Stars, Part 1: Fundamental Properties

Stars are fundamental to building fictional locations.  There may be an outstanding question about whether or not planets can form in locations other than protoplanetary disks surrounding stars, but it is obvious from experience and observation that planets do form around stars.

At any rate, if your primary goal is to build a planet that is habitable by human-like creatures, you'll need at least one star to put in its local neighborhood.
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This section is about the characteristics of stars, their fundamental properties and the relationships between them, and how they dominate their local neighborhoods.

Stellar Neighborhoods

Stars can exist all on their own perfectly happily.  Until the first confirmed discovery of planets beyond the Solar System in 1992, it seemed that perhaps most stars were loners.  Now, we're finding that planets are a pretty common commodity and their characteristics are much more varied than even our own Solar System hinted.
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Nevertheless, our own neighborhood provides a good idea of the kinds of things one can reasonably expect to find tagging along with a star as it orbits in its galactic home.  Planets we have already mentioned, and a whole section of this website is dedicated to them.  The Sun has the asteroid belt, the Kuiper belt, and the Oort cloud, Small Solar System Bodies, dwarf planets, and comets.  There's no reason not to suppose that other stars won't have at least one feature similar to these, and quite probably something similar to all of them.

Fundamental Properties of Stars

In determining the fundamental properties for a star, one can start just about anywhere.  If you know, for instance, that you want your star to be a spectral class K3.2, you can determine its temperature from that, then use the temperature to discover the mass, and use that to determine most everything else about the star—perhaps most importantly whether it falls in the mass range that makes it likely to possess habitable planets.  The equations below express the fundamental properties in terms of one another.  In most cases, I have expressed the operation on the known value as an exponent, but I have used a radical expression wherever doing so avoids a repeating decimal in the exponent.

For all the equations below, the following apply:

 

R = radius of the star in solar units;
M = mass of the star in solar units;
T = temperature in solar units;
V = lifetime in solar lifetimes;
L = luminosity of the star in solar units

 

Note that all of the equations below are approximations; in exactly the same sense as the BMI chart on your doctor’s wall is “generally” applicable, every star—just like every person—is unique and seldom fits any standardized scheme of measurement.  You are likely to find other approximations elsewhere.  What I have provided here isn’t necessarily always “accurate”—especially for spectral types other than G—but it is internally consistent.

​Note: "Lifetime" in the table above denotes total lifetime of the star; see later in this post for an equation to determine the amount of time a star spends on the Main Sequence.

Luminosity

Luminosity is less straightforward than the other properties.  However, if you know the radius and surface temperature of your star, you can use an equation based on Boltzmann's Law to calculate the luminosity:

Where:
     L = absolute luminosity of the star;
     LSUN = absolute luminosity of the Sun;
     R = absolute radius of the star;
     RSUN = absolute radius of the Sun;
     T = absolute temperature of the star;
     TSUN = absolute temperature of the Sun

​​If all values are in solar units, the equation simplifies to:

​... and the derivations are:

Below is a table of some well-known stars, with their measured masses, radii, temperatures (in terms of the temperature of the Sun), and measured luminosities.  Using the above equation, the luminosity is calculated in the next to the last column.  In the last column the calculated value is divided by the measured value, showing how close the calculated value comes to reflecting the measured value. 

Looking at the statistics of the final column:

          Minimum: 85.83%
          Maximum: 105.55%
          Median: 97.40%
​          Average: 96.94%
          Standard Deviation: 4.52%

The median and average values are very close to 100%, and the standard deviation of 4.52% means that 68% of the calculated values fall in the range [92.88%, 101.92%] of the actual measured values.

"What If I Don't Know the Radius and/or Temperature Of My Star"?

Well, at the risk of seeming dictatorial—you should.

I've done a lot of number-crunching in Excel™, and--as I describe in this later blog—calculating the luminosity using the Boltzmann equation is fairly accurate across most of the spectral classes and Luminosity Classes I-V.

Over the entire data set, the statistics were as follows:

          Minimum: 21.86%
          Maximum: 132.44%
          Median: 96.27%
          Average: 89.50%
          Standard Deviation: 21.43

Which means that 68% of the calculated values fall in the range [68.07%, 110.93%].

So, my recommendation is to start by selecting radius and temperature values and calculating luminosity and mass from those.

An Alternative System

If you don't want to have to select the temperature and radius of your star in order to calculate the luminosity, there is an alternative system (though it is not as accurate—more on that below).

For a luminosity result that is at least in the ballpark (especially for solar-analog stars), you can use the generalized formula:

Where:

    L = the luminosity of the star in solar units;
    k = a constant factor related to the mass regime of the star in question;
    M = the mass of the star in solar units;
    a = an exponent related to the mass regime of the star in question

The appropriate values for k and a are determined by the mass regime of the star in question, per the following table:

Use of this equation and table depend only on the mass of the star already being known.  Look up the regime into which the mass of the star falls, and insert the appropriate values for k and a into the equation to calculate a reasonably accurate value for the luminosity.  For the most Sun-like stars, the mass range is [0.6, 1.4] solar masses, so Row 2 is the one that should be used.

Mass-based vs Boltzmann-based Methods

Using 42 stars with known quantities for mass, radius, temperature, and luminosity, I compared the accuracy of the mass-based calculation of luminosity with the accuracy of the Boltzmann-based calculation, and found the latter to be more accurate.

In the graph below, ​L1 represents the mass-based calculation, and L2 represents the Boltzmann-based calculation.  The percentages were determined by dividing the values calculated for the luminosity by the measured value for each star.

​For the mass-based calculation the mean was 69.68% of the actual measured value, with a standard deviation of 46.85, so ±1σ covered values between 22.83% and 116.53%.  For the Boltzmann-based calculation, the mean was 118.11% of the measured value, with a standard deviation of 103.4, so ±1σ covered values between 14.71% and 221.51%.

The important things to note are that:

1. For the mass-based calculation, only 13% of the calculated values fell within 90%-110% of the measured value, whereas 53% of the Boltzmann-based calculated values met the same criterion.
2. For the mass-based calculation, 43% of the calculated values fell below 50% of the measured value, compared to only 6% of the Boltzmann-based calculated values.
3. For the mass-based calculation, 9% of the calculated values fell above 150% of the measured value, compared to 11% of the Boltzmann-based calculated values.

For main sequence stars, the numbers were somewhat stronger:​

In the graph below, again L1 represents the mass-based calculation, and L2 represents the Boltzmann-based calculation and, again, the percentages were determined by dividing the values calculated for the luminosity by the measured value for each star.

For the mass-based calculation the mean was 93.99% of the actual measured value, with a standard deviation of 38.93, so ±1σ covered values between 55.05% and 132.92%.  For the Boltzmann-based calculation, the mean was 98.69% of the measured value, with a standard deviation of 20.74, so ±1σ covered values between 77.95% and 119.44%.

So, here again, the important things to note are that:

1. For the mass-based calculation, only 23% of the calculated values fell within 90%-110% of the measured value, whereas 77% of the Boltzmann-based calculated values met the same criterion.
2. For the mass-based calculation, 14% of the calculated values fell below 50% of the measured value, compared to 0% of the Boltzmann-based calculated values.
3. For the mass-based calculation, 14% of the calculated values fell above 150% of the measured value, compared to 5% of the Boltzmann-based calculated values.

For the same 13 well-known stars analyzed at the start of this blog, comparing the luminosities calculated by the mass-based method vs the Boltzmann-based method, the graph below compares the results:

Clearly, it is best to use the Boltzmann-based method whenever possible, though for solar-analog stars, the mass-based calculation will produce reasonably accurate numbers.

Under the mass-based system, finding the mass based on the luminosity requires a different equation than that presented at the opening of this blog.  Instead of mass simply being the cube-root of the luminosity, use the equation:

… and the values for k and a now must be determined from the following table of luminosity regimes, calculated by running the mass values from the previous table through the equation:

​Mass Calculation Methods

There are two ways to calculate the mass of a star from known values.

Above, I list the equations for calculating the mass of star based on its luminosity.  Using these equations on the same 42 known stars mentioned above, and calculating the mass based on the measured luminosity yields the following statistics:

Over the entire data set:

          Minimum: 29.91%
          Maximum: 354.09%
          Median: 115.01%
          Average: 122.91%
          Standard Deviation: 53.81

Which means that 68% of the calculated values fall in the range [69.11%, 176.73%].

For the fifteen (15) solar-analog stars in the list:

          Minimum: 62.62%
          Maximum: 142.27%
          Median: 96.11%
          Average: 95.58%
          Standard Deviation: 20.88

Which means that 68% of the calculated values fall in the range [74.70%, 116.46%].

Using Boltzmann Luminosities

In the Boltzmann equation, the luminosity is calculated from the known radius and temperature of a star:

Substituting the quantity R² ⨉ T⁴ for L into the equations based on luminosity regimes yields:

Using these equations on the 42-star data set yields the following statistics:

​​Over the entire data set:

          Minimum: 31.20%
          Maximum: 327.37%
          Median: 104.33%
          Average: 113.61%
          Standard Deviation: 45.74

​Which means that 68% of the calculated values fall in the range [67.87%, 159.35%].

For the fifteen (15) solar-analog stars in the list:

          Minimum: 89.29%
          Maximum: 114.27%
          Median: 100.53%
          Average: 99.09%
          Standard Deviation: 9.69

Which means that 68% of the calculated values fall in the range [89.40%, 108.77%].

The following table shows the side-by-side comparison of the statistics for the entire data set:

... and for the 15 solar-analog stars:

To sum up the above, over the entire dataset, the Boltzmann-based mass calculation was between 90% and 110% of the actual measured mass value 45.24% of the time, compared to 19.05% of the time for the luminosity based calculation, even though the latter used the measured luminosity value.

In the 15-star solar analog set, the Boltzmann-based mass calculation was between 90% and 110% of the actual measured mass value 86.67% of the time, compared to 4.76% of the time for the luminosity based calculation, even though the latter used the measured luminosity value.

Conclusion

Thus, whenever possible, it is better to calculate the mass based on the known radius and temperature values using the Boltzmann-based mass calculation equations listed above.

​Main Sequence Lifetime

Above, I introduced equations for calculating the ​total lifetime of a star.

In general, a main-sequence star will spend about 90-95% of its total lifetime on the main-sequence, so a good rule-of-thumb is to take 92.5% of the total lifetime as the mains-sequence lifetime.

... so, for a star with a mass of 0.852 solar masses, its total lifetime can be expected to be something in the vicinity of:

... and its time on the main-sequence:

... or (taking the Sun's expected lifetime to be 10 billion years) about 13.4321 billion years.

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1. https://en.wikipedia.org/wiki/Aldebaran
2. www.space.com/22437-main-sequence-stars.html​​