Stellar System Generator, Part 3

Copyright RMF Runyan © 2011

Edited by Aaron Smalley for The Guild Companion

Editor's Note: The following is the third chapter of the documentation provided by the author. This as well as the other installments (five in all) are intended to provide background information as to the science and reasoning behind the Stellar System Generator application that RMF Runyan has put together. Once we publish the last installment (tentatively planned for December of 2011), we will also provide the entire documentation in the form of a pdf, which will include all of these article installments as well as a table of contents. In the meantime we are also publishing the "SSG Users Guide" put togehter by RMF Runyan which includes a link to download the application that he is developing. He is continuing to work on the application to improve it and create more functionality within it, and as such he is looking for comments and suggestions from users, which can be posted on the appropriate forums at RealRolePlaying.com. We will release each updated and improved version of the application through the end of the series of articles in December of this year.

Chapter 3: Stellar Primaries

Now that you have decided on the type of galaxy and system, it is time to determine what kind of stellar primary you will have. First, we will need to look into the different types of stars there are.

For the purposes of this SSG, I have grouped stars into three broad category types: giants, major, and minor. I do not include dwarfs or minors because they are usually poor stars for harboring life. Of course, dwarf stars may have life, but the rarity is great enough that I chose to not include them. If you want a dwarf star to have life, then do so. You can still use this SSG to generate the data.

The Stellar Type of a star comprises three elements: Spectral Class, Spectral Level, and Luminosity Class. For an example, our star, Sol, is a G2V Stellar Type. The "G" is the Spectral Class, "2" is the Spectral Level, and "V" is the Luminosity Class. Here is an interactive Hertzsprung-Russell Diagram from University of Nebraska at Lincoln .

Spectral Class

There are seven main Spectral Classes: O, B, A, F, G, K, and M. A nice mnemonic: Oh Be A Fine Girl/Guy Kiss Me. And, 3 colder and less massive "brown dwarf" classes: L, T, Y. There are also 8 additional spectral subclasses: C, D, N, P, Q, R, S, W. Then, there are a number of "slash" class stars. Also see this Wikipedia article for further discussion, external links, and web portals. The major problem with this Wikipedia page, and I have been contacting Wikipedia about it, is that the temperature ranges for the spectral classes are wrong. At least when compared to the temperature ranges I have seen all my life and as listed in the astronomy books I have and keep (expensive) updated versions. This webpage at search.com has the correct temperature ranges . At least the temperature ranges I have been familiar with all my life. Use the below tables to determine a star randomly. Or, just simply choose. The best stars for Earth-like planets are the F, G, and K class stars.

Table 1: Spectral Class
Roll Result
01 Special; roll Table 2
02-04 F
05-12 G
13-24 K
25-00 M

Table 2: Spectral Class Specials
Roll Result
01-40 reroll Table 1
41-75 A
76-90 B
91 O
92-93 L
94-95 T
96 Y
97 White Dwarf
98 Neutron Star
99 Black Hole
00 Special

Specials from table 2 are reserved for those who wish to come up with a pseudo-science object such as gravimetric expulsor, cosmic string, temporal rift, interspatial flexor, etc. This SSG does not cover any of these types of objects.

The Deadly Stars

This is a discussion specifically within the Main Sequence stars (Luminosity Class V).

Spectral O and B Class stars are most likely to not have habitable planets. Not to say that they cannot have them, just very unlikely. O class stars range from 16 to 60 times the mass of our sun and 6.5 to 15 times the radius. B class stars range from 2.1 to 18 times the mass and 1.8 to 6.5 times the radius. The image below shows the relative mean sizes of Luminosity Class V stars.

Image borrowed from Young Astronomers

Although they could have stellar systems, class O and B stars are so massive that they will have pulled all of the stellar material into themselves and left almost nothing for planets to form, except perhaps the lower end mass B stars. And what little bit may have remained would have been blown away by the star's immense radiation and particle winds. Another main problem with these stars that may have terrestrial planets is the immense amount of radiation they pump out. Any terrestrial planet would have to have a density so great in order to have an atmosphere thick enough to block the star's radiation that the planet would end up being a radioactive fireball (vesuvian or furian, q.v.). Of course, the planet could have a life form that relies on the radiation, but the planet surely would not be amicable for life as we understand it. More than likely, it would be a mineral-based life form. Protoplasmic, carbon-based life forms surely could not survive in such a radioactively hostile world. Or the planet would have to be far enough away from the star that it would be an arctic wasteland, supporting only the most primitive of life forms.

Spectral class A stars can have terrestrial planets that can be Earth-like. However, they are very rare. Currently, of all the stars we have catalogued, more than 99% of all stars are F, G, K, and M class stars. And this is including ALL Luminosity Classes. Thus, all A+ spectral class stars are on Table 2 above.

Spectral Level

Depicted by a number from 0 to 9 indicating tenths of the range between two star classes, so that A5 is five-tenths between A0 and F0, but A2 is two-tenths of the full range from A0 to F0. Another way of looking at this number is the number indicates the number of tenths away from the 0 end of the scale. Thus, the A2 would be two-tenths away from being A0. Lower numbered stars in the same class are hotter. This number also helps to determine the temperature of the star. For random determination, simply roll a d10, reading a 0 as zero.

Luminosity Class

Since the radius and mass of a star is proportional to luminosity, determining the luminosity class will later aid in determining the star's radius and mass. As always, you may either use the below table, or simply choose.

Roll Class Description
01 O Hypergiant
02 Ia Luminous Supergiant
03 Ib Supergiant
04-05 II Bright Giant
06-10 III Giant
11-20 IV Subgiant
21-90 V Main Sequence
91-97 VI Dwarf
98-00 VII White Dwarf/Neutron Star

NOTE: If you roll a subgiant (Luminosity Class IV) for an O, K, or M class star, simply reroll. There are no O, K, or M class subgiants. Reason is still unknown. However, if you want a Subgiant O, K, or M class star, go for it. I have never seen any reason why there cannot be one.

Surface Temperature

Knowing the star's surface temperature will help in determining the star's luminosity and some other parameters.

Temperature Ranges
Spectral Class Temperature (K)
Minimum Maximum Difference Mean
O 30,000 60,000 30,000 45,000
B 10,000 30,000 20,000 20,000
A 7,500 10,000 2,500 8,750
F 6,000 7,500 1,500 6,750
G 5,000 6,000 1,000 5,500
K 3,500 5,000 1,500 4,250
M 2,000 3,500 1,500 2,750
  1. Roll d100, closed, reading "00" as 00, to get a number 00-99. This is referred to as Temperature Rating.
  2. Concatenate result of Step 1 with the Spectral Level to get a number ranging from 000 to 999. Remember, the higher this number, the lower the temperature.
  3. Subtract result of Step 2 from 1000.
  4. Divide result of Step 3 by 1000.
  5. Multiply result of Step 4 by Difference in above table.
  6. Add result of Step 5 to Minimum in above table.
  7. This is the surface temperature in Kelvins (K).
  8. Divide result of Step 7 by 5778 to get Temperature Factor.

Conversions
C = K - 273.15
F = ((K -273.15) 1.8) + 32
R = K 1.8

Luminosity

Luminosity is the measure of a star's power output. This may be listed in Watts or Joules/second. Luminosity is measured in Watts, or for spectral luminosity, Watts/meter2. Be careful, though, there is a difference between power and energy. See this webpage stardestroyer.net (look for header "Force, Energy, and Power") for an excellent discussion on the difference.

\[L = \sigma T^4 \left(4 \pi r^2\right) \] Where \(L\) = luminosity in Watts (÷3.839e26 for Sol units); \(\sigma\) = Stefan-Boltzmann Constant (5.6704e-8 \(\mathrm{\frac{Watts}{meter^2Kelvin^4}} \)); \(T\) = surface temperature in Kelvins; \(r\) = volumetric mean radius in meters.

Absolute Magnitude

This is the measure of a star's brightness at a distance of 10 parsecs (32.61688071 light years). Also see this Wikipedia page for further information.

\[M = 4.83 - 2.5\log L\] Where \(M\) = absolute magnitude; \(L\) = luminosity in Sol units.

Radius

This is the size of the star. Since stars tend to vary greatly in their shape from rotation period to rotation period, this is the volumetric mean radius.

\[R = \sqrt{\frac{L}{4\pi\sigma T^4}} \] Where \(R\) = volumetric mean radius in meters (÷6.955e8 for Sol units); \(L\) = luminosity in Watts; \(\sigma\) = Stefan-Boltzmann Constant (see above); T = surface temperature in Kelvins.

Mass

This is the mass of the star. Need I say more? Most often, it is measured in kilograms.

\[ M = L^{1/3.5} \]

Where \(M\) = mass in Sol units ( 1.98892e30 kilograms); \(L\) = luminosity in Sol units.

Volume

This is the star's volume. Volume units are dependent upon units used for radius. If radius units are kilometers, then volume will be cubic kilometers. Meters would equal cubic meters. Etc.

\[ V =\frac{4}{3}\pi r^3 \]

Where \(V\) = volume; \(r\) = volumetric mean radius.

Mean Density

This is the mean density of the star. Note that is the overall mean density. Density in stars, as with planets, will vary from layer to layer as one works towards the inner core. Also note that density units will be dependent upon units used for mass and volume. Example: If mass uses kilograms and volume uses cubic meters (this is the standard), then density will be in kilograms/meter3.

\[ D = M/V \]

Where \(D\) = mean density; \(M\) = mass; \(V\) = volume.

Description

This is the apparent color of the star. Main color is highlighted in green italic s, with adjectives following.

Spectral Class Description
O extremely bright blue (very blinding)
B very bright blue-white (blinding); from blue (hottest) to whitish (coolest)
A white (fairly blinding); from bluish (hottest) to white (coolest)
F yellow-white ; from whitish (hottest) to yellowish (coolest)
G yellow ; from whitish (hottest) to orangish (coolest)
K orange ; from yellowish (hottest) to reddish (coolest)
M red ; from orangish (hottest) to dim (coolest)

Here is a link about the apparent color of a star. And Mitchell N. Charity's home page .

Biosphere Radii

This is the "magic zone" for planets to be conducive for life. It is also referred to as the Goldilocks Zone, Comfort Zone, and Habitability Zone. I chose Biosphere Radii since this region is determined by the innermost radius and outermost radius. If not recorded, then convert the star's luminosity into Sol units by dividing luminosity by 3.839e26. Take the square root of this number then multiply by 0.7 and 3.0. Results are the innermost and outermost biosphere radii in AUs. To convert AUs to meters, multiply by 149,597,870,691. Most often, only two planets will fit in this region.

\[ E_i = 0.7 \sqrt{L} \] \[ E_o = 3.0 \sqrt{L} \]

Where \(E_i\) = innermost radius; \(E_o\) = outermost radius; \(L\) = luminosity in Sol units.

Eccentrics

Eccentrics are captured planets, rogue comets, or other stellar objects that did not form in the initial protoplanetary disk of the stellar system. These objects are exceptionally hazardous since they usually orbit the stellar primary in orbits out of the ecliptic, but within the orbit of the furthest orbital path.

Eccentric Determination: Convert the star's mass into Sol units by dividing its mass by 1.98892e30, round off. This is the Mass Factor.
First Roll: Roll d100, closed. Add the Mass Factor and +30. If 101+, then there is at least one eccentricity. Go to Second Roll.
Second Roll: Roll d100, closed. Add Mass Factor 0.5 and +20. If 101+, then there is a second eccentricity. Go to Third Roll.
Third Roll: Roll d100, closed. Add Mass Factor 0.25 and +10. If 101+, then there is a third eccentricity. Go to Fourth Roll.
Fourth Roll: Roll d100, closed. Add Mass Factor 0.125 and +0. If 101+, then there is a fourth eccentricity.

Remember, this SSG only determines the major Eccentrics, as shown above. There may be other Eccentrics. If desired, repeat this method to determine other minor Eccentrics. And keep repeating for further eccentrics. However, remember this: The more Eccentrics there are, the less likely a planet may be conducive for complex life, meaning there would be a greater possibility of more extinction level event impacts. Either choose the orbital paths of the eccentrics or determine them randomly, the choice is yours. Eccentrics can be the same orbital paths determined above or can be extra ones. For instance, you may choose that orbit 4 has the focus planet on the ecliptic but also has an eccentric with the same orbital radius but has an orbital inclination of -73. See image below for a visual representation. Of course, an eccentric as shown in the below image can cause another cataclysmic event known as orbital perturbation. In other words, the two objects can come close to colliding, but instead perturb each other out of their orbits. Such a perturbation can cause the two objects to never come close to colliding ever again, but the perturbation can cause cataclysmic effects such as completely changing the orbital radius and thus the global climate.

If you really think about it, you just might decide to eliminate all eccentrics. In fact, if desired, you can just skip this step. Remember, I said to not be a slave to the dice. Only, if you want eccentrics do you need to place them. Most often, I just skip this step. This is mainly due to the fact that I do not want to have to go through the brutal mathematics for recalculating any major orbital perturbations. Also remember that perturbing even one orbit will also perturb all others in the system. Reiteration: The mathematics are brutal. Although there are computer programs that will do this for you, there are very few laymen out there who could even understand how to use those programs.

Blue orbit is the focus planet. Red is the eccentric.
Yellow circle is the primary (star). Of course, scale is exaggerated.

For the orbital inclination, either choose, or roll 1d100, open-ended. The result of the roll is the orbital inclination in degrees off the ecliptic.

Also, eccentrics will also tend to have more elliptical orbits. In the above image, both orbits have the same radius. However, even an eccentric with an elliptical orbit could still intersect a planet's orbit at one or two points. And this could lead to a situation similar to the Threadfalls in Dragonriders of Pern.

System Resources

This is usually listed as adjectives such as extremely poor, very rich, exceptional, etc. This is usually determined after all the planets and moons have been generated. The greater the overall density within the system is, the greater the resources.

Stellar Comparisons

The images below offer some comparison between the different stellar primaries. The below images are true to scale. Note that these depict the median size of each star type, not the minimum or maximum size.

Comparison of Our Sun and Smaller Objects
Comparison of Main Sequence Stars by Spectral Class
Comparison of Stars by Luminosity Class
Main Sequence star is that little dot above the "q" in the text "Main Sequence."
For a very excellent video on size comparison, watch this video by morn1415 at YouTube.

Approximate Lifetime of a Star

The below equation will give you the approximate lifetime of a star. Remember, this is the entire lifetime of the star, not how much lifetime is left. Also remember, this is only an approximation with a margin of 15%. And when you are dealing with numbers with exponents of 10, this means 1,500,000,000. And the numbers just get larger after that.

\[ T = M^{-2.5} \times 10^{10} \]

Where \(T\) = number of years; \(M\) = star's mass in Sol units

Modified Hertzsprung - Russell diagram

Most Hertzsprung - Russell Diagrams plot stars using only the temperature and luminosity of a star. The modification I made is the addition of radii. The previous version of the HR Diagram was a program that plotted the curves using a star's Spectral Class/Level (temperature) and luminosity. When I sent the generated graph to my friend, he let me use a new program (over the WWW) that also plotted the star's radius with the star's luminosity. Both programs he allowed me to use as a favor to a friend. No, I cannot get him to let others use them. There is an additional appendix showing how to read the HR Diagram.

After using the below diagram, you can use the equations to further refine the luminosity and radius parameters.

There is also a Cepheid Instability Zone where stars can become Cepheid Variable stars. These variables are not very amicable towards the development of life. Remember, stars in this zone have only a possibility of becoming a Cepheid Variable.

Both of the programs I used discount any star below the M9 class, thus no curves plot into the L class region.

The curves for the 0-Hypergiants and Ia-Luminous Supergiants end up going off the scale towards the M9 end of the curves. Just for example, the M9-0 Hypergiant ended up with a radius of 232,000 Sol Units (1.61356e11 kilometers). That is a radius 1079 times larger than the Earth's orbit!! Or, it is a radius almost 9 times larger than our entire stellar system, including the Kuiper Belt.

It still amazes me how the curves for those greater than Main Sequence generally follow a curve where the stars get larger as they get colder.

As far as I know (off the top of me head), we have found only two Hypergiants: the Pistol Star and VY Canis Majoris (VY). There is a great debate on the actual size of VY. Most say it has a radius of about 2100 Sol Units, making its size larger than the orbit of Saturn. Some few are saying its actual radius may only be about 600 Sol Units, making it slightly larger than the orbit of Mars. Read the article on Wikipedia linked above. At the current believed size of VY Canis Majoris, its density is only 0.000010 kg/m3. That is less dense than the Earth's outer atmosphere. Even the believed smaller size of 600 Sol Radii still only makes its density about 0.0002 kg/m3, which puts its density much closer to the average mean density of other red supergiants. However, it is still mind-boggling to think of a star that huge. As provided earlier in this document, here is an excellent video created by morn1415 on size comparison .

Individual Curves

Herein follows images of the individual luminosity class curves since they merge in the O spectral class.