The Astro Navigation Resource

See the latest article: Stars For All Seasons Part 2 – Revolution

Although this website aims to promote the Astro Navigation Demystified series of books, it is hoped that it will also provide a useful resource for navigators, scholars and students of the subject.

EARTH AND SUN IN THE SPHERE updateA wealth of information on the subject of astro navigation can be found under the various headings on the menu bar at the top of the page and in the archives listed down the right. The images below give links to various pages which may be of interest.

Why Astro Navigation?  There is rapidly growing interest in the subject of astro navigation or celestial navigation as it is also known. It is not surprising that, in a world that is increasingly dominated by technology and automation, there is an awakening of interest in traditional methods of using the celestial bodies to help us to navigate the oceans.

Astro navigation is not just for navigators; the subject is an interwoven mix of geography, astronomy, history and mathematics and should appeal to both mariners and scholars alike.

altitude and azimuth mod

Russia is one of the few countries in the world to acknowledge the educational value of astro navigation and to include it as an important part of the school curriculum. In other countries, institutions such as nautical schools and maritime colleges include the subject in their curricula as a subject in its own right while for some independent schools, it provides the perfect theme for integrated studies and open-ended project work.

The question is often asked: ‘how could seafarers navigate the oceans if the global positioning system (GPS) failed? The answer is quite simple; they could revert to the ‘fail-safe’ art of astro navigation. The problem here though, is that we have become so reliant on automated navigation systems that traditional methods are being forgotten.  Even so, there is a very real
pillars danger that the GPS could be destroyed.  
During periods of increased solar activity, massive amounts of material erupt from the Sun. These eruptions are known as coronal mass ejections and when they impact with the Earth they cause disturbances to its magnetic field known as magnetic storms. Major magnetic storms have been known to destroy electricity grids; shut
down the Internet, blank out communications networks and wipe out satellite systems
(includin
g the global positioning system).

azimuth and azimuth angle

Couple this danger with that posed by cyber terrorists who could block GPS signals at any time, then it can easily be seen that navigators who rely solely on electronic navigation systems could be faced with serious problems.

 

 

crossUnfortunately, many sea-goers are deterred from learning astro navigation because they perceive it to be a very difficult subject to learn. In fact, it is very interesting and easy to learn but sadly, some writers and teachers of the subject attempt to disguise its simplicity by cloaking it in an aura of mystery.

 

 

 

I am throughly enjoying working through the wonderful book, ‘Astro Navigation Demystified’. At last a well written book on the subject. I was also very pleased to find this accompanying website.

 

Books of the Astro Navigation Demystified Series:

Astro Navigation Demystified.

Applying Mathematics to Astro Navigation

Astronomy for Astro Navigation

Celestial Navigation.  Theory and Practice

email: astrodemystified@outlook.com

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Stars For All Seasons Part 2 – Revolution

In the last article, we discussed rotation, the reason why stars rise earlier each night; in this article we discuss revolution, the reason why the stars in the night sky change from season to season.

 In the diagram below we see the Earth as it orbits the Sun or to put it another way, we see it as it revolves around the Sun.  The positions of some of the more well known stars in relation to our Sun are also shown and it can be seen that, as the Earth follows its orbital path, different stars will gradually come into and out of view in the night sky.  For example, we will see Sirius in the night sky during the Northern Hemisphere’s winter but we won’t see it during the summer nights.  Its not that it’s in a different place, its just that it is now on our daylight side.

So, in the Northern Hemisphere, we have our Winter Stars such as Aldebaran, Rigel and Betelgeuse and we have our Summer Stars such as Nunki and Kaus Australis; of course, it is the other way round for the Southern Hemisphere.

Books of the Astro Navigation Demystified Series:

Astro Navigation Demystified.

Applying Mathematics to Astro Navigation

Astronomy for Astro Navigation

Celestial Navigation.  Theory and Practice

email: astrodemystified@outlook.com

Posted in astro navigation, astronomy, celestial navigation | Tagged , ,

Stars For All Seasons Part 1

Often asked questions:

Why do the stars seem to rise earlier each night?

 Why do the stars that we see in the night sky change from season to season?

 There are two separate reasons for these phenomena, Rotation and Revolution.

I.e. The Earth rotates about its axis while it revolves around the Sun.

Rotation.

The Earth rotates from west to east about its axis of rotation which is a line joining the celestial poles and if this axis is produced far enough, it will cut the celestial sphere at a point marked by the North Star (Polaris) as shown in the diagram.  Facing north from the Earth, the Pole Star appears stationary, and the other stars appear to rotate from east to west around the Pole Star although in fact the positions of the stars are fixed and it is the Earth which is rotating from west to east.

 

The time taken for a star to complete a circuit around the Pole Star is called a star’s day or sidereal day.  If the sidereal day were to be exactly 24 hours, as is the Mean Solar Day, then the stars would rise and set at the same times every day.  However, the Earth completes each rotation about its axis in 23 hours, 56 minutes and 4 seconds so the stars will take the same amount of time to circuit the Pole Star and that is the length of the sidereal day. Therefore, if a star rises in the east at a certain time on a certain day, it will next do so  23 hours, 56 minutes and 4 seconds later. In other words, the star in question will rise 3 minutes and 56 seconds earlier each day (usually rounded off to 4 minutes).

For example, Say that Arcturus (the brightest star in the northern celestial hemisphere) rises at 18.00 mean time on a certain day; we know that it will rise again 23 hrs. and 56 mins. later so we can easily calculate that it will rise at 17.56 mean time the next day (4 minutes earlier).

Circumpolar Stars.  Depending on the latitude of the observer, some stars will never rise or set because they will always be above the horizon, these are known as circumpolar stars.

Example.  The diagram below shows the constellations Ursa Major (Great Bear) and Cassiopeia which are both circumpolar to observers throughout the northern hemisphere and down to 20o South in the southern hemisphere.   As Ursa Major revolves about the Pole Star so do the five stars of Cassiopeia.

bear to cassiopeia update

There are many other circumpolar constellations such as Ursa Minor, Auriga and Perseus in the Northern Hemisphere and Centaurus and Crux in the Southern Hemisphere; we will be looking at these more closely in future articles of this series.

Watch out for the next article in this series which will explain Revolution and  the stars that we can see as the night sky changes from season to season.

Books of the Astro Navigation Demystified Series:

Astro Navigation Demystified.

Applying Mathematics to Astro Navigation

Astronomy for Astro Navigation

Celestial Navigation.  Theory and Practice

email: astrodemystified@outlook.com

Posted in astro navigation, Astro Navigation Demystified, astronomy, celestial navigation | Tagged , ,

Accuracy of Sight Reduction Methods.

In my recent article ‘Why Astro‘, I highlighted the risks in using the GPS.  Since writing that article, I am frequently asked “if astro / celestial navigation is to be used, which of the many systems is the best”.  Sight reduction methods tend to fall under two categories, Formula and Tabular. (Computerised  sight reduction systems will involve a combination of these methods i.e. mathematical calculation of data contained within a matrix held in a database).  Therefore, in this article, I will discuss the relative merits of these two methods.

Please note that the terms ‘Astro Navigation’ and ‘Celestial Navigation’ are synonymous but for the rest of this article I will use the term astro navigation.

 Sight Reduction. This is the process of reducing the data gathered from observations of celestial bodies down to the information needed to establish an astronomical position line.  The two essential items of data that we need to begin the process of sight reduction are the azimuth and the altitude of the celestial body in question.

The azimuth gives us the direction of the celestial body from the calculated position.

When we measure the altitude, what we are really trying to establish is the zenith distance (Zenith Distance = 90o – Altitude) that is the distance to the geographical position of the body.  We measure the altitude at our true position and we calculate the altitude at the DR position (or assumed position); this enables us to calculate the zenith distances at the two positions.  The difference between the two zenith distances will give us the distance from the DR position to the true position measured along the direction line of the calculated azimuth.

Measuring the altitude and azimuth at the true position with a sextant and azimuth compass is relatively straightforward but calculating what they would have been at the DR or Assumed position is the real work of sight reduction.

Formula Methods. The traditional way of calculating the azimuth and zenith distance at the DR position is by spherical trigonometry. Before the advent of the electronic calculator, this would have been a very lengthy and time consuming method involving the use of tables of logarithms to make calculations involving the Haversine Formula.  However, these days we can still make use of spherical trigonometry with the use of a scientific calculator and with the application of just two formulas derived from the Cosine Rule, one for the azimuth and one for the zenith distance.  With just a little practice, it will be found that this method is quick and easy to apply.  We usually refer to these methods as Formula Methods.  Accuracy is the greatest advantage of formula methods; calculations are usually made to 3 or 4 decimal places but this can be extended if greater accuracy is required.   Of course there is always the risk of human error when making mathematical calculations but with an electronic calculator, it takes very little time to double check.

TABULAR METHODS.  During the twentieth century, tabular sight reduction methods were first devised and today there is such a proliferation of these methods that choosing one can be very confusing.  Tabular methods do not require a knowledge of spherical trigonometry; they involve the use of sets of pre-computed tables of data from which the altitude and azimuth can be interpolated.  The disadvantage of these tables is that they have to be entered with the latitude and Local Hour Angle rounded to the nearest degree so that calculation of the altitude and azimuth depends on interpolation and extrapolation.  To achieve this, an ‘Assumed Position’ has to be chosen.  This is a position where the latitude and longitude closest to the DR position have the following properties: The assumed latitude is the DR latitude rounded up to the nearest whole degree and the assumed longitude is the longitude closest to the DR longitude that makes the local hour angle a whole degree. In comparison, when we solve the problem directly by spherical trigonometry, we use the latitude and longitude of the DR or EP position and we make exact calculations without the inaccuracies of interpolation methods.  Of course the greatest advantage of tabular methods is that the navigator does not require a knowledge of trigonometry and the only mathematical calculations needed involve simple arithmetic.

Comparison.  Below, we compare the accuracy of calculations made to establish astronomical position lines using two different methods, one a formula method and the other a tabular method. We use identical input data for both examples.  The first example shows the calculations made using the cosine formula method and the second shows those made using the Rapid Sight Reduction Method (NP303).  Please note that the sight reduction forms used in these examples are not standard but are designed as learning aids for use with exercises in my books.

Cosine Formula Sight Reduction Method

Rapid Sight Reduction Method

Findings.  There is a difference of 1.558 nautical miles in the calculation of the intercepts produced by the two methods, that is to say there is a difference of 1.558 arcminutes in the two sets of calculations.  In terms of distance, 1.558 nautical miles seems quite a lot but in terms of mathematical calculation, 1.558 arcminutes does not seem so great a difference (unless you are a rocket scientist of course).  So how do we decide which method is the more accurate?

  • On the one hand, it could be argued that the formula method is the more accurate of the two methods for the following reasons: There are accumulative and unavoidable errors caused by the addition and rounding-off of quantities taken from sight reduction tables whereas with formula methods; calculations are usually made to three or more decimal places thereby providing a greater degree of accuracy.
  • On the other hand, it could be argued that sight reduction by the use of spherical trigonometry is time consuming and gives considerable scope for mathematical error. Because time and accuracy are of the essence in practical navigation, it is an advantage to be able to calculate altitude and azimuth by relatively simple table operations.

Summary.  The arguments above are really inconclusive and it would seem that, from the point of view of accuracy, there is not a great deal of difference between the two methods.  Yet, we are told that the accuracy of astronavigation position fixing is only plus or minus 1 nautical mile, so if it’s not the method, where does this level of inaccuracy stem from?

Errors That Occur No Matter Which Sight Reduction Method Is Used.  If we are concerned about accuracy in astro navigation, it matters not which sight reduction method we use, the real danger of inaccuracy lies in other areas.   Inaccuracy in calculations may be introduced by a number of contributory errors irrespective of the sight reduction method being used; these errors are summarized below.

Errors in the observed altitude.   Even when the sextant altitude has been corrected for index error, semi-diameter and parallax, the resultant altitude reading may still be incorrect owing to a combination of other errors such as incorrect calculated values for dip and refraction.  An error in the observed altitude will lead to an error in the observed zenith distance.

Refraction.  A pronounced error in refraction is likely to occur when the altitude is below 15o.  The dip being affected by refraction is the most likely cause of error; when atmospheric conditions are abnormal, the actual value of dip may differ from the tabulated value by up to 10′.

Deck-watch error.  If the deck-watch error is incorrect, the GMT and the LHA will be incorrect.  An error in the LHA will lead to an error in the calculated altitude and this will cause the position line to be displaced.

Errors in the D.R. position.  Errors in the course and distance laid down on the chart may result from a combination of inaccurate plotting, compass error. the effects of wind and tidal stream and incorrect calculation of speed made good over the ground.  An error in the DR position and resultant assumed position will lead to errors in the estimated longitude and hence the local hour angle and this in turn will lead to an error in the calculated altitude.

Nautical Almanac.  There are accumulative and unavoidable errors caused by the addition and rounding-off of quantities taken from the almanac.

Errors In Observed Positions Derived From More Than One Position Line.  Position lines obtained from two or more astronomical observations are not likely to pass through a common point.  The reasons for this are firstly, the observations are not likely to be taken simultaneously since it is not possible to take sextant readings of three several celestial bodies at the same instant.  The faster a vessel travels, the greater the movement of the observer between the three observations and the more significant this error becomes even when special methods of calculation such as ‘MOO’ are used.  Secondly, observed altitudes are very seldom correct and therefore, the resultant observed zenith distances will not be correct.  For these reasons, the resultant position lines will be displaced and a ‘cocked-hat’ will be formed and because the position within the triangle of the cocked-hat is arrived at by guess-work, it is unlikely to be correct.

Conclusion:  Arguments concerning the relative accuracy of different sight reduction methods are not important, the real cause of inaccuracy in astro navigation is more likely to stem from the types of error described  above and these can occur irrespective of the method being used.  For the average yachtsman sailing in the vast expanse of the ocean, an accuracy level of plus or minus 1 nautical mile is probably nothing to worry about but for those engaged in activities that require a greater level of accuracy such as surveying and naval operations, it is obviously a matter of concern.

Wish to learn more?  The cosine formula method, it is comprehensively taught in my book ‘Celestial Navigation – Theory and Practice’.  The Rapid Sight Reduction Method is comprehensively taught in my book ‘Astro Navigation Demystified’.

Other Books by Jack Case:

Applying Mathematics to Astro Navigation

Astronomy for Astro Navigation

email: astrodemystified@outlook.com
Posted in astro navigation, celestial navigation, global positioning system, gps, Marine Navigation, mathematics, navigation, spherical trigonometry, trigonometry | Tagged , , , ,

In Defence of Mer Pas

The noon sight for latitude is a method of calculating latitude from the altitude of the sun at the instant it crosses your meridian and for this reason, the method is also known as ‘Meridian Passage’ or ‘Mer. Pas’.  I am often asked “what’s the point of this when we already know our latitude”; my reply is “unless we are using GPS, we won’t know our exact latitude and if we are using GPS, why are we bothering with astro navigation anyway?  The whole point is that, as I have pointed out in previous posts, we can no longer depend of GPS for a variety of reasons so prudent navigators will keep up their skills in astro navigation.

When we calculate a position fix using astro navigation, our starting points are our DR position or our EP which are only approximate positions so how can we say that there is no point in taking a noon sight when we already know our latitude.  The fact is that we don’t know our exact latitude and the noon sight is a way of calculating just that.

Why stop at latitude?  In the past, before the advent of the chronometer, the noon sight would enable the navigator to only calculate latitude but of course we can now use it to calculate longitude as well.  At the instant the sun crosses our meridian, we will know the exact local mean time (LMT) from the time of Mer Pas as listed in the Nautical Almanac for the day and by combining this with the Greenwich Mean Time which we take from the deck watch at the same instant, we are able to calculate our longitude as the following example taken from my book ‘Celestial Navigation’ shows:

Step 8. Calculate time difference between LMT and GMT of Local Mer Pas.

(Latitude West: GMT Best.  Latitude East: GMT Least).

GMT at Local Mer Pas: 16h 08m 20.1s        (Subtract from LMT if DR East)
LMT Mer Pas: 12h 02m 00.0s      (Subtract from GMT if DR West)
Time Diff: 04h 06m 20.1s
 
Step 9. Calculate Longitude at Local Mer Pas.  (Convert time difference to Arc)
Time Difference =

04h 06m 20.1s

Multiply the hours by 15 and divide the minutes and seconds by 4. Convert decimals to units of arc.

degs      mins      secs

Convert hours: 4= 4 x 15  = 60o  60o        00’     00”
Convert minutes: 6m = 6 ÷ 4  = 1o.5     1o     30’     00”
Convert seconds: 20.1s = 20.1 ÷ 4 = 5m.025     0o        05′     01″.5
Total   61o        35′     01”.5
Calculated Longitude: 6135’ 01″.5 W

So the noon sight can give us a position fix in terms of both latitude and longitude.

In the summer months, in the middle latitudes, there can be up to 16 hours from morning nautical twilight to evening nautical twilight so that means that we can go up to 16 hours between star and planet sights.  However, a noon sight will give us an accurate fix between twilight times and the good news is that there is only one celestial body to shoot and no ‘cocked-hat’.

Of course, there is also the ‘Intercept Method’ or ‘Sun Run Sun’ which will give us another fix during daylight hours.  The whole point is that we never know when poor visibility will prevent us from taking morning and evening sights so the more tricks that we have up our sleeves the better.

If you wish to learn about the Meridian Passage method try this link:

To learn about the Intercept method, click here.

To read why we cannot rely on the GPS read this:

Click here to read about the optimum time for star and planet sights.

Books of the Astro Navigation Demystified Series:

Astro Navigation Demystified.

Applying Mathematics to Astro Navigation

Astronomy for Astro Navigation

Celestial Navigation.  The Ultimate Course

Posted in astro navigation, astronomy, celestial navigation, gps, navigation | Tagged , , ,

Celestial Navigation – Theory and Practice

I have received a number of messages asking why my book ‘Celestial Navigation – The Ultimate Course’ is not currently available.  The truth is that I have been spending the winter months revising and updating this book and it will shortly be available on Amazon with a new sub-title: ‘Celestial Navigation – Theory and Practice’.

Celestial_Navigation_Cover_for_Kindle

The traditional method of celestial navigation involving the use of spherical trigonometry to calculate a vessel’s position is comprehensively taught in this book.  At first sight, the term ‘spherical trigonometry’ might seem quite daunting but with the knowledge of just two formulas and with a little practice of the methods explained in this book, it will be found to be quick and easy to apply as well as very accurate. With this method, we make accurate calculations using data taken directly from a vessel’s DR position and so avoid the inaccuracies of sight reduction methods that involve interpolation from tables using data based on an ‘assumed position’.

Although the prime aim of this book is to teach the practical skills of celestial navigation, it is emphasised that without knowledge, skill is nothing; at the same time, it is recognised that students quickly lose interest if they are expected to plough through reams of theory before they can get down to the business of learning the skills.  With this in mind, my book has been uniquely designed to teach the important skills from the outset while ‘tying-in’ the relevant theory as progress is made.  There are numerous examples and self-test exercises which enhance the learning process and help to embed the knowledge and skills needed to practise the art of celestial navigation.

Although it is a large book (containing 410, letter size pages) it is thoroughly cross-referenced and its layout enables the reader to move from one section to another without having to read it from beginning to end.

With regard to the mathematical aspects of the subject, I have adopted a language style which allows the text to flow smoothly and makes for enjoyable reading which is a departure from the stilted, academic language of many text books.

Note.  The terms celestial navigation and astro navigation are generally regarded as synonymous.

Books of the Astro Navigation Demystified Series:

Astro Navigation Demystified.

Applying Mathematics to Astro Navigation

Astronomy for Astro Navigation

Celestial Navigation.  Theory and Practice

email: astrodemystified@outlook.com

Posted in astro navigation, astronomy, celestial navigation | Tagged , ,

Why Astro?

In a recent article the discussion centred on our over-reliance on GPS for navigation at sea and the need for back-up systems.  The conclusion drawn was that we already have a back-up system, one that has been tried and tested over hundreds of years and that is astro navigation or celestial navigation as it is also known.  Was this the correct conclusion though?  In this article, we set out to explore other alternatives to GPS and to examine the pros and cons of astro navigation.

(Note. The terms astro navigation and celestial navigation are synonymous but for simplicity’s sake, we shall stick to astro navigation for the rest of this article).

What are the risks to the GPS? 

Spoofing – misdirecting  GPS navigation receiver so that it thinks it is somewhere it isn’t.

Jamming – the intentional emission of radio frequency signals to interfere with the operation of GPS receivers by saturating them with noise or false information.

Hacking – breaking into GPS software to discover a receiver’s location or to corrupt it..

Malicious viruses causing GPS to malfunction.

Magnetic storms can put power grids out of action, blank out communications systems and the GPS.

Electro-magnetic interference – can disrupt radio signals causing distorted GPS readings.

Damage to aerials and equipment can leave a vessel without access to the GPS

What are the alternatives?

Sebastion Anthony suggests creating a ground-based system which would involve blanketing the Earth with hundreds or thousands of radio transmitters at an immense cost. Surely though, that would be a waste of money and time; any system that is based on radio signals would be susceptible to the risks of spoofing, jamming and hacking in the same way that the GPS is.

There has also been talk of re-commissioning some of the electronic navigation systems that were in use before the advent of the GPS such as Omega, CONSOL, DECCA, and LORAN but once again, we are back to the problem of re-introducing radio based systems that are susceptible to the same risks as GPS.

George H Kaplan of the US Naval Observatory talks of using the Stellar Reference Frame as an alternative to GPS but this system also relies on electro-magnetic signals to communicate with satellites and so it is susceptible to exactly the same risks as the GPS.

Kaplan also talks of employing inertial navigation systems which are used in guided missiles, spacecraft, submarines and other naval ships and aircraft; however, he points out that these are simply sophisticated dead-reckoning systems that need to be aligned to a reference point, usually provided by GPS.  So, we come back to the problem of reliance on GPS.  However, he does suggest that where radar plots and weapon control systems in naval ships and aircraft need some sort of electronic input of position, inertial navigation systems may fit the bill during short periods of GPS malfunction.  However, none of this matters much to ‘yachties’ and small merchant ships unless small and cheap versions of such equipment becomes available to them.

The Only Real Alternative  It seems that the only real alternative to GPS is Astro Navigation and that is probably why the US Navy has recently re-introduced it in its training programmes while the Royal Navy continues to keep it in the curriculum for specialist navigating officers.

Advantages of Astro Navigation:

  1. It has global coverage.
  2. Does not require expensive equipment.
  3. Does not require a ground based support infrastructure.
  4. Does not emit electro-magnetic signals that can be detected by an enemy.
  5. Cannot be jammed, spoofed or hacked..
  6. Is not susceptible to disruption by solar storms or other electro-magnetic disturbances.

Disadvantages of Astro Navigation:

  1. Can be hampered by cloud cover except in aircraft.
  2. Inherent Errors in data and calculations. U.S. Navy and Royal Navy navigators are taught that the accuracy of astro navigation is ±1 minute of arc or 1 nautical mile. For details of inherent errors in astro navigation click here.
  3. Even with a highly skilled navigator it can take several minutes to obtain a celestial fix whereas a GPS fix is more or less instantaneous.

Links:

  1. What’s the point of Astro Navigation when we have the GPS?
  2. Could the Global Positioning System fail?
  3. The accuracy of astro / celestial navigation.
  4. Royal Navy officers are still trained to navigate by the stars.
  5. Celestial Navigation: U.S. Navy resurrects ancient craft.
  6. Ships fooled by GPS spoofing attack.
  7. Our terrifying reliance on GPS.
  8. New technology for celestial navigation.

Books of the Astro Navigation Demystified Series:

Astro Navigation Demystified.

Applying Mathematics to Astro Navigation

Astronomy for Astro Navigation

Celestial Navigation.  The Ultimate Course

email: astrodemystified@outlook.com

Posted in astro navigation, celestial navigation, electronic navigation systems, global positioning system, gps, Marine Navigation, navigation | Tagged , , ,

Astro Navigation in the Forests of the Iroquois

Link: History of the Mason Dixon line

Jeremiah Dixon and Charles Mason plotted the famous Mason Dixon Line in 1765, long before the days of GPS or any other electronic navigation equipment. How was it then that they were they able to fix positions from the midst of the forests of the Iroquois?

They would not have been able to survey the land using triangulation methods because suitable landmarks would have been hidden by the trees.  They would not have been able to measure the altitude of celestial bodies because there would not have been a visible horizon.  All they would have been able to see would be a small circle of sky through the canopy above them and therein lies the clue.

They used an instrument known as a zenith sector which is a fixed vertical telescope through which an observer is able to view a small circle of sky centred at the zenith of his geographical position.  By using this device, they were able to accurately measure the zenith distance of celestial bodies that came within the telescope’s field of view.

In the diagram below, Z marks the zenith of the observer, X is the position of a celestial body and O is the Earth’s centre.  The zenith distance is the angular distance ZX which is subtended by the angle XOZ .  In other words it is the angular distance from the observer’s zenith to the celestial body.  (For a fuller explanation of zenith distance follow this link:)

 

 

 

 

 

 

 

 

By measuring the zenith distance of a celestial body at the instant that it crosses the observer’s meridian, the observer is able to determine the latitude of his position because the zenith distance will be equal to the distance from the latitude of the geographical position of the body to the latitude of the observer in nautical miles measured north or south (click here for an explanation of this).

Mason and Dixon plotted their line in this way choosing stars whose declinations were close to the latitude 39o 43′ N, the east/west boundary between Pennsylvania and Maryland.  Because the chronometer had not yet been invented, they were not able to calculate longitude which partly explains why their line ran along a parallel of latitude.

They chose to use only stars for their observations because the declination of a star changes very slowly and can be considered to be fixed for short periods of time.  Furthermore, the magnification of the zenith sector telescope is far greater than the telescope of a sextant and so they were able to use many faint stars that we would not normally be able to use for navigation.

To establish a north/south boundary they would have followed a line bearing true south or true north from a known landmark such as a hill or small town. It is interesting to note that the majority of the boundaries between American states, which were established before the advent of the chronometer, also ran east/west or north/south.

The boundary between Delaware and Pennsylvania, which was also fixed by Mason and Dixon, involved mainly conventional surveying techniques because it followed an arc known as the ‘twelve mile circle’ around the town of New Castle.  Similarly, the Delaware-Maryland boundary was based on conventional surveying because it was designed to bisect the Delmarva Peninsular instead of following a meridian.

A more detailed treatment of the topic of zenith distance can be found in Astro Navigation Demystified’.

Books of the Astro Navigation Demystified Series:

Astro Navigation Demystified.

Applying Mathematics to Astro Navigation

Astronomy for Astro Navigation

Celestial Navigation.  The Ultimate Course

email: astrodemystified@outlook.com

Posted in astro navigation, astronomy, celestial navigation | Tagged , , ,

Translating A Celestial Position Into A Geographical Position.

This follows the previous post which discussed the Celestial Co-ordinate System.

If we wish to use a celestial body’s position in the celestial sphere to help us to navigate on the Earth’s surface, we must be able to translate that body’s celestial co-ordinate position into a geographical position expressed in terms of our earthbound geographical co-ordinate system.

As we discussed in the previous post, we express a celestial body’s position in the celestial sphere in relation to its angular distance east or west of the celestial meridian that passes through the ‘First Point of Aries’.  Similarly, in the geographical co-ordinate system, we express a position on the Earth’s surface in relation to its angular distance east or west of the Greenwich Meridian.  The ‘Declination’ of a body expresses its angular distance north or south of the Celestial Equator in the same way that we use latitude to define a position north or south of the Equator.

The following explanation requires a little imagination.  In this diagram, the Greenwich Meridian is projected onto the Celestial Sphere.

Point X denotes the position of star Acamar in the celestial sphere and PRP1 represents the meridian running through the position of Acamar.

The Sidereal Hour Angle (SHA) is the angular distance from the meridian of the First Point of Aries to the meridian of the celestial body (R) measured westwards.

The Greenwich Hour Angle of Aries (GHA Aries) is the angular distance, measured westwards, from the projected Greenwich Meridian to the meridian of the First Point of Aries.

The Greenwich Hour Angle of Acamar (GHA Acamar) is equal to the sum of the SHA Acamar and the GHA Aries.

GHA Aries to the nearest second can be interpolated from tables in the Nautical Almanac as can the SHA and Declination of the Navigational Stars. Accordingly, for the example below, the following values have been taken from Nautical Almanac:

GHA Aries:  026o  39’.30     SHA Acamar: 315o  20’.50   Declination Acamar:  S40o 14′.3

Using these values, the GHA of a star can simply be calculated as shown below:

GHA Acamar = GHA Aries + SHA Acamar = 026o  39’.30 + 315o  20’.50 = 341o  59’.80

Longitude of the Geographical Position of Acamar.  The GHA is equivalent to the longitude; however, we must remember that GHA is measured westwards from the Greenwich Meridian from 0o to  360o whereas Longitude is measured either east or west from Greenwich from 0o to 180o.  Therefore, in this case, since the GHA of Acamar is greater than 180o, the longitude will be East so we must subtract it from 360o to convert it to an easting as follows:  Longitude = 360o – 341o 59’.80 = 18o  00’.2 East.

Declination of Acamar = S40o 14′.3  (Note that the declination of the stars can be regarded as constant and so no further calculation is necessary).

Latitude of the Geographical Position of Acamar.  Since the declination is equivalent to the latitude, we can state that the latitude of the GP is 40o 14′.3 South.

We can now state that we have translated the celestial co-ordinates of Acamar from SHA:315o 20’.50, Declination:S40o 14′.3 to a Geographical Position of 18o 00’.2 East, 40o 14′.3 South.

Note.  It is not necessary for the navigator to calculate the GHA of the Sun, Moon and planets because the Nautical Almanac tabulates these for you.

This topic is explained in far greater depth in my book ‘Astro Navigation Demystified’.

Using The Geographical Position (GP) Of A Celestial Body To Determine Our Own Position.  By measuring the altitude of a celestial body, we are able to calculate the zenith distance which will give us the distance in nautical miles from the observer’s position to the geographical position of the body.  The azimuth will give us the direction of the GP of the body from the observer’s position. This explains why measuring the altitude and azimuth are the first steps in determining our position in celestial navigation.  Those who wish to learn how we do this can follow this link to find a brief explanation: Astro Navigation in a Nutshell. However, ‘Astro Navigation Demystified’ contains a more comprehensive explanation.

 Books of the Astro Navigation Demystified Series:

Astro Navigation Demystified.

Applying Mathematics to Astro Navigation

Astronomy for Astro Navigation

Celestial Navigation.  The Ultimate Course

email: astrodemystified@outlook.com

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The Celestial Co-ordinate system.

In astronomy, we need a celestial co-ordinate system for fixing the positions of the celestial bodies in the celestial sphere.

We express a celestial body’s position in the celestial sphere in relation to its angular distance north or south of the Celestial Equator and east or west of the celestial meridian that passes through the ‘First Point of Aries’.

Declination.  The Declination of a celestial body is its angular distance north or south of the Celestial Equator.  The declinations of the stars change very slowly and can be considered to be almost constant for up to a month at a time.  The declination of the Sun changes relatively fast from 23.43N. to 23.43S. and back again during the course of a year. The Moon’s declination is more difficult to predict because the rate of change is even more rapid than that of the Sun and the pattern of the changes is less uniform. The declinations of the planets are complicated by the facts that they are at varying distances from the Sun, have different orbital patterns and travel at different speeds.

Declination can be summarised as the celestial equivalent of latitude since it is the angular distance of a celestial body north or south of the Celestial Equator.

Note.  The latitude of the tropic of Cancer is currently drifting south at approximately 0.5’’ per year while the latitude of the tropic of Capricorn is drifting north at the same rate.

The First Point of Aries is usually represented by the ‘ram’s horn’ symbol shown on the left.  Just as the Greenwich meridian has been arbitrarily chosen as the zero point for measuring longitude on the surface of the Earth, the first point of Aries has been chosen as the zero point in the celestial sphere.  It is the point at which the Sun crosses the celestial equator moving from south to north (at the vernal Equinox in other words).  The confusing thing is that, although this point lay in the constellation of Aries when it was chosen by the ancient astronomers, due to precession, it now lies in Pisces.

Note.  Because of the difficulty of inserting the symbol for Aries into text we substitute it with the character in the text below

 Right Ascension (RA).   This is used by astronomers to define the position of a celestial body and is defined as the angle between the meridian of the First Point of Aries and the meridian of the celestial body measured in an easterly direction from Aries.  RA is not used in astro navigation, Sidereal Hour Angle is used instead.

 Sidereal Hour Angle (SHA). This is similar to RA in as much that it is defined as the angle between the meridian of the First Point of Aries and the meridian of the celestial body.  However, the difference is that SHA is measured westwards from Aries while RA is measured eastwards. This is illustrated in the following diagram:

X is the position of a celestial body in the celestial sphere.

R is the point at which the body’s meridian crosses the celestial equator.  PXRPis the meridian of the celestial body.

 is the First Point of Aries.  PYPis the meridian of the First Point of Aries.

The Sidereal Hour Angle is the angle YPR.  That is the angle between the meridian running through the First Point of Aries and the meridian running through the celestial body measured at the pole P.   It can also be defined as the angular distance YR, that is the angular distance measured westwards along the Celestial Equator from the meridian of the First Point of Aries to the meridian of the celestial body.

Right Ascension can also be defined as the angle between the meridian of the First Point of Aries and the meridian of the celestial body but the difference is that it is measured in an easterly direction from Aries.

From this, we can conclude that

RA    =  360o – SHA and

SHA  = 360o – RA.

In Astro Navigation, we make use of our knowledge of the positions of the celestial bodies to help us to navigate on the surface of the Earth.  However, in order to do so we must first relate their positions in the celestial sphere to positions on the Earth’s surface.  The next post in this series explains how this is done.

A more detailed treatment of this topic can be found in Astro Navigation Demystified’.

Books of the Astro Navigation Demystified Series:

Astro Navigation Demystified.

Applying Mathematics to Astro Navigation

Astronomy for Astro Navigation

Celestial Navigation.  The Ultimate Course

email: astrodemystified@outlook.com

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What’s The Point of Astro Navigation When We Have The GPS?

Imagine that you are driving through Birmingham when, suddenly, your ‘Sat Nav’ starts to tell you that you are in Manchester.  In such a situation, you would quickly realise that the GPS had gone haywire; however, if you were in a ship, out of sight of and beyond radar contact of land, it would not be immediately obvious that you were being given false positions.  If you are one of those people who depend heavily on the GPS and believe that it will never let you down, then you might be in for a nasty shock.  The New Scientist reports that Russia may be experimenting with methods of interfering with GPS signals and that these methods could quite easily be copied by other organisations including rogue nations and terrorists.

Sebastian Anthony talks of our terrifying reliance on GPS and our need to develop back-up systems.  Imagine the devastating effects that a GPS failure would have on land, air and sea navigation, air traffic control, communications, power grids, radar, defence and a host of other systems very few of which have ‘back-ups’ in place.

Things can easily go wrong with the GPS even without malicious interference. For example, magnetic storms can put power grids out of action, blank out communications systems including the internet and destroy satellites (including those that serve the GPS).

I warned of these dangers on this website in 2008 with my post Could The GPS Fail when I made the point that fortunately, when it comes to navigation at sea, we do have a back-up system; a system which has been tried and tested over hundreds of years; of course I speak of Astro / Celestial Navigation.

It is all very convenient to find our way by GPS but what would we do without it when we are far out to sea where there are no roads, signposts or other landmarks to guide us? Prudent navigators keep up their skills in astro / celestial navigation by taking at least one astro fix a day when on passage. The reason they do this, is not only to practise their skills but also to keep a check on the GPS. In fact, many experienced yachtsmen and women do not employ GPS at all when on ocean passage but rely solely on their skills in astro / celestial navigation instead.

If Astro / Celestial Navigation is new to you or you just want to brush-up your skills, you might be interested in the following.

Books of the Astro Navigation Demystified Series:

Astro Navigation Demystified.

Applying Mathematics to Astro Navigation

Astronomy for Astro Navigation

Celestial Navigation.  The Ultimate Course

email: astrodemystified@outlook.com

Posted in astro navigation, Astro Navigation Topics, astronomy, celestial navigation, coronal mass ejections, electronic navigation systems, gps | Tagged , , , ,