Draw a ray diagram for a Cassegrain telescope - AQA - A-Level Physics - Question 1 - 2020 - Paper 4

To determine the suitability of the telescope on the spacecraft for studying a 1 km crater on Pluto, we must consider its resolution:

1. Resolution Calculation: The resolution of a telescope can be approximated using the formula: $ext{Resolution} = 1.22 \frac{\lambda}{D}$ where (\lambda) is the wavelength of light (in meters) and (D) is the diameter of the aperture (in meters).

2. Calculating for the Telescope: For the spacecraft telescope with (\lambda = 450 \text{ nm} = 450 \times 10^{-9} ext{ m}) and (D = 0.21 ext{ m}):
   $ext{Resolution} = 1.22 \frac{450 \times 10^{-9}}{0.21} \approx 2.62 \times 10^{-6} ext{ m} \approx 2.62 \text{ mm}.$

3. Comparison to Crater Size: Since the telescope can resolve features down to about 2.62 mm, it is capable of studying objects larger than this size. A 1 km crater corresponds to 1000 m, which is significantly larger than the resolution limit.

4. Conclusion: Therefore, the telescope is well-suited for observing the 1 km crater on Pluto.

The collecting power of a telescope, which determines its ability to gather light, can be given by the area of its aperture:

1. Collecting Power Formula: The formula for collecting power is given by: $C \propto \pi \left( \frac{D}{2} \right)^2 = \frac{\pi D^2}{4}$ Therefore, comparing the collecting power of the Hubble telescope and the spacecraft telescope:

2. Hubble Telescope: Hubble's diameter is 2.4 m, so its collecting power is:
   $C_{Hubble} = \frac{\pi (2.4^2)}{4} \approx 4.52 \text{ m}^2.$

3. Spacecraft Telescope: The spacecraft telescope's diameter is 0.21 m, so its collecting power is:
   $C_{Spacecraft} = \frac{\pi (0.21^2)}{4} \approx 0.0346 \text{ m}^2.$

4. Comparison: To compare, calculate the ratio of their collecting powers: $\frac{C_{Hubble}}{C_{Spacecraft}} = \frac{4.52}{0.0346} \approx 130.1.$ This indicates that the Hubble telescope has significantly greater collecting power than the spacecraft's telescope.

When deciding whether to use a reflecting or refracting telescope for the spacecraft, several factors need to be considered:

1. Reflecting Telescopes: Reflecting telescopes use mirrors to gather and focus light. They are often larger and can gather more light for deep-space observations. They are better suited for large apertures and can reduce chromatic aberration since they do not use lenses.

2. Refracting Telescopes: Refracting telescopes use lenses to focus light, which can lead to chromatic aberration—a distortion that happens when different colors are focused at different points. This can affect image clarity, especially in larger lenses.

3. Conclusion: Given the harsh conditions of space and the need for high-quality images with minimal distortion, a reflecting telescope would be preferable for the spacecraft. It allows for larger apertures and avoids the complications of chromatic aberration, making it more suitable for astronomical research.