Equations for developing a hypothetical Solar System I am currently going down the rabbit hole of writing a story but I would like it to be set in a universe which is believable. Therefore I am trying to create a hypothetical solar system in which the world where the story takes place exists. However, as I have little to no background in astrophysics I am struggling to understand the basics.
After some research I cam across this video on how to create a star which is believable:
https://www.youtube.com/watch?v=x55nxxaWXAM&list=PLduA6tsl3gygXJbq_iQ_5h2yri4WL6zsS&index=4
The maths in this video however are not explained and when applied to existing stars...simply don't work.
It begins by determining the mass of the star as anywhere between 0.6 - 1.4 solar masses -that I understand. I further understand from the video that the star would need to be of F or G classification, thus have a temperature between 5000 - 7000 K approx.
However it then goes on to calculate luminosity as $\text{Mass}^3$ (it says $4$ but it was corrected later). From further research this doesn't make any sense as luminosity would be determined by the mass-luminosity equation of $L = R^2 \cdot T^4$ where $R$ is the radius and $T$ is the temperature (though this equation seems to require the Stefan-Boltmann Constant - and sometimes not...). The video further calculates the temperature as $\text{Mass}^{0.505\ldots}$. There are more equations though the figures already break down at this point. Plus from running my own calculations I am struggling to see any meaningful relationship between a star's mass and its temperature other than bigger stars generally appear to be hotter than smaller stars.
So my question to you much more learned than I in this topic is basically:
Is there a (relatively) straight forward method of creating a hypothetical star for my solar system that is mathematically and physically feasible?
Any more info on this matter would be greatly appreciated.
Many thanks in advance to anyone would can provide some clarity"
Shane.
 A: OK, so I am not an astrophysicist, or anywhere near one, but the following is cobbled together from memory and some looking things up on Wikipedia (links below).
First of all if you naively assume that a star is (a) spherical and (b) a black body then you have this important relation:
$$L = 4\pi R^2\sigma T^4\tag{1}$$
Where $R$ is radius, and $T$ is temperature, $\sigma$ is the Stefan-Boltzmann constant and $L$ is the total luminosity (power output).  The $4\pi R^2$ is the formula for the surface area of a sphere, of course.
Secondly there is a horrible thing called the mass-luminosity relation, which is an observed relation for main-sequence stars.  It can (I am sure) be derived from models of how stars work, but here it is in a form which is suitable for 'engineering use':
$$\tag{2}
\frac{L}{L_\odot} \approx \begin{cases}
  0.23\left(\frac{M}{M_\odot}\right)^{2.3}&M < 0.43M_\odot\\
  \left(\frac{M}{M_\odot}\right)^{4}&0.43M_\odot \le M < 2M_\odot\\
  1.4\left(\frac{M}{M_\odot}\right)^{3.5}&2M_\odot \le M < 55M_\odot\\
  32000\frac{M}{M_\odot}&M \ge 55M_\odot
\end{cases}
$$
Where $L$ is total luminosity, $M$ is mass, and $L_\odot, M_\odot$ are the luminosity and mass of the Sun respectively.
This comes directly from this Wikipedia page.
Finally there is another 'good enough for engineering' relation for main-sequence stars:
$$R \propto M^{0.78}\tag{3}$$
So, the first things to start from are knowing $M_\odot$, $L_\odot$ and $R_\odot$, which you can look up.  Given these you can work out the proportionality constant in (3) easily enough.
Then given the mass of the star you can use (2) to work out its luminosity.  And finally you can use the $R$ you get from (3) together with (1) to work out its surface temperature.
There is another important thing to bear in mind: the spectrum of the light emitted by a black body depends on the surface temperature, and Wein's displacement law says that the wavelength for the peak intensity is
$$\lambda_p = \frac{b}{T}$$
where $b \approx 2.9\times 10^{-3}\,\mathrm{m\,K}$ (see Wikipedia again).
If you are planning on carbon-based biological life in your imagined stellar system, you don't want the wavelength of the light emitted by the star to be too short, and that means you don't want the temperature too high.  If the temperature is too high then you'll get a lot of ionizing radiation and this will pull complex organic molecules to bits.  So this places some kind of upper bound on $T$ (picking something similar to the Sun's temperature is probably a safe idea).

So the place to start is probably with the temperature of the star,and the assumption that it is main-sequence.  If you want carbon-based life you can assume that $T$ in some suitable range: you don't want it too low because you want some UV, you don't want it too high because you don't want too much UV or x-rays or whatever.  And stars spend most of their time on the main sequence, so that's where you have time for life to evolve.
If you additionally assume that $0.43M_\odot \le M < 2M_\odot$ (just to make using (2) easier), then, from (1):
$$\frac{L}{R^2} = 4\pi\sigma T^4$$
And then we can use the relation from (2): with the assumption that $0.43M_\odot \le M < 2M_\odot$, we have that $L \approx (M/M_\odot)^4 L_\odot$, and we can substitute this expression for $L$ into the previous equation to get this:
$$\frac{M^4 L_\odot}{M_\odot^4 R^2} \approx 4\pi\sigma T^4$$
or, moving constant terms to the RHS:
$$\frac{M^4}{R^2} \approx \frac{4\pi\sigma M_\odot^4}{L_\odot} T^4$$
Now, writing (3) as $R \approx K M^{0.78}$, where we need to work out $K$ by looking at the Sun later, we have $R^2 \approx K^2 M^{1.56}$, so
$$\frac{M^{2.44}}{K^2} \approx \frac{4\pi\sigma M_\odot^4}{L_\odot} T^4$$
And finally this gives us an expression for $M$ in terms of $T$:
$$M \approx \left(\frac{4\pi\sigma M_\odot^4 K^2}{L_\odot} T^4\right)^{0.41}$$
So this is an expression for $M$ in terms of $T$.  So, given $T$ you can work out $M$, and then given $M$ you can work out $R$ and thus $L$.
There may be mistaked in the above maths, and the expression is clearly dimensionally a bit mad, probably because it's an approximation to something much more complicated.

Finally some more pointers (all of this is to Wikipedia, sorry):

*

*the above relations hold for main-sequence stars, which probably are the stars you want to consider;

*the Hertzprung-Russell diagram is an important thing to know about;

*the page on stellar evolution may be a good place to start;

*the page on stellar classification is also interesting.

Obviously there are both better and more in-depth places to find this information, but Wikipedia is not hopeless in my experience.
