Understanding the 5km Horizon: How Far Can We See at the Beach?

[jfoldbar]

i recently learned that according to maths, the horizon for a human is about 5klms, and if this human was 100meters tall, horizon would be 40 klm away.

in that case, if we are standing at the beach, how can we see a boat (or flat land) that is more than 40 klms away?

 

[Bystander]

https://en.wikipedia.org/wiki/Fata_Morgana_(mirage)

 

[Ibix]

Also, the calculation depends on the height of the object you are looking at. You'll be able to spot the masts of a ship while its hull is still below the horizon.

Edit: a ship whose masts you can see but not her hull is said to be "hull down", if I remember my Hornblower correctly.

 

[jfoldbar]

sure. this stuff makes sense.
the 'fact' that makes me question it is. bout 3 years ago i was in korea. i was at the beach, and i was extremely surprised that i could see islands out to sea. google maps puts one of the island that i could clearly see at about 50klm away. i read about this island, most of its elevation is about 30 meters above sea level, with a higher part about 100 meters above sea level. the whole island can be seen, including a substantial part of the 30 meter high parts. it was not a mirage.
what i saw directly conflicts with the maths.

 

[Bystander]

Is there a density gradient at work? An atmospheric refraction?

 

[Ibix]

A quick bit of geometry suggests that the horizon for an observer a height h meters above sea level is about $80 \sqrt{h/500}$ km away. This does indeed work out to 5km for a 2m tall person and about 20km for a 30m elevation chunk of rock. Horizons are additive in this sense, so the island should be just visible at 25km. Something else is going on.

You can see further if you are higher up - but you'd have to be up a large cliff to have your horizon at the necessary 30km. I presume you weren't.

Google Maps could be wrong. Do you have photos from different locations you could use to triangulate? Or access to navigational charts?

Finally, @Bystander's explanation is quite likely. If there is cool air near the sea and warmer air above light can be curved downwards, so light that "should" pass over your head ends up entering your eye and you can see over the horizon. If so, I would tend to expect the islands to be more visible at some times of the day and in some weather conditions than others. I won't venture a guess what those would be, though.

By the way, this thread would be better in one of the technical forums, since a lot of members don't look in General Discussion. I've reported it and suggested it be moved.

 

[sophiecentaur]

If the observer were in a boat and the tide happened to be low, at the time, there could be a difference in the effective height of the tops of the islands relative to sea level (@boat and @horizon). If the observer is standing "at the beach" it would be quite possible for their eye level to be four of five metres above sea level whilst they might think they were at 'sea level. In the UK, there is a tidal range of around 5m in many locations yet, standing at the high water mark. it is hard to appreciate that the sea is actually 5m below that level (plus the extra 1.5 metres for eye height.
Also, there are a number of different chart datums used for maps and charts in different administrations. There can be a disagreement of a couple of metres there.
As there has to be a rational explanation, it would be worth while re-doing the sums with modified heights for observer and the upper bits of the island and see if those numbers could explain the 'anomaly'.

 

[Ibix]

5m extra height for both 2m high person and 30m high island gets us up to ~35km, which is still a fair way short of 50km. We'd be able to see the 100m high part though - I make that visible at 57km. That opens up another possibility - how sure are you that you saw the 30m high part and not just some higher shoulder?

 

[nasu]

What island was that? And from where did you observe it?

 

[DarkBabylon]

There is also the effect of temperature gradients. The refractive index of the air changes with temperature, which causes sometimes the effect of being able to see objects further than what they should be even when the object should be below the horizon entirely. Basically the light rays sometimes can curve with the Earth's curvature. It doesn't happen all the time though, but it is typical behavior of light when you look over water.

 

[jfoldbar]

34°02'58.0"N 127°19'07.4"E is the island
i was on jeju island. on a few separate occasions on separate days i could see 3 islands to the north. this island was one of them. i done remeber the exact beaches i was on when i saw these islands, but they were on the north and east side of jeju. i was there for 1 month and basically any clear day these islands could be seen easily.

i see the point about the light bending from certain weather conditions. it does make sense. i geuss i could try to investigate if there are certain clear days that the islands can not be seen.

i remember speaking to an old guy once who was on a battle ship. he said the guns could fire about 50 klms because that was the distance they could see an enemy ship

 

[A.T.]

34°02'58.0"N 127°19'07.4"E is the island
i was on jeju island

That's like 70km away.

 

[sophiecentaur]

There is also the effect of temperature gradients. The refractive index of the air changes with temperature, which causes sometimes the effect of being able to see objects further than what they should be even when the object should be below the horizon entirely. Basically the light rays sometimes can curve with the Earth's curvature. It doesn't happen all the time though, but it is typical behavior of light when you look over water.

That is true. If there is a negative density gradient with height (less dense as you go up), then the wave speed will increase with height and that will refract the light on a downwards curved path. In an ideal atmosphere there will be a small gradient but I don't think it amounts to much of an effect but a positive temperature gradient can increase the rate of change of density enough to extend the optical range.
Optical mirages are caused by total internal reflection at the interface between warm and cold air. But the image gets inverted so it is relatively easy to spot the difference - unless you are on a life raft and loopy with hunger and thirst.
At the other end of the frequency scale, Medium Frequency radio waves (vertical polarisation) travel way beyond the horizon due to a forward tilt of the wave. This is caused by the resistivity of the ground. Early experimenters were very surprised by the range that they could communicate over, using this Ground Wave effect. Reflection by the Ionosphere is another phenomenon and so is the 'ducting' effect that you can get with UHF and VHF signals which can be trapped in a warm layer of air (acting like a waveguide due to total internal reflection).

 

[jbriggs444]

i remember speaking to an old guy once who was on a battle ship. he said the guns could fire about 50 klms because that was the distance they could see an enemy ship

Shooting at an enemy ship at that range is a waste of ammo. The two longest recorded hits in WWII were at 24 km.

 

[russ_watters]

i recently learned that according to maths, the horizon for a human is about 5klms, and if this human was 100meters tall, horizon would be 40 klm away.

in that case, if we are standing at the beach, how can we see a boat (or flat land) that is more than 40 klms away?

If the object is low itself, you can't. but for a boat that has some size or a building on land, the visible distance is your horizon distance plus its horizon distance.

 

[jfoldbar]

good point about 'their/its' horizon distance. that still puts 6klms way short of 50.

interesting that the longest recorded ww2 hit was 24 klms. how did they see what they hit if it was past the horizon.? or do the maths allow to see from bridge height to top of other ship?

another example i suddenly remembered was melbourne, australia harbour. I've been there a few times (im from sydney). from the queenscliff- sorrento ferry and wharves, melbourne city can easily be seen on a clear day. its some 70-80 klms away.
while true big buildings would increase the visible distance, i reckon the a ferry that is about 6 meters high and 2 story houses on the other size (their roofs can just be seen) is still way past the maths.im sure throughout the world there would be many other places where people can see things that conflict with the horizon maths, but those 2 place are just my experience that's all

 

[sophiecentaur]

many other places where people can see things that conflict with the horizon maths,

Not if you do the correct Maths! You really can't argue with simple school geometry - which is all that's necessary.
You can get any answer of you do the wrong sums.

 

[A.T.]

interesting that the longest recorded ww2 hit was 24 klms. how did they see what they hit if it was past the horizon.? or do the maths allow to see from bridge height to top of other ship?

Also the smoke from the chimneys.

 

[A.T.]

im sure throughout the world there would be many other places where people can see things that conflict with the horizon maths,

Yep, the Earth is flat.

 

[A.T.]

34°02'58.0"N 127°19'07.4"E is the island

That one is 70 km from JeJu, and has peaks above 200m, according to Google Earth.

But there are closer islands:
33°58'58.73"N 126°55'44.50"E : 50km with peaks around 330m
33°43'45.28"N 126°21'28.44"E : 25 km with cliffs around 25m (probably an underestimate - below height map resolution)

 

[DaveC426913]

You must take into account the bending of lines of sight over water.

I live on the shore where the far side of the lake is 25km away. On certain days I cannot see the far shore whereas on other days I can easily see some of the taller buildings.

It is dependent on the temperature of air and the temperature of the water, both of which are dependent on the time of year. (Water temp lags behind the seasonal temp by several months, so you can have warm water under cool air and cool water under warm air.) The water temp causes a temp/density gradient in the air very near the water's surface.

The upshot is that, in the right conditions, it is possible to see quite a bit farther beyond the "ideal" horizon.

 

[jbriggs444]

interesting that the longest recorded ww2 hit was 24 klms. how did they see what they hit if it was past the horizon.? or do the maths allow to see from bridge height to top of other ship?

The gun director is located in the superstructure, not on the bridge. Naval gunnery is (or was) done by a series of salvos directed at the target's anticipated position. One of the important components of the initial salvos was range finding. One observes the fall of shot in a salvo to see if all shots are long, all are short or if you have found the range and have a "straddle". Accordingly, it is not good enough merely to see the target. One must be able to see one's own shell splashes.

 

[russ_watters]

interesting that the longest recorded ww2 hit was 24 klms. how did they see what they hit if it was past the horizon.? or do the maths allow to see from bridge height to top of other ship?

I was a navigator in the Navy and we kept tables handy on the bridge to calculate horizon and visual range to objects. It was mostly for identifying when we should see navigation markers (based on their height and ours) but also applied to sighting other ships.

 

[Bystander]

if it was past the horizon.?

Google "indirect fire."

 

[A.T.]

This is what happens when the job is given to an artist, who misses the entire point. Most of the ship, especially the chimney for which the light path is shown, are not beyond the horizon anyway.

 

[A.T.]

I live on the shore where the far side of the lake is 25km away. On certain days I cannot see the far shore whereas on other days I can easily see some of the taller buildings.

How high are those building tops and you above the water?

 

[sophiecentaur]

And how high are your eyes above sea level?

 

[Sue Rich]

I can't really see how an accurate distance can be defined, since the horizon moves with you.

 

[DarkBabylon]

I can't really see how an accurate distance can be defined, since the horizon moves with you.

We can move our point of origin with us, such that if you'd look at us with this new point of origin, you would see everything else move even if in reality we are the ones moving. Of course point of origin is where you define a point to be in the co ordinates (0,0,0) and if you were to move, say 1 meter towards the north, and it is symbolized by the second number, you would be (0,1,0) if the north is positive.
Once we establish that, we can then use an informed assumption that the Earth is a sphere and move on thanks to Eratosthenes who has tested it, and proofs piled up ever since.
With a bit of trigonometry, you can find how far can you see with respect to you. If you want to know what a person 20 kilometers away can see, you just shift the circle of the horizon you would get from the calculation to move with him. Of course changing the height if needed.
Because it moves with us, we can define an accurate distance of the horizon with respect to us. If you want to shoot a canon at a certain point, you will have to take distance and velocity into account sure.

 

[trainman2001]

Gun directors on later model battleships (Iowa class) were at the very top of the highest structure on the ship, the air defense tower. They were about 100 feet above the waterline, so their apparent horizon was quite far off, although the guns could fire farther than that. That's what the spotting planes were for (drones today). They were sent out so they could observe the target even when over the horizon and report on shell impacts where the aimers could then adjust the weapons. The 16", 50cal guns on the Missouri had an effective range of 25 miles. There were very few ship to ship battles in WW2 since aircraft were so much more effective. None of the Iowas fired any of their big guns on other capital ships. They were used for shore bombardment where the targets were generally stationary.

 

[DaveC426913]

And how high are your eyes above sea level?

Total distance of 51km.

Water tower is about 80 feet tall. (at least it was. It seems to be missing now.)

On a good day I can see it from the Islington bridge, which is maybe 40 feet high.
On a very good day, I see see it from the shoreline (15 feet).

For a long time I toyed with the idea of going out every few days for a year to the exact same spot and taking a pic of the horizon with some referencible landmarks in view - just to show a montage of the horizon in varying temperatures and seasons. (I may have waited too long.)

 

[DaveC426913]

This is what happens when the job is given to an artist, who misses the entire point. Most of the ship, especially the chimney for which the light path is shown, are not beyond the horizon anyway.

It gets the point across.

It would be very difficult - and not very informative - to try to render this to-scale.

And, I confess, there is an extreme paucity of such diagrams out there. This is the best I could find.