How Far Away Can You See Mount Fuji?
― It Has Been Seen from a Mountain in Wakayama, Over 300 km Away
Tokyo is about 100 km from Mount Fuji. On a winter morning the mountain stands out sharply, yet for days on end in summer you can hardly see it. Two things decide how far you can see: the murk of the air and the curve of the Earth. Work out each one, and both limits settle at roughly 300 km.
On a clear winter morning, you look west from a station platform. A small white Mount Fuji floats on the horizon.
In summer you stand on the same platform, and the sky is milky and hazy. There is not even a shadow of the mountain.
Mount Fuji has not moved. What has changed is the 100 km of air between you and the mountain.
Only Two Things Decide How Far You Can See
Light from the mountain hits air molecules and tiny particles on its way and scatters. The farther away the mountain is, the more it melts into the colour of the sky, until you can no longer pick it out. In humid summer, the particles soak up water and swell, and the haze gets worse.
A distant mountain sinks below the horizon. Even from the summit of Mount Fuji, your sight line grazes the ground only about 219 km away. To see it from farther, the viewer must also be up high.
The first decides "whether you can see it today". The second decides "how far you could ever see it, however clear the day". Let's take them in turn.
Why Do Distant Mountains Look Pale and Hazy?
Light from a mountain travels through tens of kilometres of air before it reaches your eye. On the way it bumps into air molecules and dust, and changes direction. The light from the mountain fades bit by bit, while bright light from the sky mixes in from the side. So the farther the mountain, the closer its colour gets to that of the sky.
When the difference in brightness between mountain and sky shrinks to about 2% of the original, the human eye is said to be unable to tell them apart. This distance is called "visibility". Look at Figure 1. The slider changes how clear the air is, so you can see how well Mount Fuji shows at 100 km.
The curve in Figure 1 does not fall in proportion to distance. Instead, it shrinks by the same fraction with every 10 km. On a day with 100 km visibility, a mountain 50 km away keeps about 14% of the difference. At 100 km, that has fallen to 2%. Double the distance, and the remaining difference is squared.
Summer is hard on views because the tiny particles in the air swell as they soak up moisture. Bigger particles scatter light much more strongly. In winter, on the Pacific side, a dry north-westerly wind blows after dropping its snow on the Sea of Japan side. With little moisture and dust, there are more days when visibility is well over 100 km.
Even perfectly dust-free air still scatters light, though, because the air molecules themselves do it. It is the same effect that makes the sky blue. So even on the clearest day, visibility is calculated to top out at roughly 300 km.
How Much Does the Curve of the Earth Hide?
Even if the air were perfectly transparent, you could not see forever. The Earth is round. Look at Figure 2. Draw a line from the summit of Mount Fuji that just grazes the ground. The point where the line touches the ground is the horizon as seen from the summit. From a low place beyond it, the summit is hidden below the horizon.
The length of the line changes a lot with the height of the viewer. For a person standing at the seaside, with eyes about 1.6 m up, the horizon is only about 4.5 km away. From the summit of Mount Fuji (3776 m), it stretches to about 219 km. If the viewer is on a 1000 m mountain, that person's own horizon (about 113 km) is added, making about 330 km in total.
The air is also denser nearer the ground. Because of that difference, light bends slightly downward as it travels. In practice, this lets you see a little under 10% farther than the calculation suggests.
The farthest place from which Mount Fuji has been photographed is a mountain pass in Nachikatsuura, Wakayama Prefecture, at a distance of about 323 km. That distance can be reached only on a day when two conditions are met: a high vantage point and extremely clear air.
The visibility limit set by air molecules alone and the sight-line limit set by the curve of the Earth both come to the 300 km range. If only one of them is good, you still cannot see farther. The 300 km record is a distance where you hit both walls at almost the same time.
Summary
How far you can see is decided by the murk of the air and the curve of the Earth. In summer, particles swollen with moisture scatter the light, and even Mount Fuji 100 km away fades into haze. In winter, dry air lets you see far. However clear the air gets, though, the curve of the Earth and the air molecules themselves put up a wall at roughly 300 km.
The air decides "whether you can see it today",
and the curve of the Earth decides "how far you can ever see".
For more on how air molecules scatter light, see "Why Is the Sky Blue, but the Sunset Red?". For the clouds that sit over Mount Fuji, see "Why Is "Rain When Fuji Wears a Cap" So Often Right?".
- On a clear day, find a place where nearby and distant mountains appear one behind the other. Check that the farther the mountain, the paler and bluer it looks.
- From the same spot, record each morning for about a week how far you can see distant mountains and buildings. Write down the humidity from that day's forecast too.
- Using a map such as those from the Geospatial Information Authority of Japan, measure the distance to the mountains you could see. Compare: did you see farther on days with lower humidity?
When looking into the distance, do it in a safe place. Do not lean out over a railing at a height, and do not stop in the road.
For Those Who Want to Know More ― Terms, Formulas and Links to TextbooksEach part is labelled with its level, from middle-school science to university specialist courses
- Middle schoolCovered in middle-school science
- High schoolCovered in high-school physics or earth science
- High school+Advanced high-school material, or a textbook sidebar
- UniversityUniversity specialist courses (atmospheric optics, geodesy), not taught in high school
- ResearchNot yet settled even at university level; researchers are still investigating
Middle schoolTerms: This Phenomenon Has Names
- Visibility: the farthest distance at which you can clearly make out the shape of an object by eye. It is one of the items recorded in weather observations.
- Light scattering: light hitting air molecules or tiny particles and bouncing off in many directions. It is why distant scenery looks hazy.
- Atmospheric refraction: light bending because of differences in air density, so objects look slightly higher than they really are. The effect is larger near the horizon.
Middle schoolHigh schoolChecking with Formulas: How Far Is the Horizon from the Summit of Mount Fuji?
We work out the main quantity, how far you can see, from the curve of the Earth. We also work out the visibility set by how clear the air is.
| In symbols | d ＝ √( 2 × R × h ) |
| In words | Distance to the horizon ＝ the square root of (2 × Earth's radius × height) |
| Where it comes from | The Pythagorean theorem. The Earth's centre, the summit and the horizon point form a right triangle, so ( R ＋ h )² ＝ R² ＋ d². Since h is far smaller than R, we drop h² |
| The air side | V ＝ 3.912 ÷ σ (visibility ＝ 3.912 ÷ how strongly the air weakens light). The brightness difference shrinks by the same fraction with distance, and the distance at which it reaches 2% is the visibility. 3.912 is the natural logarithm of 50 |
What the symbols mean: d is the distance to the horizon (km), R is the Earth's radius (km), h is the height of the viewing point (km), V is visibility (km), and σ is how strongly light is weakened per kilometre (unit: 1/km).
| Earth's radius R | 6371 km |
| Height of Mount Fuji h | 3.776 km |
| Eye height of a person at the seaside | 0.0016 km (1.6 m) |
| σ of dust-free air (green light, near the ground) | said to be about 0.0116 /km |
| σ on a hazy summer day (one example) | about 0.2 /km |
| 2 × R | 2 × 6371 ＝ 12742 |
| Summit of Mount Fuji: 2 × R × h | 12742 × 3.776 ≒ 48114 |
| Its square root (summit to horizon) | The square root of 48114 is about 219 km |
| Person at the seaside: 2 × R × h | 12742 × 0.0016 ≒ 20.4 |
| Its square root (to the sea horizon) | The square root of 20.4 is about 4.5 km |
| Person on a 1000 m mountain: square root of (12742 × 1) | about 113 km |
| Add the two horizons | 219 ＋ 113 ＝ 332 km |
| Visibility of dust-free air | 3.912 ÷ 0.0116 ≒ 337 km |
| Visibility on a hazy summer day | 3.912 ÷ 0.2 ≒ 19.6 km |
Considering the curve of the Earth alone, the summit of Mount Fuji and a person on a 1000 m mountain can see each other across about 330 km. The visibility of dust-free air is also about 340 km. The numbers confirm that the two walls stand at almost the same height. On a hazy summer day, visibility is about 20 km, far too short to reach Mount Fuji 100 km away.
High schoolHigh school+Why Do Distant Mountains Look Blue-Hazy, and Humid Days White-Hazy?
High schoolScattering by things much smaller than the wavelength of light, such as air molecules, is called Rayleigh scattering. Its strength is inversely proportional to the fourth power of the wavelength, so blue light scatters most. Distant mountains look bluish because blue light scattered by the air in between is added on top. On the other hand, scattering by particles that have swollen with moisture to a size close to the wavelength of light is called Mie scattering. It is not very selective about colour, so the scene turns whitish-hazy.
High school+In surveying, the bending of light travelling near the ground is expressed by the "refraction coefficient", which averages about 0.13. Including this bending is equivalent to making the Earth's radius apparently about 7/6 times larger. The distance to the horizon grows by the square root of that, which is about 7%. The "a little under 10%" in the main text is based on this value.
UniversityKoschmieder's Law and the Hygroscopic Growth of Particles
The curve in Figure 1 is based on Koschmieder's law, which combines the Beer–Lambert law (light weakens by a constant fraction as it travels) with the effect of background sky light mixing in. The formula for visibility, 3.912 ÷ σ, is derived from it. The "meteorological optical range" defined by the World Meteorological Organization (WMO) puts the threshold at 5% rather than 2%, which gives a coefficient of about 3.0. The process by which particles take up water and grow with humidity is explained by Köhler theory, and it is known that they swell sharply as relative humidity rises.
📖 Derivations and further reading: Visibility (Wikipedia) ／ Atmospheric refraction (Wikipedia)
ResearchWhat Is Still Not Clear
- How differently each particle takes up water Airborne particles are a mix of sea salt, combustion particles, soil dust and more. Their ability to take up water differs greatly by composition, so predicting visibility accurately from humidity is still said to be difficult.
- Day-to-day changes in how light bends near the ground How light bends varies a lot with the temperature profile near the ground. Distant land and ships can appear to float upward, which adds uncertainty to sight-line calculations.
- The eye's "threshold of discrimination" The 2% and 5% thresholds are rules of thumb, and vary with brightness, object size and the observer. How to put real-world visibility into numbers is still debated.
In other words, this article too is "an explanation within what is known today". In particular, the distance for the farthest record is a value based on terrain calculations and a photographic report.
Links to Textbooks (by Level)
| Level | Subject / unit | Where in this article |
|---|---|---|
| Middle school | Maths "Pythagorean theorem", science "Properties of light" | The horizon-distance formula, light scattering |
| High school | Physics "Scattering of light", earth science "Structure of the atmosphere" | Rayleigh and Mie scattering, the dry winter north-westerly |
| High school+ | Physics "Refraction of light", advanced | The refraction coefficient, and why you see a little under 10% farther |
| University | Atmospheric optics, geodesy | Koschmieder's law, meteorological optical range, Köhler theory |
| Research | Aerosol science, atmospheric radiation | Differences in particle hygroscopicity, the threshold of discrimination |
| ― | Links to daily life | Why distant mountains show on winter mornings; caution when driving in poor visibility |
- Wikipedia (Japanese), "Visibility" (視程)
- Wikipedia (Japanese), "Atmospheric refraction" (大気差)
- Wikipedia (Japanese), "Rayleigh scattering" (レイリー散乱)
- Wikipedia (Japanese), "Mie scattering" (ミー散乱)
- Koschmieder, H. (1924) Theorie der horizontalen Sichtweite. Beiträge zur Physik der freien Atmosphäre.
- World Meteorological Organization, Guide to Instruments and Methods of Observation (WMO-No. 8), chapter on visibility observation.
*This article is a popular-science explanation for general readers. The figures given are rough guides to help you understand how things work. The distance of the farthest record is a value based on reports and calculation.