We're going to be putting a CubeSat into orbit next year and want to use a hyperspectral imaging device onboard.
My questions relates to the maths needed to calculate the imaging lens needed to effectively photograph different swaths of ground on the Earth's surface.
For example, if I want to see with enough detail forest fires and the front line of the fire, how do I calculate that.
Any help would be appreciated.
Here is a way to start.
Choose an altitude or height $h$, say 400 km (if it's deployed from the ISS)
Choose a required spatial resolution $d$ on the ground sufficient to see what it is you'd like to see, say 2 meters.
Converted to an angle that's $d/h = $5E-06 rad or about 1 arcsec.
Choose a wavelength $\lambda$. Middle of the visible band would be around 500 nm (5E-07 meters).
Use a simple expression for the resolution of a telescope or camera $1.22 \lambda / D$ where $D$ is the aperture diameter.
Set them equal:
$$\frac{d}{h} = 1.22 \frac{\lambda}{D}$$
Solve for aperture $D$:
$$D = 1.22 \frac{\lambda h}{d} = 0.122 \text{meters}$$
Notice that this is roughly 10 cm or the width of 1U of a cubesat.
Compare to the resolution of a 3U Dove satellite which is about 3-5 meters and note that they are higher, about 600 to 700 km. See also Spaceflight101 and Planet Labs.
This is just the beginning. You'll have to estimate your exposure time and then realize that the surface of the Earth will appear to be moving about 7000 meters/second when viewed from LEO. So you'll have to take very short exposures and perhaps do shifting and adding to build up a bright enough image.
Hyperspectral imaging separates (one way or another) the image into many more than three wavelength "bins" which means you have a lot less light to work with in each bin. So exposure time is going to be a real issue for you if you want to push the magnification to the diffraction limit. I think you'll find that you'll be limited to a lower resolution than the diffraction limit in order to get reasonably low noise hyperspectral images.
That means you'll either have to very carefully slew your spacecraft during each exposure to null out the streaking from that 7000 m/s apparent ground speed, or you'll have to take a lot of short exposures and shift and stack them in your on-board computer, which would require some simple correlation algorithm to line them all up nicely.
You may want choose a bandwidth (say 5nm or 10 nm) and then ask a new question about exposure times required based on Earth albedo and pixel size (say 2 or 20 meters) on the ground.
You'll also need to estimate your data size and then figure out how you'll get all of that data to Earth. How does the Satellite Networked Open Ground Station operate? How is it used?
Here are some other questions and answers here that touch on other issues related to ground resolution and aperture. Browsing them and all related comments may be helpful.
You can get an idea what things on the ground actually look like from the 3U Doves in these two videos:
and learn more about them here:
