Tonight I am starting a Celestial Navigation Course at Naval Point Club Lyttelton. I am a bit apprehensive as I don’t enjoy working with numbers and calculations, so we shall see how we go!

I am planning on taking notes and writing them up here on my blog so that I can refer back to them at any time. So if you are interested – check back here tomorrow, and each week after. I will just add them on to this post as opposed to having a whole heap of separate posts on the topic…

We are doing six sessions for two hours each, every Tuesday starting from tonight (September 2014).

We have been asked to bring along

• dividers
• protractor (square version recommended)
• short clear plastic ruler
• scientific calculator (Casio fx82 or similar)
• B pencil
• eraser
• pad of 2mm graph paper
• lined ‘topless’ pad
• A sextant if we have one – thankfully I have one that my friend Ivan gave me a few years ago. It is a Davis and I have just found the user guide for this very sextant online as well as a replacement mirror and other parts on the Davis website.

Each week has a different topic scheduled of what we are going to learn. I have noted the topic at the start of each bold section. At this stage I understand absolutely nothing even in the course explanation notes – this is going to be interesting! My comments are in italic and my notes are written below that.

Session 1: Introduction; the ‘how’ and ‘why’ of it 

– Medieval cosmology, and the assumptions that underlie its present use
– Co-ordinates – terrestrial and celestial
– Time, and the equation of time
– Arc / Time
– The concept of ‘angular distance’
– The ‘shape’ of both the problem and its answer
– The line (or circle) of position
– Obtaining an approximate solution be construction (using globe of the earth)
o Exercise: Declination to latitude; GHA to longitude; Longitude of Sun by calculation (i) ignoring equation of time and (ii) incorporating (given) equation of time

So the first session went really well. There are about 23 people on the course – so a good number of people are still interested in learning this ancient art! My brain was hurting by the end of the two hours, but I think I grasped the concepts discussed, and I hope they will all begin to make sense by the end of the course. Anyway here are some of my notes – I am not expecting that they will mean much to you – but this post is going to be a place for me to brain dump the things I am learning!

• Why learn? – Well GPS systems are incredibly reliable and cheap these days, so you can easily have a couple of back ups on board. But what if it all goes wrong? Just this week a solar storm threatened to disrupt GPS signals and in addition to that, last week we were narrowly missed by a meteor which narrowly missed hitting our satellites – this could be a major issue if you are out at sea with no other way of finding your way home other than your GPS! Other than these disastrous things happening, I need to learn how to navigate celestially for my Ocean Yachtmaster qualification. So I have no intention of throwing out the GPS – but I am interested to also know how to use my sextant!
• What is the Celestial Sphere? – Well waaaay back a long time ago, it was assumed that the earth was the centre of everything, and that the universe revolved around us. So the celestial sphere was like a huge balloon and all the stars were stuck on the inside of it and this balloon/sphere revolved around the earth. The stars were all mapped on this balloon and were given positions – similar to how we use latitude and longitude – but instead of lat and long their position is measured in declination and hour angle. There is also a celestial equator and poles.
• Dec – Declination – is the equivalent of the sun’s latitude – measured in degrees north or south of the celestial equator. Declination changes as the sun moves around the ecliptic. It is 0 degrees when the sun passes overhead at local noon at the equator – the equinox. It reaches a maximum of 23.5 degrees north or south at the solstices.
• The first point of Aries is the fictitious point in the sky where the sun passes through the plane of the earth’s equator in its journey North – in March (vernal point). Greenwich Hour Angle is measured to this point and then Sidereal Hour Angle is measured from this point to the position of the star.
• LHA is Local hour angle – so LHA measures the distance from where you are standing to the star.
• Accuracy – there are a number of variables here – the observers skill & experience, the conditions – clear view of the body and horizon, accuracy of the sextant, which bodies are observed, the time and much more. The best you can hope to get is +- 2 or 3 miles.
• Time – GMT (Greenwich Mean Time) or UT (Universal Time) is the standard time which the system uses. You can get an accurate time check from – a GPS, the pips on NZ National radio or a continuous time signal on the SSB radio – 1, 5, 10 & 15 MHz. 4 seconds of time is the equivalent of 1 nautical mile at the equator, so that is why it is vitally important to have the correct time to get a good position. Get a decent digital watch which is set to GMT – date hour minute and second.
• If it is cloudy then you won’t be able to get a fix at all!
• When the sun/star or celestial body is directly overhead of a certain point on the earth it is said to be at “zenith”. So this means the star is 90 degrees overhead. Or a zenith is the point directly over the head of the observer – altitude of 90 degrees.
• The position of the sun moves westward over the surface of the earth at 15 degrees each hour – 1 degree every 4 minutes or 360 degrees each day.
• If you take a sight of the sun with your sextant, and measure the angle at which you view it, you can then figure out exactly where at that particular time the sun’s zenith is on the earth. The difference between your angle and 90 degrees can then be measured as a distance and from there you can get a position circle from the plotted ground position of the sun – you are somewhere on that circle!

• The next bits of information you require are the Declination and Greenwich Hour Angle of the body at that time. These can be found on tables, computer programmes and almanacs.
• The tables give the Declination and GHA for an identified year, date and time, and then provide a means of adjusting the given values immediately preceding the moment of observation so that those at that time can be derived. In the Nautical Almanac, the values are given at intervals of one hour and correction factors are found at the bottom of the table.
• So in summary – If you know the time, the altitude of the celestial body (as measured with your sextant), the co-ordinates of that particular body and its relation to the earth – then you can get a position line.

Session 2: The sextant (The Sextant Handbook – Bauer)

– What it is and how it works
– Choosing a sextant
– Checking adjustment and dealing with errors (particularly index error)
– Sight taking and recording
– Averaging – and identification of rogue result
– Altitude correction – Hs to Ho
o Exercise: Applying index error, Sight averaging (with rogue site), Applying altitude corrections (table supplied)

Well thankfully I wasn’t the only person to have trouble with the homework! However when we were going through the examples everyone got very confused, and so the tutor is going to come up with some better worked examples and email them through to us to work through. We had a good look at our sextants, and mine needs a replacement horizon mirror, which I have been able to order online via the Davis Instruments website. We also discussed the various errors that sextants can have which need to be taken in to account when you are doing your calculations. Here are my notes:

• We had a good look at our sextants tonight. The mirror at the top is the index mirror – this is what the body is reflected in. There is also a horizon glass on the front of the sextant – which is half clear and half mirrored. You look at the horizon through the telescope in the clear bit and line it up with the sun reflection in the mirror bit. There are shades to use to protect your eyes from the sun’s glare. To get the two mirrors to line up you move the index arm along. The figures at the bottom are the measurement angle – the number you use for your calculations.
• When you are lining up a star – you line up the centre of the star with the horizon
• With the sun you line up either the bottom or top edge of the sun with the horizon (and you need to make a calculation depending on which edge you use)
• If the edge of the sun isn’t visible, then you can line up the centre of the sun’s glow – if it is obscured by haze.
• With the moon, you can either line up the top or bottom edge of the moon with the horizon depending on the shape of the moon.
• If you swing your sextant slightly from side to side you can make sure that you are lined up correctly on the horizon.
• Have two people to take the sight – one to check the angle reading on the sextant at the same second as the other person checks the time. Get a waterproof stopwatch or digital watch set to GMT date, time (down to the second)
• It is good to take at least five sights in a row one after the other – it will then become evident if there is an error. The time and altitude should progress steadily. You can average the time and angle out and that should equal pretty much the same as your middle – or 3rd sight.

Here is a brilliant YouTube tutorial on how to take a measurement with your sextant

• • There are a number of ways of calculating the distance once you have the measurement – pre-computed sight reduction tables, using a scientific calculator and trigonometrical formulae, using a pre-programmed navigational calculator, using a computer programme. (for Ocean Yachtmaster you need to know how to do it via tables or calculator)
  • Adjustments – you need to make provision for certain errors and they strictly need to be made in this order

  1. Instrument Correction – unique to your sextant (these should be noted on a certificate of accuracy)

  2. Index Correction IC – used to deal with an imprecise adjustment of the horizon mirror

  3. Dip D or dip – A height of eye correction – there is a table to use – the value is always subtracted from the product of sextant reading and the preceeding corrections. Estimates of height above sea level are usually ok.

  4. Total, third or combined correction – that can be obtained from the appropriate tables and combine all of the other significant corrections

  Once you have made all these adjustments you end up with a figure of Ho or true altitude and this is the first value you need to calculate your sight reduction.

• Hs is the angle as noted on your sextant. To this you apply your sextant error adjustments – IC and Dip.
• That gives you Ha – and to this you apply some other adjustments to give you
• Ho the figure you use in your next calculation

We also learnt how to use a button on the scientific calculator! (this is a breakthrough as I had absolutely no idea how to use any of those buttons prior to tonight!)

To add degrees, minutes and decimals of a minute in to the calculator you can use a button which has (° ’ ”) on it…

• So 201°21’30” is entered as – 201 (° ’ ”) 21 (° ’ ”) 30 (° ’ ”) =
• this then puts the figure down in the bottom right hand side of the screen.
• To convert this to a time you divide by 15 (because this is the number of degrees the sun moves in an hour)
• The result is displayed as hours, minutes and seconds

To add hours, minutes and seconds in to the calculator you do the same – so 1 minute 14 seconds

• 0 (° ’ ”) 1 (° ’ ”) 14 (° ’ ”) =

(you put the 0 in first as you are saying it is 0 hours, 1 minute 14 seconds)

If you multiply that number by 15 you translate the time in to degrees.

When the local longitude is East you need to convert it to GHA (this is because GHA is measured through 360 degrees, but longitude is measured from 0-180° E or W of Greenwich. So to convert an East longitude, you just need to deduct the angle from 360°

Remember that Declination and hour angle is an angle measured from the centre of the earth.

A star with a N15°27′ declination will be above North latitude

A star’s angle from you has to be between 0-90° or 270°-360° as anything other than that won’t be visible to the observer.

An equation of time is taking in to account the difference between the mean/average sun and where the sun actually is in the sky.

The mean sun moves at 15° per hour. The actual sun moves depending on the actual speed of the earth’s rotation and it varies depending on what time of the year it is. This equation of time is measured in + or – minutes. So put this equation of time figure in to your calculator as described above – convert it to degrees, and then add (or subtract) that to your longitude.
Session 3: Extracting the ephemeral data – GHA and Dec

– Nautical Almanac
– ‘Long term’ almanac
o Exercise: Determining GHA and Dec for Sun and star for supplied date and time, using both Nautical Almanac (page supplied) and Long Term Almanac;

I think things are starting to sink in a bit now. I managed to have a good crack at my homework. Today we went through some of the almanac tables. Here are my notes…

When you are taking a sight, you need to take it when you are at the top of a wave – to get a greater accuracy for where the real horizon is. You also need to take in to account your height of eye. This error adjustment is called Dip and it is always subtracted from the figure you got once you made any index error corrections on your sextant. The information on what to subtract is found on a page in the almanac – but to save you looking that up – here are the adjustments to make for an average yacht. Of course if you are on a big ship then you will need to look them up!

Height of Eye        Correction

• 1m                -1.8′
• 1.5m             -2.2′
• 2m                -2.5′
• 2.5m             -2.8′
• 3m                -3′

Once you have taken the reading from your sextant, and then you have applied your index error correction +/- you then subtract your Dip – as per the table above. This then gives you a figure called Ha – or Apparent Altitude

To this figure you need to apply Total Correction – which is taken from the same table that you got the Dip figure from. This table doesn’t change – the figures are fixed – so this doesn’t need to be updated annually. Total correction includes a whole lot of different errors like refraction, parallax etc etc.

Here is the table from the almanac – you can see on the right hand side is the table for Dip. There are various different columns but I have highlighted the ones that we are most likely to use on a small boat.

On the left hand columns are the ‘Total Correction’ tables. So say for example as per my notes in the book – your Ha is 18°10′

• Check the top column to see which month period you fall in to – Oct-Mar or Apr-Sep – and you will be reading down that table
• Read down the table until you get to the 18°06′ and 18°42′ – our reading of 18°10′ lies between these two numbers.
• Was your reading for the upper limb or the lower limb of the sun? Ours was lower limb – so our correction is +13.4′
• So using my snazzy scientific calculator – I add 18°10′ + 13.4′ = 18°23’24”

Once you have got that bit worked out, then you need to have a look at another table to see where the sun actually is at that time of the day.

The time on the watch that you use to take your sights with should be set to GMT day, hour, minute, second – so NOT local time. The reason for this is that the tables are all listed in UT – Universal Time or GMT – make sure you get your day right – it can get a bit confusing for us on the opposite side of the world.

There are two kinds of almanacs – one is the annual almanac which is as you might expect – published annually! This is more accurate, more expensive, and easier to use.

The perpetual almanac is less expensive, doesn’t need to be replaced annually, it is also less accurate and harder to use…

You can also download for free the sight reduction tables for marine navigation for free online – however there are six volumes, and presumably if you are using your sextant, then your computer is possibly not working either… anyway they would be good to use for practice anyway.

The other free download is American Practical Navigator. Another great book which has been recommended to me.

Anyway the one we used this night was from the annual almanac – so you flip to the page with the month and date that you are interested in, and look down the column that says UT and Sun.

Sorry about all the scribbles – but if say the time we took our sight was Tuesday 7 October at UT 11 hours, 2 minutes and 27 seconds.

We read down the column for the 7th October until we get to 11 hours – Read along there and the GHA for sun at that time is 348°02.2′ and the Declination was S (for South) 5°33.7′

But that was just for 11.00 on the dot! – from here we need to find out the minutes! And you guessed it – there is another table called ‘Increments and Corrections’ – find the page for 2 minutes – labelled 2m and read down the 2m column until you get to 27 seconds…

So the figure in that column is 0°36.8′

Add this figure to your original figure for 1100 hours 348°02.2′ + 36.8′ – and you get 348°39′ – this is the GHA for the Sun.

That previous table also had the Declination for the Sun – and we read on the column for that at 1100 was S5°33.7′

At the bottom of the table with that figure on there was a small italic d with the figure 1.0 beside it. Look above I have even circled it for you. If you now look at the table below you will notice that there is a separate 3 columns labelled ‘v or d Corrn’ You take the 1 from the bottom of the previous table and scan down until you get to 1 on this table – and if you look along there you will see that it is 0.0′ so we don’t need to apply any corrections to Declination – yeehaa!

So in summary what we have got so far…

1. We have taken a sight of the sun with our sextant and the angle was for example 18°17.5′
2. We then applied our index correction (which is unique to each sextant) for example  – 5′
3. We subtracted our Dip (we are at 2m high so subtracted 2.5′)
4. This gave us an Ha number of 18°10′
5. We then checked out the Altitude Correction Tables again to find the Total Correction figure (depending on the month and whether we were looking at the upper or lower limb of the sun) which we applied to the Ha figure and we ended up with 18°23’24”
6. Then we used our time of this sight to figure out the GHA and Dec of the sun by looking in the almanac for the day and hour of our sight and then the increments and corrections page for the minutes and the seconds. We also got a figure for Declination at that time and used the minutes and seconds page to apply any other variations – of which there were none in our example
7. And we were left with 3 figures – the angle that we took with all the calculations applied, and the plotted position of the Sun on the celestial sphere GHA and Dec
8. The next step is to somehow use those figures to actually figure out where we are! (and I don’t know that yet – but watch this space!)

The details above were for finding the details from the annual almanac. This is the one that is more expensive, more accurate and easier to use. The cheaper option is the Long Term Almanac – which is most likely the one you will have on board a boat unless you are planning on doing lots of celestial navigating. The Long Term Almanac requires a few different steps to get the data you need:

“Mean Sun”  is not a sun that is being a bully…!! It relates to a fictional sun which is permanently above the equator moving westward at a constant rate, and completes a circuit of the earth every 24 hours.

The “True Sun” is where the actual sun is at any time. It differs by two regular and predictable variables – equation of time and changes to declination. This cycle is repeated exactly every solar year – every 365 days, 5 hours, 48 minutes and 45.22 seconds…

The values – expressed as arc for both the equation of time and the declination change slowly +- 8′ per day for time and +- 1′ per hour for declination. So we need to work to the nearest hour for these variables.

So here are the steps!

1. Go to the Ephemeris of the Sun – Argument ‘orbit’ time table
2. At the bottom – go to the year correction to GMT to produce ‘orbit time’ table
3. Take your current year date and time GMT and look at the year on the table – so 2014 the correction is +1
4. Apply this hour adjustment to your GMT – so say the time is 7 Oct 2014 0520 – you add +1 hour to get 0620
5. Now look at the Ephemeris of the Sun – Argument ‘orbit’ time table and look down the O.T 00h table
6. Look down that table to get your date – in our case the 7th and then look across the table to get to your month – in our case October
7. The results in the table are E 8°.00′ and Dec S 5.23
8. E = Equation of time expressed as arc to which 5° has been added to always make it a positive figure – Dec is Declination
9. The next column has a triangle at the top – this is the rate at which the Declination is increasing each day. In our case – +23′ per day (or nearly 1′ per hour) In our example our actual time is 0520 so we add 5′ to our Declination = S 5.28′
10. If you look at the E column you can see that the daily change in those figures is about 4′ – so in this case – aprox 1′ every 6 hours. As we are nearly 6 hours through our day we add to our E to get 8°.01′
11. So now we need to convert our time – 0520 to an arc so that we can add the E figure to it. To do this you use your scientific calculator the keys to hit are 5 (° ’ ”) 20 (° ’ ”) = x 15 = 80° (the x 15 is how you change time in to arc)
12. Then we add our result of 80° to the E figure of 8°.01′ = 88°01′
13. Then we add 175° (remember that there is already 5° loaded in to the E figure) = 263°1′ – this is our GHA

My homework is to work out a whole lot of different GHA’s and Dec’s for Sun for a variety of different dates and times… so I had better get going – the next session starts in 2 hours and it will probably take me that long to get it sorted! 

Session 4: Tabular Solutions
– A conceptual explanation of the St Hilaire Method
– The Air Tables
– HO 249
– HO 229 (Modified form – Sight Reduction Tables for Small Boat Navigation)
o Exercise: Construct 3 plotting sheet appropriate to given DR (pp37/8 ‘Guide’); deriving and plotting lines of position for a given date, time, GHA and Dec – all methods, pages supplied

Tonight we looked at how to get the data from the tables for stars as opposed to the sun. The difficulty with stars is that you need to firstly correctly identify the star you are looking at and it also needs to be light enough to still see the horizon – so you can take your sextant measurement…

Here are the steps!

1. Go to the Long Term Stars Almanac
2. To figure out which year column to use – take the last two digits of your year – 2014 = 14. Divide that number by 4 to obtain a whole number and a remainder – so 14 divided by 4 = 3 for the whole number of 12 and 2 remainder. The 2 remainder is the number of the column that you read down… so for us October in column 2 is 8°34.6′
3. Then go to table ‘a’ Quadrennial Corrections – Aries – look down the table until you get to your year – 2014… the correction for 2014 is 0°05.5′
4. Then look at table ‘b’ Daily Corrections Aries – find your GMT day – in our case 14th – 13° 47.9′
5. Then look at table ‘c’ Hourly Corrections Stars & Aries – head down to your hour – in our case 0600 – 90°16.0′
6. Then we need to calculate the minutes and seconds – in our case 43min 16 seconds

Add 1 degree for every additional 4 minutes of time, 15 minutes of arc for each complete minute of time and 1 minute of arc for every additional 4 seconds of time…

So in our case 40 minutes = 10 degrees + 3 minutes = 45′ + 16 seconds = 4′ – a total of 10°49′

• Add all the underlined numbers to get your GHA Aries – in our case 123°33.0′
• Then select which star you have taken a sight of – Altair – check the Selected Stars table
• Note the SHA – is 62°18.3′ with an annual correction of -.73′ for each year. October 2014 is 14.8 years (as October is 1/8 of the way through the year) so in your calculator go .73 x 14.8 = -10.8′ and subtract this from the SHA figure – to get your corrected SHA of 62°7.5′
• Add your corrected SHA to your GHA Aries figure to get your GHA of the star = 185°40.5′
• Note the Dec for Altair is N8°52.1′ with an annual increase of +0.16′ – so in your calculator go 0.16 x 14.8 = 2.4′. Add this to your Declination to get your corrected Declination – N8°52.1′ + 2.4′ = N8°54.5′

Session 5: Calculator solutions
– St Hilaire method by calculator
– Sumner line of position
o Exercise – as above (with same data) – both methods
Session 6
– Practical work – beach exercise
o take and work a timed sight – ‘Guide’, workings shown), plot position line on appropriate plotting sheet (2mm graph paper, pp 37-8 ‘Guide’)
End result: A certificate that the participant has satisfactorily completed a course in basic celestial navigation

Glossary

• Altitude – the height of a celestial object above the horizon – measured in degrees, minutes and decimals of a minute.
• Azimuth – Zn or Z – the horizontal direction of a celestial body from a terrestrial point – usually expressed in terms of the true compass Zn – although it may also be expressed in terms or the angle from North or South – Z
• Celestial Sphere – an imaginary sphere of near infinite radius concentric with the earth, on which all celestial bodies except the earth are imagines to be located
• Circle of equal altitude – a circle on the surface of the earth at every point on which the altitude of a given celestial object is the same at a given instant
• Declination – Dec – the angular distance North or South of the celestial equator. The celestial equivalent of Latitude
• Dip – the effect arising from (and the correction for) height of eye above sea level. The correction increases with an increase in height and is always subtracted from Hs
• DR – Dead reckoning – the process and result of determining the position of a vessel by applying the effect of course and speed to a last known position – often with corrections for such things as drift and leeway
• Ecliptic – The apparent annual path of the sun amongst the stars – its terrestrial equivalent
• Equation of time – the difference at any instant between local apparent time and local mean time. A measure of the difference in position on the celestial sphere between the actual sun and a fictitious (mean) sun
•  First point of Aries – that point of intersection of the ecliptic and the celestial equator, occupied by the sun as it changes from North to South declination on or about 21 March
• GHA or Greenwich Hour Angle – Angular distance West of the Greenwich celestial meridian 0 degrees
• GMT or Greenwich Mean Time – Local mean time at the Greenwich meridian (long 0 degrees) Also called Universal time or Zulu time. May be regarded as the equivalent of co-ordinated Universal Time.
• Great Circle – the intersection of a sphere and a plane passing through its centre – all meridians of longitude are parts of great circles – the terrestrial equivalent of the ecliptic. The equator is the only parrallel of latitude that forms a great circle.
• Ha or Altitude – The apparent altitude of an onserved body. The sextant reading after correction for index error (IC and Dip
• He – a computed altitude derived by calculation for an assumed position
• Ho – The observed altitude of an observed body – after all necessary corrections have been made
• Hs – the sextant altitude before corrections
• Hour Angle – The angular distance west of a celestial meridian. GHA – Greenwich Hour Angle – an anglar distance measured westard from the prime meridian of a celestial meridian can be converted to its terrestrially equivalent longitude.
• SHA – Sidereal Hour Angle – is an angular distance measured Westward from an imaginary celestial meridian passing through the first point of Aries.
• LHA – Local Hour Angle – The angular distance measured westwards from the position of an observer.
• IC – Index Correction – a correction applied to deal with imprecise adjustment of the horizon mirror of a sextant – usually preferable to an attempt to remove the imprecision by adjustment.
• Local apparent noon – the instant in which the sun is over the meridian of an observer – twelve o’clock local apparent time. The difference between local apparent noon and that calculated by reference to GMT is called the equation of time.
• Longitude (lo or long) the angular distance measured eastward or westward from the prime meridian measured through 180 degrees and labelled E or W to indicate the direction of measurement – longitudes form parts of great circles converging at the poles
• Meridian – a North-South reference line. All meridians are parts of great circles. The Prime Meridian is at Greenwich
• Meridian Angle (t) – the angular distance east or west of the local celestial meridian. The GHA of a celestial body may be converted in to a meridian angle (referenced to the observers meridian) for ease of calculation
• Nautical Mile – Within the conventions of celestial navigation, the distance on the Earth’s surface that is subtended by one minute of arc at the Earth’s centre. Aprox 1852 Metres
• Position Line – a plotted line on which a vessel is located. EG the plotted bearing of a known object of known position
• Sidereal Day – the period of a daily rotation of the earth with respect to the first point of Aries – or practically speaking any star. A mean sidereal day is about 3 minutes 56.6 seconds less than a mean solar day – which accounts of the shift of the stars nearly 1 degree westward each night.
• Small circle – the intersection of a sphere and a plane that does not pass through its centre – all parallels of latitude with the exception of the equator form small circles.
• Zenith – the point on the celestial sphere that is directly overhead

Do you have any celestial navigation tips you could give me? Comments are welcome! 🙂