A spy satellite can consist of a large-diameter concave...

A spy satellite can consist of a large-diameter concave mirror forming an image on a digital-camera detector and sending the picture to a ground receiver by radio waves. In effect, it is an astronomical telescope in orbit, looking down instead of up. (a) Can a spy satellite read a license plate? (b) Can it read the date on a dime? Argue for your answers by making an order-of-magnitude calculation, specifying the data you estimate.

Key Concepts

Diffraction Limit

This is a fundamental constraint in optics which determines the smallest detail that can be resolved by any imaging system. It is reliant on the wavelength of light and the diameter of the aperture, meaning that even the most perfect optical system cannot resolve details smaller than that limit, as dictated by the wave nature of light.

Angular Resolution

Angular resolution is the smallest angular separation between two objects that an optical system can distinguish as separate. It is directly influenced by the diffraction limit and plays a critical role in determining how much detail can be captured in an image from a great distance.

Rayleigh Criterion

The Rayleigh criterion provides a quantitative way to define the angular resolution of an optical system. It sets the minimum resolvable angle based on the diffraction of light, stating that two point sources are just resolvable when the principal diffraction maximum of one image coincides with the first minimum of the other.

Order-of-Magnitude Estimation

Order-of-magnitude estimation is a problem-solving technique used to determine an approximate scale or value without needing to calculate exact numbers. In the context of imaging and resolution, it allows one to assess whether specific details (like small text or fine features) can be resolved by comparing estimated sizes, distances, and the capabilities imposed by diffraction limits.

Transcript

00:01 So we can say that d, the distance apart of the license plate letters divided by the height of the satellite above earth, it should be closely related to 1 .22 times the wavelength of the light using divided by the diameter of the dish.

00:24 Okay, so i just did some quick estimation.

00:28 So let's say a license plate is 2 centimeters.

00:31 So that's 2 times 10 to the negative second.

00:36 And the height of a low -ish, right between low and medium -height satellite, is going to be 2 ,000 kilometers, which is 2 times 10 to the 6 meters.

00:49 I'm going to set that equal to 1 .22.

00:51 And i'm just going to use the average wavelength of light, which is around 550 times 10 to the negative 9th, and then all over d.

01:04 And so this is all going to, we're going to solve for what.

01:06 The diameter of this dish would have to be.

01:10 Okay, so plug that into a calculator.

01:12 We're going to give that d is equal to around 67 .1 meters in diameter.

01:27 So that's relatively large.

01:30 So maybe like with a lower hanging satellite, it could be possible.

01:33 It would make that diameter smaller and smaller, the lower that could get it in orbit.

01:40 But discerning something that's two centimeters apart and getting the letters right...