Spiral Density Waves and Their Role in Disk Galaxies Nathan Sanders

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$ 0 cos(!t) (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.

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