2.1.1 Dispersion relations of tight-winding density waves

Lin and Shu derived dispersion relations of fluid and stellar discs under the following approximations and assumptions:

1. 1. Linear perturbations: They linearized the equation of continuity, the equations of motion (Euler equation), the equation of state, and Poisson equation, and then studied the behaviour of the linear perturbations. In this case, it is assumed that the unperturbed disc is axisymmetric and has no radial motions.

2. 2. Tight-winding approximation (short wavelength or WKBFootnote ¹ approximation): Deriving the dispersion relation for a general spiral wave is extremely complicated because of the long-range force nature of gravity (see Section 2.1.4). They assumed the spiral arm has a small pitch angle in order that distant density perturbations can be neglected. In other words, the response of the matter to the gravity perturbations becomes local. In this approximation, the dispersion relation of density waves can be written down in terms of local quantities. If we write the radial dependence of any perturbation quantity (Equation 1) in terms of an amplitude and phase as

   (2) \begin{equation} A(R) = \Phi (R) e^{if(R)}, \end{equation}
   the tight-winding approximation corresponds to the assumption that the phase f(R) varies rapidly in comparison with amplitude Φ(R), i.e.,
   (3) \begin{equation} \left| \frac{df}{dR}\right| \gg \left| \frac{1}{\Phi }\frac{d\Phi }{dR}\right|. \end{equation}

3. 3. Quasi-stationary spiral structure hypothesis (QSSS hypothesis): They hypothesized that ‘global’ spiral arms hardly change their shape during many galaxy rotational periods (‘quasi-stationarity’) based on empirical grounds (Lin & Shu Reference Lin and Shu1964; Bertin & Lin Reference Bertin, Lin, Bertin and Lin1996; Bertin Reference Bertin and Bertin2000). This is equivalent to assuming that the rotation of global spiral arms is rigid-body rotation with a specific angular velocity and pitch angle. This fixed angular speed is called the pattern speed. In the inner parts of galaxies, stars and gas rotate faster than the pattern speed, and overtake the spiral arm. In the outer parts of galaxies, stars and gas rotate slower than the pattern speed, and the spiral arms overtake the stars and gas.

Taking into account these approximations and assumptions, the Lin-Shu theory is often called a linear tight-winding, or quasi-stationary density-wave theory.

Since the stars occupy most of the mass of the galactic disc, the dispersion relation of the density wave of a stellar disc is important in understanding the spiral arms. However, the dispersion relation for a fluid disc is simpler, so we introduce this first, before discussing the dispersion relation for a stellar disc. We refer the reader to Binney & Tremaine (Reference Binney, Tremaine, Binney and Tremaine2008) and Shu (Reference Shu1992) for the mathematical details on derivation of dispersion relations, as well as Bertin & Lin (Reference Bertin, Lin, Bertin and Lin1996) and Bertin (Reference Bertin and Bertin2000) for further discussion on the concept of quasi-stationary density wave theory.

The dispersion relation for linear tight-winding perturbations in the razor-thin fluid disc (Lin-Shu dispersion relation; LS dispersion relation) is given by

(Lin & Shu Reference Lin and Shu1964). Here Ω, Σ₀, c_s and κ are the angular frequency, surface density, sound speed and epicyclic frequency

of the fluid disc, respectively. These quantities depend on the galacto-centric radius R and define the axisymmetric basis state. ω, k, and m are the angular frequency, radial wave-number, and the number of spiral arms, respectively. The radial wave-number k is related to the phase of the radial dependence of the perturbations f(R) (Equation 2) via

We define k > 0 for a trailing spiral arm and k < 0 for a leading spiral arm. Note that the so-called pattern speed Ω_p is defined as ω/m.

In the inertial frame, ω on the left hand side of Equation (4) is the angular frequency of the density wave. In the rotating frame at some radius R in the disc, (ω − mΩ) is the angular frequency of the density wave experienced by a star at R. A perturbation to the disc will be of the form exp [− i(ω − mΩ)t]. Positive (ω − mΩ)² means that the perturbations to the disc will be stable. However negative (ω − mΩ)² means that the perturbations will be of the form exp [± |ω − mΩ|t] and there is a perturbation whose amplitude grows exponentially, thus the disc is unstable. Therefore, the right-hand side of equation (4) indicates the stability of the density waves. Here, we can introduce a dimensionless parameter

known as the Toomre Q parameter, such that if Q > 1, (ω − mΩ)² > 0 for all radial wave-numbers k, while if Q < 1, (ω − mΩ)² becomes negative for a range of radial wave-numbers. Therefore Q gives us a criterion whether the disc is unstable or not to tight-winding perturbationsFootnote ² . We can also define a critical unstable wavelength λ_crit = 2π/k _crit, where (ω − mΩ)² = 0 for a cold fluid disc (i.e., c_s = 0). In this case, all perturbations with wavenumber |k| < k _crit or wavelength λ > λ_crit are unstable, where k _crit = κ²/(2πGΣ₀) or λ_crit = 4π² GΣ₀/κ². Note k _crit = k _min/2, where k _min is defined such that (ω(k _min) − mΩ)² = 0 for a neutrally stable fluid disc (Q = 1).

Figure 2 a shows the Lin-Shu dispersion relations for different Q values. Figure 2 a shows that high Q values (stability) occur for density waves with large wavelengths and angular frequencies. The physical meaning of each term of the right-hand side of equation (4) is as follows. The first term, c ² _s k ², expresses the effect of pressure, which being positive stabilizes the fluid against perturbations. This is the same as the dispersion relation of sound waves. The second term, κ² represents rotation, which again stabilizes the disc. The third term, which incorporates the self gravity of the disc, promotes the growth of instabilities. When the effect of self-gravity exceeds the limit where (ω − mΩ)² is non-negative, a real root does not exist and it is impossible for a stable density wave to exist. Thus, the LS dispersion relation shows that the (gaseous) spiral density wave can be considered to be a sort of acoustic wave taking into account the effects of rotation and self-gravity.

Figure 2. Dispersion relations for tight-winding density waves in a fluid disc (left) and stellar disc (right). Waves of a wavenumber smaller than that at the minimum frequency (|k| ≪ k _crit) are called long waves, while those with |k| ≫ k _crit are called short waves. The critical wavenumber k _crit is defined as κ²/(2πGΣ₀).

Consider now a stellar disc. The dispersion relation for linear tight-winding perturbations a in razor-thin stellar disc with a modified Schwarzschild distribution (Lin-Shu-Kalnajs dispersion relation; LSK dispersion relation) is given by

(Lin & Shu Reference Lin and Shu1966; Kalnajs Reference Kalnajs1965). Here, σ_R is the radial velocity dispersion of the stellar disc, and I _n is a modified Bessel function. Figure 2 b shows the LSK dispersion relations for different Q values defined by

The behaviour of the dispersion relation is similar to the LS relation for smaller radial wave-number (larger wavelength), but in the the larger radial wave-number (shorter wavelength) regime, the behaviour of the two is decidedly different. For the short-wave regime, the LSK dispersion relation approaches (ω − mΩ)²/κ² = 1 asymptotically, but the LS dispersion relation extends to (ω − mΩ)²/κ² > 1. This difference originates in the essential difference between the pressure for a fluid disc, and the velocity dispersion for a stellar disc: In the case of fluid discs, pressure will become large at small wavelengths. In contrast, since the stellar disc is collisionless, there is no such repelling force. Instead the frequencies of perturbations cannot become larger than the epicyclic frequency κ.

2.1.2 Propagation of tight-winding density waves

Although we have discussed waves as being quasi-stationary in the previous section, in reality if a wave is induced in the disc, it will propagate radially with some group velocity, rather than being stationary. In this and the next sections, we will discuss the group velocity and describe the further developments in density wave theory (the setting up of suitable boundary conditions) which allow the possibility, at least theoretically, of setting up a standing wave. The propagation of the tight-winding density waves is reviewed throughly in Grosbøl (Reference Grosbøl, Contopoulos, Spyrou and Vlahos1994) and Binney & Tremaine (Reference Binney, Tremaine, Binney and Tremaine2008).

The original density wave theory (Lin & Shu Reference Lin and Shu1964) is based on the QSSS hypothesis which assumes that the amplitude and shape of the spiral arm are independent of time. However, since the angular frequency ω depends on radial wave-number k via the LS or LSK dispersion relations (eqs. 4 and 8), the energy and angular momentum of the density waves propagate radially as wave packets (Toomre Reference Toomre1969). This propagation velocity is the group velocity, given by v_g = ∂ω(k, R)/∂k. If we consider propagation of density waves in a fluid disc, following the LS dispersion relation (Equation 4), the group velocity of a wave packet is

where positive and negative signs indicate trailing (k > 0) and leading (k < 0) spiral waves, respectively.

The sign of the numerator of this equation is negative for short waves and positive for long waves, and the sign of the denominator is negative and positive for R < R _CR and R > R _CR, respectively. Thus, short trailing and long leading spiral waves will propagate away from the coronation (CR) radius, while the short leading and long trailing spiral waves will approach the CR radius (propagation directions are indicated by arrows in Figure 3). Note that if the disc has a large Q, a forbidden region emerges in the vicinity of the CR, where due to the pressure or random motions the density waves diminish.

Figure 3. Propagation diagram for tight-winding stellar density waves following the LSK dispersion relation (Equation 8). The disc is assumed to have a flat rotation curve and constant Toomre’s Q = 1.2. The horizontal dashed lines are the OLR radius (upper), CR radius (middle), and ILR radius (lower), respectively. The arrows indicate the directions of group velocities. Long waves (|k/k _crit| ≪ 1) are reflected at the Lindblad resonances, while short waves (|k/k _crit| ≫ 1) are absorbed there due to Landau damping.

The behavior is essentially same for stellar density waves except for regions around the inner and outer Lindblad resonances (ILR and OLR). The difference around the ILR/OLR originates in the difference in dynamical behavior between stars and fluid (Section 2.1.1). The propagation digram for stellar density waves following the LSK dispersion relation (Equation 8) is shown in Figure 3. Long stellar density waves (|k/k _crit| ≪ 1) are reflected at the Lindblad resonances (Goldreich & Tremaine Reference Goldreich and Tremaine1978, Reference Goldreich and Tremaine1979) while short waves (|k/k _crit| ≫ 1) are absorbed there due to Landau damping (Lynden-Bell & Kalnajs Reference Lynden-Bell and Kalnajs1972). Thus, both long and short stellar density waves cannot pass through the Lindblad resonances. Therefore, the permitted region for stellar density waves is restricted between the ILR and OLR radii (with the exception again of the forbidden region). However, this does not necessarily imply that stationary density waves will exist here.

If we apply the group velocity formula to the solar neighborhood, v _g ~ 12 km s^{− 1} the stellar density wave takes ~ 400 Myr to propagate 5 kpc radially. This timescale is comparable to the rotation period of the Galaxy. Therefore, the stellar density waves will have a short lifetime of order < 1 Gyr (Toomre Reference Toomre1969).

This problem can be solved if the density waves are reflected in the central region before reaching the ILR, and amplified by some mechanism. An absorption of the short stellar density waves at the ILR can be avoided if the Toomre’s Q parameter increases significantly (forming a so-called Q-barrier) refracting the density wave outside the ILR. Short trailing stellar density waves can be excited near the CR from long trailing stellar density waves by ‘the wave amplification by stimulated emission of radiation’ (WASER) in lighter discs (Mark Reference Mark1974, Reference Mark1976), or from short leading density waves by the swing amplification mechanism in heavier discs (Goldreich & Lynden-Bell Reference Goldreich and Lynden-Bell1965b; Julian & Toomre Reference Julian and Toomre1966; Goldreich & Tremaine Reference Goldreich and Tremaine1978; Toomre Reference Toomre, Fall and Lynden-Bell1981). With these assumptions, ‘standing-wave’ patterns Footnote ³ can exist between a reflecting radius in the inner part of the galaxy and CR radius, where the waves can be amplified (Bertin et al. Reference Bertin, Lin, Lowe and Thurstans1989a, Reference Bertin, Lin, Lowe and Thurstans1989b). The spiral density waves should be located between, but not reaching the ILR and OLR.

2.1.3 Swing amplification

The quasi-stationarity of spiral arms requires wave amplification mechanisms such as WASER (Mark Reference Mark1974, Reference Mark1976) or swing amplification (Goldreich & Lynden-Bell Reference Goldreich and Lynden-Bell1965b; Julian & Toomre Reference Julian and Toomre1966; Goldreich & Tremaine Reference Goldreich and Tremaine1978; Toomre Reference Toomre, Fall and Lynden-Bell1981). In the WASER (swing amplification) mechanism, a trailing (leading) wave is turned into a trailing wave when crossing CR and is greatly amplified in the process. It is noted that there is no conflict between the swing amplification and WASER (Lin & Thurstans Reference Lin, Thurstans, Guyenne and Hunt1984; Bertin, Lin, & Lowe Reference Bertin, Lin, Lowe, Guyenne and Hunt1984; Bertin et al.Reference Bertin, Lin, Lowe and Thurstans1989b). The reason they have been considered differently is largely historical, reflecting opposing views at the time. However, if discs have non-negligible self-gravity at CR, the swing amplification mechanism can greatly dominate amplification by the WASER mechanism. On the other hand, in a system where the disc mass is only a small fraction that supports the rotation curve, the WASER mechanism can underline the growth of the most important spiral mode, as long as Q ≃ 1 at CR (Shu Reference Shu1992; Bertin & Lin Reference Bertin, Lin, Bertin and Lin1996; Bertin Reference Bertin and Bertin2000).

We focus on the swing amplification mechanism as a wave amplification mechanism for sustaining quasi-stationary density waves between the ILR and OLR. The so-called swing amplification works when short leading waves are reflected to short trailing waves at the CR radius, or when a density enhancement formed by self-gravity is stretched out by differential rotation. The dynamical response takes the form of wavelets in the surrounding medium, each amplified by its own self-gravity through the swinging of leading features into trailing ones due to shear.

The swing amplification operates through a combination of three ingredients: the shearing flow, epicyclic motions, and the disc self-gravity. Toomre (Reference Toomre, Fall and Lynden-Bell1981) interpreted the swing amplification mechanism in terms of the wave-particle interaction between spiral arms and stars. Since the direction of epicyclic motion of a star is the same as the direction which the spiral arm is sheared by differential rotation, stabilisation by rotation is reduced, and the perturbation can grow via the usual Jeans instability (Goldreich & Lynden-Bell Reference Goldreich and Lynden-Bell1965b; Julian & Toomre Reference Julian and Toomre1966; Goldreich & Tremaine Reference Goldreich and Tremaine1978). The timescale of epicyclic motion (κ^{− 1}) is comparable to the timescale of involvement with the spiral arm (A ^{− 1} where A is Oort’s constant), and unless Q ≫ 1, the structure can grow in a short time comparable to κ^{− 1}. The resulting spiral structure from this process is generally expected to be chaotic (Sellwood Reference Sellwood2011) rather than lead to a symmetric spiral pattern.

Consider a local region of a galactic disc away from the galaxy center. Since the galactic rotation is parallel to this local region (curvature can be ignored), we set an x and y-axis aligned with the radial and rotational directions of the galaxy respectively. In this case, the equations of motion of the stars are given by

(using the Hill approximation). Here, x = R − R ₀, y = R ₀(ϕ − Ω₀ t), and Ω₀ and $A_{\rm 0} \equiv -\frac{1}{2} R_{\rm 0}(d\Omega /dR)_{\rm 0}$ are the angular velocity and Oort’s constant at R ₀, respectively. f _⊥ indicates the gravitational force perpendicular to the spiral arm. γ is an angle between the spiral arm and radial direction of the galaxy: γ = 90°, γ < 0, and γ > 0 correspond to a ring, leading, and trailing structures, respectively.

Defining the normal displacement of the star perpendicular to the spiral arm,

as a new variable, the equations of motion reduce to an equation

where the squared spring rate is given by

and Figure 4 shows dependence of the spring rate on galaxy parameters of (Γ, Q, X). In the case of Γ = 0 (i.e, the galaxy has rigid-body rotation), the spring rate is always positive. Thus, the stars cannot be trapped by the spiral arm, and then the spiral arm does not amplify. In other words, the swing amplification cannot work without differential rotation.

Figure 4. Squared spring rate S(γ) as a function of the angle γ between the spiral arm and radial direction of the galaxy for Γ = 0.0 (rigid rotation) and Γ = 1.0 (flat rotation). Different lines indicate Q = 1.0 (black), 1.2 (red), and 1.5 (green), respectively. Spring rates are calculated based on the equations of motion in Toomre (Reference Toomre, Fall and Lynden-Bell1981) and Athanassoula (Reference Athanassoula1984). The squared spring rate is always positive in the case of Γ = 0.0, but it can be negative in the case of Γ = 1.0. Thus, the normal displacement of the stars around the spiral arm ξ can grow exponentially as the spiral arm is sheared by differential rotation.

By the transformation of variables to γ, instead of time t, Equation 17 becomes

Numerical integration of this differential equation gives the dependence of the swing amplification factor on the galaxy parameters (Γ, Q, X) shown in Figure 5. The effect of self-gravity and the winding of the spiral arm work in synergy, so that a star comes to stay at the spiral arm for a long time, and spiral arms are amplified temporarily. Note that the above argument is based on the linear analysis by Toomre (Reference Toomre, Fall and Lynden-Bell1981) and Athanassoula (Reference Athanassoula1984). Fuchs (Reference Fuchs2001) solved linearised collisionless Boltzmann and Poisson equations self-consistently and showed that the result is essentially the same. Further, non-linear effects are studied in N-body simulations of local regions of stellar discs (Toomre Reference Toomre and Wielen1990; Toomre & Kalnajs Reference Toomre, Kalnajs and Sundelius1991; Fuchs, Dettbarn, & Tsuchiya Reference Fuchs, Dettbarn and Tsuchiya2005), as well as N-body simulations of global stellar discs (Sellwood & Carlberg Reference Sellwood and Carlberg1984; Carlberg & Freedman Reference Carlberg and Freedman1985; Bottema Reference Bottema2003; Fujii et al. Reference Fujii, Baba, Saitoh, Makino, Kokubo and Wada2011; Baba, Saitoh, & Wada Reference Baba, Saitoh and Wada2013; D’Onghia, Vogelsberger, & Hernquist Reference D’Onghia, Vogelsberger and Hernquist2013). D’Onghia et al. (Reference D’Onghia, Vogelsberger and Hernquist2013) carefully demonstrated the growth of spiral arm features by swing-amplification and found a nonlinear evolution that is not fully consistent with the classic swing-amplification picture of Julian & Toomre (Reference Julian and Toomre1966) and lasted longer than predicted by swing amplification (Toomre & Kalnajs Reference Toomre, Kalnajs and Sundelius1991) (see also Section 2.1.5).

Figure 5. The maximum amplification factor is shown as a function of the X, Γ and Q parameters. The amplification factor is calculated based on the equations of motion given in Toomre (Reference Toomre, Fall and Lynden-Bell1981) and Athanassoula (Reference Athanassoula1984).

In order for the swing amplification mechanism to work continuously (Toomre & Kalnajs Reference Toomre, Kalnajs and Sundelius1991), we need to understand how leading waves are generated. One possibility is the case where there is no ILR. A trailing wave does not suffer from Landau damping at the ILR, instead the wave turns into a leading wave as it crosses the galaxy center. This is the so-called feedback loop proposed by Toomre (Reference Toomre, Fall and Lynden-Bell1981).

2.1.4 Global mode theory

Although linear density wave theory was successful in demonstrating the existence of a tight-winding spiral wave, the tight-winding density wave theory has room for improvement. Firstly, since they utilized the WKB approximation, this theory cannot be applied to very long waves (or open spiral arms) strictly. Secondly, the presence of neutral spiral density waves itself is theoretically questionable. It is critically problematic that a density wave propagates through a galactic disc radially in a few galactic rotations, and eventually disappears by absorption at the inner/outer Lindblad resonances (ILR/OLR). Thus, the ‘quasi-stationarity’ hypothesis is not ensured (Toomre Reference Toomre1969, Section 2.1.2). Finally, the tight-winding theory cannot predict the number of spiral arms m and sign of the wave-number k (i.e., trailing or leading). In other words, the theory cannot explain why actual spiral galaxies prefer to have trailing two-armed spirals (k > 0 and m = 2) and what determines the angular frequency of the spiral density wave. In response to these criticisms, the tight-winding density wave theory developed into a global mode theory (e.g., Lau, Lin, & Mark Reference Lau, Lin and Mark1976; Bertin et al. Reference Bertin, Lau, Lin, Mark and Sugiyama1977; Aoki, Noguchi, & Iye Reference Aoki, Noguchi and Iye1979; Iye Reference Iye1978; Bertin Reference Bertin1983; Bertin et al. Reference Bertin, Lin, Lowe, Guyenne and Hunt1984, Reference Bertin, Lin, Lowe and Thurstans1989a, Reference Bertin, Lin, Lowe and Thurstansb; Bertin & Lin Reference Bertin, Lin, Bertin and Lin1996).

A key nontrivial step at the basis of the derivation of the dispersion relation is the reduction of the long-range gravity law to a WKB dispersion relation between the perturbed potential and the perturbed density. Numerical integration of the basic perturbed equations is required. Since the first global mode analysis was applied to rotating fluid discs by Hunter (Reference Hunter1965), there have been many studies, mainly in the 1970s-80s (e.g. Bardeen Reference Bardeen and Hayli1975; Aoki et al. Reference Aoki, Noguchi and Iye1979; Iye Reference Iye1978; Takahara Reference Takahara1978), as well as extending the analysis to rotating stellar discs (Kalnajs Reference Kalnajs1972). In order to analyse the eigen-value problem of a stellar system, it is necessary to solve the density perturbations and the responsive orbital perturbations in satisfying the linearized collisionless Boltzmann equation and the Poisson equation, self-consistently. Although Kalnajs (Reference Kalnajs1972) solved the eigenvalue problem of the Maclaurin disc using the so-called ‘matrix method’, numerical integration is required to solve the eigen-value problems of stellar discs (e.g. Athanassoula & Sellwood Reference Athanassoula and Sellwood1986; Sellwood & Athanassoula Reference Sellwood and Athanassoula1986; Sellwood Reference Sellwood and Sellwood1989; Earn & Sellwood Reference Earn and Sellwood1995; Vauterin & Dejonghe Reference Vauterin and Dejonghe1996; Pichon & Cannon Reference Pichon and Cannon1997; Polyachenko Reference Polyachenko2004, Reference Polyachenko2005; Jalali & Hunter Reference Jalali and Hunter2005). However, these studies are somewhat limited due to the mathematical complexity.

Global mode analysis is based on a point of view that the spiral arms are manifestations of the gravitationally ‘unstable’ global eigen-oscillations of disc galaxies Footnote ⁴ . This eigen-oscillation problem of the galactic disc resembles the problem of oscillating patterns of the skin when a drum is struck. Similar to the way oscillation patterns are controlled by how to stretch and how to strike the skin, oscillation of a galactic disc is controlled by the density and velocity-dispersion distributions of a galactic disc. However, there are two differences between eigen-oscillation problems of the drum and galactic disc. First, changes of the gravity from the oscillation should be taken into account to solve the eigen-oscillation problem of the galactic disc. This makes the problem very complicated. In the case of a drum, there is only a traverse wave, but for the oscillation of the galactic disc, there is also a longitudinal wave as well as a transverse wave. The transverse and longitudinal waves in the galactic disc are equivalent to the bending (warp) of a galactic disc and spiral arms, respectively.

Before explaining numerical results of the global mode analysis, let us consider the stability of fluid and stellar discs. Consider first the case of axisymmetric perturbations (Toomre Reference Toomre1964; Goldreich & Lynden-Bell Reference Goldreich and Lynden-Bell1965a). Qualitatively, we consider the case where an axisymmetric disc receives a perturbation illustrated by the small arrows shown in Figure 6(a). The fluid and stars in the disc move radially, and try to make a ring structure, but the pressure (or velocity dispersion), centrifugal, and Coriolis forces suppress growth of this ring perturbation. Left panel of Figure 7 shows the neutral stability curves for tight-winding spirals ((ω − mΩ)² = 0) showing Q-values as a function of λ/λ_crit. As Q for the disc is lowered, the disc moves from the stable to unstable regime. The wavelength which becomes unstable first is pλ_crit, where p = 0.5 in the case of a fluid disc and p = 0.55 in a stellar disc.

Figure 6. Axisymmetric perturbations (a) and bar-like perturbations (b) on an axisymmetric disc. The disc rotates anti-clock wise. Directions of the perturbations are indicated by small arrows.

Figure 7. Left: Neutral stability curves for tigiht-winding spiral instabilities based on the LS dispersion relation (red; Equation 4) and LSK dispersion relation (black; Equation 8). The region below the curve is stable against tight-winding spiral instabilities. Right: Neutral stability curves for open spiral instabilities based on the BLL dispersion relation (Equation 21) with $\mathcal {J} = 0, 0.4, 0.6, 1.0$ , and 1.414.

In the case of spiral perturbations with a finite pitch angle (i.e., open spiral perturbations), the effects of the gravitational perturbation parallel to the spiral arm, and shear originating from differential rotation need to be taken into account. Again, we consider the case where an axisymmetric disc receives a perturbation given by the small arrows shown in Figure 6(b). In contrast to the case of Figure 6(a), the fluid and stars move in the azimuthal direction, and centrifugal and Coriolis forces do not appear. Thus, growth is not suppressed. The effect of the excessive centrifugal and Coriolis forces stabilises the perturbation with a long wavelength (Section 2.1.1 and Figure 2), it is expected that the stabilization effect will become weak in a long wavelength regime and that open spiral arms will become unstable.

This qualitative expectation is checked quantitatively below. Lau & Bertin (Reference Lau and Bertin1978) derived the asymptotic dispersion relation of open spiral density waves in the fluid disc (Bertin-Lau-Lin dispersion relation; BLL dispersion relation):

where $k = \sqrt{k_R^2 + k_\phi ^2}$ , k_R , and k _ϕ = m/R are the wave-number, radial wave-number, and azimuthal wave-number, respectively Footnote ⁵ . We define two quantities

where $\mathcal {J}$ indicates a stability parameter which depends on the disc mass relative to the total mass and a shear rate of the disc, and ε₀ is a parameter which relates to the degree of the self-gravity. In the LS dispersion relation, as described in Section 2.1.1, the disc is stabilized by the rotational (κ²) and pressure effects (k ² c ² _s) against self-gravity (2πGΣ₀ k). In addition to these effects, the BLL dispersion relation includes within $\mathcal {J}$ the rate of shear, $\frac{d\ln \Omega }{d\ln R}$ , as well as the self-gravity term, ε₀.

Based on the BLL dispersion relation, a neutral stability curve for spiral instabilities is given by

where λ = 2π/k is a wavelength (right panel of Figure 7). If we set $\mathcal {J}=0$ , the neutral stability curve is equivalent to ones for the LS dispersion relation. From this neutral stability curve, a value of Q larger than unity is required for stability against spiral disturbances with a larger value of $\mathcal {J}$ . This means that open spiral arms are difficult to stabilise and will, more often than not, result in growth.

Figure 8 shows the numerically integrated density contours of the global unstable modes with different $\mathcal {J}$ and Q values (Bertin et al. Reference Bertin, Lin, Lowe and Thurstans1989b). The pitch angle in the mode becomes smaller as the value of $\mathcal {J}$ decreases (panels (a), (b), and (c), respectively). For the case where $\mathcal {J}$ and Q are large (panel (a)), the bar mode becomes unstable. In the case of large $\mathcal {J}$ but small Q (panel (d)), only the spiral mode is unstable. This behaviour is in agreement with that expected from the BLL dispersion relation (right panel in Figure 8). The right panel in Figure 8 shows curves of constant pitch angle α in the $(\mathcal {J},Q)$ -plane. The pitch angle here (for the more general definition see Equation (27)) is given by

where k_R and k _ϕ are the radial wave-number and azimuthal wave-number, respectively. Thus, the unstable spiral mode is determined by two parameters, $\mathcal {J}$ and Q, given by the rotation curve of the galaxy disc. $\mathcal {J}$ controls the shape and growth rate of the unstable mode. The spiral mode appears for smaller $\mathcal {J}$ , and the bar mode for larger $\mathcal {J}$ (Lau & Bertin Reference Lau and Bertin1978).

Figure 8. Left: Density contours of global unstable modes for a rotating fluid disc where (a) $\mathcal {J} = 0.604$ and Q = 1.500, (b), $\mathcal {J} = 0.538$ and Q = 1.096, (c), $\mathcal {J} = 0.492$ and Q = 1.002, and (d) $\mathcal {J} = 0.858$ and Q = 1.004. Right: Curves of constant pitch angle $\alpha = \cot ^{-1} \frac{k_\phi }{k_R}$ in the $(\mathcal {J},Q)$ -plane. These curves are derived from the BLL dispersion relation (Equation 21) for the neutral stability condition (Equation 24) with Γ = 0 (flat rotation curve). From Bertin et al. (Reference Bertin, Lin, Lowe and Thurstans1989b).

The number of spiral arms and their pattern speed cannot be determined in the framework of the tight-winding density-wave theory of Lin-Shu-Kalnajs (Section 2.1.1). On the other hand, for global mode theory, if the unstable mode with the highest growth rate (trailing spiral modes) is assumed to be the spiral arms actually observed, then the spiral arms can be uniquely predicted from the equilibrium state of a galactic disc. Therefore, it turns out that global unstable mode theory is a self-contained theory.

However, there are some limitations in the global mode theory. Firstly, it is assumed that the spiral mode rotates as a rigid body, without changing its shape in the global unstable mode analysis. Iye et al. (Reference Iye, Aoki, Ueda and Noguchi1983) studied the global unstable modes of the fluid disc without the rigid-body rotation of the spiral modes, and reported the appearance of a global leading mode as well as global trailing modes, which is a different result from the rigid-body case (Aoki et al. Reference Aoki, Noguchi and Iye1979). Recently, N-body simulations also show that spiral arms wind up over time by the shear of the galaxy disc (Wada, Baba, & Saitoh Reference Wada, Baba and Saitoh2011; Baba et al. Reference Baba, Saitoh and Wada2013; Grand et al. Reference Grand, Kawata and Cropper2013, see Sections 2.2.1 and 4.2 for details). Secondly, because global spiral modes grow up exponentially with time, the global mode theory requires self-regulated mechanisms such as damping effects in the stellar disc (e.g., Landau damping) and/or a gas component (Lin & Bertin Reference Lin, Bertin, van Woerden, Allen and Burton1985; Bertin & Romeo Reference Bertin and Romeo1988; Bertin et al. Reference Bertin, Lin, Lowe and Thurstans1989a, see also Section 3.1). Finally, it is unclear that the global modes really accomplish a neutrally stable state. The global mode theory hypotheses that the spiral arms are global neutral stability modes, which are accomplished by regulation mechanisms for the growth of density waves. However, Lynden-Bell & Kalnajs (Reference Lynden-Bell and Kalnajs1972) showed that spiral waves transport angular momentum by the gravitational torque which changes the distributions of angular momentum and mass (i.e., induces migration of stars and gas).

2.1.5 Simulations of long-lived spiral patterns

Various studies have tried to reproduce non-barred quasi-stationary density waves using numerical simulations, but so far, no convincing isolated, long-lived grand design spirals have been produced. The m = 2 case is of particular interest because low m modes are most likely to be stable (see Section 4.5), hence simulations have tended to focus on trying to model galaxies with a 2 armed spiral structure. As discussed in the previous sections, attaining a stable m = 2 perturbation involves modelling a disc which is sufficiently self gravitating to be unstable to the m = 2 mode, but stable to bar formation, and whereby density waves are able to be maintained by the presence of a Q barrier before the ILR, where waves can be reflected back towards corotation (Section 2.1.2). Two armed spirals associated with bars, or interactions, are of course common outputs from simulations, and we discuss these in Sections 2.3 and 2.4.

Early simulations of stellar discs all tended to form a bar (or oval distortion), and develop a strong m = 2 spiral mode (Miller, Prendergast, & Quirk Reference Miller, Prendergast and Quirk1970; Hohl Reference Hohl1971). The formation of a bar was also predicted analytically in the case of a uniformly rotating disc (Kalnajs Reference Kalnajs1972; Kalnajs & Athanassoula-Georgala Reference Kalnajs and Athanassoula-Georgala1974). However with the adoption of an extended, massive (comparable to or more massive than the disc) dark matter halo, the bar mode was both predicted, and found to be suppressed (Ostriker & Peebles Reference Ostriker and Peebles1973; Hohl Reference Hohl1976). Since then, simulations of isolated, non-barred galaxies have only produced multi-armed galaxies with transient spiral arms, as we discuss in Section 2.2. These types of galaxies can be produced readily with an N-body code. By contrast, trying to model an m = 2 spiral requires a long list of criteria to satisfy, and even then, m = 2 spirals still appear to be transient, evolving to m = 3 spirals and back again.

Thomasson et al. (Reference Thomasson, Elmegreen, Donner and Sundelius1990), and also Elmegreen & Thomasson (Reference Elmegreen and Thomasson1993), performed calculations of a galaxy, where in addition to the conditions above, they also enforced that the stellar velocity had to be maintained at a low value, and included a gas component. As will be mentioned in Section 3.2, and stated in Section 2.1.4, gas is likely required to allow energy from the spiral waves to dissipate. The galaxy tends to exhibit a pattern changing between 2 and 3 arms, and consequently has an asymmetric, rather than symmetric m = 2 pattern at many time frames. Without a Q barrier, the spirals are shorter lived, whilst without cooling or gas, higher m patterns become more prominent. Donner & Thomasson (Reference Donner and Thomasson1994) found similar results with a more consistent star formation scheme for the gas, and gas cooling and heating. Zhang (Reference Zhang1996) used the same setup as Donner & Thomasson (Reference Donner and Thomasson1994), and found similar spiral patterns, but without including a gas component. As well as changing to a 3 armed pattern, spirals which transition between an m ≥ 2 spiral and a barred spiral are likewise feasible to simulate by including gas accretion (Bournaud & Combes Reference Bournaud and Combes2002). But as yet no simulated galaxy retains a steady m = 2 spiral.

Sellwood (Reference Sellwood2011) tested some of the models claiming to find m = 2 spirals. He performed N-body simulations designed to test directly whether a galaxy model corresponding to panel (c) of Figure 8 can in fact survive to support the slowly growing mode they predict should dominate. He argued that this model evolves quickly due to multi-arm instabilities originating from swing-amplified noise (see Section 2.2.1) instead of producing quasi-stationary, two-armed spiral modes. This result suggests that dynamical evolution associated with shearing of spiral arms which is not considered in quasi-stationary density wave theories is important for generating the spiral arms in real galaxies. Sellwood (Reference Sellwood2011) also tested some of the above models, which have proposed to exhibit long-lived spirals (Donner & Thomasson Reference Donner and Thomasson1994; Zhang Reference Zhang1996). He showed that the bisymmetric spiral arm represented as a mode is not a single long-lived pattern, but the superpositions of three or more waves that each grow and decay.

Whilst simulations have been unsuccessful in reproducing a stationary spiral pattern, m = 2 or otherwise, recent work by D’Onghia et al. (Reference D’Onghia, Vogelsberger and Hernquist2013) and Sellwood & Carlberg (Reference Sellwood and Carlberg2014) do report the existence of longer-lived ‘modes’, which survive multiple rotations, and thus more resemble density wave theory. However these authors still state that their results are inconsistent with the idea that spirals are quasi-stationary density waves because the arms in their simulations still fluctuate in time. By contrast in global mode theory we would expect the arm shape to be unchanging for a number of rotations. For these long-lived spiral arms, the disc is required to be fairly gravitational dominated (Sellwood & Carlberg Reference Sellwood and Carlberg2014 adopt Q = 1) or include some perturbation(s) (D’Onghia et al. Reference D’Onghia, Vogelsberger and Hernquist2013, see also Salo & Laurikainen Reference Salo and Laurikainen2000b, Section 2.4.1).

2.2.1 Swing amplified spirals

We introduced swing amplification in Section 2.1.3 as a means of sustaining quasi-stationary density waves between the ILR and OLR. Here we describe the generation of dynamic spirals by swing amplification, but unlike the quasi-stationary density waves described in Section 2.1, there is no need for the waves to be reflected and maintained.

Full N-body simulations of stellar discs can test the predictions from swing amplification theory (Sellwood & Carlberg Reference Sellwood and Carlberg1984; Carlberg & Freedman Reference Carlberg and Freedman1985; Bottema Reference Bottema2003; Fujii et al. Reference Fujii, Baba, Saitoh, Makino, Kokubo and Wada2011; D’Onghia et al. Reference D’Onghia, Vogelsberger and Hernquist2013; Baba et al. Reference Baba, Saitoh and Wada2013). We can estimate the dominating number of spiral arms, m, as

where X ≈ 2 (Equation 19) is assumed, and find this is roughly consistent with the results of N-body simulations. As described above, spiral arms typically develop most effectively when 1 < X < 2, so taking X ≈ 2 is appropriate (Figure 5). More generally, for a galaxy with a flat rotation curve (Γ = 0.5) and a total mass within the disc of M _tot (mainly dark matter and stars), since κ² = 2Ω² ~ GM _tot/R ², we also obtain the number of spiral arms as m ~ Σ_tot/Σ₀ = 1/f _disc. Carlberg & Freedman (Reference Carlberg and Freedman1985) performed N-body simulations of galactic discs with various disc mass fractions and found that the number of spiral arms is strongly correlated with the disc mass fraction f _disc. A similar result is also obtained by N-body simulations with much higher resolution (Bottema Reference Bottema2003; Fujii et al. Reference Fujii, Baba, Saitoh, Makino, Kokubo and Wada2011; D’Onghia et al. Reference D’Onghia, Vogelsberger and Hernquist2013). Also since $\kappa = \sqrt{2} \Omega \propto 1/R$ for the galaxy with a flat rotation curve, m∝1/(RΣ₀) and the number of spiral arms tends to increase with radius in outer regions of exponential-discs. This agrees qualitatively with observations. As a specific example, we show the radial distributions of the number of spiral arms obtained by N-body simulations (Bottema Reference Bottema2003) and observations of NGC1288 (Fuchs & Möllenhoff Reference Fuchs and Möllenhoff1999) in Figure 9.

Figure 9. (top) Radial distribution of the number of spiral arms obtained by N-body simulations (Bottema Reference Bottema2003). (middle) Same as the top panel, but for observations of NGC 1288 (Fuchs & Möllenhoff Reference Fuchs and Möllenhoff1999). (bottom) I-band face-on view of NGC 1288 (Fuchs & Möllenhoff Reference Fuchs and Möllenhoff1999).

The pitch angle of the spiral arm is in reasonable agreement with the predictions of swing amplification theory. Figure 10 shows the evolution of a stellar spiral arm along the pitch-angle - density-contrast ( $\alpha -\bar{\delta }$ ) plane. Due to differential rotation, one arm becomes more tightly wound as time goes by, and eventually disappears. In the meantime, new spiral arms with larger pitch angles start to grow. As the pitch angle of the spiral arm decreases from α ≈ 40° (T _rot = 12.0) to α ≈ 32° (T _rot = 12.20), the density contrast increases to a maximum, and the density contrast subsequently decreases with a decrease in the pitch angle. Thus, the spiral arm has a maximum amplitude when α ~ 32°. This value is consistent with the expectation from swing amplification theory (hatched region in Figure 10). This behavior is similar to that reported in Sellwood & Carlberg (Reference Sellwood and Carlberg1984). We compare the pitch angles of spiral arms produced by swing amplification theory with observations in Section 4.2.

Figure 10. Evolution of spiral arm on $\alpha -\bar{\delta }$ plane for T_rot = 12.0 − 12.5. The hatched region corresponds to the predicted maximum pitch angle around the analysed region (Q ≈ 1.4 and Γ ≈ 0.8) due to swing amplification (refer to Equation (98) in Fuchs (Reference Fuchs2001)). From Baba et al. (Reference Baba, Saitoh and Wada2013).

Figure 11. Evolution of spiral arms with N = 30M. Top panels show the surface density, middle panels show the surface density normalized at each radius, and bottom panels show the Fourier amplitudes. From Fujii et al. (Reference Fujii, Baba, Saitoh, Makino, Kokubo and Wada2011).

In addition to the non-stationarity of stellar spiral arms, recent N-body simulations have shown that the pattern speed of the spiral arms decreases with radius in a similar manner to the angular rotation velocity of the disc (See also Section 4.1; Wada et al. Reference Wada, Baba and Saitoh2011; Grand et al. Reference Grand, Kawata and Cropper2012b, Reference Grand, Kawata and Croppera; Baba et al. Reference Baba, Saitoh and Wada2013; Roca-Fàbrega et al. Reference Roca-Fàbrega, Valenzuela, Figueras, Romero-Gomez, Velazquez, Antoja and Pichardo2013). Thus, the spiral arms are considered to be rotating with the rest of the disc at every radius, and are material arms. In the above models, the evolution of the spiral arms is governed by the winding of the arms, which leads to breaks and bifurcations of the spiral arms. Sellwood & Lin (Reference Sellwood and Lin1989) and Sellwood & Kahn (Reference Sellwood and Kahn1991) instead argued that the dynamics originate from multiple wave modes of different pattern speeds constructively and destructively interfering with one another (Sellwood Reference Sellwood2011; Quillen et al. Reference Quillen, Dougherty, Bagley, Minchev and Comparetta2011; Roškar et al. Reference Roškar, Debattista, Quinn and Wadsley2012; Sellwood Reference Sellwood2012).

Finally, Sellwood & Carlberg (Reference Sellwood and Carlberg1984) also investigated the amplitudes of swing amplified spiral arms. They found that the growth of perturbations is in reasonable quantitative agreement with the prediction of swing amplification theory, although the growth factor was slightly larger than a naive expectation from the level of particle shot noise (i.e., swing-amplified noise). This discrepancy between numerical simulations and theoretical expectations is also seen in local stellar discs (Toomre & Kalnajs Reference Toomre, Kalnajs and Sundelius1991). It may relate to non-linear effects of swing-amplified spiral instabilities. Toomre & Kalnajs (Reference Toomre, Kalnajs and Sundelius1991) attributed the discrepancy to additional correlations between the particles that developed over a long period, i.e., the polarized disc response to random density fluctuations. By contrast, Sellwood (Reference Sellwood and Sellwood1989) showed that the amplitudes of spiral arms in global simulations of stellar discs seem to be independent of the particle number, rather than declining as N ^{− 1/2} as would be predicted from the swing-amplified noise (Toomre & Kalnajs Reference Toomre, Kalnajs and Sundelius1991). It should be noted that star clusters and GMCs in real galaxies can seed much larger fluctuations than shot noise from equal-mass particles (D’Onghia et al. Reference D’Onghia, Vogelsberger and Hernquist2013). Sellwood (Reference Sellwood2011) also argued that spiral arms originating from swing-amplified shot noise are too low compared to observed spiral amplitudes. Instead, Sellwood (Reference Sellwood2000) suggested that spiral arms are vigorous large-scale modes originating from groove mode instabilities (Sellwood & Lin Reference Sellwood and Lin1989; Sellwood & Kahn Reference Sellwood and Kahn1991) (see Section 2.2.3).

2.2.2 Corotation scattering and radial migration of stars

Since the dynamic spiral arms do not have a single pattern speed but roughly follow the galactic rotation, or multi-wave patterns with different pattern speeds exist, these arms scatter stars everywhere in the disc via the corotation resonance (Sellwood & Binney Reference Sellwood and Binney2002; Grand et al. Reference Grand, Kawata and Cropper2012b; Roškar et al. Reference Roškar, Debattista, Quinn and Wadsley2012; Baba et al. Reference Baba, Saitoh and Wada2013; Grand, Kawata, & Cropper Reference Grand, Kawata and Cropper2014). Figure 12 shows the evolution of stars along the ϕ − R plane and the azimuth angle (ϕ)-the angular momentum (L_z ) plane. The stars evolve in this plot due to changes in their angular momenta. When the stars are captured by the density enhancement (T _rot ≃ 11.8 − 12.0), they radially migrate along the spiral arms. The stars approaching from behind the spiral arm (i.e., inner radius) tend to attain increased angular momenta via acceleration along the spiral arm, whereby they move to the disc’s outer radius. In contrast, the stars approaching ahead of the spiral arm (i.e., outer radius) tend to lose their angular momenta via deceleration along the spiral arm, and they move to the disc’s inner radius. Along the ϕ − L_z plane, the stars oscillate both horizontally as well as vertically. Moreover, the guiding centers of the oscillations do not remain constant at the same value of L_z . This is essentially different from the epicycle motion in which L_z is conserved.

Figure 12. Orbital evolution of stars in the spiral arm. The stars associate around the spiral arm within a distance of ± 0.5 kpc at T _rot = 4.0. Left columns: orbits on ϕ − R plane. Middle columns: orbits on ϕ − L_z plane. Right columns: orbits on E − L_z plane. The colours denote the angular momentum at the time instants when the stars are associated with the spiral arm. From Baba et al. (Reference Baba, Saitoh and Wada2013).

The panels in the right column of Figure 12 show the so-called Lindblad diagram, where the angular momentum L_z of each star is plotted against its total energy E. The stars oscillate along the curve of circular motion by undergoing change in terms of both angular momentum and energy (Sellwood & Binney Reference Sellwood and Binney2002; Grand et al. Reference Grand, Kawata and Cropper2012a; Roškar et al. Reference Roškar, Debattista, Quinn and Wadsley2012; Baba et al. Reference Baba, Saitoh and Wada2013). This is because stars around the corotation point change their angular momenta without increasing their random energy (Lynden-Bell & Kalnajs Reference Lynden-Bell and Kalnajs1972).

Grand et al. (Reference Grand, Kawata and Cropper2012a) also noticed the slight heating of negative migrators and the slight cooling for positive migrators (their Figure 12). Roškar et al. (Reference Roškar, Debattista, Quinn and Wadsley2012), Minchev et al. (Reference Minchev, Famaey, Quillen, Di Matteo, Combes, Vlajic, Erwin and Bland-Hawthorn2012), and Baba et al. (Reference Baba, Saitoh and Wada2013) also reported a similar effect of the radial migration of stars around the spiral arms upon disc heating. Thus a non-negligible fraction of the particles that migrate outward have their orbits cooled by the spiral arm. This ‘dynamical cooling’ can be important for recurrent spiral instabilities.

2.2.3 Recurrent mechanisms for dynamic stellar spiral arms

The mechanism by which spiral arms recur is unclear. Sellwood & Kahn (Reference Sellwood and Kahn1991), and Sellwood & Lin (Reference Sellwood and Lin1989) proposed a feedback cycle whereby narrow features in the angular momentum density of stars drive large-scale dynamic spiral arms. The arms in turn lead to resonant scattering of stars, which serves as a seed for the next spiral arm formation. This large-scale spiral instability, which originates from the deficiency of stars over a narrow range of angular momenta (also corresponding to a change in the surface density for stars on a circular orbit), is called the ‘groove’ instability. This feedback cycle was observed in N-body simulations of a low-mass disc with a near Keplerian rotation curve (Sellwood & Lin Reference Sellwood and Lin1989). The phase space density is depopulated near the OLR of one wave, inducing a new large-scale spiral instability with a CR near the OLR of the first wave. Sellwood (Reference Sellwood2000) also reported that the distribution of the solar neighborhood stars on in angular momentum phase space has similar fine structures (Sellwood Reference Sellwood, Morrison and Gilmore1994, Reference Sellwood2010a), suggesting that this recurrent mechanism cycle may occur in real spiral galaxies. Scattering of stars by spiral arms at the ILR, in such a way to form a new spiral arm, is also observed in more massive discs with near flat rotation curves (Sellwood Reference Sellwood2012). However, Sellwood (Reference Sellwood2012) concluded that some other mechanism may be required for recurrent spiral instabilities, because he was not able to find evidence to support the groove-type cycle such as observed in less massive discs with a near Keplerian rotation (Sellwood & Lin Reference Sellwood and Lin1989).

Baba et al. (Reference Baba, Saitoh and Wada2013) showed that oscillating stars successively undergo aggregation and disaggregation in energy-Lz space, thereby leading to the formation of structures referred to as ‘swarms of stars’ along the ϕ − L_z and R − ϕ planes (the right column of Figure 12). The non-steady nature of the spiral arms originates in the dynamical interaction between these swarming stars with a nonlinear epicycle motion, and the high-density regions, i.e., the spiral arms moving with galactic rotation. This is entirely different from what is expected in stationary density waves, where these changes are limited to the CR and Lindblad resonances (Lynden-Bell & Kalnajs Reference Lynden-Bell and Kalnajs1972). Thus, the gravitational interaction between the stars in the spiral arm and the spiral density enhancement changes the angular momentum and random energy of the stars, and this process in turn changes the structure of the spirals. During this process, the random energy of individual stars in the system does not increase monotonically. In other words, local interactions between the non-steady arms and stars increase or decrease the total energy of individual stars locally; however, the energy remains around its value for circular motion with the occurrence of a small dispersion. This is because the interaction causes the migration of the guiding centers of the stars without increasing their eccentricity or random energy. This ‘dynamical cooling’ mechanism (Grand et al. Reference Grand, Kawata and Cropper2012a; Roškar et al. Reference Roškar, Debattista, Quinn and Wadsley2012; Minchev et al. Reference Minchev, Famaey, Quillen, Di Matteo, Combes, Vlajic, Erwin and Bland-Hawthorn2012; Baba et al. Reference Baba, Saitoh and Wada2013) is essential to preventing heating of the stellar disc and erasure of the spiral arms, and the mechanism produces ‘swarms’ of stars moving between non-steady spirals. The non-linear epicycle motion of the stars and their non-linear coupling with the density perturbation is the fundamental physics of the recurrently formed, non-steady spiral arms in a stellar disc.

2.3.1 Spirals corotating with bars

The first interpretation, which the bar and spiral arms have a common pattern speed, is intuitive from observations that most spiral arms connect to the ends of the bar. Sanders & Huntley (Reference Sanders and Huntley1976) studied the response of gas to a steady bar perturbation using hydrodynamical simulations. They found that the gas eventually settled into a steady state with a prominent trailing spiral structure. Gaseous spiral arms driven by a bar have since been seen in many further simulations (e.g. Schwarz Reference Schwarz1981; Combes & Gerin Reference Combes and Gerin1985; Athanassoula Reference Athanassoula1992; Wada Reference Wada1994; Englmaier & Gerhard Reference Englmaier and Gerhard1999; Bissantz, Englmaier, & Gerhard Reference Bissantz, Englmaier and Gerhard2003; Rodriguez-Fernandez & Combes Reference Rodriguez-Fernandez and Combes2008). The gas arms are a direct response of the bar forcing. Figure 13 show the stellar closed orbits (left) and gaseous closed orbits (right) in a weak bar potential. Stellar orbits are always parallel or perpendicular to the bar, whilst the gas orbits change their orientation with radius, due to the effects of dissipation. The elliptical gaseous orbits are inclined to the bar potential in a trailing sense outside corotation. Thus, dissipation associated with the gas viscosity plays a critical role in driving gaseous spirals. Note that simulations in which a gas disc embedded in a ‘steady’ bar potential is replaced by a collisionless disc of test star particles also gives rise to a prominent trailing spiral structure but the stars never settle into a steady spiral structure. But stellar spiral arms can be excited by a ‘growing’ bar (Hohl Reference Hohl1971).

Figure 13. Stellar closed orbits (left) and gaseous closed orbits (right) in a weak barred potential. The radii of the inner ILR, outer ILR, CR, and OLR are at 0.8, 2.4, 4.6, and 6.0, respectively. The gaseous closed orbits are calculated based on the damped orbit model by Wada (Reference Wada1994) who added the damping term (emulating the collisional nature gas) to equations of stellar orbits in a weak bar from Section 3.3 of Binney & Tremaine (Reference Binney, Tremaine, Binney and Tremaine2008). Note that Wada (Reference Wada1994) only showed a solution for radial direction. See the appendix of Sakamoto et al. (Reference Sakamoto, Okumura, Ishizuki and Scoville1999) for a full set of the solutions. A similar introduction of a damping term was also made by Sanders & Huntley (Reference Sanders and Huntley1976) and Lindblad & Lindblad (Reference Lindblad and Lindblad1994). The stellar response to forcing by a steady bar cannot form spiral arms. In contrast, the phase delay of epicycle motion in terms of the bar perturbation naturally takes place as does in a damped oscillator affected by a periodic external force. This phase delay determines direction of spirals (i.e. trailing or leading) around the Lindblad resonance (Wada Reference Wada1994).

Manifold theory or manifold flux-tube theory is proposed as a way of determining the orbits of stars in spiral arms driven by a bar (Romero-Gómez et al. Reference Romero-Gómez, Masdemont, Athanassoula and García-Gómez2006, Reference Romero-Gómez, Athanassoula, Masdemont and García-Gómez2007; Athanassoula, Romero-Gómez, & Masdemont Reference Athanassoula, Romero-Gómez and Masdemont2009b; Athanassoula et al. Reference Athanassoula, Romero-Gómez, Bosma and Masdemont2009a, Reference Athanassoula, Romero-Gómez, Bosma and Masdemont2010; Athanassoula Reference Athanassoula2012; Voglis, Stavropoulos, & Kalapotharakos Reference Voglis, Stavropoulos and Kalapotharakos2006a; Voglis, Tsoutsis, & Efthymiopoulos Reference Voglis, Tsoutsis and Efthymiopoulos2006b; Tsoutsis, Efthymiopoulos, & Voglis Reference Tsoutsis, Efthymiopoulos and Voglis1980; Tsoutsis et al. Reference Tsoutsis, Kalapotharakos, Efthymiopoulos and Contopoulos2009). According to this theory, the backbone of barred spirals are bunches of untapped stars (so-called Lyapunov orbits) escaped from the unstable Lagrangian pointsFootnote ⁶ , which are located on the direction of the bar major axis, outside the bar but near its ends. This means that, contrary to the quasi-stationary density waves, the stars do not cross the barred spiral arms but they move along them or they are spatially well confined by the manifoldsFootnote ⁷ (Figures 1 and 4 of Athanassoula Reference Athanassoula2012). Athanassoula et al. (Reference Athanassoula, Romero-Gómez, Bosma and Masdemont2010) and Athanassoula (Reference Athanassoula2012) compared the properties of spiral arms predicted by manifold theory with N-body simulations. They found good agreement between the manifold theory and the simulations (see Figure 12 of Athanassoula et al. Reference Athanassoula, Romero-Gómez, Bosma and Masdemont2010), and in the simulations stars moved along the spiral arms as predicted (see Figure 4 of Athanassoula Reference Athanassoula2012). The manifold theory predicts that the relative strength of the non-axisymmetric forcing in the region around and beyond CR influences the winding of spiral arms, in the sense that in strongly barred galaxies the spirals will be more open than in less strongly barred ones (Athanassoula et al. Reference Athanassoula, Romero-Gómez, Bosma and Masdemont2010). This trend was corroborated in observed barred spiral galaxies (Martínez-García Reference Martínez-García2012).

One observational indication of the bar-driven spiral scenario is that grand-design spirals are more frequent in barred galaxies than in unbarred galaxies (Elmegreen & Elmegreen Reference Elmegreen and Elmegreen1982). However there are still many multi-armed (e.g. NGC 1232, NGC 3344, NGC 3953, NGC 6946, IC 342, Figure 14), and flocculent (e.g. NGC 1313, NGC 5068) spirals that exhibit barsFootnote ⁸ . Early type barred galaxies tend to have stronger bars and grand-design or multiple spiral arms, while late type barred galaxies have weaker bars and flocculent spiral arms (Figure 13 of Elmegreen et al. Reference Elmegreen, Elmegreen and Yau2011). Several studies have examined correlations between bar strengths and spiral arm strengths, with some finding clear evidence of bar driven spirals (Block et al. Reference Block, Buta, Knapen, Elmegreen, Elmegreen and Puerari2004; Salo et al. Reference Salo, Laurikainen, Buta and Knapen2010), and others finding little or no evidence (Seigar & James Reference Seigar and James1998; Durbala et al. Reference Durbala, Buta, Sulentic and Verdes-Montenegro2009; Kendall et al. Reference Kendall, Kennicutt and Clarke2011). Thus observations suggest that the bar-driven spiral scenario is not necessarily valid for all barred spiral galaxies.

Figure 14. B-band images of NGC 3953 (left), NGC 3124 (middle) and NGC 3450 (right). From The de Vaucouleurs Atlas of Galaxies (Buta et al. Reference Buta, Corwin and Odewahn2007).

2.3.2 Decoupling between spirals and bars

The second possibility is that bars and spiral arms can be independent patterns. In this case, spirals in barred galaxies are associated with a spiral density wave, but probably with a pattern speed different from that of the bar (Sellwood & Sparke Reference Sellwood and Sparke1988; Rautiainen & Salo Reference Rautiainen and Salo1999). Indeed, Sellwood & Sparke (Reference Sellwood and Sparke1988) have demonstrated N-body simulations of a stellar disc, and shown that multiple pattern speeds are quite common in disc galaxies, with the spiral structure typically having a much lower pattern speed than the bar. In other words, bars and spiral arms can be independent features. This implies a more or less random distribution of the phase difference between the bar and the start of the spiral arms, which seems to be in conflict with observations. However, as pointed out by Sellwood & Sparke (Reference Sellwood and Sparke1988), contour plots of the non-axisymmetric density in their simulations show that the spiral arms appear to the eye to be joined to the ends of the bar for most of the beat frequency. This suggests that the observed correlation between bars and spirals might simply be an illusion.

2.3.3 Non-linear coupling between spirals and bars

A third interpretation for the origin of spiral arms in barred galaxies is a non-linear coupling between bars and spiral density waves, where the bar and spiral arm have different pattern speeds (Tagger et al. Reference Tagger, Sygnet, Athanassoula and Pellat1987; Sygnet et al. Reference Sygnet, Tagger, Athanassoula and Pellat1988; Masset & Tagger Reference Masset and Tagger1997; Minchev et al. Reference Minchev, Famaey, Quillen, Di Matteo, Combes, Vlajic, Erwin and Bland-Hawthorn2012). This mechanism assumes some small overlap between the corotation (CR) of the bar and the inner Lindblad resonance (ILR) of the spiral density wave. Using the tight-winding and epicyclic approximations for density waves, Tagger et al. (Reference Tagger, Sygnet, Athanassoula and Pellat1987) and Sygnet et al. (Reference Sygnet, Tagger, Athanassoula and Pellat1988) showed that this overlap enables the transfer of energy and angular momentum between the bar, spiral density wave and beat (m = 0 and m = 4) waves. The bar is stabilized at a finite amplitude by transferring energy and angular momentum to the spiral density wave, and the non-linear coupling drives beat waves. This theoretical argument on the non-linear coupling was also studied by N-body simulations of stellar discs (Masset & Tagger Reference Masset and Tagger1997; Rautiainen & Salo Reference Rautiainen and Salo1999), where the predicted strong beat waves were observed. This scenario is similarly supported by more recent N-body/SPH simulations of galactic discs (Minchev et al. Reference Minchev, Famaey, Quillen, Di Matteo, Combes, Vlajic, Erwin and Bland-Hawthorn2012). Notably though, the derived beating waves exhibit chaotic behaviour rather than a stationary spiral pattern.

Sometimes the spiral arms do not start from the ends of the bar but exhibit a clear phase difference (e.g., NGC 1365). Similarly, some observed barred galaxies, such as NGC 3124 (Buta, Corwin, & Odewahn Reference Buta, Corwin and Odewahn2007; Efremov Reference Efremov2011) and NGC 3450 (Buta et al. Reference Buta, Corwin and Odewahn2007), show the curved, leading ends of the stellar bar (Figure 14). Similar morphology can be seen in N-body simulations of barred galaxies due to the oscillations between trailing and leading ends of the bar (e.g. Fux Reference Fux1997; Rautiainen & Salo Reference Rautiainen and Salo2000; Martinez-Valpuesta & Gerhard Reference Martinez-Valpuesta and Gerhard2011). Martinez-Valpuesta & Gerhard (Reference Martinez-Valpuesta and Gerhard2011) suggested that the oscillations could be related to the oscillations seen in the bar growth in N-body simulations (e.g. Dubinski, Berentzen, & Shlosman Reference Dubinski, Berentzen and Shlosman2009) through angular momentum transfer to disc stars (e.g. Sellwood Reference Sellwood1981) and to non-linear coupling modes between the bar and spiral density wave as mentioned above.

2.3.4 Non-stationary spiral arms in barred galaxies

Grand et al. (Reference Grand, Kawata and Cropper2012b) performed N-body/hydrodynamic simulations of a Milky Way-sized barred galaxy and analysed the spiral pattern speed. They found that the spiral arms are transient features and their pattern speeds decrease with radius in a similar manner to the angular velocity, but the pattern speed is slightly higher than the angular velocity of the disc. These results suggest that spiral arms in barred galaxies could be neither rigid-body rotating patterns predicted by the quasi-stationary density wave theory nor independent features, but transient features boosted by the bar. The non-stationarity of spiral arms in barred galaxies is also reported by other N-body/hydrodynamics simulations (Fux Reference Fux1997; Baba et al. Reference Baba, Asaki, Makino, Miyoshi, Saitoh and Wada2009). Baba et al. (Reference Baba, Asaki, Makino, Miyoshi, Saitoh and Wada2009) argued that non-stationary, winding spiral arms in a simulated barred spiral galaxy originate via swing amplification (Section 2.1.3). In contrast, Roca-Fàbrega et al. (Reference Roca-Fàbrega, Valenzuela, Figueras, Romero-Gomez, Velazquez, Antoja and Pichardo2013) reported that simulated spiral arms in strongly barred galaxies have a pattern speed almost constant in radius. More interestingly, they reported that the spiral pattern speed is close to disc rotation only when the bar is weak, as obtained by Grand et al. (Reference Grand, Kawata and Cropper2012b), but becomes almost constant when the bar has fully formed. These results suggest that the relation between bars and spiral arms can change during the evolutionary stages of bars, although there is no observational evidence to support, or contradict this picture.

2.4.1 Historical overview

Tidal encounters are frequent across all astronomy, with interacting galaxies providing some of the clearest examples. Early attempts to categorise interacting, and other more unusual galaxies, showed many examples of galaxies with tidal tails, bridges and clear spiral structure (Vorontsov-Velyaminov Reference Vorontsov-Velyaminov1959; Arp Reference Arp1966) prompting the morphology of galaxies to be associated with tidal effects (e.g. van den Bergh Reference van den Bergh1959; Lindblad Reference Lindblad1960; Hodge & Merchant Reference Hodge and Merchant1966; Toomre Reference Toomre1969). The idea that tidal interactions may be responsible for spiral arms was in fact first demonstrated 20 years earlier, by Holmberg (Reference Holmberg1941). In a now famous experiment, Holmberg (Reference Holmberg1941) modelled the interaction of two galaxies by representing the galaxies by a series of lightbulbs. The lightbulbs have initial velocities associated with them due to the initial velocities of each galaxy assumed for the interaction, and their rotation curves. A photocell is used to measure the total amount of light at any particular point in the galaxies. Since light obeys a inverse square law the same as gravity, the total light received by the photocell is equivalent to the total gravitational force at that point in the galaxy. This force, or rather acceleration, is then used to calculate how far to move the given lightbulb. This step is then repeated for all the lightbulbs used, and the whole process repeated for many steps. The results of this experiment showed clearly the development of tidal spiral arms.

From the 1960s, actual numerical calculations of interacting galaxies were able to be performed (Pfleiderer & Siedentopf Reference Pfleiderer and Siedentopf1961; Pfleiderer Reference Pfleiderer1963; Tashpulatov Reference Tashpulatov1970; Toomre & Toomre Reference Toomre and Toomre1972; Eneev, Kozlov, & Sunyaev Reference Eneev, Kozlov and Sunyaev1973), although they were still limited to test particle simulations using a restricted 3 body approach, which neglects stellar self gravity. These simulations focused mainly on the origin of tails and bridges in galaxies, rather than spiral arms themselves. Nevertheless, Toomre & Toomre (Reference Toomre and Toomre1972) still represents one of the most comprehensive studies of galaxy interactions, spanning over all possible alignments of the two galaxies in space, unequivocally showing that bridges and tails were indeed the result of tidal interactions. These simulations also reproduced a number of known systems remarkably well, including M51 and the Mice.

2.4.2 Tidally induced arms: stationary, kinematic or material arms?

Since the 1970s, full N body simulations, were able to model interactions with much higher resolution, and demonstrate that tidal interactions could account not only for tails and bridges at large galactic radii, but also spiral arms penetrating to the centre of a galaxy (Hernquist Reference Hernquist and Wielen1990; Sundelius et al. Reference Sundelius, Thomasson, Valtonen and Byrd1987; Donner & Thomasson Reference Donner and Thomasson1994; Salo & Laurikainen Reference Salo and Laurikainen2000b; Dobbs et al. Reference Dobbs, Theis, Pringle and Bate2010) (see Figure 15, left panel). Whilst these studies clearly demonstrate m = 2 spiral arms, a more critical question is whether the spiral arms are representative of the quasi-stationary spiral arm picture, are kinematic density waves, or material arms. For material arms there is no difference between the spiral arms and the underlying differential rotation of the stellar disc – the pattern speed of the arms is that of the disc, i.e. Ω_p(R) = Ω(R). Whilst material arms may describe the outer arms, or tidal tails of galaxies (e.g. Toomre Reference Toomre1969; Meidt et al. Reference Meidt, Schinnerer and Garcia-Burillo2013) they are not found to characterise the arms over the main part of the stellar disc. Sundelius et al. (Reference Sundelius, Thomasson, Valtonen and Byrd1987) demonstrated using numerical simulations that tidally induced spiral arms are density waves rather than material arms, whilst observations have shown that the velocity fields of interacting galaxies do not correspond to material arms (e.g. Rots & Shane Reference Rots and Shane1975).

Figure 15. Simulation of M51 (left panel) showing the present day appearance of the galaxy, the orbit (dashed line) and the position of the perturber (white dot). The pattern speeds of the two spiral arms are shown on the right hand panel, with error bars (dotted lines). The angular velocity of the stars is also shown (red dashed line) and Ω ± κ/2 (blue dashed lines). From Dobbs et al. (Reference Dobbs, Theis, Pringle and Bate2010).

Secondly the arms may be kinematic density waves. Kinematic density waves are not actually waves, in the sense that they don’t propagate through the disc, and have zero group velocity. But gas and stars do flow through the arms, although unlike quasi-stationary density waves, the spiral pattern is not fixed. For purely kinematic density waves, self gravity of the stars can be ignored (kinematic density waves can be induced even when Q = ∞), the stars behaving simply as test particles. The influence of a perturber can be treated by the impulse approximation, and induces elliptical stellar orbits. Such orbits are not generally closed, but we can choose an angular speed such that the orbit is closed. For an m = 2 perturbation, we can choose a rotating frame such that the orbit is closed after half the epicylic frequency (or after each time an arm is encountered) i.e.

to a first order approximation (see Binney & Tremaine Reference Binney, Tremaine, Binney and Tremaine2008). If the orbits are aligned along the same major axis, then the perturbation produces a bar. If however, the orbits are offset as a function of radius, then they naturally produce a spiral pattern (Kalnajs Reference Kalnajs1973). Increasing the offset makes the spirals more tightly wound. In the case of a moving (prograde) perturber, the orbits are not aligned, and a trailing spiral pattern develops with a pattern speed given by equation 24 (trailing since Comment 39: $\Omega (R)-\thinspace \thinspace \kappa (R)/2<\Omega$ ). The aphelia of the ellipses corresponding to the densest parts (arms) in the disc. Calculations of a perturber passing a galaxy in the non self gravitating case by Oh et al. (Reference Oh, Kim, Lee and Kim2008) demonstrate that the induced arms do indeed exhibit this pattern speed. Thus the pattern speed decreases with radius, but less so than the material arms case. The locations of the spiral arms can also be determined analytically, by applying the impulse approximation to the stellar orbits, and computing the Jacobian matrix from the derivatives the resultant orbit equations (Struck-Marcell Reference Struck-Marcell1990; Donner, Engstrom, & Sundelius Reference Donner, Engstrom and Sundelius1991; Gerber & Lamb Reference Gerber and Lamb1994; Appleton & Struck-Marcell Reference Appleton and Struck-Marcell1996). The surface density of the response to a tidal perturbation is then

(Gerber & Lamb Reference Gerber and Lamb1994) where R ₀ is the original (unperturbed) radius of the (circular) orbit and J is the Jacobian. The points where J = 0 are caustics, and correspond to the locations of the induced arms.

Alternatively the spiral arms may be quasi-stationary density waves. In the self gravitating case, the effect of self gravity is to make the spiral pattern more rigid, increasing Ω_p. Swing amplification may also act to enhance the density of the arms still further. However it is not established whether self gravity is sufficient to make the spiral pattern fully self gravitating, and develop into a quasi-stationary density wave. Both Oh et al. (Reference Oh, Kim, Lee and Kim2008), and Dobbs et al. (Reference Dobbs, Theis, Pringle and Bate2010) find that although the pattern speed is higher in their models than given by Equation 27, and decreases less with radius, the arms are not completely rigid and still wind up with time (see Figure 15, right panel). Sundelius et al. (Reference Sundelius, Thomasson, Valtonen and Byrd1987) also find the development of spiral density waves in the absence of swing amplification, whilst Oh et al. (Reference Oh, Kim, Lee and Kim2008), and Dobbs et al. (Reference Dobbs, Theis, Pringle and Bate2010) find swing amplification only has a minor effect. Salo et al. Reference Salo and Laurikainen2000b also find that the pattern speed is radially decreasing and again slightly higher than given by Equation 27, and again generally suppose that swing amplification has only a minor role in generating the arms. They do however find a more constant pattern in the centre kpc or so of their simulation of M51, and suppose that here Lin-Shu-Kalnajs (LSK) waves operate, the lack of an ILR in their simulation meaning waves can penetrate to the centre without being absorbed (in a number of their models, a bar forms in the centre, similar to the actual M51).

Overall the nature of spiral arms in tidally interacting galaxies likely represent the behaviour of the underlying disc. Galaxies with dynamic spiral arms likely do not exhibit fixed spiral patterns when tidally interacting as they are not gravitationally dominated. Galaxies with more massive discs, and likely bars, may well exhibit fixed patterns, at least in the central gravitationally dominated regions.

2.4.3 Prograde and retrograde encounters, and the orbit of the perturber

In addition to the nature of the spiral arms induced, we can also consider how the orbit of the perturber affects the tidal perturbation. The simulations of Toomre & Toomre (Reference Toomre and Toomre1972), and later Howard et al. (Reference Howard, Keel, Byrd and Burkey1993) demonstrated that retrograde encounters have a relatively small effect on a galactic disc, whereas prograde encounters are very effective at producing spiral arms, naturally of a trailing nature. Other analysis showed that tidal interactions could produce a leading spiral pattern, with one predominant leading arm (Kalnajs Reference Kalnajs1971; Athanassoula Reference Athanassoula1978; Thomasson et al. Reference Thomasson, Donner, Sundelius, Byrd, Huang and Valtonen1989). For the prograde case, as discussed earlier with respect to kinematic density waves, the angular speed of the perturber at closest approach during its orbit will likely be nearest to the Inner Lindblad resonance (Ω(R) − κ(R)/2), and hence particularly effective at inducing an m = 2 perturbation. For the retrograde case, the angular speed exhibits the opposite sign, has little correspondence to any resonance (Toomre Reference Toomre1969), but will likely be closest to an m = 1 perturbation (Ω_p = Ω(R) − κ(R)) corresponding to one leading (generally κ(R) > Ω(R)) arm (Byrd et al. Reference Byrd, Thomasson, Donner, Sundelius, Huang and Valtonen1989). Leading spiral arms are found to be rare in simulations, requiring a strong perturbation, and a large halo mass (Thomasson et al. Reference Thomasson, Donner, Sundelius, Byrd, Huang and Valtonen1989). The latter is important to prevent swing amplification, which would convert any leading perturbation into a trailing one. Observationally, NGC 4622, a ring galaxy, is the only galaxy found to exhibit leading spiral arms (Buta, Crocker, & Byrd Reference Buta, Crocker and Byrd1992; Buta, Byrd, & Freeman Reference Buta, Byrd and Freeman2003).

Various simulations have also investigated the impact on the galaxy from perturbers of different masses (Byrd & Howard Reference Byrd and Howard1992; Oh et al. Reference Oh, Kim, Lee and Kim2008; Struck, Dobbs, & Hwang Reference Struck, Dobbs and Hwang2011). The simulations find that a perturber typically needs to be at least 0.01 times of the mass of the main galaxy to have an effect, ideally closer to 0.1 times the mass to produce a clear grand design pattern, although naturally there is a degeneracy with the pericenter of the orbit (Oh et al. Reference Oh, Kim, Lee and Kim2008). Toomre & Toomre (Reference Toomre and Toomre1972) showed that a perturbing galaxy has greatest impact when orbiting in the plane of the main galaxy, but changing the angle of inclination of the perturbing galaxy has little qualitative effect.

Another factor determining the dynamics of tidally induced spirals is the number of orbits of the perturber, and thus whether it is bound. For M51, the best matched orbit of M51 and NGC5195 currently involves two orbits of the NGC 5195 around M51, after which the two galaxies merge (Salo & Laurikainen Reference Salo and Laurikainen2000a; Theis & Spinneker Reference Theis and Spinneker2003). Consequently the dynamics are quite chaotic, whilst the orbit induces different sets of spiral arms resulting in noticeable kinks along the spiral arms as observed today (Salo & Laurikainen Reference Salo and Laurikainen2000b; Dobbs et al. Reference Dobbs, Theis, Pringle and Bate2010). The spiral arms consequently show clear departures from logarithmic spirals. Oh et al. (Reference Oh, Kim, Lee and Kim2008), and Struck et al. (Reference Struck, Dobbs and Hwang2011) present simulations where a perturber is on an unbound orbit, and passes the galaxy only once. In this case, the dynamics are less chaotic, and the arms smoother.

2.4.4 Longevity of tidally-induced spirals

If m = 2 spirals are difficult to produce except by tidal interactions or bars, as we conclude from Section 2.1.5, their lifetime is an important characteristic. Grand design m = 2 spirals are common, hence tidal interactions must induce relatively long-lived spiral arms if they are the main source of such galaxies.

Assuming their pattern speed is not fixed, tidal arms are expected to have a pattern that winds up slower than local transient arms discussed in the previous section, but to be shorter lived than quasi-stationary spiral arms. For tidally induced spiral arms, there are two main questions regarding how long they last. The first is how much they wind up over time, the second is how long the arms take to decay or damp. To think about the first issue, we can consider the pitch angle of the arms, defined as the angle between the tangent of the spiral arm and a circle, i.e.

where the derivation is evaluated along the spiral arm. If we consider the change in ϕ at a later time as ϕ(R, t) = ϕ₀ + Ω_p(R)t then the pitch angle can be written as:

(Binney & Tremaine Reference Binney, Tremaine, Binney and Tremaine2008). For material arms, Ω_p = Ω and for a flat rotation curve of v_c = 200 km s^{− 1}, the pitch angle will be ~ 1° after about 1 Gyr. This is considerably lower than observed pitch angles. For material arms, the pattern winds up on a timescale of order t ~ |dR/(RdΩ)| = 1/|dΩ/dln R|, in the above example ≲ 100 Myr.

For kinematic tidal arms, in the absence of self gravity, Ω_p(R) = Ω(R) − κ(R)/2 (Section 2.4.1), and so |dΩ_p/dln R| ≪ |dΩ/dln R|. Hence the spiral pattern is expected to last somewhat longer. In the presence of self gravity, Ω_p versus R can become even shallower. For example, if we take Figure 15 (right panel), in 1 Gyr, the pattern winds up at a rate ~ 4 times slower than the above example for material arms, and is expected to have a pitch angle of 5 or 6° after 1 Gyr. These values are, as would be predicted, at the lower end of observed values (Seigar & James Reference Seigar and James1998; Seigar et al. Reference Seigar, Bullock, Barth and Ho2006).

Simulations of tidally induced spiral arms confirm this behaviour, with the pattern winding up and simultaneously decaying on a timescale of ~ 1 Gyr (Oh et al. Reference Oh, Kim, Lee and Kim2008; Struck et al. Reference Struck, Dobbs and Hwang2011). Similar to the case of dynamic spiral arms (Section 2.2), Struck et al. (Reference Struck, Dobbs and Hwang2011) also found that the arms persisted for longer with higher resolution simulations. Struck et al. (Reference Struck, Dobbs and Hwang2011) also supposed that galaxy encounters in groups and clusters are likely frequent, and with spiral arms persisting for ~ 1 Gyr, tidally induced spiral galaxies common.

3.5.1 Resonances

Resonances are one means to generate gaseous substructure along spiral arms, specifically for the case of quasi-stationary density waves. Resonances occur when the epicyclic frequency, κ of the stellar orbits are some integer multiple of the angular frequency in the rotating frame of the spiral potential, or vice versa, thus

where Ω_p is the pattern speed of the spiral, and m is the number of spiral arms in the stellar disc. In this case stars perform n radial oscillations every encounter with the m-armed spiral pattern. Determining the presence and location of resonances in the disc implicitly assumes that the pattern speed, Ω_p, does not vary with radius or time. In the vicinity of resonances, the behaviour of stellar and or gas orbits are abruptly altered and become nonlinear (e.g. Contopoulos & Grosbol Reference Contopoulos and Grosbol1986, Reference Contopoulos and Grosbol1988). The primary resonances at the ILR and OLR (n = 1) are, as discussed in Section 2.1.2, associated with the boundary of where the spiral density waves exist. Thus we are predominantly interested in resonances within these radii. Shu et al. (Reference Shu, Milione and Roberts1973) supposed that the gas would be expected to experience perturbations due to resonances, and showed analytically that gas undergoes a secondary compression to a spiral potential at the ultraharmonic resonance (n = 2) Footnote ¹⁰ . Woodward (Reference Woodward1975) demonstrated the nonlinear response of gas at the location of the ultraharmonic resonance with 1D calculations, and there have since been many further 2D and 3D (Smoothed Particle Hydrodynamics (SPH) and grid code) calculations (Bertin Reference Bertin1993; Patsis et al. Reference Patsis, Hiotelis, Contopoulos and Grosbol1994; Patsis, Grosbol, & Hiotelis Reference Patsis, Grosbol and Hiotelis1997; Chakrabarti et al. Reference Chakrabarti, Laughlin and Shu2003). In particular Patsis et al. (Reference Patsis, Hiotelis, Contopoulos and Grosbol1994) showed the bifurcation of the spiral arms at the 4:1 resonance (as also predicted by Artymowicz & Lubow Reference Artymowicz and Lubow1992), provided there is a spiral forcing of F ≳ 5 %.

Chakrabarti et al. (Reference Chakrabarti, Laughlin and Shu2003) showed the development of more varied features, including branches (bifurcations) and shorter leading and trailing features (spurs / feathers), again occurring primarily near the 4:1 resonance, with the morphology of the feature dependent primarily on the level of forcing of the spiral potential. Chakrabarti et al. (Reference Chakrabarti, Laughlin and Shu2003) suppose that flocculence in spiral galaxies could be due largely to such resonant features, an idea recently followed up by Lee & Shu (Reference Lee and Shu2012), where they investigate the possibility that higher order resonances lead to the formation of multiple spurs along the arms. There is a notable difference between the work of Lee & Shu (Reference Lee and Shu2012), and GMC formation by gravitational instabilities in the gas or cloud-cloud collisions (which are subsequently sheared into spurs). For the former, the location of the spurs does not change over time, the GMCs always forming and dispersing in the same place in the spiral arms (seemingly less likely in a dynamic environment). For other GMC formation mechanisms there is no expectation that clouds form in the same place.

3.5.2 Stability and structure of the shock

Even in the non-magnetic, non self gravitating regime, several authors have questioned the stability of spiral shocks. From analytical work, Mishurov & Suchkov (Reference Mishurov and Suchkov1975) first proposed that the flow through a spiral shock could be unstable. In contrast Nelson & Matsuda (Reference Nelson and Matsuda1977) solved the fluid equations numerically in 1D, and predicted that the flow should be stable (see also Dwarkadas & Balbus Reference Dwarkadas and Balbus1996) although their solutions indicate some asymmetric features. Wada & Koda (Reference Wada and Koda2004) pointed out that the latter studies adopted a tightly wound pattern, and a flat rotation curve. They performed 2D numerical simulations with different pitch angles and rotation curves, and found the spiral arms to be Kelvin-Helmholz unstable when a more open spiral pattern was used. The instability is most readily seen as spurs along the spiral arms. Kim & Ostriker (Reference Kim and Ostriker2006) found that in 3D numerical models, Kelvin-Helmholtz instabilities were suppressed, although Kim, Kim, & Kim (Reference Kim, Kim and Kim2014) suggest an alternative ‘wiggle instability’ mechanism.

Dobbs & Bonnell (Reference Dobbs and Bonnell2006) (see also Dobbs Reference Dobbs2008) supposed a different mechanism for producing structure, and spurs in particular, along the shock in the purely hydrodynamical, non self-gravitating case. They supposed that any substructure in the gas gets amplified as it passes through a shock. Thus, like the cloud collisions in the Toomre (Reference Toomre1977) model, clouds, or structure in the gas, get forced together by orbit crowding in the spiral shock. A similar idea was shown in Roberts & Stewart (Reference Roberts and Stewart1987). Although they do not perform hydrodynamic calculations, clouds in their models undergo dissipative collisions. Like Toomre (Reference Toomre1977), clouds can be forced together and move apart after the shock, but unlike Toomre the presence of dissipation means some clouds are effectively compressed together and retain structure after the shock. Dobbs & Bonnell (Reference Dobbs and Bonnell2006) showed that this process was only valid in the presence of cold gas, when the ISM is subject to thermal instabilities (Dobbs et al. Reference Dobbs, Glover, Clark and Klessen2008) unless there is very large spiral forcing. For a warm medium, the spiral shock is relatively weaker and the pressure smoothes out any structure in the gas. The same process could have also plausibly operated (rather than Kelvin Helmholtz instabilities) in the calculations by Wada & Koda (Reference Wada and Koda2004) and Kim & Ostriker (Reference Kim and Ostriker2006).

In the presence of self gravity, Lubow, Cowie, & Balbus (Reference Lubow, Cowie and Balbus1986) showed using 2D calculations that the gas experiences a reduced shock from the stellar potential. In an extreme case, where all the gas is situated in self gravitating clouds, the behaviour of the clouds would resemble the zero pressure case, similar to billiard balls entering the potential. Wada (Reference Wada2008) performed full 3D hydrodynamical simulations with self gravity and a multi-phase medium, finding that the intermittency of dense gas entering the spiral potential leads to a non steady state, where the gas spiral arms are neither continuous, nor exhibit a constant offset from the arms (see Figure 18, and e.g. also Dobbs & Pringle Reference Dobbs and Pringle2013). Rather the arms switch back and forth with time. Consequently Wada (Reference Wada2008) does not call the response of the gas a shock in this context. The behaviour of the gas is quite different from the original Roberts (Reference Roberts1969) picture largely because the gas is far removed from a homogenous flow. Also, simulations with a multiphase medium typically do not exhibit a shock or peak in density before the spiral potential. Typically the gas density peaks after, or coincident with the minimum of the potential, because the cold gas (within a multi-phase medium) shocks later.

Figure 18. The response of gas to an m = 2 fixed spiral potential is shown, from Wada (Reference Wada2008). The minima of the spiral potential are indicated by the white lines. The simulation include a multiphase medium, and stellar feedback, so the response of the gas is highly complex. No clear continuous shock is found, and the density peak of the gas does not have a continuous offset from the minimum, although typically the density peak is after (on the trailing side of) the potential minimum.

Figures 16 and 18 illustrate a number of the points made in Section 3.5, for a section of spiral arm in M51 and a numerical simulation respectively. The various processes in the ISM, including instabilities, turbulence and feedback lead to a ‘shock’, or dust lanes that are very much more structured and broad than the simple analytic case. Nevertheless the response of the gas is still much sharper than the underlying potential or old stellar population. With gravity and cooling, trailing spurs are very easy to make from arm GMCs. As discussed in the next section, we would expected these features regardless of whether the arms are tidally induced and slowly winding up or truly stationary, the only difference for the dynamic arms being the absence of trailing spurs.