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Refraction

Astronomical refraction is coupled in a great deal with terrestrial refraction. A celestial objects is during many times observed over a terrestrial object, and thus both the terrestrial object and the celestial object are refracted and have a common light path, and that light path can be close to the ground; certainly in case of a vast plane surface. Even if a celestial object is not observed over a terrestrial object the light still travel through the atmosphere.
The largest influence of refraction is due to temperature gradients, and these are normally the largest near the ground surface. As the light path is the longest when the light travels over a vast plane surface, that situation produces the largest (variation) of refraction.

Astronomical refraction

The astronomical refraction is depending on the altitude of a celestial body.
From an old book (De sterren hemel, Kaiser, F., 1845) one can get information about the astronomical refraction. The conditions for this table are:
Astronomical refraction
Apparent altitude Astronomical
refraction
o '
-0.5 40
0 35
0.16 33
0.32 31
0.5 29
0.66 28
0.83 26
1 24
2 18
3 14
5 10
10 5
20 2.5
30 1.5
50 .5
90 0

Or using the Sinclair (Bennett ([1982, page 257], formula B) at 10 C and 1010 mbar:
refrac. = (34.46 + 4.23*app.alt + 0.004*app.alt2) / (1 + 0.505*app.alt + 0.0845*app.alt2)/60
 
altitude = app.alt. - refrac.

The inverse function (with a sigma of 0.03o) has been determined using SIMPLEX optimisation algorithm (by Victor Reijs):
inv.refrac. = (34.73 - 0.7711*altitude - 0.0004693*altitude2) / (1.119 + 0.2979*altitude+ 0.0316*altitude2)/60
app.alt. = altitude + inv.refrac.

With: For more theoretical information on astronomical refraction see: Astronomical refraction and this article.

Refraction measurements

The following picture is based on refraction measurements by Schaefer&Liller (SL) [1990], Seidelmann (KS) (1968 as quoted in Schaefer&Liller [1990], page 800-801), Sampson (SLPH) [2003] and the Sinclair formula. All of these used observations that were due to a vast plane surface.
Refraction observations
The calculated curves are: Sinclair is using USA76 atmospheric model and Stability A and G use the MUSA76 atmospheric model (all calculated values are based on: 1.5m eye elevation, 15C, 1013.25mbar and 0%).
One can see an overall large standard deviation (1 sigma: ~0.15o) in actual live and the slight difference between sun rise and set events (~0.1o).

It is not very clear if the standard deviation will increase with lower apparent altitudes (although that is expected). For instance the standard deviation for observations around -0.4o is 0.05o (n=75) has a smaller standard deviation then observations at 0o with 0.15o (n=260), and at -1.3o the standard deviation is higher (twice as high as at 0o being 0.3o (n=25). The variation (between min. and max.) looks to be the same for observation at apparent altitude of -1.3o and 0o: around 0.8o degrees. Looks to be some limiting going on. Looking at the below calculations the deviation might indeed stay more or less the same for negative apparent altitudes. The possible calculated limiting stability classes A and G are given.

Not enough data points are available to see if there is a significant difference, the below calculations might give some insight.

Refraction calculation

If using the computation method of RGO (Hohenkerk, [1985]) under the circumstances T(0)=15 [C], P(0)=1013.25 [mbar], RH(0)=0%, latitude observer = 50o, remote sea horizon elevation = 1.5 [m]  (equivalent with windspeed 7.5 [m/sec]), a lapse rate at 0 [m] level compatible with the stability class, a type of surface layer Hc=1000 [m] (van der Werf, [2003], formula (52)) and MUSA76 atmosphere (van der Werf, [2003], Table 1).
The following astronomical refraction results are due to a vast plane surface (only possible below a certain apparent altitude) as the viewing apparent altitude:
Refraction calculated
        with RGO
with the following further conditions:

Terrestrial refraction

Two case will be presented here:

Apparent altitude of terrestrial object

The terrestrial refraction changes the apparent altitude of a terrestrial body.
If one needs to calculate the apparent altitude, seen from local point, of a distant object (like the top of a mountain), the following formula is needed (from Thom, A., 1973, page 31 and changing it to metric and keeping air pressure explicit):
app. alt. = 0.0573*Elev/L-0.00447*L+0.00830*K*L*P/(273.15+T)^2
topo alt. = 0.0573*Elev/L-0.00447*L

With: Remember that the above does not include atmospheric conditions like extraordinary convective and inversions boundary layers. For more theoretical information on terrestrial refraction see: Astronomical refraction.

Apparent altitude of a vast plane surface

The angular depression of the apparent horizon is known as (apparent) dip (also referred to sometimes as levelling refraction). According to Thom, A., 1973 (page 32, and changing it to metric and making air pressure and temperature explicit) the apparent altitude of a vast plane surface is:

app. alt. = -ACOS(1 / (1 + Elev / Ra))*SQRT(1-1.8480*K*P/(273.15+T)^2)


topo. alt. = -ACOS(1 / (1 + Elev / Ra))

With: Remember that the above does not include atmospheric conditions like extraordinary convective and inversions boundary layers. For more theoretical information on horizon dip see: Dip of the Horizon.

Calculating the effects of refraction on apparent altitude

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General parameters

Air pressure:   [mbar] (at eye- orsea-elevation reference)
Temperature:    [oC] (using same elevation reference as Air pressure)
Time of day:  sun set/rise or night (latitude 53 and windspeed 4 [m/sec] (at 10 m height))
  noon at equinox (latitude 53 and windspeed 4 [m/sec] (at 10 m height))
standard atmosphere
  sun set/rise or night at average windspeed of +/-  [m/sec] (at 10 m height)
  own K +/-  [-]
There are some useful conversions available!

Altitude due to astronomical refraction

Apparent altitude:  [o]
The apparent altitude value could be the apparent altitude of a distant object or the vast plane surface.

altitude of celestial object: [o

Apparent altitude of distant object due to terrestrial refraction

Eye elevation: , distant object elevation:  +/-  [m]
Distance:    +/-  [km]
apparent altitude of distant object:  +/- [o]
    apparent altitude of vast plane level:  +/- [o

Useful conversions

  [oF]      [feet]   [mile]     [in Hg]      [mm Hg]   
of the above :
[oC]  [m]  [km]  [mbar]  [mbar]

Measurements in the above environment

Remember that if one measures the altitude with a clinometer, altimeter or sectant; one always measures the apparent altitude! If using values from maps or GPS, one has to calculate the apparent altitude using the above terrestrial refraction.
For determining the declination of the celestial body one needs the (astronomical) altitude.
So when combining the above things (like when calculating the declination), one has to remember the above formulae for celestial and terrestrial bodies.

Acknowledgments

I would like to thank the following people for their help and constructive feedback: Catherine Hohenkerk, Geoffrey Kolbe, Russell Sampson, Bradley Schaefer, Miles Standish, Marcel Tschudin, Siebren van der Werf and Andrew Young and all other unmentioned people. Any remaining errors in methodology or results are my responsibility of course!!! If you want to provide constructive feedback, let me know.

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Last content related changes: July 22, 2000