Exploring the universe

How do we calculate the distance to the stars

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


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 
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).

Andromeda Hubble

Hability around K-type stars

A new study suggests that the best strategy for discovering signs of life in the atmosphere of exoplanets is to study K-dwarf stars. Link to the study: looking for life try around k dwarfs

One way to identify life on other planets is by detecting atmospheric biosignatures. If Earth-like exoplanets both have the presence of methane and oxygen in the atmosphere it would be a possible proof that there exists life on the planet, but it is very difficult to detect it. Because the methane is destroyed by chemical reactions driven by the host stars light. Smaller and dimmer stars than our G-type Sun is better candidates. A problem with small red dwarf M-type stars is their stellar activity and radiation could be deadly for living organisms. The best candidate stars are K-type stars which are in size between G-type and M-type stars.

We have so far discovered several possible habitable candidates exoplanets around K-type stars. I have picked some of the most interesting planets for this article.

HD 85512 b is an exoplanet that is orbiting the K-type star HD 85512 was discovered in 2011. It was considered along with Gliese 581 d (that probably does not exist) to be one of the best candidates for habitability. It is 3.6 times the mass of Earth on is just outside the inner edge of the Goldilocks zone of its star, where it could be too hot and it possible that the planet is tidally locked. HD 85512 is 36 light-years away from Earth in the constellation Vela.

HD 40307 g is located 42 light-years away in the constellation Pictor. The planet was discovered in 2012 by using the ESO HARPS telescope that is using the radial velocity method to find exoplanets. It is inside the Goldilocks zone of its star. The discoverer Hugh Jones said that the longer orbit of the planet means that it has the correct climate and atmosphere to support life. Later studies suggest that the planet could be a mini Neptunian that has migrating inward.

Image credit: NASA Ames/JPL-Caltech/T. Pyle

The most interesting K-type star with a planet system is Kepler-62. It is located about 1200 light-years away in the constellation Lyra. The system has two potentially habitable candidates Kepler-62f and Kepler-62e. The planets were announced in 2013. Studies have suggested that the planets are water worlds with oceans that cover the entire surface. That suggests that life on Kepler 62e and Kepler 62f would be different than on our planets. Life could exist there but technology advanced civilizations would have problems to evolve with no access to metals or fire for metallurgy. Kepler-62f orbits its host star every 267.29 days and is a Super Earth with 1.41 of Earth radius and it is possible that it could have a moon. Kepler-62 system is much older than our solar system. Intelligent life would have 3 billion years more than us to evolve. A study released in a June 2018 suggests that Kepler-62f may have seasons and a climate similar to Earth. Kepler 62 is being specially targeted being targeted by the SETI program in search of extraterrestrial life.


Kepler-62 e Kepler-62 f HD 85512 b HD 40307 g

Researchers have taken a picture of a black hole

For the first time, researchers have managed to take a picture of a black hole. The sensational news was announced at press conferences sent simultaneously from six different places in the world. The discovery was made by the Event Horizon Telescope ETH that consist of a global network of radio telescopes that combined data from eight stations located on different places in the world with help of a technique called very long baseline Interferometry. The black hole is 6.5 billion times the mass of the sun. It is located 55 million light-years from Earth in the constellation Virgo at the center of supergiant elliptical galaxy Messier 87 also known as Virgo A or NGC 4486.

Credit: Event Horizon Telescope Collaboration

Source: eventhorizontelescope

Black holes were first discovered as a theoretical solution in the theory of general relativity published by Albert Einstein in 1915. General relativity describes gravity as a geometric property of space and time. Heavy objects like our Earth will curve space-time with its mass. And a black hole is such massive object that at a point nothing can escape its gravitational pull not even light this is called the event horizon. If you would approach the event horizon of the black hole some bizarre things would happen. An outside observer watching you fall into the hole will see your time slow down until you stop completely.

Stephen Hawking studied black holes theoretically. In empty space, quantum fluctuations occur. A particle and its antimatter particle can arise from nothing to later collide and become annihilated. It may happen that the anti-particle is sucked into the black hole, while the other particle succeeds in leaving the event horizon. This is a violation of the conservation of energy an explanation could be that the particles time travel in the time dimension to later collide. That would mean that black holes would evaporate after a long time. This is not proven. Even if we have known about and been fascinated by black holes a long time. We have never actually seen a black hole until now. How awesome is that!


How do locate the planets and the Moon

A new feature in the Exoplanethunter android app is that the planets of our solar system and the Moon has been added to the star map. By using the phone's sensor (GPS sign) it is possible to pinpoint the objects on the night sky

So how is it possible to locate the planets and the moon in the sky. Here are a short mathematical explanation and a historical background to the theory.
Claudius Ptolemy was a Greek astronomer born around 90 and died around 170. He lived in the city of Alexandria in Egypt that was a Roman province at the time. He summarized and extended the ancient astronomical knowledge. He is known for the Ptolemaic system that was based on the ancient belief that the Earth was in the center of the universe and that the Sun, Moon, stars, and planets orbited Earth. A model that also was his predecessor Aristotle (384 BC-322 BC) position.  A problem with the Geocentric model was to explain the weird movement of the planets. Ptolemy described the movements of the planets with epicycles. It means that the planet was moving in a circle and the center point on that circle was moved along the periphery on another circle with Earth near at center. Ptolemy introduced the concept of equant. It was the point from which the movement looked uniform. The equant was inside the circle and directly opposite to Earth from the center of the circle. The Ptolemaic system was the dominating astronomical system during antiquity and middle ages. The first astronomy book based on a heliocentric worldview was the Revolutions of the Heavenly Spheres. The book was written by the Polish astronomer Nicolaus Copernicus but was first published shortly before his death in 1543. In his model, Earth and the other planets were orbiting the Sun and that explained the weird movements of the planets. This theory was later supported by Galileo Galilei's work and the church banned the book.

The astronomer and mathematician Johannes Kepler studied the movement of the planet and made accurate calculations. He formulated three scientific laws that describe the motion of planets around the Sun. It was published between 1609 and 1619 and improved Copernicus heliocentric theory. These laws also laid the ground to Isaac Newton's laws of gravity.

Kepler's three laws are the following:

  1. The orbit of a planet is an ellipse with the Sun at one of the two foci.
  2. A radius vector joining any planet to the Sun sweeps out equal areas in equal lengths of time.
  3. The expression \(\frac {T ^ {2}} {r ^ {3}}\) gives the same constant value for all planets that orbit the Sun, where T is the planet's orbit period and r is half the major axis of the ellipse.

To calculate the position of the planets on the sky seven steps are needed. A celestial object has coordinates right ascension and declination. These coordinates are fixed points with reference origo where the earth's celestial equator is crossed by the ecliptic line. How these coordinates could be calculated from a position on earth can be found here: How do locate stars

Step 1: Calculate the mean anomaly. The mean anomaly is the angular distance from the point where the planet was closest to the Sun (perihelion) which the planet would have moved if it had a circular orbit and with the same orbital period as the real planet moving on the ellipse. If we use a reference point in time where we know the value of the mean anomaly


where a is the length of the semimajor axis of the orbit and \(\mu\) is the mean angular motion  of the object

Step 2: calculate the true anomaly. The true anomaly is the real angle between the perihelion and the planet, seen from the sun and measured in the direction of movement of the planet. The true anomaly can be calculated from the mean anomaly by using a Fourier expansion

$$\nu =M+\left(2e-{\tfrac {1}{4}}e^{3}\right)\sin {M}+$$

$${\tfrac {5}{4}}e^{2}\sin {2M}+$$

$${\tfrac {13}{12}}e^{3}\sin {3M}+....$$

Step 3: calculate the distance from the sun. Eccentricity measure how an orbit deviates from circular by using the value of the planet eccentricity the distance from the sun could be calculated using this formula

$$r=\frac{a(1-e^{2})}{1+e\cos \nu }$$

Where e is the eccentricity

Step 4: calculate the rectangular heliocentric ecliptic coordinates these coordinates is

$$x= r(\cos\Omega\cos(\omega+\upsilon)-$$$$i\sin\Omega\sin(\omega+\upsilon))$$

$$y= r(\sin\Omega\cos(\omega+\upsilon)+$$$$\cos i\cos\Omega\sin(\omega+\upsilon))$$

$$z= r(\sin i\sin(\omega+\upsilon))$$

i is the inclination the angle between a plane of reference and the orbit of the planet. \(\omega\) is the angle between the periapsis (the closest distance from the sun) and its ascending node. \(\Omega\) is the ecliptic longitude. These four steps need to be done for both the Earth and the planet we are investigating

Step 5: calculate the coordinates of the planet relative to the Earth




Step 6 calculate the geocentric ecliptical longitude and latitude by using the coordinates in step 5.

$$\lambda = \arctan (y,x)$$

$$\beta =\arcsin\left (\frac{z}{\sqrt{x^{2}+{y^2}+z^{2}}}\right)$$

Step 6  Earth's tilt angle with respect to the ecliptic line (Earth's path around the sun) is called obliquity of the ecliptic \(\epsilon\). By using \(\lambda\) and \(\beta\) we can now calculate the right ascension \(\alpha\) and declination \(\delta\) of the planet with these formulas.

$$\delta = \arcsin(\sin\beta\cos\epsilon +$$$$ \cos\beta \sin\epsilon\sin\lambda)$$

$$\alpha = \arctan(\sin\lambda\cos\epsilon-$$$$\tan\beta\sin\epsilon\cos\lambda) $$

The moon's position need to be calculated in a different  way as it revolves around Earth.

Geocentric ecliptical coordinates  that were calculated in step 6 for planets could be calculated for the Moon using these formulas:

$$\lambda = L +6.289\sin M$$

$$\beta =5.128\sin F$$

Where L is the mean geocentric ecliptic longitude M mean anomaly and F mean distance   

More details and example calculations can be found here aa.quae.nl

Is there life on Mars

Life on Mars is a song by David Bowie from 1971 on the album Hunky Dory it was also released as a single. But David Bowie was not the first one considering it could be life on Mars. The past 300 years people have believed it could be life on Mars. Christiaan Huygens was a Dutch astronomer that is known for the discovery of Saturn rings he also studied Mars with a telescope.
He discovered a dark spot on the planet known as Syrtis Major and included it in a drawing of Mars in 1659. He used repeated observations of the feature to estimate the length of a day on Mars and that happens to be 24 hours the same as Earth. We know now that the dark color comes from the basaltic volcanic rock and the lack of dust. He also estimated that Mars is about 60 percent of the size of Earth. In 1672 he makes a drawing of Mars that includes the south polar cap.

Later Sir William Herschel that is also known for discovering Uranus in 1780 was studying Mars. He believed that the dark areas on Mars were oceans and the lighter regions land. He speculated that Martian inhabitants have the same climate and conditions as people on Earth.
The famous mathematician Friedrich Gauss was so convinced that intelligent life existed on Mars that he proposed that we should draw huge figures in the snow to signal the Martians.
In 1877, astronomer Giovanni Schiaparelli saw several lines crossing each other on the Martian surface. He believed this was canals built by the Martians to lead water from the polar regions. Percival Lawrence Lowell fueled speculation that there were canals on Mars. The belief of Martians inspired H.G Wells to write the book “The War of the Worlds” that was published in 1897.

The Mariner Program was a series of teen NASA built interplanetary space probes that were sent to several planets in our solar system Venus, Mercury, and Mars. Mariner 4 made the first successful Mars passage in 1965 and sent pictures on the Mars surface. Both Mariner 6 and 7 passed Mars in 1969 and sent data and pictures back to Earth. Mariner 9 became the first artificial satellite in orbit around Mars in 1971. There was no sign of canals. The canals Giovanni saw was just an optical illusion. The Mariner saw indications that there could have been water on Mars in the past. These evidence on ancient water on Mars was not entirely confirmed by the Mariner mission and a closer look was needed. 

The Viking 1 lander was the first spacecraft ever to land successfully on Mars on July 20, 1976. Two months later, the Viking 2 lander landed on Mars. The landers took images of the surface and studied soil samples for biosignatures and one of the tests was positive. But the result could not be reproduced and the positive test was probably the result of a non-biological reaction.

When I was young in 1996 I heard the sensational news on television that the scientist finally found evidence on life on Mars. A billions years old Martian meteorite that was found in Antarctica contained alien bacterias. But this was probably also a false alarm. These features on the Martian meteorite could also be the result of non-biological chemical reactions.

The spacecraft Odyssey Orbiter reached Mars on October 24, 2001, and its scientific mission began on February 19, 2002. Its three main measure instruments were a thermal emission imaging system, a gamma-ray spectrometer, and an energetic particle spectrometer. In addition to these, a neutron spectrometer was also included. The Orbiter found a large amount of hydrogen, a sign that water exists less than one meter below the surface. Two rovers named Opportunity and Spirit were sent to Mars to investigate. Opportunity landed in Meridiani Planum on January 25, 2004, Spirit landed on the other side of the planet three weeks later. Spirit got stuck in a sand trap in 2010. Opportunity continued being operational and last heard of in June 2018. After a planet-wide storm NASA was hoping that it would come back online but it never did. Both rovers found geological evidence that Mars used to be more liquid having oceans. Phoenix lander was the first vessel to investigate Martian water in the Northern polar region. It found perchlorate in the water it is salt that acts like antifreeze and that could be bad news for life.

Image credit: NASA/JPL-Caltech/MSSS

The latest rover Curiosity landed on Mars 5 August 2012. The rover is still operational and confirmed the presence of water in the soil. Recent studies show methane in the atmosphere and organic molecules on the surface. These could still be the result of non-biological processes but it got the scientists very excited as it could come from the decay of once living matter. One of the recent discoveries is that it is a 20 km wide underground lake on Mars. Curiosity will continue its mission until 2020, but more missions are planned. ExoMars is a project to search for signs of past life on Mars and is run by the European Space Agency (ESA) and the Russian space agency Roscosmos. They are planning to land the Rosalind Franklin rover on Mars on July 2020.

Finding evidence on past life on Mars will tell us that if the conditions are right on a planet life will evolve. So far we only know one place in the universe where life developed, on Earth. If it happened twice in our solar system then it probably will be life on Exoplanets that have similar conditions as us.


Visit Mars in the catalog here


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