What is the angular resolution (in arc minutes) of (a) a \(100 \mathrm{m}\) diameter telescope operating at \(1 \mathrm{cm}\) wavelength, and (b) a \(30 \mathrm{m}\) telescope operating at \(1 \mathrm{mm}\) wavelength?

Angular resolution is a measure of a telescope's ability to distinguish between two closely spaced objects. The higher the resolution, the better the detail the telescope can capture. This is crucial for astronomers because it allows the observation of fine details in distant celestial objects. In telescopes, angular resolution is usually measured in radians or arc minutes, where smaller values indicate better resolution. Understanding and calculating angular resolution helps in assessing the quality and performance of telescopes.

The diameter of a telescope is a key factor in determining its angular resolution. A larger diameter allows the telescope to gather more light and resolve finer details. To understand how telescope diameter impacts resolution, consider the formula for angular resolution: \[ \theta = \frac{1.22 \times \text{wavelength}}{\text{diameter}} \]In this formula, as the diameter (D) increases, the angular resolution (\theta) decreases, leading to a sharper image. Therefore, larger telescopes, such as those used in major observatories, often have superior resolution capabilities.

Wavelength conversion is critical when applying the angular resolution formula, as it ensures that all units are consistent. Light waves, used in telescopes, are often measured in various units such as centimeters, millimeters, or meters. For example, if the wavelength is given as 1 cm, it needs to be converted to meters for use in the formula:\[ 1 \text{ cm} = 0.01 \text{ m} \]Similarly, 1 mm (millimeter) converts to meters as:\[ 1 \text{ mm} = 0.001 \text{ m} \]This step is essential to avoid any miscalculations and ensure accurate results when determining angular resolution.

Arc minutes are a unit of angular measurement, commonly used in astronomy. One degree equals 60 arc minutes. When dealing with angular resolutions in telescopes, results often need to be converted from radians to arc minutes to be more interpretable. This makes understanding and comparing telescope capabilities easier. For example:\[ 1 \text{ radian} = 3437.75 \text{ arc minutes} \]Using this conversion factor helps in translating the radian value of angular resolution into a more familiar unit.

To convert radians to arc minutes, you need to know the relationship between these units. One radian is equal to approximately 3437.75 arc minutes. Applying this conversion helps make the angular resolution values more practical and understandable. For instance, if the angular resolution in radians is given as:\[ \theta = 1.22 \times 10^{-4} \text{ radians} \]Multiplying this by the conversion factor will give the resolution in arc minutes:\[ \theta \times 3437.75 \text{ arc minutes/radian} \ = 1.22 \times 10^{-4} \times 3437.75 \ \theta \text{ (arc minutes)} \ \theta \text{ ≈ 0.42 arc minutes} \]Understanding this conversion is vital for interpreting and communicating astronomical measurements.