Calculating the Minimum Telescope Diameter for a Spy Satellite
What is the minimum telescope diameter that a spy satellite orbiting 445 km above Earth must carry to count individual people in a crowd in visual-wavelength images?
Based on the data provided, how can we estimate the telescope diameter required for this task?
Answer:
The minimum telescope diameter that the satellite must carry to count individual people in a crowd in visual-wavelength images is approximately 20.8 cm. To answer the question, we need to find the minimum telescope diameter. For this, we need to convert the linear size (0.6 m) to angular size using the small-angle formula. The angular size (in arc seconds) is equal to (linear diameter / distance) * 206,265 arcsec.
To calculate the minimum telescope diameter, we first need to convert the orbit height to the same units as the size of a person (from km to m) by multiplying by 1,000. The distance from the satellite to the surface of the Earth is calculated to be 445,000 m. By substituting the given values into the formula, we get an angular size of 0.278 arc seconds.
Next, we use the formula for resolving power in a telescope. The minimum angular size that can be distinguished by an optical system is given by Δθ = 1.22 λ/D, where λ is the wavelength, and D is the telescope diameter. By rearranging the formula and substituting the values, we find that the minimum telescope diameter required is approximately 20.8 cm.
In conclusion, the calculation of the minimum telescope diameter for the spy satellite orbiting 445 km above Earth to count individual people in a crowd involves converting linear size to angular size and applying the resolving power formula in optics. This estimation helps determine the necessary specifications for the satellite's telescopic equipment to achieve the desired imaging capabilities.