Worked: compass error by amplitude at sunrise
1The lesson in plain words
At rising or setting, the true bearing needs only latitude and declination: sin A = sin Dec ÷ cos Lat. Time it to when the Sun's centre is on the celestial horizon.
Amplitude
The amplitude (A) is the angle between due east or west and the body's bearing at rising or setting. It is named E when rising, W when setting, then N or S with the declination's name. With the body's centre on the celestial horizon (true altitude 0°): sin A = sin Dec ÷ cos Lat. No exact time is needed, only latitude and declination.
Which moment
A limb is the edge of the disc. When the Sun's centre is on the celestial horizon, its lower limb is about two-thirds of a diameter above the visible horizon. For the Moon, observe when its upper limb touches the visible horizon; for stars and planets, about one Sun diameter above it.
Visible horizon instead
At higher latitudes, a misjudged observing moment causes a larger bearing error. Instead, take the bearing with the Sun's centre on the visible horizon. Apply Bowditch Table 23 to the observed bearing, away from the elevated pole (the north pole in northern latitudes; the south pole in southern latitudes). Then compare the corrected compass bearing with the calculated celestial-horizon bearing. Near 16°N in this example, the correction is about 0.2°.
Worked: sunrise
4 December 2026, about 0859 UT, DR 16°07′N 040°25′W, heading 262°C. Dec S22°15.3′. sin A = sin 22°15.3′ ÷ cos 16°07′ = 0.3787 ÷ 0.9607 = 0.3942, so A = E 23.2° S. True bearing = 090° + 23.2° = 113.2°T.
Worked: the error
With the Sun's lower limb about two-thirds of a diameter above the visible horizon, its centre bears 126.6°C. Error = 113.2 − 126.6 = −13.4°: 13.4°W. WMM2025 gives variation 14.5°W here. Deviation = −13.4 − (−14.5) = 1.1°E on 262°C, close to the 0.9°E found by azimuth that afternoon.
2Key data
- Formula
- sin A = sin Dec ÷ cos Lat (centre on the celestial horizon)
- Naming
- E rising, W setting; then N or S with the declination
- Sun's centre on celestial horizon
- Lower limb about two-thirds of a diameter above the visible horizon
- Moon; stars and planets
- Moon: upper limb on the horizon; stars: about one Sun diameter up
- Worked amplitude
- Dec S22°15.3′, lat 16°07′N: E 23.2° S = 113.2°T
- Worked error
- 113.2 − 126.6 = 13.4°W; variation 14.5°W → deviation 1.1°E
3Key points
- 1
sin A = sin Dec ÷ cos Lat: latitude and declination only, no exact time.
- 2
Name A from east (rising) or west (setting), towards the declination's pole.
- 3
Bear the Sun when its lower limb is about two-thirds of a diameter above the horizon.
- 4
At higher latitudes, observe the Sun's centre on the visible horizon and apply Bowditch Table 23.
4Frequent mistakes
Wrong: Take the bearing just as the Sun's upper edge appears.
Right: Then the centre is still below the celestial horizon. Wait until the lower limb is about two-thirds of a diameter up.
Wrong: A rising Sun with southerly declination bears 090° − A.
Right: E 23.2° S means 23.2° south of east: 090° + 23.2° = 113.2°T.
Wrong: The amplitude is the same at every latitude for the same declination.
Right: It grows with latitude: for Dec 22°15′, about 22.2° at the equator, 25.9° at 30° and 36.1° at 50°.
5Practice case & exam keys
Evening, heading 280°C, latitude 50°N, Dec N20°00′. The Sun's centre is on the celestial horizon and bears 312.0°C. Variation is 2.0°W; the card says 0.5°W on 280°C. A mate wants to replace the card value. Calculate the amplitude, true bearing, error and deviation, and decide.
Show the answer
A = W 32.1° N, so Zn = 302.1°T. Error = 302.1 − 312.0 = 9.9°W. Deviation = −9.9 − (−2.0) = 7.9°W on 280°C. Neither this single check nor the old card is yet confirmed. Repeat the observation, check inputs and magnetic objects, and cross-check with another method. Use an independently checked heading source meanwhile. Log the confirmed deviation before adopting it.
Exam keys
- Amplitude: sin A = sin Dec ÷ cos Lat, with the body's centre on the celestial horizon.
- Named E (rising) or W (setting), then N or S like the declination.
- Sun: lower limb about ⅔ of a diameter above the visible horizon; Moon: upper limb on it.
- Sun's centre on the visible horizon: correct the observed bearing with Table 23, away from north in N latitudes or away from south in S latitudes.
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