But the ray curvature depends on the atmospheric
*lapse rate*,
as described on the
bending page.

This page provides a simple JavaScript program to estimate the apparent
*altitudes*
of distant objects.

To begin with, we know that the 6.5°/km lapse rate of the Standard Atmosphere is unrealistic. So let's do the calculation for any specified lapse rate, γ.

We can either set the lapse rate directly, or calculate it from the temperatures at two different heights. The scripts used below allow both options.

Assuming a constant lapse rate greatly simplifies the problem.
Then, provided that the rays are nearly horizontal, and that the
thickness of the layer they span is much smaller than the 8 km
reduced height
of the atmosphere, the rays are nearly arcs of circles; so
we can make use of many simple results for this case, like Lambert's
**(1759)**
demonstration that such refraction produces a nearly undistorted image of
distant objects that is simply displaced in the vertical direction.
(Cf. the
bending page,
and the discussions of
terrestrial refraction,
and of
looming and sinking.)

Therefore, if we can find the apparent altitude of *any* point
of the distant object, we can easily find the altitudes of all the others.
(See the simulations of
looming
and
sinking
for ray-traces of examples.)

A second simplification is that good approximations for the
dip
and
distance of the horizon
are available in closed form for this case.
Both can be calculated from simple geometry if we adopt an effective
curvature for the Earth that's the *difference* of curvatures of
the actual surface and the circular ray.
It's as if the Earth's radius of curvature *R* had been replaced by
*R _{eff}* =

where *d* is the distance to the horizon, and
*h* is the height of the eye above the level surface that
forms the
*apparent horizon*.

The dip- and distance-formulae apply to any ray tangent to a spherical
surface of radius *R* ; such a ray is horizontal at the point of
tangency. The same geometric relations apply to any spherical surface; it
doesn't have to be the Earth's surface.
We can choose the surface to be a level surface through the observer, so
that the point of tangency is at the observer's eye. Then the height
*h* is the height of this ray above the eye, at a distance *d*
from the observer.

So let's consider a horizontal ray at the observer: everything along this
ray appears at the observer's *astronomical* horizon. The following
diagram shows the details.

Here, the arc AB represents the surface of the Earth, with center at C.
The observer at O is a height *H*_{obs} above A. The dashed
arc OD represents a level surface at the height of the observer.

The heavy arc OH is the ray that's horizontal *at the observer*;
so any point on it is seen at the observer's astronomical horizon.
It's usually less curved than the Earth, so that's how I've drawn it.
Q is the center of curvature of the ray OH, whose radius of curvature is
(*R / k*).

The observer sees a target point, T, located a height
*H*_{target} above the point B on the Earth's surface.
(I omit the ray connecting O to T, as we don't really need it.)
The line CBDHT is the local vertical through T, just as CAO is the local
vertical at the observer.
The angle θ between these two verticals is rarely as large as 2
or 3 degrees; it has been **grossly exaggerated** in the diagram.
(Likewise, the heights of all the points above the Earth's surface —
again, grossly exaggerated in the drawing — are
negligible, not only in comparison to the Earth's radius, *R*,
but also in comparison to the horizontal distance between A and B,
or O and D.)

Because θ and the heights are all so small, we can regard all the
points B, D, H and T as being practically the same distance *d*
from O.

If we use the level surface OD instead of the Earth's surface AB, the
distance formula
still applies, provided that we use the height difference between H and O
(or H and D, which is the same height above B as O is above A)
as the height *h* in the formula; that is,
*h* = DH =
*H*_{H} − *H*_{obs}.

So, solving the distance formula for *h*, we have:

And as this is just the difference between
*H*_{H} and *H*_{obs}, we find that
*H*_{H} = *H*_{obs} + *h*.

But, as every point on OH appears at the observer's astronomical horizon, in particular the point H is seen on that horizon; so its altitude is exactly zero. So the height of H provides a reference point of known altitude, to which we can refer every other point on the vertical BDHT.

That means that the angular altitude of the target T above the
observer's astronomical horizon is just the angle subtended by the height
difference
(*H*_{target}− *H*_{H})
at a distance *d*. This angle is
(*H*_{target}− *H*_{H}) */ d*
radians, because the (assumed) constant lapse rate produces no image
distortion.

So, to sum up: we first use the lapse rate to determine the ray curvature.
We then use the ray curvature to find the height of the observer's
astronomical horizon, *H*_{H}, at the distance *d*
of the target. Finally, we find the *angular* height of the target
point above the observer's astronomical horizon from the height of the
target above the horizon point, H.

Having determined the lapse rate, we find the ratio of
curvatures of horizontal rays and the Earth's surface, using the
equation
above. (Actually, I include an additional factor of
exp(−*H / *8 km) to allow for the decrease in
atmospheric density with height.)

**Results**: Your values gave *k * =
___
,
corresponding to a radius of curvature for the ray of
___
km.

All that remains is to find the apparent altitude of the target —
which was the original goal. To do this, we need to know the *distance*
from the observer to the target.

Finally, we find the apparent altitude of the target:
**
___
minutes** of arc =
**
___
degrees** of arc.

To find the distance to the observer's sea horizon, we use the
distance formula
from the top of this page, being careful to use
*H*_{obs} for the value of *h* in the formula;
this assumes that the same ray curvature found above extends down to the
sea surface.
For your values, the **distance** to the sea horizon is
**
___
**, which puts the target on the
___
side of the horizon.
...
.

In your case, the **dip** of the sea horizon is
**
___
minutes** of arc, or
**
___
degrees**.
These values place the target
___
minutes of arc, or
___
degrees,
**
___
**the apparent sea horizon.
___

But this upper surface is missing from the oversimplified model adopted
here: we have assumed a constant lapse rate throughout the boundary layer.
So there's no way to determine the width of the blank strip. However,
the Earth's surface appears concave to an observer within the strong
inversion, so everything is visible — much as was predicted by
**Kummer**
in 1860.

Obviously, you can't set both the heights and both the temperatures, and then also type in a value for the lapse rate, without expecting some change to keep things consistent. I've written the scripts to keep the heights and the observer's temperature fixed, and re-compute the target temperature automatically, if you set the lapse rate manually.

If you change any of the values for the
observer
or the
target,
you'll need to re-set the lapse rate, either by clicking the
*Compute lapse rate*
button,
or by entering something in the lapse-rate entry box below it.
If you want to re-use the same lapse rate after changing a height or a
temperature, just click in the lapse-rate box and add a space —
being careful *not* to hit the “Enter” key (which would
clear all your entries, because of a quirk in JavaScript). The ray
curvature will be re-computed when you release a key in this box; some
browsers will act even if it's a <Shift> key or a <Control>
key.

If you just change the target distance, everything that depends on it will be recomputed automatically.

Clearing an entry box is equivalent to entering zero there.

Notice that the running text that reports the results will change when you change the critical input data. This only works with browsers introduced after Netscape 4; if you have problems with a modern browser, please let me know.

Copyright © 2009 – 2010, 2020 Andrew T. Young

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dip of the horizon page
**

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