What is the rate of turn at this speed and radius?

Formula, assumptions, and limits

Rate of turn in degrees a minute equals 3/π, or 0.955, times the speed in knots divided by the turning radius in nautical miles. At 12 knots a ship covers 370.4 m a minute. On a circle of 1,852 m radius, one nautical mile, that is 0.2 radians or 11.46° of arc, so the heading has to change 11.46° every minute for the ship to stay on the circle. That figure is the rate of turn.

Formula

The constant $3/\pi$ is 0.954930.

• $ROT$: rate of turn, in degrees a minute; positive to starboard, negative to port.
• $V$: speed along the turn, in knots; 1 knot is 1,852 m an hour exactly.
• $R$: radius of the turn, in nautical miles, from the center of the circle to the ship’s track.
• $\Delta$: course alteration, in degrees, from 1° to 180°.
• $t$: time to alter course, in minutes, at a steady rate.
• $d_{WO}$: distance from the wheel-over point to the intersection of the old and new courses, in nautical miles, along the old course.

The distance sailed on the arc is $R\,\Delta\,\pi/180$ nautical miles. In degrees a second the rate is $ROT/60$; in radians a second it is $V/(3600\,R)$.

Derivation

A ship moving at speed $V$ on a circle of radius $R$ turns through $V/R$ radians per unit of time. With $V$ in knots and $R$ in nautical miles the unit is radians an hour. Multiplying by $180/\pi$ gives degrees an hour, and dividing by 60 gives degrees a minute, the unit IMO resolution A.526(13) sets for the indicator scale. The two factors combine to $3/\pi = 0.954930$.

The wheel-over distance is plane geometry. A circle that meets both courses at a tangent has its center on the bisector of the angle between them; the turn starts and ends at the two tangent points, each $R\tan(\Delta/2)$ from the intersection. The time follows from the definition of a rate: $\Delta$ degrees at $ROT$ degrees a minute takes $\Delta/ROT$ minutes.

Assumptions

• The rate is reached at the wheel-over point and held to the end of the turn. A real ship takes time and distance to build it up.
• The speed is constant through the turn. Most ships lose speed in a turn, and at a constant radius the rate falls with the speed.
• The radius is measured to the ship’s track, and the rate of turn is the same at every point on the hull, since the whole hull yaws together.
• The turn is through the water. In a current the circle drifts with the water, so the track over the ground is not a circle.

Worked example

Take 12 knots, a radius of 1.0 nm and a 60° alteration to starboard from a course of 000°.

1. Rate of turn: 12 ÷ 1.0 = 12 radians an hour; × 180 ÷ π ÷ 60 = 11.459° a minute, shown as 11.5°/min.
2. Time to alter course: 60 ÷ 11.459 = 5.236 minutes, or 5 min 14 s.
3. Arc: 1.0 × 60 × π ÷ 180 = 1.047 nm.
4. Wheel over: 1.0 × tan 30° = 0.577 nm, or 5.8 cables, before the intersection.
5. New course: 000° + 60° = 060°.

Working back from a held rate: at 15 knots and 30° a minute, the radius is 0.9549 × 15 ÷ 30 = 0.477 nm, or 884 m.

Edge cases and limits

A 180° alteration has no intersection, because the two courses are parallel, so the calculator gives the time and the arc and no wheel-over distance. An alteration above 180° is refused: the same heading is reached sooner by turning the other way.

Above 30° a minute the result passes the end of the ±30° a minute scale that A.526(13) section 2.4.1 requires as a minimum. An indicator with only that scale shows full-scale deflection, which section 2.3.2 requires the display to make clear. The calculator flags a radius under 0.1 nm or over 10 nm, a screening band of this site’s own, because a radius typed in meters moves the result by a factor of 1,852. A radius in cables typed as nautical miles is 10 times too large and passes the band unless it goes over 10 nm, so check the unit.

Where the formula comes from in IMO instruments

No IMO instrument prints the rate of turn formula; it follows from the definition. IMO resolution MSC.74(69), Annex 2, defines rate of turn as the “change of heading per time unit” and the wheel-over line as “the line where the ship has to initiate a curved track”, and its section 5.1.8 has track control systems turn on a preset radius or on a radius calculated from a preset rate of turn. MSC.64(67), Annex 3, section 3.2 sets the same two modes for heading control systems, and its section 8.2 connects the system to a speed source in radius mode. SOLAS Chapter V Regulation 19.2.9.1 makes a rate of turn indicator, or other means, compulsory on ships of 50,000 gross tonnage and upwards, and A.526(13) sets the indicator’s performance.

Common errors

• Degrees a second read as degrees a minute, a factor of 60. MSC.74(69) states its track control limit as 10° a second; A.526(13) scales are in degrees a minute.
• Turning diameter entered as the radius, which halves the rate of turn.
• Radius in cables or meters entered as nautical miles.
• The wheel-over line drawn at the geometric point, with no allowance for the time the ship takes to start turning.
• Service speed used for a turn the ship makes after slowing down.

Rate of turn by speed and radius

Each figure in this table is the rate of turn in degrees a minute, $3/\pi \times V/R$, rounded to one decimal:

Doubling the speed on the same radius doubles the rate, and doubling the radius at the same speed halves it. The first column shows why a tight bend is taken slowly: at 8 knots a quarter-mile radius already needs more than the 30° a minute on the minimum indicator scale.

The wheel-over factor $\tan(\Delta/2)$ for common alterations:

Degrees a minute, degrees a second and the AIS field

The bridge indicator reads in degrees a minute, and A.526(13) section 2.4.1 marks its scale every 1° a minute with figures every 10°. MSC.74(69) states the track control limit in degrees a second, a maximum rate of turn of 10° a second; 11.5° a minute is 0.191° a second.

AIS sends the rate in a third form. The USCG Navigation Center’s description of the Class A position report, after ITU-R M.1371-5, gives the field as 4.733 × √ROT rounded to the nearest whole number, from 0 to +126 to starboard and 0 to −126 to port. The code is not linear:

The value +127 or −127 means a turn of more than 5° in 30 seconds with no indicator connected, and −128 means no turn information. The Navigation Center adds that the rate “should not be derived from COG information”. To read a rate back from the field, divide by 4.733 and square: a field of 16 is 11.4° a minute, so the decoded figure carries the rounding of the code.

Where to find each input

The radius comes from the passage plan, where it is chosen for each alteration and drawn on the chart as the arc between the legs, with the wheel-over line across the old course. The ship’s turning ability sets its lower limit. IMO resolution MSC.137(76), recommended for ships of 100 m and over and for all chemical tankers and gas carriers built from 1 January 2004, counts a tactical diameter of no more than five ship lengths at 35° rudder on full-load trials as satisfactory, and the maneuvering booklet on board gives the trial results for the ship.

The speed is the log speed expected during the turn. The rate of turn, when it is the known figure, is read from the indicator or set on the heading or track control system. A course before the turn, entered in degrees true, gives the new course. Bowditch (NGA Publication 9, Chapter 10, Article 1002) plans a turn the other way, from the advance and transfer in the ship’s turning data, and its worked example is a 60° turn with an advance of 920 yards and a transfer of 350 yards.