A binary star system in the constellation Orion has an angular separation between the stars of 10-5 radians. Assuming a wavelength of 500 nm, what is the smallest aperture (diameter) telescope that will just resolve the two stars? (1 nm = 10-9 m). Assume the telescope operates at the Rayleigh resolution criterion.
Question
A binary star system in the constellation Orion has an angular separation between the stars of 10-5 radians. Assuming a wavelength of 500 nm, what is the smallest aperture (diameter) telescope that will just resolve the two stars? (1 nm = 10-9 m). Assume the telescope operates at the Rayleigh resolution criterion.
Solution
To solve this problem, we can use the Rayleigh criterion for resolution, which states that the minimum resolvable angle for a telescope is given by:
θ = 1.22 * (λ/D)
where:
- θ is the minimum resolvable angle (in radians),
- λ is the wavelength of light (in meters), and
- D is the diameter of the telescope's aperture (in meters).
We are given that θ = 10^-5 radians and λ = 500 nm = 500 * 10^-9 m. We want to find D.
Rearranging the Rayleigh criterion equation to solve for D gives:
D = 1.22 * (λ/θ)
Substituting the given values into this equation gives:
D = 1.22 * ((500 * 10^-9 m) / (10^-5 radians))
Solving this equation gives:
D = 0.061 m or 61 mm
So, the smallest aperture (diameter) telescope that will just resolve the two stars is 61 mm.
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