Resolving and collecting power of telescopes (AQA A-Level Physics): Revision Notes

Resolving and collecting power of telescopes

Introduction to resolving power

When we use a telescope to observe distant objects in space, one of the most important characteristics we need to consider is its resolving power. This refers to the telescope's ability to distinguish between two closely spaced objects and produce separate, distinct images of them. Without adequate resolving power, two stars that are close together might appear as a single blurred point of light.

The limitation on resolving power arises from the wave nature of electromagnetic radiation. When light (or other electromagnetic waves) passes through the opening of a telescope, the waves don't simply travel straight through. Instead, they undergo diffraction, bending around the edges of the aperture. This diffraction causes the waves to interfere with each other, creating a characteristic pattern.

Diffraction and Airy discs

Because of diffraction effects, even when a telescope observes a perfect point source like a distant star, the image formed is not a perfect point. Instead, the telescope produces a pattern called an Airy disc. This consists of a bright central disc surrounded by progressively fainter concentric rings.

The Airy disc is a diffraction pattern that results from constructive and destructive interference of light waves as they pass through the circular aperture of the telescope. The central bright region contains most of the light energy, with the first dark ring (the first minimum) marking the boundary of this central maximum.

Angular width of the Airy disc

The angular position of the first dark fringe (first minimum) in the Airy disc can be calculated using the following relationship:

$\sin \theta = \frac{\lambda}{D}$

Where:

• $\lambda$ (lambda) is the wavelength of the electromagnetic radiation in metres
• $D$ is the diameter of the objective (the primary lens or mirror) in metres
• $\theta$ (theta) is the angular position in radians

In astronomical observations, the angles involved are extremely small. This allows us to make a useful approximation: for small angles measured in radians, $\sin \theta \approx \theta$. This simplifies our equation to:

$\theta \approx \frac{\lambda}{D}$

This simplified formula tells us two important things about improving the resolving power of a telescope:

• Increasing the diameter $D$ of the objective makes $\theta$ smaller, which reduces the width of the central maximum. This means a larger telescope can resolve finer details.
• Using shorter wavelengths (smaller $\lambda$) also reduces $\theta$, producing a narrower Airy disc. This is why ultraviolet or X-ray telescopes can potentially achieve better resolution than optical telescopes, assuming the same aperture size.

Overlapping Airy discs and the Rayleigh criterion

When we observe two point sources that are close together (such as two nearby stars or a binary star system), each produces its own Airy disc. If the objects are sufficiently separated, the two Airy discs will be distinct and the objects are easily resolved as separate sources. However, as the objects get closer together, their Airy discs begin to overlap.

At the critical separation, where two objects are just resolvable, we can write:

$\theta = \frac{x}{L} \approx \frac{\lambda}{D}$

Where:

• $x$ is the physical separation between the two objects
• $L$ is the distance from the telescope to the objects

The Rayleigh criterion provides a precise definition for the minimum separation needed to resolve two point objects. It states that two point sources can be resolved as separate objects if their angular separation is at least:

$\theta \approx \frac{\lambda}{D}$

The imaging process operating at this limit is described as being diffraction-limited, meaning that diffraction is the primary factor limiting the telescope's performance.

Angular measurements in astronomy

Astronomers use angular measurements to describe the apparent size of objects in the sky (their angular size) and the separation between objects. These measurements use a system based on degrees, with subdivisions into smaller units.

The angular measurement system works as follows:

• 1 degree (1°) = 1/360 of a complete circle
• 1 arcminute (1') = 1/60 of a degree
• 1 arcsecond (1") = 1/60 of an arcminute = 1/3600 of a degree

These units allow astronomers to express very small angles precisely.

Collecting power of telescopes

The collecting power of a telescope is another vital performance parameter, distinct from resolving power. It measures the telescope's ability to gather incoming electromagnetic radiation from astronomical sources.

The collecting power depends on the area of the objective that is exposed to incoming radiation. For a circular aperture with diameter $D$, the area is proportional to $D^2$. Specifically, the area of a circle with diameter $D$ equals:

$A = \pi \left(\frac{D}{2}\right)^2 = \frac{\pi D^2}{4}$

This leads to the fundamental relationship:

$\text{Collecting power} \propto D^2$

Where $D$ is the diameter of the objective.

For optical telescopes specifically, the collecting power is often called the light-gathering power (LGP), measured in square metres (m²). The LGP provides a relative comparison between different telescopes - it indicates how much light each telescope can collect from astronomical sources. A telescope with greater light-gathering power will produce brighter images, making it possible to observe fainter objects or gather data more quickly.

Benefits of large-aperture telescopes

Both resolving power and collecting power improve significantly with larger objective diameters. This explains why astronomers continually seek to build telescopes with larger and larger apertures:

For resolving power: Since minimum angular resolution $\theta \approx \frac{\lambda}{D}$, increasing $D$ reduces $\theta$, allowing the telescope to distinguish finer details and separate closer objects.

For collecting power: Since collecting power $\propto D^2$, doubling the diameter quadruples the light-gathering ability. A telescope with a 4-metre mirror collects four times as much light as a 2-metre mirror.

These advantages make large-aperture telescopes extremely valuable for astronomy, enabling observations of fainter and more distant objects while resolving finer details in astronomical images.