Question

The telescopes on some commercial surveillance satellites can resolve objects on the ground as small as 92 cm across (see Google Earth), and the telescopes on military surveillance satellites reportedly can resolve objects as small as 12 cm across. Assume first that object resolution is determined entirely by Rayleigh's criterion and is not degraded by turbulence in the atmosphere. Also assume that the satellites are at a typical altitude of 408 km and that the wavelength of visible light is 537 nm. What would be the required diameter of the telescope aperture for (a) 92 cm resolution and (b) 12 cm resolution? (c) Now, considering that turbulence is certain to degrade resolution and that the aperture diameter of the Hubble Space Telescope is 2.4 m, what can you say about the answer to (b), i.e. is the military surveillance resolution accomplished?

Answer 1

From Rayleigh’s criterion, we can determine the minimum angular separation subtended by the sources at the slit for which the images are just resolved.This equation indicates that the first minimum in a single-slit diffraction pattern occurs at the angle for which

where a is the width of the slit. In our case, the width of the object for wich we want to know the minimum angle for resolving the image.

So, in order to resolve the required diameter of the telescope aperture for an object of 92cm of diameter, first we need to know the if at the distance of 408km. For this we have:

This is the minimum angle needed for the telescope to resolve an object of 92cm when it is at 408km of height.

Then we need to know wich is the minimum angle that the telescope needs so it can resolvethe object if the lengthwave of the light is 537nm, and it would be given by:

For the telescope to resolve the object this condition must be true:

As you can see, this condition is fulfilled, so, now we can calculate the diameter of the aperture for a 92cm resolution, using this equation:

Where is the wavelenght of the Light and is the aperture.

Then we have:

    This is the aperture needed to resolve an object of 92cm at 408km of distance.

Now we must repeat the process to see if the telescope can resolve an object of 12cm across.

and,

As you can see the codition is not fulfilled, wich means that, at that distance the telescope can not resolve an object that small. We can prove this by calculating the aperture needed, given by:

An aperture this big cannot resolve an object that small.

So, when revising that Hubble Space Telescope has an aperture diameter of 2.4m we can say that this aperture is not able to resolve objects that small at close distance as 408kms, for a telescope laike the Hubble wich is capable of taking picture at much farther distances. A military telescope that could be able to resolve objects this small would had to have to be closer or have a much bigger lens to allow apertures that big without loosing resolution of the object.