Oleg Zabluda's blog
Friday, June 08, 2012
 
Let's calculate and sanity-check Hubble/KH-11 resolution and stuff.
Let's calculate and sanity-check Hubble/KH-11 resolution and stuff.
https://plus.google.com/112065430692128821190/posts/Zi6xF5cTdZo

A plane wave, say from a point source at infinity, viewed through a a circular aperture creates diffraction pattern called Airy disk. When 2 such points are close together, their Airy disks overlap. People disagree what counts as "points are resolved", and it's different for camera vs human eye, and eye-to-eye. But it's on the order of θ = C * λ/D, where for astronomy, C is somewhere in between 1.220 (Raleigh limit 25% MTF), and 1.011 (Dawes limit, 5% MTF), and 0.61 ( Sparrow limit 0% MTF). For terrestrial observations, people often take 50% MTF, for which I estimate C=2, since I am too lazy to type Bessel functions into Wolfram Alpha.

Violet light is 380nm. 2*380e-9/2.4=3.2e-7 radians or 0.06 arcsec. Red light is 750nm, for resolution of 2*750e-9/2.4=6.25e-7 radians or 0.12 arcsec. KH-11 may or may not be shooting in ultraviolet and/or infrared. In red light, from 250km high, 250,000*6.2e-7=0.15m.

Let's compare it to me, as a spy satellite. When standing up, and looking down on the ground [1] (comparisons to Hubble/KH-11 are in parenthesis), my aperture is ~2.4mm (1000x), and altitude is ~2.5m (100,000). So my resolution is ~100x smaller, or 0.15cm. Which is exactly right (just checked). 

For illustration, see Hubble's image of Betelgeuse, which is 0.05 arcsec:
http://imgsrc.hubblesite.org/hu/db/images/hs-1996-04-a-print.jpg

It also checks nicely with current crop of private Earth imaging satellites, which sell 1.0-0.5 m resolution images with mirrors 0.5-1.0 m. I even worked briefly for one, which shall remain unnamed until the info is public.  

http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/raylei.html
http://www.rocketmime.com/astronomy/Telescope/ResolvingPower.html
http://en.wikipedia.org/wiki/Diffraction-limited_system
http://en.wikipedia.org/wiki/Angular_resolution
http://en.wikipedia.org/wiki/Dawes_limit
http://en.wikipedia.org/wiki/Sparrow%27s_resolution_limit
http://www.normankoren.com/Tutorials/MTF.html

[1] In non-equatorial geosynchronous orbit, no less.
http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/raylei.html

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