The dip diagram is a graph based on the
refractive invariant,
the quantity **I = (n R sin z)**, which has
the same value, **I**, at all points along a ray of light.
Here, **n** is the refractive index of air at a distance **R**
from the center of the Earth, and **z** is the zenith distance of the
ray at that point.

Symmetry
makes a ray horizontal at its lowest point, or
*perigee*.
There, **z** = 90°, so (sin **z**) is unity,
and **nR** = **I**.
Then the ratio of (**nR**) at any *other* height to the value
of (**nR**) at the perigee gives (sin **z**) at that
other height.
This means we can determine that ray's slope at any height from a plot of
(**nR**) vs. height — which is the dip diagram.

Because the zenith distance **z** is the complement of the altitude,
**h**,
sin **z** = cos **h**. So we can also say that
**nR**/**I** = cos **h**.
In particular, the ray at an observer's sea horizon is horizontal at the
sea surface; so the cosine of the
dip
of the horizon seen from any height is just (**nR** at sea level)
divided by (**nR** at the observer).

The dip diagram is just a plot of (**nR**) as a function of height
or **R**.
Here's an example, based on the
Standard Atmosphere:

The solid line in the graph shows **nR** as a function of **R**,
for the bottom 22 km of the Standard Atmosphere.
The dashed line is the function **nR** = **R**; the solid line
approaches it asymptotically at large heights, because
n → 1 at large heights.

The scale at the right side of the diagram is graduated in degrees of
dip for an observer placed 19 km above the Earth's surface. That is,
the graduation marked “0” is at the value of **nR** for
a height of 19 km; the graduation marked “1” is at the value
of **nR** for a height of 19 km, multiplied by the cosine of 1°;
the “2” marks the observer's value of **nR** multiplied
by the cosine of 2°; and so on.
The dip of the horizon for this highly elevated observer is a little more
than 4°, because the surface of the Earth (at the left end of the
model curve) is a little below the 4° mark on the scale.

This example showed a big interval of height on the diagram, to illustrate
the fact that **nR** → **R** at the top of
the atmosphere, where the refractive index, **n**, approaches unity.
Most of the time, it's more useful to show a much smaller
piece of the diagram, just above the surface of the Earth. And it's more
convenient to use height for the horizontal scale, instead of distance
from the center of the Earth.

Here's the dip diagram for a duct model I've used repeatedly on these pages; the 2° thermal inversion that produces the duct extends from 50 to 60 meters. As I proposed above, height is the horizontal axis; the dip scale on the right is now graduated in minutes of arc rather than degrees, because only the bottom 100 m of the atmosphere is shown.

Because the temperature gradient is so steep in the inversion, the **nR**
product actually **de**creases with height there. So there is a
minimum in the diagram at the top of the inversion (60 m).

I've drawn a dashed horizontal line tangent to this minimum. You can see
that it intersects the curve for the model atmosphere a little below
the inversion, at about 45 m. At that height, **nR** has the same
value as at the top of the inversion. Any ray with a slightly larger value
of **nR** within the region between the maximum and the dashed line is
trapped, oscillating forever between two limiting heights.
(You can see a picture of such rays on the
mirage page.)

If you slide the dip scale at the right over to the curve, so that the zero of the dip scale is at the maximum, you'll see that the dip of the duct edges (where the dashed line meets the solid curve) — as seen from the base of the inversion — is 3 or 4 minutes of arc. That's the half-width of Wegener's “blank strip”, which is centered on the astronomical horizon.

Finally, because **nR** has the *same* value at the top and
bottom of the duct, any ray passing obliquely across the duct has
the *same* inclination to the horizon at both those heights.
(This follows from the refractive invariant.)

Dip diagrams are discussed in more detail in

A. T. Young & G. W. Kattawar

Sunset Science. II. A Useful Diagram

Applied Optics37,3785-3792 (1998).

Copyright © 2003 – 2010, 2012, 2013 Andrew T. Young

Back to the ...

or the
**
astronomical refraction page
**

or the
**
GF home page
**

or the
**
Overview page
**