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Influence of air pressure and temperature on refraction

Warning: Firefox renders the formulas correctly (Chrome and Microsoft Edge can't properly render the formula [MathM] correctly)!!!
The influence of air pressure and temperature for different topocentric altitudes is determined by using ray tracing integration along the apparent altitude (Hohenkerk&Sinclair, 1985). The reference used is StdAtm, being MUSA76 at lapse rate=0.0065K/m, temperature ground level 15C, RH=0% and air pressure ground level 1013.25mbar.

Influence of Air Pressure (P) on refraction is expressed as:

\begin{matrix} κ = \frac{\ln (\frac{R e f r a c t_{P}}{R e f r a c t_{P S t d A t m}})}{\ln (\frac{P}{P_{S t d A t m}})} & (1) \end{matrix}

The P at ground level is varied between 871.4mbar and 1074mbar.

Influence of Temperature (P) on refraction is expressed as:
$\begin{matrix} λ = \frac{\ln (\frac{R e f r a c t_{T}}{R e f r a c t_{T S t d A t m}})}{\ln (\frac{T}{T_{S t d A t m}})} & (2) \end{matrix}$

The T at ground level is varied between -40C and 40C.

In below evaluation the topocentric instead of apparent altitude is used as variable, this because in levelling and terrestrial refraction the topocentric altitude is quite determined due to geometry. For astronomical refraction the topocentric altitudes was determined by subtracting the refraction (result) from the apparent altitude (input).
The three types of refraction are checked by varying one parameter (P or T) while keeping the other (T or P) at the StdAtm value:

Astronomical refraction
The refraction is determined for a light path from the eyes' position to the end of the Earth's atmosphere (in direction of a celestial object).
Levelling refraction
The refraction is determined for a light path that has a large part level with the Earth's geoid, aka towards/from a vast level surface below the eyes' position. Also known as dip.
Terrestrial refraction
The refraction is determined for a light path that goes from the eyes' position to a terrestrial object higher or lower than the eyes' position.

Astronomical refraction

The Near Height (at the observer) is 10m.

$Pressure influence in astornomical refraction$
So there is air pressure dependency and larger topocentric altitude dependency on the κ for astronomical refraction. Remember that for altitudes below 0deg, levelling refraction plays a part (but does not have to be down to the ground level).

$Temperature influence in astornomical refraction$
So there is temperature dependency and larger topocentric altitude dependency on the λ for astronomical refraction. Remember that for altitudes below 0deg, levelling refraction plays a part (but does not have to be down to the ground level).

Bessel's results (1830, page LIX-LXI and 539) at apparent altitude of 0.5deg (10C, 1002mbar) compare well with the ray tracing results (see the green crosses in above graphs).

For high altitudes (>10deg) the λ and κ are getting close to 1. This maps on observations of others, such as Bennett (1982, page 258).

The air pressure and temperature influence has also been extensively evaluated by Tschudin (unpublished yet).

Levelling refraction

The Near Height (at the observer) varied between 11 and 750m and Distant Height is 10m. This changes the topocentric altitude.

$Pressure influence in levelling refraction$
So there is slight air pressure and topocentric altitude dependency on the κ for levelling refraction.

$Temperature influence in levelling refraction$
So there is temperature and topocentric altitude dependency on the λ for levelling refraction.

Terrestrial refraction

The Near Height (at the observer) is at 10m and the Distant Height varied between 11 and 1500m, while the Distance to Distant Height varied from 85 to 8000m. These change the topocentric altitude.

$Pressure influence in terrestrial refraction$
So there is almost no air pressure or topocentric altitude dependency on the κ for terrestrial refraction, it is quite stable at 1.000

$Temperature influence in terrestrial refraction$
So there is almost no temperature dependency (between 0 and 40 degrees) on the λ for terrestrial refraction. But λ is depending on the topocentric altitude.

The resulting graphs will change if different Near and Distant Heights are used (even resulting in same topocentric altitude), but this evaluation gives at least some idea of the influences

For terrestrial refraction the expected values for κ=1 and for λ=-2 (following the formula of Thom [1958]), and the above ray tracing results map that.

General observations

In general the influence on the κ and λ due to topocentric altitude and temperature (in order of influence) are much lager than air pressure. In case of levelling refraction the influence of temperature is larger than topocentric altitude.

References

Bennett, G.G. 1982. 'The calculation of astronomical refraction in marine navigation', Journal of Inst. navigation, Vol 35: pp. 255-59.
Bessel, F. W. 1830.'Tabulae regiomontanae reductionum observationum astronomicarum ab anno 1750 usque ad annum 1850 computatæ', in https://archive.org/stream/tabulaeregiomon00bessgoog#page/n614/mode/2up [accessed January 1 2018].
Hohenkerk, Catherine Y., and A.T. Sinclair. 1985. "The computation of angular atmospheric refraction at large zenith angles." ed. by HM nautical almanac office. Cambridge.
Thom, Alexander. 1958. 'An empirical investigation of atmospheric refraction', Empire Survey Review, Vol 14: pp. 248-62.
Tschudin, Marcel E. to be published. 'Refraction near the horizon—an empirical approach. Part 2: Variability of astronomical refraction at low positive altitude (LPAAR)'.

Acknowledgements

I would like to thank people, such as Marcel Tschudin and others for their help and constructive feedback. 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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Major content related changes: January 1, 2018