Problem 45 - University Physics with Modern Physics

Question

You are asked to design a space telescope for earth orbit. When Jupiter is 5.93 \(\times\) 10\(^8\) km away (its closest approach to the earth), the telescope is to resolve, by Rayleigh's criterion, features on Jupiter that are 250 km apart. What minimum-diameter mirror is required? Assume a wavelength of 500 nm.

Step-by-Step Solution

Answer

The minimum diameter of the mirror should be approximately 1.45 meters.

Step 1: Understanding Rayleigh's Criterion

Rayleigh's criterion for resolving two points is given by the formula \( \theta = 1.22 \frac{\lambda}{D} \), where \( \theta \) is the angular resolution in radians, \( \lambda \) is the wavelength of light, and \( D \) is the diameter of the telescope's aperture.

Step 2: Convert Wavelength to Consistent Units

The wavelength \( \lambda \) is given as 500 nm. Convert this into kilometers to match the distance units: \( 500 \text{ nm} = 500 \times 10^{-9} \text{ m} = 500 \times 10^{-12} \text{ km} \).

Step 3: Calculate Angular Size of Features

The angular size \( \theta \) of the features on Jupiter is calculated by the formula \( \theta = \frac{\text{feature size}}{\text{distance to object}} = \frac{250 \text{ km}}{5.93 \times 10^8 \text{ km}} \). This simplifies to \( \theta = 4.217 \times 10^{-7} \) radians.

Step 4: Apply Rayleigh's Criterion to Find Diameter

Rearrange Rayleigh's criterion formula to solve for \( D \): \( D = 1.22 \frac{\lambda}{\theta} \). Substitute \( \lambda = 500 \times 10^{-12} \) km and \( \theta = 4.217 \times 10^{-7} \): \[ D = 1.22 \times \frac{500 \times 10^{-12}}{4.217 \times 10^{-7}} \approx 1.45 \text{ meters} \].

Key Concepts

Angular Resolution

Angular resolution is an important concept used in astronomy to determine how much detail a telescope can see. It refers to the smallest angle between two points that the telescope can clearly distinguish as separate entities. An easy way to relate it is through eyesight: the higher the angular resolution, the more detail you can perceive. When designing telescopes, achieving high angular resolution is crucial. It is mainly influenced by the diameter of the telescope's aperture and the wavelength of light it collects. In essence, larger apertures result in finer angular resolution, allowing astronomers to see more detailed images of celestial bodies. Rayleigh's criterion provides the formula for calculating angular resolution: \( \theta = 1.22 \frac{\lambda}{D} \) This equation tells us that the angular resolution \( \theta \) depends inversely on the diameter \( D \). When constructing telescopes, scientists aim to minimize \( \theta \) to improve the clarity of the observations.

Telescope Design

The art of telescope design is a blend of engineering and science, aiming at maximizing the telescope's performance by carefully selecting its components. The focus is on creating a structure that captures as much light as possible and processes that light to produce clear and high-resolution images of distant celestial objects. Key considerations in designing a telescope include:

• The aperture size: Larger apertures provide better angular resolution and image quality.
• Optical components: These include mirrors and lenses that may be used to focus the light into an image.
• The type of telescope: Choices include refracting telescopes (using lenses) and reflecting telescopes (using mirrors). Telescopes for space, like the one in the exercise, often prefer mirrors due to their lighter weight.

By understanding how each component contributes to the overall capabilities of the telescope, designers can better meet the needs of specific observational goals. For instance, observing Jupiter requires accounting for its distance and the desired resolution to see fine details on its surface.

Wavelength Conversion

In the context of telescopes, understanding and managing wavelength conversion is vital for accurate calculations. Light behaves differently at various wavelengths, impacting the observation capabilities of a telescope. Most astronomical observations prioritize visible light, though some telescopes also observe in infrared or ultraviolet spectrums depending on their mission. In the exercise, the wavelength of light is given as 500 nm (nanometers), a value typically used in optical astronomy. Converting this to match the units used for other measurements is necessary for calculations like Rayleigh's criterion. To do this conversion:

• Convert nanometers to meters: 500 nm = 500 × 10^-9 m
• Convert meters to kilometers: 500 × 10^-9 m = 500 × 10^-12 km

This allows for consistent unit usage, ensuring accuracy in determining the specifications needed for telescope design, including the mirror diameter.

Minimum Diameter Mirror

The minimum diameter of the mirror in a telescope is a critical factor in achieving the desired angular resolution based on Rayleigh's criterion. In general, larger mirrors mean better resolution. In this specific exercise, the goal is to resolve features on Jupiter that are 250 km apart while using a light wavelength of 500 nm. Achieving this means calculating the smallest possible diameter \( D \) that satisfies the resolution requirement. By rearranging Rayleigh's criterion, we solve for \( D \): \[ D = 1.22 \frac{\lambda}{\theta} \] Where \( \lambda \) is the converted wavelength and \( \theta \) is derived from dividing the distance between the features by the distance to Jupiter. Substituting the values from the exercise provides \( D \approx 1.45 \text{ meters} \). This showcases how crucial accurate calculations are to designing efficient telescopes. These mirrors must be built with precision engineering to achieve the necessary resolution aimed. Properly designed, these mirrors capture more light, leading to clearer, sharper images of distant objects.