Question

11. Can a 12 cm telescope distinguish two stars that are 1.5 arcseconds away from each other? Justify your answer numerically. (You can assume we're using light in the middle of the visual band.)

Answer 1

Solution-

Yes, we should be able to distinguish between stars that are further than 1.5 arcseconds away from each other.

The Bessel function can be written as below

Jɴ(x) = (1/π) ∫(0,π) cos[Nt − x sin t] dt − [sin(Nπ)/π] ∫(0,∞) exp[−x sinh t − Nt] dt

If N is an integer, then the term to the right of the minus sign is zero, and

Jɴ(x) = (1/π) ∫(0,π) cos[Nt − x sin t] dt
(N: integer)

The coefficient, k, in the wavelength-dependent form of the Dawes Limit is the first positive root of the of the Bessel Function of order one (N=1), divided by π.

sin θ = k λ / D

[ λ is the wavelength of the light in meters
D is the aperture diameter of the telescope in meters
θ is the resolution limit in radians.]

Now when J₁(x) = 0 for the first time rightward of x=0, then x=3.831705970207281.

k = x/π = 1.219669891266431
sin θ = 1.219669891266431 λ / D
λ = 5.5e-7 m
D = 0.12 m
sin θ = 5.5901536683e-6
θ = 5.5901536683e-6 radians
θ = 1.1530519633