Refracting Telescopes

Astrophysics | AQA A-Level Physics

Key Definitions

Refracting telescope (refractor): A telescope that uses two converging lenses to produce a magnified image of a distant object.

Normal adjustment: The arrangement where the final image is formed at infinity, achieved when the focal points of the objective and eyepiece lenses coincide.

Angular magnification: The ratio of the angle subtended by the image at the eye to the angle subtended by the object at the unaided eye.

How a refracting telescope works

A refractor uses two converging lenses: an objective lensThe larger lens at the front of the telescope. It has a long focal length and collects light from distant objects, forming a real intermediate image at its focal point. and an eyepiece lensThe smaller lens closest to the observer's eye. It has a short focal length and acts as a magnifying glass on the intermediate image..
The objective lens collects light from a distant object and brings it to a focus at its focal length $f_o$.
The eyepiece lens is placed at a distance of its focal length $f_e$ from that intermediate image, producing parallel rays of light for the observer.
Fundamentally, a refractor in normal adjustment has the focal points of both lenses meeting at the same place, so the total length of the telescope is $f_o + f_e$.

Drawing the ray diagram

In the exam, you will be expected to draw a ray diagram with at least 3 non-axial rays passing through both lenses. The key steps are:
Step 1: Draw and label the two lenses and the principal axis. The objective focal length must be longer than the eyepiece focal length ($f_o > f_e$).
Step 2: Mark and label the common principal foci where $f_o$ and $f_e$ meet.
Step 3: Draw an off-axis ray through the centre of the objective to the eyepiece. Draw the intermediate image at the common focal point.
Step 4: Draw a construction line from the end of the intermediate image through the centre of the eyepiece.
Step 5: Draw the remaining rays to the eyepiece, crossing at the focal point. All rays emerging from the eyepiece must be parallel to the construction line.

Angular magnification

For very distant objects, it makes more sense to use angular magnification rather than linear magnification, because we cannot easily measure an object's physical size or distance.
The angular sizeThe angle, in radians, subtended by an object at the observer's eye (or telescope). Calculated as $\theta = h / d$ where $h$ is the physical size and $d$ is the distance. of an object is given by $\theta = h / d$, where $h$ is the object's size and $d$ is its distance.
Angular magnification $M$ is defined as:

$$M = \frac{\beta}{\alpha}$$

Where $\beta$ is the angle subtended by the image at the eye and $\alpha$ is the angle subtended by the object at the unaided eye.

Deriving the magnification equation

From the ray diagram in normal adjustment, the rays inside the telescope form similar triangles. Using the small angle approximation ($\tan\theta \approx \theta$ for small angles in radians):

$$\alpha = \frac{h}{f_o} \quad \text{and} \quad \beta = \frac{h}{f_e}$$

Combining these gives the angular magnification in normal adjustment:

$$M = \frac{\beta}{\alpha} = \frac{f_o}{f_e}$$

This tells us that to achieve greater magnification, you need a longer objective focal length and a shorter eyepiece focal length. The consequence is that refractors must be very long, since the total length is $f_o + f_e$.

Common Mistake

When drawing the ray diagram, students often draw axial rays (parallel to the principal axis) rather than non-axial rays (at an angle to the principal axis). They also frequently bend the rays at the intermediate image instead of at the lenses. Make sure the rays emerging from the eyepiece are parallel to each other and to the construction line, not converging or diverging.