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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:
- it is calculated over a (vast plane) 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 astronomical
refraction increased with some 5%.
- at normal air pressure.
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.alt^{2}) / (1 + 0.505*app.alt + 0.0845*app.alt^{2})/60
altitude = app.alt. - refrac.
The inverse function (with a sigma of 0.03^{o}) has been
determined using SIMPLEX optimisation algorithm (by Victor Reijs):
inv.refrac. = (34.73 - 0.7711*altitude - 0.0004693*altitude^{2}) / (1.119 + 0.2979*altitude+ 0.0316*altitude^{2})/60
app.alt. = altitude + inv.refrac.
With:
- refrac: astronomical refraction [^{o}]^{}
- inv.refract: 'inverse astronomical refraction' [^{o}]
- app.alt.: apparent altitude [^{o}] of a celestial
body, > -3.5^{o}
- (topocentric) altitude [^{o}] of a celestial body,
> -4.5^{o}
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.
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: ~0.15^{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 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 H_{c}=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:
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
elevation 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] above ground.
These cases would be close to actual astronomical refraction
measurements as done above. The interesting is that the limiting of
astronomical refraction
has a somewhat comparable behavior as in the measurements.
- The variation in astronomical 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 astronomical 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}).
- It might be good to also implement the computation method of van der Werf (2003). That is a slightly different method than Hohenkerk&Sinclair and Sinclair's refraction formula. This can overcome possible circular argumentation.
- One could distinguish three apparent altitude regions that are resulting due to variability of the temperature gradient of observation for a vast plane surface.
- It is important to realise that in the above calculations, the height of the observer has been adjust to make sure that the light path passes over the (vast plane) surface. So for negative altitudes the light path is normally (but does not have to be the case! Depending on the relative elevation of observer and surface/object) over a vast plane surface, for positive ones it is not coming that close to the ground and thus the temperature gradients variations are also smaller.
- Also the variation of the refraction does not change that much with changing negative apparent altitudes. This looks to be due to the fact that the length of the light path close the vast plane surfaces is some what independent of the value of this (negative) angle.
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:
- app. alt.: the apparent altitude [^{o}] of Distant
object
- topo. alt.: the topocentric altitude [^{o}] of Distant
object
- Elev: elevation difference [m] between point Eye elevation and Distant object elevation (Distant object elevation - Eye elevation). Both eye and
distant object elevation must have same reference/datum.
- L: distance [km] between Eye
elevation
and Distant object
elevation (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°)
K = 990/2* (1/R + g) (formula (5) in Thom, 1985: with l=L)
R =53.3 ft
g =F/ft (temp. gradient)
gamma=C/m (temp. gradient)
K = 495 * (1*0.304/53.3 + (9/5)*0.304*gamma)
K = 495*0.304 * (1/53.3 + (9/5)*gamma)
K = 15058*9/5 * (1/53.3*(5/9) + gamma)
K = 270.9 * (0.0337 + gamma) (0.0342: according to Bomford, 1980, Geodesy)
K = 41.7 * (0.0342 + gamma)/0.154
K = 41.7 * k
K = k/0.0240
- P: air pressure at Eye
elevation [mbar]
- T: temperature at Eye
elevation [°C]
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:
- app. alt.: the apparent altitude [^{o}] of the vast
plane surface
- topo. alt.: the topocentric altitude [^{o}] of the vast
plane surface
- Elev: elevation in [m] between observer's eyes (Eye elevation) and vast plane
(Distant object elevation)
(Eye elevation - Distant object elevation).
Both eye and distant object elevation 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
elevation [mbar]
- T: temperature at Eye
elevation [°C]
- Ra: radius of earth = 6378137 [m]
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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