Density wave theory or the Lin–Shu density wave theory is a theory proposed by C.C. Lin and

Frank Shu Frank Hsia-San Shu (; born June 2, 1943), is a Chinese-American astrophysicist, astronomer and author. He is currently a University Professor Emeritus at the University of California, Berkeley and University of California, San Diego. He is be ...

in the mid-1960s to explain the spiral arm structure of

spiral galaxies Spiral galaxies form a class of galaxy originally described by Edwin Hubble in his 1936 work ''The Realm of the Nebulae'' The Lin–Shu theory introduces the idea of long-lived quasistatic spiral structure (QSSS hypothesis). In this hypothesis, the spiral pattern rotates in a particular angular frequency (pattern speed), whereas the stars in the

galactic disk A galactic disc (or galactic disk) is a component of disc galaxies, such as spiral galaxies and lenticular galaxies. Galactic discs consist of a stellar component (composed of most of the galaxy's stars) and a gaseous component (mostly composed ...

are orbiting at a different speed depending on their distance to the galaxy center. The presence of spiral density waves in galaxies has implications on the

star formation Star formation is the process by which dense regions within molecular clouds in The "medium" is present further soon.-->interstellar space

, since the gas orbiting around the galaxy may be compressed and form shock periodically. Theoretically, the formation of global spiral pattern is treated as an

instability In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable are unstable; systems can also be mar ...

of the stellar disk caused by the self-gravity, as opposed to

tidal interactions The tidal force is a gravitational effect that stretches a body along the line towards the center of mass of another body due to a gradient (difference in strength) in gravitational field from the other body; it is responsible for diverse phenomen ...

. The mathematical formulation of the theory has also been extended to other astrophysical disk systems, such as

Saturn's rings The rings of Saturn are the most extensive ring system of any planet in the Solar System. They consist of countless small particles, ranging in size from micrometers to meters, that orbit around Saturn. The ring particles are made almost entirel ...

Galactic spiral arms

Originally, astronomers had the idea that the arms of a spiral galaxy were material. However, if this were the case, then the arms would become more and more tightly wound, since the matter nearer to the center of the galaxy rotates faster than the matter at the edge of the galaxy. The arms would become indistinguishable from the rest of the galaxy after only a few orbits. This is called the winding problem. Lin & Shu proposed in 1964 that the arms were not material in nature, but instead made up of areas of greater density, similar to a

traffic jam Traffic congestion is a condition in transport that is characterized by slower speeds, longer trip times, and increased vehicular queueing. Traffic congestion on urban road networks has increased substantially since the 1950s. When traffic de ...

on a highway. The cars move through the traffic jam: the density of cars increases in the middle of it. The traffic jam itself, however, moves more slowly. In the galaxy, stars, gas, dust, and other components move through the density waves, are compressed, and then move out of them. More specifically, the density wave theory argues that the "gravitational attraction between stars at different radii" prevents the so-called winding problem, and actually maintains the spiral pattern. The rotation speed of the arms is defined to be

\Omega_

, the global pattern speed. (Thus, within a certain

non-inertial reference frame A non-inertial reference frame is a frame of reference that undergoes acceleration with respect to an inertial frame. An accelerometer at rest in a non-inertial frame will, in general, detect a non-zero acceleration. While the laws of motion are ...

, which is rotating at

\Omega_

, the spiral arms appear to be at rest). The stars ''within'' the arms are not necessarily stationary, though at a certain distance from the center,

R_

, the corotation radius, the stars and the density waves move together. Inside that radius, stars move more quickly (

\Omega > \Omega_

) than the spiral arms, and outside, stars move more slowly (

\Omega < \Omega_

). For an ''m''-armed spiral, a star at radius ''R'' from the center will move through the structure with a frequency

m(\Omega_ - \Omega (R))

. So, the gravitational attraction between stars can only maintain the spiral structure if the frequency at which a star passes through the arms is less than the

epicyclic frequency In astrophysics, particularly the study of accretion disks, the epicyclic frequency is the frequency at which a radially displaced fluid parcel will oscillate. It can be referred to as a " Rayleigh discriminant". When considering an astrophysica ...

\kappa (R)

, of the star. This means that a long-lived spiral structure will only exist between the inner and outer

Lindblad resonance A Lindblad resonance, named for the Swedish galactic astronomer Bertil Lindblad, is an orbital resonance in which an object's epicyclic frequency (the rate at which one periapse follows another) is a simple multiple of some forcing frequency. Res ...

(ILR, OLR, respectively), which are defined as the radii such that:

\Omega (R)=\Omega_ + \kappa /m

and

\Omega (R)=\Omega_ - \kappa /m

, respectively. Past the OLR and within the ILR, the extra density in the spiral arms pulls more often than the epicyclic rate of the stars, and the stars are thus unable to react and move in such a way as to "reinforce the spiral density enhancement". File:Galaxy rotation rigid.ogv, Animation 1: If the spiral arms were rigid mass concentrations, the galaxy must rotate as a whole around its center in order to maintain its spiral structure. According to Lindblad's observation and the laws of physics this is not the case. File:Galaxy rotation wind.ogv, Animation 2: Differential rotation as observed by Lindblad would dissolve spiral arms in a short period of time were they composed of fixed mass concentrations. File:Galaxy rotation wave.ogv, Animation 3: Orbits predicted by the density wave theory allow the existence of stable spiral arms. Stars move in and out of the spiral arms as they orbit the galaxy.

Further implications

The density wave theory also explains a number of other observations that have been made about spiral galaxies. For example, "the ordering of H I clouds and dust bands on the inner edges of spiral arms, the existence of young, massive stars and

H II region An H II region or HII region is a region of interstellar atomic hydrogen that is ionized. It is typically in a molecular cloud of partially ionized gas in which star formation has recently taken place, with a size ranging from one to hundreds ...

s throughout the arms, and an abundance of old, red stars in the remainder of the disk". When clouds of gas and

dust Dust is made of fine particles of solid matter. On Earth, it generally consists of particles in the atmosphere that come from various sources such as soil lifted by wind (an aeolian process), volcanic eruptions, and pollution. Dust in homes ...

enter into a density wave and are compressed, the rate of star formation increases as some clouds meet the Jeans criterion, and collapse to form new stars. Since star formation does not happen immediately, the stars are slightly behind the density waves. The hot

OB star OB stars are hot, massive stars of spectral types O or early-type B that form in loosely organized groups called OB associations. They are short lived, and thus do not move very far from where they formed within their life. During their lifeti ...

s that are created ionize the gas of the

interstellar medium In astronomy, the interstellar medium is the matter and radiation that exist in the space between the star systems in a galaxy. This matter includes gas in ionic, atomic, and molecular form, as well as dust and cosmic rays. It fills interstella ...

, and form H II regions. These stars have relatively short lifetimes, however, and expire before fully leaving the density wave. The smaller, redder stars do leave the wave, and become distributed throughout the galactic disk. Density waves have also been described as pressurizing gas clouds and thereby catalyzing star formation.

Application to Saturn's rings

Beginning in the late 1970s,

Peter Goldreich Peter Goldreich (born July 14, 1939) is an American astrophysicist whose research focuses on celestial mechanics, planetary rings, helioseismology and neutron stars. He is the Lee DuBridge Professor of Astrophysics and Planetary Physics at Cal ...

, and others applied density wave theory to the rings of Saturn. Saturn's rings (particularly the

A Ring The rings of Saturn are the most extensive ring system of any planet in the Solar System. They consist of countless small particles, ranging in size from micrometers to meters, that orbit around Saturn. The ring particles are made almost entirel ...

) contain a great many spiral density waves and spiral bending waves excited by

Lindblad resonances A Lindblad resonance, named for the Swedish galactic astronomer Bertil Lindblad, is an orbital resonance in which an object's epicyclic frequency (the rate at which one periapse follows another) is a simple multiple of some forcing frequency. Res ...

and vertical resonances (respectively) with

Saturn's moons The moons of Saturn are numerous and diverse, ranging from tiny moonlets only tens of meters across to enormous Titan (moon), Titan, which is larger than the planet Mercury (planet), Mercury. Saturn has 83 natural satellite, moons with confirmed ...

. The physics are largely the same as with galaxies, though spiral waves in Saturn's rings are much more tightly wound (extending a few hundred kilometers at most) due to the very large central mass (Saturn itself) compared to the mass of the disk. The ''Cassini'' mission revealed very small density waves excited by the ring-moons Pan and

Atlas An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth. Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geographic ...

and by high-order resonances with the larger moons, as well as waves whose form changes with time due to the varying orbits of

Janus In ancient Roman religion and myth, Janus ( ; la, Ianvs ) is the god of beginnings, gates, transitions, time, duality, doorways, passages, frames, and endings. He is usually depicted as having two faces. The month of January is named for Janu ...

and

Epimetheus In Greek mythology, Epimetheus (; grc-gre, Ἐπιμηθεύς, , afterthought) was the brother of Prometheus (traditionally interpreted as "foresight", literally "fore-thinker"), a pair of Titans who "acted as representatives of mankind". The ...

References

External sources

* Bertin, Giuseppe. 2000. ''Dynamics of Galaxies.'' Cambridge: Cambridge University Press. * Bertin, G. and C.C. Lin. 1996. ''Spiral Structure in Galaxies: A Density Wave Theory.'' Cambridge: MIT Press. * C.C. Lin, Yuan, C., and F.H. Shu
"On the Spiral Structure of Disk i Galaxies III. Comparison with Observations"
Ap.J. 155, 721 (1969). (SCI) * Yuan, C
"Application of Density-Wave Theory to the Spiral Structure of the Milky Way System I. Systematic Motion of Neutral Hydrogen"
Ap.J., 158, 871 (1969). (SCI)

External links