Re: Visibility of very tall structure?

Peter Knutsen wrote:
Assuming a planet the size of Earth, how far away can
a 1 km tall pillar-like structure be seen, if it's on
a flat plain? The viewpoint is that of a roughly
1.7-1.8 m tall human standing at sea level, but the
height of that person probably won't make a huge
difference.

Is there an easy formula for calculating this?

Draw a right triangle: The right angle is at the
surface of the Earth where the line-of-sight from
the mega-tree just kisses the horizon. One leg is
from the horizon-point down to the center of the
Earth (length R, the radius of the Earth). The
second leg is from the horizon to the top of
the tree (length to be announced). The hypotenuse
is from the top of the tree down to the center
of the Earth (length R + h, with h the height of
the tree).

Label the angle at the center of the Earth q.
Label the distance along the surface of the Earth
from the horizon-point to the mega-tree
(which is what we want to know) d.

The arc-length d of a circle of radius R, subtending
an angle q is R*q, so
d = R*q

The cosine of the angle at the center of the Earth,
q, is the near leg (R) over the hypotenuse (R + h), so
cos(q) = R/(R + h)
or
cos(d/R) = R/(R + h)

We can simplify this a lot if we assume that h << R
(which is still true, even if h is for a 1 kilometer
tall tree).

cos(d/R) = 1 - 1/2*(d/R)^2 + 1/4!*(d/R)^4 - ...

R/(R + h) = 1 - h/R + (h/R)^2 - (h/R)3 + ...
Thus
1 - 1/2*(d/R)^2 ~= 1 - h/R

d ~= sqrt(2*h*R)


(Does it matter that the structure is about 70 meters
wide? It's a giant tree, with leaves, and I imagine
its "crown" is something like 300 meters wide.)

I'm not sure, off hand, how to answer this.

If we take R = 6371 km, I get tree-to-horizon of
113 km and human-to-horizon of 4.7 km.

You could pick a solitary tree, let's say 10 meters
tall, 70 cm thick with a 3 meter crown and back away
about 1.13 km. That tree would look very much like
the mega-tree to you -- as near as could be easily
simulated.

But how the mega-tree looked would depend a lot
upon viewing conditions, which could be very different.
It would look redder, for example, just like the setting
or rising sun. Air molecules scatter bluer wavelengths
of light more.

Just before sunrise or just after sunset, there should
be places to stand where the Tree, just poking
over the horizon, is backlit by the Sun, but you
are still in night. That would be very visible, I
would think. Very moving, too.

Jim Burns
.

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