1) Two stars are photographed utilizing a telescope with a circular aperture of
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Question
1)
Two stars are photographed utilizing a telescope with a circular aperture of diameter of 2.36 m and light with a wavelength of 483 nm. If both stars are 1022 m from us, what is their minimum separation so that we can recognize them as two stars (instead of just one)?
d =
m
2)
A car passes you on the highway and you notice the taillights of the car are 1.16 m apart. Assume that the pupils of your eyes have a diameter of 6.8 mm and index of refraction of 1.36. Given that the car is 13.8 km away when the taillights appear to merge into a single spot of light because of the effects of diffraction, what wavelength of light does the car emit from its taillights (what would the wavelength be in vacuum)?
=
nm
Explanation / Answer
a) I'm not clear what 1022m means for star distance ..
I will assume D = 10^22 = 1.0^23m
Minimum angular resolution (min) given by the Rayleigh criterion ..
sin (min) = 1.22 /a .. (a = lens width)
sin (min) = 1.22 (483^-9m) / 2.36m
sin (min) = 2.4968^-7
Applying (min) to sources separated by distance x at 1.0^23m
sin (min) = x / 1.0^23 = 2.4968^-7m .. .. x = 2.49^16 m
[if D = 1.0^22m .. .. x = 2.49^15 m]
b) sin (min) = 1.22 /a
Within eye ' = /n .. (= wavelength in air, n=ref.index 1.36)
sin (min) = 1.22 '/ (6.80^-3m) = 1.22 / 1.36(6.80^-3m) ..
sin (min) = 131
Outside the eye..
sin (min) = source sep. / distance = 1.16m / 13.8^3m = 8.407^-5
sin (min) = 8.407^-5 = 131 .. .. = 6.41^-7m .. (642 nm)
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