Celestial Navigation using last years Almanac

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skinicod

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Hello all,

I am about to do a cruise over to Portugal from Plymouth, and I thought it would be fun to try my hand at some celestial navigation. I believe the skipper only has last years almanac, and has borrowed a friends looseleaf almanac pages for the particular ports we'll be heading to, but I wanted to know if last years almanac will give me accurate enough information to make my first attempt at sextant usage, sight reductions etc.

Thanks in advance for your advice - and I should hasten to say, that we will not be relying on my celestial navigation skills unless something goes wholly amiss!!
 
skinicod said:
Hello all,

I am about to do a cruise over to Portugal from Plymouth, and I thought it would be fun to try my hand at some celestial navigation. I believe the skipper only has last years almanac, and has borrowed a friends looseleaf almanac pages for the particular ports we'll be heading to, but I wanted to know if last years almanac will give me accurate enough information to make my first attempt at sextant usage, sight reductions etc.

Thanks in advance for your advice - and I should hasten to say, that we will not be relying on my celestial navigation skills unless something goes wholly amiss!!
Click to expand...

The 2012 almanac can be used in 2013 as follows:
For the Sun, take GHA and Dec from the same date but (for March to December, I assume you're not interested in Jan and Feb) for a time 5h 48m 00s earlier than the UT of the observation. and add 87°00'.0 to the GHA. The error should be within 0'.4.
For example, Observation 12:00:00 on 1/8/13. 1/8/12 06:12:00 Sun GHA = 268°25'.2 + 3°00'.0 = 271°25'.2. Adding 87° gives 358°25'.2. Sun GHA at 12:00:00 on 1/8/13 from 2013 almanac is 358°25'.2. Declination was N17°54'.0 - 0'.1 at 06:15:00 1/8/12 and (will be) the N17°53'.9 at 12:00:00 on 1/8/13.
For stars subtract 15'.1 from the GHA. Again the error should be within 0'.4, although the error is from different sources.
You can't use last year's almanac for the Moon or the planets.

There are various almanacs available on line, and you can print off the relevant pages if you wish - some of them are designed to look exactly like the commercial / UKHO almanac. I can let you have a spreadsheet almanac - PM me if you wish.

Your comment about the skipper having last year's almanac and having borrowed the pages for the ports you'll be using brings several questions to mind. The two that relates to Astro are:
- are you sure that the almanac you have is an astro almanac?
Almanacs such as Reeds and the Cruising almanac are really combination tide tables / pilot books / skipper's handbooks. They are not almanacs. Almanacs tell you the positions (GHA and Declination) of the various celestial bodies and also include various tables for altitude correction.
- do you have the required sight reduction tables (or formulae and calculator)? several tables are available on the internet (Ageton's and the Aero tables certainly are, the full tables are probably available from NOAA but it will be a lot of printing...).

Have fun with it all, and post again if you have any more questions...
 
I'm not proposing the OP does this, but it is interesting to know that the officers of Captain Cook's Third Voyage computed the lunar ephemeris during the last year of their voyage; it was only available for two years on their departure! However, Cook himself was a Fellow of the Royal Society, and no mean astronomer, and there was also a professional astronomer aboard. They taught the officers, who in turn became first-rate navigational astronomers (e.g. Bligh, Vancouver and others). I don't know, but I'd imagine that the junior officers were used to perform the repetitive and tedious calculations required.

In theory you can follow the same process by knowing the elements of the orbits of the various celestial bodies (including the Earth's). In practise, it's easier to buy the almanac, or use an online version! Some downloadable systems do actually calculate the internal ephemeris from orbital elements, and don't rely on pre-computed ephemeridae.
 
AntarcticPilot said:
In theory you can follow the same process by knowing the elements of the orbits of the various celestial bodies (including the Earth's).
Click to expand...

It is almost so but not exactly: the problem is that the celestial bodies interact among each other so that their orbits are not described for ever and exactly by few constant parameters given at a certain epoch. For what I know, it seems still convenient to assume that they are so during a relatively short period (i.e. one year) but the astronomers in charge of the almanacs are constantly updating the parameters according to actual observations; although the corrections from one year to the following are not large, they can be relevant for navigational or astronomical purposes.

Daniel
 
danielefua said:
It is almost so but not exactly: the problem is that the celestial bodies interact among each other so that their orbits are not described for ever and exactly by few constant parameters given at a certain epoch. For what I know, it seems still convenient to assume that they are so during a relatively short period (i.e. one year) but the astronomers in charge of the almanacs are constantly updating the parameters according to actual observations; although the corrections from one year to the following are not large, they can be relevant for navigational or astronomical purposes.

Daniel
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The positions of the planets in the Solar System can be computed with good accuracy for hundreds of thousands of years at least; it gets iffy beyond that because the system has been proved to be chaotic, and tiny errors in the known position of the planets get magnified in an exponential manner at geologically long time periods. They can be computed with an accuracy that allows spacecraft to be launched on decades long unpowered flights with precision far exceeding that required for marine navigation. The relatively low precision required for navigation poses no problem that requires freqent updates of the orbital elements, which are not "a few parameters" but rather a complex computer model accounting for all the planets. Yes, if we were still using simple models of the Solar System that did not allow for gravitational interaction we'd be in the position you describe, which was the position perhaps 20-30 years ago. But computational power has grown to the extent that we can now model gravitational interactions with high accuracy, and there have been many studies of the evolution and stability of the Solar System that have modelled the orbits of the plnets for billions of years - though as I said, chaotic effects mean that past a few million years the results tend to be of statistical value only.
 
AntarcticPilot said:
The positions of the planets in the Solar System can be computed with good accuracy for hundreds of thousands of years at least; it gets iffy beyond that because the system has been proved to be chaotic, and tiny errors in the known position of the planets get magnified in an exponential manner at geologically long time periods. They can be computed with an accuracy that allows spacecraft to be launched on decades long unpowered flights with precision far exceeding that required for marine navigation. The relatively low precision required for navigation poses no problem that requires freqent updates of the orbital elements, which are not "a few parameters" but rather a complex computer model accounting for all the planets. Yes, if we were still using simple models of the Solar System that did not allow for gravitational interaction we'd be in the position you describe, which was the position perhaps 20-30 years ago. But computational power has grown to the extent that we can now model gravitational interactions with high accuracy, and there have been many studies of the evolution and stability of the Solar System that have modelled the orbits of the plnets for billions of years - though as I said, chaotic effects mean that past a few million years the results tend to be of statistical value only.
Click to expand...

VSOP87, the standard solar system model, is generally considered to be accurate to 0".01 from about year -4,000 to year +8,000. However, it contains (in some cases)over a thousand periodic terms for each body for each of ecliptic longitude, ecliptic latitude and radius vector. You can reduce the number of terms (and reduce the accuracy to about 1") to about 200 or fewer for each body (not all bodies are of interest for navigation), put these in a spreadsheet, convert ecliptic positions to geocentric positions, and deliver positions suitable for astro navigation at an accuracy at least as good as the Nautical Almanac, all in about 300Kb - and you can run it on a phone.

It's possible, but not reasonable, to do it on a calculator.

The Moon needs ELP2000, not VSOP87, but the same applies. About 200 periodic terms provide GHA and Declination for the Moon to an accuracy of about 0'.17 and 0'.07 respectively.

Stars are comparatively easy, as there's only the J2000.0 position and proper motion needed (in addition to elements already covered, such as precession, nutation, etc), and accuracy is excellent, although as radial velocity is not normally included the accuracy reduces over time (a thousand years or so).

The trickiest thing with all this is deltaT, which is the amount by which the Earth's rotation doesn't match dynamical time (which is the time the universe keeps). The Earth's rotation is not uniform, and is changing at an unpredictable rate (although current predictions are considered suitable for navigational purposes for the next hundred years or so). This means that although it is possible to tell where the planets and stars are in relation to each other and to the Earth, it's not possible to tell exactly which way the Earth will be pointing.

If you're interested in the 'definitive' model, look at the JPL Horizons system (available on line). That uses the DE405 model and is slightly more accurate than VSOP87, particularly over longer time periods. If you want to use this system you need to be clear about your reference frame and coordinates system. Horizons also gives all the asteroid, comet and spacecraft positions.

No comment as to how long it took me to write the spreadsheet though.
 
It is great to read from informed people, I mean it. I wonder if one of you can help in this question: I have an old HP programmable pocket calculator (41CV) that I like very much and still use for most of my simple computing and sight reduction. It has a memory card with an almanac for navigation that uses relatively simple formulas and the constants in the series expansions given by a paper appeared in AJS by Van Flandern and Pulkkinen (1979). I expect the constants to need updating and I would be able to reprogram it accordingly but... where can I find their values? I do not want to get involved in a best fit computation myself and constants valid for, say, another 20 years will surely cover my navigation life span.

Daniel
 
danielefua said:
It is great to read from informed people, I mean it. I wonder if one of you can help in this question: I have an old HP programmable pocket calculator (41CV) that I like very much and still use for most of my simple computing and sight reduction. It has a memory card with an almanac for navigation that uses relatively simple formulas and the constants in the series expansions given by a paper appeared in AJS by Van Flandern and Pulkkinen (1979). I expect the constants to need updating and I would be able to reprogram it accordingly but... where can I find their values? I do not want to get involved in a best fit computation myself and constants valid for, say, another 20 years will surely cover my navigation life span.

Daniel
Click to expand...

The Van Flandern and Pulkkinen figures are available here: http://articles.adsabs.harvard.edu//full/1979ApJS...41..391V/0000391.000.html
They produce planetary (including Earth) positions that are accurate to about 1'.0. Not sure of the accuracy for the Moon, but I suspect it's quite a bit worse, and you need SHA and Dec (and change) for the stars. Not sure if they include details of local siderial time (GHA Aries), but I can give you a formula for that if required. I have their data in a spreadsheet somewhere, that i can send if you wish - it may be easier to copy it rather than to transcribe it.

John
 
John Barry said:
VSOP87, the standard solar system model,...
Click to expand...

Thanks for your expert knowledge. I knew of the models and their general range of validity from my geological background; they are used in modelling the early history and evolution of the Solar System.Of course, missions to the outer planets would be completely impossible without the fantastic accuracy of the current models. I think the most amazing testimony to their accuracy is the ability to land a vehicle on Mars within a landing ellipse about 10km by 5km! And a substantial part of the error there was in modelling the Martian atmosphere.
 
I thought a year or so ago that I'd like to know how accurately one can model the solar data (DEC and GHA) plus GHA Aries by as few parameters as possible.

I used only:

- mean length of the tropical year
- eccentricity of the earth's orbit
- obliquity of the ecliptic
- date and time of the preceding winter solstice
- date and time of the perihelion.

5 parameters only, of which the first 3 do not vary for decades. To do this is fairly simple trig with the only difficulty solving (numerically) Keppler's trancendental equation for calculating eccentric orbital speeds.

I found I could get within about +/- 1', with the bulk of the error having a strong 28 day period. I then added:

- distance from the centre of the earth of the earth-moon's centre of rotation (baricentre)
- date and time of the first full moon of the year (I think the date and time of the last one of the old year would have done equally well)

These improved the accuracy to about +/- 0.1'.

So for navigation you need nowhere near 200 parameters. An interesting thing I noticed is that the largest residual error does seem to be a simple time offset - ie the deltaT John Barry refers to.

A graph of that error for 2013 using only this v simple model is here:
equivalent_chronometer_error_2013.png
.

None of this helps the original poster much! Hopefully this might:

In the old days when I was too mean / poor to buy tables every year I used to make a table of GHA and DEC for every day of the year (at 12:00 UTC) and print and laminate it. On the back I put tables of the correction to GHA by hour and the total correction angle for the sun (which doesn't vary year to year, and besides is always 12.5' anyway!). I carried reduction tables for the range of latitudes and time of year (ie range of DEC) for the intended passage. I also made an A4 laminated sheet of logs and antilogs of sines and cosines so I could work out departure if all else failed, but this is optional.

Nowadays I use a spreadsheet or a JavaScript ephemeris calculator: lots available on the web, several by inhabitants of this parish, including me.
 
John Barry said:
The Van Flandern and Pulkkinen figures are available here: http://articles.adsabs.harvard.edu//full/1979ApJS...41..391V/0000391.000.html
They produce planetary (including Earth) positions that are accurate to about 1'.0. Not sure of the accuracy for the Moon, but I suspect it's quite a bit worse, and you need SHA and Dec (and change) for the stars. Not sure if they include details of local siderial time (GHA Aries), but I can give you a formula for that if required. I have their data in a spreadsheet somewhere, that i can send if you wish - it may be easier to copy it rather than to transcribe it.

John
Click to expand...
Thank you but, I am sorry, I did not explain myself clearly.
I already did have the paper by Van Flandern and Pulkkinen with the formulas and the constants of the series expansions. What I was trying to say is that their constants were computed by "best fitting" the observations in 1979 and probably need updating. The first question is: is there a simple way to update them? The second question is: are there alternate formulas as simple as theirs which give the planetary positions (possibly including the Moon) good for navigation and for the next 20 years? Please notice that their formulas could be implemented on a pocket programmable calculator with only few hundred program lines and few tens memory registers.

Daniel
 
danielefua said:
Thank you but, I am sorry, I did not explain myself clearly.
I already did have the paper by Van Flandern and Pulkkinen with the formulas and the constants of the series expansions. What I was trying to say is that their constants were computed by "best fitting" the observations in 1979 and probably need updating. The first question is: is there a simple way to update them? The second question is: are there alternate formulas as simple as theirs which give the planetary positions (possibly including the Moon) good for navigation and for the next 20 years? Please notice that their formulas could be implemented on a pocket programmable calculator with only few hundred program lines and few tens memory registers.

Daniel
Click to expand...

As intimated by jdc's post above, the method used to determine positions is set by the accuracy you want. All of the methods can be made less accurate by reducing the number of periodic terms used, or by reducing the number of significant digits used at each calculation stage.

Van Flandern and Pulkkinen's process used about 45 periodic terms for the Sun (Earth) position to give accuracy of about 0.02°. They use this method because it is the same method used for the Moon and other planets which required a (fairly) similar number of terms. I'm not aware of any update done to their work, or corrected figures.

VSOP87 uses 2425 terms to give an accuracy of 0.000003°. Again, similar accuracy from a similar number of terms is available for other planets.

For both processes, you can reduce the number of terms and reduce the accuracy. For VSOP87 this gives (for the Sun (Earth)) 0.0003° with 160 terms (this is what I use) and 0.01° with 30 terms. However, if you only want accuracy of 0.01° (about 0'.5) which you may consider is sufficient for navigation, you don't need to do anything like so much calculating.

1. For 0.01° accuracy you don't need to allow for the Moon's or planets' (mainly Venus, Mars and Jupiter) effects - you can just calculate an elliptical orbit. For this you need (Meeus formulae):
T = days (and fractions of a day) since 2000 Jan 1 at 12:00. Time here is dynamical time, which is currently 68.25 seconds greater than UT. The rate of change is currently about +.49s/year
L0 = 280.46646 + 36000.76983T + 0.0003032T^2
M = 357.52911 + 35999.05029T - 0.0001537T^2
e = 0.016708634 - 0.000042037T - 0.0000001267T^2
C = (1.914602 - 0.004817T - 0.000014T^2)sin(M) + (0.019993 - 0.000101T)sin(2M) + 0.000289sin(3M)
R = 1.000001018(1-e^2)/(1+e.cos(M+C))
O = L0 + C
L = O - 0.00569 - 0.00478sin(125.04 - 1934.136T)
p = 23°26'21".448 - 46".8150T - 0".00029T^2 + 0".001813T^3
RA = atan(cos(p)sin(L)/cos(L))
Dec = asin(sin(p)sin(L))
The atan for RA should be resolved in the correct quadrant (ATAN2 function, or polar coordinate conversion).

All you need then is local sidertal time at Greenwich (GHA Aries): GHA body = GHA Aries - Body RA
GHA Aries = 280.46061837 + 360.98564736629(36525t) + 0.000387933t^2 - t^3/38710000
where t is calculated in the same way as T but using UT rather than dynamical time (not correcting for deltaT)

2. Just in case you thought that my nerdiness was over (and in case there's anyone still reading down here) I have a process for the Sun (Earth) that includes about 30 steps rather than the 10 above, for an accuracy of about 0.001°, that I can post if required.

3. Other solar system bodies are tricky, and I doubt you'll find anything better than Van Flandern and Pulkkinen unless you go to VSOP87. The Moon is a nightmare, and again reducing ELP2000 to the number of elements Van Flandern and Pulkkinen use will probably not give you improved accuracy.

4. Hope the OP has fun with all this - all he wanted was to know how to use last year's almanac!
 
Thank you John, you were very clear and I followed you perfectly.
But you are right, the development of the discussion is becoming OT for this forum and I might contact you by PM.

Just a curiosity by another nerd: what is the formula in your signature? It does not ring a bell in my washed out brain but in particular: if y, x and H have the dimension of a length, w is dimensionless. Isn't it?

Daniel
 
danielefua said:
Thank you John, you were very clear and I followed you perfectly.
But you are right, the development of the discussion is becoming OT for this forum and I might contact you by PM.

Just a curiosity by another nerd: what is the formula in your signature? It does not ring a bell in my washed out brain but in particular: if y, x and H have the dimension of a length, w is dimensionless. Isn't it?

Daniel
Click to expand...

The formula in my signature is the catenary curve. H is the horizontal force, and w is the weight per metre, X and Y are the horizontal and vertical distances.

I'll probably have to change it now : what about (g.lamda/(2.pi).tanh(2.pi.d/lamda))^(1/2) Multiple by 3600/1852 to get knots.

John
 
John Barry said:
what about (g.lamda/(2.pi).tanh(2.pi.d/lamda))^(1/2) Multiple by 3600/1852 to get knots.

John
Click to expand...


wave speed in the transitional state, possibly ?



add

Probably not, there should be a T instead of the first Lambda, and I can't see why the square root

lambda = g.T^2/2pi . tanh(2pi.d/lambda)

divide by T

speed = g.T/2pi . tanh(2pi.d/lambda)

Looks similar but it must be something totally different :)
 
Last edited:
Thank you all so much for your answers - that is very helpful, particularly John Barry - you are quite correct, I was under the impression that Reeds almanac is what was needed (clearly confused by the word Almanac!).
 
In case anyone is interested I searched the internet on John Barry's recommendation, and found the following very useful links:

Excel Spreadsheets for calculations and Almanac: http://www.nauticalalmanac.it/en/eng-lic-ast/

A downloadable 2013 Almanac and Corrections tables: http://www.navsoft.com/downloads.html

Thanks again for everyone's replies - and here's hoping for some clear skies over the next week, so I can actually try and put theory into practice!
 
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