Astrophysics is an extraordinary scientific discipline because most of the phenomena it studies do not take place (and cannot take place) in a lab on Earth. This is why gathering astronomical data is an essential subject that relies heavily on the characteristics of the devices used to perform measurements. These devices are usually called telescopes or astronomical telescopes.
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Jetzt kostenlos anmeldenAstrophysics is an extraordinary scientific discipline because most of the phenomena it studies do not take place (and cannot take place) in a lab on Earth. This is why gathering astronomical data is an essential subject that relies heavily on the characteristics of the devices used to perform measurements. These devices are usually called telescopes or astronomical telescopes.
The making of astronomical telescopes is a very old discipline. In the seventeenth century, Galileo Galilei became the first person to point a telescope towards the sky. His own telescope was able to magnify objects by up to twenty times, which allowed him to observe distant objects with his eyes. Nowadays, telescopes do not require us to look with our eyes because machines can gather data more precisely than we do (digital astronomical telescopes). Also, if we only observe with our eyes, we are limited to measurements in the visible region of the electromagnetic spectrum.
Despite this, it is still relevant to study how telescopes (based on the first design by Galileo) function to understand how the magnification of other telescopes works.
Before we look at how telescopes allow us to magnify distant objects, we need to understand the basic operation of lenses.
A lens is an optical, physical device that focuses or disperses light due to refraction.
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.
The focal distance is the distance at which we can place an object to form its image at an infinite distance.
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.
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:
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.
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.
Converging lenses obey the following equation:
\[\frac{1}{x_0} + \frac{1}{x_i} = \frac{1}{f}\]
Here, xo is the distance of the object to the lens, f is the focal distance of the lens, and xi is the distance of the image of the object. We always take xo to be positive, so if xi is negative, it will be a virtual image, and if xi 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, yi is the vertical size of the image, yo 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.
Take a converging lens whose focal distance f is 10cm. Calculate the characteristics of the image of an object if it is placed at:
a) 15cm from the lens.
We can calculate the distance where the image is produced by applying the formula for converging lenses:
\(\frac{1}{x_0} + \frac{1}{x_i} = \frac{1}{f} \rightarrow x_i = \frac{1}{\frac{1}{f} - \frac{1}{x_0}} = \frac{1}{\frac{1}{10cm} - \frac{1}{15cm}} = 30 cm\)
Since the quantity is positive, the image is formed on the opposite side of the lens and upside down (as we know from the first diagram). We can now calculate the magnification:
\(M = \frac{y_i}{y_0} = \Big| \frac{x_i}{x_0} \Big| = \Big|\frac{30 cm}{15cm} \Big| = 2\)
This means that the image produced is double the size of the object.
b) 30cm from the lens
We can calculate the distance where the image is produced by applying the formula for converging lenses:
\(\frac{1}{x_0} + \frac{1}{x_i} = \frac{1}{f} \rightarrow x_i = \frac{1}{\frac{1}{f} - \frac{1}{x_0}} = \frac{1}{\frac{1}{10 cm} - \frac{1}{30 cm}} = 15 cm\)
Since the quantity is positive, the image is formed on the opposite side of the lens and upside down. We can calculate the magnification as
\(M = \frac{y_i}{y_0} = \Big|\frac{x_i}{x_0} \Big| = \Big| \frac{15 cm}{30 cm} \Big| = 0.5\)
This means that the image produced is half the size of the object.
c) 5cm from the lens.
We can calculate the distance where the image is produced by applying the formula for converging lenses:
\(\frac{1}{x_0} + \frac{1}{x_i} = \frac{1}{f} \rightarrow x_i = \frac{1}{\frac{1}{f} - \frac{1}{x_0}} = \frac{1}{\frac{1}{10 cm} - \frac{1}{5 cm}} = -10 cm\)
Since the quantity is negative, the image is formed on the same side of the lens and upright (as we know from the second diagram). We can calculate the magnification as
\(M = \frac{y_i}{y_0} = \Big|\frac{x_i}{x_0} \Big| = \Big| \frac{-10 cm}{5 cm} \Big| = 2\)
This means that the image produced is double the size of the object.
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.
A two-lens astronomical refracting telescope is a device that magnifies the images of distant objects by combining two converging lenses one after the other.
These two lenses (the objective lens and eyepiece lens) fulfil different roles, and their combination allows us to use telescopes as powerful observation tools.
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, fo is the focal distance of the objective lens, and fe 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.
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.
If we have a telescope whose objective lens has a focal distance of 1m and whose eyepiece lens has a focal distance of 1mm, the magnification is 1000. This is the power of telescopes, and they can be made of more complex combinations of lenses to increase their power even.
Refracting telescopes – telescopes using large lenses for their objective. https://www.schoolphysics.co.uk/age16-19/Optics/Optical%20instruments/text/Telescopes_/index.html
Astronomers use both kinds of telescopes since they allow us to gather data. However, the resolution and power of reflecting telescopes are greater, so they are the ones used for science.
An astronomical telescope receives light and processes it via a set of lenses and/or mirrors to magnify it and process it with digital or visual means.
An astronomical telescope is a device that allows us to gather data on distant objects in the universe.
Choose the correct answer.
A converging lens focuses light rays and a diverging lens disperses them.
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A converging lens can produce real and virtual images.
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Virtual images are always bigger than the objects that produce them for converging lenses.
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The magnification power of two-lens telescopes depends only on the focal distance of the lenses it is made of.
Choose the correct answer.
The objective lens has to be as large as possible and have the biggest focal distance possible.
Why is gathering data relevant in astrophysics?
Gathering data is relevant in astrophysics because we cannot reproduce astronomical bodies in labs on Earth.
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