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?

Verified step by step guidance

1. Understand Rayleigh's criterion for resolution, which states that two sources are resolvable if the angular separation between them is greater than the angular resolution limit of the observing instrument. The formula for the angular resolution limit \( \theta \) is given by \( \theta = 1.22 \frac{\lambda}{D} \), where \( \lambda \) is the wavelength and \( D \) is the diameter of the dish.
2. Convert the given wavelength from centimeters to meters. Since 1 cm = 0.01 m, the wavelength \( \lambda \) is \( 3.6 \text{ cm} = 0.036 \text{ m} \).
3. Calculate the angular separation \( \theta \) between the two satellites using the formula \( \theta = \frac{s}{h} \), where \( s \) is the separation between the satellites (28 km converted to meters) and \( h \) is the altitude of the satellites (1200 km converted to meters).
4. Substitute the values into the angular resolution formula \( \theta = 1.22 \frac{\lambda}{D} \) and solve for \( D \), the diameter of the receiving dish. Rearrange the formula to \( D = 1.22 \frac{\lambda}{\theta} \).
5. Ensure all units are consistent and solve for \( D \) using the calculated \( \theta \) and the given \( \lambda \). This will give the minimum diameter needed to resolve the two transmissions according to Rayleigh's criterion.

Key Concepts

Rayleigh's Criterion

Rayleigh's criterion is a standard for determining the resolution limit of an optical system. It states that two point sources are resolvable when the principal diffraction maximum of one image coincides with the first minimum of the other. This criterion is crucial for calculating the minimum diameter of a receiving dish needed to resolve two separate signals.

Diffraction

Diffraction refers to the bending of waves around obstacles and openings, which affects the resolution of imaging systems. In the context of satellite communications, understanding diffraction is essential for determining how the wavefronts of microwaves spread and overlap, impacting the ability to distinguish between two closely spaced sources.

Microwave Wavelength

The wavelength of microwaves, in this case, 3.6 cm, is a critical factor in calculating the resolving power of a dish. The wavelength determines the scale of diffraction effects and is used in conjunction with Rayleigh's criterion to compute the minimum dish diameter required to resolve two signals separated by a given distance.