How can the Hertzsprung–Russell diagram be used in star building?

As you can probably guess from the title, I have some questions about the Hertzsprung–Russell diagram, especially how to use it to make plausible stars.

Some questions:

• Can stars exist in the black areas? For example, take a star of 30,000 K and $10^2$ solar luminosities.
• How likely is a star to be a supergiant, main sequence star, etc?

HR-diagram. Source: Wikipedia

• I saw over in Worldbuilding Chat that HDE is planning on answering this, so you might want to wait for a day or two for him to do so--he's really good at doing in-depth stuff with this kind of astronomy stuff. Just a tip – FoxElemental Aug 1 '18 at 19:55
• Oh, and welcome to Worldbuilding! If you have a moment, please take the tour and visit the help center to learn more about the site. You may also find Worldbuilding Meta and The Sandbox (both of which require 5 rep to post on) useful. Here is a meta post on the culture and style of Worldbuilding.SE, just to help you understand our scope and methods. Hope you get good answers--and have fun! – FoxElemental Aug 1 '18 at 19:56
• This can be a good question for astronomy.stackexchange.com – Alexander Aug 1 '18 at 21:03
• @Alexander, it's a much better question to ask here as the answer will be useful to a great many worldbuilders (and is unlikely to be found by them on Astronomy). – JBH Aug 10 '18 at 4:24

To accurately answer your question, you might need to use a stellar evolution code, either doing your own modeling or looking up existing data tables. I'd recommend the MESA code for the former approach, and the Geneva grids for the latter (see Eggenberger et al. (2008) for details). Numerical simulations are excellent, and as the results are often available to the public, you can get some nice results with a bit of effort.

That said, simply using these doesn't tell you anything about the population as a whole - in other words, how the masses of stars are distributed. This can be computed easily using an initial mass function, or IMF. An IMF is something of the form $\xi(m)\Delta m$, which tells you how many stars were born with masses between $m$ and $m+\Delta m$.1 This is usually in the form of a power law, i.e. $$\xi(m)\Delta m=\xi_0m^{-a}\Delta m$$ where $m$ is in solar masses and $\xi_0$ is a constant. Essentially, if you want to find out the number of stars which have masses between $m_1$ and $m_2$, you integrate: $$N(m_1,m_2)=\xi_0\int_{m_1}^{m_2}m^{-a}dm$$ where $\xi_0$ is a normalization constant such that $N(m_1,m_2)=N$ for a sample of $N$ stars. A common IMF is the Kroupa IMF, represented as a broken power law - in other words, $a$ is different for different mass ranges. This is really the answer to your question, if you're talking about the likelihood of finding a star in a given mass range.

I've written some code for this answer (see on Github) that generates a few plots. The first one is the Kroupa IMF, for a sample where $N=100$:

Here is the cumulative distribution function, the number of stars of mass less than $m$:

However, stars are born and die at different rates. We can divide up their lifetimes into four stages:

1. Pre-main sequence evolution2
2. Main sequence evolution
3. Post-main sequence evolution
4. Stellar remnant

If you want to find the time it takes a star to pass through each phase, you can use some simple analytical approximations. We can estimate timescales for pre-main sequence evolution ($\tau_{\text{PMS}}$) and main sequence evolution ($\tau_{\text{MS}}$). These give the amount of time a star spends in a stage as a function of its initial mass: $$\tau_{\text{PMS}}=10^7\left(\frac{M}{M_{\odot}}\right) ^{-2.5}\text{ years}, \quad\tau_{\text{MS}}=10^{10}\left(\frac{M}{M_{\odot}}\right)^{-2.5}\text{ years}$$ The second exponent is often picked between $-2.5$ and $-3$; I've chosen $-2.5$ as a conservative estimate. Then, seeing that $\tau_{\text{PMS}}\ll\tau_{\text{MS}}$, we can see that at time $t$ stars of a given minimum mass won't have reached the main sequence yet, and stars of a given maximum mass will have already evolved off of it. Therefore, the total number of main sequence stars at a time $t$ is $$N_{\text{MS}}(t)=\xi_0\int_{m_\text{min}}^{m_\text{max}}m^{-a}dm$$ where $$m_{\text{min}}(t)=\left(\frac{t}{10^7\text{ yrs}}\right)^{-2/5},\quad m_{\text{max}}(t)=\left(\frac{t}{10^{10}\text{ yrs}}\right)^{-2/5}$$ I've used these to create plots of the number of stars on the pre-main sequence, main sequence, and post-main sequence tracks over 100 billion years. Notice that even after 100 billion years, many of the stars still have not left the main sequence; they're low-mass red dwarfs.

The answer to your second question - can stars be found in the black regions - is, essentially, a yes. All of the H-R diagrams here show the two large areas above the main sequence populated by giants and supergiants, but stars evolve back and forth along that area. For instance, a star may change from being a red supergiant to a blue supergiant (or vice versa), often including periods of violent activity. The evolved stars above the main sequence fall in a number of places.

You're not likely to see quite as many stars below the main sequence - the notable exception being white dwarfs - but some stars, called subdwarfs, do lie in that area. However, they're usually closer to the main sequence than to the white dwarfs. I don't think any are clearly visible on the H-R diagrams above, but they do exist. Kapteyn's star is a good example of a fairly cool subdwarf.

If you want to get a better idea of which gaps are filled and which ones aren't, you can try looking at a star catalogue. Gaia recently released a slew of new data this spring, and you should be able to determine temperature and luminosity for many stars. Assuming you can query that (or another) database, perhaps you can find some stars in unusual places.

1 Bear in mind that a star may lose mass during the course of its lifetime, often through stellar winds (or, in a select few cases, violent eruptions).
2 You've heard of a protostar; a protostar is a pre-main sequence star that is still enshrouded by material from the molecular cloud that formed it, and is likely not visible.

• Great answer, but allow me to ask some additional questions to fully understand your answer. |IMF: what is an initial mass, Is it just the mass a star had when it was 'born'? What is $\xi_{0}$? |Just to confirm $\tau_{PMS}$ is a function given an (initial?) mass which gives the time a star spends in the pms-fase. similarly for $\tau_{MS}$ (also could you give sources for the formula of $\tau_{PMS}$ & $\tau_{MS}$) |Other: How does mass change during the life time (no full answer expected, more looking for 'google terms')? What is a PMS star & the difference between a protostar? – Sam Coutteau Aug 2 '18 at 10:35
• @SamCoutteau I've edited the answer - those are some good questions. I'd linked to some notes deriving those timescales but messed up the formatting; it's clickable now, and a good read if you have the time. – HDE 226868 Aug 2 '18 at 13:48
• @SamCoutteau Just to make sure it is clear: as HDE said, stars can change over time... and when they do they can literally move around on the diagram. So a star is not fixed in one spot for all time. Further, that means that some of the black areas on the picture you provide will have stars moving through them from one point on the diagram to another point as their properties change over time. Some stars will even move around a lot, going in some general direction but swaying back and forth, making a snake-like pattern that covers a relatively large portion of the diagram through its life. – Loduwijk Aug 2 '18 at 18:36
• For an example of something moving around the diagram, I'm not finding any great images easily on google image search, but this is the best I found so far (do a search on the page for the section titled "stellar evolution", it's the picture in that section a little over halfway down the page) and note that some move around even more than that cefns.nau.edu/geology/naml/Meteorite/Book-GlossaryS.html – Loduwijk Aug 2 '18 at 18:43
• @Aaron Yes, I'm well aware. My ultimate goal is to make a usable model for star building (on a galactic scale). Hence my interest in probabilities and distributions. However as I never studied anything related to astronomy, I'm having a hard time understanding let alone making models of everything I read. – Sam Coutteau Aug 2 '18 at 18:49

I've grabbed an observational Hertzsprung–Russell diagram from the wikipedia article about the same.

In this case it's a plot of 22,000 stars and you can see that there's a lot more flexibility in where the stars lie, in practice a not insignificant number lie in some of the "black bits" of the simplified diagram you've posted. However there are still clear black areas.

1. Yes, stars can exist in the "black" area of your diagram. Here is a more detailed diagram, and this only plots 22 000 stars. There are more than a million times more stars just in the Milky Way galaxy.

1. That depends entirely on what region of space you are looking at.

And why is this even important? Are you going to be quoting statistics at your readers? You will make them fall asleep, just sayin'...