Artificial Stars for the Stady Shell

Suppose we wanted to create an artificial night sky for the Earth, on the inner surface of the stady shell?

At 50 AU, the stady shell will be illuminated with 1366/50² or about 0.5 watts per square meter of sunlight. The shell will have a finite albedo; presume 0.01, so that it reflects 5 mW/m². That will be the illumination of the Earth's night sky, 4 ppm of noontime sunlight ( 98,000 lux or lumens/m²) or 0.4 lux. Maximum full moonlight in the tropics is 0.32 lux. Too bright. Try again ...

Assume an albedo of 0.001. That gets us down to 1/8 full moonlight. The optical structures of the Stady must be better than this, even with some percentage failing. There will be lots more thermal infrared, though not enough to increase Earth temperature significantly.

But what about stars? Humans and animals use stars for navigation. The shell should simulate those, too.

The magnitude of the Sun is -26.74; the magnitude of Sirius is -1.46, a difference of 25.38 magnitudes or a ratio of 1.3e10. The Sun delivers 1.7e17 watts to Earth, so a "fake Sirius" on the inner surface of the shell should deliver 13 megawatts of reflected sunlight to the Earth, or about 27 square kilometers of shell area at perfect optical efficiency. That would be a 6 kilometer diameter focusing mirror.

At that distance, the angular diameter of the Sun is 0.01 degrees, or about 170 microradians. 50 AU is 7.5e12 meters, and an optical wavelength is about 0.5 micrometers. The "spot size" of this mirror is about 1.4 × 7.5e12 * 5e-7 / 6000 meters, around 900 meters, so it can be defocused and still deliver it's reflected light to the 6e6 meter diameter Earth. The mirror surface will be imperfect, and some of the light will scatter, but a 10 kilometer mirror should be adequate - less than 1 part in a quintillion of the stady shell surface.

The sum of luminousities of the 92 next brightest stars, down to magnitude 2.5, requires 6.3 times more light than Sirius. Simulating all the visible stars may require less than 20 times as much reflected light as fake Sirius.

Beyond that, peering through telescopes (with all that ambient light) will be frustrating and frustrated. Very large telescopes will orbit beyond the shell, in the Oort cloud, and the shell can deliver those images to astronomers (professional and amateur).