Why meridians converge
Meridians are great semicircles that meet at both poles. Their changing direction creates earth convergency, and that convergency explains why a great-circle track changes as an aircraft crosses successive meridians.
Diverging, parallel and converging meridians
Starting at the North Pole, two meridians diverge as they run towards the equator. At the equator their local directions are parallel. They then converge through the Southern Hemisphere and meet again at the South Pole. The separation in longitude is angular and remains fixed, but the inclination between the local north directions changes with latitude.
| Latitude | Relationship of the meridians | Convergency for longitude difference λ |
|---|---|---|
| Equator, 0 degrees | Locally parallel | Zero |
| Intermediate latitude | Inclined towards the nearer pole | Between zero and λ |
| Either pole, 90 degrees | Meet at the pole | Equal to λ |
Definition
Earth convergency is the angle of inclination between two selected meridians at a stated latitude. The same geometry can be described as the change in great-circle track direction between those meridians, because true direction at each position is measured from the local meridian.
Convergency is the angle of inclination between two selected meridians measured at a given latitude.Oxford ATPL Book 10, chapter 14
Inclination between two meridians at a given latitude is called convergency.R.K. Bali, Air Navigation, chapter 4
The limiting cases
For meridians 60 degrees apart, convergency is zero at the equator and 60 degrees at either pole. At 30 degrees latitude it is 30 degrees because sine 30 degrees is 0.5. These limits are the quickest check on any calculated answer.
Earth convergency on Mercator charts
A Mercator chart draws every meridian as a parallel straight line, so the graticule does not display earth convergency directly. A rhumb line is straight on the chart. A great circle away from a meridian or the equator is curved, with its arc concave towards the equator and its middle lying on the poleward side of the rhumb line. The track still changes by earth convergency even though the chart meridians look parallel.
Calculating earth convergency
Convergency is proportional to both the change of longitude and the sine of latitude. For two points on different parallels, use mean latitude for the operational calculation.
Same-latitude case
When both points lie on the same parallel, that latitude goes directly into the formula. For a 40 degree longitude difference at 30 degrees north, convergency = 40 × sin 30 degrees = 20 degrees.
Different-latitude case
When the points have different latitudes, the exact formula needs mean latitude along the great circle. A simple arithmetic mid-latitude is normally used because the numerical difference is insignificant for the moderate longitude changes found in these problems. Mean latitude is slightly nearer the pole than mid-latitude except in special symmetrical cases.
| Step | Action | Example, 40 N 002 W to 50 N 010 E |
|---|---|---|
| 1 | Find the smaller change of longitude | 12 degrees |
| 2 | Find the working mean latitude | 45 degrees north |
| 3 | Apply λ × sin mean latitude | 12 × 0.7071 |
| 4 | Round sensibly | 8.5 degrees convergency |
Signs and magnitudes
The formula normally gives the magnitude of convergency. Hemisphere and direction of travel determine whether the numerical great-circle track increases or decreases. Keep the arithmetic positive first, then apply the geometry from a sketch.
Convergency and great-circle track
A great circle is fixed on the Earth, but its true track is measured from a different local meridian at every position. The change in its true track between two meridians equals earth convergency.
Track change along a great circle
Suppose a great circle crosses one meridian on 060 degrees true and the next meridian is inclined by 8.5 degrees. In the Northern Hemisphere while travelling east, the track increases to 068.5 degrees true. Its reciprocal direction from the second point is 248.5 degrees true.
| Travel and hemisphere | Change in great-circle track number | Sketch cue |
|---|---|---|
| Eastbound, Northern Hemisphere | Increases | Meridians converge ahead to the north |
| Westbound, Northern Hemisphere | Decreases | Reverse the eastbound change |
| Eastbound, Southern Hemisphere | Decreases | Meridians converge ahead to the south |
| Westbound, Southern Hemisphere | Increases | Reverse the eastbound change |
The D I I D memory pattern
Read the sequence as westbound north, decrease; eastbound north, increase; westbound south, increase; eastbound south, decrease. A drawn pair of meridians is safer than memory alone, particularly near the 180 degree meridian.
Finding longitude from track change
A great-circle track changes from 300 degrees true to 295 degrees true between 36 degrees north and 42 degrees north. Convergency is 5 degrees and mean latitude is 39 degrees north. Change of longitude = 5 ÷ sin 39 degrees = about 8 degrees. If the second point is west of 015 degrees east, its longitude is 007 degrees east.
Intermediate track
If track changes steadily for a modest longitude span, the track at the mid-longitude is approximately midway between the endpoint tracks. In the previous example, longitude 011 degrees east lies halfway between 015 degrees east and 007 degrees east, so the track there is about 297.5 degrees true.
Conversion angle
Conversion angle is the angular difference, at one endpoint, between the great-circle direction and the rhumb-line direction joining the same two positions.
Why it is half
For equal-latitude endpoints the geometry is symmetrical about the mid-longitude. The great-circle track is parallel to the constant rhumb-line direction at the midpoint. It differs by one conversion angle at departure and the equal opposite conversion angle at arrival. The total endpoint change is therefore two conversion angles, which is the full convergency.
Between 30 degrees north, 020 degrees west and 30 degrees north, 020 degrees east, convergency = 40 × sin 30 degrees = 20 degrees. Conversion angle is 10 degrees. The eastbound rhumb-line track is 090 degrees true, while the great circle starts on 080 degrees true and reaches the destination on 100 degrees true.
| Quantity | At departure | At midpoint | At arrival |
|---|---|---|---|
| Rhumb-line track | 090 degrees | 090 degrees | 090 degrees |
| Great-circle track | 080 degrees | 090 degrees | 100 degrees |
| Difference | 10 degrees | Zero | 10 degrees |
Converting great-circle and rhumb-line bearings
Conversion is a geometry problem before it is an arithmetic problem. Draw the meridians, mark the nearer pole, draw the rhumb line and great circle, then decide whether conversion angle is added or subtracted.
The poleward rule
The shorter great-circle arc lies on the poleward side of the rhumb line, except where the two paths coincide. In the Northern Hemisphere it bows north of the rhumb line. In the Southern Hemisphere it bows south. This decides the sign at each endpoint.
| Case | Initial great-circle track compared with rhumb-line track | Final great-circle track compared with rhumb-line track |
|---|---|---|
| Eastbound, Northern Hemisphere | Smaller by CA | Larger by CA |
| Westbound, Northern Hemisphere | Larger by CA | Smaller by CA |
| Eastbound, Southern Hemisphere | Larger by CA | Smaller by CA |
| Westbound, Southern Hemisphere | Smaller by CA | Larger by CA |
Worked Northern Hemisphere conversion
A is 55 degrees north on the prime meridian and B is 54 degrees north, 010 degrees east. Mean latitude is 54.5 degrees north. Conversion angle = one half × 10 × sin 54.5 degrees = about 4 degrees. If the initial great-circle track from A is 100 degrees true, the corresponding rhumb-line track at A is 104 degrees true because the eastbound Northern Hemisphere great circle starts poleward of the rhumb line.
Worked Southern Hemisphere conversion
For a westbound route at about 42.5 degrees south with 16 degrees change of longitude, convergency = 16 × sin 42.5 degrees = 10.8 degrees, so conversion angle is 5.4 degrees. If the initial great-circle track from the eastern point is 250 degrees true, it increases to 260.8 degrees at the western point. The reciprocal great-circle bearing back from the western point is 080.8 degrees true.
Reciprocals and hemisphere clues
When two initial great-circle bearings are compared, first reverse one by 180 degrees so both directions refer to the same passage through an endpoint. A difference of exactly 180 degrees can occur at the equator. Bali also uses the unadjusted separation of the two initial reciprocal bearings as a hemisphere clue: greater than 180 degrees indicates Northern Hemisphere geometry in the stated east-west arrangement, while less than 180 degrees indicates Southern Hemisphere geometry. A sketch must confirm the arrangement before applying the shortcut.
Worked route problems and checks
The chapter closes with the problem patterns most likely to be examined: endpoint bearing, waypoint track change, missing longitude, rhumb-line conversion and hemisphere recognition.
Endpoint and return bearings
| Given | Calculation | Result |
|---|---|---|
| A 40 N 002 W to B 50 N 010 E, initial GC 060 degrees | EC = 12 × sin 45 = 8.5 degrees | GC at B is 068.5 degrees, return initial is 248.5 degrees |
| H 40 S 170 W to G 45 S 174 E, initial GC 250 degrees westbound | EC = 16 × sin 42.5 = 10.8 degrees | GC at G is 260.8 degrees, return initial is 080.8 degrees |
| J and K at 58 degrees 12 minutes north, longitude difference 10 degrees | EC = 10 × sin 58 degrees 12 minutes = 8.5 degrees | CA = 4.25 degrees; RL eastbound is 090 degrees |
Waypoint changeover
An aircraft follows great-circle legs from 53 north, 030 west to 53 north, 020 west, then to 53 north, 010 west. Each 10 degree leg has conversion angle = one half × 10 × sin 53 degrees = about 4 degrees. Each leg begins near 086 degrees and ends near 094 degrees. At the middle waypoint the new leg begins again near 086 degrees, so the track command decreases by about 8 degrees.
Recovering position
At 30 degrees north, the great-circle bearing of A from B is 266 degrees while the eastbound rhumb line is 090 degrees. The corresponding eastbound great-circle direction at B is 086 degrees, giving conversion angle 4 degrees and convergency 8 degrees. Change of longitude = 8 ÷ sin 30 degrees = 16 degrees. From A at 012 degrees east, B is therefore at 028 degrees east.
Bali calculation set
| Problem pattern | Key operation | Verified result |
|---|---|---|
| Angle between GC and RL for 60 S 165 W to 60 S 177 E | CA = one half × 18 × sin 60 degrees | 7.8 degrees |
| GC track 255 degrees at A from 60 N 120 W to 60 N 140 W | EC = 20 × sin 60 degrees = 17.3 degrees, westbound Northern Hemisphere so the track number decreases | GC track at B about 238 degrees; initial GC track back from B about 058 degrees |
| GC track joining 59 S 141 W and 61 S 148 W | EC is about 7 × sin 60 degrees | The GC track number increases by about 6 degrees westbound in the Southern Hemisphere |
| RL track from 45 N 010 W to 48 N 015 W | Plot north and west components, or use the mean-latitude triangle | Approximately 315 degrees |
| GC from P to Q measured at P as 095 degrees in the Southern Hemisphere, CA 7 degrees | GC starts equatorward of the eastbound RL | RL track 102 degrees |
| A 70 S 60 W to B 70 S 30 E | EC = 90 × sin 70 degrees and CA is half | Initial GC track about 312 degrees |
| Mercator GC from 25 S 100 W to 35 S 40 W is 242 degrees | Convert the endpoint GC bearing using Southern Hemisphere geometry | RL track from B to A about 075 degrees |
Final answer checks
- Convergency must lie between zero and the change of longitude.
- Conversion angle must be exactly half convergency.
- The great circle must lie poleward of the rhumb line.
- Endpoint great-circle tracks must differ by the full convergency.
- A reciprocal requires 180 degrees, normalised to the range 000 to 359 degrees.
- Crossing 180 degrees longitude must use the smaller longitude difference unless the question says otherwise.