Exoplanets

Already antique astronomers used their curiosity and innovative engineering abilities to determine the large distances in our Solar system. When humans start sailing on the oceans they saw how the airframe disappeared before the mast when a boat was passing the horizon which leads to the speculation that Earth was round. This notation was established by 3rd century BC by Greek astronomy. In 240 B.C the Greek astronomer Eratosthenes that also is known as the father of geography as he introduced the concept of longitude and latitude and draw a map over at that time the known world. He made a very accurate measurement of the circumference of the Earth. In the city of Syene 800 kilometers south of Alexandria (Egypt) there where a famous well. Precisely at summer solstice once a year the Sun's rays shone straight down into the well. At the same time in Alexandria. Eratosthenes measured the length of the shadow from a stick and calculated the angle to:    

$$\tan^{-1} \frac{d_{shadow}}{d_{stick}}\approx 7.2^{\circ }$$
7 degree is 1/50th the circumference of a circle and knowing the distance to Syene is 800 kilometers the earth circumference should be 50 times that distance 40 000 kilometers.

Or using trigonometry: 
$$2\pi \frac{800000}{\tan 7.2^{\circ }}\approx 40241005$$

Another Greek astronomer Aristarchus of Samos at the same time calculated the distance to the Moon (R). By looking at a lunar eclipse and calculating how long time it took for the Earth shadow to cross over the Moon that takes 3 hours and 40 minutes (t) it will take 29 days for the Moon to orbit an entire revolution around the Earth (T). He estimated the distance to 60 earth radii (r) that is correct.

$$\frac{\pi R}{r}=\frac{T}{t}$$

$$\Rightarrow R\approx 60r$$

He also estimated the distance to the Sun. During a solar eclipse, the Moon covers almost the entire disc. This tells us that the Sun is larger than the Moon and farther away. During half moon he assumed that the Moon forms a right angle with the Sun and Earth he measured that angle to 87 degrees.

$$\frac{R_{\odot}}{60r}=\cos 87^{-1}\approx 20$$
He came to the conclusion that the Sun is 20 times farther away from the Moon. This is wrong the Sun is 400 times farther away from the Moon as the angle is closer to 90. 
On a side note: trigonometric functions had not yet been invented the ancient greek used geometrical relations to find proportions. 

The first measurement of the distance to a planet was made by Gian Domenico Cassini. In 1672, He used a technique called parallax to measure the distance to Mars. If you hold up your thumb at one arm distance look at with just the left eye and then the other you will see that object farther away is shifting position that is caused by the separations of your eyes. You are watching the object from two different positions. The distance the thumb seems moving is its parallax. If you know the distance between your eyes and the angle by which your thumb moved against the background, you can calculate the length of your arm. By making an observation on two different places at Earth one can calculate the distance to objects far away in the same way.
To measure the distance to a star like Proxima Centauri that is 4.24 light-years away. One could take pictures of the star from two points when Earth is at one side of the Sun and then six months later when Earth is on opposite sides of the Sun and then calculating the parallax angle that more distant stars seem moving. The parallax angle Proxima shifting is 0.77 arc second one arc second is 1/3600 of a degree. A distance to a star was calculated for the first time in 1838 by Friedrich Bessel who measured the parallax of 61 Cygni as 0.314 arc second 11.4 light-years away. To measure large distances to stars the unit parsec (pc) is often used instead of light-years. A parsec is a distance that the parallax angle is 1 arc second that is 3.26 light-years. Parallax can only be used to find distances under 100 parsecs away

To measure the luminosity that is the total amount of energy emitted per time by an astronomical object or the brightness a logarithmic scale are used that is called the absolute magnitude. The sun has a magnitude of -27 and the dimmest objects visible with the naked eye has a magnitude of 6. The apparent magnitude is the magnitude of the object seen at 10 parsecs away. The brightness of a star is inversely proportional to the square of its distance.
$$L\sim \frac{1}{D^{2}}$$

French astronomer Charles Messier cataloged 110 astronomical objects the closest large galaxy was cataloged M31 in 1764. He thought it was a nebula within our galaxy. The object is also known as Andromeda and is visible with the naked eye. When astronomers discovered a variable star called novae in Andromeda in 1917 they noticed that it was 10 times less bright than similar stars in our galaxy. A Cepheid variable star is a very bright star that pulsates in a predictable way.
once the period has been measured its luminosity can be estimated. Then the distance to the object could be calculated in parsec with this formula

$$d=10^{(m-M+5)/5}$$  

where m is the apparent magnitude and M the absolute magnitude of the Cepheid. Edwin Hubble in 1925 calculated that the galaxy 1.5 million light-years away. Modern calculations show it is 2.5 million light-years away or 778 000 parsec.

Image credit: NASA/JPL-Caltech

Andromeda galaxy is blueshifted it moving towards the milky way due to gravitational forces but all distant galaxies are redshifted they are moving away because the universe is expanding. The velocity of a galaxy is proportional to its distance from us by the equation 
$$v=Hd$$
Where H is the Hubble constant that is estimated to be 70.0 km/sec/Mpc 
Objects like quasars that are the ultraluminous nuclei of galaxies are extremely redshifted. For example, the quasar 3C 273 has a redshift of 0.158 which means it moving away at a speed of 44000 km/s (0.158 * speed of light)
using Hubble's law its distance could be calculated to 2 billion light-years or 620 Mpc.
The most distant object GN-z11 has a redshift on 11.09 and is 13.39 billion light-years away (actually it is much further away as space has been expanding during the time it takes the light to reach us).

The riddle of the size of the universe has involved scientists ever since the childhood of cosmology. Newton believed that the universe is infinite, while Kepler believed in a finite.
Albert Einstein was the first physicist that gave the concept of a finite universe a sustainable theoretical foundation. The gravity can make space curve so much that the overall structure closes itself in the same way as the surface of a globe. The shape of the universe is depending on how much mass it is in the universe. The density parameter was derived by Alexander Friedmann in 1922 from Einstein's field equations.
$$\Omega =\frac{\rho}{\rho_{c}}$$
where ρ is the actual density of the Universe and ρc is the critical density.
The critical density is according to Friedmann equations
$$\rho_{c} =\frac{3H^{2}}{8\pi G}$$

where G is the gravitational constant 6.674×10−11 m3/(kg⋅s2)  and H is the Hubble parameter a function of time that tells us how fast the universe is expanding it may be derived from the same equations as

$$H^{2}= \frac{8\pi G \rho }{3}-\frac{kc^{2}}{a}$$

then the density parameter becomes

$$ \Omega = \frac{H^{2}+\frac{kc^{2}}{a}}{H^{2}}$$
where c is the speed of light in vacuum and  k is the curvature constant and a is the scale factor

If the density parameter: 

• Is bigger than 1 and k equals 1, then the universe is finite and has a spherical shape
• Is smaller than 1 and k equals -1, then the universe is infinite or finite and has a hyperbolic shape
• Is equal to 1 and k equals 0, then the universe is infinite and is flat 

Credit: NASA / WMAP Science Team

The Planck space observatory was a spacecraft operated by the European Space Agency (ESA) from 2009 to 2013 and mapped the cosmic microwave background CMB. CMB is the radiation leftover from the big bang. 
Several observations have indicated that the universe is flat and that fits very well with our current theoretical models, but re-analysis of the Planck data shows that we live in a finite spherical universe where the density parameter is bigger than 1
Here is the paper: nature.com
Here is an article about it quantamagazine.org

The density of the universe is according to the article calculated to be about 6 hydrogen atoms per cubic meter of space and the critical density is 5.7 hydrogen atoms per cubic meter of space, which gives the density parameter value of 6/5.7 =1.05.