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The telescopes on some commercial surveillance satellites can resolve objects on the ground as small as \(85 \mathrm{~cm}\) across (see Google Earth), and the telescopes on military surveillance satellites reportedly can resolve objects as small as \(10 \mathrm{~cm}\) across. Assume first that object resolution is determined entirely by Rayleigh's criterion and is not degraded by turbulence in the atmosphere. Also assume that the satellites are at a typical altitude of \(400 \mathrm{~km}\) and that the wavelength of visible light is \(550 \mathrm{~nm}\). What would be the required diameter of the telescope aperture for (a) \(85 \mathrm{~cm}\) resolution and (b) \(10 \mathrm{~cm}\) resolution? (c) Now, considering that turbulence is certain to degrade resolution and that the aperture diameter of the Hubble Space Telescope is \(2.4 \mathrm{~m}\), what can you say about the answer to (b) and about how the military surveillance resolutions are accomplished?

Step by step solution

01

Understanding Rayleigh's Criterion

Rayleigh's Criterion is given by the formula \( \theta = 1.22 \frac{\lambda}{D} \), where \( \theta \) is the angular resolution, \( \lambda \) is the wavelength of light, and \( D \) is the diameter of the telescope's aperture. Here, the angular resolution \( \theta \) can be related to the object size \( R \) and the distance \( h \) as \( \theta = \frac{R}{h} \).

02

Calculate Required Aperture for 85 cm Resolution

Set \( R = 0.85 \) m, \( h = 400,000 \) m, and \( \lambda = 550 \times 10^{-9} \) m. Substitute \( \theta = \frac{R}{h} \) into \( \theta = 1.22 \frac{\lambda}{D} \). So, \( D = 1.22 \frac{\lambda \cdot h}{R} = 1.22 \frac{550 \times 10^{-9} \cdot 400,000}{0.85} \). Calculating gives \( D \approx 0.316 \) m.

03

Calculate Required Aperture for 10 cm Resolution

Set \( R = 0.10 \) m, \( h = 400,000 \) m, and \( \lambda = 550 \times 10^{-9} \) m. Substitute \( \theta = \frac{R}{h} \) into \( \theta = 1.22 \frac{\lambda}{D} \). So, \( D = 1.22 \frac{\lambda \cdot h}{R} = 1.22 \frac{550 \times 10^{-9} \cdot 400,000}{0.10} \). Calculating gives \( D \approx 2.68 \) m.

04

Consider the Effect of Atmospheric Turbulence

Atmospheric turbulence increases the required aperture to achieve the same resolution. Given the Hubble Space Telescope's aperture is 2.4 m, which exceeds the previous calculated aperture for 10 cm resolution, it's likely that military satellites use additional techniques like adaptive optics or synthetic aperture radar to enhance resolution.

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Key Concepts

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

Telescope Aperture

In simple terms, the telescope aperture is the diameter of the primary lens or mirror of a telescope. It's crucial because it determines how much light the telescope can collect. More light means brighter and more detailed images. The larger the aperture, the better the capacity of the telescope to resolve finer details due to increased light collection.

Aperture impacts the telescope’s ability to distinguish between two close objects, also known as resolving power. The diameter directly affects this ability through a principle called Rayleigh's criterion, which is used to calculate the minimum angular resolution. Essentially:

• The larger the aperture, the smaller the angular resolution, and the greater the detail seen.
• Smaller telescopes generally have more difficulty resolving fine details.

In practice, this means that for highly detailed tasks, such as surveillance from space, a large aperture is essential. The calculations done in the exercise show the specific diameter needed for detailed observation of objects as small as 85 cm or 10 cm wide from a 400 km altitude.

Angular Resolution

Angular resolution is a telescope's ability to see detail and distinguish between two closely spaced objects. According to Rayleigh’s criterion, angular resolution is inversely related to the aperture size. Mathematically, it is expressed as: \[ \theta = 1.22 \frac{\lambda}{D} \] where \( \theta \) is the smallest angular separation, \( \lambda \) is the wavelength of light, and \( D \) is the diameter of the aperture.

The smaller the value of \( \theta \), the finer the detail the telescope can resolve. For instance, resolving objects 0.85 meters and 0.10 meters across at 400 km altitude requires specific aperture sizes as computed earlier. Therefore, larger telescopes have a smaller \( \theta \), enabling them to separate points that are extremely close together in the sky.

Remember:

• Better resolution means seeing finer details, like reading small text from far away.
• Resolution is pivotal in applications ranging from astronomy to spy satellites.

Atmospheric Turbulence

Atmospheric turbulence significantly affects the viewing quality from ground-based telescopes. It happens because the atmosphere is filled with moving air at varying temperatures, which can bend the path of light and blur images, similar to how heat waves cause the shimmering of objects on a hot day.

This turbulence can limit the practical angular resolution of telescopes. A clear and stable atmosphere is ideal for sharper images, but in reality, turbulence can degrade resolution despite large apertures. That’s why adaptive optics and other techniques are often used. In space, atmospheric turbulence is not a factor, which is one reason space telescopes like the Hubble can achieve excellent resolution without the atmosphere’s blurring effects. Still, military satellites might employ additional technology like synthetic aperture radar to further improve resolution in less than ideal conditions. Key points:

• Ground-based telescopes are often hampered by atmospheric effects.
• Space telescopes and advanced techniques help overcome these challenges.