Decide
Question 1/1
At latitude 40°N, what is the side PZ?
The lesson in plain words
- Three points
- Join three points of the celestial sphere with great-circle arcs: P, the elevated pole; Z, your zenith; X, the body.
- This is the PZX triangle, also called the astronomical or position triangle.
- On the Earth the same triangle joins the pole, your position and the body's GP.
- The sides
- PZ = 90° − latitude, the co-latitude.
- PX, the polar distance, is 90° − Dec when latitude and Dec are both N or both S, or 90° + Dec when one is N and the other S. Use positive angle sizes.
- ZX = 90° − Ho, the zenith distance, where Ho is corrected observed altitude.
- PZ comes from your DR, PX from the almanac; corrected observed altitude gives ZX.
- The angles
- The angle at P is the meridian angle (t): equal to LHA when LHA is under 180°, named west; otherwise 360° − LHA, named east.
- The azimuth angle at Z runs 0–180° from the direction of the elevated pole towards the body.
- Name its reference north or south and its side east or west.
- Zn expresses the same direction clockwise from true north.
- Solving it
- From your DR latitude, the Dec and the LHA, spherical trigonometry gives Hc, the altitude calculated for your DR, and the azimuth Zn.
- Your raw sextant reading is Hs; after correction it becomes Ho, your corrected observed altitude.
- Comparing Ho with Hc gives the intercept, a difference used to find your position line (Celestial: sight reduction).
- A special case
- At upper meridian passage, LHA is 0° and P, Z and X all lie on one great circle, your meridian.
- The triangle collapses into a line.
- This is the special geometry used for the noon sight (Celestial: the noon sight).
- At lower passage, LHA is 180° and the points also lie on your meridian.
Key data
- PZ
- Co-latitude = 90° − latitude
- PX
- 90° − Dec if lat/Dec both N or both S; 90° + Dec if N/S differ
- ZX
- Zenith distance = 90° − Ho (corrected observed altitude)
- Angle at P
- Meridian angle t, from LHA
- Angle at Z
- Azimuth angle, converted to true bearing Zn
- Spanish terms
- Triángulo de posición: colatitud, codeclinación, distancia cenital
Key points
- P elevated pole, Z zenith, X body.
- PZ = 90° − lat; PX = 90° ∓ Dec; ZX = 90° − Ho (corrected observed altitude).
- Angle at P comes from LHA; angle at Z gives the azimuth.
- Your DR, Dec and LHA give Hc and Zn; Ho is your corrected observed altitude.
- At upper passage (LHA 0°), the triangle lies along the meridian: the noon sight.
Check what you have read
Question 1/3
Your latitude is 30°N and the body's Dec is S 20°. What is the polar distance PX?
True or false?
Decide each statement before you see the answer.
PZ = 90° − lat; PX = 90° ∓ Dec; ZX = 90° − Ho (corrected observed altitude).
Polar distance is always 90° − Dec.
The sextant gives the azimuth angle at Z.
Your DR, Dec and LHA give Hc and Zn; Ho is your corrected observed altitude.
Practice case & exam keys
Situation
Your DR latitude is 21°10′N. A body's Dec is S 21°00.2′, and your corrected altitude is 47°45.8′. Find PZ, PX and ZX.
Show the answer
PZ = 90° − 21°10′ = 68°50′. Latitude is north and declination is south, so PX = 90° + 21°00.2′ = 111°00.2′. ZX = 90° − 47°45.8′ = 42°14.2′.
Exam keys
- PZX: elevated pole, zenith, body.
- Sides: co-latitude, polar distance (90° ∓ Dec), zenith distance.
- Angle at P from LHA; angle at Z is the azimuth angle.
- At upper passage (LHA 0°), the triangle collapses: the noon-sight case.