It's often said that the flat-Earth (or, to be more pedantic, the plane-parallel) model of the atmosphere is satisfactory near the zenith. To show just how nearly flat the Earth is, compared with the depth of the atmosphere, here's a picture, drawn to scale:

The heavy line at the bottom represents the surface of the Earth. The light lines are the levels at which the pressure is (from the top down) 10%, 20%, … , 90% of the pressure at the surface. All these lines are arcs of circles, concentric with the Earth's center (which is far below the bottom of your screen). The top (10% pressure) level is about 16 km above the surface.

Looks pretty flat, doesn't it?

The picture should convince you that the flat-Earth approximation is good enough to deserve a closer look.

Now let's consider refraction in strictly plane-parallel geometry.

Initially, we assume a single homogeneous layer of refractive index n.
The observer is at O; the observed zenith distance of an object is
z_{0} .
Applying the
sine law of refraction
at the refracting upper surface gives

But because all the verticals are parallel in this model, the angles
z_{0} and z_{1} are equal; so

As **Newton showed**
in his famous “Opticks”, this can be extended to a
second layer above the first, and then to a third. The product
(n_{i} sin z_{i})
at every horizontal interface remains equal to the sine of the zenith
distance at the top of the whole stack, where n = 1 exactly;
in particular, n sin z_{0} at the bottom remains equal
to sin z at the top of the stack. It is as though there were only
the bottom layer, of index n.

The refraction, r, is the difference of the angles z_{0} and
z_{2} in the diagram above. That is,

But the previous equation gave us sin z_{2} as a function of
z_{0} ; taking its arcsine gives

Consequently, the **exact** expression for the refraction r in the
plane-parallel model is

However, this exact result isn't very informative. More insight into the behavior of the refraction can be obtained from a useful approximation.

Now we can use the trigonometric identity for the sine of a sum of angles to write

But we know from observation that r is generally less than about half a degree, or 0.01 radian. So, to better than a part in ten thousand, we can set sin r ≈ r, and cos r ≈ 1, making

Plugging this expression for sin z_{2} back into
the refraction-law result
above,
we get

solving for r gives

or

This is the flat-Earth **approximation** for the refraction.
It's just proportional to the refractivity at the observer,
and the tangent of the apparent (refracted) zenith distance.

The Sun would appear to set on a surface about 1.4° *above *
the astronomical horizon. The drawing shows the Sun's exact shape at
the moment when its lower limb touches this false apparent horizon;
the whole disk of the Sun is shown. Everything between the false horizon
and the astronomical horizon would be filled with a gigantic refracted
image of the (flat) Earth's surface, which would appear concave.

Qualitatively, this highly flattened sunset image resembles what's seen by an observer inside a duct; see the second image in the simulation showing a wide blank strip. But quantitatively, the negative dip here is an order of magnitude larger than in that case, which is already very unusual.

Although the details of the mirage — or, more likely,
*looming*
— in this huge blank strip would depend on the density structure
of the flat atmosphere, Newton's proof mentioned
above
guarantees that *all* sunsets in the flat-Earth model must have
exactly this bizarre appearance.
In particular, the enormous elevation of the false horizon depends only on
the refractive index of air at the observer — a quantity known to
many decimal places from
laboratory measurements.

The fact that no real sunset on Earth ever has these characteristics can be taken as observational evidence that the Earth is round, not flat.

Copyright © 2003 – 2008, 2018, 2020, 2021 Andrew T. Young

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