold dark matter.

Some Quantum Gravity theories also give alternative explanations, see alternative theories to dark matter.

• Vera Rubin
• Unsolved problems in physics
• Nonsymmetric gravitational theory
• Dark matter
• Long-slit spectroscopy

• A nice animation about dark matter
• Dark Matter Candidates for Rotation Curves

• V. Rubin, W. K. Ford, Jr (1970). "Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions". Astrophysical Journal. 159: 379. Bibcode:1970ApJ...159..379R. doi:10.1086/150317.{{cite journal}}: CS1 maint: multiple names: authors list (link)

  This was the first detailed study of orbital rotation in galaxies.

• V. Rubin, N. Thonnard, W. K. Ford, Jr, (1980). "Rotational Properties of 21 Sc Galaxies with a Large Range of Luminosities and Radii from NGC 4605 (R=4kpc) to UGC 2885 (R=122kpc)". Astrophysical Journal. 238: 471. Bibcode:1980ApJ...238..471R. doi:10.1086/158003.{{cite journal}}: CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link)

  Observations of a set of spiral galaxies gave convincing 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 1968.