Rainer C. Gaitzsch

"EAAE Summerschools" Working Group

Teacher's Academy of Bavaria (Germany)

When we look into the universe beyond the milky way, we find that galaxies exist in many shapes and sizes. In this workshop we want to turn our attention to the spiral galaxies, which are quite numerous, and besides - we ourselves live in such a kind of a galaxy.

Spiral galaxies are certainly among the most beautiful objects in the universe. If only we lived long enough, we could watch the majestic spiral arms rotating around the centers of galaxies. But of course it takes too long. The rotation periods are measured in hundreds of millions of years. The direction of the rotation is "normal", like in a firewheel.

Fig. 1: Spiral galaxy M 83 Hya, 22 Mio. ly away.
Fig. 1: Spiral galaxy M 83 Hya, 22 Mio. ly away.


Most galaxies like ours are flat circular disks of radius a, and so they appear to us when they are seen face-on. But the galaxies which we see spread all over the sky randomly have any kind of aspect to us between the two extreme positions face-on and edge-on, mostly we see them in an elliptic shape, depending on the angle of inclination between their galactic plane and our line of sight. So, if we want to investigate the real shape of the spiral curves, we have to take in account this distortion caused by the angle i. One axis is foreshortened to b where b / a = sin i.

Fig. 2: Aspects of a galaxy depending on the angle of inclination to our line of sight.
Fig. 2: Aspects of a galaxy depending on the angle of inclination to our line of sight.


As a first exercise we want to find out the angle of inclination of the galaxy M 81, shown in appendix 1. You should first recognize the axes and then measure the values of a and b.

We get about b / a = 0.59 which means an angle i = 36° or something like it.

In order to obtain a simple model which can be applied at school illustrating some of the obvious dynamics in a spiral galaxy, we will investigate the spiral arms. By measuring certain details we try to find out what the formula of the spiral curves might be.

There are three different mathematical kinds of spirals: (Notice that j and r are polar coordinates)

  1. The spiral of Archimedes: r = a × φ, where a is a constant of structure, and so it has to be: Δr ~ Δφ
  2. The hyperbolean spiral: r = a / φ, where a is also a constant of structure.
  3. The logarithmic spiral: r = r0 × e respectively ln (r / r0 ) = c × φ, where c defines the constant angle δ between the spiral and any radial straight line:
    tan δ = 1 / c
Fig. 3: Logarithmic spiral.
Fig. 3: Logarithmic spiral.


Look at appendix 2 to see by the example of our own milky-way-galaxy how the investigation should be made and how the table is filled with corresponding measured and calculated values. Notice that the logarithmic spiral fits best. The hyperbolean spiral cannot be completely refused, while the spiral of Archimedes does not fit at all (besides: there does not exist any physical model to fit the last).

When we use the photographs in appendix 1 to determine the right formula for the spiral arms, we should also first draw a convenient basic-line corresponding to the starting-angle φ = 0. We shall now prove our skills on the following exercise:

Try to find out the best fitting spiral curve for the face-on galaxy NGC 628. Proceed as in appendix 2 and fill in a similar table. You should get to a resembling conclusion as in appendix 2.

The galaxy NGC 4535 stands for an additional voluntary exercise.

To do the same with galaxy M 81 is more complicated because this galaxy is not face-on. So you have to stretch it first to a circled disc before determining the values of j and r.

Which way you ever try, it is a nice challenge.

Although we do not actually see any motion, we can detect and measure the rotation of galaxies by means of the Doppler effect: Δλ / λ = v / c. The higher the speed, the greater the shift (mainly measured is the 21-cm radiation from different parts of the galaxy).

The final result of the observations is a rotation curve v (r), where r marks the distance from the galaxy's center. The outcome is: In the inner regions of a spiral galaxy rotation speed increases with distance: v ~ r , which is typical for a rigid body's rotation. In the outer regions the radial velocity is about constant (!), surprisingly, because in lack of the expected velocity-decreasing according to Kepler, there is a need for additional mass to cause this phenomenon ("dark matter"?).

Fig. 4: Rotation curve v(r).
Fig. 4: Rotation curve v(r).


We have a differential rotation, however, and a possible reason for the spiral arms to exist.

Exercise: Create a "spiral galaxy" (appendix 3a).

Suppose that there is material stretching out from the center of the concentric circles (big black dots). Draw subsequently new dots, corresponding to the constantness of radial velocity: v = r × ω => v × t = r × φ.

So for any given time you like (i.e. t = 1, 2, 4) the product r × φ is constant, which means, the angle φ to be set for the dots goes like this: φ ~ 1 / r.

In the end you will get three spiral curves proceeding by time.

Look at appendix 4a. There you can see one result of a computer simulation which shows more examples of that kind. And besides: Whatever structure you have at the beginning, it always ends up in a "spiral galaxy" (even a cow, or so...).

O.K., nice spirals. But are they true? Is this the way matter in spiral galaxies really arranges? Sorry, our model was too simple, there are two obvious mistakes:

1 - If time increases, after some orbital revolutions the spiral curves tightly wind up that they disappear; this is not the case in real galaxies! Spiral arms persist in spite of the differential rotation. How can this be?

2 - We can prove that the formula for the spiral curve is a hyperbolean one, and we remember that this is not very likely (φ = const. / r; appendix 4a).
So, what is the real nature of these spiral arms ? And how might this structure possibly have evolved then?

The solution came in the early seventies when Lindblad and others recognized that the spiral arms in galaxies are merely a pattern that moves among the stars, not a lasting material structure. Think about waves on the sea. As the waves move across the surface of the water, the individual water molecules and other particles (think of a cork) just only bob up and down in little circles. The wave is only the pattern that moves beyond the almost locally fixed material (appendix 5a). Lindblad spoke of density waves in discussing the possible cause of spiral structure. A spiral arm is therefore simply a temporary enhancement or compression of the material in a galaxy. In fact, the distances between stars are so far(model: like raindrops in a distance of 60 km!) that collisions never happen; they simply orbit along nearly circular orbits. But the stars do interact by gravity. If there is some sort of disturbance, the star is pushed slightly out of its equilibrium orbit, so the star will oscillate back and forth about its original orbit like in a kinematic waterwave. The real path of the star is a precessing ellipse. Of course the star effects also the motion of its neighbours, and so on ... This leads to self-sustaining star formation which is caused by the Supernova-produced shock waves and indicated by the evolving spiral arms.

Exercise: Notice how a beautiful spiral pattern emerges in appendix 3b, when you again draw dots one after another like in appendix 4b, 5b and 5c).

To manage this you begin at the innermost ellipse, turning round an angle of a = 15° and mark the point of contact on the second inner ellipse. Continue this way by always turning round the same angle of 15° and dotting the touching point of the subsequent ellipse. This game simulates the precessing ellipses. All ellipses differ only by a constant factor k, so it can be proved that the emerging spiral is a logarithmic one: φ → φ + n × α ⇒ r → kn × r bei r = r0 × e.


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