Astronomical Telescopes

How do lenses work?

Before we look at how telescopes allow us to magnify distant objects, we need to understand the basic operation of lenses.

A lens is characterised by its focal distance. The position of objects in relation to this distance determines what happens with the rays of light that are refracted by the lens.

There are two main types of lenses: converging and diverging. While the first kind focuses the incident light towards a point, the latter disperses it. In this explanation, we are only concerned with converging lenses since these are the ones that can be used to build the simplest model of an astronomical telescope.

Below we look at the diagrammatic functioning of converging lenses and their mathematical description to understand the power of the amplification of telescopes.

Diagrams of converging lenses

See the two images below:

These diagrams summarise the possible images formed with a converging lens. Here are the basic rules to construct these diagrams:

• Draw the object on one side of the lens (choice is arbitrary, we chose left).
• Draw the lens and mark its focal distance as f. The distances do not have to correspond to the real distance, but you do need to choose a scale so that the relation between the focal distance and the object’s distance to the lens is proportional to the real one.
• A ray travels horizontally from the object to the lens and, after being refracted by the lens, travels to a point on the horizontal middle line that is a focal distance away from the lens (focal point F).
• A ray travels from the object and crosses the lens through its middle point. This ray is not refracted and continues undisturbed.
• If the solid lines do not cross on the right of the lens, you need to extend them to the left (denoted by the dashed line) in order for the lines to cross and form a virtual image

By following these simple rules, we can find the two rays with undotted lines in the images above. It is important to note that the formation of the image depends on the object’s position with respect to the focal point of the lens.

• If the object is beyond the focal point, the light rays meet on the other side of the lens and an image is formed there. This image is turned upside down. The image will be bigger if the object is placed at less than twice the focal distance. If it is beyond that distance, the image will be smaller.
• If the object is between the focal point and the lens, the undotted light rays do not meet on the other side of the lens. By continuing the lines with the dotted lines as shown in figure 2, we can see that the image is formed on the same side of the lens where the object is. The image has the same orientation as the object and is always bigger than the object.
• If the object is on the focal point, the image is formed on the other side of the lens at an infinite distance with the same size.

When an image of an object is formed on the opposite side of a lens, the image can be projected on a screen. These images are called real images. When an image of an object is formed on the same side of a lens, the image cannot be projected on a screen, but it is observed where it forms. These images are called virtual images.

Mathematical description of converging lenses

Converging lenses obey the following equation:

\[\frac{1}{x_0} + \frac{1}{x_i} = \frac{1}{f}\]

Here, x_o is the distance of the object to the lens, f is the focal distance of the lens, and x_i is the distance of the image of the object. We always take x_o to be positive, so if x_i is negative, it will be a virtual image, and if x_i is positive, it will be a real image.

Since lenses are considered ideal systems, there is also a straightforward relationship between the vertical size and the horizontal distances. This translates into the fact that the magnification of lenses (the amount of growth of the image of an object with respect to its actual size) follows this equation:

\[M = \frac{y_i}{y_o} = \Big|\frac{x_i}{x_0} \Big|\]

Here, y_i is the vertical size of the image, y_o is the vertical size of the object, and the absolute value appears because we take magnification to be a positive quantity. The absolute value indicates the amount of growth, and the sign indicates whether the orientation of the image is the same as the object’s or if it is the opposite.

What is an astronomical refracting telescope?

Telescopes have been used since Galileo’s invention to explore the universe. Galileo’s first design used the combined refraction of two lenses to amplify images, but there have been many developments since then to enhance the observation properties of these devices. Nevertheless, the study of simple refracting telescopes provides a good understanding of the general functioning of telescopes.

Definition and characteristics of an astronomical telescope of two lenses

These two lenses (the objective lens and eyepiece lens) fulfil different roles, and their combination allows us to use telescopes as powerful observation tools.

• The objective lens collects the incident light and creates an image by focusing light. Ideally, it should be very large to collect more light (more resolution and intensity), and it should have a considerable focal distance.
• The eyepiece lens magnifies the image and creates a virtual image at infinity so that it can be seen by an observer. It should have a small focal distance.

Below is a diagram of the disposition of the two lenses and how rays of light are refracted.

The angles α and β are very small for astronomical observations, so they can be neglected. In the image, f_o is the focal distance of the objective lens, and f_e is the focal distance of the eyepiece lens. The distance between the lenses has to be the sum of their focal distances so that the virtual image is formed at an infinite distance.

Astronomical telescope magnification and formula derivation

Finally, let’s investigate how powerful astronomical telescopes are. For a telescope, the usual calculations with lenses do not yield helpful information because we are working with objects so far away that we can consider their light rays to be parallel. Furthermore, we are also working with images that are placed exactly at the focal point of a lens.

As we only offer a simplified version of lenses in the first sections, we apply these general ideas to obtain the magnification. Since magnification is the ratio of the size of the image to the actual object, we can obtain this information by using the angles α and β. It turns out that for astronomical measurements, these angles have the following expressions:

\[\alpha = \frac{\gamma}{f_e} \qquad \beta = \frac{\gamma}{f_e}\]

To find the ratio of the sizes of the image of the object and the image we observe, we have to divide the two angles, which yields:

\[M = \Big| \frac{\alpha}{\beta} \Big| = \frac{f_0}{f_e}\]

We can now see why it is useful that the objective lens has a very large focal distance and the eyepiece lens has a small focal distance.

Refracting telescopes – telescopes using large lenses for their objective. https://www.schoolphysics.co.uk/age16-19/Optics/Optical%20instruments/text/Telescopes_/index.html