Horizon
Most commonly, the horizon is the border between the surface of a celestial body and its sky when viewed from the perspective of an observer on or above the surface of the celestial body. This concept is further refined as -
- The true or geometric horizon, which an observer would see if there was no alteration from refraction or from obstruction by intervening objects. The geometric horizon assumes a spherical earth. The true horizon takes into account the fact that the earth is an irregular ellipsoid. When refraction is minimal, the visible sea or ocean horizon is the closest an observer can get to seeing the true horizon.
- The refracted or apparent horizon, which is the true horizon viewed through atmospheric refraction. Refraction can make distant objects seem higher or, less often, lower than they actually are. An unusually large refraction may cause a distant object to appear above the refracted horizon or disappear below it.
- The visible horizon, which is the refracted horizon obscured by terrain and, on Earth, by life forms such as trees and/or human constructs such as buildings.
In perspective drawing, the horizon line is the point of view from which the drawn scene is presented. It is an imaginary horizontal line across the scene. The line may be above, level with, or below the center of the drawing, corresponding to looking down, straight at, or up to the drawn scene. Vanishing lines run from the foreground to one or more vanishing points on the horizon line.
Etymology
The word horizon derives from the Greek ὁρίζων κύκλος 'separating circle', where ὁρίζων is from the verb ὁρίζω ' divide, separate', which in turn derives from ὅρος 'boundary, landmark'.True horizon
The true horizon surrounds the observer and it is typically assumed to be a circle, drawn on the surface of a perfectly spherical model of the relevant celestial body, i.e., a small circle of the local osculating sphere. With respect to Earth, the center of the true horizon is below the observer and below sea level. Its radius or horizontal distance from the observer varies slightly from day to day due to atmospheric refraction, which is greatly affected by weather conditions. Also, the higher the observer's eyes are from sea level, the farther away the horizon is from the observer. For instance, in standard atmospheric conditions, for an observer with eye level above sea level by, the horizon is at a distance of about.When observed from very high standpoints, such as a space station, the horizon is much farther away and it encompasses a much larger area of Earth's surface. In this case, the horizon would no longer be a perfect circle, not even a plane curve such as an ellipse, especially when the observer is above the equator, as the Earth's surface can be better modeled as an oblate ellipsoid than as a sphere.
Distance to the horizon
Formula
The distance to the true horizon from an observer at height above the surface of a celestial body assumed to be perfectly spherical can be calculated using the formula:Where:
- is the distance to the true horizon;
- is the radius of the assumed body, e.g., the Earth's arithmetic mean radius of 6,371,008.77138 meters ;
- is the height of the observer above the surface, e.g., the Earth's orthometric height.
Examples
- For an observer standing on the ground with h =, the horizon is at a distance of.
- For an observer standing on the ground with h =, the horizon is at a distance of.
- For an observer standing on a hill or tower above sea level, the horizon is at a distance of.
- For an observer standing on a hill or tower above sea level, the horizon is at a distance of.
- For an observer standing on the roof of the Burj Khalifa, from ground, and about above sea level, the horizon is at a distance of.
- For an observer atop Mount Everest, the horizon is at a distance of.
- For an observer aboard a commercial passenger plane flying at a typical altitude of, the horizon is at a distance of.
- For a U-2 pilot, whilst flying at its service ceiling, the horizon is at a distance of.
Other planets
Derivation
If the Earth is assumed to be a featureless sphere with no atmospheric refraction, then the distance to the horizon can be calculated. using the Pythagorean theorem.At the horizon, the line of sight is a tangent to the Earth and is also perpendicular to Earth's radius.
This sets up a right triangle, with the sum of the radius and the height as the hypotenuse.
With
- d = distance to the horizon
- h = height of the observer above sea level
- R = radius of the Earth
which may be solved to yield
where R is the radius of the Earth. For example,
if a satellite is at a height of 2000 km, the distance to the horizon is ;
neglecting the second term in parentheses would give a distance of, a 7% error.
Approximation
If the observer is close to the surface of the Earth, then h is a negligible fraction of R and can be disregarded the term, and the formula becomes-Using kilometres for d and R, and metres for h, and taking the radius of the Earth as 6371 km, the distance to the horizon is
Using imperial units, with d and R in statute miles, and h in feet, the distance to the horizon is
If d is in nautical miles, and h in feet, the constant factor is about 1.06, which is close enough to 1 that it is often ignored, giving:
These formulas may be used when h is much smaller than the radius of the Earth, including all views from any mountaintops, airplanes, or high-altitude balloons. With the constants as given, both the metric and imperial formulas are precise to within 1%.
If h is significant with respect to R, as with most satellites, then the approximation is no longer valid, and the exact formula is required.
Related measures
Arc distance
Another relationship involves the great-circle distance s along the arc over the curved surface of the Earth to the horizon; this is more directly comparable to the geographical distance on a map.It can be formulated in terms of γ in radians,
then
Solving for s gives
The distance s can also be expressed in terms of the line-of-sight distance d; from the second figure at the right,
substituting for γ and rearranging gives
The distances d and s are nearly the same when the height of the object is negligible compared to the radius.
Zenith angle
When the observer is elevated, the horizon zenith angle can be greater than 90°. The maximum visible zenith angle occurs when the ray is tangent to Earth's surface; from triangle OCG in the figure at right,where is the observer's height above the surface and is the angular dip of the horizon. It is related to the horizon zenith angle by:
For a non-negative height, the angle is always ≥ 90°.
Objects above the horizon
To compute the greatest distance DBL at which an observer B can see the top of an object L above the horizon, simply add the distances to the horizon from each of the two points:For example, for an observer B with a height of hB1.70 m standing on the ground, the horizon is DB4.65 km away. For a tower with a height of hL100 m, the horizon distance is DL35.7 km. Thus an observer on a beach can see the top of the tower as long as it is not more than DBL40.35 km away. Conversely, if an observer on a boat can just see the tops of trees on a nearby shore, the trees are probably about DBL16 km away.
Referring to the figure at the right, and using the [|approximation above], the top of the lighthouse will be visible to a lookout in a crow's nest at the top of a mast of the boat if
where DBL is in kilometres and hB and hL are in metres.
As another example, suppose an observer, whose eyes are two metres above the level ground, uses binoculars to look at a distant building which he knows to consist of thirty stories, each 3.5 metres high. He counts the stories he can see and finds there are only ten. So twenty stories or 70 metres of the building are hidden from him by the curvature of the Earth. From this, he can calculate his distance from the building:
which comes to about 35 kilometres.
It is similarly possible to calculate how much of a distant object is visible above the horizon. Suppose an observer's eye is 10 metres above sea level, and he is watching a ship that is 20 km away. His horizon is:
kilometres from him, which comes to about 11.3 kilometres away. The ship is a further 8.7 km away. The height of a point on the ship that is just visible to the observer is given by:
which comes to almost exactly six metres. The observer can therefore see that part of the ship that is more than six metres above the level of the water. The part of the ship that is below this height is hidden from him by the curvature of the Earth. In this situation, the ship is said to be hull-down.