Outside Dyson shells
Habitable solid Dyson spheres without gravity generators.

Outside Dyson shells.
Solid, rigid Dyson spheres ("Dyson shells") have certain disadvantages. The main problem is that the gravitational attraction inside a uniform spherical shell is zero. Since Dyson shells have the biosphere on the inside, this presents a problem: You need some form of gravity generators to keep the biosphere from drifting on up into the sun.

What if we put the biosphere on the outside?

Ideally we'd like Earthlike conditions on the outside surface; that is, one gee of gravity and a temperature of about 300 K. Can we get this with main sequence stars?

The mass-luminosity relation is [reference 1]

L = k Mnu equation 1
where k is a constant of proportionality and nu is a constant somewhere between 3.5 and 4.0. (k naturally depends on the choice of nu, clearly.)

The gravitational acceleration g for a mass M at a distance R is:

g = G M/R2. equation 2

Finally, the radiation power law relates the luminosity L to the area A and temperature T:

L = e sigma A T4; equation 3
here, e is the emissivity of the sphere, which is a constant which varies between 0 (for a perfect reflector) to 1 (for a perfect absorber, also called a blackbody).

We can then combine these three equations into a single one relating M (the mass of our star), T (the temperature of our Dyson shell), and g (the gravitational acceleration at the surface of our sphere). First we start by equating L via the luminosity of the star and the radiation power law:

L = L equation 4
k Mnu = e sigma A T4. equation 5

The area A of a sphere is simply 4 pi R2, so we can write

k Mnu = 4 pi e sigma R2 T4. equation 6

We can use our gravitational acceleration equation to relate R2 to M:

g = G M/R2 equation 7
R2 = G M/g equation 8
and substituing that into our equation above, we find
k Mnu = 4 pi e sigma G M T4/g equation 9
and combining like terms on both sides, we reach the final equation
k Mnu - 1 = 4 pi e sigma G T4/g. equation 10

Assuming our sphere appoximates a blackbody (e ~= 1) and then substituting ideal conditions (g = 9.81 m/s2, T = 300 K), we find that M must vary between 0.054 and 0.079 masses solar (the variance is caused by 3.5 <= nu <= 4.0).

By comparison, the "end of the main sequence" -- that is, the theoretical point at which a main sequence star is unable to sustain itself by hydrogen fusion -- is at about 0.08 masses solar, but is not precisely known. Thus it might be possible to have Dyson shells with Earthlike conditions on their outside surfaces around the smallest hydrogen-burning stars in the Universe.

One other advantage to using red dwarfs is their immense lifespan -- the least-massive red dwarfs can last hundreds of billions or trillions of years.

Principles of stellar evolution and nucleosynthesis
Donald D. Clayton
p. 40
University of Chicago Press; 1983
reference 1
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