9.1.1 Astronomical telescope (two converging lenses)

Refracting Telescopes

Refracting telescopes use two converging lenses to gather light and create magnified images of distant objects, such as stars or planets.

The two main components are:

1. Objective Lens:

• This lens is designed to collect light from a distant object and form a real image.
• The objective lens should have a long focal length and a large diameter to gather as much light as possible. This is crucial for observing faint objects in the night sky.
• The collecting power of a telescope is proportional to the square of the radius of its objective lens, meaning larger lenses collect significantly more light.

2. Eyepiece Lens:

• This lens magnifies the image created by the objective lens, so the observer can see it clearly.
• The eyepiece produces a virtual image at infinity, allowing parallel light rays to enter the observer's eyes, which reduces eye strain. This means that the viewer doesn't need to refocus their eyes as they look back and forth between the telescope and the actual sky.

Normal Adjustment:

• A telescope is in normal adjustment when the distance between the objective and eyepiece lenses equals the sum of their focal lengths $(f_o + f_e)$.
• In this setup, the principal foci of both lenses are in the same position.

Ray Diagram in Normal Adjustment

The ray diagram for a refracting telescope in normal adjustment shows light from a distant object passing through the objective lens, which brings it to a focus and forms a real image. The eyepiece then magnifies this image.

Magnification and Angular Magnification

The magnifying power or angular magnification $(M)$ of a telescope is given by the formula:

$$M = \frac{\text{angle subtended by the image at the eye}}{\text{angle subtended by the object at the unaided eye}}$$

This can also be simplified to:

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

Where:

• $\alpha$ is the angle subtended by the image at the eye.
• $\beta$ is the angle subtended by the object at the unaided eye. If the angles are small (less than 10°, this formula can be further simplified:

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

Where:

• $f_o$ is the focal length of the objective lens,
• $f_e$ is the focal length of the eyepiece lens.

Diagram Explanation for Small Angles

When both angles $\alpha$ and $\beta$ are less than 10°, they are approximately proportional, allowing for this simplified calculation. In the diagram:

• $\alpha$ is the larger angle, and $\beta$ is the smaller angle,
• Using this approach, the telescope can provide a clear and magnified view of distant astronomical objects.