Horizon

For other uses of horizon, see Horizon (disambiguation)

Horizon.

View of Earth's horizon as seen from Space Shuttle Endeavour, 2002.

The horizon (Ancient Greek ὁ ὁρίζων, /ho horídzôn/, from ὁρίζειν, "to limit") is the line that separates earth from sky.

More precisely, it is the line that divides all of the directions one can possibly look into two categories: those which intersect the Earth, and those which do not. At many locations, the true horizon is obscured by trees, buildings, mountains and so forth. The resulting intersection of earth and sky is instead known as the visible horizon.

Appearance and usage

When on a ship at sea, the true horizon is strikingly apparent. Historically, the distance to the visible horizon has been extremely important as it represented the maximum range of communication and vision before the development of the radio and the telegraph. Even today, when flying an aircraft under Visual Flight Rules, a technique called attitude flying is used to control the aircraft, where the pilot uses the relationship between the aircraft's nose and the horizon to control the aircraft. A pilot can also retain his or her spatial orientation by referring to the horizon.

In many contexts, in particular perspective drawing, the curvature of the earth is typically disregarded and the horizon is considered the theoretical line to which points on any horizontal plane converge (when projected onto the picture plane) as their distance from the viewer increases. Note that, for viewers near the ground, the difference between this geometrical horizon (which assumes a perfectly flat, infinite ground plane) and the true horizon (which assumes a spherical Earth surface) is typically imperceptibly small. That is, if the Earth were truly flat, there would still be a visible horizon line, and, to ground based viewers, its position and appearance would not be significantly different from what we see on our curved Earth.

In astronomy the horizon is the horizontal plane through (the eyes of) the observer. It is the fundamental plane of the horizontal coordinate system, the locus of points which have an altitude of zero degrees. While similar in ways to the geometrical horizon described above, in this context a horizon may be considered to be a plane in space, rather than a line on a picture plane.

Three types of horizon.

Distance to the horizon

The straight line of sight distance d in kilometers to the true horizon on earth is approximately

$d={\sqrt {13h}},$

where h is the height in meters of the eyes. Examples:

• Standing on the ground with h = 1.70 m (eye-level height), the horizon is at a distance of 4.7 km.
• Standing on a hill or tower of 100 m height, the horizon is at a distance of 36 km.

To compute to what distance the tip of a tower, the mast of a ship or a hill is above the horizon, add the horizon distance for that height. For example, standing on the ground with h = 1.70 m, one can see, weather permitting, the tip of a tower of 100 m height at a distance of 4.7+36 ≈ 41 km.

In the Imperial version of the formula, 13 is replaced by 1.5, h is in feet and d is in miles. Examples:

• Standing on the ground with h = 5 ft 7 in (5.583 ft), the horizon is at a distance of 2.89 miles.
• Standing on a hill or tower of 100 ft height, the horizon is at a distance of 12.25 miles.

The metric formula is reasonable (and the Imperial one is actually quite precise) when h is much smaller than the radius of the Earth (6371 km). The exact formula for distance from the viewpoint to the horizon, applicable even for satellites, is

$d={\sqrt {2Rh+h^{2}}},$

where R is the radius of the Earth (note: both R and h in this equation must be given in the same units (e.g. kilometers), but any consistent units will work).

Another relationship involves the arc length distance s along the curved surface of the Earth to the bottom of object:

$\cos {\frac {s}{R}}={\frac {R}{R+h}}.$

Solving for s gives the formula

$s=R\cos ^{-1}{\frac {R}{R+h}}.$

The distances d and s are nearly the same when the height of the object is negligible compared to the radius (that is, h<<R).

As a final note, the actual visual horizon is slightly farther away than the calculated visual horizon, due to the slight refraction of light rays due to the atmospheric density gradient. This effect can be taken into account by using a "virtual radius" that is typically about 20% larger than the true radius of the Earth.

Curvature of the horizon

From a point above the surface the horizon appears slightly bent. There is a basic geometrical relationship between this visual curvature $\kappa$, the altitude and the Earth's radius. It is

$\kappa ={\sqrt {\left({\frac {R+h}{R}}\right)^{2}-1}}\ .$

The curvature is the reciprocal of the curvature angular radius in radians. A curvature of 1 appears as a circle of an angular radius of 45° corresponding to an altitude of approximately 2640 km above the Earth's surface. At an altitude of 10 km (33.000 ft, the typical ceiling altitude of an airliner) the mathematical curvature of the horizon is about 0.056, the same curvature of the rim of circle with a radius of 10 metres that is viewed from 56 centimetres. However, the apparent curvature is less than that due to refraction of light in the atmosphere and due to the fact that the horizon is often masked by high cloud layers that reduce the altitude above the visual surface.

See also

• Dawn: the time right before sunrise
• Dusk: the time right after sunset, yielding to twilight
• Landscape
• Landscape art
• Aerial landscape art
• Sextant

External links

• Derivation of the distance to the horizon

Acknowledgments

The first version of this article originates from Jason Harris' Astroinfo which comes along with KStars, a Desktop Planetarium for Linux/KDE. See http://edu.kde.org/kstars/index.phtml