Prominent astronomer Galileo Galilei designed a variant of refracting telescope in the year 1609 that is known as the Galilean telescope. The telescopic design incorporated a convergent (plano-convex) lens as the objective and a divergent (plano-concave) lens as the eyepiece. The Galilean telescope produced a non-inverted and upright image because the design does not have any intermediary focus.
Initially, the telescope designed by Galileo could magnify objects only about 30 times. This initial design was not devoid of flaws like the narrow field of view and the shape of the lens. This produced blurry and distorted images. However, in spite of these flaws, Galileo efficiently used the telescope for studying and exploring the sky. The discovery of the four moons of Jupiter and the study of the phases of Venus were some of the notable works of Galileo using this telescope.
How does a Galilean telescope work?

The telescopic design incorporated a convergent (plano-convex or biconvex) lens as the objective and a divergent (plano-concave or biconcave) lens as the eyepiece. The eyepiece is positioned in front of the objective’s focal point, having a distance equal to the eyepiece’s focal length. The converging lens has a positive optical power, and the diverging lens has a negative optical power. Therefore, the algebraic sum of the lenses’ focal length is equal to the distance between the objective and the eyepiece.
The diverging eyepiece lens intercepts the converging rays that are redirected from the objective and render them parallel, producing an image located at infinity that is virtual, magnified, and erect. The non-parallel rays of light falling at an angle of α1 to the optic axis travel at an angle α2 larger than α1 after passing through the eyepiece. The ratio between the focal length of the eyepiece and that of the objective determines the system’s magnification. The Galilean telescope has an extremely narrow field of view, and hence they can magnify only up to 30 times in practice.
Improvement in Galilean telescope design
The Galilean telescope had several drawbacks. It provided limited magnification, had a narrow field of view, formed blurry and distorted images. So, Johannes Kepler decided to devise ways of improving the pre-existing telescopic design and proposed the idea of the Keplerian telescope in 1610. The Keplerian telescope was a relatively new type of telescope, having a converging lens as the eyepiece. This design produced a higher degree of magnification with comparatively less distortion than a Galilean telescope. This telescope formed images upside down, but that is not a matter of concern in astronomy. At present day, the Galilean telescope design can only be seen in inexpensive low power binoculars.
Discoveries made by the Galilean Telescope
Jupiter’s four moons

source: NASA/JPL/DLR, Jupiter and the Galilean Satellites, marked as public domain, more details on Wikimedia Commons
One of the most important discoveries in the field of astronomy was the four moons of Jupiter (Io, Europa, Ganymede, and Callisto). Galileo discovered the four brightest moons of Jupiter (now called the Galilean moons) with his telescope’s help. These moons were the first objects to be known to orbit a planet other than the Earth.
Moon’s Appearance

Galileo observed how the Moon was lit and how it varied with time. After his observations, he deduced that the variations occur due to the lunar mountains’ shadows and the Moon’s craters.
Milky way’s Clouds
Galileo discovered that the Milky Way comprised of a massive number of stars. Most of these stars were too faint to be perceived discretely with the naked eye. These stars packed together appeared to be similar to a cloud when seen from the Earth.
Phases of Venus

image source: Nichalp 09:56, 11 June 2006 (UTC), Phases-of-Venus, marked as public domain, more details on Wikimedia Commons
Galileo discovered that Venus also shows a similar set of phases like the Moon when seen from the Earth. But unlike the Moon, Venus’ phases can be observed only with the help of a telescope as it appears smaller in size from Earth. Galileo became the first person to observe these phases.
Galileo’s time believed that the Earth lies at the center and all the other planets, the Moon and the Sun, orbited around it. When Galileo discovered the phases of Venus, he knew that this could be explained only if the Sun was being orbited by all the planets, including Earth and Venus. This created a controversy. Galileo claimed the geocentric theory to be incorrect based on his findings and advocated heliocentric theory.
The heliocentric theories were not accepted by the Catholic Church and banned Galileo to study or defend heliocentrism. When Galileo refused to do so, he was sentenced to prison till his death in 1642.
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