Distance to the Sea Level Horizon

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Equation / Last modified by KurtHeckman on 2019/05/01 23:44
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ID
MichaelBartmess.Distance to the Sea Level Horizon
UUID
795af1f9-18dd-11e4-b7aa-bc764e2038f2

The Distance to Horizon at Sea Level calculator computes the distance to the horizon at sea level based on the height of the observation.

INSTRUCTION: Choose units and enter the following:

  • (h) Height of the observation

Distance to Horizon (d): The calculator returns the distance in meters.  However, this can be automatically converted to compatible units via the pull-down menu.

The Math / Science

The Distance to the Horizon equation provides an approximation for the distance you can see to the horizon from a given height(h).  This is an approximation because, for one thing, the Earth is not a perfect sphere.  Also, this is likely to be used for instances where you are not looking at the horizon at sea level. But if we think of the Earth as a perfect sphere for this approximation and use one of our multiple vCalc values from the Constants taxonomy for the Earth's radius -- Yes, Flat Earth Society, the Earth is quite definitely round -- we can do a good approximation for the distance we can see at various elevations to the horizon. 

The equation uses the WGS-84 estimate of the Earth equatorial radius (RE) for these purposes, just because it is a well-known value for the Earth's radius.

This works most accurately for looking at the ocean horizon from some vantage point higher than the shoreline but we'll derive the equation so that you can use it also as an approximation for elevations above the local (spherical) plane of the Earth.  That way I can make an approximation for looking at the horizon from Mesa Verde, looking far south at what was probably a New Mexico horizon.

Input to this equation:

  •  (h) height = the height or difference in altitude of your viewing point above the surface of the Earth near the horizon

Output of this equation:

  • `"Distance"_"horizon"` = approximate distance to point on the horizon

 

DistancetotheSeaLevelHorizon-illustration.png

Notes

This equation approximates the distance to the horizon, assuming the following among many things:

  • the Earth is spherical
  • the WGS-84 equatorial radius is a good approximation for a spherical Earth model's radius
  • we neglect any affect the atmosphere may have on our seeing a point that far in the distance
  • the point we see at the horizon may be above sea level but it's distance above sea level is negligible when compared to the Earth's radius
  • our viewing location is at the same general altitude above sea level as the point we're viewing on the horizon

If we can accept all these assumptions, our approximation can be based on a very simple application of the Tangent-Secant Theorem

For a version of this calculation which takes our altitude and the altitude of the horizon into account, see Distance to Horizon at Altitude.

A test case:

Living in Colorado, where there are large open spaces unencumbered by trees (although mountains do seem to get in the way) and having just returned from a trip to the four corners area near Durango when I was creating this equation, I was wondering if I had been looking into New Mexico when I was looking to the horizon from atop the Mesa Verde. The view to the south looks far past the San Juan River but I was wondering if I was seeing as far as New Mexico.

While visiting Mesa Verde, I was wondering how far I could see from the top of the mesa looking South.  There is another mesa to the South which has a break and beyond the break in that mesa I could see to the horizon.  So, was I seeing far enough from the Mesa Verde elevation to see into New Mexico?

If I take my assumed elevation difference above the San Juan River to the South (around 4,600 feet) and my altitude on Mesa Verde (around 8,400 feet) -- so 3,800 feet elevation difference--  this equation's approximation tells me I can see a distance in the neighborhood of 75.5 miles (121.5 km).  That is easily enough to be seeing into New Mexico.  Even if this is a rough approximation, I know I am looking beyond the Colorado border.

Try this in central Wyoming too, where there are many places you can look all the way to the horizon in every direction -- 360 degrees around -- and not see a house, a power line or any vestige of human habitation. There you can truly see to the horizon in the crisp dry summer air.