# Apparent Magnitude

The apparent magnitude of a star is negatively proportional to the logarithm of the amount of light it sends to earth. The magnitude is

m_1 - m_2 = -2.5 log( I_1/I_2 )

The apparent magnitude of Vega is 0.0 . The apparent magnitude of the Sun is -26.74 . The sun's illumination of the earth, above the atmosphere, is 1367 watts per square meter, so Vega's illumination is 28 nanowatts per square meter. The illumination through the atmosphere might be 1000 watts per square meter for the sun, and 20 nanowatts per square meter for Vega.

The dimmest apparent magnitude for a good eye is under good conditions is around 6, about 250 times dimmer than Vega. Iridium flare has an apparent magnitude of -9.5, 6000 times brighter than Vega; indeed, 16 times brighter than the whole star field with an apparent magnitude of -6.5 . The maximum full moon (I assume in syzygy ) has an apparent magnitude of -12.92, 150,000 times brighter than Vega, about 2.9 milliwatts per square meter on a clear night at the equator.

A good

## Tumbling Thinsat

A V2.0 server sky thinsat has an area of A = 0.024 m^{2}. Normally, it is oriented to greatly reduce or eliminate light pollution. The worst case is that it is broken, abandoned, sandblasted by space debris to a higher albedo, and tumbling or oriented randomly, with either the front or back side reflecting some sunlight onto the earth.

If the orientation is truly random, then the average of all orientations is a hemisphere pointed towards the sun, with the area of the hemisphere equal to the area of the thinsat. The scattering from a reflective hemisphere illuminated from a distant point source is the incoming light scattered uniformly in all directions. So if the projection of the hemisphere is half the thinsat area, and that projection is scattered uniformly over 4 pi steradians, then the sphere acts as an isotropic radiator scattering with an intensity proportional to the average light capture area ( in this case A/2 ) divided by 4 π steradians. The intensity per tumbling thinsat at distance R is e I A / 8 \pi R^2 . A thinsat illuminated by sunlight I = 1367 W/m^{2}, with an albedo of 0.5, scatters 8.2 watts in all directions. At 6411 kilometers from M288 the equator, that is 16 femtowatts per square meter hitting the top of the atmosphere, an apparent magnitude of 15.6 . At 10000 kilometers from M288 to 45 N latitude, that is magnitude 16.6 . 64,000 thinsats in a large array will be 12 magnitudes brighter, 3.6 and 4.6 respectively, visible to the naked eye.

At 45N, about 1/6 of the entire M288 orbit will be visible, and about half of the nighttime ones will be shaded. If there are 30 billion thinsats in M288 ( perhaps 100 gigawatts of computing ), then 2.5 billion visible, tumbling thinsats will have a visual magnitude of -6.9, one tenth as bright as an Iridium flare, but continuous through the night, 0.4% of the maximum full moon.