creating structure in disk galaxies

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CREATING STRUCTURE IN DISK GALAXIES Sethanne Howard US Naval Observatory, retired

Abstract There are four main theories of spiral structure in disk galaxies with density wave theory as the leading favorite. This paper investigates the theory of tidal triggering as a source for spiral structure and shows that it, as well as the more popular density wave theory, acts to produce the beautiful spiral structures we see.

GALAXIES IN OUR UNIVERSE COME IN 3 MAIN TYPES: elliptical (E), spiral (S), and irregular (I). Irregular galaxies are those without a well defined shape. Figure 1 shows the irregular galaxy NGC6822.

Figure 1 – NGC6822

E galaxies are tri-axial in shape; i.e., football or hard boiled egg shapes. They do not show much internal structure and mostly will look the same when viewed from any angle in the optical range. Figure 2 shows an optical image of the galaxy M87.

Figure 2 – M87, an E galaxy, seen in the optical

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However, some E’s show extensive jet structure when viewed in the radio regime. Figure 3 shows the huge jets of plasma and gas ejected from the central black hole in M87. The structure in this image is approximately 200,000 light-years across.

Figure 3 – The environs of M87 seen in the radio, M87 is in the center F.N. Owen, J.A. Eliek and N.E. Kassim, National Radio Astronomy Observatory, Associated Universities, Inc.

Fascinating as such objects are they are not the subject of this paper. It is the S’s that are the subject of this paper. S’s have a disk like shape with a central bulge – rather like a fried egg. Our own Milky Way is an S galaxy. Gas, dust, and stars make up the matter in S’s. Spirals look considerably different if viewed from different directions. Figure 4 shows the S galaxy M101. It is almost face on to us, so we can see the disk with all its structure.

Figure 4 – M101, Courtesy NASA Hubble Space Telescope

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Figure 5 shows the S galaxy NGC 4013 as seen edge on to us. We cannot see the spiral structure, but can see the dust lane marking a line through the center of the disk. Note how thin these S galaxies are.

Figure 5 – NGC 4013, a spiral galaxy seen edge on to us Courtesy Dr. Chris Howk and Dr. Blair Savage, NOAO

Many S’s have a slight bulge in their center. Figure 6 shows an infrared image of our Milky Way as if seen edge on. The infrared shows the dust. Note the center of the image; it has a box-like bulge shape.

Figure 6 – Infrared image of the Milky Way

There is an amazing variety of galactic structure in S’s, and, in most cases, the structure is ordered along star-lit spiral arms as illustrated in Figure 4. The number of arms can vary from one to many. S’s with two well defined arms (like M51) are called grand design spirals. The underlying disk is there but usually overshadowed by the brightness of the stars in the arms. Even though the arms are optically quite bright, they comprise a mere 5−10% of the matter in the galaxy. Fainter stars, gas, and dust form the rest of the matter. Figure 6, for example, shows the smoothly distributed dusty matter in the Milky Way as seen in the infrared. Spiral arms can be classified by their orientation with respect to the direction of rotation of the galaxy. It is thought that most spiral arms trail the disk rotation although S’s with leading arms are known. A trailing arm has an outer tip that points opposite to the direction of the motion. A

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leading arm has an outer tip that points in the direction of rotation. Figure 7 illustrates the concept.

Figure 7 - Two clockwise rotating disks are shown. The one on the left has a pair of leading arms; the one on the right has trailing arms.

What are the common properties of spirals? • They rotate. They have a well ordered rotational velocity about their centers. For many S’s the rotation velocity is constant over the disk. This means S’s rotate differentially as opposed to solid body rotation; i.e., the outer parts take longer to orbit the center than the inner parts. This is often represented by the formula Ω = V/r, where Ω is the angular velocity about the center, V is the constant orbital velocity, r is the distance from the center. As r increases Ω decreases. Not all S’s have a constant V with radius. • They group/cluster. Spirals tend to have at least a few companion galaxies typically small in mass clustered about them (Byrd and Howard 1992). Among other types, these companions can be dwarf spheroidal galaxies. • They interact. There are enough small companion galaxies around most S’s to keep the S “triggered”; i.e., tidally harassed by the companions. A companion with mass just 1/100 of its primary spiral can tidally trigger global spiral structure in the primary (Byrd and Howard 1992). These gravitational tides are similar to the lunar tides raised on the Earth by the Moon – the passage of one gravitating body by another. Some interactions are spectacular, which happens when the masses are nearly equal (Figure 8).

Figure 8 – NGC4038 Courtesy Bill Keel Washington Academy of Sciences

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They are deceiving. Spiral arms are sites of active star formation. They tend to be optically prominent in bright blue light (signature of young or massive stars) rather than faint red light. They, therefore, tend to deceive us because the bright stars we see in the arms represent only a small portion of the mass in the galaxy. Some spirals (probably about 50%) have a bar like structure stretching through the center (Figure 9).

Figure 9 - NGC1300, a barred spiral

What causes the spiral structure? There are four hypotheses: • Quasi-stationary spiral structure; i.e., density waves • Stochastic Star Formation • Bars • Tidal interactions Let us consider bars first. The explanation is straightforward. The bar’s gravitational potential (which is not circularly symmetric) as it rotates in the disk can drive the development of spiral arms, usually dangling off the end of the bar. Interstellar gas that is subject to the periodical perturbations by the bar potential can develop a wave-like condensation of matter that attracts neighboring stars and gas. The local density increases and once a critical value is reached, star formation is ignited in this area. Bars are thought to be a temporary phenomenon in the life of some S’s, the bar structure decaying over time, transforming the galaxy from a barred S to a “regular” S pattern. Also, past a certain size the accumulated mass of the bar compromises the stability of the overall bar structure. So, although bars can explain their spiral structure, only about half the S’s have bars; therefore, bars cannot explain the entirety of structures. We do not know what allows a bar to form in a disk in the first place. Second, let us consider stochastic star formation, SSF. A stochastic system is one in which only the energy integral is conserved. The word stochastic is from Greek and means pertaining to chance. Unlike a deterministic system, for example, a stochastic system does not always produce the same output for a given input. When applied to galaxies it means when a random gas cloud collapses (by happenstance) and begins Spring 2009

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to form stars it can trigger nearby clouds into collapse, subsequent star formation, and so forth, leading to an extended star forming region that defines a spiral arm. SSF works but has difficulties in mimicking the vast variety of spiral structure. Third, there is the density wave theory. The early density wave theory was put forth by Bertil Lindblad (1925) in a classic but very difficult series of papers beginning in 1925. The current density wave theory was taken from the quantum mechanics WKB (Wentzel–Kramers–Brillouin) approximation in which the wave function is recast as an exponential function, semi-classically expanded, and then either the amplitude or the phase is taken to be slowly changing. Lin and Shu (1964) introduced the current density wave theory to astronomy. They proposed that the spiral arms are areas of greater density, similar to a traffic jam on a highway. The stars and gas clouds move through these congested areas; they collide to produce spiral arms, and then move out. The enhanced density region rotates with a pattern speed that is different from the rotation velocity of the disk. Some spiral structure of S galaxies can be matched by this theory, especially the inner structure. However, the theory does not provide a “seed” for the density wave; i.e., a progenitor. The density wave is ab initio. This theory tends to be the most popular among astronomers mainly because it has well developed mathematical equations and has been taught for a few scholastic generations as THE cause of spiral structure. The theory has some caveats. It assumes that the arms are tightly wound (the pitch angle is small). Yet many spirals have arms that are not tightly wound. Density wave arms can last for a few galactic rotations but not forever. Yet spiral structure is seen as far back in time as we can distinguish spirals. Therefore this theory needs to be used with some caution. Fourth and last, there are gravitational tides as a resource for spiral structure. Tides are often given short shrift in textbooks. As an example of tides generating the spiral structure in the particular S galaxy M51 see the paper by Howard (2008). The close passage of an S galaxy by another galaxy will raise gravitational tides in both galaxies similar to the Moon raising tides on the Earth. Galaxy tides trigger spiral structure by perturbing gas clouds into collapsing and forming stars, thus forming the bright spiral arms. Occasionally a tidal passage is strong enough to yank a gas cloud completely out of its circular orbit, but typically the perturbation is not that strong. A tidal perturbation, therefore, usually triggers a material arm. In a material arm, the components of the arm remain with it as it differentially rotates. The arm moves with the rotation speed of the Washington Academy of Sciences

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galaxy. The main objection to tides has been the winding phenomenon. Because of differential rotation the material arm winds up over time. Figure 10 illustrates the concept.

Figure 10 - illustrating the winding up of a material arm, first triggered when particles A, B, C, and D are co-aligned in the disk galaxy.

Over time the arm defined by these particles winds up and dissipates. However, a material arm can last almost as long as a density wave arm. The winding problem exists for both types.

So which way do we go from here? It all depends upon your point of view. Byrd and Howard (1992) found that there are small companion galaxies around most S’s, sufficient in number to trigger tidally induced structure on a continuing basis. It does not take much to do this. A companion with mass only 1/100 of the primary galaxy can trigger global structure. Most big S’s have masses about 1011 – 1012 Solar masses. A mass 1/100 of that makes a tiny galaxy; get much smaller in mass and one does not form a galaxy. So I shall take the approach that gravitational tides from a small companion can stimulate most spiral structure. The initially formed tidal material arms can trigger any subsequent density waves (typically through gravitational resonances). The tidal arm is thus the Spring 2009

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“seed” for a density wave. When the waves wind up, all it takes is another passage by a small companion to trigger another set of arms. So both approaches – tides and density waves – work together to form the beautiful structures we see. Galaxy tidal interactions can also trigger bars to form. Add SSF to the mix, and one can probably reproduce most spiral structure. Material arms can tell us a lot about their galaxy. Consider a tidal perturbation as a small companion crosses the rotating disk plane from above to below the disk. The time of disk crossing is the time of maximum tidal perturbation on the disk. The tidal perturbation will give the gas and stars, but especially the gas, directed 0° and 180° from the point of closest approach, velocity components toward one another. Call this the line of tidal impulse. Gas clouds along a radial line directed 90° and 270° from the line of tidal impulse will receive velocity components away from their radial line. This is analogous to high and low tides on the Earth. Figure 11 illustrates this concept with a schematic of a rotating disk with angular velocity Ω. Circles at two radii are shown. The companion crosses the disk plane about 0100 (one o’clock) on the outer circle thus forming the line of tidal impulse.

Figure 11 - illustrating collision sites along the line of tidal impulse – the arrows point in the direction of the velocity perturbation – the length of the arrows indicate the magnitude of the perturbation.

Material orbiting near the line of tidal impulse will receive a kick towards this line in addition to its unperturbed counterclockwise orbital motion. As one moves towards the center of the disk the magnitude of the Washington Academy of Sciences

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kick decreases. Near the line of tidal impulse all the material within a section about twice the induced transverse epicyclic (see below) amplitude that results from the tidal disturbance will aggregate along the line of impulse and possibly gravitationally clump. This material will tend to collide with speeds at least two times the transverse epicyclic velocity amplitude. On the other hand, the surface density of material near 90° and 270° from the line of tidal impulse will tend to decrease (rarify). I suspect that the line of intense clumping will eventually create dust lanes and young stellar associations. Initially there is no arm along the line of tidal impulse; it appears gradually with time as clouds collide. The word epicycle (Greek for “on the circle”) is used in the theory of galaxy structure. When creating a mathematical theory of gas and star motion in a disk galaxy one typically uses the Boltzmann Equation. This equation describes the temporal and spatial variation of the probability distribution for the position and momentum of a density distribution of a cloud of points in single-particle phase space. For galaxies, the collisionless Boltzmann equation is

df ∂f ∂f ∂Φ ∂f = + vi − ⋅ =0 ∂xi ∂xi ∂vi dt ∂t where f(v,r) is the distribution function of matter, and Φ is the gravitational potential. The Boltzmann equation is notoriously difficult to integrate; therefore, one typically considers average quantities. This is acceptable for thin S galaxies where stars and gas move on nearly circular orbits. It is sufficient to derive approximate solutions based on perturbed circular orbits. In that case one obtains for the motion of stars and gas the epicyclic approximation – a retrograde epicycle orbit superimposed on the circular motion (similar to what the ancients used when describing the motions of the planets across the sky). In other words, the star or gas cloud performs radial oscillations about its circular orbit. This can be thought of as a little ellipse sitting on a large circle. The star or gas cloud travels around the ellipse which then slides around the circle. The orbit is seldom “closed”; i.e., the star or gas cloud does not make a complete orbit about the ellipse before completing a revolution about the galactic center. One can obtain a simple form for the epicyclic frequency κ by using a three dimensional coordinate system with x as the coordinate in the radial direction, y as the coordinate in the direction of circular motion, and z as the coordinate perpendicular to the x, y plane. The approximate planar solution then has the form Spring 2009

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x = −κ 2 x dΩ + 4Ω 2 κ2 = R dR

where κ is the frequency, R is the radial distance from the galaxy center, and Ω is the angular velocity. This is the equation for a simple harmonic oscillator with frequency κ. So the star or gas cloud performs radial oscillations with frequency κ around the circular orbit. In the case where one has a constant rotation velocity (Ω = V/R, where the velocity V is a constant), the frequency becomes κ = 2Ω . This is the case for spirals whose rotation velocity is a constant. For Keplerian orbits (such as those in our Solar System), Ω varies as R-3/2, and therefore, κ = Ω. Thus, Ω ≤ κ ≤ 2Ω

is the case for almost all spirals. In the z direction the stars and gas clouds move up and down like horses on a carousel. For the Milky Way at the Solar radius κ is about 42 km/s/kpc (Lepine et al 2007). Figures 12 and 13 are schematics of how an epicycle evolves when tidally stimulated. The small ellipses represent the epicycles; the arrows indicate the direction of the perturbation; the dotted lines represent two distances from the galactic center. The ellipses are smaller at the inner distance. Note how the impulse drives the material outward from the ellipse. Figure 13 shows the same schematic 70 million years after the impulse. Note how the material is now clustering together. Figure 14 shows a computer simulation of how a material arm will form as it is swept up in the counterclockwise turning disk as a companion drops by the outer edge. Time steps 0, 40, and 100 are shown. The companion (the small circle) crosses the disk plane at time step 100. The particles initially are aligned along radii spaced equally around the disk. Note how even by time step 40 the radial lines cluster along what will become the line of tidal impulse at time step 100. By time step 100 a strong clustering has already occurred, and the companion is pulling some material outward from the disk. The far side arm is weaker and broader than the near arm. The implication is that a large fan shaped chuck of the disk near the companion is swept into the developing arm during the passage.


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Figure 12 - a tidally stimulated epicycle at the time of impulse, kpc stands for kiloparsecs.

Figure 13 - 70 million years later Spring 2009

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Figure 14 - the development of a material arm, shown as three time steps in a simulation, steps 0, 40, and 100. The companion is marked by a small circle. The disk particles are initially aligned along radial lines. One such radial line is marked with small stars indicating particular particles. The particles are followed through the simulation.


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As an interesting sidelight, when a material arm is tidally triggered into forming, the material will gather together along the line of tidal impulse, eventually, however, as the arm continues forming inward towards the center it will meet that part of the arm that initially was rarefying. This transition produces a kink in the arm, often seen in grand design spirals (those with two well defined arms). An example of this is M51, the Whirlpool Galaxy, shown in Figure 15. The kink is interesting. Condensations begin with material moving outward in tidally induced epicyclic motion (see Figure 12). Rarefications begin with inward epicyclic motion. Since the epicyclic period is less at shorter radii, at any given time after arm formation there will be a radius where ½ an epicyclic period will have elapsed. Within this radius the ensuing epicyclic motion will have dissolved the original condensations of the near and far-side arms changing them to rarefications. This is the kink – the abrupt change in pitch of the arm is the termination of outer material arms where ½ an epicyclic period is equal to the time since arm formation. The material arm can tell us when the tidal impulse last occurred. Material arms are a rich source of information about their galaxy’s structure as well as its past. Perhaps we can restore tidal triggering to an equal status with density wave theory as the source of those beautiful spiral patterns we see in the Universe.

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kink

Figure 15 - M51 with the kink in the arm indicated by the arrow, the arms are marked by small squares and circles.

References Byrd, G. G., and Howard, S. 1992, Astronomical Journal, 103, 1089 Howard, S., 2008, WASJ, 94, No 1, page 21 Lepine, J., Dias, W.S. and Mishurov, Y., 2007, http://arxiv.org/pdf/0706.1811 Lin, C. C., and Shu, F. H., 1964, 1940, Astrophysical Journal 140, 646-655 Lindblad, B., 1925, 1941, 1954, 1956, 1958, Stockholm Observatory Annals
