Assume that you have a telescope with an aperture of 1 meter. Compare the telescope's theoretical resolution when you are observing in the near-infrared region of the spectrum \((\lambda=1,000 \mathrm{nm})\) with that when you are observing in the violet region of the spectrum \((\lambda=400 \mathrm{nm})\).

The Rayleigh criterion is a key concept in optics that helps determine the resolution of a telescope. It defines the smallest angular separation between two point sources that can be resolved by the telescope. The formula for resolution \( \theta \) in radians is given by \( \theta = 1.22 \frac{\lambda}{D} \). Here, \( \lambda \) is the wavelength of light, and \( D \) is the diameter of the telescope's aperture. This criterion highlights that shorter wavelengths or larger apertures lead to better resolution, meaning we can distinguish finer details in the observed objects.

By calculating the resolution using the Rayleigh criterion, astronomers can understand the limits of what their telescopes can observe. This is crucial for tasks such as distinguishing between closely situated stars or other celestial objects in the sky.

Wavelength conversion is the process of changing the units of wavelength to make calculations more manageable. In the given exercise, wavelengths are converted from nanometers (nm) to meters (m). This step simplifies the use of wavelengths in formulas. For instance, 1 nanometer is equal to \( 1 \times 10^{-9} \) meters.

Let's look at the conversions used in our example:

• For near-infrared light, \( \lambda = 1,000 \text{nm} \rightarrow 1,000 \times 10^{-9} \text{m} = 10^{-6} \text{m} \).
• For violet light, \( \lambda = 400 \text{nm} \rightarrow 400 \times 10^{-9} \text{m} = 4 \times 10^{-7} \text{m} \).

These conversions are essential because they allow us to plug the values directly into the Rayleigh criterion formula.

Near-infrared observing refers to observing light in the near-infrared region of the electromagnetic spectrum. This region typically covers wavelengths from about 700 nm to 1,000 nm. Due to longer wavelengths, infrared light can penetrate dust clouds that visible light cannot, allowing astronomers to study regions of space that are otherwise obscured.

In the given problem, the wavelength used for near-infrared is \( 1,000 \text{nm} \). When we calculate the resolution for near-infrared using the Rayleigh criterion:
\( \theta_{\text{infrared}} = 1.22 \frac{10^{-6}}{1} = 1.22 \times 10^{-6} \text{radians} \).

This value indicates the theoretical resolution for a 1-meter aperture telescope observing near-infrared light. Since the wavelength is larger, the resolution is larger (poorer) compared to shorter wavelengths.

Violet light observing involves analyzing light in the violet region of the spectrum, which typically covers wavelengths between 380 nm and 450 nm. Violet light has shorter wavelengths compared to infrared, leading to a better resolution according to the Rayleigh criterion.

In our example, we use a wavelength of \( 400 \text{nm} \). When plugged into the Rayleigh criterion formula:
\( \theta_{\text{violet}} = 1.22 \frac{4 \times 10^{-7}}{1} = 4.88 \times 10^{-7} \text{radians} \).

This smaller angle (better resolution) means that the telescope can distinguish finer details when observing violet light compared to near-infrared light. This superior resolution is essential for capturing detailed images of celestial objects.