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Massimo Capaccioli » 9.Origin and stability of spiral arms


The winding dilemma

Let us now explore the structures characterizing spiral galaxies: the spiral arms. A simple consideration shows that a disk galaxy (and its spiral arms) cannot be solid. [What is the observational evidence against this model?]

But spiral arms cannot even be “material” systems, that is, structures with a given body of matter. The Swedish astronomer Bertil Lindblad (1895-1965) was the first to describe, in 1925, the so called “winding problem”, the main difficulty in this model.

As a disk galaxy has a nearly constant linear velocity V, the angular velocity decreases with radial distance: \omega \propto 1/r.  Therefore, a radial arm would become curved as the galaxy rotates. After a few galactic rotations, we would observe a winding of the arms, as the stars and gas closer to the center rotate faster than those at the edge of the disk. A galaxy rotate in less than 0.5\ Gyr, while the age of the Universe is about 14\ Gyr: almost all spirals would be wounded up. Therefore, differential rotation would disrupt material spiral arms.

This is indeed, for instance, what happens to the spiral pattern in our morning cappuccino. We will see later that a traffic jam model (where cars play the role of the stars) is much more appropriate than a milky cappuccino.

What’s the difference between the spiral pattern in a cappuccino and in a disk galaxy?

What's the difference between the spiral pattern in a cappuccino and in a disk galaxy?


The winding dilemma

The winding dilemma. The figure shows how a disk galaxy with a constant rotational velocity V (dV/dr = 0) would form a spiral pattern (while the inner ring completes a rotation), ending with a wound up structure.

The winding dilemma. The figure shows how a disk galaxy with a constant rotational velocity V (dV/dr = 0) would form a spiral pattern (while the inner ring completes a rotation), ending with a wound up structure.


The pitch angle

We can quantify the amount of winding in spiral arms by defining the pitch angle. The pitch angle \psi is the angle between the tangent at the spiral arm at distance r from the center and the tangent at the circle of radius r passing at that position. If \phi is the azimuthal angle, the following relation holds:

\displaystyle{\tan\psi = \frac{1}{r}\frac{dr}{d \phi}}.

Tight arms are defined by small pitch angles. Most spirals have pitch angle approximately constant along the disk. In this case, we have:

\displaystyle{ r (\phi) = r_0 \exp[(\phi - \phi_0) \tan\psi]},

where: r_0 = r(\phi_0).

These spirals are the so-called logarithmic spirals. We can now quantify the effect of differential rotation on the winding of spiral arms.

Definition of the pitch angle.

Definition of the pitch angle.


The pitch angle

As mentioned above, in disk galaxies the linear circular velocity is approximately constant: v_c \simeq {\rm const}. The angular velocity is:

d\Omega=\left(v_c/r^2\right) dr.

Since at the time t (see figure): \phi=\Omega t,

so that:

d\phi=t~d\Omega=\frac{v_c}{r^2}t~dr,

then:

\displaystyle{tan\psi=\frac{1}{r} \frac{dr}{d\phi} = \frac{dr}{[(v_c/r)dr~t]}=\frac r {v_ct}=\frac 1 {\Omega~ t}=\frac 1 {\phi}.

Therefore, the following relation holds for the pitch angle:

tan~ \psi=\frac r {v_ct}=\frac 1 {\Omega t}=\frac 1 \phi.

The pitch angle of a material spiral arm becomes smaller as the galaxy rotates.

The pitch angle of a material spiral arm becomes smaller as the galaxy rotates.


The pitch angle

After one full rotation the pitch angle is:

\psi=\frac{1}{2\pi}=\simeq 9\ {\it deg}.

After two full rotations, the pitch angles goes down to about 4.5 degrees. We therefore should observe very tight arms. The observed pitch angles changes from early- to late-spirals.

The average value of the pitch angle in Sa spiral is about 5 degrees, while for Sc spirals it ranges between 10 and 30 degrees. This apparent dichotomy is probably related to a different nature of the observed spiral in disks galaxies.

The pitch angle of a material spiral arm becomes smaller as the galaxy rotates.

The pitch angle of a material spiral arm becomes smaller as the galaxy rotates.


The pitch angle

Correlation between the pitch angles and the galaxy morphological type (de Vaucouleurs T index). From Garcia Gomez & Athanassoula, A&AS.S, 100, 4, 1993.

Correlation between the pitch angles and the galaxy morphological type (de Vaucouleurs T index). From Garcia Gomez & Athanassoula, A&AS.S, 100, 4, 1993.


Short-lived and long-lived arms

The simple analysis of the variation of the pitch angles allows us to identify long-lived and short-lived spirals and to reveal their different nature.

Short-lived spiral arms probably origin from temporary patches (of gas and stars) pulled out by differential rotation. The patches might arise from local disk instabilities, leading to star formation which propagates along the disk thanks to the differential rotation. This scenario suggests that spirals with short-lived arms correspond to the flocculent spirals.

On the other hand, long-lived spiral arms are not material structures in the disk. They must be a pattern in the disk, through which stars and gas move. Long-live spiral should therefore correspond to grand design spirals. We need now to address the question of the physical nature of the spiral pattern in grand-design spiral.

Manifold of spirals

The galaxy cluster SDSS J1004+4112, discovered in the Sloan Digital Sky Survey and pictured by HST, is one of the more distant known, and is seen as it appeared when the universe was half its present age.

The galaxy cluster SDSS J1004+4112, discovered in the Sloan Digital Sky Survey and pictured by HST, is one of the more distant known, and is seen as it appeared when the universe was half its present age.


Evolution of spiral arms morphology

Numerical simulations of stochastic self-propagating star formation in spirals (from Gerola & Seiden, Ap.J., 223, 129, 1978). Originally roundish star-forming regions get stretched out by differential rotation.

Numerical simulations of stochastic self-propagating star formation in spirals (from Gerola & Seiden, Ap.J., 223, 129, 1978). Originally roundish star-forming regions get stretched out by differential rotation.


Mass distribution in the galaxian disk

Optical bands observations show that enhancement of spiral arms is often related to strong star-formation rates. Indeed, HII regions and molecular clouds are good tracers of spiral arms. However, near-infrared imaging shows that spirals are present also in the density distribution of old stars: therefore, spirals are related to a real density fluctuation of the disk.

The 2D matter distribution in a disk galaxy can be described by the following parametrization:

\Sigma(R, \phi)=\Sigma_0(R)+\Sigma_1(R) \cos [m\phi+f(R)],

where R and \phi are the usual polar coordinates in plane, \Sigma_0 is the underlying, axisymmetric mass distribution, that is independent of the azimuthal angle, \Sigma_1 is the amplitude of the perturbation (that can be as large as 10%) and the adimensional function f describes the spiral pattern.

The pitch angle reads now:

\tan \psi =m \biggl|r\frac{\partial f}{\partial R}\biggr|^{-1}.

A logarithmic spiral (i.e., spiral pitch angle does not dependent on the azimuthal angle) are obtained with the following choice of the function f:

f( R ) = f_0 \ln ( R) + \phi_0

Density wave model

Lindblad (1925) was the first to propose the idea of  “stellar density waves” to explain the origin and stability of spiral arms. The Lindblad’s idea was further developed in a seminal paper (published in 1964) by C.C. Lin and Franck Shu. They proposed that arms in grand-design spirals are quasi-stationary density waves, with a constant pattern speed in the stellar disk. Quasi-static density waves are sections of the galactic disk with larger mass density (about 10-20% larger). Therefore, spiral arms are structures which material components change in time, without altering the overall morphology.

Before discussing quantitatively this idea, it is worthy to mention a very similar situation in every-day life: car traffic on an high-way. Traffic produces naturally stationary density waves (at traffic lights or cross roads). Indeed, while the traffic jam does not move, the single cars move passing through the jam itself. Ocean waves also share the same underlying mechanism.

Lin & Shu model

Lin & Shu (1964; see also the paper in 1966) proposed a model in which the spiral arms are the manifestation of a quasi-stationary pattern in the stellar disk. As the pattern rotates with angular speed \Omega_p different form the stellar angular velocity \Omega, stars and spirals do not corotate (in other words: spiral are not material systems of the fixed ensemble of stars).

Stars enter and come out of spiral arms at different times of their rotation around the galactic center.

This model can also explain in a natural way the fact that spiral arms show much higher star-formation rate with respect to the stellar disk. Cold clouds of interstellar medium are compressed when swept by a spiral density wave, thus enhancing the formation of new stars.

The density-wave theory has to meet two challenges. First, in order to consider it as a viable theory for the observed grand-design spiral arms, it is necessary to show that a self-consistent model can be constructed. In other words, we have to find a set of orbits in the perturbed gravitational potential that, stacked together, reproduce the density distribution of the perturbed disk. Secondly, we have to find a mechanism to create a spiral-shaped density perturbation that can survive a large number of galactic orbits.

An example od density waves

We now consider in some detail the density waves in a galactic disk. First, we need to identify the mechanisms originating a long-lived spiral-shaped density perturbation.

Let us start from an unperturbed disk, where stars move at angular velocity \Omega(R) along circular orbits. If a global perturbation is introduced with l=m=2, then the stellar orbits are not anymore circular, but become elliptical.The perturbation has in general a non-zero pattern speed, \Omega_p, causing the orientation of the elliptical orbits to rotate with the same angular velocity.

If we consider a spiral-shaped perturbation, the orientation of the elliptical orbits is aligned with the perturbing potential. This causes a crowding of the orbits along a two-armed spiral pattern that, more or less, follows the shape of the m=2 density perturbation, and has a pattern speed equal to \Omega_p.

As shown in the figure, the consequence is a crowding of the orbits along a two-armed, symmetric spiral pattern that approximately follows the shape of the m=2 density perturbation. Its pattern angular speed is \Omega_p.

This example shows density waves characterized by a two-armed spiral can be constructed self-consistently. It also explains why the spiral pattern is also visible in the NIR light emitted by the old stars that trace the underlying stellar mass density.

Density waves produced by two types of perturbations. From Kalnajs, Proc. Astron. Soc. Australia, Vol. 2, 1741, 1973.

Density waves produced by two types of perturbations. From Kalnajs, Proc. Astron. Soc. Australia, Vol. 2, 1741, 1973.


Density waves and ISM

We have seen that in grand-design spirals, the density waves are also seen in the underlying old stellar populations. However, it is very rare to meet a galaxy in which we observe a spiral pattern and no interstellar gas. The few cases in which this is is seen (see, e.g., Strom et al., Ap.J., 206, L11S, 1976) are explained as galaxies which recently lost their gas because of a close encounter with another galaxy.

As the ISM mass is at most only about 10% of the local mass density in disks, we can model the disk potential as due to only the stellar components. In other words, ISM clouds are regarded as test particle moving in the gravitational potential of the stars. The potential can be write as the sum of an axisymmetric, unperturbed component,\Phi_0 ( R),and a perturbation that reads:
\Phi (R, \phi) = {\Re}\left[\Phi_a \exp \left(i m \phi\right)\right],

where, as before, \phi is the azimuthal angle in a frame rotating at angular speed \Omega_p. If the wave is tightly wound, it is possible to write:

\Phi_a ( R) = F \exp(i k R) ,

where F is a constant. By choosing the origin of the azimuthal angle in order to obtain a real expression for F, we can write:

\Phi_1 (R, \phi) = F \cos (k R + m \phi).

Motion of the ISM clouds can be traced by using the equations of a star in a weak non-axisymmetric field.

Tidal arms in spiral galaxies

We mentioned above that spiral arms can also be formed during galaxies close encounters. As a fact, many spiral galaxies showing symmetric m=2 arms are observed with smaller companion galaxies located near the tip of one of the main arms.

The most famous example is the galaxy Messier 51; see picture. In order to determine whether the encounters with these small companions are to be held responsible for the origin of the spiral pattern, numerical simulations are necessary.

Alar Toomre and Jury Toomre (1972) performed a simple simulation of the M51 system, with two galaxies modeled as sets of N mass-less points (treated as test particles) in circular orbits around massive central points which in turn move in hyperbolic orbits. By this simplification the complex problem of the multiple interactions reduces to the solution of N (per galaxy) restricted three body problems [go deeper into this].

The galaxy at the RHS is the spiral M51 with its satellite galaxy. M51 is the proto-type galaxy with spiral arms induced by a strong tidal interaction. Image from the Hubble Space Telescope.

Credit: NASA (HST).

Credit: NASA (HST).


Tidal arms in spiral galaxies

Their first simulations were very promising (see RHS figure with results from the seminal paper with the numerical simulations of a M51-like system, where a simple fly-by makes spiral arms. Numbers at the bottom show the units of time).

Spiral arms could be formed, but they could not be traced up to very center of the galaxy. This was later recognized as due to the fact that the stars in the disk cannot be treated as “test particles”, and also their self-gravity has to be considered in the simulations. As later demonstrated numerically by Toomre (1981), encounters with small companions can indeed give origin to density waves which are quasi stationary and stable for many orbits. However, it is clear that close encounters cannot explain the totality of observed grand-designs spirals, as these events are not very frequent.

A. Toomre & J. Toomre, Ap.J., 178, 623, 1972.

A. Toomre & J. Toomre, Ap.J., 178, 623, 1972.


Simple numerical simulations of encounters

Abstract of the paper by Toomre & Toomre (1972) with the results of the first numerical simulations applied to close encounters between galaxies. Their simple reduced three body model is sketched on the right.

Abstract of the paper by Toomre & Toomre (1972) with the results of the first numerical simulations applied to close encounters between galaxies. Their simple reduced three body model is sketched on the right.


Bars in spiral galaxies

Almost half of the disk galaxies shows the presence of a bar, including our Galaxy and its two major satellites, the Large and Small Magellanic Clouds. Generally speaking, bars are very elongated, with axis ratios in their equatorial plane between 2.5:1 to 5:1. Their thickness is not directly measured, as it is hard to determine the presence of a bar in edge-on galaxies, but from statistical analysis is inferred that on average they are as thick as the host disk. While very luminous, emitting about 1/3 of the optical total luminosity of the galaxy, their brightness profile is nearly constant along the major axis.

Bars are likely rotating in a rigid pattern: they are straight structures and stars stay within the bars. Therefore, bars are not density waves as those described by the Lin and Shu theory. Stars belonging to the bar move along the bar itself on closed orbits in frame rotating at the same angular speed of the bar.

While bars are not related to quasi-stationary density waves, they can drive a density wave in disk, helping in maintaining a stable spiral pattern.

It is interesting to investigate how the fraction of barred spirals changes over cosmic time. In recent deep surveys it has been observed a rapid increase of the fraction of barred spirals with cosmic time. Comparing the results from two surveys, the SDSS (local universe, z\simeq 0) and COSMOS (at higher redshifts), it has been found that in the local universe about 65% of luminous spiral galaxies contain bars (SB+SAB), at redshift about 0.8 this fraction drops to about 20%. This is a strong indication that, when the universe was half its present age, the census of galaxies on the Hubble sequence was fundamentally different from that of the present day.

NGC 1365

Barred spiral galaxy NGC 1365 also showing a dust lane. Credit: ESO; R. Gendler, J-E. Ovaldsen, C. Thone, and C. Feron.

Barred spiral galaxy NGC 1365 also showing a dust lane. Credit: ESO; R. Gendler, J-E. Ovaldsen, C. Thone, and C. Feron.


Nearer (older) vs. more distant (younger) spirals

Barred spiral galaxies at high redshift, observed by the Hubble Space Telescope and the Japanese telescope Subaru (located on the Manua Kea, Hawaii) within the COSMOS survey. Barred disks were much less frequent in the young Universe. Credit: NASA

Barred spiral galaxies at high redshift, observed by the Hubble Space Telescope and the Japanese telescope Subaru (located on the Manua Kea, Hawaii) within the COSMOS survey. Barred disks were much less frequent in the young Universe. Credit: NASA


Epicycles in spiral galaxies

In the last part of this lecture, we investigate the stellar orbits in the disk of a spiral galaxy. Disk stars have approximately circular orbits with small deviations: therefore, the local motions of stars can be studied by means of the equations of motion for small perturbations from a circular orbit. As a result, we will obtain a description of stellar motion in terms of epicycles.

Just like in the Ptolemaic system, star orbits can be described by superposition of two motions: a circular orbit along a guiding center (i.e. the deferent), characterized by a radius R_g and angular velocity \Omega_g, and a smaller elliptical epicycle, with angular velocity \kappa, and a retrograde motion.

Epicyclical motion

Let us consider a star at guiding center (GC; see next image), with a small, radially outwards impulse. Since the motion holds on a plane, the zcomponent of the angular momentum is conserved: L_z = m r v_{\phi}={\rm const.}, thus a radially outwards impulse makes r grow and the velocity v_\phi decreases. Hence, with respect to the GC, the star moves backwards. Consider now the new balance between gravitational and centrifugal forces: f_g\propto r^{-2}f_c = \frac{v_{\phi}^2}{r} \propto r^{-3} and f_g>f_c, and the star is pulled back inwards. While r decrease, the velocity now increases and the star moves forward relative to the GC. But at this point, we have f_c>f_g; the star moves outwards again. As the cycle repeats, we observe a small retrograde epicycle. The image in the next slide shows the four phases:

  1. As r increases, the velocity v_{\phi} decreases;
  2. Now f_g>f_c,  and the star is pulled inward;
  3. Thus r decreases and the velocity increases;
  4. f_g < f_c and the star goes forward.

Epicyclical motion

Schematic view of the epicyclical motion.

Schematic view of the epicyclical motion.


Keplerian potential

In a Keplerian gravitational potential, the orbit and epicycle frequencies are equal: \Omega_g = \kappa_g.
As a consequence, stellar orbits are closed, and we observe off-centered Keplerian ellipses, see picture. In the general case, the frequencies \Omega_g and \kappa_g are not equal, and stellar orbits are not closed. However, they do appear as closed ellipses (and centered on galactic center) when they are considered in a frame of reference rotating at angular speed:

\omega=\Omega_g-\frac 1 2 \kappa.

The figure shows an elliptical Keplerian orbit (blue) well approximated (red) by the composition of a retrograde motion at angular frequency \kappa around a small ellipse with axis ratio b/a=1/2 with a prograde motion of the ellipse center at angular frequency \Omega around a circle (dashed).

Epicyclical motion in a Keplerian gravitational potential.

Epicyclical motion in a Keplerian gravitational potential.


Axisymmetric potential

Let us derive some of the characteristics of epicycles. In particular, we focus on a smooth axisymmetric flattened mass distribution with potential \Phi (R,z). The angular momentum is still a conserved quantity, since there are no azimuthal forces. Hence, we can write the following equations:

{\bf \ddot{r}}=- \nabla\Phi(R,z),

L_z = R^2 {\dot \Phi}=const.

By separating the equations of motion into components in cylindrical coordinates (R, \Phi, z), we obtain the three equations:

\displaystyle \ddot{R} - R {\dot \Phi^2} =- \frac{\partial \Phi}{\partial R},

\dot L_z=0,

\ddot{z}=-\frac{\partial \Phi}{\partial z}.

Vertical motion

Since the disk is axisymmetric respect to z=0, the gravitational potential is axisymmetric and thus the force along z at z=0 must be null, that is:

\displaystyle{ \left(\frac{\partial \Phi}{\partial z}\right)_{z=0}=0}.

For small motions above and below the plane, we can use a linear approximation of the z-force linearly for small displacement in z:

\displaystyle{\ddot{z} = -z \left( \frac{\partial^2 \Phi}{\partial z^2}\right)_{z=0} = - \nu^2 z}.

This can be immediately recognized as the equation of the simple harmonic motion, with frequency given by the expression \nu, where:

\displaystyle{\nu^2 = \left(\frac{\partial^2 \Phi}{\partial z^2}\right)_{z=0} .

The motion along the given axis is therefore: z(t) = Z \cos (\nu t + \psi_0) .

For the Galaxy disk, in proximity of the Sun, \nu = \sqrt{ 4 \pi G \rho} \sim 0.072\ Myr^{-1}.

Therefore, the vertical oscillation period is: 2 \pi / \nu \sim 87  Myr, that is about 1/3 of the whole circular period.

Radial motions

Let us study now the motion along the radial direction. The GC moves along a circular orbit (R = R_g = {\rm const.}), with circular velocity V_c and angular velocity \Omega_g defined by equation:

\displaystyle{\left(\frac{\partial \Phi}{\partial R}\right)_{R_g} = \frac{V^2_c}{R_g} = R_g \Omega_g^2}.

When the orbit is not circular, the radial acceleration is given by:

\displaystyle{\ddot{R} = R \dot{\phi^2} - \frac{\partial \Phi}{\partial R}.

Using the relation  L_z = R^2 \dot{\Phi}, we can write:

\displaystyle{\ddot{R} = -\frac{\partial \Phi_{\rm eff}}{\partial R}}, where we introduce the effective potential \Phi_{\rm eff}:

\displaystyle{\Phi_{\rm eff} \equiv \Phi(R, z) + \frac{L_z^2}{2 R^2} }.

The effective potential is used to reduce the problem to a one-dimensional problem.

Radial motions

The \frac{L_z^2}{2R^2} term imposes an angular momentum barrier at small R. At the minimum in the effective potential, we recover the CG orbit, with radius R_g:

\displaystyle{(\frac{\partial \Phi_{\rm eff}}{\partial R})_{R_g} = 0 \rightarrow (\frac{\partial \Phi}{\partial R})_{R_g} - \frac{V_c^2}{R_g} = 0} .

In general, orbits not located at the minimum of the effective potential, will oscillate in radius, around R = R_g.

Let us consider the potential at R = R_g + x:

\displaystyle{\ddot{R} = \ddot{x} = - (\frac{\partial \Phi_{\rm eff}}{\partial R})_{R_g} - x (\frac{\partial^2 \Phi_{\rm eff}}{\partial R^2})_{R_g} =- x (\frac{\partial^2 \Phi_{\rm eff}}{\partial R^2})_{R_g} = - \kappa^2 x}.

Again, we observe an harmonic motion, now about the guiding radius R_g, with frequency:

\displaystyle{ \kappa^2 \equiv  (\frac{\partial^2 \Phi_{\rm eff}}{\partial R^2})_{R_g} = (R \frac{d \Omega^2}{d R} + 4 \Omega^2 )_{R_g}}.

Azimuthal motion

Finally, let us consider the azimuthal component of the motion in an axisymmetric gravitational potential. As the quantity L_z is conserved, any change in R yields a change in \Omega:

\displaystyle{ \dot{\phi} = \frac{L_z}{R^2} = \frac{L_z}{(R_g + x)^2} \simeq \frac{L_z}{R_g^2} (1 - \frac{2x}{R_g}) = \Omega_g (1 - \frac{2x}{R_g}) }.

By integrating, we obtain:

\phi(t)=\Omega_g-\frac{2\Omega_g X}{\kappa R_g}\sin(\kappa t+\phi_0).

Therefore, the \phi(t) follows the motion of the GC, with small harmonic oscillations superposed. With the y-axis in the forward direction and origin on the GC, we have:

y(t)=-\frac{2\Omega_g}{\kappa} X\sin(\kappa t+\phi_0).

The frequency is again \kappa, but the phase is now different. Putting things together, with \phi_0 = 0

x(t) = X \cos(\kappa t) and  \displaystyle{ y(t) = - \frac{2 \Omega_g}{\kappa} X \sin(\kappa t)}.

Properties of the epicyclic motion

Let us have a closer look at some properties of this motion. While the Ptolemy’s epicyclic motion is prograde, in spiral galaxies, the epicyclic motion is retrograde with respect to the orbit. When we have a simple Keplerian potential: \kappa = \Omega. In this case, the epicycle axis ratio is 2:1 (it is interesting to recall that Ptolemy’s is 1:1, that is, circles). Moreover, the full orbit is a closed ellipse, centered at the ellipse focus.

For a flat rotation curve, usually observed at large radii in spiral galaxies, we have \Omega \propto R^{-1}. Under this condition, \kappa = \sqrt{2} \Omega.

In the case of a rigid rotation, like for a solid body, \Omega = {\rm const.}, we have \kappa = 2 \Omega , giving circular epicycles and closed oval orbits.

In general, it is easy to show that \kappa>\Omega.  Hence, epicycles are completed before a whole rotation, and, when observed from an inertial frame, orbits don’t close, but regress.

Epicyclic motion in the Solar neighborhood

Finally, let us have a look at the Solar neighborhood. We can express the epicyclical and orbital frequencies at the Solar radius, \kappa_0, \Omega_0, in terms of Oort’s constants:

\kappa_0^2 = - 4 B (A - B) = - 4 B \Omega_0 \rightarrow \kappa \sim 37\ km/s/kpc ,

\Omega_0 = A - B \sim 27\ km/s/kpc .

As the ratio  \kappa_0 / \Omega_0 \sim 1.3, we have that stars in the Solar neighborhood make 1.3 epicyclic rotations per orbit.

I materiali di supporto della lezione

G. Bertin, Dynamics of Galaxies. Cambridge Univ. Press, 2000.

J. Binney & S. Tremaine, Galactic Dynamics, Capter 6, Cambridge Univ. Press, 1987.

J. Binney & S. Tremaine, Galactic Astronomy, Capter 10, Cambridge Univ. Press, 1998.

C. C. Lin & F. H. Shu 1964, Ap.J., 140, 646, 1964.

A. Toomre & J. Toomre, Ap.J., 178, 623, 1972.

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