Perihelion 2013

At around 0500 GMT on 2 January 2012 the Earth was at perihelion, its closest approach to the Sun this year.

If that sounds confusing to you, and has you wondering why it’s so cold given that the Earth is at its closest to the Sun, then this belies (a) a northern-hemisphere-centric attitude (in the Southern Hemisphere it’s summer right now), and (b) a misunderstanding of what causes the seasons.

The Earth orbits the sun in a nearly circular orbit called an ellipse. The degree by which an orbit differs from a perfect circle is called the eccentricity, e. If e = 0 then the orbit is circular; if e = 1 then the orbit is parabolic, and therefore not gravitationally bound to the Sun. The Earth’s orbital eccentricity is 0.0167, meaning that it is very nearly circular, with the short axis of the ellipse being around 96% the length of the long axis. Thus, during perihelion Earth is 0.983AU from the Sun, while during aphelion (its furthest distance from the Sun, occurring this year on 4 July) Earth is 1.017AU from the Sun. (1AU = 1 astronomical unit = the average distance between the Earth and the Sun = 150 million km).

The seasons on Earth have really nothing to do with how close the Earth is to the Sun at different times of year. Indeed how could they, given that the difference in distance between closest and furthest approach is only a few per cent? The seasonal differences we experience are of course caused by the tilt of the Earth’s axis, which is inclined by 23.5 degrees from the vertical.

This tilt means that, as Earth orbits the Sun, for six months of the year one hemisphere tips towards the Sun, so that it experiences longer days than nights, becoming most pronounced at midsummer, at which point the Sun reaches its highest in the sky at noon. Simultaneously the other hemisphere tips away from the Sun, and experiences shorter days than nights, becoming most pronounced at midwinter, on which day the Sun is at its lowest noontime altitude.

The further you are from the equator the more pronounced the seasonal effects. In fact equatorial countries don’t experience seasonal variations, while the poles experience extremes with six-month-long winters and summers. The timing of perihelion and aphelion relative to our seasons is entirely random. The fact the southern hemisphere midsummer (21 Dec) almost coincides with perihelion (2 Jan) is simply that; a coincidence. Given that fact, there is no reason to be surprised that perihelion occurs so close to northern hemisphere midwinter: it has to happen some time and it’s a coincidence that it happens to occur within two weeks of midwinter / midsummer.

Leap Year 2012

Thirty days hath September,

April, June, and November;

All the rest have thirty-one,

Save February, with twenty-eight days clear,

And twenty-nine each leap year.

This year is a leap year, when the month of February has 29 days in it, rather than the usual 28. The rhyme above is a mnemonic to help us remember the days in each month, but it doesn’t explain why we need leap years, and why they occur once every four years.

To understand the reason for leap years we have to look to astronomy, and in particular to the orbit of the Earth around the Sun. The Earth orbits round the Sun in 365.256363 days. This is a bit awkward as it means that the year cannot easily be divided into a whole number of days. If we round the year down to the nearest whole number of days we get 365 days in a year, which is indeed what we have in most calendar years.

So why not just leave it at that? Isn’t 365 days close enough to 365.256363? After all it’s 99.93% of the actual year, which is nearly 100% right, yes?

Actually; no. In ancient Egypt, where they lived with a calendar year of 365 days, the seasons began to drift at a rate of one day every four years. If we had stuck with the Egyptian calendar of 365 days every year then the longest day, which we take to fall on 21 June in most years would fall on 20 June four years later, then 19 June four years after that, until over the course of 730 years or so the longest day would occur when our calendars said it was the middle of winter.

Obviously something needed to be done to fix this problem. Enter Julius Caesar who, in 46BC, introduced what is known as the Julian Calendar. In this calendar Caeser recognised what Greek astronomers had long known; that the year is closer to 365¼ days long. They didn’t know that the Earth went round the Sun in 365¼ days, but they knew that the seasons repeated themselves on a 365¼ day cycle, and not a 365 day cycle as the Egyptians thought.

To account for this more accurate measure of the changing seasons, and to align the calendar better with the real world, Julius Caeser announced that every fourth year would have an extra day in it, to occur at the end of February. This would allow the calendar to keep in line with the real changing seasons, so that the longest day would always fall on the same day of the calendar.

But by 46BC the seasons had already drifted a lot; in fact the Roman calendar was about 80 days behind the actual seasons, so Caesar proclaimed that 46BC would have extra days in it, and be 445 days long, so that the calendars would be aligned on 1 January 45 BC, at which point the new calendar of leap years would begin.

The Romans didn’t call these leap years though; that name came along about 1400 years later. They were called “leap years” because the occurence of them every four years caused festive days (like Christmas), which usually advanced one weekday per year, to suddenly leap forward by two days. For example, Christmas Day in 2009 fell on a Friday, in 2010 on a Saturday, in 2011 on a Sunday, but this year, in 2012, it will leap forward to a Tuesday.

Not the Whole Story

Of course things are never that simple, are they? In fact the year is not 365¼ days long either, it’s 365.256363 days long if you measure it in terms of how long it takes the Earth to go round the Sun, or 365.242189 if you measure it in terms of how long it takes the Sun to return to the same part of the zodiac (which is indeed what we need to measure if we want to track the seasons).

We no longer have a Julian Calendar of 365 days each year with 366 every fourth leap year. Instead we have adopted the Gregorian Calendar where:

Every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100; the centurial years that are exactly divisible by 400 are still leap years.

So 1900CE wasn’t a leap year (nor was 1700 or 1800), event though it was due to be, but 2000CE was. This is to fine-tune our year to fit even better with the changing seasons. Without this slight tweak then even the Julian calendar would drift with the seasons, albeit not as drastically as the Egyptian fixed 365 day year.

Leap Seconds

Today the International Telecommunications Union are voting on whether to abolish the leap second. This miniscule measure of time is added in to our time-keeping systems every so often to make sure they align more accurately with the time as measured by the spin of the Earth.

The original definition of the second was 1/86400 of a mean solar day which is related to the speed of the Earth’s spin about its axis. We might call this the “Earth second”. However the Earth’s spin in not regular. To begin with the Earth is slowing down by a couple of milliseconds per century due to tidal breaking. This breaking action is as a result of the drag of the Earth spinning beneath the tides created by the Moon. In effect the Moon is “stealing” energy from the Earth, increasing in its orbit about us while our spin slows.

In addition to this discrepancy the Earth is occasionally wobbled off course by major geological events, such as earthquakes. The 2004 Pacific earthquake which resulted in the Boxing Day Tsunami actually caused the Earth to speed up by over 2 milliseconds.

To avoid the problem of an irregular length of day – and therefore an irregular length of second – scientists adopted the much more regular SI second, which is the length of time it takes for 9,192,631,770 cycles of vibration of atomic caesium. This “atomic clock second” is accurate to one part in ten billion, and since 1972 this has been the international standard in timekeeping.

But time kept using the the SI second doesn’t match exactly with time kept based on the spin of the Earth, which after all is the time we experience every day. In order to make these two time signals match leap seconds are added every so often. Since 1972 25 leap seconds have been added. The last leap second was added at 23:59:59 on 31 December 2008, and the next one is due to be added at 23:59:59 on 30 June 2012. But leap seconds themselves are irregular, and are decided on by the ITU whenever the two time signals drift by more than 0.9 seconds.

The argument for abolishing these additional leap seconds is that it creates problems for modern computing and navigation systems that use the atomic clock second. Every time one of these irregular leap seconds is added the world’s hi-tech time keeping devices need to check and adjust by one second. It would be far simpler for us to use only “atomic clock seconds”.

However if we were to ditch the leap second then our civil time keeping would begin to drift with respect to “real” Earth time, so that in thousands of years time our clocks might read 8am just as the Sun is setting. This might seem like a minor concern right now – after all a millennium is a long time – but it’s something that astronomers and scientists do need to consider to avoid future problems. One alternative would be to introduce a “leap hour” to be introduced every few hundred years to keep the clock aligned with the real world.

Aphelion 2011

At 1600 BST (1500 GMT) on 4 July 2011 the Earth will be at aphelion, its furthest from the Sun this year.

The Sun

If that sounds confusing to you, and has you wondering why the Earth is at its furthest from the Sun in Summer, then this belies (a) a northern-hemisphere-centric attitude (in the Southern Hemisphere it’s winter right now), and (b) a misunderstanding of what causes the seasons.

The Earth orbits the sun in a nearly circular orbit called an ellipse. The degree by which an orbit differs from a perfect circle is called the eccentricity, e. If e = 0 then the orbit is circular; if e = 1 then the orbit is parabolic, and therefore not gravitationally bound to the Sun. The Earth’s orbital eccentricity is 0.0167, meaning that it is very nearly circular, with the short axis of the ellipse being around 96% the length of the long axis. Thus, during perihelion (its closest approach to the Sun, which occurred on 3 January 2011) Earth is 0.983AU from the Sun, while during aphelion Earth is 1.017AU from the Sun. (1AU = 1 astronomical unit = the average distance between the Earth and the Sun = 150 million km).

The seasons on Earth have really nothing to do with how close the Earth is to the Sun at different times of year. Indeed how could they, given that the difference in distance between closest and furthest approach is only a few per cent? The seasonal differences we experience are of course caused by the tilt of the Earth’s axis, which is inclined by 23.5 degrees from the vertical.

This tilt means that, as Earth orbits the Sun, for six months of the year one hemisphere tips towards the Sun, so that it experiences longer days than nights, becoming most pronounced at midsummer, at which point the Sun reaches its highest in the sky at noon. Simultaneously the other hemisphere tips away from the Sun, and experiences shorter days than nights, becoming most pronounced at midwinter, on which day the Sun is at its lowest noontime altitude.

Earth

The further you are from the equator the more pronounced the seasonal effects. In fact equatorial countries don’t experience seasonal variations, while the poles experience extremes with six-month-long winters and summers. The timing of perihelion and aphelion relative to our seasons is entirely random. The fact the northern hemisphere midsummer (21 Jun) almost coincides with aphelion (4 Jul) is simply that; a coincidence. Given that fact, there is no reason to be surprised that aphelion occurs so close to northern hemisphere midsummer: it has to happen some time and it’s a coincidence that it happens to occur within two weeks of midwinter / midsummer.

To take this explanation even further, we can calculate how much variation in incident sunlight there would be in two scenarios: 1. an imaginary scenario where the seasonal variations in temperature are due to the tilt of the Earth’s axis but where the Earth goes round the Sun in a perfectly circular orbit and 2. an imaginary scenario where the Earth’s axis isn’t tilted, but where its orbit is elliptical in the same degree as ours actually is.

1. The Sun appears at its highest point in our sky each day at noon. The highest it ever gets is at noon on midsummer. The lowest noontime altitude occurs at noon on midwinter. In Scotland the Sun is around 55 degrees above the horizon at noon on midsummer, and only 10 degrees above it at noon on midwinter. The amount of energy from the Sun radiant on a fixed area is proportional to the sine of the altitude, so the ratio of the solar energy radiant on a square metre of Scotland between midsummer and midwinter is

sin(55) / sin(10) = 4.72

So here in Scotland we get 372% more energy from the Sun in summer than we do in winter, due to the tilt of the Earth’s axis.

2. If the Earth’s axis was not tilted, then we would only experience temperature differences from the Sun depending on how far or near we are from it. In this case, the amount of energy from the Sun radiant of a fixed area is proportional to the square of the distance from the Sun, so the ratio of the solar energy radiant on a square metre of Scotland between perihelion and aphelion is

(1.017/0.983)^2 = 1.07

So we get 7% more energy from the Sun at perihelion than we do at aphelion, due to the differing distances to the Sun. From this you can see that, while the distance to the Sun has some effect on how much heat we receive, it is a very small effect compared to that produced by our axial tilt.

Twice Around the Sun

Today is my little boy’s second birthday. He was born at 0940 on Wednesday 17 June 2009, and so today is the day that the Earth completes two orbits of the Sun since his birth.

However at 0940 today the Earth will not quite have completed its two orbits of the Sun; it will have a little bit further to go. Around 1.3 million km further to go to be precise, and it’ll cover that distance in 12h 18m 20s…

Happy Second Birthday!

Our calendar year is divided up into 365 days, the closest whole number of days to the actually time it takes to Earth to orbit the Sun, called the sidereal year, which is equal to 365.256363004 days.

Therefore the year (the time it takes the Earth to orbit the Sun) isn’t exactly 365 days long, and so every four years we add on one extra day in February to get a 366 leap year to make up for the difference.

If we didn’t do this, and instead insisted that the year was always 365 days long, the seasons would begin to drift until midsummer’s day (21 June on our calendars) would fall in the middle of winter within 753 years, a problem for our descendents. *

So, since my son was born there have been 2 x 365 days marked off on our calendar (730 days), but in order for the Earth to make two complete orbits of the Sun we have to wait 2 x 365.256363004 = 730.512726008 days, or an extra 0.512726008 days, which equals 12h 18m 20s.

My son was born at 0940, and so he will be two sidereal years old at 2158 this evening.

If I was interested in the instant when the Sun returned to exactly the same point within the cycle of the seasons I would have to carry out the same calculation except this time using the tropical year of 365.24219 days. In this case two tropical years equals 730.48438 days, or an extra 0.48438 days, which equals 11h 37m 30s.

My son will be two tropical years old at 2117 this evening.

Happy birthday, little spaceman!

* We have to calculate the drift based on the tropical year, the length of time it takes for the Sun to return to the same place in the cycle of the seasons, which is equal to 365.24219 days. A basic 365 day year differs from this by 5h 48m 45s, which would result in the seasons drifting so that midsummer’s day falls in midwinter in 753 years.

Even the 365.25 day leap year differs from this tropical year by 11m 15s, resulting in a drift which happens much more slowly than 753 years, but it still happens, taking 23,383 years for midsummer’s day to fall in the middle of winter. To correct for this we fine-tune our leap year formula so that we do not add in an extra day in a year which is divisible by 100 unless it is also divisible by 400!

This makes the average length of a year 365.2425 days, which still differs from the tropical year by 26.8 seconds. This barely matters, and no further corrections to our calendar are needed since the seasonal drift in this case will take over 500,000 years to happen, and by that time the Earth’s year will have changed in length anyway!

Year	Length of Year (days)	Difference from Tropical Year (seconds)
Tropical Year	365.24219	0
Basic Year	365	20,925s (5h 48m 45s)
Simple Leap Year	365.25	675s (11m 15s)
Fine-tuned Leap Year	365.2425	26.8s

The Equation of Time

Today, 13 June, is one of only four days in the year when the time as read on a sundial will be exactly correct.

Sundials usually tell the time using the shadow of the gnomon as cast by the Sun. This is possible as the Sun appears to move across the sky at an approximately constant speed, and so the shadow of the gnomon also moves at an approximately constant speed. The inconstancy of the Sun’s apparent motion in the sky – and therefore of the gnomon’s shadow on a sundial – is the subject of this article, and is calculated using the Equation of Time.

A Sundial

If you look at the shadow of a sundial’s gnomon it will fall onto a curve of numbers, along hour lines indicating local solar time. This is not equal to the official clock time until three important corrections are made:

1. Correct for daylight saving time. “The Sun reaches it highest point due south at noon” is only true in the UK between the last Sunday in October and the last Saturday in March, when the UK operates on Greenwich Mean Time (GMT). For the rest of the year we use British Summer Time, or BST, which is GMT+1. Therefore if you’re using a sundial in the UK during BST you will need to add one hour one to the sundial time to correct for this.

2. Correct for longitude (the G in GMT). In the UK our time is based on the time in Greenwich, London, hence the G in GMT. At noon in Greenwich the Sun will be due south at 12 noon GMT. At locations east of Greenwich (Norwich, Ipswich, Cambridge) the Sun will have reached its highest point due south earlier than 12 noon, and for locations west of Greenwich (which is most of the UK) the Sun will reach its highest point due south later than 12 noon. Given that most sundials are fixed in location this correction can be built into the sundial itself (i.e. the hour lines can be moved through an angle equal to the difference between the sundial’s longitude and the meridian that mean solar time is calculated).

3. Correct for the average (“mean”) position of the Sun (the M in GMT). If the Earth orbited the Sun in a perfectly circular orbit, and if its axis of rotation was perfectly perpendicular to the plane of its orbit, then the Sun would always be in the same point of the sky at noon each day. The fact that the Earth’s orbit is not a perfect circle, and the Earth’s axis is titled, means that the position of the noon Sun varies over the year.

Orbital Eccentricity

The Earth orbits the Sun in an elliptical path, meaning that at some times of the year it is further from the Sun than at other times. Crucially though, this also means that the Earth orbits the Sun at different speeds depending on where it is in its orbit. At its closest approach to the Sun (called perihelion) in early January the Earth is moving fastest in its orbit, and at its furthest from the Sun (called aphelion) in early July it is moving slowest. The Earth’s orbit isn’t drastically elliptical (it is almost circular) and so the differences are not great, but at perihelion we’re moving at 30.3 km/s and at aphelion we’re moving at 29.3 km/s (a 3.4% difference).

This means that if we view the Sun in the sky it will appear to move against the background stars at different speeds depending on where in its orbit the Earth is. At perihelion, in early January, the Sun speeds up by 7.9 seconds per day from the average, or mean; at aphelion is slows down by 7.9 seconds per day. These few seconds per day accumulate over the year, and mean that the amount by which the Sun might deviate from being at due south at noon, due solely to the effects of our elliptical orbit, varies from zero (early Jan and early July) up to 7m 40s (early April, when it is negative, and early October, when it is positive). It can be drawn as a sine wave with an amplitude of  7m40s and a period of 1 year.

The component of the Equation of Time due to Earth's Elliptical Orbit (y-axis in minutes)

Axial Tilt

There is a second factor that affects the speed at which the Sun appears to move in the sky: the tilt of the Earth’ s axis with respect to the plane of the Earth’s orbit. This means that the Sun is sometimes north of the celestial equator (the projection of the Earth’s equator out into space) and sometimes south of it. Clock time is measured along the celestial equator; in effect an imaginary Sun is drawn on this line and time calculated from that. The speed of this imaginary Sun varies over the year, and reaches a maximum at the solstices, when the major component of the real Sun’s motion due to the Earth’s orbit is parallel to the celestial equator, and reaches a minimum at the equinoxes when the major component of the real Sun’s motion due to the Earth’s orbit is mainly perpendicular to the celestial equator. At the solstices (late June and late December) the Sun appears to speed up by 20.3 seconds every day, and at the equinoxes (late March and late September) it appears to slow down by the same amount. These few seconds per day accumulate over the year, and mean that the amount by which the Sun might deviate from being at due south at noon, due solely to the effects of our axial tilt, varies from zero (solstices and equinoxes) up to 9m 52s (early February and early August, when it is negative, and early May and November when it is positive). It can be drawn as a sine wave with an amplitude of 9m52s and a period of 0.5 years.

The component of the Equation of Time due to Earth's Axial Tilt (y-axis in minutes)

 The Equation of Time

Adding these two effects together gives you the Equation of Time, that is the amount by which you need to adjust the time as read on a sundial in order for it to read the correct clock time. Or in other words the amount of time by which the Sun is off being due south at noon.

The Equation of Time (y-axis in minutes)

The maximum amount of time that a sundial is ever ahead of clock time occurs on 03 November, when the sundial will read 16m25s fast. The maximum amount of time that a sundial is ever behind clock time occurs on 11 February, when the sundial is 14m15s slow.

There are four dates each year when the Equation of Time is zero, and the sundial reads correct clock time, and those dates are 15 April, 1 September, 25 December, and today 13 June.

Perihelion 2011

At 1900 GMT on 3 January 2011 the Earth will be at perihelion, its closest approach to the Sun this year.

The Sun

If that sounds confusing to you, and has you wondering why it’s so cold given that the Earth is at its closest to the Sun, then this belies (a) a northern-hemisphere-centric attitude (in the Southern Hemisphere it’s summer right now), and (b) a misunderstanding of what causes the seasons.

The Earth orbits the sun in a nearly circular orbit called an ellipse. The degree by which an orbit differs from a perfect circle is called the eccentricity, e. If e = 0 then the orbit is circular; if e = 1 then the orbit is parabolic, and therefore not gravitationally bound to the Sun. The Earth’s orbital eccentricity is 0.0167, meaning that it is very nearly circular, with the short axis of the ellipse being around 96% the length of the long axis. Thus, during perihelion Earth is 0.983AU from the Sun, while during aphelion (its furthest distance from the Sun, occurring this year on 4 July) Earth is 1.017AU from the Sun. (1AU = 1 astronomical unit = the average distance between the Earth and the Sun = 150 million km).

The seasons on Earth have really nothing to do with how close the Earth is to the Sun at different times of year. Indeed how could they, given that the difference in distance between closest and furthest approach is only a few per cent? The seasonal differences we experience are of course caused by the tilt of the Earth’s axis, which is inclined by 23.5 degrees from the vertical.

This tilt means that, as Earth orbits the Sun, for six months of the year one hemisphere tips towards the Sun, so that it experiences longer days than nights, becoming most pronounced at midsummer, at which point the Sun reaches its highest in the sky at noon. Simultaneously the other hemisphere tips away from the Sun, and experiences shorter days than nights, becoming most pronounced at midwinter, on which day the Sun is at its lowest noontime altitude.

Earth

The further you are from the equator the more pronounced the seasonal effects. In fact equatorial countries don’t experience seasonal variations, while the poles experience extremes with six-month-long winters and summers. The timing of perihelion and aphelion relative to our seasons is entirely random. The fact the southern hemisphere midsummer (21 Dec) almost coincides with perihelion (3 Jan) is simply that; a coincidence. Given that fact, there is no reason to be surprised that perihelion occurs so close to northern hemisphere midwinter: it has to happen some time and it’s a coincidence that it happens to occur within two weeks of midwinter / midsummer.

To take this explanation even further, we can calculate how much variation in incident sunlight there would be in two scenarios: 1. an imaginary scenario where the seasonal variations in temperature are due to the tilt of the Earth’s axis but where the Earth goes round the Sun in a perfectly circular orbit and 2. an imaginary scenario where the Earth’s axis isn’t tilted, but where its orbit is elliptical in the same degree as ours actually is.

1. The Sun appears at its highest point in our sky each day at noon. The highest it ever gets is at noon on midsummer. The lowest noontime altitude occurs at noon on midwinter. In Scotland the Sun is around 55 degrees above the horizon at noon on midsummer, and only 10 degrees above it at noon on midwinter. The amount of energy from the Sun radiant on a fixed area is proportional to the sine of the altitude, so the ratio of the solar energy radiant on a square metre of Scotland between midsummer and midwinter is

sin(55) / sin(10) = 1.84

So here in Scotland we get 84% more energy from the Sun in summer than we do in winter, due to the tilt of the Earth’s axis.

2. If the Earth’s axis was not tilted, then we would only experience temperature differences from the Sun depending on how far or near we are from it. In this case, the amount of energy from the Sun radiant of a fixed area is proportional to the square of the distance from the Sun, so the ratio of the solar energy radiant on a square metre of Scotland between perihelion and aphelion is

(1.017/0.983)^2 = 1.07

So we get 7% more energy from the Sun at perihelion than we do at aphelion, due to the differing distances to the Sun. From this you can see that, while the distance to the Sun has some effect on how much heat we receive, it is a very small effect compared to that produced by our axial tilt.