The spiral arms of galaxies G. Contopoulos1 Research Center for Astronomy and Applied Mathematics

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The spiral arms of galaxies

G. Contopoulos1

Research Center for Astronomy and Applied Mathematics

of the Academy of Athens

Soranou Efesiou 4, GR-11527 Athens, Greece

Abstract

The most important theory of the spiral arms of galaxies is the

density wave theory based on the Lin-Shu dispersion relation. How-

ever, the density waves move with the group velocity towards the

inner Lindblad resonance and tend to disappear. Various mechanisms

to replenish the spiral waves have been proposed. Nonlinear eﬀects

play an important role near the inner and outer Lindblad resonances

and corotation. The orbits supporting the spiral arms are precessing

ellipses in normal galaxies that extend up to the 4/1 resonance. On

the other hand, in barred galaxies the spiral arms extend along the

manifolds of the unstable periodic orbits at the ends of the bar and

they are composed of chaotic orbits. However these chaotic orbits can

be found analytically.

1 Introduction

A large proportion of galaxies have spiral arms. But how these spiral arms

were formed? Today I will present a historical account of the various theories

of spiral arms with emphasis on the density wave theory.

An obvious theory of spiral arms is based on the diﬀerential rotation of

the galaxies. Namely, the rotational velocity Ω around the center of the

galaxy decreases (Fig. 1) as the distance from the center increases. Thus, an

initially elongated structure (Fig. 2a) is transformed soon into a spiral, as the

regions closer to the center rotate faster, leaving the outer parts behind (Fig.

2b) (trailing spiral areas). These arms are called material arms, because they

are composed of the same material always.

However, if we take the data from our Galaxy and similar galaxies, we

ﬁnd that the material arms of the Galaxy were straight lines only 300 million

years ago and they should be wrapped tightly after a few rotations (Fig. 2c).

Thus the galaxies that are much older should not show the beautiful spiral

arms that we observe.

Then some people considered a very diﬀerent mechanism of generating

spiral arms. As B. Lindblad (1926, 1936) has shown, if a galaxy is quite

1gcontop@academyofathens.gr

arXiv:2210.13632v1 [astro-ph.GA] 24 Oct 2022

Figure 1: Rotational velocities of galaxies.

Figure 2: The evolution of material spirals

ﬂat (beyond the ﬂatnem of E7 galaxies) the circular motions at its bound-

ary are unstable and matter is ejected outwards in the direction of rotation

(leading spiral arms). However, most galaxies are not so ﬂat and the obser-

vations show that the spiral arms are trailing, therefore this mechanism is

not satisfactory.

Other mechanisms, like magnetic ﬁelds acting on the gas have been shown

to be insuﬃcient because the observed magnetic ﬁelds are very weak.

2 The density wave theory

B. Lindblad considered the mechanism of density waves (Lindblad 1940, 1941,

Lindblad and Langebartel, 1953). Namely the stars move through the spiral

arms but stay longer in their neighborhood (Fig. 3). Therefore, the spiral

arms are waves. They keep their form, but they are not composed of the

same matter for a long time. This mechanism was shown to be very eﬀective

after the numerical experiments of Miller et al. (1970) and Hahl (1970).

It is remarkable that Lindblad developed the theory of density waves well

before this mechanism was used in the waves of plasma, which has been very

successful in later years.

B. Lindblad noticed that the theory of density waves was applicable both

for trailing and leading waves. And as he was preoccupied with leading spiral

arms, he applied the theory to them. But later the son of B. Lindblad made

numerical experiments with a relatively large number of stars and he always

found trailing spiral arms. Thus Prof. B. Lindblad devoted his last two

papers (1961, 1963) on trailing spiral arms.

The work of B. Lindblad on density wave spiral arms did not attract the

attention it deserved because of his emphasis on leading spiral arms and its

diﬃcult style.

Figure 3: The orbits in the density waves.

Then in 1963 I met Prof. C.C. Lin in MIT, who told me that he wanted to

develop a theory of density waves to explain the spiral arms. I told him that

such a theory was already developed by B. Lindblad. So, we went together to

the library and borrowed some volumes of the Stockholms Observatoriums

Annaler that contain the work of Lindblad. But when I met again Prof. Lin

after a few days, he told me that he had diﬃculties to understand the papers

of Lindblad. Thus, he decided to work the theory himself from scratch. In

fact, one year later C.C. Lin and F. Shu published their ﬁrst paper on the

“grand design” density wave theory of spiral arms. A similar theory was

developed by Kalnajs (1969, 1971).

Their theory deals with the perturbations of the 3 basic functions of galac-

tic dynamics, the potential V, the density ρand the distribution function f.

These functions are related by 3 equations:

1. The collisionless Boltzmann equation

dt =∂f

∂t + ¯v∂f

∂¯x−∂V

∂¯x

∂f

∂¯v= 0,(1)

where ¯x= (x, y, z) and ¯v= (vx, vy, vz), that gives the distribution

function f(integral of motion) for a given potential V.

2. The integral

ρ=Zfd ˙

¯x, d ˙

¯x=d˙xd ˙yd ˙z, (2)

that gives the density and

3. the Poisson equation

∇2V= 4πGρ (3)

that connects the density with the potential.

We consider a ﬂat model ρ=σδ(z), where σis the surface density, an

axisymmetric background V0, f0, σ0and perturbations of ﬁrst order V1, f1, σ1

and higher orders V=V0+V1+V2+. . . etc.

The perturbation V1of the potential is assumed to be of spiral form

V1=Aexp[(i(φ(r) + ωt −mθ],(4)

where Ais the amplitude and mis the number of spiral arms (usually m= 2

and ω= 2Ω).

The solution of the Boltzmann equation gives f1. Then the integral (2)

gives the “response density” σresponse

1(linear theory). On the other hand the

Poisson equation gives the imposed density σimposed

1and we have to solve the

self consistency equation

σimposed

1=σresponse

1.(5)

This is an integral equation (Kalnajs 1965, 1971). On the other hand

Lin and Shu (1964) found under some simplifying assumptions a dispersion

relation

1−2πGσ0

|k|h ˙r2i1−v

2 sin(νπ)Zπ

−π

dγ cos(νγ) exp(−x(1 + cos(γ))= 0,(6)

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ThespiralarmsofgalaxiesG.Contopoulos1ResearchCenterforAstronomyandAppliedMathematicsoftheAcademyofAthensSoranouEfesiou4,GR-11527Athens,GreeceAbstractThemostimportanttheoryofthespiralarmsofgalaxiesisthedensitywavetheorybasedontheLin-Shudispersionrelation.How-ever,thedensitywavesmovewiththegroupveloci...

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