Collecting Power of Telescopes

Astrophysics | AQA A-Level Physics

Key Definitions

Collecting power: A measure of the amount of light energy a telescope collects per second. It is equivalent to the power per unit area, or intensity, of the incident radiation collected.

Collecting power and aperture diameter

Telescopes are designed to gather as much light as possible. The more light energy a telescope can gather, the brighter the images it will be able to produce.
The collecting power of a telescope is directly proportional to the square of the diameter of its objective:

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

This is because:
- Intensity is proportional to surface area.
- The surface area of a circular object of diameter $D$ is equal to $\frac{\pi D^2}{4}$.
The consequence is that objects at greater distances can also be seen, because the intensity of light from a point source decreases inversely with the square of the distance from the inverse square lawThe law stating that the intensity of radiation from a point source is inversely proportional to the square of the distance from the source: $I \propto \frac{1}{d^2}$..

Advantages of large-diameter telescopes

Larger aperture diameter telescopes are advantageous for two main reasons:
They have a greater collecting power, so images are brighter.
They have a greater resolving power (a smaller minimum angular resolution $\theta = \frac{\lambda}{D}$), so images are clearer.

Comparing collecting power

The collecting power of two telescopes can be compared using the ratio:

$$\frac{\text{collecting power of telescope 1}}{\text{collecting power of telescope 2}} = \left(\frac{D_1}{D_2}\right)^2$$

The key part is that if one telescope has double the diameter of another, its collecting power is $2^2 = 4$ times greater.

Comparing resolving power

The resolving power of two telescopes operating at the same wavelength can be compared using the ratio:

$$\frac{\theta_1}{\theta_2} = \frac{\frac{\lambda}{D_1}}{\frac{\lambda}{D_2}} = \frac{D_2}{D_1}$$

Crucially, the ratio of $\theta$ values is the inverse of the ratio of diameters. A telescope with double the diameter resolves angular separations that are half as small.

Common Mistake

Students often confuse which ratio is which. Collecting power scales with $D^2$ (so doubling the diameter gives 4 times the collecting power), while resolving power scales with $\frac{1}{D}$ (so doubling the diameter halves the minimum angular resolution, which means better resolution). Remember: the smaller the value of $\theta$, the greater the resolving power.