Question

The Hubble Space Telescope (Figure 34.33 ) is capable of resolving optical images to an angular resolution of $2.80\cdot {10}^{-7}$ rad with its 2.40 -m mirror. How large would a radio telescope have to be in order to image an object in the radio spectrum with the same resolution, assuming the wavelength of the waves is $10.0\mathrm{cm}?$

Short answer

Answer: The radio telescope would need to be approximately 43.6 meters in diameter.

Step-by-step solution

Find the wavelength of light used by the Hubble Space Telescope

To find the wavelength at which the Hubble Space Telescope operates, we need to rearrange the angular resolution formula to solve for $\lambda$: $$\lambda =\frac{D\cdot \theta }{1.22}$$ Plug in the given values for the Hubble Space Telescope's angular resolution ($\theta =2.80\cdot {10}^{-7}$ rad) and the diameter of its mirror (D = 2.40 m): $${\lambda }_{Hubble}=\frac{2.40\cdot (2.80\cdot {10}^{-7})}{1.22}$$ Calculate the wavelength: $${\lambda }_{Hubble}=5.49\cdot {10}^{-7}\mathrm{m}$$

Find the diameter of the radio telescope

Now we want to find the diameter of the radio telescope to get the same angular resolution as the Hubble Space Telescope at a different wavelength. We can again use the angular resolution formula, rearranging for the diameter: $$D_{radio}=\frac{1.22\cdot {\lambda }_{radio}}{\theta }$$ We are given the wavelength of the radio waves (${\lambda }_{radio}=10.0\mathrm{cm}=0.1\mathrm{m}$) and the angular resolution of the Hubble Space Telescope ($\theta =2.80\cdot {10}^{-7}$ rad). Plug in the values: $$D_{radio}=\frac{1.22\cdot 0.1}{2.80\cdot {10}^{-7}}$$ Calculate the diameter of the radio telescope: $$D_{radio}=43.6\mathrm{m}$$ The radio telescope would have to be approximately 43.6 meters in diameter to achieve the same angular resolution as the Hubble Space Telescope.

Key concepts

Hubble Space Telescope

The Hubble Space Telescope is an incredible piece of technology orbiting outside the Earth's atmosphere. It allows astronomers to view the universe with unprecedented clarity. One of its remarkable features is its ability to achieve high angular resolution, which refers to the capability to distinguish between two slightly separated objects in the sky.
With a 2.40-meter mirror, the Hubble can resolve images to an angular resolution of approximately $2.80\cdot {10}^{-7}$ radians. This means it can differentiate between objects that are very close together. Astronomers take advantage of this to study distant galaxies, nebulae, and other exotic cosmic phenomena in detailed focus.
Angular resolution depends on both the diameter of the telescope's mirror and the wavelength of light it observes. For the Hubble's size and the visible light spectrum, these characteristics help it function as a powerful tool for exploring diverse cosmic events.

Radio Telescope Diameter

When considering how radio telescopes work, it is important to compare them to optical telescopes like the Hubble Space Telescope. Radio telescopes need to be much larger to achieve similar angular resolution because they observe at longer wavelengths.
For instance, in attempting to equal the Hubble's angular resolution using radio waves with a wavelength of $10.0$ cm, the telescope must have an extremely large diameter. The calculations show that a radio telescope must have a diameter of about 43.6 meters to resolve images similarly to Hubble. This requirement stems from the difference in the frequencies and wavelengths that the radio waves cover, which are much longer than those of visible light.
Building such large dishes is a feat of engineering, exemplifying endeavors like the Arecibo Observatory or the MeerKAT array, which track phenomena across the radio spectrum and perform tasks like identifying far-off quasars and mapping distant galaxies.

Wavelength of Light

Understanding the wavelength of light is crucial in astronomy, as it affects how telescopes operate and their potential to resolve fine details in their observations.
The wavelength refers to the distance over which the wave's shape repeats, and different colors of that light correspond to different wavelengths. For the Hubble Space Telescope, functioning predominantly in the visible part of the spectrum, it operates around a wavelength of about $5.49\cdot {10}^{-7}$ meters. This allows it to represent what we can perceive directly with our eyes.
Comparatively, radio telescopes deal with much longer wavelengths. For example, the wavelength of $10.0$ cm used in radio observations is significantly longer than optical wavelengths. This property necessitates a drastically different approach to designing and utilizing those telescopes, resulting in much larger dishes to gather and focus radio waves effectively.
All of these wavelengths, whether short or long, help astronomers piece together the cosmic puzzle, broadening our understanding of the universe's vast tapestry.