Lunar eclipse 27 July 2018

On 27 July 2018 there will be a total eclipse of the Moon, which will be viewable from many areas of the world. This will be the first total lunar eclipse able to be observed in the UK for nearly three years and it will be worth making the effort to see, especially since, for viewers in Europe, Africa and eastern Asia, it will occur at a sociable hour in the evening.

NASA Image Lunar Eclipse

The Moon during a recent total lunar eclipse – image from NASA

 

What happens during a lunar eclipse?

A lunar eclipse occurs when the Earth prevents some or all of the Sun’s light from hitting the Moon’s surface. This is shown in the diagram below:

 

Image from Wikimedia Commons

In this diagram in the region marked Umbra the Earth completely blocks the Sun. In the region marked Penumbra the Earth partially blocks the Sun.

 

The stages of the July 27 lunar eclipse

The next diagram below shows how, to someone on Earth, the Moon will move through the Earth’s shadow on 27 July. The six points labelled P1, U1, U2, U3, U4 and P4 are known as the eclipse contacts and are the times when the eclipse moves from one stage to the next.

 

 

Diagram from NASA

 

At point P1 the Earth will start to block some of the Sun’s light from reaching the Moon.  This will start at 5:15 pm GMT and is the start of the penumbral phase. The Moon’s brightness will dim a little, but this will be quite difficult to notice with the naked eye.

 

As the Moon continues in its orbit, more and more of the Sun’s light is obscured, until after about an hour some of the Moon will get no direct sunlight.  This is known as the partial phase. It will start at point U1 which will occur at 6:24 GMT. The part of the Moon which receives no direct sunlight will appear dark, as shown in the picture below.

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Lunar_eclipse_ Partial

The partial phase of a lunar eclipse – Image from Wikimedia Commons

 

After a further hour the Earth will block all direct sunlight from reaching the entire Moon. This is shown as U2 is the diagram and this total phase will start at 7:30 PM. In the total phase, rather than disappearing completely, the Moon goes a dull red colour as shown in the picture at the top of this post. This is because, even though no direct sunlight can reach the Moon, some light from the Sun is bent round the Earth’s atmosphere towards the Moon. This light appears red because visible light from the Sun is a mixture of different wavelengths – red light has the longest wavelength and violet the shortest. Most of the light of the shorter wavelengths  (orange, yellow, green, blue, indigo and violet) is removed from this light bent by the Earth’s atmosphere by a process called scattering, which I discussed in an earlier post https://thesciencegeek.org/2015/09/30/why-is-the-sky-blue/ . The same effect causes the western sky to be red after sunset on a clear day.

Interestingly, if we could stand on the surface of the Moon and view the eclipse we would see a red ring around the Earth.

The Moon will emerge from the total phase (point U3) at 9:13 GMT, the partial phase (point U4) will end at 10:19 PM and the eclipse will finish (P4) at 11:29 PM.

Which areas of the world can see the eclipse?

The eclipse timings are summarised below

Data from NASA (2009)

Not all areas of the world will be able to see the eclipse. This is because the Moon will have already set after the eclipse starts or will not have risen before it finishes. Other places will only be able to see part of the eclipse.

  • In Manchester where Mrs Geek and I live, the Moon will rise at 9:06 PM local times which is 8:06 PM GMT, so when the Moon rises the total eclipse will already be underway.
  • In Manila, the Moon will set at 5:44 am on July 28, Philippine Standard Time (PST) which is 9:44 PM GMT, so viewers will miss part of the final partial phase because this will occur after the Moon has set.

I have adapted the diagram below from NASA (2009) and this shows where in the world the eclipse can be seen.

The regions labelled A to L are as follows

 

 

 

How often do lunar eclipses occur?

Even though the Moon takes roughly a month to orbit the Earth, lunar eclipses do not occur every month. The Moon’s orbit around the Earth is tilted at about five degrees with respect to the Earth’s orbit around the Sun, as shown below.

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Moon Tilt

This means that during most lunar months, as seen from the Moon, the Earth passes just below or just above the Sun rather than obscuring it. There are only two time windows in a year when a lunar eclipse can occur.  These two points are known as the nodes (See note 2). Even then most lunar eclipses are partial eclipses where the Earth only partially covers the Moon.

 

Notes

 

  1. GMT versus UTC

Although the term Greenwich Mean Time (GMT) is often used in popular writing it is no longer used by astronomers.  Instead, they use two different times which agree with each other to within 1 second.

  • Universal Time, often abbreviated to UT1, is the mean solar time, the time determined by the rising and setting of the Sun at the Greenwich Meridian, zero degrees longitude.
  • Co-ordinated Universal Time, usually abbreviated to UTC, is the time measured by atomic clocks and is kept to within 1 second of UT1 by the addition of leap seconds.

 

In common use, GMT is often taken to be the same as UTC, which is the approach I have taken for this post. However, it can also be taken to mean UT1. Owing to the ambiguity of whether UTC or UT1 is meant, and because timekeeping laws usually refer to UTC, the term GMT is normally avoided in precise writing.

 

  1. Nodes when eclipses can occur

The two nodes when a lunar eclipse can occur aren’t the same dates every year but change from year to year due to an astronomical effect called precession of the line of nodes.

References

NASA (2009) Total lunar eclipse of 2018 July 2017, Available at: https://eclipse.gsfc.nasa.gov/LEplot/LEplot2001/LE2018Jul27T.pdf (Accessed: 8 July 2018).

June 21 2018 – the solstice

This year, the June solstice will fall on 21 June.  In the northern hemisphere, it is the day when there is the most daylight and when the Sun is at its highest in the midday sky.

 

Sunrise at the solstice at Stonehenge, England – image from Wikimedia commons

The origin of the word solstice is from two Latin words:  sol, which means Sun, and sistere, to stand still. This is because, at the time of the solstice, the Sun stops getting higher, appears to stand still at the same height for a few days, and then gets lower in the midday sky.

 

The graph below shows the maximum height, or elevation, of the Sun, measured in degrees above the horizon, during the month of June. The graph is for a location 50 degrees latitude North, which is the same latitude as the southern tip of the British Isles.  

The fact that the Sun’s elevation changes gradually means the amount of daylight also changes very little around the solstice. This is shown in the table below, which gives the sunrise and sunset times and the amount of daylight in hours, minutes and seconds for June in London.

Table of sunrise and sunset times for London (Time and Date 2018).

 

Precise definition of the solstice

The diagram above shows the Earth’s orbit around the Sun. For clarity the sizes of the Earth and Sun have been greatly exaggerated.

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Seasons

 

  • During June, marked as Ain the diagram, the Earth’s North Pole is tilted towards the Sun and the days are longer in the northern hemisphere.
  • During December, marked as Cin the diagram, the Earth’s South Pole is tilted towards the Sun and days are longer in the southern hemisphere.
  • At points Band D, known as the equinoxes, neither pole is tilted towards the Sun and the amounts of daylight in the northern and southern hemisphere are equal.

The precise astronomical definition of the June solstice (also called the summer solstice in the northern hemisphere) is the exact point in time when the North Pole is tilted furthest towards the Sun. The times for this event for the years 2016-2020 are given in the table below – in GMT, in Tokyo time (which is 9 hours ahead of GMT) and in Hawaiian time (which is 10 hours behind GMT).

June Solstice Times

 

 

As you can see, the time of the solstice varies from year to year. It can fall on 20, 21 or 22 June, depending on your longitude (and thus your time zone).

Importance of the solstice to early man

The solstice was of great importance to early man, and many prehistoric sites appear to have been built to celebrate it. The most famous of these is Stonehenge, which is located in Wiltshire, England. It is a set of concentric stone circles built between 4000 and 5000 years ago. It was an amazing feat of construction for stone age man. The stone circle is over 30 metres in diameter. The largest stones are more than 9 metres tall, weigh over 25 tonnes and were hauled over 30 km to the site. It is reckoned that the smaller stones were moved from western Wales, a distance of 225 km (Jarus 2014).

Stonehenge

Image from Wikimedia commons 

At the centre of Stonehenge is a horseshoe arrangement of five sets of arches called triliths, each containing three stones.  The open side of the horseshoe points North East towards a large stone 80 metres away from the main circle. Today this large stone is given the name  ‘The Heel Stone’.

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Stonehenge_Heelstone

Image from Wikimedia commons

The monument is arranged in such a way that, for a few days either side of the June solstice and only at those dates, someone standing in the centre of the horse shoe and facing North East will see the Sun rise over the Heel stone.

Heel Stone Sunrise

How sunrise at the summer solstice at Stonehenge would have looked after the monument’s construction.

It is amazing that prehistoric man built such a large monument to line up with the June solstice. It clearly must have been a major event for a people living outdoors with only natural daylight, and in fact the solstice is still celebrated at Stonehenge today. Modern groups with ancient origins, such as Druids and Pagans, who revere the natural world more than many modern humans, join approximately 30,000 people who flock to Stonehenge to watch the Sun rise at the solstice each year.

Interestingly, to prevent damage to such an important ancient monument it is not normally possible to get right up to the stones. However, the charity which manages the site English Heritage open it up every year for the solstice, giving people a rare chance to get up close.

For the BBC report on the 2017 Stonehenge solstice celebrations click on the link below.

https://www.bbc.co.uk/news/uk-england-wiltshire-40352528

The southern hemisphere

To those of you who live in the southern hemisphere the June solstice is the winter solstice, when the midday Sun is at its lowest in the sky. After the solstice the days start getting gradually longer and the nights gradually shorter, although the change doesn’t really become noticeable until July.

Note

Strictly speaking it isn’t true that for the whole northern hemisphere the midday Sun is at its highest in the sky on the solstice. At the Tropic of Cancer, which is 23.5 degrees north, and is shown as the upper red line in diagram below, the Sun is directly overhead at midday on the June solstice. At low latitudes between the equator and the Tropic of Cancer the Sun is directly overhead at midday on two dates either side of the solstice. For example, in San Juan, Puerto Rico, which lies 18.5 degrees North of the equator, the Sun is overhead at midday on May 13 and July 30.

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Tropic of cancer

References

Jarus, O (2014) Stonehenge: Facts & Theories About Mysterious Monument, Available at: http://www.livescience.com/22427-stonehenge-facts.html(Accessed: 10 June 2016).

 

Time and Date (2018) London, ENG, United Kingdom — sunrise, sunset, and daylength, June 2018, Available at: http://www.timeanddate.com/sun/uk/london(Accessed: 4 June 2018).

 

20 March 2018 – the equinox

Now that we are in the month of March, it is only a short time until 21 March, the first day of spring (or first day of autumn if you’re one of my readers in the southern hemisphere).

There is a commonly held view that 21 March is an equinox and that the equinoxes are the two days in the year when all places on the Earth have exactly 12 hours of daylight and 12 hours of darkness. In fact, as I’ll explain later, both these statements are only approximately correct.  In reality the situation is as follows.

  • 21 March can sometimes be the date on which the spring equinox falls but its date varies from year to year and also depends upon where you are located.  In 2018 it will fall on 20 March for most places in the world.
  • At the equinoxes there is actually nowhere on the Earth where there are exactly 12 hours of daylight and 12 hours of darkness.

What is an equinox?

The origin of the word equinox comes from two Latin words aequus (equal) and nox (night). This definition suggests that at an equinox the length of the day and night are equal. However the precise astronomical definition of an equinox is slightly different.

Earths Orbit

The diagram above shows the Earth going around the Sun in its orbit

  • At the December solstice (point A in the diagram) the North Pole is tilted further away from the Sun than at any other time of the year, and the South Pole is tilted nearest the Sun.  In the northern hemisphere the period of darkness is longest compared with the period of daylight, and in the southern hemisphere the reverse applies.
  • At the summer solstice in June (point C) it is exactly the opposite of the winter solstice – it is the North Pole which is now tilted nearest to the Sun so the northern hemisphere experiences the longest period of daylight.
  • There are two times a year (B and D in the diagram) when the neither the North Pole nor the South Pole are tilted towards the Sun and these times are the equinoxes.  If we take two places with the same latitude, one of which is North of the equator and the other one South of the equator, (for example Istanbul, Turkey 41oand Wellington New Zealand 41oS ) they will both have the same amount of daylight at the equinox.

On what date do the equinoxes occur?

The diagram also shows that the Earth moves in an elliptical orbit around the Sun. This means that it has further to travel in its orbit between the March equinox and the September equinox than in the return leg of its journey from September to March. The two equinoxes are therefore not exactly half a year apart: from the March equinox to the September equinox is around 186 days, whereas from the September equinox to the March equinox is only 179 days.

The tables below give the times of the two equinoxes from 2016 to 2021  for three Locations: London (Greenwich Mean Time or GMT), Honolulu (GMT -10 hours) and Tokyo (GMT +9 hours).  As you can see, the northern hemisphere spring equinox can occur on 19, 20 or 21 March and the autumn equinox on 22 or 23 September.

spring equinox times

autumn equinox times

(Data TimeandDate.com 2016a)

On what dates in a year are there are exactly 12 hours of daylight?

The first thing we need to think about when we answer this question is what do we mean by the word ‘daylight’? Do we consider twilight, the time just after sunrise or just before sunset when it is not completely dark, to be daylight? Or do we consider daylight as being the time when the Sun is above the horizon?

If we use the definition of ‘daylight’ as being the interval between sunrise and sunset then there are actually slightly more than 12 hours of daylight at the equinox everywhere in the world.  The first reason for this is that the definition of sunrise is the time when the first light from the Sun’s rays reaches above the horizon, not when the centre of the Sun is above the horizon. The diagram below shows the path of the Sun’s disc around sunrise at the equinox in London.  In the early morning, the time when the half of the Sun is above the horizon and half below the horizon is 6:03 am, shown as B in the diagram, but sunrise is about a minute before this time.

Sun path sunrise

Similarly, in the early evening the time when half of the Sun is above the horizon and half below the horizon is 6:13 pm,  B in the diagram, but sunset is when the last light from the Sun’s rays are above the horizon and is about a minute after this time.

Sun path sunset

The second reason for there being more than 12 hours of daylight at the equinox is that when the Sun is just below the horizon the Earth’s atmosphere bends the Sun’s rays, causing it to appear just above the horizon. This bending of light is known as refraction and has the effect of slightly extending the hours of daylight.

Taken together, these two effects mean that there are slightly more than 12 hours of daylight at the equinox. The table below shows the  amount of daylight for dates around the equinox in London and Wellington.  It shows that the date on which there are exactly 12 hours of daylight and 12 hours of darkness in London is 17 March, three days earlier but in Wellington it is 3 days later on 23 March.

 

(TimeandDate.com 2016b)

References

TimeandDate.com (2016) Solstices & Equinoxes for London (Surrounding 10 Years).  Available at: http://www.timeanddate.com/calendar/seasons.html?n=136 (Accessed: 5 March 2016).

TimeandDate.com (2016) London, ENG, United Kingdom — Sunrise, Sunset, and Daylength, March 2016, Available at: http://www.timeanddate.com/sun/uk/london(Accessed: 1 March 2016).

 

Kepler’s other achievements

As discussed in my previous post, Kepler’s improvement of Copernicus’s heliocentric system led to its more general acceptance, and his three laws describing the way planets move are fundamental laws of astronomy. However, this wasn’t his only contribution to science. He was one of the greatest thinkers of the seventeenth century scientific revolution and in this post I’ll outline some of his other major achievements.

Statue of Kepler in Linz, Austria – image from Wikimedia Commons

The Keplerian telescope

The Italian astronomer Galileo Galilei (1564-1642) was the first person to take observations of celestial objects with a telescope . However, Galileo’s telescope could only magnify objects 30 times before the image became distorted. It also had a narrow field of view

In 1610 Kepler began theoretical and experimental investigations of the way that different combinations of lenses could work together to produce a magnified image. He published his finding in a book called Dioptrice, which laid the foundation of modern optics.  Using the results of his investigations, he invented a new type of telescope with a different combination of lenses than that which Galileo had used. This new design became known as the Keplerian telescope.  It is still in use today and enables a higher magnification to be achieved with less distortion than a Galilean telescope.

Keplerian telescope – image from Wikimedia Commons

For more details on the differences between the two types of telescope see the notes at the bottom of this post.

The supernova of 1604

In October 1604 Kepler took observations of a new object which had appeared in the constellation Ophiuchus. Although Kepler was not the first to see it, he took accurate measurements of its position and brightness over a period of year.

He observed that the new object did not move with respect to background stars, so wasn’t an object revolving around the Sun like a planet or a comet. Also, the fact that it did not show any parallax meant that it must be a great distance away and wasn’t a nearby object in front of the stars. This is shown in the diagram below

It it were closer than the background of fixed stars then, at  different times of year, the new star would appear to be in a different positions with respect to the more distant background of fixed stars. As this shift in position was not seen,  the new star must be the same distance as the fixed stars.

The appearance of a new star which increased in brightness and then gradually faded over time contradicted an important belief, which had been held since ancient times, that all the stars were fixed in position respect to each and were unchanging. In 1606 he published his results in a book called ‘De Stella nova in pede Serpentarii’, which like most scientific literature of the time was written in Latin.

Its title translated into English is ‘On the new star in Ophiuchus’s foot’. For those of my readers able to read Latin, it can be downloaded for free from the following website: http://www.univie.ac.at/hwastro/rare/1606_kepler.htm

Today this object, rather than being a new star, is known to be a supernova, a massive star which exploded at the end of its life. The explosion completely destroyed the star, blowing the outer layers into space in a massive glowing gas cloud, which is what Kepler observed. The remnant of the supernova is officially known as SN 1604 but is  more commonly called Kepler’s supernova and is 20,000 light years away, which is well within our Milky Way galaxy.  It is the last time that a supernova exploded close enough to be visible to the naked eye.

Remnants of Kepler’s supernova – image from NASA

Kepler’s contribution to mathematics

Kepler’s contributions weren’t restricted to astronomy either. In 1611 he produced  a pamphlet entitled Strena Seu de Nive Sexangula (A New Year’s Gift of Hexagonal Snow). In this he published the first description of the hexagonal symmetry of snowflakes.

All snowflakes when freshly formed have a hexagonal symmetry such that shown above.

Kepler discovered a series of regular solid shapes, which are known as ‘the Kepler solids’.  The term ‘regular’ means that all the faces are the same.

The Kepler solids

In 1611 he posed a mathematical problem, which became known as the Kepler Conjecture. It deals with the most efficient way to pack spheres together in a large container, so there is as little empty space as possible. It can be summarised as follows:

Imagine filling a large container with small equal-sized spheres. The packing density is equal to the total volume of the spheres divided by the volume of the container. So a packing density of 1 would mean that there was no free space at all.

The Kepler conjecture states that the maximum packing possible density is:

∏ /(3√2), (which is roughly  equal to 0.7405).

∏ /(3√2)  is the packing density we get if we pack together spheres in layers, as shown in the picture below.

So what the Kepler conjecture in saying is that there is no other arrangement out of the very large number of possible ways to pack spheres together which gives a higher packing density than that shown in the picture. Although Kepler and other subsequent mathematicians believed this statement to be true, they were unable find a way to prove it. For over 400 years, it remained as one of the greatest unsolved problems in mathematics. It wasn’t until 2017 that a team led by the American mathematician Thomas Hales proved it to be true (phys.org 2017).

Thomas Hales – image from Wikimedia Commons

Somnium

Interestingly, Kepler also has the distinction of writing what the astronomer and science educator Carl Sagan called the first ever work of science fiction. It was written in 1608 in Latin and is called Somnium (The Dream) and is about a man who travels to the Moon. In this book he describes how the Sun, Planets and the Earth would appear to an observer on from the viewpoint of the Moon. In Kepler’s time is was not known how harsh and barren the Moon was as an environment and some writers had speculated that there might be creatures on the Moon similar to those found on the Earth and even lunar civilisation.

To Kepler it was clear that the dual effects of the lunar climate and the irregular, hostile terrain would produce plants and animals far different from those that inhabit the Earth. in Kepler’s Lavania  (which was the name he gave the Moon in Somnium) there are no men and women, no civilizations.

And finally…

I hope you have enjoyed reading this post. I have tried to outline some of Kepler’s achievements, but in a 35 years scientific career he made many additional contributions, which I’ve not had time to mention, including discovering how the human eye works. What is clear to me is that he is one of most important thinkers of the scientific revolution which took place in Europe during the seventeenth century.

The Science Geek

Notes

These additional notes give a brief overview of the differences between the Galilean and Keplerian telescopes. I will discuss them in more detail in a subsequent post on how telescope work.

To understand how these telescopes are constructed it is necessary to understand a little about lenses.

There are two types of lens:

  • A converging lens, shown in the top of the diagram above, causes parallel light rays from a distant object, shown in red, to converge at point known as the focus. The focal length of the lens is the distance between its centre and the focus.
  • A diverging lens, shown in the bottom of the diagram, causes parallel light rays from a distant object to spread out so they appear to come from the focus. Like a converging lens the focal length of the lens is the distance between its centre and the focus. By convention the focal length of a a diverging lens is negative.

A Galilean telescope consists of a converging lens of long focal length (known as the objective) and an eyepiece which is a diverging lens of a shorter focal length.

If FO is the focal length of the objective and FE the focal length of the eyepiece, then  the magnification is given by FO/FE. So a Gallilean telescope with an objective with focal length of 50 cm and and eyepiece of focal length of -10 cm, would have an magnification of -5.  The minus sign just means the objective is the right way up.

A Keplerian telescope consists of a converging lens of long focal length (known as the objective) and an eyepiece which is a converging lens of a shorter focal length. It can achieve higher magnifications than the Galilean telescope and has a larger field of view.

As with the Galilean telescope, if FO is the focal length of the objective and FE the focal length of the eyepiece, then the magnification is given by FO/FE. So a Keplerian telescope with an objective with focal of length 300 cm and and eyepiece of focal length of 5 cm would have an magnification of 60.

References

phys.org (2017) Mathematicians deliver formal proof of Kepler Conjecture, Available at: https://phys.org/news/2017-06-mathematicians-formal-proof-kepler-conjecture.html(Accessed: 25 January 2018).

Johannes Kepler

My latest post is about the work of the German astronomer Johannes Kepler (1571-1630).  He is most famous for his improvement to the earlier model of Copernicus by introducing the idea that the planets move in elliptical, rather than circular, orbits and that their movements in these orbits are governed by a set of laws, which became known as Kepler’s laws of planetary motion. However, as I’ll talk about later, he also made many other major contributions to astronomy and mathematics.

Johannes Kepler – Image from Wikimedia Commons

As readers of a previous post will be aware, in 1543 the Polish astronomer Nicolas Copernicus (1473–1543) published a theory in which the Earth and all the planets orbited the Sun. Prior to Copernicus, the generally accepted view was that the Earth was the centre of the Universe and the Sun, the stars and the planets were all in motion around it. However, like all astronomers before him Copernicus believed that the planets’ orbits must be perfect circles. So, to make his theory fit the observations of the positions of the planets, each planet moved in a small circle called an epicycle and the centre of the epicycle was in orbit around the Sun. This is shown in the diagram below.

 

The fact that Copernicus’s model, like the geocentric model, still required epicycles  gave it a rather complex and unsatisfactory feel. This complexity and the religious objections that Earth was not at the centre of the Universe were reasons why the model was not adopted more widely. Kepler made a huge advance by improving Copernicus’s model into the one we use today.

However, before I go onto talk about Kepler’s work, I’ll just give some background on ellipses, as I assume it may have been some time since many of my readers studied mathematics at high school. An ellipse is shown below and can be defined as a set of points where the sum of the distances from two other points, known as the focuses, is a constant.

In the diagram above: points  A, B and C all lie on an ellipse, which is coloured purple. The two focuses of the ellipse are marked as F1 and F2. So:

  • the sum of the lengths of the two green lines, which link A to each focus, is equal to
  • the sum of the lengths of the two black lines ,which link B to each focus, and
  • the sum of the lengths of the two red lines ,which link C to each focus.

The long axis (known as the major axis) passes through the centre of the ellipse and the focuses and is shown as the blue dashed line.  The short axis (known as the minor axis) is at a right angle to this is and is shown as the red dashed line.

 

The other key feature of an ellipse is its eccentricity, which is a measure of how elongated an ellipse is. This is defined as the distance between the centre of the ellipse and either focus (the green line in the diagram above) divided by half the major axis (the brown line in the diagram above). The eccentricity is always between zero and one. An eccentricity just above zero is close to a perfect circle and an eccentricity just below one is a highly elongated ellipse.

 

 

 

Kepler’s work

Johannes Kepler was born in the town Weil der Stadt near Stuttgart in Germany.  At the age of 18 he went to Tübingen University, which in Kepler’s time was one of the best universities in Germany. I understand that this is still the case today.

At Tubingen, Kepler studied philosophy, theology and astronomy. He learned both the  Earth-centred Ptolemaic system (which was favoured by the church) and the rival heliocentric Copernican system. After considering the merits of both, he became a firm believer in the Copernican system.

When he left university, he began teaching mathematics and astronomy. In 1595 he published a book ‘Mysterium Cosmographicum’ defending Copernicus’s system. Interestingly although it was published over 50 years after Copernicus’s death, it was the earliest book defending this system. In 1600, as Kepler’s reputation as an astronomer spread, he was appointed as an assistant to the Holy Roman Emperor’s Imperial mathematician Tycho Brahe (1546-1601), based at the Prague observatory. Brahe was one of greatest astronomers before the invention of the telescope and had painstakingly gathered observations of the stars and planets over decades.

Tycho Brahe –  Image from Wikimedia Commons

Unlike Kepler, Brahe didn’t accept the Copernican system. Instead he invented a hybrid, a ‘geo-heliocentric’ system, in which the Sun and Moon orbited the Earth, while the planets orbited the Sun.

The year after Kepler’s appointment, Brahe died. This was good news for Kepler in two ways. Firstly, the Holy Roman Emperor appointed Kepler as the new imperial mathematician, which boosted both his prestige and his income. Secondly, it gave him unrestricted access to Brahe’s data.

Kepler tried to fit Brahe’s observations to Copernicus’s model. However he felt it was unwieldly that epicycles and multiple epicycles (for some planets) were needed to made the model work. He came up with the radical idea that, instead of moving in perfect circles, planets moved in ellipses. By doing this he was able to remove the need for epicycles altogether. To me this idea appears so simple that it is surprising no one had thought of it earlier. However, I suspect that earlier astronomers were constrained into believing that the heavenly bodies must move in perfect circles. In 1609 Kepler published his first two laws of planetary motion

Kepler’s first law – The planets move in ellipses with the Sun at one of the focuses of its ellipse.

His first law is shown in the diagram below.

  • The point where the planet is closest to the Sun is called the perihelion.
  • The point where it is furthest away is the aphelion.
  • Half the length of the major axis is known as the semi-major axis. It is equal to the average of the planet’s closest distance from the Sun (perihelion) and its furthest distance from the Sun (aphelion).

Kepler’s second law – A line between the Sun and a planet sweeps out equal areas at equal times.

This is shown in the diagram below. As the planet moves around its orbit, it moves faster when it is closer to the Sun and slower when it is further away. However, the relationship between its speed and its distance from the Sun is such that the area of the triangle is always the same.

In 1619 Kepler published his third law of planetary motion, which relates the size of a planet’s orbit to its period, which is the time is it takes to complete an orbit around the Sun

Kepler’s third law – The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

The third law is useful because it allows the distance to a planet to be calculated. For example, if we take the Earth and Jupiter and measure the periods in years and distances in astronomical units.

  • The Earth’s semi-major axis (RE) is 1 astronomical unit from the Sun
  • The time it takes the Earth to orbit the Sun (PE) is 1 year
  • The time it takes Jupiter to orbit the Sun (PJ) is 11.86 years

Where an astronomical unit is the mean distance between the Earth and the Sun.

So, if we want to calculate the average distance between the Sun and Jupiter (RJ), then from Kepler’s third law.

(PJ/PE)2 = (RJ/RE)3

Putting the actual values in gives.

(11.86/1)2 = (RJ/1)3

So the average distance between the Sun and Jupiter (RJ) is the cube root of 11.86 squared which is 5.2 astronomical units.

The Importance of Kepler’s laws

Kepler’s three laws have shown to be fundamental laws of astronomy. Although Kepler derived them by analysing the planets’ orbits around the Sun, they apply to any object orbiting a larger body. This includes the orbits of moons around planets and artificial satellites around the Earth.

The orbit of the International Space Station is governed by Kepler’s laws

One thing Kepler was unable to provide was an adequate explanation as to the underlying reason about the planets’ motions and why they obeyed his laws. However, his work provided a springboard for Isaac Newton to develop his theory of gravitation later in the seventeenth century.

Kepler’s‘ Legacy

Kepler’s improvement of Copernicus’s system lead to the more general acceptance of the heliocentric system. However this wasn’t his only contribution to science. He made many other contributions including: a new design of telescope, taking observations of a supernova and even has the distinction of writing the first ever science fiction story.  I will talk about Kepler’s other work in my next post.

The plural of focus is it focuses or foci?
On a final note…

When I studied mathematics and physics at high school, we were taught that the plural of focus is foci. This is still the case in formal written British English. However, language changes over time and  focuses is now much more commonly used nowadays.  Certainly  to use ‘foci’ in spoken English can sound  affected. Therefore (after discussing the matter with Mrs Geek) I decided to use  ‘focuses’ for the plural of focus.

Geocentric Cosmology

Today it is generally accepted as a scientific fact that the Earth is one of eight planets which revolve around the Sun, that the Sun is one of 400 billion or so stars in our Milky Way galaxy and that the Milky Way is one of hundreds of billions of galaxies in the observable Universe.

Location of our solar system in the Milky Way galaxy

However, for most of human history a geocentric model was the standard explanation of the cosmos. In this model the Earth is the the centre of the Universe and all the planets and stars revolve around it. Although it has been long superseded, this model could actually still be used to predict the motion of the Sun, Moon and the planets to a reasonable accuracy.

The basis of the geocentric model

Anyone watching the sky over a period of time will notice that the Sun rises in the east, is at its highest in the sky at midday and the sets in the west. Like the Sun, the Moon and the stars also rise in the east, get higher in the sky and set in the west. It  was perfectly natural for ancient astronomers to assume a geocentric model in which the Earth doesn’t itself rotate but that the Sun, Moon and stars all rotate around it. After all, people on the Earth do not actually feel that it is rotating.

Ancient astronomers noticed that the time at which a given star rises wasn’t the same but was roughly 4 minutes earlier each day and that virtually all of the thousands of stars in the sky didn’t move with respect to each other but were fixed. So, they put them together in structures  which became known as constellations. However, five of the brightest stars weren’t fixed but moved around a narrow band of sky which became known as the zodiac. These five objects: Mercury, Venus, Mars, Jupiter and Saturn became known as the wandering stars or planets. (The outermost two planets Uranus and Neptune were only discovered in the last 250 years)

The geocentric model

The name most associated with the geocentric model  is the ancient Greek astronomer Claudius Ptolemy. He lived in Alexandra, Egypt in the second century CE, although no one knows the exact dates of his birth and death. Ptolemy didn’t invent the geocentric model – no single astronomer did.  Ptolemy’s contribution was to refine the model which had been built up by Greek astronomers over the previous centuries,  so that it could accurately predict the positions of the stars and planets.

The geocentric model 

An early version of the geocentric model is shown in the diagram above. At the centre of the Universe was the Earth. The Earth did not rotate and was surrounded by a set of eight invisible spheres to which the Sun, Moon, planets and stars were attached. The outermost of these spheres was a sphere of fixed stars. All the stars were attached to this sphere. Moving inward from the sphere of fixed stars were Saturn’s sphere, Jupiter’s sphere, Mars’s sphere, the Sun’s sphere, Venus’s sphere, Mercury’s sphere and finally, closest to the Earth, the Moon’s sphere. To simplify the diagram, only the sphere of fixed stars, the Sun’s and Mars’s spheres are shown.

To a viewer in the northern hemisphere, the sphere of fixed stars rotated anticlockwise around a point in the sky directly above the North pole, known as the North celestial pole, once every 23 hours 56 minutes (see note 1).

As viewed from Earth,  the Sun’s, Moon’s and the planets’ spheres rotate more slowly than the sphere of fixed stars. For example, the Sun’s sphere takes 24 hours to complete a single rotation ,compared to the 23 hours 56 minutes for the sphere of fixed stars. So, compared to the sphere of fixed stars, the Sun’s sphere slipped back by around 4 minutes giving it a slow backwards rotation.  The diagram above also shows that axes about which the Sun’s, Moon’s and planets’ spheres rotated didn’t go through the North pole but was inclined at an angle to it.

The diagram below shows a two dimensional slice through the model with the Sun, Moon and all the planets shown.

The uneven motions of the planets

If you observe how a planet moves over a period of time you will notice that it does not move at a steady rate. At times it will move more slowly against the background of stars, stop altogether,  move in the opposite direction, stop again and then continue moving in the original  direction.  The reason for this is shown for one of the planets, Mars, in the diagram below.

 

As you can see from the diagram, the Earth is closer to the Sun than Mars, and moves faster in its orbit. At point 3, the Earth approaches Mars; to an observer on the Earth, between points 3 and 5 Mars will appear to change direction and move in a reverse direction against the background of stars. When it reaches point 5 the Earth is receding from Mars and so Mars will continue in its normal direction.

In the geocentric model the apparent uneven speed and change of direction of the planets’ motion was explained by introducing a concept known as an epicycle.

Epicycles and deferents – for simplicity only Mars is shown

Rather than than revolve directly round the Earth, a planet moves at constant speed in a small circle called an epicycle. The centre of the planet’s epicycle, marked with an ‘x in the diagram above, moves in a circular path around a larger circle called a deferent. For each planet the Earth lies at the centre of its deferent. At times the planet will be moving in the same direction as the orbital motion of the deferent and other times it will moving in the opposite direction. This will make the motion of the planet appear to be uneven – in keeping with the observations taken by the ancient astronomers.

The inferior planets Mercury and Venus

Mercury and Venus differ from the other planets in that, to an observer on the Earth, they never stray too far away from the Sun and, to viewers at low latitudes  they can only be seen for a few hours after sunset or a few hours before sunrise. For this reason they were known by the Greek astronomers as the the inferior planets. The superior planets (Mars, Jupiter and Saturn) are not tied to the position of the Sun and at certain times were visible in the middle of the night.

The reason why Venus and Mercury always appear in the same part of the sky as the Sun is that their orbits lie inside the Earth’s orbit. This is shown in the diagram below.

Venus cannot appear more than 46 degrees away from the Sun.  The green line shows the limits of Venus’s apparent position from the Sun.  

In the geocentric model this was explained by locking the Mercury and Venus’s motion to that of the Sun. The centres of their epicycles took exactly 1 year to orbit the Earth and were always in a direct line between the Earth and the Sun. This is shown in the diagram below.

As Venus moves around the Earth the centre of its epicycle always lies on the red line between the Earth and the centre of the Sun’s epicycle. This ‘locks’ the position of Venus to the Sun so that, to an observer on Earth. it cannot appear too far from the Sun. Mercury is locked in a similar manner.

Fine tuning the geocentric model.

The geocentric model which I’ve described so far had been developed by astronomers in the centuries before Ptolemy, although very little of the early astronomers’ original writings survive. Ptolemy’s contribution was to make two further modifications to make it more accurately fit observations of stars and planets. The first one I’ll describe is to deal with a phenomenon known as the ‘precession of the equinoxes’.  Even though the telescope hadn’t been invented, Greek astronomers had been taking precise observations of stars and measuring their positions for hundreds of years. Although the stars appeared fixed with respect to each other, over hundreds of years the alignment of the entire celestial sphere was slowly moving with respect to the Earth.

An effect of the precession of the equinoxes is that the pole star gradually changes. Today the bright star Polaris in the constellation Ursa Minor (the Little Bear) lies near the celestial North pole and all the other stars appear to rotate around it. In Ptolemy’s time another star in Ursa Minor, β Ursae Minoris, was the pole star and in about 13,000 years time the bright star Vega will lie close to the celestial North pole.

Ptolemy explained procession of the equinoxes by giving the celestial sphere an additional very slow rotation once every 26,000 years about a different axis in addition to its daily rotation around the Earth’s North-South axis.

Ptolemy had to make one more adjustment to the model to allow it to fit historic observations of the stars and planets and thus be able to accurately predict their future positions. Even when epicycles were added, the position of the planets was not where the model predicted they would be. The motion of the planets was still uneven compared to the background stars. The reason for this is that the planets move around the Sun  in elliptical (oval-shaped) orbits and the speed of a planet is fastest when it is closer to the Sun (see note 2). This is shown in an exaggerated form in the diagram below. In reality none of the planets’ orbits are actually this elliptical.

However Ptolemy, like all other Greek astronomers before him, believed that the all the planets’ deferents and epicycles must be perfect circles. This was for philosophical reasons, as the heavens were the epitome of perfection and the circle was seen as a perfect shape. So what he had to do was to modify the geocentric theory in a way summarised in the diagram below.

Firstly, the Earth didn’t lie at the centre of the deferent. He introduced a new point which he called the ‘eccentric’ for the centre of the deferent. The eccentric was some distance away from the Earth. Secondly, the centre of the epicycle didn’t move at a constant speed around the deferent but an uneven speed:

  • when it was closer to the Earth it moved faster – the dashed line
  • when it was further away it moved more slowly – the solid line.

The uneven speed was such that for each planet there was a point called the equant, at which the deferent would appear to move with a constant speed. A further complexity was that distance between the Earth, the equant and the eccentric varied for each planet.

So, strictly speaking, Ptolemy’s theory wasn’t truly geocentric, because the planets didn’t orbit the Earth, but each of their deferents orbited a hypothetical point called its eccentric which was offset from from Earth and was different for each planet. However, the theory fitted the observations in that it was possible to work out the past and the future position of the stars and planets using it. In around 150 CE Ptolemy published his theory in a book called ‘The Almagest’, which over the succeeding centuries was translated into many languages including Arabic and Latin and became the most influential astronomy textbooks for the next 1500 years.

A sixteenth century Latin translation of The Almagest -Image from Wikimedia Commons 

Next post

I hope you enjoyed this rather long post. In my next post I’ll talk about how Ptolemy’s theory was swept away by the scientific revolution

Notes

(1) To a viewer In the southern hemisphere, the sphere of stars wold appear to to rotate clockwise around a point in the sky directly above the South pole.

(2) For the more technically-minded reader the motions of the planets around the Sun are governed by Kepler’s laws of planetary motion, which state:

  • The orbit of a planet is an ellipse with the Sun at one of the two foci.
  • A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
  • The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

 

Solar eclipse 21 August 2017 – America on the move

Around 200 million Americans live within a day’s drive of the total eclipse path, the narrow band of territory from Oregon to South Carolina.  According to an article in The Atlantic, up to 7 million people, perhaps even people reading this, will travel to see Monday’s total eclipse. They will join the 12 million who are lucky enough to see the total eclipse without needing to travel.  If 7 million people do travel then it will be the biggest movement of people in human history to see a natural phenomenon.  Even if the fear of being stuck in a traffic jam for many hours puts people off and a lower number of people actually make the journey, the growth of social media means that the number of people following the eclipse on-line in real time will be unprecedented.

Headline from the Atlantic

To mark this event I’ve decided to publish an updated version of my eclipse post from July 9.

Update of the original post below———-

As nearly all of my readers, particular those who live in the US, will know, there will be a solar eclipse on 21 August. For lucky viewers in a narrow band of territory running West to East across the US, it will be visible as a total eclipse – when the Moon completely obscures the Sun and it suddenly goes dark for a short period of time.

Solar_eclipse_1999

Image from NASA

Although there is, on average, a total eclipse somewhere on Earth every 18 months, for each eclipse the region of on the world where a total eclipse can be seen is relatively small. Unless you are a so-called eclipse chaser – people who travel long distance to see solar eclipses, you will probably only get a change to see a total eclipse within 1000 km of where you live once or twice in your lifetime.  The last time that one has been visible anywhere in the contiguous US was back in February 1979, when most people currently alive weren’t even born (see note 1).

Image from NASA

The path of totality, where a total eclipse can be seen is shown as the blue band in the map below. Places either side of the blue band will only see a partial eclipse, when the Sun is only partially obscured and the further away from it you are, the smaller the fraction of the Sun that will be covered. Unfortunately for people like me who live in the UK the eclipse will be barely noticeable, only a small fraction of the Sun will be obscured, just before sunset.

 

Image from timeanddate.com

The path of the eclipse as it moves across the US is shown below. If you click on the diagram it will show the map in greater detail. The total eclipse will begin over the Pacific Ocean and will reach the Pacific coast of America in Oregon just West of the state capital Salem (shown below) at 10:16 am local time.

Salem Oregon, the first major city to see the total eclipse

The table below shows the eclipse times from some selected cities in its path.

Data from timeanddate.com

Why do we have eclipses?

It is common knowledge that solar eclipses occur when the Moon passes in front of the Sun, obscuring some or all of its light. This is shown in the diagram below, although the distances and sizes of the Earth, Moon and Sun aren’t to scale.

Moon Sun Earth

In the diagram above, the regions labelled A see a partial eclipse. Only in the small region labelled B does the Moon fully obscure the Sun and a total eclipse is seen.

From the diagram above you would expect the Moon to pass in front of the Sun every month and there to be an eclipse every month. This is clearly not the case.

The Moon’s obit around the Earth is actually tilted at about five degrees with respect to the Earth’s orbit around the Sun, as shown in the diagram below.

 

This means that during most lunar months the Moon will pass just below or just above the Sun rather than obscuring it. There are only two time windows in a year when it is possible for a solar eclipse to occur (see note 2). It also only possible for a lunar eclipse, when the Earth blocks sunlight hitting the Moon, to occur in the same time window.

The other reason why we can have eclipses is that, although the Moon is much smaller than the Sun – roughly 400 times less in diameter, by a strange coincidence it is also roughly 400 times closer to the Earth than the Sun. This means that, when viewed from the Earth, the Sun and the Moon appear to be almost exactly the same size. If the Moon were closer to the Earth, or larger in size, then solar eclipses would be more frequent and would last longer. If the Moon were further away, or smaller, then we would only have what is called an annular eclipse where the disc of the Moon is too small to fully cover the Sun.

Variation in the apparent size of the Sun and Moon throughout the year

The Moon, rather than moving in a circular orbit, moves in an elliptical (oval-shaped) orbit around the Earth. This means that its apparent size as seen from Earth varies. It appears largest when it is closest to the Earth and smallest when it is furthest away. The apparent sizes of large objects in the sky are measured in degrees. 1 degree is roughly how big a British 1 pence or US 1 cent coin would appear to be if you held it up at a distance of approximately 1.2 metres (4 feet) away from your eye. When it is at its closest, the Moon is 0.558 degrees in diameter and at its furthest away it is 0.491 degrees in diameter.

The Earth also moves in a elliptical orbit around the Sun. So the apparent size of the Sun varies as seen from the Earth, but the variation is not as a great as the apparent variation in the size of the Moon. When the Earth is at its closest to the Sun, the Sun is 0.545 degrees in diameter and when the Earth is at its furthest away the Sun is 0.526 degrees in diameter.

This variation means that there are times when the Moon lies directly in front of Sun, but, because it appears to be slightly smaller, we only see an annular eclipse.

annualar eclipse

An annular eclipse- image from NASA

How long can a total eclipse last?

The maximum time that anyone will see a total eclipse on 21 August 2017 is just over 2 and half minutes.  The are a number of factors leading to a longer eclipse. The main ones are

  • the Moon should be as close to the Earth as possible making its apparent size as large as possible
  • the Earth should be as far as possible from the Sun making the Sun’s apparent size as small as possible.

Another less important factor is that the eclipse should occur at a point on the Earth when the Sun and the Moon are directly overhead. This gives an additional but very small boost to the apparent size of the Moon relative to the Sun.

Moon Overhead

This diagram, which is greatly exaggerated, shows that when the Moon is directly overhead (A) it  is very slightly closer to the Earth then when it is at the horizon (B). This causes the moon to appear to be slightly (1.8%) larger in diameter.

All of these conditions will be achieved for an eclipse which is predicted in South America near the equator on 16 July 2186 and will be 7 and half minutes at its greatest duration (Espinak and Meeus 2011).  I will not be around to see it, and neither will you!

 

Notes

(1) This is because, according to the CIA world fact book, for the United States  the median age (the age at which 50% of the population is under this age and 50% of the population is over this age ) is 37.9 years. So half the American population were born less than 37.9 years ago, i.e on on after Sept 1979.

(2) The two times in a year when a total eclipse can occur aren’t the same every year but change from year to year. This is due to an astronomical effect called precession of the line of nodes.

References

Espinak, F and Meeus J (2011) Five millennium canon of solar eclipses: -1999 to +3000, Available at: https://eclipse.gsfc.nasa.gov/SEpubs/5MCSE.html (Accessed: 9 July 2017).