91Ó°ÊÓ

\(\bullet$$\bullet\) Resolution of telescopes. Due to blurring caused by atmospheric distortion, the best resolution that can be obtained by a normal, earth-based, visible-light telescope is about 0.3 arcsecond (there are 60 arcminutes in a degree and 60 arcsec- onds in an arcminute).(a) Using Rayleigh's criterion, calculate the diameter of an earth-based telescope that gives this resolu- tion with \(550-\) nm light. (b) Increasing the telescope diameter beyond the value found in part (a) will increase the light- gathering power of the telescope, allowing more distant and dimmer astronomical objects to be studied, but it will not improve the resolution. In what ways are the Keck telescopes (each of 10 -m diameter) atop Mauna Kea in Hawaii superior to the Hale Telescope \((5-m\) diameter) on Palomar Mountain in California? In what ways are they not superior? Explain.

Step by step solution

01

Understand Rayleigh's Criterion

Rayleigh's criterion provides a formula to calculate the minimum resolvable angle (resolution) for a telescope. The formula is given by \( \theta = 1.22 \frac{\lambda}{D} \), where \( \theta \) is the resolution in radians, \( \lambda \) is the wavelength of light, and \( D \) is the diameter of the telescope's aperture.

02

Convert Resolution to Radians

Convert the given resolution from arcseconds to radians using the conversion factor \(1\text{ arcsecond} = \frac{1}{60 \times 60 \times 180/\pi} \) radians. The given resolution is 0.3 arcseconds, so \( \theta = 0.3 \times \frac{\pi}{180 \times 60 \times 60}\text{ radians}\).

03

Calculate the Diameter

Solve the Rayleigh's criterion formula for \( D \) using the converted \( \theta \) and \( \lambda = 550 \) nm (which is \( 550 \times 10^{-9} \) meters): \( D = 1.22 \times \frac{\lambda}{\theta} \). Calculate \( D \) and find its value.

04

Compare Telescope Features

Keck telescopes, each of 10 m diameter, can gather more light compared to the Hale telescope, which has a 5 m diameter. The larger aperture of Keck increases its light-gathering power, enabling observation of fainter objects. However, due to atmospheric distortion, both Keck and Hale have similar resolution limits for visible light on Earth.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Telescope Resolution

Telescope resolution refers to the ability of a telescope to distinguish between two closely spaced objects in the sky. This ability is crucial for observing the fine details in astronomical objects, such as the surface features on planets or the tight grouping of stars in a distant galaxy.

Rayleigh's Criterion is typically used to determine a telescope's resolution. It states that the minimum angle between two distinguishable sources depends on the wavelength of light (\( \lambda \)) and the diameter of the telescope's aperture (\( D \)). The formula is: \[ \theta = 1.22 \frac{\lambda}{D} \] where:

• \( \theta \) is the smallest resolvable angle in radians.
• \( \lambda \) is the light wavelength, generally given in meters.
• \( D \) is the aperture diameter of the telescope.

Understanding these components helps astronomers decide how to build more effective earth-based telescopes for clear observations.

Atmospheric Distortion

Atmospheric distortion is a significant hindrance to telescope observations from Earth. As light from celestial bodies travels through the Earth's atmosphere, it encounters varying air densities that cause the light to scatter and blur. This effect is known as "seeing" and essentially limits the resolution of ground-based telescopes.

Factors influencing atmospheric distortion include:

• Temperature variations causing air turbulence.
• Humidity and particle content in the air.
• Weather patterns affecting air stability.

Telescopes are often placed in high altitude locations to reduce these effects, as thinner air at higher elevations typically provides a clearer view. However, despite these measures, the turbulence still imposes a limit on the optical clarity achievable.

Light Wavelength

The wavelength of light is a measure of distance between successive peaks of a wave and it significantly affects telescope resolution. Visible light has wavelengths ranging approximately from 400 nm (violet) to 700 nm (red). In astrophotography, using specific wavelengths provides different levels of detail.

In the context of Rayleigh’s Criterion, using shorter wavelengths (e.g., blue or violet) generally yields better resolution because:

• \( \theta \) decreases with shorter wavelengths, enhancing detail detection.
• This is crucial when observers need to discern small details in bright celestial objects.

Choosing the optimal wavelength depends largely on the type of observation and the specific features astronomers are interested in studying.

Astronomical Observation

Astronomical observation using telescopes allows scientists to study planets, stars, galaxies, and other celestial phenomena. The choice of telescope specifications, like resolution and aperture size, can significantly affect the outcome and quality of these observations.

Key considerations include:

• Larger telescopes, like the 10-meter Keck telescopes, boast better light-gathering ability, allowing astronomers to observe fainter objects.
• However, beyond a certain point, increasing the aperture size does not enhance resolution due to atmospheric limitations.
• Space-based telescopes bypass the issue of atmospheric distortion altogether, offering superior resolution for the same size due to the pristine environment of space.

Overall, careful design and location choice, along with advancements in technology, drive the continuous improvement of astronomical observations.