When Mars is at its closest approach to Earth,the distance is approximately 0.75 AU (1 AU = 1.51 * 10¹¹ m).What aperture is required if a telescope is to be able to resolve a 16 km-wide canyon on Mars's surface? Assume a wavelength in the red-orange,or 610 nm.

Question

Quizplus · Accepted Answer

We can use the Rayleigh criterion to determine the aperture required to resolve the canyon on Mars. The Rayleigh criterion states that two point sources are just resolved when the maximum of one point source's central diffraction maximum falls on the first diffraction minimum of the other point source's diffraction pattern. For this problem, we can approximate the canyon as two point sources separated by a distance of 16 km. The angular separation between the canyon's two edges can be found using the small angle approximation:

θ = tan⁻¹((16 km) / (0.75 AU))

Converting the distance to meters and the AU to meters gives:

θ = tan⁻¹((16 km * 1000) / (0.75 * 1.51 * 10¹¹ m)) = 3.51 x 10⁻⁶ radians

We can now use the Rayleigh criterion:

θ = 1.22 λ / D

where λ is the wavelength and D is the aperture diameter. Solving for D gives:

D = 1.22 λ / θ = 1.22 (610 nm) / (3.51 x 10⁻⁶ radians) = 5.30 m

Therefore, the aperture required to resolve the canyon on Mars is 5.3 m. The answer is B.

When Mars Is at Its Closest Approach to Earth,the Distance