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).

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).

 

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.

 

The equinox March 20 2017

Now that we are in the month of March, for most of us in the northern hemisphere the worst of the winter is over, and it is only a few days until 21 March, the first day of spring.

March 21

 

There is a commonly held view that March 21 is the spring 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, this is only approximately correct. March 21 can sometimes be the date on which the spring equinox falls but the precise date varies from year to year and also depends upon where you are located.  In fact, 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), suggesting 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

Because the axis of the Earth is tilted rather than perpendicular to its orbit around the Sun, different parts of the Earth are closer to the Sun at different times of the year.

  • At the winter solstice in December (point A in the diagram) the North Pole is tilted furthest 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 are longest compared with the period of daylight, and in the southern hemisphere the reverse applies.
  • At the summer solstice in June (point C in the diagram) 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. At any given latitude, whether north or south of the equator, there will be the same amount of daylight.

On what date do the equinoxes occur?

The diagram also shows that the Earth moves in an oval, or 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 March 19, 20 or 21 and the autumn equinox on Sept 22 or 23. Over a longer time span there is an even greater range of dates (see notes).

spring equinox times

autumn equinox times

(Data TimeandDate.com 2016a)

What date of the year are there are exactly 12 hours of daylight?

The first point to consider is what to we mean by daylight? Do we consider twilight, the time just after sunrise or just before sunset when it is not completely dark, to be daylight? If we use the common definition of “hours 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. There are two reasons for this. First the definition of sunrise is actually the point at which 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 disk at sunrise at the equinox in London.

Sun path sunrise

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

Sun path sunset

In addition, 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 dates around the equinox in London and Wellington (in the northern and southern hemispheres respectively) and it is clear to see that date on which there are exactly 12 hours of daylight and 12 hours of darkness is not 20 March. In London it is 3 days earlier on March 17 but in Wellington it is 3 days later on March 23.

Length of day march

(TimeandDate.com 2016b)

 

Notes

(1) The table shows that there is a pattern in that the times of the two equinoxes in a given year are just under six hours later than the previous year, unless the year is a leap year, in which case they are just under 18 hours earlier than the previous year.  Thus, the equinoxes will occur at roughly the same date and time every four years. For example the March equinox will occur at:

  • around 4 am (GMT) on March 20 in the years  2020, 2024 and 2028
  • around 10 am (GMT) on March 20 in the years 2017 , 2021 and 2025
  • around 4 pm (GMT) on March 20 in the years 2018 , 2022 and 2026
  • around 10 pm (GMT) on March 20 in the years 2019 , 2023 and 2027

However this four-year pattern doesn’t always hold because leap years don’t always occur every four years. Century years which are not divisible by 400 e.g. 1800, 1900, 2100 are not leap years. So for example in the year 1903, where there had not been a leap year for 7 years, the equinoxes occurred relatively late. On this year the equinoxes occurred at 7:15 pm (GMT) on 21 March and 5:45 am (GMT) on 24 September. So in Tokyo Japan, which is 9 hours ahead of GMT, in 1903 they occurred at 4:15 am on 22 March and 2:45 pm on 24 September

(2) The exact day on which there is 12 hours of daylight will vary with latitude.

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).

 

Satellite navigation – the next ten years

Satellite navigation is such a vital part of day-to-day life that other countries of the world are planning to build up their own network of navigation satellites and move away from total reliance on the American system, GPS.

America

As mentioned in my previous post, the GPS system is operated and funded by the US government. The first GPS satellite was launched in 1978 and, over the following years, more were launched – covering more and more of the Earth’s surface. In 1993 the system become fully operational when it had a full set of 24 satellites covering the entire surface of the globe. However the GPS signals were scrambled, making them available only to US military users who had the equipment to unscramble the signals. In 2000, following a decision by president Bill Clinton four years earlier, the signals stopped being scrambled and since then have been available for civilian use anywhere on the Earth.

gps_satellite

Image from NASA

The lifetime of  a GPS satellite is 12.5 years (Wall 2015) and the cost to build and launch a satellite is $200 million (see notes at the bottom of the post for more details). Satellites need to be launched constantly because once a satellite stops functioning it must be replaced, as it cannot be repaired. The total cost of keeping the GPS network up and running is around $1 billion per year, which works out at only $3 for each person in the US. I think that the US government will continue to operate and fund GPS as a free civilian service for the whole world, for the foreseeable future. However, they have the ability to deny use to a hostile country, should the need ever arise. One way in which access could be denied is by switching off the GPS satellites’ transmitters when they pass over particular areas of the world.

Russia

Towards the end of the cold war the Soviet Union developed its own version of GPS called GLONASS, an abbreviation for “GLObal NAvigation Satellite System”. Like the American system, it was for military use only.

glonass_logo

Image from Wikimedia Commons

GLONASS wasn’t fully completed by the Russian government until 1995 – four years after the breakup of the Soviet Union. The satellites only had a lifetime of 7 years and in the 1990s there was an economic crisis in Russia. So, as they began to fail they were not replaced. By the end of 2001, there were only 6 working satellites, and a minimum of 24 is needed to give full coverage of the Earth. In the 2000s the Russian economy improved and the funding to reestablish the service became available.  The Russian government made the commitment that, like GPS, GLOSNASS would be freely available to civilian users all over the globe (Sputnik 2006). In December 2015, the system became fully operational again, with 24 satellites covering the globe. In fact. even before it was fully completed many smart phones already had ‘dual satellite chips’ which can receive both GPS and GLONASS signals. This provides greater accuracy because if a phone can’t receive a strong signal from GPS it can supplement this with signals from the GLONASS satellites.

Europe 

Like the US government with GPS, Russian governments have stated that their policy is that GLONASS will be freely available for civilian use for the foreseeable future. This cannot be absolutely guaranteed and it is possible that, if the political climate were to change radically, then either the US or Russia could restrict the availability of their systems.

Current estimates suggest that 6 to 7% of the EU’s GDP depends on satellite navigation applications, including critical sectors such as energy, telecommunications and financial services. Therefore, if satellite navigation services in Europe were to be disrupted for even just a few days, the large-scale economic losses would be massive.

To address this, the EU announced back in 2002 that it would build its own satellite navigation system called Galileo after the famous Italian astronomer (shown below).

Galileo_Galilei

Image from Wikimedia commons

Galileo was planned to be fully up and running by 2008 at a cost of 3 billion euros, but the project has been subject to numerous delays and cost overruns. Currently there are only 11 Galileo satellites operating, so the coverage is only partial, and it won’t be fully operational until 2020 (European GSA 2017). At the moment most smartphones can’t receive a signal from the Galileo satellites, but this will change over the next 5-10 years as Galileo becomes established and manufacturers move towards multi-satellite chips.

galileo_logo

 

In order to make it independent of the US, Russia and Europe, China too is building its own satellite navigation system called BeiDou, which it hopes to have fully operational in 2020.

beidou_navigation_satellite_system

Jamming signals

As also discussed in my last post, the typical power of the transmitter of a GPS satellite is only about 500 watts, similar to that from a mobile phone mast. However, a GPS satellite is at least 20,000 km away from a GPS receiver on the Earth’s surface. Therefore, as shown in the diagram below, by the time it reaches the Earth’s surface the signal has spread out over such a huge area that it is extremely weak.

gps-weakening

The signal received from a GPS satellite on an area 10 centimetres square on the Earth’s surface is about 0.0000000000000003 watts  (or 300 billion billionths of a watt).

This means that if someone were to transmit on the same frequencies as used by GPS satellites the transmitter power would need to be quite low, a small fraction of a watt, to block out or jam the much weaker signal from GPS satellites over a large area. If GPS signals were jammed, the effects could be very severe.  Satellite navigation systems, including those used by the military and emergency services, would no longer work and many aircraft would not accurately know their position. The potential impact is so serious that the US government has made the selling, marketing or use of any GPS jamming devices illegal, punishable by a fine of tens of thousands of dollars or imprisonment. Since October 2012 there has even been a US government hot-line where the public can report anyone they suspect of using jamming equipment. However, it is unlikely that this would prevent jamming by a hostile foreign power.

gps-jamming

 

jammer-hot-line

 

References

European GSA (2017) European GNSS Service Centre FAQs, Available at: https://www.gsc-europa.eu/helpdesk/faqs (Accessed: 23 January 2017).

Sputnik (2006) Russia to lift Glonass restrictions for accurate civilian use, Available at: https://sputniknews.com/russia/2006111355588641/ (Accessed: 18 February 2017).

Wall, M (2015) US Air Force Launches Advanced GPS Satellite into Orbit, Available at: http://www.space.com/28926-air-force-launches-gps-satellite.html (Accessed: 18 February 2017).

 

Notes

Over the past 40 years there have been three generations of GPS satellites. Known as Block I, Block II and Block III. Each generation is larger, more sophisticated and has a longer lifetime than the previous one.

  • Block I satellites were launched between 1978 and 1985. They were only designed to last 5 years and none of them are still operational.
  • Block II satellites were launched between 1989 and 2016 The cost of $200 million quoted refers to a Block II-F satellite, the latest versions of the Block II generation.
  • The launch of the latest GPS satellites, Block III, is planned for mid 2018.

 

GPS

The Global Positioning System, better known as GPS, has come to affect countless aspects of our daily lives, from directing our holiday aeroplanes to enabling us to drive round an unfamiliar city without any map other than the one on our mobile phone.  At the risk of making myself sound like a scary stalker, I can check even up on Mrs Geek’s whereabouts at any time using Find Friends on my iPhone.

There have been GPS satellites in position orbiting the Earth for many years, but only the US military had the equipment able to unscramble the signals and in the mid 1990’s even the lightest GPS receivers still weighed a pretty chunky 1.25 kg.  When President Clinton announced in 1996 that the system would be opened up for civilian use by the year 2000, few could have anticipated that this would kick start a massive industry and that by 2017 almost a third of the population of the world would have a GPS receiver in their pocket, on their smartphone, which together with a mapping application such a Google maps allows people to get around unfamiliar places without the fear of getting lost.  Today, Sat Nav devices using GPS signals come as standard in many new cars.

bill-clinton

 

What is GPS?

GPS is a network of satellites in orbit around the Earth. In total, since 1978, the US has launched a total of 72 GPS satellites. The satellites are in a region of space called Middle Earth Orbit, roughly half way between Low Earth Orbit satellites, such as the International Space Station, and the higher Geostationary orbits used by satellites which transmit TV signals.

gps_satellite

Image from NASA

Since the first launch, there have been two launch failures and most of the earlier satellites launched before 1997 are no longer transmitting. So, the GPS system currently consists of 31 working satellites.  The satellites aren’t all in the same orbit, as to give full coverage of the Earth’s surface there are six separate orbits.  Each of these six orbits is at a height of 20,200 km above the Earth’s surface, but the orbits are at different angles to the equator, as shown in the diagram below .

As each satellite moves in its orbit, it will appear to an observer on the Earth to: rise, move slowly through the sky, and then set a few hours later. Because there are 31 satellites in total, from a given point on the Earth there will typically be 7 or 8 visible at a given time and, as I’ll explain later, you only need to get a good signal from four satellites to get an accurate position.

gps-satellites

The satellites which are above the horizon are shown in red. The satellites which are below the horizon and thus cannot be seen are shown in black – Image from wikimedia commons

How does GPS work?

Each GPS satellite has a set of atomic clocks, accurate to a few billionths of a second per day. These clocks are kept in step with atomic clocks on the ground, so the clocks on the GPS satellites all show the same time. The satellites have a transmitter which sends a signal to Earth. The power of the signal is around 500 watts, which is roughly similar to the power radiated by the transmitter on a mobile phone mast (see note 1). There are three parts to the signal: the first part identifies which satellite is sending it, the second is a very accurate time signal, and the third part, known as almanac data, tells the GPS receiver the position of both itself and every other satellite. This enables the receiver to know exactly where the signal is coming from.

When a GPS receiver receives the signal it can compare the time at which it received the signal with the time at which it was transmitted and thus work out how long the signal has taken to travel.  The distance to the satellite can therefore be worked out as using the simple formula  distance = time x speed.  The speed of the radio waves measured here is the same as the speed of light, since radio-waves and light are both forms of electromagnetic radiation.  Therefore, if a GPS signal takes 0.1 seconds to reach us then the satellite must be 29,979.2458 km away, as the speed of light is 299,792.458 kilometres per second

Measuring how long the signal takes to get from you from a single satellite only tells you your distance from that satellite.  In order to get your position on the Earth you would need to know your distance from three satellites. For example the red sphere shows all the points that are a certain distance, say 23,000 km, from satellite 1. Similarly, the yellow sphere shows all the points that are at distance of 23,500 km from satellite 2 and the green sphere shows all the points that are at distance of  22,500 km from satellite 3. There is only one point, labelled with an ‘x’, which is on all three spheres (23,000 km from satellite 1, 23,500 km from satellite 2 and 22,500 km from satellite 3).

gps-distance

 

In fact it is a little bit more complicated than this because the GPS receiver does not have an accurate atomic clock so it cannot know the current time as accurately as the GPS satellites. The clock on the GPS receiver will almost certainly happen to be fast or slow by a small fraction of a second.  However, in order to get an accurate measure of the time taken for the signal to arrive, the clock on the GPS receiver must synchronise perfectly with the clocks on the GPS satellites. The way it gets around this is by using the time signal from a fourth satellite to work out the amount of time by the which the receiver is fast or slow. For those of my readers more mathematically inclined, I have put the details of how it does this in the notes at the bottom of this post

Limitations of GPS

There are some limitations to the accuracy of position provided by GPS.  One is that the speed of radio waves is not constant, as conditions in a region of the upper atmosphere, known as the ionosphere, can cause them to slow down slightly. This makes the time taken for the signal to arrive at the receiver slightly longer, implying that a satellite is further away than it actually is. Another factor is that large objects such as trees between the receiver and the GPS satellite may weaken the signal or even block it altogether.

gps-ionosphere

 

An additional false signal can also be generated when a GPS signal is reflected off tall buildings. This effect, which is known as multi-path error, is shown in the diagram below. The receiver receives two signals from the same satellite: a direct signal (the pink line) and a reflected signal (the yellow line). The two signals have travelled different distances, so the GPS receiver calculates two different distances to the same satellite. This can result in an incorrect distance if the wrong signal is chosen by the receiver. Alternatively, the two signals can interfere and cancel each other out so no signal is received.

gps-building

 

Errors could also be caused by slight inaccuracies in the atomic clocks on the satellites or by the satellite making an error when calculating its own exact position. Nevertheless, even taking all these factors into account, a GPS receiver should be able to work out its position to an accuracy of within 10 metres.  This, however, only applies to locations outdoors and above ground, as the signal cannot travel through large and complex buildings.

Given the current state of world politics, I wonder whether today’s US politicians would be so willing to make military technology available to the masses.

And finally.. the future of GPS

GPS satellites are managed by the US government. In order to make themselves independent of America other countries have introduced, or are planning to introduce, their own networks of navigation satellites. I will talk about this in my next post.

Notes

(1) This is actually an oversimplification the satellites also transmit on a number of different wavelengths. The power of the transmission at the different wavelengths varies.

(2) The GPS calculation in the receiver uses four equations in the four unknowns x, y, z, tc, where x, y, z are the receiver’s coordinates, and tc is the time correction for the GPS receiver’s clock. The four equations are:

where

  • c  is the speed of light  = 299,792.458 kilometres per second
  • tt,1, tt,2, tt,3, tt,4 are the times that GPS satellites 1, 2, 3, and 4, respectively, transmitted their signals (these times are provided to the receiver as part of the information that is transmitted).
  • tr,1, tr,2, tr,3, tr,4 are the  times that the signals from GPS satellites 1, 2, 3, and 4, respectively, are received (according to the inaccurate GPS receiver’s clock)
  • x1, y1, z1  are the coordinates of GPS satellite 1 (these coordinates are provided to the receiver as part of the information that is transmitted); similar meaning for x2, y2, z2, etc.

The receiver solves these equations simultaneously to determine x, y, z, and tc.