OTHER CONCEPTS | EMPIRE OF A THOUSAND SUNS & MULTI-STAR SYSTEMS

'HOW TO BUILD 1000 STARS'

You may have watched Artifexian's video about How to Build 1000 stars, if not, go watch it now.

Done?

Well, so why am I bringing this up?

Realistic night skies, actually, this is something I wanted to do for a long time - actually, tried once with a 3D environment, plotting the positions of stars in space, in order to get the sky map from the Ran system (eEridani). It didn't worked out well since my hard drive fried - by experience I will warn you that this is extremely wearing work, for little pay out other than a well thought environment that is rather cool to look at.

TL;DR:

Milkyway-wise, it can be about 900 Main-Sequence, 96 Degenerates, and 4 Giant stars.

This can be further expanded as containing:

O 1

B 1

A 5

F 30

G 70

K 100

M 700

WD 100

III&IV 4

And maybe 1 Neutron star and 1 Black Hole - we will not talk about them over here though.

Remember that Main Sequence stars have masses between 0,08Msol and 50Msol.

Stellar masses roughly distributed as:

O +16

B 2,1~16

A 1,4~2,1

F 1,04~1,4

G 0,8~1,04

K 0,45~0,8

M 0,08~0,45

Humans, and life as we know it, could only appear, and survive around K, G, and F-type stars.

You can define your stars using the automatic STAR&PLANET calculator in our [CALCULATORS and SPREADSHEETS] section later on.

INTRODUCING MULTI-STAR SYSTEMS

One thing Edgar didn't touch, is about the multiplicity of stars, ~1/3rd of all stars have at least 1 companion, and at least 5% of wide binaries and 50% of close binaries do have the stability conditions for habitable planets.

Even though only 30-ish percent of binaries have planets, we are still talking about (100*0,3*0,3*0,5= 4,5) ~4,5% of all stars with suitable conditions to habitable worlds, compared with the figures of 2,8% for single stars like the Sun, for that we can take that at least half of our habitable planets are around double stars.

So, let's take a look at how those multi-sun systems work:

You can pretty much calculate the percentage of multiplicity for a star of a given mass using:

Mindex% = 75 * (-0.92^Msol) + 100

For example, by entering 0,94Msol, we get that ~30,6% of stars with this mass have at least 1 companion.

As a rule of thumb for double stars:

• If your component B is as bright as component A, it is more likely that the two are from the same spectral class.

• If B is considerably dimmer than A, the component B is more likely to be hotter

How cold? You can use this, derived from MSC graph data:

TB = T A - (0,6 * (1,00149 ^ TA))

Temperatures given in Kelvin, for example; If component A's temperature is around 4750K (K1V), then is more likely that component B's temperature would be about 707K cooler, or around 4043K (K6V).

The temperature difference gets larger as the primary get's hotter, a sun-like star at 5800K would have an M5V red dwarf as companion by this trend.

The lower the mass of the primary star, the more similar the secondary will be to the primary.

This formula also will yield a brown-dwarf compatible temperature for primaries between 5900~6200K, above 6200K the correlation states no companion is possible, you will point out there is binaries hotter than 6200K, - although I believe that is due some star systems are not formed within the same molecular cloud, at least, not so close they would interfere with the other's formation, or within a certain mass limit, the correlation becomes less clear above 5000K, but is a good overview so far.

ORBITAL STABILITY AROUND BINARIES

You will see that there are certain limits for stable planetary orbits around double stars, Edgar's Forbidden Zone.

I would like to point out that it's more of a rule of thumb than actual physics - it works, but works the same way it's easier to hit a basketball hoop from across the Atlantic ocean if you use the Moon as a basket ball.

In Build your own Tatooine, the Makkel system is formed by two stars, one F-type 1,3Msol, and one G-type 0,9Msol, separated by 0,3AU.

Now, he calculates the inner and outer edges using 1/3*min Sep and 3*max Sep, what yields 0,05AU and 1,32AU.

Using an actual study on planet stability around binaries, we see that the actual edge - the minimum distance a planet can be from the close-binary pair, is about ~1,09AU.

For reference, if Edgar wanted Makkel to be an S-type system around both stars, he would have to put the stars 21AU apart.

• Is worth noting that circumbinary planets (P-type) found to the date of the paper seem to form closer to the inner limit the wider the binary is, and also form not much further than 10AU.

• And when the binaries are separated further than 10AU, then is way more probable the planets in the system are actually circumstellar (S-type).

Both figures for A~1,0Msol and B~0,5Msol.

PLANETS AROUND BINARIES

FORMATION

P-type planets are somewhat rare, current findings put a low-estimate of 2,8~3%, while simulations put the frequency at ~10%.

This may be limited by our current detection technology, which works fine for planets around single stars because they orbit close enough to cause visible perturbations in the system.

Planet formation around binaries is also narrower than around single stars, the planetary disk's inner edge tend to be wider, and the outer edge tighter in close binaries.

Three items should be met in order for planets to form in stable and habitable configurations.

1. The planetary disk must exist either inside the S-limit for wide binaries, or beyond the P-limit for close binaries.
2. The habitable zone must exists either inside the S-limit for wide, and beyond the P-limit for close binaries, on top of the planetary disk. 
3. The planet should exist within the habitable zone, but not closer than 1/3rd the minimum separation between the stars for S-type orbits.

The close binary pair on top can host a habitable planet, while the wide binary at the bottom can only host a habitable planet on it's dimmer component

FREQUENCY

Current technology lay a frequency of planets of: ~5Re at 10~15%, 6~9Re are as low as 5%, and planets larger than 10 Earth radii at ~1%

In this perspective, Jupiter and Saturn-sized planet frequency is <0,5%.

Most systems are also flat, ~4º in inclination, but the larger the planet, the more inclined it's orbit gets, planets with 10Re are found to be ~40º from the system's plane.

Speaking of large planets, hot-jupiters are more common in binaries wider than 100AU, at 10~15%, compared to <5% in binaries wider than 1AU and closer than 50AU - there is a zero frequency gap between 50AU and 100AU.

Stars that are 10AU apart are the more varied in planet abundance, having from as little as 12% and up to 80% as much planets as single stars, and average at 20%.

For pairs which combined masses are around 1,5Msol, the most common orbital distance is between 0,25~2,5AU, the second most common is between 2,5~3,8AU.

The most common orbital periods are in decreasing order: 5, 6, 4, 8, and 7 days.

The remaining ~30% of binary stars have orbital periods of: 3, 2, 9, 1, 10, and less than 1 day.

See [CALCULATORS and SPREADSHEETS] for all the essential calculations about binary stars.

HOW MANY COMPANIONS?

This paper, cites the proportion of multi-star systems as (470 :108 : 27 : 5 : 2), for 3, 4, 5, 6, and 7 stars systems, respectively - although "is not necessarily representative of the true proportion [...]".

But assuming that's is somewhat similar to the true proportion, a regression of y~ab^x gives us a sample of ~16.078 stars, we then have 1302 binaries, and the previously stated numbers for multiple systems, which unfortunately gives us only ~11,3% of stars as multiples.

If we maintain those proportions but raise the numbers by 3x, we should get 33,4% of stars as multiples - however that may be unnecessary, one of the authors of this paper, recently in 2008 published another study regarding that matter.

Where we find this table:

We end up with multiplicities of: 59.62%, 31.52%, 6.25%, 1.88%, 0.44%, 0.24%, and 0.04%.

Table 6, from page 8, A catalogue of multiplicity among bright stellar systems (P.P. Eggleton, A.A. Tokovinin), colored for visual clarity

Applying that to our 1000-sun empire, we have the following amount of stars available:

This chart above shows the available components for each formation, for example, the 26 double G-type stars means that, there are about 26 stars of type G available to be part of a binary system - regardless of what the other components are

This way, you are free to choose if you'll have 24 GG binaries, 1 GK, and 1 GM binaries - or pair each of the 26 G

You could group an FG-KM quadruple system, and pile nested hierarchical systems with doubles and triples.

Systems with 100+ stars are better defined as dynamical open clusters - it would make sense to have your most massive stars as part of a 

THOUSAND SUN EMPIRE AS AN OPEN CLUSTER

THE HOT SINGLES IN YOUR AREA

Place your high-mass stars at least 50~100Ly away from your main habitable systems.

Supernovae up to 1000Ly distance can have still noticeable effects on a planet like Earth, with global warming effects of 3~4ºC.

Historically, about 20 supernovae happened within 1000Ly from Earth in the last 11Myr, and 1 supernova happens within <30Ly every 240 Myr.

The problem with supernovae is that the gamma-ray bursts they emit can destroy a considerable part of the ozone layer, exposing the planet's surface to dangerous UV light and cosmic rays, damaging life on the surface, reefs, and phytoplankton communities.

In worst case scenarios, a type-II supernova happening less than 30Ly could happen as rarely as 1 every 2Gyr, up to 10 per Gyr, within less than 26Ly and half of your planet's ozone layer (the side facing the star) is destroyed.

Only stars with more than 8Msol go supernova, which means that we have at least 1~2 stars which can go supernova, however, if we consider the other 4 giant stars which can easily be put into the supernovae mass requirements, we can have as much as 6 stars ready to explode in our empire.

We shouldn't expect to ever be within safe distance from any star. Given the dimensions of our empire (63x63x63), the core worlds would be at high risk (30-ish Ly from the outer reaches), the external colonies would be at moderate risk (30~60Ly from other stars), it would be safer to put the supernovae candidates in the surrounding

CLUSTERING MODEL: PLEYADES

For instance, the Pleyades Cluster nucleus is about 16Ly wide, and the cluster itself is about 86Ly wide. While the Pleyades can easily weigh out 800Msol, our empire could be a relatively small cluster with mass topping at 550~600 Msol.

If the galactic average distance between stars is little more than 1pc (3~5Ly), at the center of our cluster the distance between stars could be of ~1Ly, and increase towards the edges to the galactic average.

Closer than ~1Ly and the gravitational interactions would disrupt the entire system, and lower the planetary stability in systems to as low as 100Myr - that's why globular clusters aren't a good place to search for planets - plus, the sheer proximity between stars, comparable to that of the size of the solar system, makes planetary accretion nearly impossible, and even if some did managed to have a permanent planetary system, it would be exposed to dangerous levels of radiation and solar winds, and night wouldn't be darker than Earths twilight, because there is a thousand nearby suns in the sky. The centers of globular clusters are also believed to host black holes, which carry all this flock of stars around with them, again, emitting more radiation.

If we divide our cluster in 6 concentric regions, each 5,25Ly thick, we increase the average distance in half a light-year per level, as much as 28% of the cluster population, lies just in the first level, in the second level, we have encompassed about 54% of the stars, and 74% in the third level, we easily have ~3/4th of our total population of stars half-way through the empire, and the other 25% spread over a volume 7x greater than on the inner half, where counter-intuitively, the true spacing between stars end up being less than half that of the galactic average (

From hundreds of light-years away, the cluster may look like this:

number of stars is heavily exaggerated for visual clarity

Still taking the Pleyades as an example, we could also include ~2% more mass by adding several brown dwarfs, if every brown dwarf we add weighs about 0,05Msol, then we have added between 11~12 thousand other places in between stars, and even companions/sub-planetary systems.

CLUSTERING MODEL: HYADES

Looking for something more solid to base our empire on, we can use Open Clusters instead.

For a typical cluster with 1,000 stars with a 0.5 parsec half-mass radius, on average a star will have an encounter with another member every 10 million years. The rate is even higher in denser clusters. These encounters can have a significant impact on the extended circumstellar disks of material that surround many young stars. Tidal perturbations of large disks may result in the formation of massive planets and brown dwarfs, producing companions at distances of 100 AU or more from the host star.- Wiki

This means that we can have a typical cluster, where half of it's mass is located within 1,63Ly from the center. However, this default cluster is too tightly packed for us, as it would imply a core density of 50 stars/Ly³.

Looking at a more sparse cluster, the Hyades, we can have an idea of what we're looking for although the group have less stars than we have (between 100~300 stars):

The core radius is 2.7 parsecs (8.8 light-years, a little more than the distance between the Sun and Sirius), while the half-mass radius, within which half the cluster's mass is contained, is 5.7 parsecs (19 light-years). The tidal radius of ten parsecs (33 light-years) represents the Hyades' average outer limit, beyond which a star is unlikely to remain gravitationally bound to the cluster core.

Multiplicity in those types of clusters seems to be concentrated in the core region due mass segregation - while typical galactic standards for binary K and A type stars are about 25% and 36%, on the Hyades, it goes up to 26% and 87%, respectively - binaries in the Hyades also show an average separation of less than 50AU.

About 90% of all open clusters dissipate early on on the first billion years, and very few survive to the age of the solar system, bear in mind we want the latter.

Mass segregation also mean that our high-mass stars will be located closer to the center, while our K and M type stars will be on the surrounding Frontier (yes that's a Titanfall reference).

Of what I could find, the Hyades are a rather unusual cluster, as they lack extreme mass stars such as K, M, and brown-dwarfs on the light side, and any stars heavier than 3Msol.

As we have defined early on this post, our empire's mass is about 60~70% contained in low-mass stars, while the remaining 40~30% in only 37~41 stars, so unless you wanna flip that over, we won't follow much of this population distribution.

Using the Hyades as model for what's a cluster dimensions would be, I've also created a star cluster calculator you can use.

See [CALCULATORS and SPREADSHEETS] section for BINARY STAR and STAR CLUSTER calculators.

- M.O. Valent, 31/07/2020