Problem 39 - University Physics with Modern Physics

Question

Two satellites at an altitude of 1200 km are separated by 28 km. If they broadcast 3.6-cm microwaves, what minimum receiving-dish diameter is needed to resolve (by Rayleigh's criterion) the two transmissions?

Step-by-Step Solution

Answer

The minimum receiving-dish diameter is approximately 1.876 meters.

Step 1: Understand Rayleigh's Criterion

Rayleigh's criterion states that the minimum angular separation \( \theta \) that can be resolved by a dish of diameter \( D \) is given by \( \theta = 1.22 \frac{\lambda}{D} \), where \( \lambda \) is the wavelength of the microwave signal. We need to calculate \( D \).

Step 2: Calculate Angular Separation

The satellites are separated by 28 km at an altitude of 1200 km. The angular separation \( \theta \) in radians can be calculated using \( \theta = \frac{d}{H} \), where \( d = 28 \text{ km} \) is the separation of the satellites and \( H = 1200 \text{ km} \) is their altitude. Thus, \( \theta = \frac{28}{1200} \).

Step 3: Convert Wavelength to Meters

The wavelength \( \lambda \) given is 3.6 cm. Convert this into meters: \( \lambda = 3.6 \times 10^{-2} \text{ m} \).

Step 4: Use Rayleigh's Criterion to Solve for Dish Diameter

Substitute \( \theta = \frac{28}{1200} \) radians and \( \lambda = 3.6 \times 10^{-2} \) meters into the equation \( \theta = 1.22 \frac{\lambda}{D} \). Rearrange to solve for \( D \): \[ D = 1.22 \frac{\lambda}{\theta} = 1.22 \frac{3.6 \times 10^{-2}}{\frac{28}{1200}} \].

Step 5: Calculate the Minimum Diameter

Perform the calculation: \[ D = 1.22 \times \frac{3.6 \times 10^{-2} \times 1200}{28} \approx 1.876 \text{ m}. \] Therefore, the minimum receiving-dish diameter needed is approximately 1.876 meters.

Key Concepts

Satellite Communication

Satellite communication involves transmitting and receiving signals relayed by artificial satellites. This technology enables long-distance information exchange across the globe. Satellites in orbit receive signals from Earth stations, amplify them, and redirect them back to other locations on Earth. There are different types of satellite orbits, including:

• Geostationary Orbit (GEO): Remain over the same spot on Earth, commonly used for broadcasting.
• Low Earth Orbit (LEO): Situated closer to Earth, used for GPS and some communication satellites.
• Medium Earth Orbit (MEO): Used mainly for navigation systems.

Satellite communication is essential for services like TV broadcasting, GPS, and internet connectivity, especially in remote areas.

Angular Resolution

Angular resolution refers to the ability of a receiving system to distinguish two closely spaced objects. In the case of radio telescopes or satellite dishes, this determines how well the system can separate signals that are close together in space. According to Rayleigh's criterion, the angular resolution is given by:
\( \theta = 1.22 \frac{\lambda}{D} \)
where:

• \( \theta \): Angular resolution in radians.
• \( \lambda \): Wavelength of the incoming signal.
• \( D \): Diameter of the dish.

A smaller value of \( \theta \) indicates better resolution. To achieve this, either a larger dish or a shorter wavelength is required.

Microwave Wavelength

Microwaves are a type of electromagnetic radiation with wavelengths ranging from one millimeter to one meter. They are widely used in satellite communications, radar, and wireless networks. In terms of satellite communication, the typical wavelength used is about a few centimeters. Microwaves, such as the 3.6 cm ones mentioned in the problem, are particularly appealing because they can effectively penetrate the atmosphere and provide reliable communication links.
The conversion of wavelength from centimeters to meters is crucial for formula applications. For instance, converting 3.6 centimeters to meters by multiplying by 0.01 results in:
\( \lambda = 3.6 \times 10^{-2} \text{ m} \).

Receiving Dish Diameter

The diameter of a receiving dish is pivotal in determining its ability to distinguish between different signals in satellite communication. A larger dish is more effective at collecting weak signals and resolving closely spaced signal sources. The Rayleigh criterion illustrates the relationship between wavelength, angular resolution, and dish diameter. Using the formula:
\[ D = 1.22 \frac{\lambda}{\theta} \]
where:

• \( D \): Receives dish diameter.
• \( \theta \): is the angular separation between sources.
• \( \lambda \): is the wavelength of the signal.

For tasks like resolving signals from two satellites separated by a distance, it's crucial to calculate the precise diameter needed. For instance, in the given problem, the equation helps determine a minimum dish diameter of approximately 1.876 meters to achieve adequate resolution.