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.

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

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

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 denotestotallifetime 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

*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:

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

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

*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:

*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%.

- 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.
- 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.
- 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.

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%.

- 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.
- 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.
- 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.

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:

## Mass Calculation Methods

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:

*R²*⨉

*T⁴*for

*L*into the equations based on luminosity regimes yields:

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:

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

*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.