My BIG World Needs More Suns

Summary version:

So we're talking people living on top of 30 miles of topsoil, fake dino fossils and water on the outer surface on a hollow (but sturdy) unobtanium sphere with a radius about 1 AU. The sphere has an intermediate sized singularity at the core providing gravity of $$1~g$$ at sphere surface.

How can I provide sufficient lighting? I am not fussy about things like day length, I just want some light both equatorial and in my polar regions. How do I get it, without further liberal use of unobtanium?

Full Description

So I've been toying with the idea of a large Niven-like world. I don't like rings however. So instead, in an act of cosmic vandalism, I've handwaved in a hollow unobtanium sphere, using its frankly unbelievable compressive strength to effortlessly resist collapsing under $$1~g$$ of acceleration from a bikini-clad singularity at the center of the sphere, massing a moderate $$1665~M_{☉}$$ (solar masses). To make it more fun, my land, seas, tectonic simulators, dinosaur bone planters and other such-stuff is on the outside of the sphere.

This spares them from withering under the gaze of the bikini-singularity (for it is written that none shall see a near-naked singularity and live).

However, it does bring up the rather annoying question of providing adequate lighting for my inhabitants. First, however, a bit about the sizes we're talking about.

Let there be: $$g=G\frac{M}{r^{2}}$$

where
$$g$$ - local ("surface") gravitational field
$$G$$ - gravitational constant
$$M$$ - mass of another body
$$r$$ - distance from another body

$$G=6.67428 \cdot 10^{-11}~\text{N}\left(\frac{\text{m}}{\text{kg}}\right)^{2}$$
$$M = 1,665~M_{☉}=3.33 \cdot 10^{33}~\text{kg}$$ - our constructors like the number 3.

Solving for $$g = 9.8~\frac{\text{m}}{\text{s}^{2}}$$

We get $$r=1.50 \cdot 10^{11}~\text{m}$$, which suspiciously conveniently is right about 1 astronomical unit (i.e Sun-Earth distance).

So we're talking people living on top of 30 miles of topsoil, fake dino fossils and water on the outer surface on a hollow unobtanium sphere with a radius about 1 AU.

Now, I'm ok with the polar regions being a bit dark, but I don't want them in eternal darkness either, as it tends to do nasty things to my sphere's vapor circulation (think 60 mile glaciers). To provide lighting, I'll have a series of planets (I'm thinking somewhere in the range of Jupiter to Brown dwarf in size), which contain sufficient artificial fusion burners to provide day-like illumination.

I was thinking about having them in a nice Klemperer rosette (circling about sharing an orbit, with my burners causing occasional flaring to maintain orbits) like thus (not to scale, obviously):

But assuming they're on the black-hole equatorial plane, this leaves my polar regions depressingly dark.

How can I provide sufficient lighting? I am not fussy about things like day length, I just want some light in my polar regions. How do I get it, without further liberal use of unobtanium?

• While not directly related to your question, it's worth mentioning that the center of a spherical shell is an inherently unstable Lagrangian point. Even minimal perturbation will result in a net acceleration of the singularity toward one point on the surface of your planet. Commented Mar 14, 2016 at 23:36
• @LBushkin, with the mass of a thousand suns, that would be quite a perturbation. Commented Mar 15, 2016 at 12:32
• Well, technically it would mostly be the shell moving, not the BH. Ringworlds have the same issue. Personally, I am more interested why an object with such a huge gravity field doesn't have an hydrogen-helium atmosphere... And wouldn't it work better without the BH? An active star would create internal pressure in addition to the gravity. And heat. So you'd only need to add light and control temperatures with below ground insulation. Would be much smaller though. Commented Mar 15, 2016 at 14:32
• Hmm, when i have some time, I'll pull in some numbers and calc the likely mass of the object, then make some assumptions about the handwavium mass. Commented Mar 15, 2016 at 17:37
• Maybe I'm missing something obvious, but what's with the bikini? Commented Mar 17, 2016 at 4:53