Resolving Power of Telescopes

Diffraction at a circular aperture

• A circular aperture, such as a lens or mirror in a telescope, is designed so that a cone of light can enter a region behind it. This allows light to act like a point source once it passes through.
• When two point sources are placed near each other, or viewed from a large distance, they will appear to be a single unresolved source of light. For example, two distant car headlights may initially appear as a single point source until the car moves close enough for your eyes to resolve them into two individual headlights.
• Light from any object passing through a circular aperture, including the human eye, will diffract and create an interference pattern upon the detector inside.
• The pattern is circular and is an approximation of the single-slit diffraction pattern for a circular aperture.
• The large central maximumThe brightest, central region of the circular diffraction pattern. It is also called the Airy disc and is twice as wide as the further maxima in the pattern. is called an Airy disc and is twice as wide as the further maxima in the pattern.

The Rayleigh criterion

• Diffraction affects how well a telescope can resolve fine detail. The resolving power or minimum angular resolution of a telescope can be determined using the Rayleigh criterion.
• The Rayleigh criterion states that: two sources will be resolved if the central maximum of one diffraction pattern coincides with the first minimum of the other.
• The resolution of a telescope can be increased (meaning finer detail can be resolved) by reducing the amount the light diffracts. This can be achieved by:
  • Increasing the diameter of the aperture.
  • Operating at a shorter wavelength of light.

Three resolution scenarios

• Unresolved: Two sources are too close together and their diffraction patterns overlap significantly, appearing as one single source.
• Just resolved: Two sources can only just be distinguished as two separate sources, as defined by the Rayleigh criterion. Their diffraction patterns do not overlap significantly.
• Clearly resolved: Two sources are far enough apart that their diffraction patterns are clearly distinct.

Deriving the resolving power equation

• The Rayleigh criterion can be mathematically described by considering angular separation and single-slit diffraction through a circular aperture.
• Angular separation can be calculated using the equation:

• Where $\theta$ = angular separation (rad), $s$ = distance between the two sources (m), $d$ = distance between the sources and the observer (m).
• In single-slit diffraction, minima in the pattern appear at angles given by:

• Where $n$ = the order of the minimum (1, 2, 3 etc.), $\lambda$ = the wavelength of the light (m), $D$ = the slit width or aperture diameter (m).
• For a telescope, the first minimum (when $n = 1$) occurs when:

• Using the small-angle approximation ($\sin\theta \approx \theta$), we obtain an expression for the minimum angular resolution of the telescope:

• Where $\theta$ = minimum angular resolution of the telescope (rad), $\lambda$ = operating wavelength of the telescope (m), $D$ = diameter of the telescope's aperture (m).

Applying the Rayleigh criterion

• The Rayleigh criterion can therefore be written mathematically as follows:
• Sources are resolvable when $\theta > \frac{\lambda}{D}$.
• Sources are just resolvable when $\theta \approx \frac{\lambda}{D}$.
• Sources are not resolvable when $\theta < \frac{\lambda}{D}$.
• For a circular aperture, the value is multiplied by a factor of 1.22:

• This is the same expression for two sources that are "just resolvable" but removes the need for the "approximately equals" ($\approx$) sign.
• Crucially, the smaller the value of $\theta$, the greater the resolving power. A telescope with a smaller minimum angular resolution can distinguish finer details.