This step is the core of the new design sequence for stars and stellar objects. Most world-design sequences found in tabletop games take no account of that fact that stars evolve and change over time – two stars of the same mass and composition can look very different if they are of different ages. Applying these changes to a randomly generated population of stars will provide increased plausibility, and reflect the greater variety of stars found in the real universe. This step is a trifle complex – it has four separate cases, and requires a bit more math.
Step Six: Stellar Evolution
At this point, we have determined the number of stars in the system, the initial mass of each, and the age and metallicity of the overall system. This step determines how the star system has evolved since its formation, and sets the current effective temperature, luminosity, and physical size of each star.
While on the main sequence, a star will evolve and change rather slowly, due to the “burning” of hydrogen fuel and the slow accumulation of helium “ashes” in the stellar interior. Stars normally grow brighter over time, changing in effective temperature and size as well. Stars which reach the end of their stable main-sequence lifespan go through a subgiant phase, then evolve more as a red giant, eventually losing much of their mass and settling down as a small stellar remnant called a white dwarf.
This step should be performed for each star in a multiple star system.
Selecting for an Earthlike world: For there to be an inhabitable world in the star system, at least one star must be on the main sequence or a subgiant. If the mass and age of the primary star were selected to allow for an Earthlike world, that star is likely to fit this criterion as well.
To begin, for any object of 0.08 solar masses or more, refer to the Master Stellar Characteristics Table on the following two pages. Record the Base Effective Temperature (in kelvins) for each star. Record the Initial Luminosity (in solar units) and the Main Sequence Lifespan (in billions of years) as well.
If any star has a mass somewhere between two of the specific entries on the Master Stellar Characteristics Table, use linear interpolation to get plausible values for the star’s base effective temperature, initial luminosity, and main sequence lifespan.
|Master Stellar Characteristics Table
||Base Effective Temperature
||Main Sequence Lifespan
Once you have determined the Base Effective Temperature, Initial Luminosity, and Main Sequence Lifespan for each star in the system being designed, examine the following four cases for each star:
- The first case applies to any brown dwarf with mass less than 0.08 solar masses.
- The second case applies to any star with mass between 0.08 and 2.00 solar masses, if the system’s age is less than the star’s Main Sequence Lifespan. Such a star is considered a main sequence star.
- The third case applies to any star with mass between 0.08 and 2.00 solar masses, if the system’s age exceeds the star’s Main Sequence Lifespan by no more than 15%. Such a star will be a subgiant or red giant star.
- The fourth case applies to any star with mass between 0.08 and 2.00 solar masses, if the system’s age exceeds the star’s Main Sequence Lifespan by more than 15%. Such a star will be a white dwarf.
Apply the guidelines under the appropriate case to determine the current effective temperature, luminosity, and radius for each star.
First Case: Brown Dwarfs
A “star” with less than 0.08 solar masses will be a brown dwarf. Such an object accumulates considerable heat during its process of formation. A very massive brown dwarf may also sustain nuclear reactions in its core (deuterium or lithium burning) for a brief period after its formation, giving rise to additional heat. This heat then escapes to space over billions of years, causing the brown dwarf to radiate infrared radiation, and possibly even a small amount of visible light.
A very young and massive brown dwarf may be hard to distinguish from a small red dwarf star. Older objects will fade through deep red and violet colors, eventually ceasing to radiate visible light at all. A “dark” brown dwarf will eventually resemble a massive gas-giant planet like Jupiter. Ironically, even a very massive brown dwarf will still have about the same physical size as Jupiter, making the resemblance even stronger.
To estimate the current effective temperature of a brown dwarf, let M be the object’s mass in solar masses, and let A be the object’s age in billions of years. Then:
Here, T is the brown dwarf’s current effective temperature in kelvins. Effective temperature for a brown dwarf can be no higher than 3000 K.
The radius of a brown dwarf will be about 70,000 kilometers, or about 0.00047 AU.
A brown dwarf’s luminosity will be negligible. To estimate its luminosity, let T be its current effective temperature in kelvins. Then:
Here, L is the brown dwarf’s luminosity in solar units.
Second Case: Main Sequence Stars
Objects with 0.08 solar masses or more will be stars. For each star, first check to see if the star system’s age is less than or equal to the star’s Main Sequence Lifespan, as determined from the Master Stellar Characteristics Table. If so, then the star is still on the main sequence and this case applies.
A main sequence star will have an effective temperature reasonably close to the Base Effective Temperature from the table for a star of that mass. The exact effective temperature will depend on the star’s exact composition, and other factors that are beyond the scope of these guidelines. Feel free to select a current effective temperature within up to 5% of the value on the table for a star of that mass.
Low-mass stars grow hotter extremely slowly, and can be considered to have the same temperature as when they formed no matter how old they are. Select an effective temperature for them without any concern for their age.
Intermediate-mass and high-mass stars change in temperature more noticeably during their main-sequence lifespan. In general, intermediate-mass or high-mass stars will tend to begin life with an effective temperature as much as 3% or 4% below the value on the table, but will reach a peak of about 2% to 3% above that value by about two-thirds of the way through their main sequence lifespan. After that, they will tend to grow cooler again, falling back to the value on the table or even slightly lower by the end of their stable lifespan. Select an effective temperature for such stars accordingly.
Round effective temperature off to three significant figures.
Main sequence stars also grow brighter over time, as temperatures in their core rise and they are forced to radiate more heat. To estimate the current luminosity for a given main-sequence star, use:
Here, L is the current luminosity for the star in solar units, L0 is the Initial Luminosity for the star from the Master Stellar Characteristics Table, A is the star system’s age in billions of years, and S is the star’s Main Sequence Lifespan from the table. Feel free to select a final value for the star’s luminosity that is within 5% of the computed value. Round luminosity off to three significant figures.
Note that very low-mass stars have main-sequence lifespans that are far longer than the current age of the universe. Such stars have simply not had enough time to grow significantly brighter since they first formed! You may choose to simply take the Initial Luminosity for such stars without modifying it with the above computation.
Once the effective temperature and luminosity of a star have been determined, its radius can be computed. If T is a star’s effective temperature in kelvins, and L is its luminosity in solar units, then:
The result R is the star’s radius in AU (multiply by 150 million to get the radius in kilometers). Its diameter will be exactly twice this value. Most main-sequence stars will have radii of a small fraction of one AU.
Third Case: Subgiant and Red Giant Stars
If a given star’s age is greater than its Main Sequence Lifespan, but exceeds that value by less than 15%, then it has evolved off the main sequence and is approaching the end of its life. Such a star first evolves through a subgiant phase, during which it loses little of its brightness but grows slowly larger and cooler. At some point its core becomes degenerate, shrinking and increasing dramatically in temperature. This sets off a new form of hydrogen fusion in a shell around the core, releasing considerably more energy and causing the star’s outer layers to balloon dramatically outward. The star becomes much brighter, cooler, and larger, becoming a red giant.
Stars in this stage of their development are often somewhat unstable, and the precise path a star will follow is highly dependent on its mass, composition, and other factors. Rather than attempting to trace the star’s evolution precisely through time, we suggest simply selecting one of the three options described below. To select an option at random, roll d% on the Post-Main Sequence Table.
|Post-Main Sequence Table
||Red Giant Branch
Subgiant stars: During this period, the star remains at about the same luminosity it had at the end of its main-sequence lifespan. Select a luminosity for the star between 2.0 and 2.4 times its Initial Luminosity from the Master Stellar Characteristics Table. The star will also cool to an effective temperature of about 5000 K. Select an effective temperature for the star somewhere between that value and the Base Effective Temperature from the table.
Red giant branch stars: At the end of the subgiant phase, a star is at the “foot” of a structure on the H-R diagram called the red giant branch. From this point, it will grow still cooler, but considerably brighter as well, swelling up to become many times its main-sequence size. The characteristics of stars at the “tip” of the red-giant branch are almost independent of the star’s mass. For stars of moderate metallicity, this implies a luminosity of about 2000 to 2500 solar units, and an effective temperature of about 3000 K.
To select an effective temperature and luminosity for a red giant branch star, select a value between 0 and 1, or roll d% to generate a random value between 0 and 1. If R is the selected value, T is the star’s current effective temperature, and L is its current luminosity, then:
Round effective temperature and luminosity to three significant figures, and feel free to select a value for each that is within 5% of the computed value.
Horizontal branch stars: Upon reaching the tip of the red giant branch, a star of moderate mass undergoes a phenomenon called helium flash. Temperatures and pressures at the star’s degenerate core have risen so high that the star can now fuse helium instead of hydrogen. A substantial portion of the star’s mass is “burned” in a few hours, releasing tremendous quantities of energy that (ironically) are almost invisible from a distance. Most of this titanic energy release is used up in lifting the star’s core out of its previously degenerate state, permitting the star to settle into a brief period of relatively stable helium burning. The star’s surface grows hotter, but it shrinks and reduces its luminosity considerably.
The temperature and luminosity of horizontal branch stars again tend to be almost independent of the star’s mass. Select a luminosity between 50 and 100 solar units, and an effective temperature of about 5000 K.
After spending a brief period on the horizontal branch, a star evolves though an asymptotic red giant phase, during which it ejects a substantial amount of its mass into space. This is the primary mechanism by which heavier elements are dispersed back into the interstellar medium, to contribute to the metallicity of later generations of stars. Asymptotic red giant stars are extremely rare, as they normally pass through that stage of their development in less than a million years. They should not be placed at random.
No matter which category the star falls into, its radius can be computed using the same formula as in the case for main sequence stars. Red giant stars are likely to be quite large, with radii of about 1 AU at their greatest extent.
Fourth Case: White Dwarf Stars
At the end of the asymptotic red giant stage, a star’s remaining core is exposed, giving rise to a stellar remnant called a white dwarf. A white dwarf is tiny, only a few thousand kilometers across, and so even if it remains extremely hot it radiates very little energy. The star’s active lifespan is now over – it no longer produces energy through nuclear fusion. Instead, the heat it retains from previous stages of its development will radiate slowly into space over billions of years.
If a given star’s age is greater than its Main Sequence Lifespan, and exceeds that value by 15% or more, then it has become a white dwarf star. As with main sequence stars, the properties of a white dwarf are strongly dependent on its mass and age.
A white dwarf star is only the remnant core of a main sequence star, which will have lost a significant amount of its mass during the transition. Let M0 be the mass of the original main sequence star, as generated in earlier steps. Then:
Here, M is the mass of the white dwarf remnant in solar masses. Feel free to select a value within 5% of the one computed. Replace the star’s mass, as generated in previous steps, with this result.
White dwarf stars are formed with very high effective temperatures, and then cool off over time as they radiate heat. To estimate the current effective temperature of a white dwarf star, let A be the age of the white dwarf (that is, the overall age of the system, minus 1.15 times the star’s Main Sequence Lifespan as taken from the Master Stellar Characteristics table). Let M be the mass of the white dwarf in solar masses, as computed above. Then:
Here, T is the white dwarf’s current effective temperature in kelvins.
The radius of a white dwarf star is almost completely determined by its mass. If M is the mass of the white dwarf, then:
Here, R is the approximate radius of the white dwarf star in kilometers. White dwarf stars are extremely small, packing a star’s mass into a sphere no larger than the Earth!
A white dwarf star’s luminosity is usually negligible, although a young (and therefore very hot) white dwarf might have a significant fraction of the Sun’s brightness. To compute a white dwarf’s luminosity, let R be its radius in kilometers and T its effective temperature in kelvins. Then:
Here, L is the star’s luminosity in solar units.
Arcadia: Alice records the values from the Master Stellar Characteristics Table for the 0.82 solar-mass primary star in the Arcadia system. It has a Base Effective Temperature of 5050 K, an Initial Luminosity of 0.28 solar units, and a Main Sequence Lifespan of 22 billion years.
The primary is 5.6 billion years old, and so is still rather early in its lifespan as a main sequence star. Alice decides to select an effective temperature for the star about 2% below the value from the table, or 4950 K. To estimate the star’s current luminosity, she uses:
She accepts this value for the star’s luminosity, about one-third that of our Sun. To compute the star’s radius, she uses:
This gives the star’s radius in AU, which equates to a radius of about 550,000 kilometers, or about 80% that of our Sun.
Beta Nine: Bob has already determined that the Beta Nine primary star is a low-mass star with 0.18 solar masses. Since low-mass stars evolve very slowly and the whole system is only 2.1 billion years old, Bob decides to take the values for effective temperature and luminosity straight from the Master Stellar Characteristics Table, modifying each of them by less than 5% to allow for a little variety. Bob then computes the radius for the primary star using the formula under the case for main-sequence stars. For the companion, Bob computes the effective temperature and then the luminosity using the formulae under the case for brown dwarfs, and notes the fixed radius. The results are as in the table.
|Beta Nine Star System
The models set out here for various stellar classes are, of course, drastically simplified for the sake of ease of use. For brown dwarfs, useful data were derived from the Burrows and Freeman papers cited below. For white dwarfs, the Catalan paper and Ciardullo’s lecture notes were most useful. Main sequence stars are the easiest to characterize, since they are the most easily observed in large numbers, so there are plenty of detailed models in existence for their properties. The Mamajek data helped to produce the Master Stellar Characteristics Table, as did the EZ-Web application for stellar modeling posted by Townsend.
Burrows, A. et al. (2001). The Theory of Brown Dwarfs and Extrasolar Giant Planets. Reviews of Modern Physics, volume 73, pp. 719-766.
Catalan, S. et al. (2008). The Initial-Final Mass Relationship of White Dwarfs Revisited: Effect on the Luminosity Function and Mass Distribution. Monthly Notes of the Royal Astronomical Society, volume 387, pp. 1692-1706.
Ciardullo, R. White Dwarf Stars. Retrieved from http://personal.psu.edu/rbc3/A414/23_WhiteDwarfs.pdf (2018).
Freeman, R. et al. (2007). Line and Mean Opacities for Ultracool Dwarfs and Extrasolar Planets. The Astrophysical Journal Supplement Series, volume 174, pp. 504-513.
Mamajek, E. A Modern Mean Dwarf Stellar Color and Effective Temperature Sequence. Retrieved from http://www.pas.rochester.edu/~emamajek/EEM_dwarf_UBVIJHK_colors_Teff.txt (2016).
Townsend, R. EZ-Web (Computer software). Retrieved from http://www.astro.wisc.edu/~townsend (2016).