Question

Calculate and compare the angular resolutions of the Hubble Space Telescope (aperture diameter \(2.40 \mathrm{~m}\), wavelength \(450 . \mathrm{nm}\); illustrated in the text), the Keck Telescope (aperture diameter \(10.0 \mathrm{~m}\), wavelength \(450 . \mathrm{nm}\) ), and the Arecibo radio telescope (aperture diameter \(305 \mathrm{~m}\), wavelength \(0.210 \mathrm{~m}\) ). Assume that the resolution of each instrument is diffraction limited.

Short answer

Answer: The order of image quality from highest to lowest is: Keck Telescope, Hubble Space Telescope, and Arecibo Radio Telescope.

Step-by-step solution

Hubble Space Telescope Resolution

To calculate the angular resolution of the Hubble Space Telescope, we will use the given values for the aperture diameter (D) as \(2.40 \mathrm{m}\) and the wavelength (λ) as \(450 \mathrm{nm}\). Plug these values into the formula and convert the wavelength to meters for consistency: $$ \theta_{Hubble} = 1.22\frac{450 \times 10^{-9} m}{2.4 m} $$

Keck Telescope Resolution

To calculate the angular resolution of the Keck Telescope, we will use the given values for the aperture diameter (D) as \(10.0 \mathrm{m}\) and the wavelength (λ) as \(450 \mathrm{nm}\). Plug these values into the formula and convert the wavelength to meters for consistency: $$ \theta_{Keck} = 1.22\frac{450 \times 10^{-9} m}{10.0 m} $$

Arecibo Radio Telescope Resolution

To calculate the angular resolution of the Arecibo Radio Telescope, we will use the given values for the aperture diameter (D) as \(305 \mathrm{m}\) and the wavelength (λ) as \(0.210 \mathrm{m}\). Plug these values into the formula: $$ \theta_{Arecibo} = 1.22\frac{0.210 m}{305 m} $$

Calculate The Resolutions

Now, calculate the angular resolutions for each telescope: $$ \theta_{Hubble} = 2.275 \times 10^{-7} radians \\ \theta_{Keck} = 5.46 \times 10^{-8} radians \\ \theta_{Arecibo} = 8.524 \times 10^{-4} radians $$

Compare The Resolutions

Now that we have the angular resolutions for each telescope, we can compare them. Lower angular resolution means better image quality. From the calculated values, we can see that the order of image quality (from highest to lowest) is: Keck Telescope, Hubble Space Telescope, and Arecibo Radio Telescope.

Key concepts

Diffraction Limit

In the fascinating world of telescopes, the concept of the **diffraction limit** is crucial. It defines the best possible resolution an optical instrument can achieve due to the wave nature of light. When light passes through an aperture (like the opening of a telescope), it spreads out. This spreading, known as diffraction, limits the detail visible.

The diffraction limit is defined mathematically by the formula: \[ \theta = 1.22 \frac{\lambda}{D} \] where:

• \( \theta \) represents the angular resolution in radians, which indicates the smallest angle over which details can be distinguished.
• \( \lambda \) is the wavelength of light.
• \( D \) is the aperture diameter of the telescope or optical device.

The factor 1.22 comes from calculations involving a circular aperture’s point spread function. Diffraction limits real-world performance, meaning even with larger apertures, the observed resolution cannot surpass this physical barrier. Understanding this limitation helps astronomers and engineers optimize telescope designs for specific wavelengths to achieve the closest to the diffraction limit possible.

Aperture Diameter

The **aperture diameter** of a telescope is the diameter of its main lens or mirror. It's a critical parameter as it determines the amount of light the telescope can gather and its resolution potential.

A larger aperture allows for a greater capacity to collect light. This results in improved clarity and detail of the observed objects. In astronomical terms, a large aperture means more brightness and better angular resolution.

Here's why aperture diameter is so vital:

• **Light Gathering:** A bigger aperture collects more light, making faint celestial objects visible.
• **Resolution Power:** As per the formula \( \theta = 1.22 \frac{\lambda}{D} \), a larger aperture \( D \) reduces the angle \( \theta \), indicating better resolution.

To illustrate, consider the Arecibo Radio Telescope, which has a massive aperture diameter of 305 meters, providing fantastic radio observation capabilities, albeit at longer wavelengths where its resolution is naturally lower.

Wavelength

Wavelength is a fundamental aspect of light and an essential concept in understanding telescope resolutions. **Wavelength** refers to the distance between successive peaks of a wave and is usually measured in nanometers (nm) for visible light or meters for radio waves.

In terms of telescopes, the light's wavelength directly affects resolution limits due to diffraction. Shorter wavelengths, like blue light (450 nm), generally provide better resolution, while longer wavelengths, found in radio waves, result in poorer resolution for the same aperture size.

Here’s how wavelength influences telescope performance:

• **Resolution Dependence:** A shorter wavelength offers higher resolution for a given aperture diameter. This is why the Hubble Space Telescope can capture sharp images using visible light.
• **Choice of Instrument:** Different telescopes are designed to operate optimally at specific wavelengths— optical telescopes focus on shorter wavelengths, while radio telescopes suit longer ones.

Understanding the influence of wavelength allows astronomers to select the right telescopes and technologies to study celestial phenomena, ensuring the clearest vision of the universe's mysteries.