# Spiral Galaxies

I only have a very, very, basic understanding of linear motion, much less so of circular motion. What I can recall is my book telling me if you spin a stone tied to the end of a string and the string breaks, the stone will fly off at a tangent to its orbit and not spiral away or anything else like radially, etc.

Gravity is that string and stars are the stone and whatever supermassive object (some say they're supermassive blackholes) is at the center of spiral galaxies exerting this pull of gravity, my knowledge (as disclosed vide supra) can't explain the spiralness of spiral galaxies.

Any simple explanation on how spiral galaxies form will be deeply appreciated.

You're right that a central mass can't explain the spiral structure (and you're also right about the tangent motion in case the string breaks). Spiral structure is indeed partially caused by gravity, but hydrodynamic forces are also important, as I will explain below:

Theories of the formation of spiral structure in galaxies go roughly 100 years back and flourished in the 1960's and 1970's, after which there hasn't been any ground-breaking advances. Spiral arms were first discovered by Lord Rosse (1845) through his homemade 1.8 m telescope, the Leviathan of Parsonstown, but it wasn't until Edwin Hubble's and others' realization, in the 1920's, that the nebulae were outside the Milky Way that their immense sizes were appreciated, and with that also the implied speeds.

## How not to form spiral arms

Astronomers soon realized that the rotation should wind up the spiral structure on timescales much shorter than the age of the stars inside them (e.g. Wilczynski 1896), so some mechanism was needed to maintain the spiral pattern. The is called the winding problem, illustrated below:

The winding problem: Stars that start out aligned and rotate at the same speed around the center of the galaxy will initially form a spiral pattern. However, the pattern will be wound up after a few 100 million years, in clear conflict with the fact that we see lots of spiral galaxies that are 10 or 100 times that age. In this simple simulation, which can be seen animated here, I assumed a radius of 10,000 lightyears and a speed of 200 km/s.

From the 1920's to his death in 1965, the Swedish astronomer Bertil Lindblad worked on solving the mystery of the formation and maintenance of spiral arms. His basic idea was to consider spiral arms as patterns made out of the trajectories of individual stars (e.g. Lindblad 1927; 1940; 1961).

The interaction between the trajectories were then supposed to create a number of "quasi-stationary" spiral patterns, i.e. patterns whose shape is "frozen", moving around with a speed which is not the same as the speed of the stars (Lindblad 1963).

The stars in the disk of the galaxy, both inside the arms and in-between, will have an overall orbit but will also oscillate slightly around this orbit. Under certain conditions, the trajectories will tend to become elliptical. If stars in different distances follow elliptical, shifted orbits, some regions will form in the disk where the density of stars is greater than in the rest of the disk. As seen below, these overdensities will be spiral-shaped:

Lindblad's kinematic spiral waves. On the left are elliptical orbits whose major axes all have the same phase. To the right, the phase is shifted increasingly more, the larger the orbit. The result is a spiral-shaped overdensity of stars.

## How to form spiral arms

Lindblad's theory could explain certain aspects of the spiral arms, but most of his works is sort of forgotten now. He made a wealth of insightful considerations, but has probably been hampered by a lack of both empirical evidence, computing power and, especially, falsifiable predictions, the cornerstone of science.

#### Density wave theory

Shortly before Lindblad's death, a breakthrough was made by Lin & Shu (1964): Considering the gas and stars as a continuous fluid and using equations known from hydrodynamics, they showed that the spiral arms could be thought of as waves in the density of gas and stars. The basic idea in this density wave theory is that the gravitational force from primarily the stars creates waves, and that these waves then move through the disk of the galaxy. Just as in Lindblad's theory, the density waves are quasi-stationary and are therefore not wound up.

When gas and stars are inside the wave, they move a little slower than otherwise, so that the density here is slightly larger than outside the wave (this effect is often compared to cars piling up behind a slow-moving truck on the highway; although all cars are moving forward, overdensities of cars are created in the vicinity of the truck). Within a certain distance of the center of the galaxy, called the corotational radius, matter overtakes the waves — further out, it is the waves that overtake the matter.

As gas clouds move into the increased density of a spiral arm, they are shocked and compressed. This effect causes the clouds to collapse and form masses of new stars. Star are born both big and small; the most massive stars shine the brightest, in blue colors, but also die quickly. On the other hand, the small stars are faint and reddish, and live long. This is the reason for the bright, bluish spiral arms: Once the stars move out of the arms, the blue one have died out, leaving only the fainter ones to occupy the space between the waves.

Blue spiral arms: When a gas cloud enters the density waves that make up a spiral arm, either by overtaking or being overtaken by the wave, it shock-compresses, fragments, and forms stars. Whereas the small stars eventually leave the wave, the massive stars — which are the ones that make the arms so prominently bright and blue — die before leaving the arms.

The theory explains how the density waves propagate in the galactic disk, and how they help form new stars, but it doesn't exactly explain how they're formed in the first place. Their origin isn't entirely clear, but can probably be caused by several different mechanisms, e.g gravitational instabilities or tidal forces from a neighboring galaxy.

Even after half a century, and despite many unclear and unexplained circumstances, the density wave theory is still one of the two prevailing theories on spiral structure. The Lin & Shu paper has more citations than all of Lindblad's collected works over 50 years, and for a good reason: The theory explains, detailed and quantitatively, many aspects of spiral galaxies, including the density in the arms, their width and lifetime, and where dust clouds are located. Nevertheless, their is more to it than this, and other theories have been proposed.

#### Self-propagating star formation

In particular one theory has gained wide acceptance, namely the "self-propagating (stochastic) star formation" (SPSF, Mueller & Arnett 1976; Gerola & Seiden 1978). This theory is completely different from the density wave theory:

When the most massive stars explode as supernovae, they send "detonation waves" through the interstellar gas in the galaxy. An explosion therefore causes nearby gas clouds to shock-collapse and form stars, and in this way the star formation spreads like "rings in the water" from the original explosion. However, due to the differential rotation of the galaxy's disk (i.e. the difference in angular speed at different distances from the center), the "rings" — which consist of young, bright stars as well gas clouds that are lit up by the UV light of the stars — are therefore quickly pulled into elongated structures.

Early computer simulation of self-propagating star formation. The three images show a galaxy 25, 200 and 500 million years after the start of the simulation. The spiral patterns created by the differential rotation are evident, although it is not easy to tell how many arms it has. Credit: Gerola & Seiden (1978).

The result is a so-called flocculent spiral galaxy. The word means "wool-like", referring to the fact that the disk consists of multiple small parts of spiral arms, together forming a fluffy spiral pattern. Individual spirally structures are quickly wound up, but new ones are continuously formed. In contrast to galaxies dominated by density waves, flocculent galaxies therefore typically do not have a well-defined number of spiral arms.

## A combination of multiple mechanisms

In other words, in the density wave theory spiral arms cause star formation, whereas in SPSF star formation causes spiral arms.

In some galaxies it is clear that one or the other theory is the explanation, while in other galaxies, for example the Milky Way, there may be a mixture of several mechanisms at play. The figure below shows the galaxies M51 (the one in which Lord Rosse found spiral structure) and NGC 4414 which are dominated by density waves and SPSF, respectively.

Credit: NASA, ESA, S. Beckwith (STScI) and the Hubble Heritage Team (STScI/AURA) (M51) and The Hubble Heritage Team (AURA/STScI/NASA) (NGC 4414).

• Asante sana (Thanks a lot). Would I be correct to say that, if I had $n$ stones tied to a string and if I spin them in a circle around me, if the string breaks individually (just imagine this were possible), stone by stone, then the stones would form a spiral pattern around me (obviously I'm the SMBH at the center of my stone galaxy). This is intriguing to say the least. What can't happen with $1$ stone (spirals) can happen with $> 1$ stones. It's an lllusion (maya) of sorts created by the forces at play (gravity?). Commented Nov 16, 2023 at 3:25
• @AgentSmith You're welcome! But no, in general the stones wouldn't form a spiral pattern. The moment a string snaps, the stone changes its circular motion to a linear motion, tangent to its former orbit, i.e. directly away. You could in principle create a temporary spiral pattern, but it would require a very specific string-snapping.
– pela
Commented Nov 16, 2023 at 11:42
• @AgentSmith I realize I sort of went out a tangent, and forgot about your initial question about the stone-on-a-string. As I wrote now in the intro, you're basically right about how spiral arms are not produced.
– pela
Commented Nov 16, 2023 at 14:03
• @AgentSmith Although the SMBH is big, it does not dominate the gravity of the galaxy. Eg, our SMBH, Sgr A* has a mass of ~4.3 million $M_\odot$ (solar masses), but the total star mass of the Milky Way is between 46 to 64 billion $M_\odot$, with gas + dust being around 6 billion $M_\odot$. But most of the mass is dark matter, with recent estimates (based on Gaia data) of 200 billion $M_\odot$. doi.org/10.1051/0004-6361/202347513 Commented Nov 16, 2023 at 15:05

# TLDR; Differential Rotation

## Part 1: Accretion disk

Now of course, I am not stating forth that spiral galaxies are the accretion disks of their supermassive black holes.

But their proto-galaxies were.

We have absolutely no idea as of now how SMBHs were formed (Certain theories posit primordial origins, or relentless consumption of mass etc). However, we have a clear picture of how supermassive black holes formed their galaxies.

It all starts as a small accretion disk around a massive black hole, that grows larger and larger in size.

But well, black holes are very sloppy eaters and tend to throw up mass in the form of astrophysical jets, and radiation. So, black holes only manage to pull in a very "small" fraction of the mass in the accretion disk into them, which causes the disk to grow bigger and bigger in size.

Eventually, there is a point where the accretion disk becomes way too unstable to actually remain a "disk" and simply splinters out, inducing regional collapse, forming stars and planets.

There is not a exact mass limit to how large such monstrous accretion disks can grow before collapsing, however, as far as we know, it seems to be in the ballpark of hundreds of millions of solar masses.

## Part 2: Dark Matter

However, it also pulls in dark matter as well, and this is where things start to take a turn. After reaching a critical limit, there is a transition from where the "accretion disk" is no longer gravitationally bound by the black hole, it's now bound by the collective mass of itself and a large fraction of dark matter. The black hole is just now absolutely dwarfed by the mass of the once- accretion disk, and is now just a puny contributor to binding it together.

BOOM! Your accretion disk has now transitioned into a proto-galaxy.

Previously, in the very early stages of galaxy formation, the galaxy "seed" was gravitationally bound by the SMBH. However, after this critical mass limit is exceeded, this is not the case. The SMBH now no longer plays a major role in keeping the galaxy together. In fact, most "developed" galaxies are bound by their collective mass of stars, gas, planets etc. $$^1$$

## Part 3: Differential Rotation

Many times, the growth stops and galactic evolution simply screws up, and you end up with a runty dwarf galaxy, or even worse, a irregular galaxy.

However, if you get much more massive, then at some point, there occurs a phenomenon where, the proto-galaxy gets so massive that the inner regions of the PG start to orbit the center faster than those in the outskirts of the galaxy. This phenomenon is known as differential rotation

In some galaxies like the Milky Way, a large central bulge can prevent the wave from reaching a resonance, which forms a large standing spiral wave with a uniform rotation rate. And the stars and gas in the central region orbit faster than those further out. This causes the gas to enter the waves at ridiculously high speeds, causing it to form a shockwave that forms thin dust lanes in each "arm" of the spiral.

And this may be the answer to your question, summed up in a nutshell: "Spin irregularly in

There are some galaxies that lack a SMBH, such as the Triangulum Galaxy (M33), which does imply that they did not need a SMBH to form in the first place. However, despite such rare occurences, we are sure of the fact that SMBHs did play a critical role in shaping galaxies during the early stages of galaxy formation

• If I understand your answer correctly, you say that black holes are needed as seeds to form a galaxy. This is not true. Rather it's the other way round, SMBHs form at the centers of galaxies where matter has lost enough angular momentum to condense.
– pela
Commented Nov 15, 2023 at 16:30
• You mention bars toward the end of this answer, and I agree that bars play a role in maintaining spiral arms. But apart from this, I don't think you really answer the question at all, and your part 1 and 2 don't seem to be really related (or correct).
– pela
Commented Nov 15, 2023 at 23:36