# Is the object mass distribution different for galaxies of different diameter?

Let's say we compare two elliptic/spiral galaxies with the bigger having a diameter 100 times the smaller.
Now we count every separated object inside them and classify them into mass categories e.g. $$10^i$$kg with $$i = -40..50$$.
(Finally we divided the number of each category by the total number of counted objects)

Q: Will the proportion of each mass category (in mean) be different for different galaxy diameter sizes?

E.g. have galaxies with a large diameter a higher proportion of more massive objects?

Or for example let's pick a random star inside the small and large galaxy. How likely will it be a red dwarf or a blue giant? Is the Probability different for a small/large galaxy?

Big galaxies might have bigger black holes in their center. Would it change if we just count the objects which are at least e.g. 10% of the diameter away from the center?

• I don't know, some elliptical galaxies are compact and have higher central densities, while others are diffuse and have lower central densities. The distribution of stars within can also vary, with concentrated central regions or uniform distributions. It depends on formation and merger history. I think they are as varied as spiral galaxies. Perhaps binary black holes also change the elliptical density and there can also be quasars. Aug 28, 2023 at 7:49

## What are the objects in a galaxy

First of all, we need to discuss what are these 'objects' composing the galaxy referred by the question. The total mass of a Galaxy will be divided between

• Dark matter
• Stars
• Gas & Dust
• Other non luminous objects such as planets, asteroids, black holes...

This list is roughly in order of mass contribution. Let's take the Milky Way as an example. According to the estimates reported on Wikipedia, the dark matter halo is the most massive component. The total mass of stars is between 1% and 10% of the dark matter mass. We don't know what dark matter is made of in the first place, so it is difficult to speculate on the mass distribution of the dark matter constituents. So, let's ignore dark matter, because the question seems to be focused on stars.

The second entry is stars, which are also the second most massive component. In fact the mass of gas and dust is between 10% and 15% of the one of the stars. For the last entry, I don't have a number, but I expect to be much smaller than that. Planets and asteroids are generally negligible, for instance more than 99% of the mass of the solar system is in the Sun alone.

Regarding black holes, they can be divided in two types, supermassive ones and stellar ones. Supermassive black holes are huge, but there are also very few of them, usually one per galaxy (or maybe two or three if there were recent mergers). The black hole at the center of the Milky Way, Sagittarius A*, is the only supermassive black hole known in our galaxy and its mass is 10000 times smaller than the stellar mass of the Galaxy. Stellar black holes are instead formed from the death of massive stars. They come from a subset of the stars and therefore their total mass must be smaller than the total mass of stars. Much smaller, because the massive stars that produce them are very rare.

## How does the mass of the object vary with the mass of the galaxy

Now that we have talked about how much each category of objects weights on the total mass contribution, let's talk about how the single object mass depend on the mass of the galaxy. I will focus on stars because it seems to be the focus of the question.

Stars form from the collapse of a cloud of gas. Perhaps counterintuitively, it turns out that the mass distribution of newborn stars almost the same everywhere, it doesn't depend on the mass of the cloud or any other parameter. It is called Initial Mass Function (IMF) and it roughly goes as $$f(m) \sim m^{-2.35}$$. This means that the larger the mass of the star, the less likely it is to find it in a population of newborn stars.

Does this mean that all galaxies have the same distribution of stellar mass? No, because the IMF is only valid for a bunch of newborn stars. If you wait some time, the more massive stars, that have a shorter life, will start to die out[^1]. Additionally, in a galaxy not all stars are born at the same time, but the star formation process can go on for a long time and may be continuing even today. So, when you look at a galaxy, you are seeing the sum of all the stars that were ever born, minus all the stars that have died. Massive stars have a short life, so you only see the ones that were formed recently. On the other hand, low mass stars live much longer and thus they keep accumulating in the galaxy, you see all the ones that where ever formed, they are all still alive.

If the star formation rate were constant in time, this would imply that older (and more massive galaxies) would tend to have a large population of low mass stars, due to the accumulation of a long time of star formation. So, in this case, "larger" (i.e. older/more massive) galaxies would have a larger proportion of low mass objects! This is in fact what usually happens with the large elliptical galaxies, they mostly contain small stars, with a mass like the Sun or smaller.

But it is not a universal rule. You could also have a very small galaxy (a dwarf galaxy) that has lost all the gas a long time ago, it has not been forming stars for a long time, and therefore it is mostly composed of old, low mass stars.

Or, you could have a large galaxy that has just undergone a major merger with another galaxy. The violence of the process has compressed the gas and triggered a new burst of star formation. In this case you would see all the newborn massive stars populating the galaxy.

In conclusion, the typical mass of a star does not really depend on the mass of the galaxy, but rather on the history of star formation.

Would it change if we just count the objects which are at least e.g. 10% of the diameter away from the center?

This also depends on the star formation history. In a spiral galaxy, like the Milky Way, the central part of the galaxy (called bulge) is composed of older stars, therefore typically less massive. We find the younger stars in the disk of the galaxy. It is different for large elliptical galaxies. For them there's not a clearly separated bulge (you may say that the whole galaxy is a bulge), so I think they should also show the same stellar mass distribution across the whole radius.

[^1] To give some numbers, a star like the Sun lives ~ 10 billion years, almost as much as the present universe, while a star 3 times the mass lives about 400 million years and a star with 10 times the mass of the Sun will only live 30 million years.

• A minor addendum: there's some evidence that really massive elliptical galaxies -- or perhaps just their central regions -- have what's called a "bottom-heavy" IMF, meaning that most of the stars formed with relatively more low-mass stars and relatively less high-mass stars. (This would amplify the "'larger' (i.e. older/more massive) galaxies would have a larger proportion of low mass objects!" effect...) Sep 2, 2023 at 13:51
• thank you for the detailed explanation. Good point with the older galaxies having more smaller stars. Sorry for the delay, thought there is no other response anymore. Oct 12, 2023 at 23:13

Via the Baryonic Tully-Fisher relation, for spiral galaxies, we find out that the Baryonic mass (one crucial problem with this is that it does not account for dark matter) is proportional to the stellar velocity dispersion. Now, the stellar velocity changes the diameter of the galaxy, just like Earth's equation, because of it's speed it is elongated the most, so technically the Baryonic mass is proportional to the diameter .

In case of elliptical galaxies, we just replace it with the Faber-Jackson relation.

So, if the diameter is proportional to mass, ideally, most of the mass is concentrated in stars, which become Supernova remnants once they explode, so if the mass is higher, I assume that the SNRs must be bigger, also much of the mass is in the form of stellar nurseries, leading to more matter for accretion and therefore more massive stars on HR diagram with a higher temperature. But one important thing to keep in mind is that it is mainly influenced by other factors

• If I understood it correctly it is saying a galaxy with a higher diameter has most likely a higher Baryonic mass. Good to know but my question was related to the mass distribution. For example let's pick a random star inside the small and large galaxy. How likely will it be a red dwarf or a blue giant? Is the Probability different for a small/large galaxy? Aug 28, 2023 at 14:23
• @J.Doe Perhaps, if the diameter is proportional to mass, ideally, the mass is concentrated in stars, which become Supernova remnants once they explode, so if the mass is higher, I assume that the SNR must be bigger, leading to more massive star on HR diagram. Aug 28, 2023 at 14:55
• Another way of increasing the mass would just a higher number of stars. What speaks against this? Is the total galaxy star count independent of the galaxy diameter? I would not expect that Aug 28, 2023 at 17:09
• "so if the mass is higher, I assume that the SNR must be bigger" -- that does not follow at all. In generally, the distribution of stellar masses when stars are formed (the Initial Mass Function) is independent of galaxy properties such as size. This means that the distribution of compact remnants from SNe (a "SNR" is the expanding shock wave in the interstellar medium) will also be independent of galaxy properties. Sep 2, 2023 at 13:44
• "we find out that the Baryonic mass (one crucial problem with this is that it does not account for dark matter) is proportional to the stellar velocity dispersion. " -- no, it's proportional to the rotational velcity; the verlocity dispersion is something different. Sep 2, 2023 at 13:45