The Astro Navigation Resource

See the latest article: In Defence of Mer Pas

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.  The Ultimate Course

email: astrodemystified@outlook.com

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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 – Formula Method

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 – Formula Method’.

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.  The Ultimate Course

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

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

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

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

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 , , , ,

Calculating Changes in Longitude and Time Along a Parallel of Latitude.

If two ships are both positioned exactly on the Equator but are separated by 900 nautical miles in an east/west direction, then in terms of longitude, they will be 15apart and in terms of Greenwich Mean Time, they will be 1 hour apart.  If however, they both move to latitude 50oN maintaining a distance of 900 nautical miles between them, they will find that their difference in longitude has increased to 23o.36. and their difference in GMT has increased to 1.5 hours.

What are the reasons for these differences?

Longitude and Distance at the Equator.  The Earth’s equatorial circumference is 21639 n.m.  Since the Equator is a great circle, 1o will subtend an arc of: 21639 ÷ 360  =  60.1  »  60 n.m.  There are 360 meridians of Longitude so it follows that, measuring from the Earth’s centre, the angular distance between adjacent meridians at the Equator is 1o.  Since, as calculated above, 1o subtends an arc of 60 n.m. it follows that the distance between adjacent meridians of longitude at the Equator is 60 n.m.

Returning to the original question, the distance between the ships when measuring along the Equator, is 900 nautical miles.  Since, as explained above, the distance between adjacent meridians of longitude at the Equator is 60 n.m, the difference in longitude between the ships must be 900 ÷ 60 = 15o.

 Longitude and Mean Time.  The Mean Sun completes its 360o revolution of the Earth in 24 hours.    So, in 1 hour, the Mean Sun moves 15o,

in 4 minutes, it moves 1o,

in 1 minute it moves 15′,

in 4 seconds it moves 1′.

From this, it becomes obvious that there is a direct relationship between arc and mean time such that 1 minute of time equals 15 minutes of arc.  We know that the angular distance between meridians of longitude at the Equator is 1o and that 1o equates to 4 minutes of time so we can conclude that the mean time difference between adjacent meridians of longitude is 4 minutes.

Why Greenwich Mean Time.  Greenwich Mean Time (GMT) is the local mean time anywhere on the meridian of Greenwich.  Since the Greenwich meridian is used as the base meridian from which the longitude of all places on Earth are identified, it follows from the discussion above, that GMT provides the link between the longitude of a place and the longitude of Greenwich. Therefore, if we know the longitude of a position on the Earth’s surface, we can easily calculate the GMT at that position since 1o of longitude equates to 4 minutes of GMT.  Alternatively, we can calculate the longitude of a place if we know the GMT there.

Returning to the original question again, we have calculated that the ships are separated by 15o of longitude which in terms of GMT, equates to a time difference of one hour.

 Note. Universal Time (UT).  The term Universal Time was adopted internationally in 1928 as a more precise term than Greenwich Mean Time, because GMT can refer to either an astronomical or a civil day. However, the term Greenwich Mean Time persists in common usage to this day and is generally considered to be synonymous with the term Universal Time.  It should be noted that the Nautical Almanac and other tables of astronomical data usually refer to UT instead of GMT.

 Longitude and Distance Along a Parallel of Latitude.  The diagram below shows that, as the meridians of longitude move away from the Equator, they draw closer together until they eventually converge at the poles.

 

 

 

 

 

 

 

 

 

 

 

In the next diagram, the arcs AC and BD lie on different meridians of longitude. The arc AB is the distance between these meridians measured along the surface of latitude 50oN and the arc CD is the distance between the same meridians measured along the Equator.  Clearly, the distance CD is much greater than the distance AB.

 

 

 

 

 

 

 

 

 

 

 

 

 

To Calculate The Distance Between Two Meridians Along A Parallel Of Latitude.  The following formulae are used for calculating the difference in distance along a parallel of latitude (Ddist) corresponding to a difference in longitude (Dlong) and vice versa.  The formulae are simply stated below without explanation but if you wish to see a full explanation of their derivation then click here).

Ddist = Dlong x Cos Lat.   and  Dlong = Ddist ÷ Cos Lat.

To return to the original question once again, we know that the two ships are separated by a distance of 900 nautical miles along the surface of the parallel of latitude 50oN.  Using the formulae given above, we calculate the difference between them in terms of longitude as follows:  Dlong = Ddist ÷ Cos Lat = 900 ÷ Cos(50) = 900 ÷ 0.64278 = 1400′.168 = 23o.33  

To calculate the difference between them in terms of GMT we proceed as follows: 23o.33 ÷ 15 = 1.5 hours (since 15o of arc equates to 1 hour of mean time).

A more detailed treatment of this topic can be found in the books listed below.

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 Importance of Azimuth and Altitude in Astro Navigation

 In the diagram above, the celestial sphere is drawn in the plane of the observer’s meridian with the observer’s zenith (Z) at the top.

 Point O represents both the observer and the Earth.

Z represents the observer’s zenith.

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

A is the point where the virtual circle running through the position of the celestial body meets the celestial horizon.

P and P1 are the north and south poles respectively.

Zenith. The Zenith is an imaginary point on the celestial sphere directly above the observer.  It is the point where a straight line drawn from the geocentric centre of the Earth, through the observer’s position and onwards, intersects with the celestial sphere.

 The Zenith Distance. The Zenith Distance is the angular distance from the zenith to the celestial body measured from the Earth’s centre. It is the angular distance ZX which is subtended by the angle XOZ and is measured along the vertical circle that passes through the celestial body.  (A vertical circle is a great circle that passes through the observer’s zenith and is perpendicular to the celestial horizon).

 The Altitude.  Altitude is the angle AOX, that is the angle from the celestial horizon to the celestial body and is measured along the same vertical circle as the zenith distance.

Relationship Between Zenith Distance And The Nautical Mile.  An angle of 1 minute at the earth’s centre will subtend an arc of length 1 n.m on the earth’s surface.  Therefore if the angle XOZ is 30o (the arc ZX) will be equal to 30 x 60 = 1800 arc minutes at the earth’s surface and so the zenith distance will be equal to 1800 nautical miles.

Relationship between Altitude and Zenith Distance. Since the celestial meridian is another vertical circle and is therefore, also perpendicular to the celestial horizon, it follows that angle AOZ is a right angle and angles AOX and XOZ are complementary angles.  From this we can deduce that:

Zenith Distance = 90o – Altitude

and Altitude = 90o – Zenith Distance

In the PZX Triangle diagram which is shown below, the arc AU is the arc joining the observer’s position to the geographical position of the celestial body.  This arc when projected onto the celestial sphere forms the arc ZX which is the zenith distance.  Therefore, from the discussion above, it can be seen that the angular distance ZX is equal to the angular distance AU which when converted to nautical miles will give us the distance from the GP of the body to the position of the observer.

Azimuth.  The angle PZX is the azimuth of the celestial body and is the angular distance between the observer’s celestial meridian and the direction of the position of the body (GP).

 Summarizing The Role Of Altitude, Azimuth And Zenith Distance In Celestial navigation.  The preceding discussion illustrates the importance of altitude and azimuth in celestial navigation.  It can be seen that by measuring the altitude of a celestial body, we are able to easily 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 from the observer’s position.  This explains why measuring the altitude and azimuth are the first steps in determining our position in celestial navigation.

Where to buy books of the Astro Navigation Demystified series:

Astro Navigation Demystified at Amazon.com

Astro Navigation Demystified at Amazon.uk

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Applying Mathematics to Astro Navigation at Amazon .com

Applying Mathematics to Astro Navigation at Amazon .uk

http://www.amazon.com/Applying-Mathematics-Astro-Navigation-Demystified/dp/1496012062/ref=sr_1_2?s=books&ie=UTF8&qid=1393696809&sr=1-2&keywords=astro+navigation

Astronomy for Astro Navigation at Amazon.com

Astronomy for Astro Navigation at Amazon.uk

http://www.amazon.com/Astronomy-Astro-Navigation-Black-Demystified/dp/1511675594/ref=sr_1_2?s=books&ie=UTF8&qid=1446153840&sr=1-2&keywords=astro+navigation+demystified

Celestial Navigation at Amazon.com

Celestial Navigation at Amazon.uk

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web: http://www.astronavigationdemystified.com

email: astrodemystified@outlook.com

 

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Bearing, Azimuth and Azimuth Angle.

For students of astro navigation, the various definitions of azimuth, azimuth angle and bearing can cause much confusion. It is hoped that the following will help to clarify this topic.

Bearing is the direction of something in relation to a fixed point. A bearing can be measured in degrees in any direction and in any plane.

For example, in marine navigation, relative bearing is measured in the horizontal plane in relation to the ships heading from 0o to 180o to either port (red) or starboard (green).  For example, the relative bearing of an object on the port beam would be Red 90o and an object on the starboard bow would be Green 45o.

Note. The definition given above is according to the Admiralty Manual of Navigation.  However, it is acknowledged that there are other definitions which state that relative bearings are measured from ‘right ahead’ in a clockwise direction from 0o to 360o.  It is little wonder that there is so much confusion surrounding this topic.

Azimuth is a specific type of bearing which measures the direction of an object in relation to true north, in the horizontal plane, clockwise from 0o to 360o.  For example, in terms of azimuth, due east is 090o and due west is 270o.

Azimuth is measured by use of either a magnetic compass or a gyro compass.  A gyro compass is a form of electrically driven gyroscope which measures azimuth in relation to true north.  Such an azimuth measurement is known as true azimuth.  A magnetic compass employs a magnetised needle which is used to measure the azimuth of an object in relation to magnetic north. Magnetic compass readings must be corrected for variation and deviation in order to convert them to true azimuth.

Azimuth Angle (see diagram above).  Another area of confusion is the difference between azimuth and azimuth angle.  In astro navigation, when we calculate the azimuth of a celestial body, the result is expressed as an azimuth angle.  Azimuth angle is measured from 0o to 180o either westwards or eastwards from either north or south.  If the observer is in the northern hemisphere, the azimuth angle is measured from north and if in the southern hemisphere, it is measured from south.   For example, if the true azimuth of an object is 225o, the azimuth angle for an observer in the northern hemisphere will be N135oW but for an observer in the southern hemisphere, it will be S045oW.

To Convert Azimuth Angle to True Azimuth.  The rules for converting azimuth angle to true azimuth are summarised in the following table:

Rules for converting Azimuth Angle (Z)  to True Azimuth (Zn)

Lat. North                              Lat. South
LHA>180o Zn = Z    Zn =  180o – Z
LHA<180o    Zn = 360o-Z    Zn =  180o + Z

Examples:

If latitude is 55oN, LHA is 145o and azimuth angle is 120o then true azimuth is 360 o – Z  i.e. 360o – 120o = 240o.

If latitude is 25oS, LHA is 245o, and azimuth angle is 075o then true azimuth is 180o – Z i.e. 180o – 075o = 105o.

Where to buy books of the Astro Navigation Demystified series:

Astro Navigation Demystified at Amazon.com

Astro Navigation Demystified at Amazon.uk

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Applying Mathematics to Astro Navigation at Amazon .com

Applying Mathematics to Astro Navigation at Amazon .uk

http://www.amazon.com/Applying-Mathematics-Astro-Navigation-Demystified/dp/1496012062/ref=sr_1_2?s=books&ie=UTF8&qid=1393696809&sr=1-2&keywords=astro+navigation

Astronomy for Astro Navigation at Amazon.com

Astronomy for Astro Navigation at Amazon.uk

http://www.amazon.com/Astronomy-Astro-Navigation-Black-Demystified/dp/1511675594/ref=sr_1_2?s=books&ie=UTF8&qid=1446153840&sr=1-2&keywords=astro+navigation+demystified

Celestial Navigation at Amazon.com

Celestial Navigation at Amazon.uk

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web: http://www.astronavigationdemystified.com

email: astrodemystified@outlook.com

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