Umbra length is about 10x diameter (was GEO already has 295 commercial satelli)

--- In spacesettlers 11210, Joe Strout wrote:
> GEO has a radius of 42,164 km. That's a circumference of
> 264,924 km. So each of those 295 commercial satellites you
> mention has about 898 km to itself (on average).

How does that compare to the umbral lengths of large solar collectors?
http://en.wikipedia.org/wiki/Solar_eclipse#Geometry

The Moon's diameter is 3,474 km, and it creates and umbra on Earth at a maximum distance of about 380,000 km. Based on that, we might expect a circular solar array of given diameter to cast an umbra as such:

array diameter umbra length
347 km 38,000 km
34.7 km 3,800 km
3.47 km 380 km

The umbra length seems to be about about 10x or 11x the diameter of the occulting body. Therefore, a 10 km diameter solar collector could be expected to cast an umbra at least 1,000 km long. Outside of the total-blackout area of the umbra is the partial-blackout area of the penumbra.
http://en.wikipedia.org/wiki/Umbra

> each of those 295 commercial satellites you
> mention has about 898 km to itself (on average).

They are bunched up in high-value real-estate, and they are not alone. As the linked-document mentions, there are also military satellites in GEO. Are you expecting the number of satellites in GEO to never grow? Are you expecting that real-estate east of Echostar 1 (248 West) will never fill up and/or that it is just as valuable a place to park an SPS satellite as the middle of the North America pack?