The formula for distance from an object on the other side of the horizon is quite correct. Buildings of different heights would have a horizon that overlaps with yours, so you would count twice From a point above the earth`s surface, the horizon looks slightly convex; it is a circular arc. The following formula expresses the basic geometric relationship between this visual curvature κ {displaystyle kappa }, the height h {displaystyle h} and the radius of the earth R {displaystyle R}: the distance to the horizon therefore depends on the height of your eyes above the water. If your eyes are 8 inches (20 cm) above the water, the distance from the horizon is about 1.6 km. A rough formula for calculating distance to the horizon is: Why does the National Weather Service give (maximum) visibility to 10.00 miles? Think that by a formula that is “given”, the eye height higher than the standard 6 feet. adult? They have the same 9-foot eye level, so your distance to the horizon is still 3.51 nautical miles. You will approach a harbor that has a lighthouse that appears on your map with a height of 81 feet. Using the same formula, you would find that 1.17 times the square root of 81 (1.17 * 9) = 10.53 nautical miles (the lighthouse can be seen 10.53 nautical miles above the horizon) This formula is more accurate. Verified with Autocad on model earth with a radius of 3,958.8 miles. Thus, the beautiful formulas for calculating the “distance to the horizon” are really only approximate approximations of the truth. You can usually think of them as accurate to a few percent. But sometimes they will be wild, especially if superior mirages are visible. Then it is common to see much further than usual – a condition known to be threatening.
For example, if the size of your eyes is 9 feet above the surface of the water, the formula would be: If the Earth were a void like the moon, the above calculations would be accurate. However, the Earth has an air atmosphere whose density and refractive index vary considerably depending on temperature and pressure. As a result, the light from the air is refracted to varying degrees, which affects the appearance of the horizon. Normally, the density of the air just above the Earth`s surface is higher than its density at higher altitudes. As a result, its refractive index is greater near the surface than at higher altitudes, causing the refracted light to move roughly horizontally downwards. [7] Therefore, the actual distance to the horizon is greater than the distance calculated with geometric formulas. Under standard atmospheric conditions, the difference is about 8%. This changes the factor from 3.57 in the metric formulas used above to about 3.86. [2] For example, if an observer stands on the coast, with eyes at 1.70 m above sea level, according to simple geometric formulas above the horizon should be 4.7 km. In fact, atmospheric refraction allows the observer to see 300 meters further and move the true horizon 5 km from the observer.
Outside the range of visual wavelengths, refraction will be different. For radar (e.B. for wavelengths of 300 to 3 mm, i.e. frequencies between 1 and 100 GHz), the earth`s radius can be multiplied by 4/3 to obtain an effective radius that gives a factor of 4.12 in the metric formula, that is, the radar horizon will be 15% above the geometric horizon or 7% above the visual horizon. The factor 4/3 is not accurate, because in the visual case, refraction depends on atmospheric conditions. For example, let`s say you`re on the water in a friend`s sport fishing boat and your eye level is 9 feet above the water surface. The formula for calculating the distance to the horizon is: If the observer is near the Earth`s surface, then it is valid to disregard h in the term (2R + h), and the formula is: At sea level, the curvature of the Earth limits the field of view to 2.9 miles. The formula for determining how many miles an individual can see at higher levels is the square root of their elevation times. Using this “typical” value means that we should only use the formula given above, but we should use an R′ value instead of R for the effective radius of the Earth, by which once you know the size of your eyes, you simply insert it into the following formula: with d, D and h, all measured in the same units. These formulas can be used when h is much smaller than the radius of the Earth (6371 km or 3959 miles), including all views of mountain peaks, planes or balloons at high altitude. For the given constants, the metric and imperial formulas are 1% accurate (see the next section on achieving greater accuracy). 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.
If you`re in a Jon boat, it would probably be about three feet (if you`re sitting as you should be in a Jon boat). .