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In these pages back in April 2013 I showed how an accurate dip formula
can be derived from an optical principle: the refractive invariant. In
the process of modifying a program, I recently revisited that material.
It wasn't easy reading. Generally I give myself good marks for the
explanations, but probably the key equations should have been summarized
in a separate message for the benefit of someone who doesn't want to
suffer through their derivation. I'll try to do that here with a worked
example.

Pobs = observer pressure (millibars) = 1010
Tobs = observer temperature (C) = 10
Psea = sea level pressure
Tsea = sea level temperature
h = observer height (meters) = 5.0
Nobs = refractivity of air at observer
Nsea = refractivity of air at sea level

Estimate air temperature at sea level from the standard atmosphere lapse
rate of -.0065 C per meter. I.e., temperature decreases as height
increases, and so Tsea = 10 + 5.0 * .0065 = 10.03 C.

For a small difference in altitude, air pressure has almost linear
variation with height, the rate depending on temperature:

Psea = Pobs * (1 + h * .034163 / (273 + Tobs))
= 1010.6 mb

Next, calculate N, the refractivity of the air, at sea level and the
observer. For small differences in temperature and pressure, N is
practically proportional to air density. Therefore a correction to
actual conditions is simple if we know its value at some specified
conditions. Well, at 10 C, 1010 mb, 50% humidity, CO2 = 400 ppm, and 550
nanometers wavelength, N = .00028161. Correct that standard value for
actual temperature and pressure with this equation:

N = .00028161 * P / 1010 * 283 / (T + 273)

which is simplified by combining the constants:

N = P / 12673 / (T + 273)

To make the example easy, I used standard conditions at the observer, so
Nobs is simply .00028161. At the sea level temperature and pressure
previously calculated, Nsea = .00028175.

Let dN = Nobs - Nsea = -1.4e-7. Compute dip from dN and h. (The constant
6,371,000 converts height of eye from meters to Earth radii.)

dip (radians) = √(2 * (dN + h / 6,371,000)), or
dip = 3438′ * √(2 * (dN + h / 6,371,000))

Dip = 3.90′. The almanac table says 3.95′.

An almost perfect match to the almanac table is possible if you change
lapse rate from -.0065 to -.009 °C/meter.

Dip is little affected by air density variation at the observer if
temperature lapse rate (degrees per meter) remains constant. However, a
small variation the latter has large effect.

For example, if you change observer temperature from 10 C (50 F) to 35 C
(95 F) but lapse rate (-.0065 °C/m) does not change, dip increases .07′.

On the other hand, if you change lapse rate from -0.0065 to -0.0265, and
observer temperature (10 C) does not change, there's a 0.29′ increase in
dip. That change in lapse rate equals an (observer - sea level)
temperature variation of only a tenth degree C!

The dip equation shows that dN, the refractivity difference between
observer and sea level, is equivalent to a change in height of eye, the
unit of measure being Earth radius. If dN has an error of one unit in
the 6th decimal place, that's equivalent to a height of eye error of a
millionth of an Earth radius: about 6 meters. Therefore dN should be
computed to 7 or 8 decimal places.

Since dN is normally negative, the effective height of eye is decreased
by refraction. If dN is too negative, the formula has no real solution
since it's the square root of a negative number. That can happen with
extreme lapse rates.

A program could have an option to input observer and sea level
temperatures, instead of assuming a fixed lapse rate. In that case you
should include defensive code to avoid the square root problem.

In the real world I don't believe my equations have a significant
advantage over the simple expression in the almanac. Any theoretical
gain in accuracy is probably lost in unpredictable temperature
variations between the observer and horizon.

The derivation of the equations is explained in my postings in the April
and May 2013 archive. The applicable messages are obvious from the
Subject fields. But note that equation 4 in my May 13 message

dn = (n0 - 1) * (dP/P0 - dT/T0)

is wrong. Also, in the old messages I use n (refractive index) instead
of N (refractivity). The difference is that n = N + 1. For example, if n
= 1.003, N = .003. I think N is easier to work with.

   

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