The Astro Navigation Resource

See the latest posts on The Moon.

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.

A wealth of iEARTH AND SUN IN THE SPHEREnformation 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.

Russia is one of the few countries in the worlaltitude and azimuthd 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 realPZX TRIANGLE 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 (including the global positioninplot 3g system).

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.

cross

 

Unfortunately, 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.

 

 

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

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

email: astrodemystified@outlook.com

  1. 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.

     

Where to buy books of the Astro Navigation Demystified series:

Astro Navigation Demystified at Amazon.com

Astro Navigation Demystified at Amazon.uk

Applying Mathematics to Astro Navigation at Amazon .com

Applying Mathematics to Astro Navigation at Amazon .uk

Astronomy for Astro Navigation at Amazon.com

Astronomy for Astro Navigation at Amazon.uk

web: http://www.astronavigationdemystified.com

e: astrodemystified@outlook.com

Posted in Uncategorized

Altitude Corrections For The Moon

Continuing the series on the Moon.

When a navigator measures the altitude of the Moon, there are several corrections that he has to make to the readings.

Corrections For The Moon’s Semi-Diameter.  The point on the Moon’s circumference nearest to the horizon is called the lower limb and the point furthest from the horizon is called the upper limb.  When the Moon is not full, sometimes only the upper limb will be visible and sometimes only the lower limb.  From the diagram below it can be seen that sometimes, depending on the phase of the Moon, either the upper or the lower limb cannot be seen.

image057

It should be noted that whether the Moon’s upper or lower limb is visible is dependent not only on its phase but also on the relative altitudes of the Sun and the Moon.  For example, if, one morning, a crescent or gibbous moon is visible in the eastern sky and the Sun is at a higher altitude, only the upper limb will be visible but if, in the evening of the same day, the Moon is visible in the western sky and the Sun has set below the western horizon, only the lower limb will be visible.

In navigational practice, the altitude that we measure is that of the lower limb; however, when the lower limb cannot be seen, we have no choice other than to measure the altitude of the upper limb.   Regardless of which limb we use, what we really need is the altitude of the Moon’s centre so we must either add or subtract the value of its semi-diameter.  The following diagram shows why the semi-diameter must be added when the altitude when the lower limb is measured. Based on the assumption that the lower limb will normally be used for altitude measurements,  30’ is added to the corrections during compilation of the Moon altitude tables to allow for semi-diameter.   Therefore, when the upper limb is used,  the 30’ must be subtracted.

 lower limbAs the Moon travels around its orbit and its distance from the Earth changes, so the value of the visible moon’s semi-diameter will change.  The value of the Moon’s semi-diameter for each day is given in the daily pages of the Nautical Almanac.

 Corrections For Refraction.  When a ray of light from a celestial body passes through the Earth’s atmosphere, it becomes bent through refraction and causes the apparent altitude to be greater than the true altitude.  Since the sextant measures the apparent altitude, a correction for refraction must be applied to find the true altitude.  Refraction is at its greatest when the altitude is small (i.e. when the celestial body is near the horizon) and becomes less as the altitude increases.  The effects of refraction are illustrated in the diagram below.

refraction

O is the observer’s position and Z is the zenith at that point.  OH is the horizon.  XOH is the true altitude of the Moon from the observer’s position.  However, due to refraction, the celestial body appears to be at Y and so YOH becomes the observed altitude and a correction will have to be made to compensate for this.

Corrections For Parallax. We measure the altitude of a celestial body from our position in relation to our visible horizon; this is known as the observed altitude.  However, when calculating the true altitude, measurements are made from the Earth’s centre in relation to the celestial horizon.  The displacement between the observed position of an object and the true position is known as parallax.

 parallax

Parallax corrections for the Moon.  Because the Sun and the Moon are relatively close to the Earth, parallax will be significant and so a correction has to be made.  These corrections are included in the altitude correction tables in the Nautical Almanac.

Horizontal Parallax. Parallax error is greatest when the celestial body is close to the horizon and decreases to zero as the altitude approaches 90o.  It is negligible except in the case of the Moon which is close to the Earth in comparison with the other celestial bodies.  Because horizontal parallax is significant in the case of the Moon, a separate correction has to be applied.  The hourly values of horizontal parallax for the Moon are listed in the daily pages of the Nautical Almanac.

Links:    Lunar Distance       Phases of the Moon     Tidal Effects of the Moon.  

Survival – Finding Direction From The Moon

Where to buy books of the Astro Navigation Demystified series:

Astro Navigation Demystified at Amazon.com

Astro Navigation Demystified at Amazon.uk

Applying Mathematics to Astro Navigation at Amazon .com

Applying Mathematics to Astro Navigation at Amazon .uk

Astronomy for Astro Navigation at Amazon.com

Astronomy for Astro Navigation at Amazon.uk

web: http://www.astronavigationdemystified.com

e: astrodemystified@outlook.com

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

The Tidal Effects of the Moon

Continuing the series on the moon.

If it were not for the gravitational attraction of the Sun and the Moon, the water level of the seas and oceans would be kept at equal levels by a combination of the Earth’s own gravity pulling it inwards and centrifugal force pushing it outwards.  However, the gravitational force of the Moon is strong enough to attract the water towards it and cause a bulge beneath it.  As the Earth rotates and the Moon orbits around it, the bulge follows the Moon causing high tides in its vicinity.

The combined effects of the Earth’s rotation and the Moon’s orbit around it, cause the ‘bulge’ to move around the Earth in 24 hours and 50 minutes and so it would seem at first sight that we would get high tides only at that time interval.  However, other forces are at play.  On the opposite side of the Earth to where the ‘bulge’ occurs, the Moon’s gravitational pull is at its weakest and this allows the Earth’s centrifugal force to push the water outwards and so cause another bulge there giving us two high tides a day.  This means that the time between high tides is approximately 12 hours and 25 minutes and the time between high tide and low tide is 6 hours 12.5 minutes in deep ocean areas. This can change dramatically owing to a variety of factors such as the topography of the ocean floor, local currents, varying water depths and the declination of the Moon.

bulge

The height of the tides vary during the course of a month because the Moon is not always at the same distance from the Earth due to its elliptical orbit.  As the Moon comes closer to the planet, its gravitational pull increases and this leads to higher tide levels.  Likewise, when the Moon’s orbit takes it further away from the Earth, the tides become lower. When the Moon is at its closest distance to the Earth, its gravitational pull increases by as much as 50% and this leads to higher sea levels on Earth.  When it is at its furthest distance, sea levels are much lower,

Tidal Effects of the Sun. The Sun also affects the rise and fall of the tides on Earth.  The gravitational attraction of the Sun pulls the ocean water towards it but at the same time, the effect of the Earth’s rotation around the Sun creates a centrifugal force which pushes the water outwards on the side facing away from the Sun.  The combined effect of these two forces creates a tidal bulge on the side of the Earth facing away from the Sun.  However, this effect of the Sun is less than that created by the Moon which is much closer to the Earth.

Spring Tides.  When the Sun, the Moon and the Earth are in syzygy, that is when they are lined up as during a Full Moon or New Moon, the combined tidal effect of the Sun and Moon is at its greatest and causes what is known as Spring Tides.  This has nothing to do with the season of Spring but to do with the saying that the water ‘springs’ higher than normal.

Neap Tides.  When the directions of the Sun and the Moon in relation to the Earth are at right angles, as during a Half Moon, the combined effects of their gravitational pull is less and so the height of the tides is much lower and are known as Neap Tides.

Links:    Lunar Distance       Phases of the Moon     Altitude Corrections for the Moon.

Survival – Finding Direction From The Moon

Where to buy books of the Astro Navigation Demystified series:

Astro Navigation Demystified at Amazon.com

Astro Navigation Demystified at Amazon.uk

Applying Mathematics to Astro Navigation at Amazon .com

Applying Mathematics to Astro Navigation at Amazon .uk

Astronomy for Astro Navigation at Amazon.com

Astronomy for Astro Navigation at Amazon.uk

web: http://www.astronavigationdemystified.com

e: astrodemystified@outlook.com

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

The Moon

My recent article ‘Finding Direction By The Moon’, sparked off a flurry of correspondence on the topic of the usefulness of this celestial body in navigation and in response, the next few articles will be on this subject.

PHASES OF THE MOON.

The diagram below shows, that as the Moon completes its 27.3 day orbit around the Earth, we see it pass through various phases of illumination.   It goes from New Moon, to Full Moon and back to new Moon again.moon phases

The Phases.

New Moon.  When the illuminated side of the Moon is facing away from the Earth.  The Moon and the Sun are lined up on the same side of the Earth, so we can only see the shadowed side.  During a new moon, we can see the reflected light from the Earth, since no sunlight is falling on the Moon – this is known as ‘earthshine’.

Waxing Crescent – The waxing crescent moon is the first sliver of the Moon that we can see after the new moon. From the northern hemisphere, the crescent moon has the illuminated edge of the Moon on the right. This situation is reversed for the southern hemisphere.  “Waxing” means that the Moon becomes more illuminated night-by-night,

First Quarter – This occurs when the Sun and the Moon make a 90-degree angle compared to the Earth.   Although it’s called a quarter moon, we actually see it as half illuminated.

 Waxing Gibbous – This phase of the Moon occurs when more than half of the Moon is illuminated but it is not yet a full Moon.

Full Moon – This is the phase when the Moon is brightest in the sky. The Moon and the Sun are lined up on opposite sides of the Earth, so from our perspective here on Earth, the Moon is fully illuminated by the light of the Sun.

Waning Gibbous – In this lunar phase, more than half of the Moon is illuminated but it is not yet a full moon. The term “waning” means that it’s getting less illuminated each night.

Last Quarter – At this point of the lunar cycle, the Moon has reached half illumination again. Now it is the left-hand side of the Moon that’s illuminated, and the right-hand side is in darkness (from a northern hemisphere perspective).

Waning Crescent – This is the final sliver of illuminated moon we can see before the Moon goes into darkness again.

Links:    Lunar Distance       Tidal Effects of the Moon.     Altitude corrections for the Moon

Survival – Finding Direction From The Moon

Where to buy books of the Astro Navigation Demystified series:

Astro Navigation Demystified at Amazon.com

Astro Navigation Demystified at Amazon.uk

Applying Mathematics to Astro Navigation at Amazon .com

Applying Mathematics to Astro Navigation at Amazon .uk

Astronomy for Astro Navigation at Amazon.com

Astronomy for Astro Navigation at Amazon.uk

web: http://www.astronavigationdemystified.com

e: astrodemystified@outlook.com

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

Survival – Finding Direction From The Moon.

Just as the Sun can help us to find our general bearings during the day, so can the Moon at night.

Moonrise and Moonset.  As in the case of the Sun, the Moon will rise between north east and south east and will set between north west and south west.  The exact bearing will depend on the observer’s latitude, and the Moon’s declination but in a survival situation, this is not important; all we need to know is that the Moon will rise in the east and set in the west.

MOON TIPSMoon Tips.   A crescent moon can provide us with a quick and easy method of finding the directions of north and south.  If you imagine a line that joins the tips of the crescent and extend this line down to the horizon, it will point roughly south for observers in the northern hemisphere.  In the southern hemisphere, the line will point to the north.

 

SHADOW STICK EW MOONShadow Stick. Sometimes, the Moon will be bright enough to cast a shadow and when this happens, we can use the same shadow stick method that we use for the Sun to find the directions of east and west.

Method

  1. Place a stick in the ground and mark the end of the Moon’s shadow with a stone or some other object.
  2. Wait for half an hour or more and then mark the point that the tip of the shadow has moved to.
  3. Scratch a line in the ground to join the two marks. (If the terrain is not suitable for this or the line cannot be seen clearly, use a piece of string or a straight pole etc.).
  4. Because the Moon moves from east to west, the line that you have made will also point east and west.

 waxing waningWaxing / Waning Moon.   If we are out in the open in a survival situation, we will have plenty of opportunity to study the night sky and so we will probably know if the moon is waxing or waning and this knowledge can help us to find the directions of east and west.

When the Moon is waxing (between the new moon and the full moon) it follows the Sun as it crosses the sky from east to west  and so, as we look at it from the Earth, its western side will face the Sun and will therefore be illuminated.

When the Moon is waning (between the full moon and the new moon) it leads the Sun across the sky and so its eastern side will face the Sun and be illuminated.

The Pointing Method. The pointing method that we use with the Sun to keep us on course during the day can be used with the Moon during the night.  The method is explained below:

  1. Face the direction in which you plan to travel and point in the direction of the Moon.
  2. Hold this position for a few seconds until the Moon’s direction relative to your body is imprinted in your mind.
  3. Set off on your chosen course keeping the Moon in the same relative position.
  4. Stop every ten to fifteen minutes to reacquaint yourself with the Moon’s relative position.

Survival Links:  Astro Navigation in a survival situation.  Latitude from the midday Sun. Find your longitude.   Latitude from the North Star  Calculating declination.   Declination table.    The Survival Sundial

Where to buy books of the Astro Navigation Demystified series:

Astro Navigation Demystified at Amazon.com

Astro Navigation Demystified at Amazon.uk

Applying Mathematics to Astro Navigation at Amazon .com

Applying Mathematics to Astro Navigation at Amazon .uk

Astronomy for Astro Navigation at Amazon.com

Astronomy for Astro Navigation at Amazon.uk

web: http://www.astronavigationdemystified.com

e: astrodemystified@outlook.com

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

Survival – The Daytime Star

Just as the stars can be your compass at night, there is another star to guide you during the day and that of course is the Sun.

Sunrise/Sunset.   We are taught that the Sun rises in the East and sets in the West; well that is true on only two days of the year, the Vernal and Autumnal Equinoxes.  During the rest of the year, sunrise will be between north east and south east while sunset will be between north west and south west; however, all we really need to know in a survival situation, is that sunrise is approximately east and sunset is approximately west.

Between Sunrise and Sunset.  Fine, we can find east and west at dawn and dusk and from there we can align ourselves with north and south but how do we manage between these times?  Well, there are several methods that we can use depending on whether we are on the march or we have time to stop.

The Watch Compass  If you are lucky enough to have an analogue watch, this can serve as a substitute compass.

watch compassMethod.

  1. Make sure the watch shows the time for your locality (standard time if on land or zone time if at sea).
  2. Point the hour hand towards the Sun.
  3. Imagine a line from the centre of the watch face to the 12 o’clock position and then imagine an angle between that line and the hour hand as shown in the diagram. When in Daylight Saving Time (Summer Time) use the 1 o’clock position instead of 12 o’clock).
  4. Next imagine a line bisecting this angle as represented by the dotted line in the diagram,
  5. If you are in the Northern Hemisphere, the line bisecting the small angle will point towards the South and the line bisecting the large angle will point towards the North. If in the Southern Hemisphere, it will be the other way round.

Notes.

  1. This method is based on the ‘True Sun’ (the Sun we see in the sky). However, clock time (mean time or standard time) does not keep in step with the True Sun and so the watch compass will not always exactly indicate true south or true north but it is close enough for survival purposes.
  2. If you are in the tropical zone (23.5o north to 23.5o south), you may have difficulty with this method because the Sun will sometimes be to the north, sometimes to the south and sometimes, exactly over your latitude.
  3. Never look directly at the Sun as this can seriously damage your eyes. A piece of smoked glass can easily be made for use as a light filter; simply hold a piece of glass in the smoke of a candle or oil lamp until it is covered in a layer of smoke residue.

Shadow Stick  If you do not have an analogue watch, you can still use the Sun to find your direction as long as you have time to stop in one place for a while and if there is sufficient sunlight to make a shadow.

East / West Shadow Stick Method.

SHADOW STICK EWMOD

Method

  1. Place a stick in the ground and mark the end of the Sun’s shadow with a stone or some other object.
  2. Wait for half an hour or more and then mark the point that the tip of the shadow has moved to.
  3. Scratch a line in the ground to join the two marks. (If the terrain is not suitable for this, use a piece of string or a straight pole etc.).
  4. Because the Sun moves from east to west at 15o per hour, the line that you have drawn will also point east and west.

North / South Shadow Stick Method  At noon, the True Sun will be over the meridian of longitude of your position and will have reached its highest altitude for the day.  Because of this, shadows cast by the Sun at this time will lie from north to south and so indicate those directions for us.

SHADOW STICK NS

Method

  1. Judge when the Sun is approaching the highest point of its path across the sky.
  2. Place a stick in the ground and mark the tip of its shadow.
  3. Draw a line from the base of the stick to the tip of the shadow.
  4. About 5 minutes later, mark the point that the tip of the shadow has moved to and draw a line along the length of the shadow as before.
  5. Repeat this process several times. The shadow will become shorter as the Sun approaches its highest altitude at noon and then it will begin to get longer again.
  6. The shortest shadow occurs when the Sun is at its highest altitude and this line will lie north/south.
  7. In the Northern Hemisphere, the end of the shadow nearest the stick will point to the South at noon and in the Southern Hemisphere, it will point to the North.

Notes

  1. Never look directly at the Sun as this can seriously damage your eyes.
  2. Again, there may be difficulties with this method in the tropics.
  3. If you have a watch but you are not sure of the time, you can set it at noon as indicated by the shortest shadow.

 Survival Sundial.  If you are camped in one place for a day or more, you can make a survival sundial which will not only serve as a clock but also as a compass.  Click here to learn more about the survival sundial.

The Pointing Method.  If you do not have time to stop and you do not have an analogue watch,  you can always use the ‘pointing’ method to keep you on course.

Method.

  1. Face the direction in which you plan to travel and point in the direction of the Sun.
  2. Hold this position for a few seconds until the Sun’s direction relative to your body is imprinted in your mind.
  3. Set off on your chosen course keeping the Sun in the same relative position.
  4. Stop every ten to fifteen minutes to reacquaint yourself with the Sun’s relative position, bearing in mind that it covers 15o per hour as it makes its arc across the sky.

Notes.

  1. If the Sun is obscured by cloud, you can adapt this method by using a tall object such as a church spire or mountain to point at.
  2. You can also adapt the method by noting the direction of your shadow instead of the position of the Sun.

Survival Links:  Astro Navigation in a survival situation.  Latitude from the midday Sun. Find your longitude.   Latitude from the North Star  Calculating declination.    Declination table.    The Survival Sundial

Where to buy books of the Astro Navigation Demystified series:

Astro Navigation Demystified at Amazon.com

Astro Navigation Demystified at Amazon.uk

Applying Mathematics to Astro Navigation at Amazon .com

Applying Mathematics to Astro Navigation at Amazon .uk

Astronomy for Astro Navigation at Amazon.com

Astronomy for Astro Navigation at Amazon.uk

web: http://www.astronavigationdemystified.com

e: astrodemystified@outlook.com

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

Survival – The Star Compass

“Know The Stars And You Will Always Have A Compass”.

dreamstime_m_5648294In a survival situation, whether at sea or on land, the chances are you may have nothing to navigate by other than the stars in the sky.

Finding the Direction of North.    The Pole Star (otherwise known by various names including Polaris, North Star, Lodestar and the Guiding Star).  As the Earth rotates, the Pole Star, which is almost exactly in line with the Earth’s celestial north pole, does not change its position in the sky unlike the other visible stars. For this reason, it will always indicate the direction of north.  The trick is to find the Pole Star in the sky and for this we need the help of the following constellations of stars.

 Ursa Major (also knowursa major separten as The Big Dipper, the Plough or The Great Bear).  The best known and easily recognizable constellation in the northern hemisphere is the constellation Ursa Major which is also known by various other names such as the Big Dipper and the Plough.  Ursa Major is a circumpolar constellation which means that it rotates around the celestial north pole and never sets below the horizon.  It is visible all year round in the northern hemisphere and in northern regions of the southern hemisphere.

UrsaUrsa minor Minor (also known as the Little Dipper or the Little Bear) contains Polaris, the Pole Star.  Ursa Minor is also a circumpolar constellation and it can be seen throughout the northern hemisphere and as far south as 10oS.

Using Ursa Major and Minor to find the Pole Star.  Ursa major and minor2As illustrated in the diagram below, Ursa Major contains a reference line known as the line of pointers.  The line joining Merak to Dubhe, when extended, will point to Polaris (the Pole Star) which is in the constellation Ursa Minor.  Polaris is not a particularly bright star although it is the brightest star in Ursa Minor.

cassiopeia aloneCassiopeia. The Queen.  Cassiopeia is another circumpolar constellation; it is quite easy to find because of its ‘W’ shape which sometimes hangs upside down as it circles the north celestial pole.  It can be observed throughout the northern hemisphere and down to 20oS.

The star Segin which lies in Cassiopeia can be located along a line of reference from the Pole Star at an angle of 135o to the line of pointers in Ursa Major as the diagram below shows.  As Ursa Major revolves around the Pole Star, so do the five stars of Cassiopeia with Segin always keeping its position 135o from the line of pointers.  Knowing that the  Pole Star always lies between these two constellations provides us with a further way of finding it.

bear to cassiopeia updateFinding The Direction Of South.   The Southern Cross (the constellation Crux)Crux (Latin for cross) it is one of the smallest constellations in the sky but also one of the brightest.  It is not visible north of 20°N in the northern hemisphere but it is circumpolar in the southern hemisphere south of 34°S which means that it never sets below the horizon there.

Crux has four main stars which mark the tips that form the ‘Southern Cross’:

Acrux, the brightest star in the cross.crux

Becrux, the second brightest.

Gacrux, the third brightest.

Palida has variable levels of brightness.

 

 

finding crux with pointers

 

How to find the Southern Cross.  The constellation Centaurus contains two bright stars which make excellent pointers to help us find the Southern Cross.  The Pointers as they are known, are Rigil Kentaurus and Hadar.

 

Finding south by using the Southern Cross.  Whereas the Pole Star coincides with the position

pointerof the north celestial pole, Crux does not coincide with the celestial south pole so we have to rely on other methods of using it to find the direction of South.  There are several methods but the simplest is as follows: Make an imaginary line between Gacrux and Acrux then extend this line from Acrux (the brightest star) for 4.5 times the length of the Southern Cross, as shown in the diagram below. This will take you to the position of the South Celestial Pole in the sky.  From the South Celestial Pole, drop a line down to the horizon. Where this line touches the horizon is the direction of south.

 

Finding The Directions Of East And West. There will be times when neither the Pole Star nor the Southern Cross can be seen for various reasons.  However, this need not be a problem, for if we can find east or west, we can find north and south.  Fortunately, there are at least two constellations that can help us in this respect.

Orion, The Hunter  This easily recognized constellation straddles the celestial equator and for this reason, it always rises in the East and sets in the West.  The stars Alnitak, Alnilam and Mintaka form the belt of the hunter and are easy to find.  Alnilam which is the middle star of the belt and the brightest of the three is almost exactly on the Celestial Equator so it will always rise atorion due east and set at due west.  The bright red star Betelgeuse will rise first and this will give you a warning when Alnilam is about to rise.

After Orion has risen  it moves across the sky in an westerly direction following the Celestial Equator and so, by noting its position at intervals, we can gauge the direction in which it is moving and so find east and west.

Orion is a Winter Constellation.  This means that it is only visible in the night sky during the northern hemisphere’s winter months (summer in the southern hemisphere).  During summer in the northern hemisphere, it is above the horizon during daylight hours so we cannot see it.  However, all is not lost; we have a summer constellation which can help us to find east and west when ‘Orion is asleep’.

Aquila,aquila The Eagle. Like Orion, Aquila sits astride the celestial equator and its brightest star, Altair, rises very slightly north of due east and sets just north of due west.  During the northern hemisphere’s summer, Altair takes the place of Alnilam and becomes our guiding star to the directions of east and west.

 

summer triangle

 

How to find Altair.  Together with Altair, the stars Deneb in the constellation Cygnus, and Vega in the constellation Lyra form an astronomical asterism known as the ‘Summer Triangle’ which is formed by imaginary lines drawn between those stars as shown in the diagram opposite.  As well as being an important navigation aid in its own right, the Summer Triangle helps us to easily find Altair which can, in turn, help us to find the directions of east and west.

 Reference for the quotation “Know the stars and you will always have a compass”: Michael Punk. 2002. The Revenant

Survival Links:  Astro Navigation in a survival situation.  Latitude from the midday Sun.   Find your longitude.   Latitude from the North Star  Calculating declination.    Declination table.    The Survival Sundial

Where to buy books of the Astro Navigation Demystified series:

Astro Navigation Demystified at Amazon.com

Astro Navigation Demystified at Amazon.uk

Applying Mathematics to Astro Navigation at Amazon .com

Applying Mathematics to Astro Navigation at Amazon .uk

Astronomy for Astro Navigation at Amazon.com

Astronomy for Astro Navigation at Amazon.uk

web: http://www.astronavigationdemystified.com

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The Demystified Astro Navigation Course Unit 6

UNIT 6 –  Calculating zenith distance and azimuth at assumed position.

We can use sight reduction tables to calculate the zenith distance and azimuth at the assumed position or else we can use the traditional method of making the calculations by spherical trigonometry.  The sight reduction method, as well as being less accurate, is quite involved and requires the provision of expensive sets of tables.  Alternatively, spherical trigonometry provides a more accurate and inexpensive method which requires only a calculator and little mathematical ability.

In this unit, we will focus on the spherical trigonometry method but an explanation of the sight reduction method can be found in ‘Astro Navigation Demystified’, the parent to this book.

Calculating zenith distance and azimuth by spherical Trigonometry. At first sight, the spherical trigonometry method might seem quite daunting and difficult but with the knowledge of just two formulas and with a little practice of the procedures explained below, it will be found to be quick and easy.

Essentially, we need to know only two formulas which are are explained below.

 To Calculate Zenith Distance (ZX).  The formula for calculating side ZX is:

Cos (ZX) =  [Cos(PZ) . Cos(PX)] + [Sin(PZ) . Sin(PX) . Cos(ZPX)]

To Calculate Azimuth (PZX)  The formula for calculating angle PZX is:

Cos PZX = Cos(PX) – [Cos(ZX) . Cos(PZ] / [Sin(ZX) . Sin(PZ)]

The use of these formulas will become clear if we study the example below which shows how the zenith distance and azimuth of a celestial body can be calculated by the use of the above formulas.

Example.   Assume that we have measured the altitude and azimuth of the star Alioth from our true position and have calculated that the intercept there is 1618.54 n.m. and that we can now wish to calculate what the intercept would be from our assumed position.

Celestial body:  The star Alioth

Assumed Position:  Lat. 30oN    Long. 45oW

Data for Alioth taken from Nautical Almanac daily pages:

SHA: 166    Declination: 56oN   GHA Aries:  250

Note.  Refer to the PZX diagram in unit 4 to identify PZX. PX, PZ and ZX in the calculations below.

 Step 1.  Calculate LHA

SHA Alioth +  GHA Aries = 166+  250 = 416

Apply Long :         416 -45 =  371

If greater that 360 then subtract 360:     371-360  = 11

∴ LHA  = 11

∴ ZPX = LHA = 11o

Step 2. Calculate PZ/PX

PZ = 90o – Lat.  = 90o – 30o = 60o

PX = 90o – Dec. = 90o – 56o = 34o

Step 3.  Calculate Zenith Distance (ZX). Use the following formula to calculate ZX:

Cos (ZX) =  [Cos(PZ) . Cos(PX)] + [Sin(PZ) . Sin(PX) . Cos(ZPX)]

To calculate zenith distance of Alioth:

Cos (ZX) =  [Cos(PZ) . Cos(PX)] + [Sin(PZ) . Sin(PX) . Cos(ZPX)]

=  [Cos(60o) . Cos(34o)] + [Sin(60o) . Sin(34o) . Cos(11o)]

=  [0.5 x 0.829} + [0.866 x 0.559 x 0.982]

=  0.415 + 0.475

Cos (ZX) =  0.89

∴ ZX      =  Cos-1 (0.89)

=  27o.13   Therefore zenith distance at assumed position =  27o.13

 = 1627.8′    = 1627.8 nm

 Step 4.  Calculate Altitude.

Altitude  = 90o – ZX

= 90o – 27o.13          = 62o.83

Step 5.  Calculate Azimuth (PZX)

The formula for calculating angle PZX is:

Cos PZX = Cos(PX) – [Cos(ZX) . Cos(PZ] /[Sin(ZX) . Sin(PZ)]

To calculate azimuth of Alioth:

= Cos(34) – [Cos(27.13) . Cos(60]/ [Sin(27.13) . Sin(60)]

= 0.829 – [ 0.89 x 0.5] / 0.456 x 0.866

=  0.829 – 0.445 / 0.394

=  0.384 /  0.394

∴ Cos(PZX) = 0.975

∴  PZX   =  Cos-1(0.975)  = 12.8  ≈ 13

Note. Azimuth is west if LHA is less than 180 o. Azimuth is east if LHA is greater than 180o.

Therefore, azimuth is N13oW.  In terms of bearing this is 347o (360o – Az.)

 Step 6.  Calculate Intercept.

Zenith Distance at assumed position  = 1627.8 nm

Zenith Distance at true position  = 1618.54 nm.

Therefore intercept = 1627.8 – 1618.54  = 9.26 nm.

Note. Since true position is closest to the GP, the intercept must be 9.26 nm away from the assumed position towards the azimuth (i.e. 347 o).

Therefore,  Alioth intercept = 1.46 nm from the assumed position towards 347 o.

A thorough treatment of this topic can be found in  ‘Astro Navigation Demystified

Where to buy books of the Astro Navigation Demystified series:

Astro Navigation Demystified at Amazon.com

Astro Navigation Demystified at Amazon.uk

Applying Mathematics to Astro Navigation at Amazon .com

Applying Mathematics to Astro Navigation at Amazon .uk

Astronomy for Astro Navigation at Amazon.com

Astronomy for Astro Navigation at Amazon.uk

web: http://www.astronavigationdemystified.com

e: astrodemystified@outlook.com

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The Demystified Astro Navigation Course – Unit 5

Unit 5 – Demonstration of a 3 point fix.

Aim.  We saw in unit 4 how a position line is obtained from a single intercept; the aim of this demonstration is to establish a 3 point fix based on the intersection of three intercepts obtained from observations of three celestial bodies as detailed below.  For the purpose of clarity, the calculations for the zenith distances and azimuths of the three bodies at the assumed position are not included at this stage. However, the next unit will explain how such calculations are made.

 Scenario:  A yacht is sailing on a course of 090  at roughly 5 knots.   Observations of 3 celestial bodies are made in order to obtain a three point fix. Of course it is not possible to take sextant readings of three celestial bodies at exactly the same time and you will see from the data below that there are small time intervals between each reading.  You will also see, from the plot diagram, that the three intercepts are drawn from separate positions along the yacht’s course (i.e. AP1,AP2,AP3).

Relevant Data.

Assumed Position:  4752’N, 4735’W.

Index error:  + 0’.53

Ht. of eye:  5.8m.  Resultant Dip: -4′.2

Celestial Bodies Chosen:

Moon’s lower limb,  Saturn,  Alphecca

Sextant Altitudes at true position:

Moon’s lower limb = 35o 28’.23

Saturn = 41o 48’.73

Alphecca = 54o 58’.06

Corrections to sextant altitudes (altitude tables not shown here but can be seen in unit 3):

                                                  Moon              Saturn           Alphecca          
GMT of sextant reading     23h 48m 08s              23h 49m 37s        23h 50m 50s
Sext Altitude                          34o 29’.2               41o 53’.50            55o 02’.43
I.E.                                                 +0’.53                  +0’.53                  +0’.53
Dip (5.8m)                                    -4’.2                      -4’.2                      -4’.2  .
Apparent Alttitude              34o 25’.53              41o 49’.83            54o 58’.76
Altitude Correction                                                    -1’.1                       -0’.7
Moon Altitude Correction     +56’.8
(Moon HP:58’.5)
Moon’s HP Correction              +5’.9
Observed (True) Altitude   35o 28’.23             41o 48’.73              54o 58’.06

Calculating Zenith Distances at the True Position

Zenith distance of Moon = 90o – 35o 28’.23 = 3271′.77 = 3271.77nm

Zenith distance of Saturn = 90o – 41o 48’.73 = 2891′.27 = 2891.27nm

Zenith distance of Alphecca = 90o – 54o 58’.06 = 2101′.94 = 2101.94nm

Calculated Altitudes at the Assumed Position.

Moon’s lower limb = 35o 50’

Saturn = 42o 03’

Alphecca = 55o 26’

Calculated Zenith Distances And Azimuths At The Assumed Position.

Zenith Distance of Moon = 3250nm.  Azimuth = 232

Zenith Distance of Saturn = 2877nm. Azimuth = 223

Zenith Distance of Alpheca = 2074nm. Azimuth = 116o

Calculating the Intercepts

Moon Intercept = 3271.77 – 3250 = 21.77nm away from 232o

Saturn Intercept = 2891.27 – 2877 = 14.27nm away from 223o

Alphecca Intercept = 2101.94 -2074 = 27.94nm away from 116o

 Plotting the fix.  The next step is to plot the fix on the chart.  Because of the small errors inherent in astro navigation techniques, the three position lines will very rarely cross at one precise point.  Usually, a small triangle known as a ‘cocked-hat’ will be produced and as long as the triangle is not too large, it can safely be assumed that the ship’s position is at the centre of this triangle.  The diagram shows how the fix would be plotted on a chart.

image025

Note.  Where position lines are derived from astronomical observations, the resultant position is not known as a ‘fix’ but is known as an observed position and is marked on the chart as ‘Obs’.

A thorough treatment of this topic can be found in  ‘Astro Navigation Demystified

Where to buy books of the Astro Navigation Demystified series:

Astro Navigation Demystified at Amazon.com

Astro Navigation Demystified at Amazon.uk

Applying Mathematics to Astro Navigation at Amazon .com

Applying Mathematics to Astro Navigation at Amazon .uk

Astronomy for Astro Navigation at Amazon.com

Astronomy for Astro Navigation at Amazon.uk

web: http://www.astronavigationdemystified.com

e: astrodemystified@outlook.com

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The Demystified Astro Navigation Course – Unit 4

Unit 4 – The Importance of Altitude, Azimuth and Zenith Distance in Astro Navigation.

 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; that is, it is the angular distance ZX in the diagram below.

PZX TRIANGLE 

The relationship between zenith distance and the distance from the observer to the GP.   Position A in the diagram is the position of the observer and U is the geographical position of the celestial body (GP).  It can be seen that the angular distance of the arc ZX is equal to the arc AU and given that 1 arc second is equal to a distance of 1 nautical mile on the Earth’s surface, it can be concluded that the angular distance ZX is equal to the distance from the observer to the GP in nautical miles.  (An explanation of the relationship between the nautical mile and angular distance can be found in the book ‘Astro Navigation Demystified’)

 Relationship between Altitude and Zenith Distance    The following is given without explanation:

Zenith Distance = 90o – Altitude

Altitude = 90o – Zenith Distance

From this, it follows that by measuring the altitude of a celestial body, we can easily calculate the zenith distance and hence the distance to the GP.  (A detailed explanation of the derivation of the relationship given above can be found in ‘Astro Navigation Demystified’).

Azimuth.  The angle PZX in the diagram 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 Astro Navigation.  The preceding discussions illustrate the importance of altitude and azimuth in astro 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 astro navigation.

 Establishing A Position Line From The Altitude And Azimuth.  Suppose we are in a yacht and we measure the altitude of the Sun and find it to be 35o; what does this tell us?  All that we know is that the yacht lies somewhere on the circumference of a circle centered at the geographical position of the Sun.  Such a circle is known as a ‘position circle’ since our position is known to lie somewhere on its circumference.  At any point on the circumference of the circle, the altitude of the Sun will be 35o and our distance from the GP will be equal to the radius of the position circle.  The problem is to establish at which precise point on the position circle the yacht lies.

At first, it might seem that all we need to do is to observe the azimuth of the Sun at the same time that we measure its altitude and then draw the line of bearing on the chart along with the position circle.  In this way, it would seem that our true position would correspond to the intersection of these lines on the chart.  However, there is a problem with this idea which makes it impracticable.  Because of the great distance of the Sun from the Earth, the radius of the position circle will be very large (approximately 3000 n.m. or so).  A chart on which such a large circle could be drawn would require such a small scale that accurate position-fixing would be impossible.  However, we know our dead-reckoning position (DR) which, although approximate, should be accurate to within a degree of latitude and longitude and this may give us another way of tackling the problem.

We know that altitude minus 90o gives us the zenith distance and that from this we can calculate the distance of our position from the GP of the celestial body.  Now, if we could work out what the altitude would have been at the D.R. position (or assumed position) at the time that the altitude was measured at the true position, we would then be able compare their respective zenith distances and so calculate the distance between them.

Example.    Calculating the Zenith Distance at the True Position

Suppose the altitude of the Sun, as measured at the true position, is 67o.85   Using this information, the calculation for finding the zenith distance at the true position would be as shown below:

Altitude of celestial body = 67o.858

Zenith Distance = 90o – Alt

= 90o – 67o.858

= 22o.142  = 1328’.52

= 1328.52n.m.

Calculating the zenith distance of the assumed position.  There are a number of methods of calculating the zenith distance at the assumed position and these include sight reduction and spherical trigonometry.  Both of these methods are explained in detail in ‘Astro Navigation Demystified’.  Suffice it to say that for this example we have calculated that the altitude at the assumed position is 67o.972 giving a zenith distance of 1321’.68 as calculated below.

Altitude of celestial body = 67o.972

Zenith Distance = 90o – Alt

= 90o – 67o.972

=  22o.028  = 1321’.68

= 1321.68n.m.

Summary.  From the above calculations, we have established that the zenith distance of the true position is 1328.52 n.m.  We have also established that the zenith distance of the assumed position is 1321.68 n.m.

To continue, we know that the assumed position lies somewhere on a position circle of radius 1321.68 n.m. from the GP.  We also know that the true position lies somewhere on a position circle of radius 1328.52n.m. from the GP.   The difference between the two position circles is 6.84 nm and this is known as the ‘intercept’.  The intercept is drawn as a straight line from the assumed position along the line of the azimuth.  Since the true position is further away from the GP than the assumed position, the intercept is drawn towards the reciprocal of the azimuth.  If the azimuth is N135oE (bearing 135o) then the intercept will be drawn towards N045oW (bearing 315o) as shown in the following diagram.

position line2 

Of course, a single position line does not constitute a fixed position; the intersection of at least two and preferably three position lines would be necessary to achieve this.  ‘Astro Navigation Demystified’ explains the three point fix and also explains other methods including the Marcq St. Hilaire’ and the meridian passage methods.

(A thorough treatment of this topic can be found in the book Astro Navigation Demystified)

Notes.    A brief explanation of the three point fix will be given in unit 5 of this series.

Units of this course are listed in numerical order under ‘Astro Nav Course’ on the menu bar.

Where to buy books of the Astro Navigation Demystified series:

Astro Navigation Demystified at Amazon.com

Astro Navigation Demystified at Amazon.uk

Applying Mathematics to Astro Navigation at Amazon .com

Applying Mathematics to Astro Navigation at Amazon .uk

Astronomy for Astro Navigation at Amazon.com

Astronomy for Astro Navigation at Amazon.uk

web: http://www.astronavigationdemystified.com

e: astrodemystified@outlook.com

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