The largest fully steerable radio telescope at the National Radio Astronomy Observatory (Green Bank, W.Va.) has a parabolic mirror \(140 \mathrm{ft}\) in diameter. Approximately what angular resolution does it have for the 1,420-M Hz \((21-\mathrm{cm})\) line radiated by interstellar atomic hydrogen? (In practice, the sensitivity of the detector at the parabola focus is not isotropic. A careful analysis would include a nonuniform aperture-illumination function analogous to Probs. \(10.4 .8\) and 10.4.9.) The mirror was constructed to conform to an ideal paraboloid within a tolerance of \(0.030 \mathrm{in}\). What minimum wavelength can be used with this instrument if the criterion is that the construction errors not exceed \(\lambda / 8\) ? What is the approximate resolution for the minimum wavelength? Answer: 21 minutes of arc; \(\lambda=6 \mathrm{~mm}, 0.6\) minute of arc.

A radio telescope is an instrument designed to observe the universe through radio waves emitted by celestial bodies. Unlike optical telescopes which detect light, radio telescopes capture radio frequencies using a large antenna, typically a parabolic dish. The size of the dish is directly proportional to the telescope's sensitivity and angular resolution—the ability to distinguish two objects that are close together in the sky.

Radio waves are much longer than light waves, and because of this, radio telescopes need to be much larger to achieve a resolution similar to that of optical telescopes. They are essential for studying phenomena such as interstellar gas, pulsars, and quasars, which might not emit much visible light but are active in the radio spectrum.

This intricate dance of capturing invisible signals from vast distances makes radio telescopes one of the paramount tools for astronomers to unravel the universe's mysteries.

The parabolic mirror is a key element of a radio telescope, designed to reflect and focus incoming radio waves to a single point where the detector is located. This mirror is shaped like a paraboloid, a three-dimensional parabola, which ensures that radio waves hitting the surface, no matter where they strike, will be reflected to the central focal point.

Accuracy is crucial in the construction of a parabolic mirror; even minor deviations from the precise shape can cause significant distortions in the data collected. The Green Bank Telescope's mirror, with a diameter of 140 feet, is a technological marvel allowing high precision and contributing to the telescope's ability to resolve fine details in the radio sky.

Interstellar atomic hydrogen is abundant in the universe and can be detected due to the characteristic wavelength it emits, known as the 21-cm line or the hydrogen line. It corresponds to the radio frequency of 1,420 MHz. This emission originates from a change in energy state of the electron in the hydrogen atom.

Understanding this signature emission is paramount as it provides astronomers with a tool to map the distribution and density of hydrogen in the galaxy. Consequently, it offers insights into the structure and dynamics of the Milky Way and other galaxies. The mapped data from interstellar atomic hydrogen can help us study the galactic spiral arms and trace the large-scale structure of the cosmos.

The wavelength of electromagnetic radiation is the distance between successive peaks (or troughs) of a wave. It determines the type of radiation, from long-wavelength radio waves to short-wavelength gamma rays. In the context of a radio telescope, the wavelength of the radiation it is designed to detect will dictate the size of its components, particularly the parabolic mirror.

In radio astronomy, the wavelength also determines the resolving power of a telescope. The shorter the wavelength, the higher the potential resolution. For a given size of the telescope's mirror, as the wavelength decreases, the finer the details that can be observed. For instance, the 6 mm minimum operational wavelength of the Green Bank Telescope allows for a higher angular resolution compared to the 21-cm line radiated by interstellar hydrogen.

Arcminutes, a unit of angular measurement, are one-sixtieth of a degree (60 arcminutes = 1°). They are used in astronomy to describe the angular size of an object in the sky or the angular distance between celestial objects. When discussing the resolution of telescopes, arcminutes determine the smallest angle of separation at which the telescope can distinguish between two distinct points of light or radio sources.

The precision of radio telescopes, like the one at Green Bank, is often expressed in arcminutes. For example, an angular resolution of 21 arcminutes for a 21-cm radio line means the telescope can separate two sources of radio waves that are at least 21 arcminutes apart in the sky. The higher the resolution (e.g., 0.6 arcminutes), the closer together two objects can be while still being distinct, allowing for more detailed observations of the universe.