Galaxy rotation curve

A rotation curve is a plot of the distance from the center of a galaxy (R) vs. the orbital speed (V) of its stars. This is the rotation curve of a typical spiral galaxy M 33 (yellow and blue points with error bars) and the predicted one from distribution of the visible matter (white line). The discrepancy between the two curves is a major problem for physics and astrophysics.

Left: A simulated galaxy without dark matter. Right: A simulated galaxy with a flat rotation curve can be constructed by an ad hoc addition of a dark matter halo.

The rotation curve of a disc galaxy (also called a velocity curve) is a plot of the measured magnitude of the orbital velocities (i.e., the speeds) of visible stars or gas in that galaxy versus their radial distance from that galaxy's center and typically rendered graphically as a plot. The rotational speeds are inferred from observing the Doppler shifts of the stars, which requires a very accurate spectrometer and a high-resolution telescope. Only the velocity component in the direction of the line of sight is measurable, the other two components are estimated.

The measured rotation rates of those stars are very different from the calculated rates, which calculation is based on the inverse-square relation of Newtonian gravity theory. A general feature of the galaxy rotation curves is that the orbital speed of stars and gas rises or is almost constant as far from the galactic center as it can be measured: that is, stars are observed to revolve around the center of the galaxy at increasing or the same speed over a large range of distances from the center of the galaxy. Given the observed mass distributions in galaxies, the orbital speed should decrease at increasing distances in the same way as do other systems with most of their mass in the center, such as the Solar System or the moons of Jupiter.

The rotation curves of spiral galaxies are also known to be asymmetric. The observational data from each side of a galaxy are generally averaged. Rotation curve asymmetry appears to be normal rather than exceptional.

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 ratios in the stellar disks, they do not match with the masses derived from the observed rotation curves and the law of gravity. It is a major, unsolved mystery of cosmology.

Dark matter was originally proposed to account for the behavior of these stars, and then to assume it is distributed in some manner from the galaxy's center out to an hypothesized halo. It is an unknown form of gravitating matter that has no electromagnetic properties. This unseen material would add an external source of gravity to account for the high speeds of the stars.

Though gravitating dark matter is the most promoted attempt to explain the galaxy rotation problem, other proposals have been offered with varying degrees of success. One of the proposed alternatives is Modified Newtonian Dynamics (MOND), which involves modifying the theory of gravity. Other classes of theories investigate plasma cosmology mechanisms and Aether flow.

History and description of the galaxy rotation problem

In 1932, Jan Hendrik Oort was the first to report that measurements of 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. 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. 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. In 1959, Louise Volders demonstrated that spiral galaxy M33 does not spin as expected according to Keplerian dynamics. In the late 1960s and early 1970s, Vera Rubin worked with Kent Ford's new sensitive spectrograph that could measure the velocity curve of edge-on spiral galaxies to a greater degree of accuracy than before. Kent Ford and 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, and that this implied that galaxy masses grow approximately linearly with radius well beyond the location of most of the stars (the galactic bulge). All of 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.

If Newtonian mechanics 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.

For unknown reasons, the rotational dynamics of galaxies are reasonably-well characterized by their position on the Tully–Fisher relation, which shows that for spiral galaxies the rotational velocity is roughly correlated with its total luminosity. A consistent way to predict the rotational velocity of a spiral galaxy is to measure its bolometric luminosity 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.

The rotation curves might be explained by hypothesizing the existence of 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. Any of these halo distributions represent an inversion of the usual understanding, that gravitating systems are more massive toward the center, not the edge.

For these and other reasons, there are a variety of theories put forth to explain the motion of the stars in the galaxies.

Invisible matter theories

Main article: Dark Matter

There are a variety of theories that propose some unknown and unseen form of matter, mostly orbiting in a ring, or halo, out past the bulk of the visible galaxy. Most of these theories are variations of Dark Matter. Each new form of proposed matter has some weakness, aside from being invisible, which then leads investigators to propose another variation. These invisible halos are claimed to be the largest mass component of the galaxy, exceeding the mass of the stars several times over.

Invisible-matter proponents claim that about 84% of the mass of the Universe is composed of dark matter, a massive component which does not emit radiation, but it dominates the gravitational potential of galaxies and cluster of galaxies. In this idea, galaxies are baryonic condensations of stars and gas (namely H and He) that lie at the centers of much larger dark haloes of dark matter, affected by a gravitational instability caused by primordial density fluctuations of the Big Bang.

The main goal has become 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.

While fitting proposed bulge, disk, and halo mass and density profiles is a complicated process of trial and error, it is straightforward to model the observables of rotating galaxies through the Tully-Fisher relation. 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 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 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.

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 center of dark-matter dominated systems, indicated a profile that differed from the NFW spatial mass distribution profile. This so-called cuspy halo problem 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.

Halo density profiles

In order to accommodate a flat rotation curve, a density profile, and therefore a mass distribution, for a galaxy and its outer environs must be radically different from one that is observed to be centrally concentrated. This is the opposite of an accretion disk. Newton's version of Kepler's Third Law states that the radial density profile ρ(r) is:

$\rho(r) = \frac{3 v(r)^2 }{4 \pi G r^2}\left(1+2~ \frac{d\log~ v(r)}{d\log~ r}\right)$

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 ρ ∝ r⁻² 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 Navarro–Frenk–White profile which is consistent both with N-body simulations and observations given by

$\rho (r)=\frac{\rho_0}{\frac{r}{R_s}\left(1~+~\frac{r}{R_s}\right)^2}$

where the central density, ρ₀, and the scale radius, R_s, are parameters that vary from halo to halo. Since r goes to zero in the denominator, 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:

$v(r)= (r \, d\Phi/dr)^{1/2}$

with Φ the galaxy gravitational potential. 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, in alternative, 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.

Gravity theories

Modified Newtonian Dynamics

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 Modified Newtonian Dynamics (MOND), originally proposed by Mordehai Milgrom 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, and the velocity dispersions of the small satellite galaxies of the Local Group. These results are surprising in the context of dark matter, which does not make the same predictions as MOND without considerable fine-tuning.

MOND is not a relativistic theory, although relativistic theories which reduce to MOND have been proposed, such as tensor–vector–scalar gravity, scalar–tensor–vector gravity (STVG), and the f(R) theory of Capozziello and De Laurentis.

Meta model

Tom Van Flandern proposes a type of gravity with a finite range.

Electrogravity

There are a variety of theories which posit that gravity is an electrostatic or electrodynamic effect.

Plasma theories

Hannes_Alfvén, Anthony Peratt, Wal Thornhill, Donald Scott, other

Aether theories

Many Aether theories have been developed over the centuries. A few of the more recent ones propose solutions to the galaxy rotation problem. Reginald T. Cahill proposes that the rotation curve problem is one of the results of interpreting gravity as an in-flowing motion of the quantum foam.

A reformulation and generalisation of the Newtonian theory of gravity in terms of a velocity in-flow field, representing at a classical level the relative motion of a quantum-foam substructure to space, reveals a key dynamical feature of the phenomenon of gravity, namely the so called ‘dark matter’ effect, which manifests ... in spiral galaxy rotation curves...

Miles Mathis states that the inverse-square law of gravity contain an additional, hidden term for a repulsive force, which partially counteracts the attractive force of gravity, and so accounts for the rotation problem as well as the numerous other gravitational anomalies. In this theory, an inflow of ubiquitous "charge" or "charge photons" causes gravity.

See also

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

External links

• Dark Matter Candidates for Rotation Curves

Bibliography

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

• 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.