The Hubble Space Telescope (Figure 34.33 ) is capable of...

The Hubble Space Telescope (Figure 34.33 ) is capable of resolving optical images to an angular resolution of $2.80 \cdot 10^{-7}$ rad with its 2.40 -m mirror. How large would a radio telescope have to be in order to image an object in the radio spectrum with the same resolution, assuming the wavelength of the waves is $10.0 \mathrm{~cm} ?$

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Key Concepts

Angular Resolution

Angular resolution refers to a telescope's ability to distinguish between two closely spaced objects in the sky. It is generally defined as the smallest angle between two point sources that can be distinctly resolved and is critical in determining the level of detail that can be observed in astronomical imaging.

Diffraction Limit

The diffraction limit is a fundamental limit imposed by the wave nature of light, which dictates that even a perfect optical system cannot resolve details smaller than a certain angle. This limit is quantified by the formula ? = ?/D, where ? is the angular resolution, ? is the wavelength of the observed radiation, and D is the diameter of the telescope's aperture.

Wavelength and Aperture Size Relationship

The resolving power of a telescope is directly affected by the wavelength of the light being observed and the size of the telescope’s aperture. Longer wavelengths, such as those used in radio astronomy, require a larger aperture to achieve the same angular resolution as shorter wavelengths used in optical astronomy. This relationship is a key consideration in telescope design and instrumentation across different regions of the electromagnetic spectrum.

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