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Astronomical refraction

The astronomical refraction is depending on the altitude of a celestial body.
From an old book (De sterren hemel, Kaiser, F., 1845) I got some information about the astronomical refraction. The conditions for this table are:

it is calculated over a surface with a temperature of melting ice.

at normal air pressure.

Astronomical refraction
Apparent altitude	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.

With:

refrac: refraction [^o]
app.alt.: apparent altitude [^o] of a celestial body
altitude [^o] of a celestial body

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) and Sampson (SLPH) [2003] (it includes the Sinclair formula).

$Refraction observations$

One can see an overall large standard deviation (1 sigma: ~0.15^o) in actual live and the slight difference between sun rise and set events (~0.1^o).

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 -0.4^o is 0.05^o (n=75) has a smaller standard deviation then observations at 0^o with 0.15^o (n=260), and at -1.3^o the standard deviation is higher (twice as high as at 0^o being 0.3^o (n=25). The variation (between min. and max.) looks to be the same for observation at apparent altitude of -1.3^o and 0^o: around 0.8^o 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 = 50^o, remote sea horizon height = 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 H_c=1000 [m] (van der Werf, [2003], formula (52)) and MUSA76 atmosphere (van der Werf, [2003], Table 1).
The following results are gotten:

$Refraction calculated with RGO$

with the following further conditions:

In practice, the stability class will vary for Sun set/rise events mostly between D (dark blue line) to F (red line) and with lower frequency A (green) and G (yellow) can happen (C and B happen even with a lower frequency).
For the lines that have 'dip' at the end (purple [C], dark blue [D], light blue [E], red [F] and yellow [G] lines), the height of the observer (H_Obs) is changed to make sure that the App. Alt. is equivalent to the dip (H_Obs between 4200 and 5 [m]). For apparent altitudes bigger then 0^o, the observer height was kept constant at 5 [m].
These cases would be close to actual refraction measurements as done above. The interesting is that the limiting of refraction has a somewhat comparable behavior as in the measurements.
The variation in refraction might be constant for 'large' negative and positive apparent altitudes; 'large' meaning abs(App. Alt.)>0.75^o
The astronomical refraction formula of Sinclair is very close to the stability class D (dark blue) line (which uses the standard atmosphere line).
No change in H_c has been included yet. This could perhaps help in explaining the change of refraction during Sun set and rise events (the H_c is somewhat related to the atmospheric boundary layer). Although the behavior seen in Seidelmann's measurements looks not explainable using a surface layer.
More study is needed with regard to the boundary layer (related to H_c).

Terrestrial refraction

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. = 0.057288*H/L-0.00447387*L+0.008296359*K*L*P/(273.15+T)^2

With:

app. alt.: the apparent altitude [^o] of Distant object
H: height difference [m] between point Eye height and Distant object height (Distant object height - Eye height). Both eye and distant object height must have same reference.
L: distance [km] between Eye height and Distant object height (measured along earth's surface)
K: refraction constant K=4.91 at noon, K=10.64 during sun set/rise and night (equinox, wind speed [4 m/sec] (at 10 m height) and latitude 53°)
P: air pressure at Eye height [mbar]
T: temperature at Eye height [°C]

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

Apparent altitude of a vast plain horizon

The angular depression of the apparent horizon is known as dip. 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 plain horizon is:

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

With:

app. alt.: the apparent altitude [^o] of the vast plain horizon
H: height in [m] between observer's eyes (Eye height) and vast plain (Distant object height) (Eye height - Distant object height). Both eye and distant object height must have same reference.
K: refraction constant K=4.91 at noon, K=10.64 during sun set/rise and night (equinox, wind speed [4 m/sec] (at 10 m height) and latitude 53°)
P: air pressure at Eye height [mbar]
T: temperature at Eye height [°C]
Ra: radius of earth = 6378137 [m]

Remember that the above does not include atmospheric conditions like convective boundary layers and inversions. 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-height reference)
Temperature:    [^oC] (using same height 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 plain horizon.

altitude of celestial object: [^o]

Apparent altitude of distant object due to terrestrial refraction

Eye height: , distant object height:  +/-  [m]
Distance:    +/-  [km]

apparent altitude of distant object: +/- [^o]
apparent altitude of sea 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