I can't seem to find correct starbuilding equations that work together anywhere and am generally confused. Which ones do you use?

In reality, if you want to build a star, you need to specify a mass and a chemical composition, and then use the equations of stellar structure. This requires some numerical integration, and it's far from simple. People have making careers out of it for generations. (One upshot of this, of course, is that there are plenty of existing stellar models out there, and you can essentially pick a star from a set of grids and find out all of its properties without having to do any calculations yourself!)

What we can do is make some analytical approximations that are valid in some specialized cases. The ones we'll use are valid for stars on the main sequence, where they'll spend the bulk of their lives. They also don't (for the most part) take into account the star's composition. These results depend solely on the mass of the star, which is arguably the single most important parameter you have to consider.

Luminosity

By making some assumptions about energy transport, we can determine that the luminosity should scale with mass approximately like $$\boxed{L\propto M^3}$$ Mass-luminosity relations are important topics of research which are actually take different forms depending on the mass of the star. The simplest are piecewise, of the form $L\propto M^{\alpha_i}$, with different $\alpha_i$ used in different mass ranges. $\alpha=3.5$ is usually a good rule of thumb for Sun-like stars, but let's work with $\alpha=3$ for now.

Radius

Using the same assumptions, we can deduce that $$\boxed{R\propto M^{(\nu-1)/(\nu+3)}}$$ where $\nu$ is a number that depends on the process by which the star produces energy. For the proton-proton chain reaction, used in stars of $M<1.3M_{\odot}$, we have $\nu=4$. For the CNO cycle, used in stars of $M>1.3M_{\odot}$, we have $\nu=20$. This gives us two different relations: $R\propto M^{3/7}$ and $R\propto M^{19/23}$.

Surface temperature

Stars are, approximately, black bodies. This means that their luminosity, radii, and surface temperatures ($T_{eff}$) are connected via the Stefan-Boltzmann law: $$L=4\pi R^2\sigma T_{eff}^4$$ We can rearrange this to get $$T_{eff}\propto\left(\frac{L}{R^2}\right)^{1/4}\implies \boxed{T_{eff}\propto\frac{M^{3/4}}{M^{(\nu-1)/2(\nu-3)}}}$$ For low mass stars, we get $T_{eff}\propto M^{15/28}\approx M^{1/2}$; for high-mass stars, we get $T_{eff}\propto M^{31/92}\approx M^{1/3}$.

Main sequence lifetime

The rate at which a star loses mass is proportional to its luminosity. We can then make a very rough guess at its main sequence lifetime by saying that $\dot{M}\propto L\propto M^3$. Integrating that differential equation gives us $$\boxed{\tau\propto M^{-2}}$$ or, if you used $\alpha=3.5$, $\tau\propto M^{-2.5}$, which is the relationship you see tossed around a lot.

Habitable zone

We can get some very, very basic bounds on the classical habitable zone by considering the temperatures at which water can exist in liquid form. This criterion is sometimes disputed, but it's what we've got to work with. Using the effective temperature of a planet - more black body models - we can see that the inner and outer boundaries are given by $R_h\propto L^{1/2}$ or $$\boxed{R_h\propto M^{3/2}}$$

Miscellaneous notes

• These mass scaling relations are applicable only on the main sequence. They won't tell you anything about post-main sequence evolution, which is arguably much harder - if not impossible - to reduce to analytical approximations. I think most folks don't bother, for worldbuilding purposes, with post-main sequence stars anyway.
• They neglect radiation pressure and convection and make some unrealistic assumptions about constant opacity. For some stars, we can ignore radiation pressure; for others, we can ignore convection. This is one reason why these scaling relations are best suited for Sun-like stars, rather than a broad range of masses.
• These are, I would say, order-of-magnitude results. In astronomy, I'm usually happy to get something right within a factor of 10, and I won't quibble with a factor of 2 or 3. To back that up: stars can differ by about a factor of 100 in mass, a factor of maybe 20 in temperature, and a factor of . . . well, quite a few orders of magnitude in luminosity.

Finally, a note on numerical grids: Your question ended with "Which [equations] do you use?" My personal answer is that I usually don't run the numbers myself; I find tables of stellar models and pick and choose the ones I want. Astronomers have already gone to the trouble of doing the detailed (and much more accurate) computations, and if the results are out there, hey, I might as well grab some.

Some quick Googling should turn up some helpful results. For a lot of answers on Worldbuilding, I've grabbed numbers from a set of main sequence models by Eric Mamajek. They're finely spaced and contain some interesting quantities (e.g. color indices) that might be useful in niche situations. But there are really plenty of other grids out there. The Geneva grids are excellent if I'm not feeling too lazy to sift through them.

posted almost 4 years ago

HDE 226868‭

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