A spy satellite is reputed to be able to resolve objects 10....

A spy satellite is reputed to be able to resolve objects 10. cm apart while operating $197 \mathrm{km}$ above the surface of Earth. What is the diameter of the aperture of the telescope if the resolution is only limited by the diffraction effects? Use $550 \mathrm{nm}$ for light.

Key Concepts

Relationship Between Aperture Diameter, Wavelength, and Resolution

This concept ties together the physical size of the optical aperture, the wavelength of light, and the system’s resolution. A larger aperture or a shorter wavelength leads to a finer angular resolution, which is crucial when designing instruments to resolve closely spaced details, such as in high-resolution imaging from spy satellites.

Angular Resolution

Angular resolution measures the smallest angle between two objects that an optical instrument can distinguish as separate. It is determined by the diffraction limit and is a critical factor in the performance of telescopes, directly impacting the ability to resolve distant or small objects.

Diffraction Limit

The diffraction limit is the inherent restriction on the resolution of any optical system due to the wave nature of light. It describes how light waves bend around corners or spread as they pass through an aperture, leading to an interference pattern that limits the ability to distinguish fine details.

Rayleigh Criterion

The Rayleigh criterion is a standard for defining the minimum resolvable detail in an optical system. It states that two point sources are just resolvable when the principal diffraction maximum of one image coincides with the first minimum of the other, linking the angular resolution to the wavelength and the aperture diameter.

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