Lighting an area of the Earth Surface from LEO

What does the math look like to calculate the size of an object in LEO that when angled off the sun, would illuminate a given portion of the Earth's surface?

For example, if I wanted to "light up" a square mile of Earth's surface, how big would the reflecting object need to be to achieve this?

I'd like to see the math that calculates this.

• How much light do you want? Aug 27, 2018 at 16:05
• Also, what is the shape of the area to be lit? Square? Circular? And is the light level to be constant over the entire area, or can it vary a little from center to edges, as long as it stays above a minimum value? Aug 27, 2018 at 16:49
• You might want to look into "iridium flares", where something like this happens accidentally. With a normal flat mirror, you're collecting a small amount of sun and spreading it out over a large portion of the Earth. You might be able to accomplish what you want with a parabolic mirror. Under unachievably perfect conditions, you could probably fry ants (or people) from space :)
– user7073
Aug 27, 2018 at 17:05
• Keep in mind that LEO isn't far away, it is fast. If you are trying to constantly illuminate a fixed point you will need a new satellite every few minutes as the previous one falls toward the horizon. Unrelated, but worth looking into is Rjukan Norway that already does what you are asking on a smaller scale with the mirrors mounted on a hillside.
– Lex
Aug 27, 2018 at 20:55
• This has been done as an experiment: smithsonianmag.com/smart-news/… Aug 28, 2018 at 9:24

A rough calculation based on a small (less than km) mirror:

Light from a small mirror will have the sam divergence as the incident light. The Sun’s angular diameter is about 10mr. So a mirror in a 300 km orbit will create a 3km diameter spot on the earth

So the intensity of the illumination, compared to Sunlight, is just the ratio of the mirror area to that of the spot:

$$I_{\rm{delivered}} = (r/1500m)^2 I_{\rm{sun}}$$

• Diffraction will increase the divergence especially for small mirrors.
– Uwe
Mar 28, 2020 at 10:20
• True, but it would have to be a very small mirror to have a divergence larger than the spread of light from the Sun. Your eye, a small optic, can resolve features on the Sun and moon unaided (well, except for attenuation of the Sun’s light) Mar 28, 2020 at 17:21

Any size of object will reflect enough light to reach the ground. The question is how much light do you want?

As a tangible near-baseline example, a Starlink satellite is visible from the ground.

They are about 2.4 m by 1.4 m in base area. (Source)

So at least an area of ~3.3 m^2 is sufficient to illuminate the ground.

• Due to the high sensitivity of the human eye a satellite with a reflecting are of some square meters may be visible. But to get a visible ilumination of the surface of Earth a much larger reflecting area will be neccessary
– Uwe
Feb 27, 2020 at 21:38
• Though not really an answer to the question as asked, +1 for introducing a new and more intuitive unit of measure for payload size!
– uhoh
Mar 28, 2020 at 4:45