Why Does a Compass North Differ from Map North?
― In the Mountains, You Drift About 100 m Sideways Every Kilometre
When hikers and surveyors open a map, the first thing they do is draw several slanted pencil lines. These are called "magnetic north lines". The north a compass points to and the north on a map are not the same north. In Japan, the difference is said to be about 5 to 10 degrees. It is a small angle, but it opens wider the farther you walk.
You are walking through a forest with a map and a compass. The hut you are heading for lies "due north" on the map.
So you follow the compass needle and set off straight north. On the map, it is about 1 km.
But when you arrive, there is no hut. You have come out on a slope a little to the west. The compass is not broken, and the map is correct.
There are just two reasons
Map north is the north that the axis of Earth's spin points to. Compass north is the north that the axis of Earth's great magnet points to. These two axes are misaligned from the start.
The magnetic force is produced deep inside Earth, and its pattern is warped from place to place. So the direction and size of the gap change with where you stand.
These two effects combine to make the gap between "map north" and "needle north". The gap has a name: magnetic declination. Everywhere in Japan the needle leans west: about 9 to 10 degrees in Hokkaido, about 7.5 degrees in the Kanto region, and about 5 degrees in Okinawa.
Reason 1: Earth's magnet is tilted
The magnetic force around Earth is thought to be produced in the core at its centre. The axis of the magnet this creates is tilted by a dozen or so degrees from the axis Earth spins on. So the point the needle aims at is not the geographic North Pole on the map.
What is more, needle north does not stay still. It has been moving for a very long time. In the 20th century it lay in northern Canada, but it is now said to have moved toward Siberia. Map north stays put, while needle north is a moving landmark. Look at the left side of Figure 1. You can see that the vertical dotted line and the tilted solid line do not line up.
Reason 2: The gap differs from place to place
If Earth were a neat bar magnet, the gap could be worked out by calculation alone. But the real magnetic pattern is warped from place to place, and both its strength and direction change under continents and under oceans.
So the gap has to be measured region by region. The Geospatial Information Authority of Japan measures Earth's magnetism across the country and publishes maps showing how many degrees west the needle leans. The compass note printed at the edge of a topographic map is also a result of this survey.
The gap also shifts from year to year. In Japan, the westward angle is growing little by little, at a rate said to be a few arcminutes a year (one arcminute is one 60th of a degree). This is why the note on an old map is slightly off if you use it as it stands.
Airport runways have numbers painted on them, such as "34" or "16". These are the heading, measured from needle north, divided by 10. When needle north moves, the numbers no longer match the real direction, so runway numbers have been repainted in various places around the world. Use a moving landmark, and even the writing on the ground has to change.
Summary
A compass honestly points along Earth's magnetic force. But that magnetic axis does not line up with Earth's spin axis, so it does not match map north. In Japan it leans about 5 to 10 degrees west, and by that angle you drift sideways the farther you walk. That is why, in the field, people draw magnetic north lines on the map before they line up the needle.
The needle does not point to "north".
It only points along Earth's magnet.
How the needle comes to face north is explained in Why Does a Compass Needle Point North?, and how to use the stars as a fixed landmark is explained in Is the North Star the Brightest Star?
- Get a paper topographic map and find the note at its edge. It says how many degrees west the needle leans, for example "7°10′ west declination".
- With a protractor, draw several lines across the whole map, each tilted to the left of the map's vertical lines by that angle. These are your magnetic north lines.
- Outdoors, set the compass on the map and turn the whole map until the needle is parallel to the magnetic north lines. The map now faces the same way as the ground.
Iron posts, cars and overhead power lines pull on the needle. Step a few metres away and read it twice in different spots, and you will spot the problem.
For Those Who Want to Know More ― Terms, Formulas and Links to TextbooksEach part is labelled by level, from junior-high science to university courses
- Junior highCovered in junior-high science
- High schoolCovered in high-school "Basic Physics and Mathematics I"
- High school+Advanced high-school material, or a textbook sidebar topic
- UniversityNot taught in high school; university specialist material (geomagnetism)
- ResearchNot yet taught as settled fact, even at university; researchers are still investigating it
Junior highTerms: this phenomenon has a name
- Magnetic declination: the angle between map north (true north) and the north a compass points to (magnetic north). In Japan the needle leans west, so it is written as "west declination".
- Magnetic north line: a line drawn on a map, tilted by the declination. It is a tool for lining up the needle with the map, and is used by hikers and surveyors.
- Inclination (dip): the angle by which the needle points below (or above) the horizontal. In Japan the north end tilts downward, by about 50 degrees.
Junior highHigh schoolCheck it with a formula: how many metres is a 7.5-degree gap?
Multiply an angular gap by the distance travelled and it becomes a length. Here we calculate the sideways drift itself, the subject of this article.
| In symbols | d ＝ L × tan θ |
| In words | Sideways drift ＝ distance travelled × tangent of the declination |
| Where it comes from | It comes from the definition of trigonometric ratios in a right triangle. Take the distance travelled as the base and the declination as the angle at the start; the sideways drift is the opposite side. |
| Distance travelled L | 1000 m (1 km on the map) |
| Declination θ (around Kanto) | 7.5 degrees west, as commonly stated |
| Tangent of 7.5 degrees | About 0.132, as commonly stated |
| Spacing from valley to valley (rule of thumb in mountains) | 300 m |
| Sideways drift after 1 km (m) | 1000 × 0.132 ＝ 132 |
| Sideways drift after 5 km (m) | 132 × 5 ＝ 660 |
| How many valley spacings | 660 ÷ 300 ＝ 2.2 |
That is about 132 m at 1 km and about 660 m at 5 km. If one valley is 300 m wide, that is 2.2 times as much. It is enough to cross two ridges and end up in a different valley. The angle is small, but multiplied by distance it becomes a length that spans the terrain.
High schoolHigh school+What decides the direction the needle faces?
High schoolThe needle wants to face the horizontal part of Earth's magnetic field (the horizontal component). Earth's field points obliquely downward, so taking out only its horizontal direction gives the "needle north" on the map.
High school+Earth's magnetism at any point can be described by three quantities: declination, inclination and total intensity. These are called the three elements of geomagnetism. In the dipole approximation, which treats Earth as a single bar magnet, inclination and magnetic latitude are linked by tan(inclination) ＝ 2 × tan(magnetic latitude). How far real values stray from this approximation shows how strongly the field is warped from place to place.
UniversityHow is the world's declination calculated?
Earth's whole magnetic field is expressed as an expansion in spherical harmonics. The standard model whose coefficients (Gauss coefficients) are fixed from observatory and satellite records worldwide is the International Geomagnetic Reference Field (IGRF). The declination shown on maps and instruments is a value calculated by putting a place and a date into this model. The theory of the flows that generate the field itself is called geodynamo theory.
📖 Derivations and further reading: Geomagnetism (three elements and the dipole approximation) ／ Dynamo theory
ResearchWhat is still not well understood
- Why needle north moves fast. The north magnetic pole is said to have moved faster since the second half of the 20th century, but what change in flow inside the core is behind it is still debated.
- The weak region over the South Atlantic. There is an area of weak field stretching from the South Atlantic to Brazil, and it is said to be growing. Whether it signals the start of a geomagnetic reversal has not been settled.
- When, and how fast, a reversal will happen. Rocks hold records of north and south swapping many times in the past, but no method has yet been established to predict the timing or course of the next reversal.
In other words, this article too is "an explanation within what is known today". Declination values are also updated with each survey, so check the latest map when you actually use one.
Links to Textbooks (by Level)
| Level | Subject / unit | Where in this article |
|---|---|---|
| Junior high | Science Field 1, "Electric Current and Magnetic Fields" | Earth is a big magnet; the needle points along the magnetic field |
| High school | Mathematics I "Trigonometric Ratios" / Basic Physics "Magnetic Fields" | Finding sideways drift with the tangent; the idea of the horizontal component |
| High school+ | Earth Science, "Earth's Magnetic Field" | The three elements of geomagnetism; deviation from the dipole approximation |
| University | Geomagnetism and Geodesy | Spherical harmonic expansion; the International Geomagnetic Reference Field (IGRF) |
| Research | Solid Earth Physics | Movement of the magnetic poles; predicting geomagnetic reversals |
| ― | Link to daily life | Drawing magnetic north lines on a hiking map; reading the compass note on a topographic map |
- Geospatial Information Authority of Japan (国土地理院), "Geomagnetic Survey and Magnetic Charts" (地磁気測量・磁気図): the distribution of declination across Japan and its year-by-year change
- Japan Meteorological Agency (気象庁), Kakioka Magnetic Observatory (柿岡地磁気観測所) observation records: long-term changes in declination, inclination and total intensity
- International Association of Geomagnetism and Aeronomy (IAGA), "International Geomagnetic Reference Field (IGRF)" coefficient tables
- Japanese Wikipedia, "Geomagnetism" (地磁気) (summary of the three elements, the dipole approximation and pole movement)
*This article is a popular-science explanation for general readers. The figures given are rough guides to help you understand the mechanism. When you go into the mountains, follow the latest topographic maps, local guidance, and information from local governments and relevant agencies.