The Earth, the celestial sphere & the PZX triangle
Celestial navigation rests on one picture: the sky as a sphere round you, each body standing over one spot on the Earth. From it comes the triangle every sight solves.
1 · The celestial sphere
The sky as a sphere round you
How high above the horizon is the celestial pole at my latitude?
As high as your latitude: the elevated pole P, the one above your horizon, stands at an altitude equal to your latitude. On the equator both poles lie on the horizon.
Latitude is the arc of your meridian from the equator to you; 1′ of it is one mile. Longitude is the arc of the equator from Greenwich to your meridian, east or west up to 180°.
Your zenith Z is the point overhead, the nadir the point underfoot. The celestial horizon is the great circle 90° from the zenith, through the Earth's centre; a sextant altitude is corrected to it first. Your meridian is the great circle through the celestial poles and your zenith.
The elevated pole P is the one above your horizon, as high as your latitude; on the equator both poles lie on the horizon.
2 · GP and hour angles
The spot under a body
How do I get the local hour angle from GHA and my longitude?
LHA = GHA + east longitude, or − west longitude, ± 360°. Under 180° the body is west of you, past your meridian; over 180° it is east of you, still climbing.
The geographical position (GP) is the spot with the body in its zenith. Its latitude is the declination (Dec); its longitude comes from the Greenwich hour angle (GHA), measured westward from Greenwich through 360°: GHA west up to 180°, beyond that 360° − GHA east.
The Sun's GP runs west at 15° an hour: a watch 4 seconds out moves it, and your fix, 1′ of longitude. A wrong Dec moves it north or south.
GHA and Dec are the same for every navigator. The local hour angle (LHA), measured westward from your own meridian, is yours alone: under 180° the body is west of you, past your meridian; over 180° it is east, still climbing.
Rule
LHA = GHA + east longitude, or − west longitude (± 360°).
LHA across 360°
GHA 300° · long 100°E
LHA = 400° − 360° = 40° (west)
GHA 20° · long 60°W
LHA = 20° + 360° − 60° = 320° (east)
3 · Aries and SHA
One point for every star
How do I find a star's GHA from the almanac?
Add the star's sidereal hour angle (SHA) to the GHA of the First Point of Aries for that hour, and take off 360° if the sum is over: GHA star = GHA Aries + SHA.
The stars keep their places, so the almanac tabulates one point, the First Point of Aries, hour by hour and gives each star a fixed sidereal hour angle (SHA), measured westward from Aries. For a planet, GHA minus GHA Aries gives its SHA.
Rule
GHA star = GHA Aries + SHA, less 360° if over.
A star's GHA and LHA
GHA Aries 200° + SHA 250° = 450°
GHA star = 450° − 360° = 90°
Long 30°W: LHA = 90° − 30° = 60°
The star is west of your meridian
Against the stars the Earth turns in 23 h 56 min: a star crosses your meridian about 4 minutes earlier each night. The Moon slips east among the stars about 13° a day and crosses some 50 minutes later.
4 · The Sun's year
Why the Sun's declination swings
Why does the Sun's declination change through the year?
Because the Sun's yearly path, the ecliptic, is tilted 23°26′ to the celestial equator. Its Dec is 0° at the equinoxes and 23°26′ north or south at the solstices, changing fastest near the equinoxes, about 24′ a day.
A star's Dec hardly changes. The Sun's does: its yearly path, the ecliptic, is tilted 23°26′ to the celestial equator. That tilt, not the distance to the Sun, makes the seasons.
About
Sun's Dec
Event
20 March, 23 September
0°
Equinoxes
21 June
N23°26′
Northern solstice
21 December
S23°26′
Southern solstice
Near the equinoxes the Dec changes fastest, about 24′ a day: the wrong day's figure costs miles. The Sun passes through your zenith only between the tropics, when its Dec equals your latitude; its bearing then swings from east to west within minutes of noon.
5 · The PZX triangle
The triangle behind every sight
What are the sides of the PZX triangle, and what does solving it give?
The PZX triangle joins the elevated pole, your zenith and the body: PZ is 90° − latitude, PX is 90° − Dec, or 90° + Dec for contrary names. Solving it gives ZX, hence Hc, and the azimuth.
Join the elevated pole P, your zenith Z and the body X by great circles. Knowing PZ, PX and the angle between them, you get ZX, hence the calculated altitude Hc, and the angle at Z, the azimuth. The angle at X is never used.
PX is 90° − Dec, or 90° + Dec when Dec and latitude have contrary names. South of the equator P is the south celestial pole. The azimuth angle Z is reckoned from the elevated pole, east or west, then turned into true azimuth Zn.
Setting up the triangle
Lat 35°S · Dec N19° · LHA 330°
PZ = 90° − 35° = 55°
PX = 90° + 19° = 109° (contrary names)
Angle P = 360° − 330° = 30°, body east
Z from the tables: S 147° E
Zn = 180° − 147° = 033°
Latitude
Body east
Body west
North
Zn = Z
Zn = 360° − Z
South
Zn = 180° − Z
Zn = 180° + Z
An absurd intercept usually means Dec named north for south.
6 · Circle of equal altitude
One sight, one circle
How far am I from a body's geographical position, given its observed altitude?
The zenith distance, 90° − Ho, at 60 miles to the degree: Ho 60° puts you 1800 miles from the GP, on a circle round it shared by everyone measuring that altitude at that instant.
The zenith distance, 90° − Ho, is also your distance from the GP, 60 miles to the degree. Everyone measuring that altitude at that instant is on one circle round the GP: Ho 60° puts you 1800 miles from it.
The circle is so large that near you its arc is a straight position line, at right angles to the azimuth. Compare Ho with Hc for an assumed position (AP): if Ho is greater you are nearer the GP, and the line moves towards the body by the difference in miles, whichever AP you chose. A body due east or west gives a north–south line: longitude.
For a very high Sun, above about 87°, you draw the circle round the GP itself. An altitude 1′ too high puts the line a mile too far towards the body.
7 · Meridian passage, rising, setting
The daily path of a body
When does a star never set at my latitude?
When its Dec has the same name as your latitude and exceeds 90° − latitude: the body is circumpolar. With that Dec and the contrary name it never rises; on the equator everything rises and sets.
A body is highest as it crosses your meridian, its GP then nearest to you. P, Z and X lie on one arc: the zenith distance is the difference between latitude and Dec, or their sum for contrary names, and the body bears north if its GP is north of you. So a noon sight gives latitude without tables.
Meridian altitude
Lat 40°N · Dec S10°: ZX = 40° + 10° = 50°
Altitude 40°, bearing south
Lat 20°S · Dec S23°: ZX = 23° − 20° = 3°
Altitude 87°, bearing south
A body whose Dec has the name of your latitude and exceeds 90° − latitude is circumpolar: it never sets, and at lower transit stands latitude + Dec − 90° high. With that Dec and the contrary name it never rises. On the equator everything rises and sets.
A body with Dec 0° rises due east; with a north Dec, north of east in either hemisphere, staying up over 12 hours in north latitude.
Practice questions
Questions from this topic, as the app asks them. Open one to see the answer.
Plotter, Carlisle Bay
Position 13°05′N 59°37′W
Barbados, at anchor
You are at anchor off Barbados and a crew member asks what the 13°05′N on the plotter actually measures. What do you tell them?
The arc of your meridian from the equator to here
The arc of your meridian from the North Pole to here
The arc of your parallel from Greenwich to here
The arc of the equator from Greenwich to here
Show the answer
A. The arc of your meridian from the equator to here
Latitude is the arc of the meridian (the angle at the Earth's centre) from the equator: 13°05′ = 785 M north of it.
All 100 questions of this topic are in the app: practice by topic with the reason after every answer, your mistakes brought back, and a timed mock exam. Free, no sign-up, in the browser or in Telegram.
