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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.
Doing some calculations with SkyMap; an increase of 5^{o} C,
decreased the apparent altitude with 1'. Thus the refraction increased
with some 5%. - 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.alt^{2}) / (1 + 0.505*app.alt + 0.0845*app.alt^{2})/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).
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:
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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