Derivation of Resolving Power of a Telescope

# Derivation of Resolving Power of a Telescope

Let us derive the expression for the limit of resolution (\( d\theta \)) and the resolving power of a telescope objectively using the principles of Fraunhofer diffraction.

1. The Setup

Consider a telescope with an objective lens of diameter (aperture) \( D \). Imagine two distant stars, \( S_1 \) and \( S_2 \), located very close to each other in the sky.

• The light rays from the first star \( S_1 \) arrive parallel to the principal axis and form a diffraction pattern centered at \( P_1 \).
• The light rays from the second star \( S_2 \) arrive at a small angle \( d\theta \) relative to \( S_1 \) and form a diffraction pattern centered at \( P_2 \).

2. Rayleigh's Criterion for Resolution

According to Rayleigh's criterion, the two stars are just resolved when the central maximum of the diffraction pattern of \( S_2 \) falls exactly on the first minimum (dark ring) of the diffraction pattern of \( S_1 \).

This means the angular separation \( d\theta \) between the stars must be equal to the angular separation between the central maximum and the first minimum of the diffraction pattern formed by the telescope aperture.

3. Path Difference Calculation

Consider the wavefront of light from star \( S_2 \) arriving at the aperture \( AB \) of diameter \( D \). Since the light arrives at an angle \( d\theta \), there is a path difference between the rays entering at the top edge \(A\) and the bottom edge \(B\) of the lens.

From geometry, if we drop a perpendicular from \( B \) to the ray at \( A \), the path difference \( \Delta \) is given by:

\[ \Delta = D \sin(d\theta) \]

4. Condition for First Minimum

We know from the theory of single-slit diffraction that the condition for the first minimum (destructive interference) is that the path difference between the extreme rays of the aperture must be equal to one wavelength (\( \lambda \)).

\[ \Delta = 1 \cdot \lambda \]

However, a telescope objective is a circular aperture, not a rectangular slit. The diffraction pattern for a circular aperture is described by Bessel functions (forming the Airy Disc). The mathematical correction factor for a circle introduces the number 1.22.

Thus, for a circular aperture, the condition for the first minimum is:

\[ \Delta = 1.22 \lambda \]

5. Equating and Solving

Equating the geometric path difference with the diffraction condition:

\[ D \sin(d\theta) = 1.22 \lambda \]

Since astronomical angles are extremely small, we can use the small-angle approximation where \( \sin(d\theta) \approx d\theta \) (in radians).

\[ D \cdot d\theta = 1.22 \lambda \]

Rearranging for \( d\theta \), we get the formula for the Limit of Resolution:

\[ d\theta = \frac{1.22 \lambda}{D} \]

6. Final Formula for Resolving Power

The Resolving Power is defined as the reciprocal of the limit of resolution. Therefore:

\[ \text{Resolving Power} = \frac{1}{d\theta} = \frac{D}{1.22 \lambda} \]

Conclusion: To resolve two stars that are very close together (small \( d\theta \)), you need a high Resolving Power, which requires a large diameter \( D \) or a smaller wavelength \( \lambda \).