How to Calculate Gyro Error by Sun Azimuth — Complete Guide for Merchant Navy Officers
Navigation · Watchkeeping · SIRE 2.0 Prep
1700 on the Bridge. "Let's Calculate the Gyro Error."
A complete step-by-step guide using the scientific calculator formula method — written for cadets who want to understand it properly, not just memorise it.
⚓ Aatish Saini, Master Mariner📖 10 min read🎯 Formula Method · SIRE 2.0 Ready
It is 1700 hours. The Chief Officer is on watch — this is his time, 1600 to 2000, the golden hour watch when the evening sun hangs low and the sea turns amber. A cadet has just stepped onto the bridge.
The Chief Officer puts down his coffee. He walks to the compass repeater on the bridge wing and picks up the azimuth circle.
"Come here," he says. "Sun is still up. Good time to show you something we do every watch without fail."
The cadet steps closer. This is how it begins.
The evening watch. The Chief Officer and the cadet — one passing knowledge, the other receiving it.
Why the Gyrocompass Cannot Be Trusted Blindly
Your gyrocompass is the foundation of everything on the bridge. ECDIS heading, radar bearings, autopilot course — every single one starts with the gyro. If it is wrong, everything built on it is wrong.
A gyrocompass drifts. Slowly. Silently. Due to the ship's speed, changes in latitude, course changes and the mechanical nature of a spinning mass trying to find true North on a moving vessel.
A 1° error sounds small. After 60 nautical miles, you are 1 mile off your intended track. In a traffic separation scheme or a channel approach — that gap is not acceptable.
The gyrocompass tells you which way is North. But only the sun can tell you whether the gyrocompass is telling the truth.
— Something you learn on the first watch
This is why we check it. Every watch. Without exception. The sun — the most predictable object in the sky — is our reference. And here is exactly how we do it.
What You Need Before Starting
Gather These First
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GPS — Exact UTC Time and PositionNote the time to the nearest second and your Latitude and Longitude from GPS at the exact moment of observation. These are your starting values for everything.
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Nautical Almanac (Current Year)You need the Sun's GHA (Greenwich Hour Angle) and Declination for your date and time. Always use the current year's almanac.
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Azimuth Circle on Bridge Wing Compass RepeaterThis is what you use to take the sun's gyro bearing. Not the radar — the azimuth circle on the compass repeater, bridge wing.
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Scientific CalculatorYou need Tan, Sin, Cos and InvTan (arctan) functions. Any standard scientific calculator works fine.
The Four Formulas — Learn These
Everything reduces to four formulas and one final comparison. Here they are, clearly.
Step 1 — Meridian Angle P
If LHA < 180° → P = LHA If LHA > 180° → P = 360° − LHA
P is the angular distance of the sun from your meridian. It is always between 0° and 180°. You get LHA from GHA ± your longitude.
Step 2 — Calculate A
A = TanLat ÷ TanP
Naming rule for A: In all practical sun azimuth observations, A is named OPPOSITE to your Latitude.
Latitude North → A is South | Latitude South → A is North
Note: If your calculator shows a minus sign for A (this happens when P > 90°), ignore the minus. The naming rule already accounts for it.
Step 3 — Calculate B
B = TanDec ÷ SinP
Naming rule for B: B is always the SAME name as Declination. Declination North → B is North | Declination South → B is South
Step 4 — Calculate C, then True Azimuth
C = A ± B → TanAz = 1 ÷ (C × CosLat)
C = A + B if same name (add, keep that name) C = A − B if different names (subtract, keep name of the larger value)
Then: Az = InvTan [ 1 ÷ (C × CosLat) ] Az is a quadrant angle — convert it to a True Bearing using the table below.
Converting Az to True Bearing (Zn)
The True Bearing has two parts. First letter is C's name. Second letter is where the sun actually is.
Quick Conversion Table
C = N, Sun in East (LHA 180°–360°)
N xx° E → Zn = Az
C = N, Sun in West (LHA 0°–180°)
N xx° W → Zn = 360° − Az
C = S, Sun in East (LHA 180°–360°)
S xx° E → Zn = 180° − Az
C = S, Sun in West (LHA 0°–180°)
S xx° W → Zn = 180° + Az
The Memory Rhyme — Burn This In
Every Officer Knows This
Error East — Gyro Least Error West — Gyro Best
Compare the True Bearing you calculated with the Gyro Bearing you read from the compass repeater.
→ Gyro Bearing is LOWER than True Bearing → Error is EAST (Gyro Least) → Gyro Bearing is HIGHER than True Bearing → Error is WEST (Gyro Best)
The difference in degrees is the magnitude of the error.
Worked Example — Follow Every Step
The Chief Officer opens his calculation book. He reads out the observation data. The cadet writes it down and they work through it together on the calculator
The chart table. Nautical Almanac open, calculator in hand, working through it step by step.
Complete Worked Example — Sun Azimuth at 1700
Formula Method
Observation Data — From GPS and Nautical Almanac
Latitude
23° 21.0' S
Longitude
47° 18.0' W
GMT
13d 13h 10m 22s
GHA (from Almanac)
018° 38.9'
Declination
03° 34.3' N
Gyro Bearing (C. Repeater)
047.5°
Step 1 — Calculate LHA then P
GHA=018° 38.9'
Long (W)−047° 18.0'
LHA=018°38.9' − 047°18.0' + 360° (add 360° since result is negative)
C = N | LHA = 331° (between 180° and 360°) → Sun is in the EAST
N xx° E → Zn = Az
True Bearing = N 49.8° E = 049.8° T
Step 7 — Find Gyro Error
True Bearing=049.8°
Gyro Bearing=047.5°
Gyro is LOWER (47.5° < 49.8°) → Apply rhyme: Gyro Least → Error East
Difference = 2.3°
True Bearing
049.8° T
N 49.8° E
Gyro Error
2.3° East
Gyro Least → Error East
Record every check in the deck log — date, GPS time and position, method, True Bearing, Gyro Bearing, Gyro Error, signature. If it is not recorded, it did not happen.
Now the Sun Drops Lower — Amplitude
It is now close to 1830. The sun is maybe thirty minutes from the horizon. The sky is deep orange, the sea reflecting it. The cadet watches it falling, slower than you would expect.
The Chief Officer steps back onto the bridge wing.
"We have one more method to learn while we have the chance. Amplitude. This one needs only one formula and no LHA. But the timing has to be exact — and I mean exact."
⚓ Amplitude Method — Sunrise and Sunset Only
The simplest and fastest gyro error check. One formula. No almanac pages after declination. But only valid at one specific moment.
Sin (Amplitude) = Sin Dec ÷ Cos Lat
Amplitude (A) = InvSin [ Sin Dec ÷ Cos Lat ]
☀️
The Critical Moment — 1 Semi-Diameter ABOVE the HorizonThe amplitude observation must be taken when the sun's lower limb is exactly 1 semi-diameter above the visible horizon. At this moment, the centre of the sun — the "real sun" — is at the celestial horizon, which is where amplitude is mathematically defined. This is the only correct moment. Too early and the sun is too high. Once the lower limb touches the horizon, you have missed it.
Lower limb is 1 semi-diameter ABOVE the visible horizon. At this exact moment the centre of the sun is at the celestial horizon — take the bearing now.
🧭
Converting Amplitude to True Bearing Prefix is E (sun rising) or W (sun setting). Suffix is same name as Declination.
At Sunrise (E): Dec N → Zn = 090° − A | Dec S → Zn = 090° + A
At Sunset (W): Dec N → Zn = 270° + A | Dec S → Zn = 270° − A
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Clear Horizon is EssentialAmplitude is only valid with a clean, sharp visible horizon. Haze, sea fog, heavy swell or tropical haze near the horizon make the observation unreliable. If the horizon is not sharp and clean — do not use amplitude. Use azimuth instead, which works at any time of day under most conditions.
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Only One Value Needed from the AlmanacFor amplitude you only need the Sun's Declination — not GHA, not LHA. Just Declination and your Latitude. This makes it the fastest calculation at sea. Always cross-check your amplitude result against your azimuth method when both are available on the same watch.
What SIRE 2.0 Inspectors Ask About This
🔍 SIRE 2.0 — Gyro Error Questions and Answers
Q: How do you check gyro error on this vessel?
"We check gyro error by taking the bearing of the sun using the azimuth circle on the bridge wing compass repeater. We calculate the true azimuth using the scientific calculator formula method — with GHA and declination from the Nautical Almanac and exact UTC time and position from GPS. At sunrise and sunset we also use amplitude when the horizon is clear. All observations are recorded in the deck log every watch."
Q: Show me your last gyro error entry in the deck log.
Have the log ready to show. The entry must show: Date, UTC time from GPS, vessel position, method used, True Bearing, Gyro Bearing, Gyro Error with direction (E or W), and officer's signature. A missing or unsigned entry is a deficiency.
Q: What is the acceptable gyro error limit on your vessel?
"As per our SMS the acceptable limit is ±1°. Any error exceeding this is reported to the Master immediately, recorded in the log, and the gyrocompass manufacturer or technical superintendent is notified. Navigation continues with the error correction applied."
Q: Gyro reads 245° and true bearing is 248°. What is the gyro error?
"Gyro reads 245°, True is 248°. Gyro is lower than True — applying the rule: Gyro Least means Error East. The error is 3° East. The gyrocompass is reading 3° less than the true bearing."
Q: When do you take amplitude and when is it not suitable?
"Amplitude is taken at sunrise or sunset when the sun's lower limb is 1 semi-diameter above the visible horizon — that is when the centre of the sun is at the celestial horizon and the observation is valid. It requires a clear, sharp horizon. If the horizon is hazy or indistinct, amplitude is not reliable and we use azimuth instead. Azimuth can be used at any time of day."
Quick Reference — Everything in One Place
⚓ Gyro Error by Sun — Complete Reference
PLHA if < 180° | 360° − LHA if > 180°
A = TanLat ÷ TanPAlways OPPOSITE to Latitude in sun observations. Ignore any minus sign — naming handles it.
B = TanDec ÷ SinPAlways SAME name as Declination
CSame name → add | Different names → subtract | Takes name of the larger
TanAz = 1 ÷ (C × CosLat)Az = InvTan result → convert using quadrant table above
Memory RhymeError East — Gyro Least (Low) | Error West — Gyro Best (High)
AmplitudeSin A = Sin Dec ÷ Cos Lat | Sunrise or sunset only | Clear horizon required
Amplitude TimingLower limb must be 1 semi-diameter ABOVE the visible horizon — not touching it
Always RecordGPS UTC time + position, method, True Bearing, Gyro Bearing, Error (E or W), signature
The sun touches the horizon. The cadet raises the azimuth circle. He has the calculator ready, the declination noted, the GPS time recorded.
He takes the bearing. He does the calculation. He checks the result against the azimuth check done an hour earlier.
They match.
The Chief Officer nods. "Write it in the log. Sign it. Now you know why we do this every watch."
That understanding — not just the steps but the reason — is what separates a careful officer from someone who merely follows a routine.
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