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    Dip and the refractive invariant
    From: Paul Hirose
    Date: 2016 Sep 07, 23:13 -0700

    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
    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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