How Far Away is the Horizon

APPARENT HORIZON: Where the sky appears to meet the Earth. (Also sea horizon.) Because of perspective effects, different observers generally have different apparent horizons. Because of refraction, even the sea horizon usually lies above the geometric horizon.

-From the Atmospheric Optics Glossary.

Knowing the distance to the horizon is handy when estimating how long it will take you to travel to a distant point. To accurately estimate to distance, understand that the apparent horizon changes based on the height of your eyes; the higher your eyes, the further you can see. That’s why it seems you can see forever when standing atop a mountain.

Estimating the Distance to the Apparent Horizon

First, determine the height of your eyes above the ground. This will vary depending on whether your standing, sitting, or if you’re high on a cliff above the water. Once you know the height of your eyes enter it into R. Langton Cole’s  formula from his 1913 article “Distance of the visible horizon,” which appeared in Nature:

distance to horizon (miles) = √ 7h(feet)/4

This formula gives you the rough distance (Imperial) of the apparent horizon accounting for normal refraction. This is the distance to where the sky appears to meet the Earth. If mirages are apparent, the distance to the apparent horizon may differ.

Because of the atmosphere, these calculations are estimates, so we can simplify to a more commonly used formula. This second formula is slightly easier to remember:

distance to horizon (miles) = sqrt [1.5h]

For paddlers living in advanced nations using the metric system, the formula is:

distance to horizon (kilometers) = sqrt [13h]

The Imperial formula uses feet for height (h), and the metric formula uses meters for height (h). Note: The abbreviation “sqrt” stands for square root.

Examples using the first horizon formula:

• Sitting in a kayak: The distance to the horizon for someone sitting in a kayak is approximately 2.1 miles. When you’re sitting in a kayak your eyes are about 2.5 feet above the surface of the sea–this is why waves appears so big when you’re in your kayak. 7 x 2.5 = 17.5 and 17.5 / 4 = 4.375 and sqrt of 4.375 = 2.09165.
• Sitting in a canoe: The distance to the horizon for someone sitting in a canoe is approximately 2.3 miles. Because your eyes are higher in a canoe–about three feet–you see further away to the horizon.
• Standing on shore (standing in a canoe): The distance to the horizon for someone standing is approximately 3.1 miles. I’m 5’10” tall and my eyes are about 68″ above my feet. When I’m on the beach and my feet are just touching the water, I can see about 3.1 miles.

Using Horizon Estimates on the Water

Quoting from Douglas Adams, “the practical upshot of all this is that” when you’re paddling and can just see on the horizon the point where the land meets the water, you know you’re seeing the apparent horizon. This means that you’re within 2.1 miles for a kayak or 2.3 miles for a canoe away. This is very handy when making long crossings. Or for knowing how far away the beach is.

Estimating Distance to Distant Objects

You can use this formula to estimate the distance to distant objects, too. Figure out the distance to the apparent horizon for the distant object and add that distance to your apparent horizon. When you can just see the top of the distant object, you are that distance away. The formula, using the second apparent horizon formula, looks like this:

distance between objects (miles) = sqrt [1.5h(distant object)] + sqrt [1.5h(your apparent horizon)]

Examples for Distance to Distant Objects:

• Grand Marais, MN lighthouse from a kayak: From the NOAA charts, we know that the Grand Marais, MN lighthouse focal plane is 48 feet above datum. The apparent horizon from the light is about 8.5 miles. When in a kayak, our apparent horizon is 2.1 miles. We add 8.5 to 2.1 and get 10.6 miles. When we can see the light, we know we’re about 10.6 miles away. This puts us near Cascade State Park or the 121 surf break. Chance are unless it’s a crystal clear day, we won’t be able to make out the lighthouse, but we may be able to see the flashing light.
• Carlton Peak from a kayak: Another northshore Lake Superior landmark is Carlton Peak. It stands out as a lone peak and rises to 1,526 feet, but it only rises 925 feet above Lake Superior. When atop the peak, your apparent horizon is about 37.2 (sqrt [925 * 1.5]) miles away. Add that to 2.1 and you should be about to see the peak just pop over the horizon at about 39.3 miles away.
• BWCA pine shoreline from a canoe: Mature pine trees range in height from 80 to 110 feet tall. So from the top, you could see about 11 miles to the horizon. In a canoe, you would see them appear above the water about 14 miles away.