Telescope Resolving Power
Let a be the diameter of the objective of the telescope. Consider the incident rays of light from two neighbouring points of a distant object. The image of each point object is a Fraunhofer diffraction pattern. Let P1 and P2 be the positions of the central maxima of the two images. According to Rayleigh, these two images are said to be resolved if the position of the central maximum of the second image coincides with the first minimum of the first image and vice-versa. The path difference between the secondary waves travelling in the directions AP1 and BP1 is zero and hence they reinforce with one another at P1. Similarly, all the secondary waves from the corresponding points between A and B will have zero path difference. Thus, P1 corresponds to the position of the central maximum of the first image.
The secondary waves travelling in the directions AP2 and BP2 will meet at P2 on the screen. Let the angle P2AP1 be dθ. The path difference between the secondary waves travelling in the directions BP2 and AP1 is equal to BC.
From the Δ ABC,
BC = AB sin dθ = AB.dθ = a.dθ (for small angles)
If this path difference a.dθ = λ, the position of P2 corresponds to the first minimum of the first image. But P2 is also the position of the central maximum of the second image. Thus, Rayleigh’s condition of resolution is satisfied if
a.dθ = λ
or, dθ = λ/a (i)
The whole aperture AB can be considered to be made up of two halves AO and OB. The path difference between the secondary waves from the corresponding points in the two halves will be λ/2. All the secondary waves destructively interfere with one another and hence P2 will be the first minimum of the first image. The equation dθ = λ/a holds good for rectangular apertures. For circular aperture, this equation, according to Airy, can be written as
dθ = (1.22 λ)/a (ii)
where λ is the wavelength of light and a is the aperture of the telescope objective. The aperture is equal to the diameter of the metal ring in which the objective lens is mounted. Here dθ refers to the limit of resolution of the telescope. The reciprocal of dθ measures the resolving power of the telescope.
∴ 1/dθ = a/(1.22 λ) (iii)
From equation (iii), it is clear that a telescope with large diameter of the objective has higher resolving power, dθ is equal to the angle subtended by the two distant object points must have, so that their images will appear just resolved according to Rayleigh’s criterion.
If ƒ is the focal length of the telescope objective, then
dθ = r/f = (1.22 λ)/a
or, r = (1.22 fλ)/a (iv)
where r is the radius of the central bright image. The diameter of the first dark ring is equal to the diameter of the central image. The central bright disc is called the Airy’s disc.
From equation (iv), if the focal length of the objective is small, the wavelength is small and the aperture is large, then the radius of the central bright disc is small. That is the diffraction patterns will appear sharper and the angular separation between two just resolvable point objects will be smaller. Correspondingly the resolving power of the telescope will be higher.
Let two distant stars subtend an angle of one second of an arc at the objective of the telescope.
1 second of an arc = 4.85 × 10-6 radian. Let the wavelength of light be 5500 Å. Then, the diameter of the objective required for just resolution can be calculated from the equation
dθ = (1.22 λ)/a
= 13.9 cm approximately
The resolving power of a telescope increases with increase in the diameter of the objective. With the increase in the diameter of the objective, the effect of spherical aberration becomes appreciable. So, in the case of large of large telescope objectives the central portion of the objective is covered with a stop so as to minimize the effect of spherical aberration. This, however, does not affect the resolving power of the telescope.
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