The Astro Navigation Resource

See the latest post – “The Star Compass”

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

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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

e: astrodemystified@outlook.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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The Demystified Astro Navigation Course Unit 3 Part 3

Unit 3 Part 3 – Altitude Correction tables for the Moon

Note.  Units of this course can be viewed in numerical order under ‘Astro Nav Course’ on the menu bar.

 The Moon’s Semi-Diameter.  When the Moon is not full, sometimes only the upper limb will be visible and sometimes only the lower limb.  In such cases, we have no choice other than to measure the altitude of the limb that we can see and to either add or subtract the semi-diameter accordingly.

When the upper limb is used, 30’ must be subtracted.  This is because 30’ is added to the upper limb corrections during compilation of the tables to keep the value small and positive.

 Parralax corrections for the Moon. Because the Moon is 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 and therefore do not have to be applied separately.

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 (abbreviated to HP) has to be applied.

The hourly values for HP are given in the daily pages of the Nautical Almanac as shown in the extract below:

na_2009-249 copycropped2

Altitude Correction Tables for the Moon There are two correction tables for the Moon in the Nautical Almanac.  One is for apparent altitudes of 0o to 35o and the other is for 35o to 90o.  You will see from the extract below, that the altitude correction table is divided into two parts.  The top part of the table is entered with the apparent altitude and the correction is read from the appropriate column.

The lower part of the table is for horizontal parallax (HP) and has different columns for lower limb and upper limb.  This part of the table is entered with the value of HP (which is obtained from the daily pages of the nautical almanac) and the correction is read from the same column as that for the apparent altitude correction above.

na_2009_moonaltcorrn_pages1 copy

Example:   At 0200 GMT on 24 December, the sextant altitude of the Moon’s upper limb is measured and the following data is obtained:

Sextant Alt. = 33o 15’.0

I.E. = +1’.8

Ht. of eye = 18 ft.

H.P. = 54’.9

Using the extracts given above, the true altitude is calculated as follows:

Sext. Alt.          33o 15’.0

I.E.                            + 1’.8

Obs. Alt. =         33o 16’.8

Dip.                           – 4’.1

App. Alt.  =        33o 12’.7

Corr. (1)                +  57’.4  (for App. Alt.)

Corr. (2)                   + 2’.1 (for HP upper limb)

=                           34o 12’.2

.                                – 30’.0 (for Upper Limb.)

True Alt.  =       33o 42’.2

Test Question.  On 6 June. Sextant altitude of Moon’s lower limb: 31o 32’.8.  Index error: +2’.4.  Ht. of eye: 15m.  HP: 59’.7.  Weather conditions standard.  What is the true altitude?

Solution

Sext. Alt.          31o 32’.8

I.E.                           + 2’.4

Obs. Alt.  =      31o 35’.2

Dip.                           – 6’.8

App. Alt. =       31o 28’.4

Corr. (1)                + 58’.2  (for App. Alt.)

Corr. (2)                  + 7’.2  (for H.P.)

True Alt. =       32o 33’.8

 (Note. This topic is covered in greater depth in the book ‘Astro Navigation Demystified’).

Unit 4 will explain the importance of altitude and azimuth in astro navigation.

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 3 Part 2

Unit 3 Part 2 – Altitude Correction tables for the Sun, Stars and Planets.

As shown in the table extract below, corrections for the Sun are divided into two parts to allow for changes in the Sun’s semi-diameter during the course of a year.  The first part is for the period October to March and is based on a semi-diameter of 16’.15.  The second part is for April to September and is based on a semi-diameter of 15’.9.

na_2009-A2-A4-A2 cropped

Example:

Date = 20 May

Apparent altitude of the Sun’s lower limb = 10o 47’.0

Correction = +11’.1 (interpolating as necessary).

Full example of altitude corrections for the Sun.  The following example shows how altitude corrections for index error, dip, semi-diameter, refraction and parallax are applied to a sextant altitude of the Sun:

On 12 Feb. a reading of the Sun’s lower limb was taken.

Sextant altitude = 14o 35’.5.  Index error = -2’.3.  Height of eye = 4.2m.

Temperature = 20oC.  Pressure = 1020 mb.

Using the extracts of tables A2 above and A4 below, the calculations for the True Altitude would be as follows:

Sext. Alt.            14o 35’.5

I.E.                            -2’.3

Observed Alt.     14o 33’.2

Dip                            -3’.6

Apparent Alt.      14o 29’.6

Sun’s correction       +12’.6  (combined correction)

True Alt.             14o 42’.2

If the upper limb had been used instead of the lower limb, the result would be:

Sext. Alt.           14o 35’.5

I.E.                            -2’.3

Observed Alt.     14o 33’.2

Dip                             -3’.6

Apparent Alt.      14o 29’.6

Sun’s correction       -19’.7

True Alt.              14o 09’.9

Corrections for the stars and planets are listed in only one column since semi-diameter corrections are not necessary.  However, a column of additional corrections for Venus and Mars is provided for use when extreme accuracy is required.

Example:

Corrections for Altitude of a Star:  Using the following data and the extracts from table A2 and table A4 on the preceding pages, calculate the true altitude of a star.

Data:  Sextant altitude = 12o 20’.4     I.E = +1’.8      Ht. of eye = 4.8m.

Temperature = -20oC.   Pressure = 1010 mb.

(Remember, the correction for a star consists of index error, dip and refraction only since parallax and semi-diameter are negligible).

Solution:

Sext. Alt.           12o 20’.4

I.E.                            +1’.8

Observed Alt.     12o 22’.2

Dip                            -3’.9

Apparent Alt.      12o 18’.3

Star’s correction      -4’.3

True Alt.             12o 14’.0

 

Altitude Correction Tables for the Planets. 

  • As in the case of the stars, because they are so far from the Earth, parallax and semi-diameter arguments for the planets are negligible and so the only corrections necessary are for dip and refraction.
  • For normal navigational practices, the navigational planets are treated as stars and the correction table for stars is used.
  • For the very rare cases that they are needed for extreme accuracy, additional corrections for phase and parallax for Venus and parallax for Mars are given in the Nautical Almanac. 

Example:  Corrections for Altitude of a Planet.

Using the following data and the extracts from table A2 table A4 on the preceding pages, calculate the true altitude of a planet.

Sextant altitude = 10o 38’.6      I.E. =  +2’.4      Ht. of eye = 13.0 ft.

Temperature = 10oC.  Pressure = 1000 mb.

Solution:

Sext. Alt.        10o 38’.6

I.E.                       +2’.4

Obs. Alt.          10o 41’.0

Dip                        -3’.5

App. Alt.          10o 37’.5

Plnt’s corr.             -5’.0

True Alt.          10o 32’.5

Additional Corrections  An additional correction for refraction may be needed if the temperature and atmospheric pressure are greatly different to the standard conditions which are assumed to be 10oC, 1010mb.  Part of Table A4 from the Nautical Almanac is shown in the extract below.  This table tabulates the additional corrections for non standard conditions:

table6c

The graph above the table is entered with the temperature and pressure to find a zone letter.   For example, entering with a temperature of 20oC. and a pressure of 1010mb. gives zone letter J.

The table is then entered with the apparent altitude and zone letter to find the additional correction for refraction.

Example.  If apparent altitude = 4o 30’, temperature = 30 oC, pressure = 1000mb, the zone letter will be L and the correction will be +0’.8.

Self Test  Using the Nautical Almanac extracts on the given above, answer these questions:

Question 1.

On 12 June, a reading of the Sun’s lower limb was taken.

Sextant altitude: 50o 45’.2.  Index error: +1’.8. Height of eye: 24 ft.

Temperature: 25oC.  Pressure: 1000mb.

What was the true altitude?

Question 2.

Sextant altitude of Sirius: 18o 08.’5.  Index error:  +2’.1;  Ht. of eye: 12m.

Temperature: -10oC.  Pressure: 980mb.

What is the true altitude?

Question 3.

Sextant altitude of Venus:  17o 43’.3.  Index error: -1’.4.  Ht. of eye: 15 ft.

Temperature: 8oC.  Pressure: 1025mb.

What is the true altitude?

Solutions

Q1.

Sext. Alt.            50o 45’.2

I.E.                             +1’.8

Observed Alt.    50o 47’.0

Dip                             -4’.8

Apparent Alt.     50o 42’.2

Sun’s Corr.                +15’.2

Add’nl. Corr.               0’.0

True Alt.             50o 57’.4

 

Q2.

Sext. Alt.            18o 08’.5

I.E.                             +2.’1

Obs. Alt.              18o 10.’6

Dip.                             -6’.1

App. Alt.             18o 04’.5

Star’s Corr.               -2’.9

Add’nl. Corr.              -0’.2

True Alt.             18o 01’.4

 

Q.3.

Sext. Alt.          17o 43’.3

I.E.                            -1’.4

Obs. Alt.             17o 41’.9

Dip                            -3’.8

App. Alt.             17o 38’.1

Plnt’s Corr.                -3’.0

Add’nl. Corr.              -0’.1

True Alt.             17o 35’.0

Altitude correction tables for the Moon will be explained in Unit 3 Part 3.

(Note. This topic is covered in greater depth in the book ‘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 3 Part 1

Unit 3  Part 1 – Altitude and Azimuth 

 altitude and azimuth 

The Azimuth is similar to the bearing in that it is the angle between the observer’s meridian and the direction of the celestial body.  However, whereas bearings are measured clockwise from north from 0o to 360o, azimuth is measured from 0o to 180o from either north or south.  If the observer is in the northern hemisphere, the azimuth is measured from north and if in the southern hemisphere, it is measured from south.

If for example, the bearing of a celestial body is 045o , in terms of azimuth it is either N45oE for an observer is in the northern hemisphere or S135oE for an observer in the southern hemisphere.  The diagram below illustrates this.

 diag 21 nonum

Altitude.  The altitude of a celestial body is the angular distance between its position in the celestial sphere and the celestial horizon as measured at the observer’s position.

Measuring the Altitude.  To measure the altitude of a celestial body, we use a sextant.

diag 25modAs shown in the diagram, the horizon is viewed directly through the sextant telescope and the celestial body is viewed via two mirrors. The upper mirror is attached to the index bar.  The index bar is moved until it reflects an image of the celestial body into the lower mirror which is fixed.  The position of the index bar is finely adjusted until the image of the celestial body appears to sit on the horizon.  As the index bar is adjusted, it moves a pointer over a graduated scale and when the images are made to coincide, the angle indicated by the pointer is the altitude.

The altitude measured by a sextant is referred to as the Sextant Altitude (Sext. Alt).  

Corrections.  A number of corrections have to be made to the sextant altitude before we arrive at the True Altitude.

Index ErrorNo matter how carefully a sextant is manufactured, there will usually be a very small error in its reading and this is known as index error.

To calculate index error, view a single object through the sextant telescope and through the mirrors; move the index bar until the two images coincide and note the reading.  If the reading is not zero, the actual reading is the index error.

For example, if the reading is 2’ too high, it is said to be 2’  ‘on the arc’ and recorded as:  Index Error – 2’.0.  If the reading is 2’ too low, it is said to be 2’ ‘off the arc and recorded as +2’.0. 

Dip.  A correction has to be made to allow for the height of the observer’s eye above the horizon; this is known as Dip.

nonum diag26

In this diagram, O is an observer’s position on the Earth’s surface and E is the position of his eye.  We can see that, as the observer’s height of eye is raised above sea level, his visible horizon ‘dips’ below the true horizon and so the altitude measured at E becomes greater than that measured at O.  Dip is the error caused by this difference and has to be subtracted from the reading.

Tables of corrections for dip are printed in the Nautical Almanac as shown in the extract below:

dip

For example, if the height of eye is 4.6m. the correction will be 3’.8 (interpolate as necessary). 

Apparent Altitude is found by applying the index error and dip to the sextant altitude.

Example:

Sextant Altitude =      48o 15’.2

Index Error =                       +1’.3

Dip   =                                       -5’.3

Apparent Alt. =           48o  11’.2

Effectively, when measuring the altitude the Sun the stars, the Moon and the planets, index error and dip are the only corrections that the navigator has to apply manually.  However, there are a number of additional corrections which are incorporated in the altitude correction tables which must be applied in order to arrive at the true altitude.  The additional corrections are fully explained in ‘Astro Navigation Demystified’ but are briefly described below:

Semi Diameter.  In practice, the altitude that we measure is that of the lower limb; however, what we really need is the altitude of the Sun’s centre and so,we must add a correction for the value of its semi-diameter.

Parallax. The observer measures the altitude in relation to the visible horizon from his position on the Earth’s surface whereas the true altitude is measured from the Earth’s centre and so a correction called parallax must be added to allow for this.

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

Additional correction for temperature and atmospheric pressure.  This may be needed if the temperature and pressure are greatly different to the standard conditions which are assumed to be 10oC, 1010mb.

True Altitude is found by applying the additional corrections for parallax, refraction, semi-diameter, temperature and atmospheric pressure which are incorporated in the altitude correction tables.

The Altitude correction tables will be explained in Unit 3 Part 2.

(Note. This topic is covered in greater depth in the book ‘Astro Navigation Demystified‘).

Watch for unit 3 part 2

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 2

Unit 2 – Local Hour Angle and Greenwich Hour Angle

 Local Hour Angle (LHA).  In astro navigation, we need to know the position of a celestial body relative to our own position.

(The following diagram illustrates the explanations given below).

PZX TRIANGLE

LHA is the angle ANU on the Earth’s surface which corresponds to the angle ZPX in the Celestial sphere.   In other words, it is the angle

between the meridian of the observer and the meridian of the geographical position of the celestial body.

Due to the Earth’s rotation, the Sun moves through 15o of longitude in 1 hour and it moves through 15 minutes of arc in 1 minute of mean time.  So the angle ZPX can be measured in terms of time and for this reason, it is known as the Local Hour Angle.

Greenwich Hour Angle (GHA).  The hour angle between the Greenwich Meridian and the meridian of a celestial body is known as the Greenwich Hour Angle.

The Local Hour Angle between an observer’s position and the geographical position of a celestial body can be found by combining the observer’s longitude with the GHA. This is demonstrated in the following diagram.

GHA and LHA

O represents the longitude of an observer;

X represents the meridian of a celestial body;

G represents the Greenwich Meridian.

Because, in this case, the observer’s longitude is east and because LHA is measured westwards from the observer’s meridian to the meridian of the celestial body, LHA is equal to the GHA plus the longitude.

So we have the rule:  Long East, LHA = GHA + LONG

Note.  If the result is greater than 360o, we must modify the rule so that the result will be between 0o and 360o.  So the rule now becomes:

Long East, LHA = GHA + LONG (-360o as necessary).

For example: If Long. is 90oE. and GHA is 300o

Then LHA = GHA + LONG -360o

= 300o + 90o = 390o – 360o = 30o

If the longitude is to be west then the rule will change so that LHA would equal GHA minus Longitude. In this case, the rule is:

Long West, LHA = GHA – LONG

Note.  If the result is greater less than 360o, we must modify the rule so that the result will be between 0o and 360o.  So the rule now becomes:

Long West, LHA = GHA – LONG (+ 360o as necessary)

For example, if Long. is 90oW. and GHA is 45,o we have:

LHA = 45o – 90o = -45o + 360o = 315o

Calculating the Greenwich Hour Angle. The Nautical Almanac contains tables of raw data concerning the Greenwich Hour Angle for the Sun, the Moon, the navigational planets and selected stars.

The extract shown below, is of the Nautical Almanac daily page displaying hourly values of the Greenwich Hour Angle and the Declination of the Sun and the Moon for the 24th.  December.  The examples following demonstrate how the daily pages are used to calculate GHA:

na_2009-249 copycropped2
Example 1.  To find the LHA and Declination of the Sun.  
At 04 hours, 32 minutes, 04 seconds GMT on 24 December 2009, the assumed position of your yacht is 40o 35.5’ South 32o 13.8’ East.   Find the LHA and Declination of the Sun.

 Calculating the Local Hour Angle (LHA).  Before we can calculate the LHA, we need to find the Greenwich Hour Angle of the Sun.  

 Step 1.  Find the GHA for 0400 GMT.   From the daily page extract, we find that the GHA of the Sun for 24 December at 0400 is 240o 07’.5.  This is written as:  GHA (04h) : 240o 07’.5

Step 2.  Calculate the increment for 32 minutes, 04 seconds:  In this step, we find the increase in GHA for minutes and seconds of GMT.

The Nautical Almanac contains tables of corrections for increments of time from 0 minutes to 59 minutes.  The following extract shows the increments and corrections tables for 32 minutes.

32 33cropped2

From the extract we see that, in the table for 32m, the increment for 04 seconds is 8o 01’.0

This is written as:

Inc. (32m 04s):  8o 01’.0

Note.  Since GHA is always increasing, the increment correction is always added.

 Step 3.  Calculate GHA at 04 hr 32 min 04 sec GMT

GHA (04h)             240o 07’.5

Inc. (32m 04s)         + 8o 01’.0

GHA Sun               248o 08’.5

 Step 4.  Find LHA 

In this step, we combine the GHA with the longitude to calculate the LHA:

GHA                       248o 08’.5

Long:                     +  32o 13’.8 East

LHA Sun                 280o 22’.3

The full procedure for calculating the LHA of the Sun can be summarised in the following format:

GHA Sun (04h)       240o 07’.5

Inc. (32m 04s)            8o 01’.0

GHA Sun               248o 08’.5

Long                        32o 13’.8 E.  (+)

LHA Sun                280o 22’.3

Click here for an exercise in this topic

 (Note. This topic is covered in greater depth in the book ‘Astro Navigation Demystified’).

Note.  Units of this course are issued weekly.

Watch for unit 3

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 1

Unit 1 Essential Astronomy for Navigators

EARTH AND SUN IN THE SPHERE

The Celestial Sphere is an imaginary sphere with the Earth located at its centre. We imagine that the ‘celestial bodies’ such as the Sun, Moon, stars and planets are placed on the inner surface of the celestial sphere just as we would see them in the sky.

Ecliptic.  Although it is the Earth that orbits the Sun, it appears to us that the Sun moves around the celestial sphere taking one year to complete a revolution. This apparent movement of the Sun is called the Ecliptic.

Earth’s Rotation.  It takes exactly 24 hours for the Earth to turn once on its axis with respect to the Sun but it takes 23 hours, 56 minutes and 4 seconds to complete one rotation with respect to the rest of the universe.  The amount of time it takes for the Earth to turn on its axis with respect to the universe is know as the sidereal day and the time taken with respect to the Sun is called a solar day.

Ecliptic Poles.  If we imagine a line taken from the centre of the Earth to the Sun, it will be at right angles to the path of the ecliptic and where this line meets the celestial sphere will mark the north and south ecliptic poles.

Celestial Poles.  These are the points where the Earth’s axis of rotation meets the celestial sphere.

Geographic Poles.  These are the points where the Earth’s axis of rotation meets the Earth’s  surface.  These are simply known as the North Pole and the South Pole.

True North.  The direction from a position on the Earth’s surface towards the Geographic North Pole is known as True North.

 Magnetic Poles.  These are the north and south poles of the Earth’s magnetic field and are offset slightly from the geographical poles.

Magnetic North.  The direction from a position on the Earth’s surface towards the Magnetic North Pole is known as Magnetic North.

Deviation.  The difference between magnetic north and true north is known as deviation.

Axial Tilt.  The Earth’s axis is not in line with the Ecliptic Poles but is offset at an angle of 23.4°.  In other words, axial tilt is the angle between the geographic/true north pole and the ecliptic north pole measured from the Earth’s centre.

The Earth’s Equator is an imaginary line on the Earth’s surface the plane of which is at right angles to the axis of rotation.  It is equidistant from the North and South Poles and divides the Earth into the Northern Hemisphere and Southern Hemisphere.

The Celestial Equator is the projection of the Earth’s equator onto the surface of the celestial sphere.

DeclinationThe declination of a celestial body is its angular distance North or South of the Celestial Equator.

The tropic of Cancer.  This is where the Sun’s declination reaches its northernmost latitude of 23.4oN.

The tropic of Capricorn.  This is where the Sun’s declination reaches its southernmost latitude of 23.4oS.

The Equinoxes.  The Sun crosses the celestial equator on two occasions during the course of a year and these occasions are known as the equinoxes.  Because the Sun is on the celestial equator at the equinoxes, its declination is of course 0o.

The Autumnal Equinox occurs on about the 22nd. September when the Sun crosses the celestial equator as it moves southwards.

The Vernal Equinox occurs on about the 20th.March when the Sun crosses the celestial equator as it moves northwards.

The Solstices.  The times when the Sun reaches the northerly and southerly limits of its path along the ecliptic are known as the solstices.  The Summer Solstice (mid-summer in the northern hemisphere) occurs on about 21st. June when the Sun’s declination reaches 23.4o North (the tropic of Cancer).

The Winter Solstice (mid-winter in the northern hemisphere) occurs on about 21st. December when the Sun’s declination is 23.4South (the tropic of Capricorn).

First Point of AriesIn astronomy, we need a celestial coordinate system for fixing the positions of all celestial bodies in the celestial sphere.  To this end, we express a celestial body’s position in the celestial sphere in relation to its angular distances from the Celestial Equator and the celestial meridian that passes through the ‘First Point of Aries.   This is similar to the way in which we use latitude and longitude to identify a position on the Earth’s surface in relation to its angular distances from the Equator and the Greenwich Meridian. The First Point of Aries is usually represented by the ‘ram’s horn’ symbol shown below:

aries-znak

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 along the ecliptic (at the vernal Equinox in other words).

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.

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.

These concepts are illustrated by the diagram below:

 RA SHA ARIES

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

PXP’ is the meridian of the celestial body.

Y is the point at which the body’s meridian crosses the celestial equator.

Note.  Units of this course will be issued weekly.

Watch for unit 2

A thorough treatment of this topic can be found in the book 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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