Final answer:
Using the Rayleigh criterion, the minimum diameter of the telescope's objective lens needed to resolve two objects 2 m apart from a height of 180 km can be calculated by first determining the angular separation and then rearranging the formula to solve for the lens diameter with the given wavelength of 550 nm.
Step-by-step explanation:
To calculate the minimum diameter of the objective lens in a telescope required to resolve columns of troops marching 2.0 m apart from a satellite at an altitude of 180 km, we'll use the Rayleigh criterion for optical resolution. The Rayleigh criterion states that the minimum angular resolution θ (in radians) can be approximated by θ = 1.22 λ/D, where λ is the wavelength of light and D is the diameter of the lens or mirror. The satellite's altitude doesn't directly affect D, but it helps to calculate θ by determining the angular size of the objects being observed from that altitude.
To proceed with the calculation, we first need to find the angular resolution necessary to resolve two objects 2 m apart at a distance of 180 km from the satellite. This angular separation in radians is θ = 2 m / 180,000 m. After finding θ, we can rearrange the Rayleigh criterion formula to solve for D, resulting in D = 1.22 λ/θ. Given that the wavelength λ is 550 nm (or 550 x 10-9 m), we plug in the values to find D, the minimum objective diameter for the telescope lens.