Spiral Density Waves and their Role in Disk Galaxies
Nathan Sanders
ABSTRACT
The existence of “grand design” spiral arms in galaxies is attributed to self-propagating density waves that rigidly rotate within a differentially-rotating disk. A brief history of observation and theory is presented, followed by an explanation of basic physical concepts developed by Bertil Lindblad in 1963. The more complete exposition developed by C.C. Lin and Frank Shu is explained as well as consequences and predictions of the theory. Excitation mechanisms are discussed, along with density wave theory’s connection to star formation. Observational tests are summarized and assessed.
INTRODUCTION
The most widely accepted explanation for the existence of spiral structure in galaxies was proposed in the late 1960s by C.C. Lin and Frank Shu [1]. The so-called Lin-Shu density wave theory, first presented in a series of three papers published over the course of several years, was built upon previous work by Bertil Lindblad and others and provides a model that explains the nature of spiral structure and, central to any scientific theory, offers predictions which can be tested observationally.
HISTORICAL CONTEXT
The observational history of spiral structure in galaxies began in the late 18th Century with the French astronomer Charles Messier. In 1781 he published the final version of what is now the famous Messier Catalogue.1 Using only a four-inch telescope, Messier was able to catalogue over one hundred objects which were then classified under the general term “nebulae” but which we now know include galaxies, globular clusters, open clusters, HII regions, planetary nebulae, and supernova remnants. Approximately fifty years later Lord Rosse, through the use of a more powerful telescope, was able to discern spiral structure within some of the Messier objects. In Rosse’s time “spiral nebulae” were understood to be just that: spiral-shaped clouds within our own Milky Way as opposed to extra-galactic stellar systems. Immanuel Kant may have been the earliest proponent of the “island universe” theory (before the term was even coined). In his 1755 treatise Allgemeine Naturgeschichte he correctly philosophized that some of the observed “nebulae” might in fact be Figure 1. Rosse's sketch of M51 (Whirlpool Galaxy) made in systems of stars and gas 1845 external to the Milky Way.2 A war of ideas was waged over this issue, culminating in the famous “Great Debate” in 1920 between Harlow Shapley and Heber Curtis. It was not until 1929 that Kant’s philosophical musings were widely accepted as true, largely as a result of the work of Edwin Hubble. Using the 100-inch Hooker Telescope atop Mount Wilson, Hubble discovered Cepheid variables in M31, thereby enabling him to reckon a distance of 275 kiloparsecs - much farther away than the Small Magellanic Cloud and well beyond the
1 http://en.wikipedia.org/wiki/Charles_Messier 2 http://en.wikipedia.org/wiki/Immanuel_Kant confines of the Milky Way.3 Once spiral nebulae became spiral galaxies, questions naturally arose regarding the nature and origin of such large scale spiral patterns.
SPIRAL THEORIES
“There are at least two possible types of spiral theories,” said C.C. Lin and Frank Shu in the introduction to their first 1964 paper [1]. The first is that spiral arms might be material arms. That is, they might be composed of the same, unchanging set of stars. It can be shown, however, that differential rotation of the disc would cause arms to wind up many times over billions of years, destroying the correlation between bulge size and openness of the spiral arms that is actually observed and systematized in the Hubble classification scheme. This is called the winding dilemma and is the motivation for finding an alternate explanation for spiral structure. One possible alternative spiral theory is one in which the spiral structure is regarded as a wave pattern which remains quasi-stationary in a frame of reference rotating around the center of the galaxy at some fixed angular speed. This is precisely the type of theory developed by Lin and Shu.
KINEMATICAL WAVES
One of the first attempts to find a solution to the winding dilemma was undertaken by considering how stars move in their orbits as they travel around the galactic center. In an axisymmetric gravitational field a star moves in uniform circular motion; its centripetal acceleration is balanced by gravitational acceleration. The equation of motion in a reference frame rotating with the star’s local standard of rest, the LSR, comes from Newton’s Second Law:
dv m r = F + m" 2 r + 2m" v dt r 0 0 # dv m # = F ! 2m" v (1) dt # 0 r dv m z = F dt z
In the first expression Fr is the gravitational force as a function of radius, which is balanced by the centripetal force experienced by the star at that radius, i.e.
2 Fr = "m! g (r) r (2)
The second term is the centrifugal force at the LSR, and the third term is the Coriolis force. Fθ is zero in the second equation because the galaxy in question is, as an idealization, considered symmetric. The third equation describes harmonic motion in the direction perpendicular to the disk.
3 http://en.wikipedia.org/wiki/Edwin_Hubble To find out what happens when a star is perturbed from its orbit at the LSR by a slight change in radius ξ, ωg(r) must be expressed as a first order Taylor series:
#" " = " + g ! (3) g 0 #r
By substituting this approximation back into the equation of motion, dropping higher order terms, performing some algebra, and solving the system, a set of equations is obtained which describes the motion of a star relative to its LSR.
vr = vr0 sin(!t) v = v cos(!t) $ $ 0 (4) # = %[vr0 /!]cos(!t)
" = [v$ 0 /!]sin(!t)
The constant κ represents the epicyclic frequency:
& 1 , r d- g )# . 2 = 4- 2 $1+ * '! (5) g 2 * dr ' %$ + - g ("!
It can also be written in terms of Oort’s constants that describe the local shearing:
" = ! 4B(A ! B) (6) where
1 &. 0 , d. ) # A = $ - * ' ! 2 R dR $ 0 + ( R0 ! % " (7)
1 &. 0 , d. ) # B = - $ + * ' ! 2 R dR %$ 0 + ( R0 "! where θ is the azimuthal velocity.
These results can be interpreted physically by imagining a star displaced inward from its LSR. Because angular momentum is conserved as its radius decreases, the star will appear to move ahead of its initial position at its LSR and will be drawn outward as the gravitational and centripetal forces become unbalanced. A similar process takes place as the star is displaced outward; the star appears to fall behind its reference position and experiences a restoring force pulling it inward. Stars, therefore, oscillate in small ellipses relative to their reference circular orbits, much like other systems which oscillate at characteristic natural frequencies, e.g. pendula and guitar strings. These are called epicycles, and while the concept of epicyclic orbits may seem archaic, it is in fact an accurate description of relative stellar motion in Figure 2. Quasi-circular orbit (left) in the presence of a flat rotation curve. The orbit is decomposed by means of an epicycle on the right the limit of small amplitudes. [2]. The relationship of epicyclic frequency to Oort’s constants show that they are ultimately dependant on the distribution of gravitating matter, made manifest by the galaxy’s rotation curve.
Imagine tracing the trajectory of star from an inertial reference frame as it orbits the galactic center. Its orbit may not be closed because its epicyclic frequency and the local galactic angular frequency are not necessarily commensurate. But one could also imagine choosing a rotating frame of reference with angular frequency ωp such that the orbit appears to be closed. This would require the angular frequency of our chosen frame, and the local galactic angular frequency to be some integer multiple of the epicyclic frequency.
! " # " = ± (8) p m
Specifically, if m = 2 then the stellar orbit would appear to close into an ellipse centered at the center of the galaxy. Lindblad observed that the quantity ω-κ/2 is approximately constant as a function of radius. So, in an appropriately-chosen rotating frame of reference, all stellar orbits would be approximately closed. If these ellipses could be somehow arranged so that the position angles of their major axes varied with radius, then from an inertial frame one would observe a spiral pattern rotating at a pattern speed ωp and traced not by material arms but by regions of enhanced stellar density caused by orbit crowding. This particular type of spiral wave pattern has been dubbed a kinematical wave.
Lindblad’s kinematical waves represent a partial victory in the search for a resolution to the winding dilemma. The insight that stellar orbits can theoretically exhibit oscillatory motion in the form of epicycles
Figure 3. Illustration of Lindblad’s kinematical waves shows that the stellar component of a galactic disk is, at least in principle, capable of supporting waves. The idea of stellar orbits, organized in such a way as to move in and out of regions of orbit crowding, creating a spiral density pattern rotating like a rigid body is elegant. In their book on spiral structure, Bertin and Lin point out some problems concerning Lindblad’s model [2]. First of all, the value ω-κ/2 is only approximately constant as a function of radius, so even if stellar orbits were nicely organized at some point so as to delineate a spiral pattern, the pattern would deteriorate over time. Spiral structure is thought to be a relatively long-term phenomenon, so kinematical waves are probably not the whole story. Second, the theory totally neglects the self-gravity of the stars. The presence of a rotating density enhancement is sure to affect the gravitational potential field in a galaxy, and hence affect the motion of the stars moving through that field. But the derivation of the equations upon which the idea of kinematical spiral waves is based does not take this into account. Moreover, it does not say anything about how stellar orbits might become organized in the first place. Their organization and the function which describes how the major axes of their orbits vary with radius are totally arbitrary. Last of all, the galactic disk in this model is assumed to be very cold, that is to say it does not account for the natural amount of velocity dispersion which stars would possess in any real galactic disk.
LIN-SHU DENSITY WAVE THEORY
In 1964, only a few years after Lindblad published his findings, C.C. Lin and Frank Shu published a paper which offered up a new theory describing spiral structure. Lin and Shu attacked the problem from a completely different angle but were able to support many of the general conclusions made by Lindblad, which may suggest an underlying relationship between the two approaches.
Lindblad began his investigation from the standpoint of stellar dynamics, whereas Lin and Shu imagined a galaxy as a fluid. They used three equations of fluid mechanics to derive a dispersion relation for the behavior of a stellar fluid. The first of the three is the continuity equation which expresses the principle of conservation of mass [1].
"# v + ! • #V = 0 (9) "t
Where ρ is the mass density and V is the fluid velocity vector. The second equation is the substantial derivative, which expresses the principle of conservation of momentum in a gravitational potential field [1].
$ v v v v (%V ) + %(V • "V ) = #"P # %"! (10) $t
Where P is the linear momentum vector and Φ is the gravitational potential scalar field. The third and final expression is the Poisson equation, which here describes gravitational potential in terms of density [1].
$ 2# = 4"G! (11)
Solving the above equations is rather complicated, so in short, they invoked a superposition of waves of the following type:
$(r,!,t) = $ˆ (r)exp[i("t # m! )] (11)
Where the radial component of the gravitational potential behaves as
"ˆ (r) = A(r)exp[i!(r)] (12)
These equations describe a spiral-shaped density perturbation which is rigidly rotating, just like Lindblad’s kinematical waves. If the pattern has sufficiently rapid spatial variation, then a local dispersion relation can be derived [2], which relates propagating speeds to pertinent frequencies.
2 2 2 2 ($ % m&) = %2#G" 0 k + k c + ! (13)
This dispersion relation is absolutely teeming with information and is referenced so much in the literature that it is probably worth talking about a little more. Quickly going over what the terms represent, Ω is the pattern frequency of the density wave as seen earlier, m is the number of arms, σ is the surface mass density of the disk, k, c, and κ are the wave number, velocity dispersion, and epicyclic frequency, respectively. The k2c2 term describes pressure or “sound” waves and, along with the κ2 term which characteries epicyclic motion in the disk is a stabilizing force, while the self-gravitation term with a negative sign is destabilizing with respect to perturbations.
RESONANCES
A number of important ideas come from the dispersion relation. The first is that of resonances. Remember that matter in galactic disks rotates differentially, ideally represented by a flat rotation curve. In such a disk with a rigidly-rotating density wave pattern there must be a radius such that the material in the vicinity of that radius rotates with the same angular velocity as the density wave pattern. This is called the corrotation radius, or co-rotation zone. Because matter in the co-rotation zone moves at the same angular velocity as the density wave pattern, its position relative to the pattern never changes: if the matter is outside the pattern it will always remain outside it, and vice versa. Obviously this is not the case for stars and gas and at other radii, which will overtake or be overtaken by the density wave if they lie inside or outside the co-rotation zone, respectively. The frequency at which some test particle might expect to encounter the wave pattern relative to the local epicyclic frequency can be expressed by a dimensionless quantity.
m(# $ #) " = p (14) !
When the two angular velocities are equal, i.e. at co-rotation, ν = 0. When ν = ±1 we say that the wave is in resonance with the epicyclic frequency. That is to say, the particle encounters the wave pattern at the same point in its epicyclic orbit each time. The gravitational force of the wave consistently gives the particle a tug at the same point in its epicycle, much like a child on a swing set pumping her legs at the same point in each swing. The amplitude of oscillation increases with each encounter of the wave pattern and boom: resonance. When this occurs outside the co-rotation circle (ν = 1) it is called the Outer Lindblad Resonance (OLR), and when it occurs inside the co-rotation circle (ν = -1) it is called the Inner Lindblad Resonance (ILR). The importance of these resonances is this: density perturbations are constrained to the annular region bounded by the OLR and ILR. Wave energy is totally absorbed by the local state at the resonance radii, thus prohibiting the transfer of energy beyond them.
D.M. Elmegreen reports a number of different features expected to be observed at various resonances [3]. As already mentioned, the OLR and ILR are expected to delineate the extent of spiral structure. Rings are also very good resonance indicators: the observed ratios of outer to inner ring radii are nearly the same as the theoretical ratio of OLR to corrotation and are reproduced in n-body simulations of ring formation [3]. Inner and outer rings can be observed in NGC 1433 and NGC 4736. Nuclear rings should theoretically appear near the ILR and can be seen in the pictured infrared image of NGC 3351 [3]. Short, symmetrically placed spurs are often observed between main spiral arms, theoretically a result of the ultra-harmonic (4:1) resonance [3].
EXCITATION MECHANISMS
Bertin and Lin place a lot of emphasis on a particular aspect of density wave theory: the idea that they are quasi-stationary, self-sustained, global modes of oscillation [2]. The idea of quasi stationary structure was in fact first proposed by Lindblad in 1963, and was adopted by Lin and Shu as the starting point for their study of spiral structure in galaxies [2]. The quasi-stationary hypothesis asserts that “the global structure for regular spiral Figure 4. IRAC image of NGC 3351 showing a nuclear ring patterns remains in a state of slow evolution; it is only long lasting in the sense that the global structure continues to be present, and is likely to vacillate between morphologies within the same Hubble type, not that it does not change in shape.” [2].
Of course, to truly be considered a quasi-stationary mode of oscillation requires the wave pattern to be sustained in some way. Indeed, in 1969 Alar Toomre published a paper which showed that with regard to waves of the type proposed by Lin & Shu “any packet of such waves propagates radially … with a group velocity that is sufficient to obliterate it within a few galactic revolutions.” [7]. Beyond showing that density waves of the type proposed by Lin and Shu were doomed to die out, Toomre later went on to propose a possible excitation mechanism dubbed “swing amplification” [8]. James Mark, in several papers written in the mid 1970s, proposed another means of internal excitation called the WASER process [9]. To summarize: the dispersion relation allows solutions for four different types of waves: short trailing, long trailing, short leading, and long leading. Both types of excitation mechanisms involve the reflection and refraction of these different wave types near the co-rotation circle. The source of excitation energy in both cases is attributed to the coupling of a negative energy region with the outside positive energy density region of the disk [2]. The general basic principle is that energy stored in the form of shear can, under the proper circumstances, be released by collective modes [2].
DENSITY WAVES AND STAR FORMATION
Density waves are expected to affect local star formation by favoring or enhancing the natural conditions that lead to Jeans collapse. In a paper written in the late 1960s, W.W. Roberts developed a two-armed shock scenario (TASS) which first addressed the theoretical challenges associated with the Lin-Shu density wave theory and its affect on star formation [10]. Roberts argued that the gaseous component of a galaxy, modeled as a thin disk, may respond non-linearly to a low amplitude smoothly-varying sinusoidal density pattern in the stars, provided that its velocity relative to the spiral pattern is supersonic. The gas is expected be sharply compressed – by a factor of five to ten - right before the imposed gravitational potential minimum. On a large scale, TASS manifests itself as roughly circular, concentric gaseous streamtubes, with kinks where the gas is compressed.
The large-scale shock scenario proposed by W. W. Roberts is an example of star formation triggering. It is expected to affect giant molecular cloud complexes, triggering Jeans collapse and subsequent fragmentation. Collapse and new star formation should cease once Figure 5. Diagram of the gas response to a spiral potential field [10]. the gas leaves the shocked area. On the basis of his solution, Bertin and Lin point out that the structure of spiral arms should be arranged in the following way: A dust lane should mark the spot of sudden HI compression, followed by a more diffuse HI arm. Allowing for a few tens of millions of years for star formation to occur, HII regions and OB associations should be observed downstream of the dust lane, with the distance from the arm varying according to the ages of the stars – older stars are expected to be observed farther from the arm. Inside co-rotation, the HII regions and OB associations would be on the convex outer side of the arm with the dust lane occurring on the concave inner side of the arm, while outside co-rotation the opposite is expected [2].
OBSERVATIONAL TESTS
One of the first observational tests of density wave theory by H.C.D. Visser in 1977 involved comparing theoretical motions of atomic hydrogen, based on the model developed by Lin and Shu and the large scale shock scenario developed by Roberts, to gas motions observed in M81 by means of 21-cm line emission [5]. This occurred almost ten years after the aforementioned paper was published by Roberts, due to the lack of available resolution of radio telescopes at the time. By 1977, several tests of density wave theory had been carried out by Burton, Lindbad, Toomre, and others, but were hampered by the fact that they were all done using the Milky Way [5]. After the construction of the Westerbork Synthesis Radio Telescope, angular resolution became good enough to enable HI studies of external galaxies.
Visser first developed an axisymmetric mass model of M81 based on an observed HI rotation curve [4,5]. The theoretical spiral pattern was then calculated based on Lin- Shu density wave theory. A pattern speed of 18 km s-1 kpc-1 was adopted so as to provide the best model, which resulted in an ILR at 2.5 kpc from the galactic center, and a corrotation radius of 11.3 kpc. The amplitude of the wave could have been calculated theoretically, but was instead Figure 6. Comparisons between predicted (symbols) and observed (full lines) isovelocity contours for the velocity along the line of sight of the cold gas in M81 [5]. determined empirically by surface photometry measurements carried out by Schweitzer [4]. Theoretical gas flow was then able to be computed based on Roberts’ theory. Comparisons with the actual HI velocity field provided the first good indication of the existence of spiral density waves in other galaxies (see Figure 6).
Dias and Lépine recently used Lin-Shu density wave theory in conjunction with the ideas of Roberts to directly determine the spiral pattern rotation speed of the Galaxy [6]. In the light of much controversy surrounding the spiral structure of our own galaxy it is, they say, “therefore an important step in the understanding of the spiral structure to firmly establish the rotation velocity of the spiral pattern in the Galaxy, as well as to verify whether different arms have the same velocity.”
Their method is relatively straight forward. Using the New Catalogue of Optically Visible Open Clusters and Candidates, they establish the birthplaces of a number of open clusters, based on their age and the observed rotation curve of the Galaxy. The birthplaces are found by assuming that the clusters moved (in circular orbits) a distance equal to their age multiplied by their velocity, given by the rotation curve [6]. Once the birthplaces were found they adopted, as an equation describing the shape of the spiral arms, a logarithmic spiral and adjusted the parameters so as to obtain a best fit for the youngest clusters. The next step was to rotate the initial phase angle so as to obtain a best fit for an older group of clusters’ birthplaces and so on in this manner, eventually obtaining a relation between mean age and pattern rotation angle, which can be converted to a pattern rotation speed.
Their results show that the pattern rotation speed is most likely 24 km s-1 kpc-1 which situates the ILR at 2.5 kpc and the OLR at 11.5 kpc. The ultra-harmonic (4:1) resonances occur at 5 kpc and 10 kpc. They report that the three main arms in the solar neighborhood present the same rotation velocity, which strongly supports the idea that the spiral pattern rotates like a rigid body [6]. They found that the majority of open clusters have birthplaces that coincide with the position of a spiral arm at the position of their birth, reinforcing Robert’s idea that spiral arms are triggering mechanisms for star formation. The corrotation radius was found to be at around 7.9 kpc from the galactic center, which when taking uncertainties into account, places the solar system in the co- rotation zone.
This particular spiral arm pattern has some potentially interesting consequences. Because the Solar System lies so close to the co-rotation radius, it seldom encounters spiral arms. That means it would rarely suffer gravitational perturbations of the Oort comet cloud resulting from crossing spiral arms. It would also be relatively unaffected by supernova explosions that result from the deaths of OB stars formed by shocked gas in the spiral. Therefore the Earth should not have experienced periodic increases in the number of long period comets entering the inner solar system, nor high-energy radiation bombarding its upper atmosphere. Both of these consequences could have led to periodic mass- extinction events on the Earth.
Other investigators reach the opposite conclusion [11]. They assume a particular spiral- arm pattern in the distribution of free electrons, as surmised from observations of pulsar scintillation timing in the disk. They then compare the Solar System’s arm-crossing history with the record of mass extinctions on Earth – showing a strong correlation. These conflicting studies underscore the difficulties of understanding spiral-arm dynamics in the Milky Way. Less-ambiguous observational tests of density-wave theory can be made in nearby spiral galaxies, where the various gaseous and stellar components – and their respective kinematics – can be adequately resolved [4,5].
References
1. Lin, C.C., Shu, F.H., On the Spiral Structure of Disk Galaxies, Astrophysical Journal, No. 140, 646-655 (1964)
2. Bertin, G., Lin, C.C., Spiral Structure in Galaxies: a Density Wave Theory, The MIT Press, Cambridge, MA, 1995
3. Elmegreen, D.M., Galaxies and Galactic Structure, Prentice Hall, Upper Saddle River, NJ, 1997
4. Visser, H.C.D., The Dynamics of the Spiral galaxy M81, I. Axisymmetric Models and the Stellar Density Wave, Astronomy & Astrophysics, No. 88, 149-158 (1980)
5. Visser, H.C.D., The Dynamics of the Spiral galaxy M81, International Astronomical Union (1977)
6. Dias, W.S., Lépine, J.R.D., Direct Determination of the Spiral Pattern Rotation Speed of the Galaxy, Astrophysical Journal, No. 629, 825-831 (2005)
7. Toomre, A., Group Velocity of Spiral Waves in Galactic Disks, The Astrophysical Journal, No. 158, 899-913 (1969)
8. Toomre, A., The Structure and Evolution of Normal Galaxies, ed. S.M. Fall and D. Lynden-Bel, Cambridge University Press, Cambridge, p. 111.
9. Mark, J.W-K. Astrophysical Journal, No. 205, 363 (1976)
10. Roberts, W.W., Astrophysical Journal, No. 158, 123 (1969)
11. Leitch, E. M. and Vasisht, G., astro-ph/9802174v1 (1998)