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.

This section is about the characteristics of stars, their fundamental properties and the relationships between them, and how they dominate their local neighborhoods.

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

Stars can exist all on their own perfectly happily. Until the first confirmed discoveries of planets beyond the Solar System, 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.

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.

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 type 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:

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.*

*Notes:*

1. "Lifetime" in the table above denotestotal lifetime

1. "Lifetime" in the table above denotes

*of the star; see later in this post for an equation to determine the amount of time a star spends on the Main Sequence.*

2. The equations in the "Luminosity" column provide a reasonable approximation for Main Sequence, Sun-like stars (I will discuss Sun-like—or Solar Analog—stars in a later blog).

2. The equations in the "Luminosity" column provide a reasonable approximation for Main Sequence, Sun-like stars (I will discuss Sun-like—or Solar Analog—stars in a later blog).

## Luminosity

Luminosity is less straightforward than the other properties; the approximations have been somewhat improved upon through observation, at least for Main Sequence stars. The equation immediately below provides a reasonable approximation of stellar luminosity, based primarily on a mass-luminosity relationship. Again, this is most effective when describing members of the Main Sequence (luminosity class V); for stars of other luminosity classes, it is frankly best to find a known star which closely approximates your fictional star, and borrow its characteristics.

The following generalized mass-luminosity relationship may be used instead to determine luminosity from mass:

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 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

For instance, assume a star of 4.0 solar masses; what would its luminosity be? A mass of 4.0 solar masses falls into the third row of the table, which gives a value for the constant of

*k*and*a*into the equation to calculate a fairly 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.For instance, assume a star of 4.0 solar masses; what would its luminosity be? A mass of 4.0 solar masses falls into the third row of the table, which gives a value for the constant of

*k**= 1.5*and for the exponent of*a = 3.5*, so the appropriate equation becomes:Inserting the mass of the star,

*M = 4.0*, and carrying out the calculation:… returns a luminosity for this star of 192 times that of the Sun. This is 300% more than the value returned by the earlier approximation of

What if you have already determined what luminosity you need your star to have, and now wish to use this equation to calculate the mass? The basic equation becomes:

*L = M³*, which returns a value of*L = 64*times the luminosity of the Sun (more on this below).What if you have already determined what luminosity you need your star to have, and now wish to use this equation to calculate the mass? The basic equation becomes:

… 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:Notice that the upper-end values for luminosity in the third and fourth lines of the table are ludicrously large; the likelihood of a fictional star system with a star of these luminosities is remote in the extreme, but the values are provided for the sake of completeness.

## Comparison of Equations for Sun-Like Stars

Below is a graph of the luminosity for solar-analog stars, comparing the returned values as calculated using the L = M³ generalized equation and the L = M⁴ mass-specific method:

As can be seen, for masses between [0.6, 1.0] solar masses, the difference is slight, but the luminosity returned by L = M³ is higher than that returned by L = M⁴; whereas, for masses between [1.0, 1.4] solar masses, the L = M⁴ relation returns higher luminosities, and the difference between the values returned from the two relations grows progressively greater as the mass increases.

For masses between [>1.4, <2.0] solar masses, the gap increases rapidly:

For masses between [>1.4, <2.0] solar masses, the gap increases rapidly:

For the third mass regime of [2.0, <20.0] solar masses, the discrepancy begins at 212% (not discernable at the scale of the graph) for the mass-specific relation L = 1.5 × M³ᣟ⁵, and grows to 671% at

*M = 20.0*:## Examples

As stated above, the reality is that when applied to real stars, most of these equations do not return real-world values much of the time. For instance, the known mass of Sigma Draconis is 0.87M⨀; using R = 0.87⁰ᣟ⁹, we arrive at a radius of 0.882R⨀, which is 113.4% of the actual measured value of 0.778R⨀.

Conversely, the known mass of 55 Cancri is 0.95M⨀; the equation R = 0.95⁰ᣟ⁹ returns 0.995R⨀, which is 82.9% of the actual measured value of 1.152R⨀.

Aldebaran (α Tauri) has a measured mass of 1.5 times that of the Sun, but a radius of 44 times the Sun’s. This mass puts Aldebaran in the second-row mass regime, so we can calculate its luminosity thus:

Conversely, the known mass of 55 Cancri is 0.95M⨀; the equation R = 0.95⁰ᣟ⁹ returns 0.995R⨀, which is 82.9% of the actual measured value of 1.152R⨀.

Aldebaran (α Tauri) has a measured mass of 1.5 times that of the Sun, but a radius of 44 times the Sun’s. This mass puts Aldebaran in the second-row mass regime, so we can calculate its luminosity thus:

… yet, Aldebaran’s measured luminosity is 518±32 [1] that of the Sun. What is the problem, here?

The problem is that Aldebaran isn’t a Main Sequence star; it is a red giant (luminosity class III), and the above relations are only remotely accurate for Main Sequence (luminosity class V) stars.

Comparatively, Bellatrix (γ Orionis) has a measured mass of 8.6 times that of the Sun, and is 5.75 times as large. This mass puts it in the third row of mass regimes, so we’d calculate its luminosity as:

The problem is that Aldebaran isn’t a Main Sequence star; it is a red giant (luminosity class III), and the above relations are only remotely accurate for Main Sequence (luminosity class V) stars.

Comparatively, Bellatrix (γ Orionis) has a measured mass of 8.6 times that of the Sun, and is 5.75 times as large. This mass puts it in the third row of mass regimes, so we’d calculate its luminosity as:

… which is only about 30% of Bellatrix’s measured luminosity of 9211 times that of the Sun. Again, Bellatrix is a luminosity class III star, so the mass-luminosity relation doesn't return real-world values.

The star Vega (α Lyrae) has a measured mass of ~2.135 the Sun’s, and a measured luminosity of 40.12 the Sun’s. This mass puts it (just barely) in the third-row mass regime, so we’d calculate Vega’s luminosity as:

The star Vega (α Lyrae) has a measured mass of ~2.135 the Sun’s, and a measured luminosity of 40.12 the Sun’s. This mass puts it (just barely) in the third-row mass regime, so we’d calculate Vega’s luminosity as:

… which is about 53% of it’s measured value, yet Vega is most definitely a Main Sequence star. Indeed, it serves as the “standard candle” for stellar apparent magnitudes.

Lastly, the measured mass of Alpha Centauri A (α Centauri A) is 1.1 times the Sun’s, and its measured luminosity is 1.519 solar luminosities. Being in the second-row mass regime, it’s luminosity would be calculated as:

Lastly, the measured mass of Alpha Centauri A (α Centauri A) is 1.1 times the Sun’s, and its measured luminosity is 1.519 solar luminosities. Being in the second-row mass regime, it’s luminosity would be calculated as:

… 96.4% of the measured value, close enough for fiction.

Clearly, the mass-luminosity relations above are most accurate for stars which are most similar to the Sun, and vary with increasing breadth from observation the further from Sun-like the star is.

Below is a table of some well-known stars, with their measured masses and luminosities, compared to that calculated using the appropriate mass regime relation:

Clearly, the mass-luminosity relations above are most accurate for stars which are most similar to the Sun, and vary with increasing breadth from observation the further from Sun-like the star is.

Below is a table of some well-known stars, with their measured masses and luminosities, compared to that calculated using the appropriate mass regime relation:

As can be seen by the examples of Wolf 359 and Betelgeuse, the less Sun-like a star is, the less the mass-luminosity relations return real-world values.

The best course is probably this: If basing a star system on a star for which the vital statistics are known, use those. If creating a star system from scratch, either 1) use the vital statistics for a known star, or 2) select a desired mass (or radius or luminosity, etc.), use the equations above, and be prepared to make frequent references to the

The best course is probably this: If basing a star system on a star for which the vital statistics are known, use those. If creating a star system from scratch, either 1) use the vital statistics for a known star, or 2) select a desired mass (or radius or luminosity, etc.), use the equations above, and be prepared to make frequent references to the

*rationale ultimum*, or the Omega Argument: "It is so because I say it is so."## 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.

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.