Astronomic Azimuth Determination
Rossen Grebenitcharsky, Nico Sneeuw
Department of Geomatics Engineering
University of Calgary
February 28, 2002
1 From planar to spherical trigonometry
The following relationships exist between planar and spherical trigonometry
PLANAR
γ
b
SPHERICAL
γ
a
α
c
b
β
a
α
β
c
planar
angles:
sides:
cosine formula:
sine formula:
sine-cosine formulae:
α+β+γ =
spherical
180◦
α + β + γ 6= 180◦
a, b, c are distances measured in linear units
a, b, c are spherical arcs measured in angular units
a2 = b2 + c2 − 2bc cos α
sin β
sin γ
sin α
=
=
a
b
c
cos a = cos b cos c + sin b sin c cos α
sin β
sin γ
sin α
=
=
sin a
sin b
sin c
sin a cos β = cos b sin c − sin b cos c cos α
sin b cos γ = cos c sin a − sin c cos a cos β
sin c cos α = cos a sin b − sin a cos b cos γ
1
2 Time systems used in astronomic azimuth determination
2 Time systems used in astronomic azimuth determination
The time scales used in astronomic azimuth determination are represented in the scheme below.
2.1 Time scales
sidereal
∆ut
αM
gmst
nutation
atomic (radiosignals)
solar
civilian (stop watch)
∆Z
UT1 ≡ UT
TZ
UTC
Eq.E.
gast
jd
with:
αm = the right ascension of the fictitious Sun
= 12h 38m 45.s 836 + 8640184.s 542 T + 0.s 0929 T 2 ,
Eq.E. = difference between the true and mean equinox.
2.2 Sidereal time and hour angle
Graphically, different sidereal times and the hour angle could be represented in the following way
according to Schwarz (1999):
HOUR ANGLE&
LOCAL SIDEREAL TIME
SIDEREAL TIMES
n
ea
Celestial meridian
M
Gr
ich
nw
ee
me
LMST
n
ia
rid
LAST
Λ
GMST
GAST
true
inox
equ
zenith
Eq.E
x
uino
mean eq
hh
North Pole
*
North Pole
zenith
lm
er
idi
an
LA
S
T
α
tia
les
Ce
2
tr
ox
in
qu
e
ue
rcle
hour ci
star
From the schemes above it is visible for the hour angle h of a celestial body that:
last
z
}|
{
h = UT + (αm − 12h ) + Eq.E. +Λ −α∗ .
|
{z
}
gast
3 Determination of the astronomic azimuth
Two main methods for azimuth determinations will be considered: the hour angle method (often applied
as observations to Polaris) and the altitude method (often applied as observations to the Sun). The
description of both methods could be found in Thomson (1978).
Definition: An astronomic azimuth is defined as the angle between the astronomic meridian plane of a
point i and the astronomic normal section through i and another point j.
North Pole
* star
Earth Object
j
A*
Aij
A
R*
(Rj -R*)
i
Rj
To determine the astronomic azimuth of a line ij on the Earth we need:
• azimuth to a celestial body at time T ;
• measure the horizontal angle between the celestial body and the terrestrial reference object.
The advantage of the hour angle method is that the observer has only to observe the star as it coincides
with the vertical wire of the telescope. No zenith distance (altitude) is necessary. In this case the
atmospheric refraction has no effect. The main disadvantage is the need for a time registering device
and a good knowledge of the longitude.
The advantage of the altitude method is that an accurate time device and precise longitude are not
necessary. The disadvantages of this method are that it is affected by the astronomic refraction and the
fact that the star should coincide both with the horizontal and the vertical wires. The altitude method
is less accurate compared to the hour angle method. They have the following precision: up to 1.00 5 for
the hour angular method and up to 5.00 0 for the altitude method.
3
3 Determination of the astronomic azimuth
All parameters and measurements, necessary for the hour angular method and altitude method are
represented graphically in the following figure.
Spherical trigonometry hour angle and altitude method
North Pole
GA
ST
∗
S
Aij
q
z=90-a
*
-R
Rj β
i
δ
A* Φ
Λ=0
Equin
j
−Φ
90
h
LAST
ox
α
Λ
3.1 Hour angle method
• From the sine formula applied to the co-latitude 90 ◦ − δ and the zenith distance z we have:
sin (90◦ − δ)
sin z
cos δ sin h
=
⇒ sin β =
sin β
sin h
sin z
• From the cosine formula applied to 90 ◦ − δ we have:
cos (90◦ − δ) = cos z cos (90◦ − Φ) + sin z sin (90◦ − Φ) cos β
⇒ sin δ = cos z sin Φ + sin z cos Φ cos β
sin δ − cos z sin Φ
⇒ cos β =
sin z cos Φ
• From both sine and cosine formulas it is valid that:
tan β =
cos δ sin h
sin z cos Φ
sin z (sin δ − cos z sin Φ)
• From another side the cosine formula for z gives:
cos z = cos (90◦ − Φ) cos (90◦ − δ) + sin (90◦ − Φ) sin (90◦ − δ) cos h
⇒ cos z = sin Φ sin δ + cos Φ cos δ cos h
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3.1 Hour angle method
• Substituting cos z in the equation for tan β and after some simplifications we have:
or
tan β =
cos δ sin h cos Φ
sin δ − (sin Φ sin δ + cos Φ cos δ cos h) sin Φ
tan β =
cos δ sin h cos Φ
sin δ(1 − sin Φ) − cos Φ cos δ cos h sin Φ
tan β =
cos δ sin h cos Φ
cos Φ(cos Φ sin δ − cos δ cos h) sin Φ
tan β =
sin h
tan δ cos Φ − sin Φ cos h
2
• Finally, taking into account that tan β = tan (360 ◦ − A∗ ) = tan (−A∗ ) = − tan A∗ , we get for the
azimuth of the star:
tan A∗ =
sin h
sin Φ cos h − tan δ cos Φ
.
This is the azimuth in terms of known Φ and δ and measured h. Since it is expressed as a tangens
function the quadrant will be determined correctly. Note that this equation does not require zenith
angle observations. So the azimuth determination is not going to be affected by atmospheric refraction.
The corresponding differential formula of tan A ∗ reads (see the derivation with the altitude method):
dA∗ = sin A∗ cot z dΦ + cos Φ(tan Φ − cos A∗ cot z) dh ,
=
cos δ cos q
sin A∗
dΦ +
dh .
tan z
sin z
The second line can be achieved after some manipulation with spherical trigonometry. This differential
formula does not contain terms in dδ since we assume the star coordinates to be known exactly—or at
least with sufficient precision. The equation can be used as a linearized observation equation. Here we
only use it to assess the effects of errors in the known latitude dΦ and timing errors, translated into
dh.
when A∗ = 0◦ or A∗ = 180◦ the effect of dΦ is eliminated. This occurs when the star is on—or
close to—the meridian.
when cos q = 0 the timing error dh has no effect on A ∗ . This happens if the parallactic angle
q is either 90◦ or 270◦ . This situation is called elongation, which means that the star is at its
westernmost or easternmost position. At these positions the star moves vertically, i.e./ it moves
over the vertical wire. You would have ample time to observe the horizontal angle.
when cos δ ≈ 0 the effect of dh is minimized too. High declination stars move through the telescope
relatively slowly. Choosing a star close to the North pole reduces at the same time the azimuth
and therefore the aforementioned effect of dΦ.
From these consideration the case for observing Polaris (α Ursae Minoris) is strong. For latitudes
Φ > 15◦ it is easily visible and not affected by the atmospheric refraction. Polaris has the following
advantages:
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3 Determination of the astronomic azimuth
A∗ ≈ 0◦ the error dΦ is minimized;
For Polaris we have δ ≈ 90◦ and the effect of the hour angle error will be eliminated.
These two advantages of Polaris make this star the best choice for the hour angle method. Remember
that in §5.1 a timing requirement, due to Earth rotation, of 0.2 s was derived for purposes of field
astronomy (∼ 300 ). This requirement is relaxed now because of the choice of Polaris.
The error budget for the hour angle method using Polaris:
• pointing errors
• timing errors in local time device → h
• longitude errors → h
• latitude errors → Φ
• errors in interpolation of the right ascension α ∗ and the declination δ of Polaris. They are tabulated
in different star catalogues and almanacs.
3.1.1 Observation procedure
The suggested observation procedure is according to Mueller (1977):
i) Direct to Earth object, record horizontal circle reading (HCR).
ii) Direct to Polaris, record HCR and time T .
iii) Repeat step ii).
iv) Reverse the telescope and repeat steps iii–i).
Repeat the entire procedure eight times, having eight sets of observations.
To get a more precise azimuth a striding level can be used. The correction of the azimuth because of
the levelling is:
∆A00 =
d00
((w + w0 ) − (e + e0 )) cot z ,
4
in which
d00 is the value of each division of the striding level in arc seconds.
e, w are the readings of both ends of striding level in direct observations.
0
e , w0 are the readings of both ends of striding level in reverse observations.
3.1.2 Computation procedure
The computation procedure is described in Mueller (1977) and Thomson (1978).
i) compute the hour angle for every mean direct and reverse readings.
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3.2 Astronomic azimuth determination by the altitude method
ii) compute the astronomic azimuth corresponding to every mean direct and reverse time readings.
iii) compute the astronomic azimuth of the terrestrial line from mean direct and mean reverse
HCR to Polaris and the mean readings to the Earth object.
iv) compute the statistics for all eight set of observations. Every one consists of two (direct and
reverse) azimuths. Mean value, standard deviation of one observation and standard deviation
of the mean value are the necessary statistics.
The astronomic azimuth by the hour angle method is used for all types of astronomic work because of
the high accuracy that can be reached.
Exercise 3.1 For Lab 5 — Astronomic azimuth determination by Polaris (hour angle method)— the
observation procedure is changed. We have one set of observations with six repetitions for direct and
reverse observations. See the table prepared for AstroLab 5 for more information about the observation
and computation procedure.
3.2 Astronomic azimuth determination by the altitude method
Applying the spherical cosine formula for 90 ◦ − δ and z = 90◦ − a we will have:
cos (90◦ − δ) = sin (90◦ − Φ) sin (90◦ − a) cos β + cos (90◦ − Φ) cos (90◦ − a) .
Taking into account that cos β = cos (360 ◦ − A∗ ) = cos A∗ we have:
sin δ = cos Φ cos a cos A∗ + sin Φ sin a .
Finally, for the astronomical azimuth of the celestial body we get:
cos A∗ =
sin δ − sin Φ sin a
cos a cos Φ
with a = 90◦ − z is the altitude of the celestial body. In this formula we use a cosine function of A,
whose inverse cannot uniquely determine the proper quadrant. We must know a priori if A ∈ [0; π] or
A ∈ [π; 2π]. This means we only need to know if we’re measuring before or after noon (if we’re doing
sun shots, that is).
The measured values are the zenith distance z or the altitude angle a. The values for the declination δ
of the celestial body are obtained from catalogues, ephemeris or almanacs. Note that h is absent from
the above formula. The time only enters because we have to take δ from an almanac. The time is only
necessary with an accuracy to the nearest minute.
Differentiating the equation for cos A ∗ leads to:
sin A∗ dA∗ = (tan a − cos A∗ tan Φ) dΦ + (tan Φ − cos A∗ tan a) da .
Now we are able to investigate the effect of different errors.
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3 Determination of the astronomic azimuth
• Let us consider one sine-cosine formula for the spherical triangle in Fig 4.
sin (90◦ − δ) cos h = cos (90◦ − a) sin (90◦ − Φ) − sin (90◦ − a) cos (90◦ − Φ) cos A∗ .
If h = 90◦ = 6h or h = 270◦ = 18h we will have:
sin a cos Φ = cos a sin Φ cos A∗
=⇒
tan a − cos A∗ tan Φ = 0 .
This fact means that the effect of dΦ will be minimized if we measure the sun close to dusk
or dawn. This is not a clever strategy, though, because of atmospheric refraction at these low
elevations.
• Let us consider another sine-cosine formula for the spherical triangle in Fig 4.
sin (90◦ − δ) cos q = cos (90◦ − Φ) sin (90◦ − a) − sin (90◦ − Φ) cos (90◦ − a) cos A∗ .
If the parallactic angle of the celestial body q = 90 or q = 270 ◦ we will have:
sin Φ cos a = cos Φ sin a cos A∗
=⇒
tan Φ − cos A∗ tan a = 0 .
This fact means that the effect of altitude errors is eliminated. It happens when the celestial body
is at elongation. For sun shots this will not occur too often. Perhaps in summer at high Northern
latitudes and in winter at high Southern latitudes.
• Both errors — dΦ and da — can not be simultaneously eliminated. But their effect could be
minimized if two stars are observed. The two stars should have a 1 = a2 and A1 = 360◦ − A2 , i.e.
symmetric with respect to the local meridian.
For azimuth determination by the altitude method the observations of the Sun are used very often.
Using a 100 theodolite the standard deviation of the determined astronomic azimuth could be 20 00 . With
special solar attachments (solar prism) the accuracy of determine azimuth could be 5 00 .
The suggested observation procedure consists of two morning and two afternoon determinations. The
best choice for the altitude a is 30◦ < a < 40◦ , but never a < 20◦ and a > 50◦ .
3.2.1 Observation procedure for azimuth determination by altitude observations of the Sun
The following observation procedure has been proposed in Mueller (1977) and Thomson (1978):
i) Direct to terrestrial object, record horizontal circle reading (HCR) and level (plate or striding
level) readings.
ii) Direct to the Sun, record the HCR and the vertical circle reading (VCR). Record the time to
the nearest minute.
iii) Reverse the telescope. Direct to the Sun, record HCR and VCR and time to the nearest minute.
Record the level readings.
iv) Reverse on the Earth object, record HCR and level readings.
v) Record the temperature and pressure.
8
3.2 Astronomic azimuth determination by the altitude method
3.2.2 Computation procedure for azimuth determination by altitude observations of the Sun
The computation procedure could be found in Mueller (1977) and Thomson (1978).
i) Compute the mean direct and reverse HCRs to the Earth object and to the Sun
ii) Correct the horizontal directions in i)) for the readings of the striding level. Use the formula:
d00
∆A00 = ((w + w0 ) − (e + e0 )) cot z ,
4
which is the same as in the hour angle method.
iii) Compute the angle between the Sun and Earth object.
iv) Compute the direct and reverse zenith distance (altitude) to the Sun. Correct the mean values
for the effect of astronomical refraction.
v) Determine the time of observation to be able to obtain the tabulated declination δ values of
the Sun.
vi) Compute the azimuth of the Sun, using the formula for cos A .
vii) Using the angle from iii) and the azimuth of the Sun from vi) to compute the azimuth to the
Earth object.
The altitude method for astronomic azimuth determination by observations to the Sun has an accuracy,
which will be sufficient for purposes requiring 5 00 .
References
Schwarz, K.P. (1999), Fundamentals of Geodesy, UCGE Reports N10014, U of Calgary, pp 134–136
Thomson, D.B. (1978). Introduction to Geodetic Astronomy, Lecture Notes No.49,Department of
Surveying Engineering, University of New Brunswick, 175pp.
Mueller, I.I. (1977). Spherical and Practical Astronomy as Applied to Geodesy, Ungar, New York,
615pp.
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