Past Blog Tie-in:

• Stars, Part 1: Fundamental Properties

• Stars, Part 2: Spectral Classes and Spectral Types

Although my general focus for this blog was on solar-analog stars for the purposes of building human-habitable star systems, based on a question I received, I undertook an exploration of the methods of calculating luminosities for Luminosity Class I-IV stars.

As I said in the previous blog, Stars, Part 1: Fundamental Properties, 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:

As I said in the previous blog, Stars, Part 1: Fundamental Properties, 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:

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

*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 SunIf all values are in solar units, the equation simplifies to:

... and the derivations are:

Using 42 stars with known quantities for mass, radius, temperature, and luminosity (list available upon request), I calculated the accuracy of the Boltzmann-based calculation against known luminosity values, and found it to be fairly accurate across all spectral classes and luminosity classes.

Dividing the calculated luminosity by the measured luminosity returned the following statistics.

For solar-analog stars (Luminosity Class V), the numbers are quite good:

Minimum: 76.68%

Maximum: 107.66%

Median: 97.18%

Average: 95.49%

Standard Deviation: 6.59%

Which means that 64% of the calculated values fall in the range [86.92%, 102.41%].

For Luminosity Class I-IV stars, the numbers were less exciting:

Minimum: 21.86%

Maximum: 132.44%

Median: 94.44%

Average: 84.51%

Standard Deviation: 28.42%

So, 64% of the calculated values fall in the range [56.10%, 112.93%].

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 64% of the calculated values fall in the range [68.07%, 110.93%].

So, while the Boltzmann equation isn't as accurate for the larger main-sequence and giant-plus stars, it is quite close. Outside the realm of solar-analog stars, I'd recommend basing your fictional star on an existing star which is close to the kind you want to use in your fictional system.

Dividing the calculated luminosity by the measured luminosity returned the following statistics.

For solar-analog stars (Luminosity Class V), the numbers are quite good:

Minimum: 76.68%

Maximum: 107.66%

Median: 97.18%

Average: 95.49%

Standard Deviation: 6.59%

Which means that 64% of the calculated values fall in the range [86.92%, 102.41%].

For Luminosity Class I-IV stars, the numbers were less exciting:

Minimum: 21.86%

Maximum: 132.44%

Median: 94.44%

Average: 84.51%

Standard Deviation: 28.42%

So, 64% of the calculated values fall in the range [56.10%, 112.93%].

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 64% of the calculated values fall in the range [68.07%, 110.93%].

So, while the Boltzmann equation isn't as accurate for the larger main-sequence and giant-plus stars, it is quite close. Outside the realm of solar-analog stars, I'd recommend basing your fictional star on an existing star which is close to the kind you want to use in your fictional system.