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What is the angular resolution of the Hubble Space Tele- scope's \(2.4-\mathrm{m}-\) diameter mirror when viewing light with a wavelength of \(550 \mathrm{nm} ?\) The resolution of a reflecting telescope is calculated exactly the same as for a refracting telescope.

Short Answer

Expert verified
The angular resolution of the Hubble Space Telescope's mirror when viewing light with a wavelength of 550 nm is approximately 0.058 arcseconds.

Step by step solution

01

Identify the values you are given

We are given the diameter \(D\) of the mirror, which is \(2.4m\), and the wavelength \(λ\) of the light, which is \(550nm = 550 \times 10^{-9} m\). We are asked to find the angular resolution, \(θ\).
02

Recall the formula for the resolution of a telescope

The formula for the resolution of a telescope is given by \(θ = 1.22 \times \frac{λ}{D}\).
03

Substitute the given values into the formula

Substituting our given values into the formula, we have \(θ = 1.22 \times \frac{550 \times 10^{-9}}{2.4}\).
04

Calculate the angular resolution

Evaluating the expression on the right, the angular resolution \(θ\) is approximately \(2.79 \times 10^{-7} radians\). This value is typically reported in arcseconds for convenience. We use the conversion factor \(1 radian = 206265 arcseconds\) to get the final answer. Thus, \(θ=2.79 \times 10^{-7} \times 206265 = 0.058 arcseconds\).

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Key Concepts

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

Hubble Space Telescope
The Hubble Space Telescope (HST) is a space-based observatory that has revolutionized our understanding of the universe. Launched in 1990, it orbits Earth at an altitude of approximately 547 kilometers (about 340 miles). This position above the atmosphere allows Hubble to capture crisp and clear images, free from the blurring effects of Earth's atmosphere.
Hubble is equipped with a variety of scientific instruments, enabling it to observe the universe in near-ultraviolet, visible, and near-infrared spectra. Some of the key contributions of Hubble include the determination of the rate of expansion of the universe and the observation of distant galaxies. Its precision and ability to photograph distant cosmic phenomena have provided insights not possible with ground-based telescopes.
Despite initial issues with its optics, which were corrected in a 1993 servicing mission, Hubble has continued to provide breathtaking images and valuable data, significantly advancing our knowledge of the cosmos.
Reflecting Telescope
A reflecting telescope is a type of telescope that uses a series of mirrors to gather and focus light. Unlike refracting telescopes that rely on lenses, reflecting telescopes use curved mirrors to collect light and form an image. The primary advantage of using mirrors is that they can be manufactured in larger sizes, which allows for the collection of more light, resulting in better observations of faint astronomical objects.
Reflecting telescopes come in various designs, such as Newtonian, Cassegrain, and Gregorian, each using different combinations of mirrors to focus light.
  • Newtonian telescopes, named after Sir Isaac Newton, utilize a parabolic primary mirror and a flat secondary mirror.
  • Cassegrain telescopes use a parabolic primary mirror and a hyperbolic secondary mirror to reflect light back through a hole in the primary mirror.
The design of reflecting telescopes helps mitigate issues like chromatic aberration, which can occur in refracting telescopes. This design also allows for a compact build, making reflecting telescopes popular for both amateur and professional astronomers.
Diffraction Limit
The diffraction limit is a fundamental limit to the resolution or clarity of images that any optical system, including telescopes, can achieve. It arises because light behaves as waves, and when these waves pass through small apertures, they spread out, an effect known as diffraction.
For telescopes, the diffraction limit determines the smallest detail that can be distinguished. This resolution is characterized by the formula \( \theta = 1.22 \times \frac{\lambda}{D} \), where \( \theta \) is the angular resolution, \( \lambda \) is the wavelength of the observed light, and \( D \) is the diameter of the telescope's aperture.
A telescope can achieve its best resolution, operating at the diffraction limit, when it has a large aperture and observes light at shorter wavelengths. For instance, the Hubble Space Telescope's 2.4-meter mirror helps it achieve a high angular resolution, allowing it to separate closely spaced objects in the sky that ground-based telescopes might see as a single blurred point.
Telescope Optics
Telescope optics covers the principles and technologies used to build systems that collect and focus light to create detailed images of celestial bodies. It's a field that combines elements of physics, especially optics, with engineering to design and construct telescopes that offer sharp and accurate visualizations of the universe.
Two primary types of optics used in telescopes are reflecting and refracting optics:
  • Refracting optics uses lenses to bend light and bring it to a focus. However, large lenses are challenging to produce and maintain.
  • Reflecting optics involves mirrors that reflect light to a focal point. Mirrors, especially large ones, are easier to manufacture and support than lenses.
Optical quality is paramount in telescopes. It involves minimizing aberrations and maximizing resolution and light-gathering ability to ensure that even faint celestial objects can be observed.
Technological advancements, like adaptive optics, help compensate for atmospheric distortions, allowing telescopes to achieve near-diffraction-limited performance. This means they approach the maximum theoretical resolution possible given their size and design, as seen with the Hubble Space Telescope's high-quality optics.

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