The Role Of Altitude, Azimuth And Zenith Distance In Astro Navigation.

altitude and zenith distance

Point Z represents the zenith of the observer’s position. 

Point X represents the position of the celestial body and this point, if projected onto the Earth’s surface, would correspond to the Geographical position of the body.

P and P1 are the north and south poles respectively.

 The Zenith Distance. The zenith distance is the angular distance ZX that is subtended by the angle XOZ and is measured along the vertical circle that passes through the celestial body.

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

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

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

Zenith Distance = 90o – Altitude

and Altitude = 90o – Zenith Distance

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

 The Role Of Altitude, Azimuth And Zenith Distance 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.

(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

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The Ecliptic, The Age of Aquarius and the Tropics.

EARTH AND SUN IN THE SPHERE

The Ecliptic.  Because of the orbital motion of the Earth, the Sun appears to us to move around the celestial sphere taking one year to complete a revolution. This apparent movement of the Sun is called the Ecliptic.  A year is approximately 365.25 days in length.  However; for the sake of convenience, the Gregorian calendar divides three years of the cycle into 365 days and the fourth (the leap year) into 366.

The Age Of Aquarius.  In a popular song, the words the ‘dawning of the age of Aquarius’ refer to the period when the vernal equinox will lie inside the constellation Aquarius.  The vernal equinox is the point where the Sun crosses the Equator on its northward movement along the ecliptic and heralds the first day of spring in the northern hemisphere on 20th./21st.March.  This point is known as the ‘First Point of Aries’ because in 150 B.C. when Ptolemy first mapped the constellations, Aries lay in that position. However, although still named the ‘first point of Aries’, due to precession, the vernal equinox now lies in the constellation Pisces, so logically, it should be named the ‘first point of Pisces’ since we are now in the ‘Age of Pisces’.  There are various predictions of when the next ‘age of Aquarius’ will begin but the most prominent of these is about 2600 A.D.

The Tropic of Cancer. These days, the Sun passes through the constellation Cancer in late July; however, in the time of Ptolemy, this occurred during the summer solstice when the Sun reached 23.4o N, the northern limit of the ecliptic.  The latitude 23.4oN is still called the tropic of Cancer even though the Sun now resides in Taurus at the summer solstice.

The tropic of Capricorn.  In a similar way,  the Tropic of Capricorn is so named because the Sun once passed through the constellation Capricornus during the winter solstice on 21/22 December when the Sun’s declination reached its southernmost latitude of 23.4oS.  However, due to precession, the Sun is now over the constellation Sagittarius at the Winter Solstice.

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

First Point of Aries

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

Posted in astro navigation, astronomy, celestial navigation

First Point of Aries.

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:

firstpointofaries

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).  This point is known as the ‘First Point of Aries’ because in 150 B.C. when Ptolemy first mapped the constellations, Aries lay in that position. However, although still named the ‘first point of Aries’, due to precession, the vernal equinox now lays in the constellation Pisces.

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

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

The following diagram illustrates the concepts discussed above.

 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.

aries is the First Point of Aries.

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

Right Ascension can also be defined as the angle ariesPY  or the angular distanceariesY but the difference is that it is measured in an easterly direction from Aries

From this, we can conclude that

RA    =  360o – SHA and

SHA  = 360o – RA.

A more detailed treatment of this topic is given in the following companion books:

Astronomy for Astro Navigation

Astro Navigation Demystified.

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The ‘Where To Look’ Method.

To determine the altitude and azimuth of a celestial body, we could make calculations by using mathematical formulae; we could compute them with the aid of sight reduction tables; we could use star globes and star charts or we could use computer software.  However, all of these procedures only tell us what the altitude and azimuth of the chosen body would be at our assumed position which is only an approximate position.  The only way that we can determine the altitude and azimuth of a celestial body at our true position is by making accurate measurements with a sextant and an azimuth-compass or similar equipment.  However, before we can do this, we must obviously be able to locate the body in the sky.

The ‘Where To Look’ Method enables us to quickly and easily calculate which stars and planets will be above our horizon at the time that we wish to make a ‘fix’.  It is quite a useful method, especially in the cramped confines of a chart-table, in a rough sea.

It is emphasised that this is not intended to be a method of accurately pin-pointing the position of a star or planet but a simple and quick method of establishing whether or not the body is likely to be visible at the time an observation is required and if so, what its approximate position will be in the sky.

The following extract is taken from ‘Astronomy for Astro Navigation’ which is a companion book to ‘Astro Navigation Demystified’.

 Method.  To find if a star or planet will be above the horizon at our position at the time of the planned observations, we need to take two things into account, its local hour angle and its declination.

 Local Hour Angle (LHA).  For a star or planet to be visible, its meridian must be within 90o east or west of our estimated longitude at the time of the planned observations.

  • If a body’s LHA is greater than 0o and less than 90o or if LHA is greater than 360o and LHA – 360o is less than 90o then body will be above the western celestial horizon.
  • If a body’s LHA is greater than 270o but less than 360o then 360o – LHA will be less than 90o and the body will be above the eastern horizon.

We can formulate the above statements as follows:

Body is above western horizon if:  LHA = 0o TO 90o or LHA – 360o = 0o TO 90o

Body is above eastern horizon if:  LHA = 270o to 360o or 360o – LHA = 0o TO 90o

In the following example, we have plotted the positions of stars X and Y in terms of their LHA and declination which are as follows:

Star X: LHA 22o, Dec. 65oN   Star Y: LHA 295oE, Dec. 20oN.

azimuth X and Y

 

 

 

 

 

 

 

Point O is the position of an observer on the Earth’s surface at latitude 50oN.

NS is the meridian of the observer and in terms of LHA is 0o.

WE is the celestial equator.

The limits of the observer’s western and eastern horizons are at LHA 90o and LHA 270o which are both 90o from the observer’s meridian.

Since the LHAs of these bodies is either less than 90o or greater 270o, they will be visible above the horizon.

Bodies whose LHAs are greater than 90o but less than 270o would be below the celestial horizon and therefore would not be visible.

To Calculate Azimuth.  Once we have ascertained the LHA and declination of a celestial body, we can make a rough approximation of the azimuth by plotting the LHA and declination in relation to the observer’s assumed position on the azimuth diagram.

If we join the positions of X and Y to the position of the observer, we will see that the approximate azimuths are as follows:

X: Approximate Azimuth = N30oW   Y : Approximate Azimuth = N110oE

The full procedures for making these calculations is shown in detail in the books Astronomy for Astro Navigation and Astro Navigation Demystified.

Declination.  To ascertain whether or not a celestial body will be above the horizon, we have to take into account its declination as well as its LHA.  To be visible above the horizon, its declination must be within 90o of the latitude of our position.  If our latitude is north, then the declination must be within the range 90o north to (90o – latitude) south.  If the latitude is south, then the declination must be within the range 90o south to (90o – latitude) north.We can formulate the above statements as follows:

Latitude North:  visible range = 90oN to (90o – Lat)S.

Latitude South:   visible range = 90oS to (90o – Lat)N.

We can explain the reason for this rule with the aid of another diagram which shows the declinations of the celestial bodies X, Y together with a third body, planet V.  Please note that the diagram is for illustrative purposes only; for this reason, it is not drawn to scale and angles are not drawn accurately.

declination visible horizon

Summary of data relating to stars X and Y and planet V which are represented in the diagram:

Assumed Position of observer:  Lat. 50oN  Long. 135oW

Star X:   Declination = 65oN   LHA = 22o

Star Y:  Declination = 20oN   LHA = 295o

Planet V:  Declination = 10oS   LHA = 190o

O is the position of the observer at latitude 50oN and Z is the zenith at that position.

The line PQ forms part of the visual horizon at point O and is tangential to the circumference of the Earth.

C is the centre of the Earth and the line CZ is perpendicular the visual horizon.

The line ACB is the celestial horizon which is perpendicular to CZ and therefore horizontal to the visual horizon.

The celestial horizon will cut the circumference of the Earth at latitudes that are 90o to the north and to the south of the observer’s latitude (in this example, latitude 40oN and latitude 40oS).

X is the position of the star X in the celestial sphere and X1 is its geographical position on the Earth’s surface.

Y is the position of the star Y in the celestial sphere and Y1 is its geographical position.

V is the position of planet V in the celestial sphere and V1 is its geographical position.

The diagram shows that although the geographical positions of X and Y are below the visible horizon and therefore out of sight to the observer, the bodies themselves are above the celestial horizon and therefore since they are within the limits for LHA and declination, they will be visible above the horizon at the observer’s assumed position.

The diagram also shows that V would have been above the celestial horizon but for one thing.  Because its LHA is 190o which is greater than 90o but less than 270o, it is outside the LHA limits and therefore will not be above the horizon.

Estimating Approximate Altitude.  This method enables us to calculate the approximate altitude of a celestial body when it lies over the meridian of the observer; in other words when LHA equals 0o or 360o.  When the LHA of a body is greater than 0/360 (i.e. when it is not over the observer’s meridian) adjustments have to be made to the calculated altitude.

Calculating the Altitude of Stars X and Y and planet V at noon.  We can modify the previous diagram to help us with these calculations.

Relevant Data:

Latitude of observer’s assumed position = 50oN

Star X:   Declination = 65oN

Star Y:    Declination = 20oN

Planet V:  Declination = 10oS

XandYapprox altitude

In the diagram, Angle ACN is the angle between the Celestial Horizon and the direction of North and NCX is the angle between the direction of North and the direction of X.  Angle ACX is equal to the sum of these angles and represents the altitude of X which is 85o.

Angle BCE is the angle between the Celestial Horizon and the Celestial Equator and ECY is the angle between the direction of East and the direction of Y.  BCY is equal to the sum of these angles and represents the altitude of Y which is 60o.

Angle BCV is equal to angle BCE minus the declination of V and represents the altitude of planet V which is 30o. However, as discussed earlier, V would not be visible because it is outside the limits for LHA.

 Summary.

Altitude X = 85o

Altitude Y = 60o

Altitude V = 30o

With a little practice, it will become an easy task to estimate the azimuth and altitude of a body without the aid of diagrams.

Calculating altitude when LHA is not zero.  We must remember that the altitudes we have calculated above would apply when the celestial bodies are over the observer’s meridian (i.e. when LHA is 0o).  Therefore, when the LHA is other than 0o, we have to re-calculate the altitude.

The altitude of a celestial body is 0o when it rises at LHA 270o and increases to its maximum when LHA is 360o/ 0o before descending again to 0o when it reaches LHA 90o

Re-calculation of altitude of star X for LHA 22o:

The diagram above shows that the altitude of star X is 85o.

Therefore, Altitude = 85o when LHA = 360o

But Altitude = 0o when LHA = 270o

Thus 90o worth of LHA = 85o of altitude

But LHA = 22o

Therefore, 22o of LHA = (85 / 90) x 22 = 20.7o of altitude.

Therefore, when LHA X = 22o, altitude = 85 o– 20.7o = 64.3o

Therefore Estimated altitude of X when LHA is 22o = 64.3o.

Re-calculation of Altitude of star Y for LHA 80o:

The diagram shows that the altitude of star Y is 60o.

Therefore, Altitude = 60o when LHA = 360o

But Altitude = 0o when LHA = 270o

Therefore 90o worth of LHA = 60o of altitude

But LHA = 280= 80o E.

Therefore 80o of LHA = (60 / 90) x 80 = 53.3o of altitude.

Therefore, when LHA  = 80o, altitude = 60o – 53.3o = 6.7o

Therefore, Estimated altitude of Y when LHA is 80o = 6.7o.

We cannot make a recalculation for planet V because its LHA is outside LHA limits.

Of course we cannot find an accurate solution to a spherical problem with straight-line, two dimensional, drawings such as these which do not take account of phenomena such as refraction and parallax or the fact that the Earth is not a perfect sphere.  However, this method does give us a good idea where to look in the sky for a celestial body that we have selected to use for an accurate position fix.

It is also stressed that the drawings above are for illustrative purposes only, are not drawn to scale and that angles are not accurately drawn.

The procedures for making these calculations are shown in detail in the companion books Astronomy for Astro Navigation and Astro Navigation Demystified.

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Jupiter’s Retrograde Motion

Jupiter moves across the sky in a very predictable pattern, but every now and then it reverses direction in the sky, making a tiny loop against the background stars – this is Jupiter in retrograde.

The following diagram shows that, as Jupiter is overtaken by the Earth, its apparent motion across the sky appears to describe a loop as its direction changes from prograde to retrograde and then back to prograde again.

Jupiter loop

 

At position 1, it appears to be moving from west to east in prograde motion.  At positions 2 and 3, its direction appears to have changed from prograde to retrograde so that it is now moving from east to west.  At position 4, it appears to have resumed prograde motion as it moves from west to east again.

Note.  Sky maps can be very confusing because they are not drawn in the conventional way with east on the right and west on the left.  They are drawn as they would appear in the sky if we were lying down with our legs pointing to the south and looking upwards so that east would be on our left and west on our right.

Jupiter’s retrograde periods last for 4 months and are then followed by   periods of nine months of prograde motion before going retrograde again.  So the time from the beginning of one retrograde movement to the beginning of the next is approximately 13 months.

 The relatively slow movement of Jupiter across the sky makes it very easy for the navigator to locate.  It appears to move from one constellation to another every 13 months describing its retrograde loop as it pauses in each one.

In January 2016, Jupiter will pause in Leo when it goes into retrograde motion and thirteen months later, in February 2017 it will be in Virgo when it begins the sequence again.  After that Jupiter is retrograde in Libra in March 2018 and so on as it follows its path through Scorpio, Sagittarius, Capricorn, Aquarius, Pisces, Aries, Taurus, Gemini, Cancer and back to Leo again.

Jupiter’s predictable path across the sky together with the fact that it is the fourth brightest celestial body in the sky explains why it is such an important navigational planet.

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Finding Stars and Constellations, Part IV

This post continues the series Finding Stars and Constellations.    

Boötes  The Herdsman   If we take a line from Alioth to Alkaid in the Great Bear and extend that line in an -imaginary curve for about roughly three hand-spans as shown in the diagram below, it will point to Arcturus, the brightest star in the constellation Bootes.

bootesBoötes is the 13th largest constellation and is located in the northern hemisphere and can be seen from +90 to -50.  The ancient Greeks visualized it as a herdsman chasing Ursa Major round the North Pole and its name is derived from the Greek for “Herdsman”.

Arcturus is the fourth brightest star in the sky and is a navigation star; it can be seen during nautical twilight in June.  The ancient Greeks named Arcturus the “Bear Watcher”  because it seems to be looking at the Great Bear (Ursa Major).

Corona Borealis, The Northern Cross   If we next take a line from Nekkar to Princepes in Boötes, and extend that line by about one and a half hand-spans, it will point to Nusakan and on to Alphecca in the nearby Corona Borealis constellation.

bootes and corona

Corona Borealis, whose name in latin means northern crown, is a small constellation in the northern hemisphere and can be seen between latitudes +90 to -50. Alphecca, the brightest star in the group, is a navigation star and is best seen during evening nautical twilight in July.

The main stars in Corona Borealis form a semi-circle which is associated with the crown of Ariadne in Greek mythology.  It said that the crown was given to Ariadne by Dionysus on their wedding day and after the wedding, he threw it into the sky where the jewels became stars which were formed into a constellation in the shape of a crown.

Virgo, the Virgin

 virgodrawingThe constellation Virgo takes its name from the Latin for virgin or young maiden.  In ancient Greek mythology, Virgo is associated with the goddess Dike, the goddess of justice and as the diagram below shows, the constellation Libra, the scales of justice, lies next to Virgo.

Virgo lies over the southern hemisphere and is one of the largest constellations in the sky, smaller in size only to  Hydra.  It is visible between latitudes +80 to -80 and for navigation purposes, it is best seen during evening nautical twilight in May.

The brightest star in Virgo is Spica, the 15th. brightest star in the sky and a navigation star.

Finding Virgo.

As we learned when studying the constellation Boötes, an imaginary curved line from Alkaid in the Great Bear leads to the bright orange star Arcturus, in the constellation Boötes.  If, as shown in the following diagram, we continue that curved line by another hand span from Arcturus we will come to the bright bluish-white star Spica, in the constellation Virgo which is to the left of Leo.

beartobootesto virgo and leo

Notes.    Star maps can be very confusing because they are not drawn in the conventional way with east on the right and west on the left.  They are drawn as they would appear in the sky if we were lying down with our legs pointing to the south and looking upwards so that east would be on the left and west on our right.

Although this series of posts show the locations of certain constellations relative to other constellations in the sky, it does not necessarily indicate whether or not they will be visible above the horizon.  That will of course depend on its times of rising and setting at the  position of the observer.  ‘Risings and Settings’ will be the subject of another post.

Watch out for the next post in this series which will be issued shortly.

Links:   

Finding Stars and Constellations Part I

Finding Stars and Constellations Part II

Finding Stars and Constellations Part III

Rising and setting times of stars.

Latitude from Polaris

Astro Navigation Demystified on Amazon

The Application of Mathematics To Astro Navigation on Amazon

About Astro Navigation Demystified

 

 

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Calculating Azimuth And Altitude At The Assumed Position By Spherical Trigonometry.

There are several ways of calculating the azimuth and altitude at the assumed position; these include the use of sight reduction methods and software solutions. However, the traditional method is by the use of spherical trigonometry which is demonstrated below.

diag 3 no number

The PZX Triangle

 

 

 

 

 

 

 

 

 

 

In the diagram above,

PZ is the angular distance from the Celestial North Pole to the zenith of the observer and is equal to 90o – Lat.

PX is the angular distance from the Celestial North Pole to the celestial body and is equal to 90o – Dec.

ZX is the Zenith Distance and is equal to 90o – altitude.

Therefore, altitude is equal to 90o – ZX

The angle ZPX is equal to the Local Hour Angle of the Celestial Body with respect to the observer’s meridian.

The angle PZX is the azimuth of the body with respect to the observer’s meridian.

Summary.

PX = 90o – Dec.

PZ = 90o – Lat.

ZX = 90o – Alt.

Alt = 90o – ZX

<PZX = Azimuth.

<ZPX = Hour angle.

In order to calculate the azimuth and altitude of a celestial body we must solve the triangle PZX in the diagram above.  Specifically, we must calculate the angular distance of side ZX so that we can find the altitude and we must calculate the angle PZX so that we can find the azimuth.

However, because the triangle PZX is on the surface of an imaginary sphere, we cannot solve this triangle by the use of ‘straight line trigonometry’; instead we must resort to the use of ‘spherical trigonometry’ which is explained here.

Examples of the use of spherical trigonometry to calculate the azimuth and altitude of celestial bodies.

Note.  Traditionally, the ‘half-haversine’ formula was used for this task but this formula does not lend itself well to solution by electronic calulator; therefore, the following solutions involve the cosine formula.

Example 1.  Star Sight.

Scenario:     Greenwich date: 30 June 18hrs 05 mins  33 secs

Assumed Position:  Lat. 30oN    Long. 45oW

Selected body: Alioth

SHA: 166

Declination: 56oN

GHA Aries:  250

Step 1.  Calculate LHA

SHA Alioth:   166

Add GHA Aries: 166 + 250 = 416

Subtract Long(W) = 416 – 45 = 371

Subtract 360 = 11

Therefore,  LHA  =  11W

(all results in degrees)

Step 2. Calculate PZ/PX/ZPX

PZ = 90o – 30o = 60o      ∴PZ = 60o

PX=  90o – 56o = 34o     ∴PX = 34o

ZPX = LHA = 11west

Step 3.  Calculate Zenith Distance  (ZX).

As explained here, the formula for calculating side ZX is:

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

Step 4.  Calculate Altitude.

Altitude  = 90o – ZX   = 90o – 27o  = 63o

Step 5.  Calculate Azimuth (PZX)

As explained here 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 PZX   = Cos(34) – [Cos(27) . Cos(60]  /  [Sin(27) . Sin(60)]

= 0.829 – [ 0.89 x 0.5]  /  0.454 x 0.866

=  0.829 – 0.445 / 0.393

=  0.384   / 0.393   = 0.977

Cos(PZX) = 0.977

∴  PZX   =  Cos-1(0.977)  = 12.31

∴  Azimuth = N12oW (since LHA is west)

In terms of bearing, the azimuth is 348o.

Step 6.  Summarize results.

LHA =  11west

Declination = 56oN

Azimuth at assumed position = N12oW

Altitude at assumed position = 63o

Links:

Spherical Trigonometry

What is Astro Navigation

Accuracy of astro navigation

Relationship between Altitude and Zenith Distance

Planning Star and Planet observations

Azimuth and Altitude

Astro Navigation Demystified

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