15.3 Solutions
759
15.61 The limit of resolution of a telescope is
dθ = 1.22
λ
D
=
1.22 × 55 × 10 −5
500
= 1.342 × 10
−7 rad
If the distance between two points is x and the moon–earth distance r , then
x = r dθ = 3.8 × 10
8
× 1.342 × 10
−7 m = 51 m
15.62 θ =
1.22λ
d
=
1.22 × 5.89 × 10 −7
30 × 10 −6
= 0.024 rad
15.63 I = I 0
2J 1 (ρ)
ρ
2
where ρ =
2π
λ
a sin θ and J 1 (ρ) is the Bessel function of the first kind.
According to the Rayleigh criterion, the separation of the peaks is equal to the
distance between the first minimum and the centre of the diffraction pattern,
that is, the first minimum of the Bessel function is at ρ = 3.83, and we have
ρ = 3.83 =
2πa
λ
R 1
X
or
γ =
R 1
X
= 0.61
λ
a
= 1.22
λ
2a
= 1.22
λ
D
where 2a = D is the diameter of the lens, γ is the angle between the two
stars, the distance between the observation screen and the lens is X (equal
to focal length of the lens) and the position of the details of the diffraction
pattern is R 1 , Fig. 15.35a, b.
Fig. 15.35 a Intensity
distribution for diffraction
from a circular aperture
described by Bessel function
Fig. 15.35 b Resolution of
diffraction patterns for two
stars, the angle between them
being γ . For the explanation
of parameters X , R 1 and ρ,
see the text