Transit of Venus 2004, Calculation of the Astronomical
Unit from the transit durations at two dierent locations
Fredi Messmer, Rütistrasse 14, CH8134 Adliswil
27. April 2004
1 Introduction
1.1 The method of Halley, as described in his dissertation
For a point on the axis of the Earth the transit duration is independent from the Sun's parallax
and can be calculated using Kepler's laws. On the other hand the sun's parallax cannot be deduced
from this transit duration. But for a point on the Earth's surface the transit durations depend on
the Sun's parallax.
In his dissertation [11] Halley calculated the transit like a solar eclipse for the banks of the Ganges,
presuming a parallax of the Sun of 12,5". From the dierence between calculation and observation
of the transit duration the parallax may be corrected. Hence an observation at one site would
suce, if the appearent diameters of the Sun and Venus, as well as the geographical longitude
are very accurately known. The clock should show the true local time but does not need to go
synchronous with any remote clock.
To improve the accuracy Halley suggested to observe the transit also in Nelson's Harbour in the
Hudson Bay. There the second contact will happen shortly before sunset, and the third contact
after sunrise. At the Ganges the transit duration is shorter than the geocentric transit duration
because of the rotation of the Earth. At the night side in the Hudson Bay, the observer's motion
is reversed and the transit duration is prolonged. Thus a dierence of the transit durations of
17 minutes results. But Halley used unreliable tables, and in his graphical solution he subtracted
the inclination of the orbit of Venus from the angle between the Earth's axis and the axis of the
ecliptic, instead of adding them. The third contact in the Hudson Bay then happened already
before sunrise. Hence that observation was not possible.
As James Ferguson [7] wrote, the transit of 1761 was observed on many places on Earth.
"James Short has taken an incredible deal of pains in deducing the quantity of the Sun's parallax
from the best of those observations which were made both in Britain and abroad, and nds it to
have been 8".52 on the day of the transit", so the AU was 95'173'127 english miles or 153'165'592
km. James Short's paper is contained in the Philosophical Transactions of the Royal Society for
the year 1763.
1.2 Halley's method
In the last sentence of his dissertation Halley mentions a method that "shall be explained on some
other occasion".
1
In "Splendour of the Heavens", [12] Vol. II, page 451 the following method quite dierent from
that what Halley suggested in his dissertation is mentioned as Halley's method:
Two observers at dierent sites measure the times of the second and third contact. The trajectories
of Venus on the Sun's disc can now be replaced by chords calculated from the transit duration. The
distance of these two chords on the Sun's disc yields the parallax. From the parallax, the rectilinear
distance of the observers, and the distance relation EarthSun to VenusSun the distance Earth
Sun can be calculated. Certain corrections have to be made due to the Earth's rotation.
1.3 Our method
We did not nd any paper of Halley that describes the above mentioned "Halley's method". Our
aim was to develop it according to the known principles.
Our calculation of the astronomical unit (AU) is thoroughly described in "Venustransit 2004" of
Prof. Dr. Heinz Blatter and Reny Montandon [1], and in "Venustransit 2004: Bestimmung der
Sonnenparallaxe" of Blatter-Messmer-Hauswirth [2].
The advantage of our method is, that the clocks at dierent locations do not need to go exactly synchronous, and the longitudes do not have to be known accurate to an arc minute; the disadvantage,
that exact measurements of the contact times are very dicult.
More succes may be expected from a modern t-method developed by Dr. Roland Brodbeck in
astro!nfo [5] and in Orion [8], measuring the distances between the centre of Venus and the centre
of the Sun repeatedly during the transit. But the measurement of the distance of two points which
cannot actually be seen requires some skills from amateurs. Our professional predecessors in the
19. century did this using specially constructed Heliometers which Andreas Verdun described in
his excellent paper in Orion [6]. With todays Megapixel CCD Chips and other digital cameras,
now aordable for amateurs, and optics calibrated using a grid (drawn on paper), a good accuracy
could be possible.
Heinz Blatter and I see our project as a performance of one of the historical methods of deducing
the AU from the measurement of the transit times at two sites , one in the north, and another
very distant in the south, what may have been applied after the transit of 1761.
The calculation of examples with parameters calculated using Calsky [9] shows that with some
additional approximations not described in detail in [1] and [2] an accuracy of better than one
percent is possible. In the following sections the calculation shall be explained step by step. The
detailed derivations of the equations follows in paragraph 9, as far as they are not already done in
the papers of Heinz Blatter [1] [2].
2 Position-vectors and base lines
The vectors of the base line between the observing sites P (in the north) and Q (in the south) shall
be calculated for the times of the second and third contact respectively. The rst and the fourth
contact could also be used without problems. I shall not consider them because the rst contact
may be very dicult to observe.
In what follows index 2 is used for the time of the 2. contact, index p for the observing site P, and
in the same sense index 3 and index q. First we calculate the distance H of the observing site from
the meridian of the Sun, i.e. from the x-axis of the coordinate system in gure 1, according to the
hour angle of the Sun at the observing site P, using the same sign, increasing in eastern direction.
Hp2 = tp2 − M ,
Hp3 = tp3 − M
(1)
Hq2 = tq2 − M ,
Hq3 = tq3 − M
(2)
2
Abbildung 1: Sun oriented cartesian coordinate system and position of the observer P.
tp2 , . . . , tq3 = true local time of the contacts,
M = true noon = 11:59:05 local time.
Working with local time does not need synchronous clocks. The geographical longitude does not
appear in our equations. The synchronization of our clocks is provided by the Sun via the true
noon, and the dierence of the geographic longitudes follows from the dierence of the hour angles
Hp , Hq , without regard to the fact that the contacts at dierent sites are not simultaneous.
∆t
To work with UT the time correction λ ∆λ
must be subtracted from the contact times and from
the true noon.
λ = geographic longitude of the observing site, positiv to east,
∆t
∆λ = time correction, + 4 minutes per degree of eastern longitude.
We calculate the positionvectors of the observing sites considering the geocentric latitude:
φ0 = φ − 695.65” sin 2φ + 1.17” sin 4φ
(3)
and the oblateness of the Earth:
R0 = R (0.99832 + 0.001684 cos 2φ − 0.000004 cos 4φ)
(4)
φ = geographic latitude,
φ0 = geozentric latitude of the observing site,
R = equator radius of the Earth.
R0 = distance of the observing site from the centre of the Earth,
See also Handbuch für Sternfreunde [10]
Position-vectors of the observing sites:


cos φ0p cos Hp2
P~2 = Rp0  cos φ0p sin Hp2  ,
sin φ0p


cos φ0q cos Hq2
~ 2 = Rq0  cos φ0 sin Hq2 
Q
q
sin φ0q
(5)
Analogous for the 3. contact.
The transit trajectories are generally curvilinear because the distance of the observer from the
ecliptic varies during the transit as more as the observer is closer to the equator.
3
The band of the curvilinear trajectories (in function of time) is the central projection of the base
line upon the Sun through the centre of Venus. The mean base line:
´ ³
´
³
~ 2 − P~2 + Q
~ 3 − P~3
Q
~m =
B
(6)
2
is the median of the trapezium that is erected by the base line at start and end of the transit. It's
projection may well t into the band of the two trajectories. See also gure 3.
Abbildung 2: Sun oriented cartesian coordinate system and the perpendicular to the ecliptic plane
3 Unit vectors
Normal vector to the ecliptic:
Line of sight EarthSun:


− sin ² sin α
~e =  − sin ² cos α 
cos ²
(7)


cos δ
~s =  0 
sin δ
(8)
Line of nodes ~k of the orbital plane of Venus:
The unit vector of the line of nodes is perpendicular to the normal vector of the ecliptic and makes
the angle κ with the vector of the line of sight EarthSun.
~k · ~s = cos κ ,
~k · ~e = 0 ,
4
~k · ~k = 1
(9)
κ = dierence between the ecliptical longitudes of the Sun and the descending node of the orbit
of Venus at half time of the transit.
Normal vector p~ to the plane of the orbit of Venus:
The normal vector to the plane of the orbit of Venus is a unit vector perpendicular to the vector
of the line of nodes and makes the angle i with the normal vector to the ecliptic.
p~ · ~e = cos i ,
p~ · ~k = 0 ,
p~ · p~ = 1
(10)
4 Distance relations and angular velocities
While we assume that the astronomical unit and hence the distances in the Sun system are unknown
I use as far as possible distance relations. But in the equations for the Earth rotation the distance
EarthSun cannot be eliminated.
For Kepler orbits I use the denotations from [2], d for distance, index s for the Sun, index v for
Venus und index e for the Earth. For elliptical orbits I add another index e.
The distance relations at the time of the transit are:
d
vse
dvse
= dese
deve
1 − ddvse
ese
dvse
1.00398 dvs
=
,
dese
1.015 des
(11)
The mean angular velocity of Venus is, as seen from
observer P
ω
~ p = k1 ·
dvse
deve
³
´


"µ
#
¶2
µ
¶2
~3 − P~2 × ~s 
 2π
P
des
~e
dvs
p~
+
[”/s]
·
−
·
 86400
dese
Te
dvse
Tv
(tp3 − tp2 ) · dese 
(12)
observer Q
³
´


"µ
#
¶2
µ
¶2
~3 − Q
~ 2 × ~s 
Q
dvse  2π
des
~e
dvs
p~
ω
~ q = k1 ·
·
−
·
+
[”/s]
deve  86400
dese
Te
dvse
Tv
(tq3 − tq2 ) · dese 
(13)
Tv und Te are the sidereal orbit times of Venus and Earth
k1 = arc seconds/radian = 206'264.81
5 Transit chords
In gure 3 distance and inclination of the transit chords are highly exaggerated. The length of ∆β
is between about 15 and 30 arcseconds. The inclination of the chords is 0.53◦ in the most extreme
example (Tiksi-Pretoria).
The hight of the curvilinear transit trajectory upon the transit chord is about 1% of the length of
the chord. The transit chords on the Suns disk
aa0 = |~
ωp | ∆ tp ,
bb0 = |~
ωq | ∆ tq
(14)
shall be erected perpendicular to the vectors ω
~ of the appearent angular velocity of Venus. Hence
we get the vectors β~ of the "smallest" angular distance VenusSun:
5
s
~p
~ = β~p − β~q = ω
∆β
|~
ωp |
µ
R2 −
aa0
s
¶2
2
−
ω
~q
|~
ωq |
R2 −
µ
bb0
2
¶2
(15)
R is the radius of the Sun, diminished by the radius of Venus, for elliptical orbits.
~ ist the dierence of the vectors β~ , which could be up to 50 times greater than ∆β
~ . These
∆β
dierence is the critical term of our calculation method!
~q are the distances of the transit chords from the centre of the Sun calculated from the
β~p und β
~ would ideally be the central projection of the base line through the centre of
transit times. ∆β
Venus upon the Sun. Generally the trajectories are slightly curvilinear because of the rotation of
the Earth, and the appearent velocity of Venus is smaller at Sunset than at noon. The calculated
~ are not "seen" at the time of the smallest angular distance, and also not simultaneous.
vectors β
For instance Zimmerwald sees the smallest angular distance about 47 seconds after halftime of the
transit. On the equator in the meridian of Zimmerwald it happens another 20 seconds later.
Abbildung 3: Transit chords and vectors of the appearent motion of Venus
6 Astronomical unit
~ are projected upon the
~ m and the vector of the distance ∆β
The vector of the mean base line B
unit vector of the appearent motion of Venus:
p~s =
ω
~p + ω
~q
,
|~
ωp + ω
~ q|
~ m · p~s ,
B=B
~ · p~s
∆β = ∆β
(16)
The astronomical unit corresponds (not abolut exactly) to the mean distance EarthSun:
AE = des =
B dvse des
∆β deve dese
(17)
The derivation of this equation and the problems with the geometry of the parallaxes are treated
by Heinz Blatter and Reny Montandon in the Orion-paper [1] . In paragraph 9 they are again
represented in detail with regard to our denotations.
6
7 Practical application
7.1 Parameter
All the parameters needed for our calculations and their origin are listed in table 1. As far as
possible I used publicly available sources.
Here we cheat a little bit: All the parameters where known to Halley, but not in todays precision!
We could of course use historical parameters.
7.2 First calculation using a historical AU
As already mentioned in paragraph 4 the astronomical unit des which we like to calculate cannot
be eliminated in equation (39). A rst calculation could be done using on of the following historical
values:
138'400'000 km, Cassini and Richer 1672
129'200'000 km, Lacaille 1751
7.3 Iteration
The result of this rst approximation may now be used for Iteration.
The function xi+1 = f ( xi ) konverges monotonous.
The limit ∆ x = 1 km is reached after about 5 to 15 Iterations.
7.4 Examples
The table 3 shows the results of the calculated examples. They show that a strong inclination of
the base line to the Earth's axis reduces slightly the precision of the results.
7.5 Inuence of the approximations
The inuences of the particular approximations on the accuracy of result are about as follows:
Earth rotation: 5-10% for base lines about parallel to the Earths axis, otherwise 30-50% and more
Elliptical orbits: 4%
Inklination of the orbital plane of Venus: 4%
Distance of the line of nodes from the line of sight EarthSun: 1%
Oblateness of Earth and geocentric latitude: 0.6%
7.6 Errors in measuring the contact times
An error of 7 seconds in one of the contact times yields an error in the result of about 1%. If both
contact times are equally 7 seconds too large or too small the errors would compensate to about
0.08%.
8 Reference
I calculated the examples using MS Excel 97 (german version). The table "Venustransit" for the
calculation as well as the le "Beobachter" with the parameters from the "Observers" are available
in Astroinfo [5]. The "Beobachterdaten" may be copied from the le "Beobachter" into the table
"Venustransit". Hints for the practical work are given at the bottom of the tables.
7
9 Derivation of the Equations
9.1 Line of nodes
The descending node is passed by Venus about 1.2 days before the transit and therefore it lies not
exactly on the EarthSun line of sight .
The unit vector of the line of nodes is perpendicular to the ecliptical normal vector and makes the
angle κ with the vector of the EarthSun line of sight.
~k · ~e = 0 ,
~k · ~s = cos κ ,
kx2 + ky2 + kz2 = 1
(18)
ky ey + kz ez
ex
(19)
κ = λSun − λN ode = 1, 1807◦ at the Transit 2004 .
kx ex + ky ey + kz ez = 0 ,
kx sx + ky sy + kz sz = cos κ ,
kx = −
ky sy = cos κ − kx sx − kz sz
ky ey + kz ez
sx + ky sy + kz sz = cos κ
ex
µ
µ
¶
¶
ey
ez
ky sy − sx
+kz sz − sx
= cos κ
ex
ex
{z
}
{z
}
|
|
−
=A
=⇒
ky =
cos κ − kz B
A
cos2 κ + kz2 B 2 − 2kz B cos κ
A2
(25)
ky2
=C
=D
cos κ +
kz2 B 2
(26)
(27)
=E
ky2 C + kz2 D + ky kz E − 1 = 0
2
(23)
(24)
2
e2y
ey ez
2 ez
+
k
+ 2ky kz 2 + ky2 + kz2 = 1
z
e2x
e2x
ex
Ã
!
µ
¶
e2y
e2
ey ez
ky2 1 + 2 +kz2 1 + 2z +ky kz · 2 2 = 1
ex
e
e
| {zx }
| {z x }
| {z }
=⇒
(22)
ky2 e2y + kz2 e2z + 2ky kz ey ez
e2x
kx2 =
kx2 + ky2 + kz2 = 1
(21)
=B
ky A + kz B = cos κ ,
ky2 =
(20)
− 2kz B cos κ
cos κ − kz B
C + kz2 D +
kz E − 1 = 0
A
µ
¶
µ
¶
B2C
BE
E
2BC
C cos2 κ
kz2 D +
−
+
k
−
cos κ +
−1=0
z
2
2
A
A
A
A
A2
A2
(28)
(29)
(30)
Because A 6= 0, we may multiply this equation by A2
¡
¢
kz2 A2 D + B 2 C − ABE + kz (AE − 2BC) cos κ + C cos2 κ − A2 = 0
M = A2 D + B 2 C − ABE , N = (AE − 2BC) cos κ , P = C cos2 κ − A2
√
cos κ − kz B
ky ey + kz ez
−N − N 2 − 4M P
, ky =
, kx = −
kz =
2M
A
ex
(31)
(32)
(33)
The quadratic equation kz in equation (33) yields two solutions. According to the orientation of κ
the minus sign in front of the square root in equation kz of (33) is valid.
8
9.2 Normal vector to the orbital plane of Venus
The normal vector to the orbital plane of Venus is a unit vector perpendicular to the vector of the
line of nodes and makes the angle i with the normal vector to the ecliptic.
p~ · ~k = 0 ,
p~ · ~e = cos i ,
p2x + p2y + p2z = 1
(34)
The equations have exactly the same form as the equations for the line of nodes. We just have to
replace ~k by p~ , ~e by ~k , ~s by ~e , cos κ by cos i . Here too the minus sign in front of the square root
of the quadratic equation is valid.
9.3 Transit velocity
Abbildung 4: Angular velocity of the appearent motion of Venus.
For slightly elliptical orbits a good approximation for Kepler's equation would be:
v r = ve r e ,
ω r2 = ωe re2 ,
v = ωr,
=⇒
ωe =
r2
·ω.
re2
(35)
v is the instantaneous velocity, r the radius vector and ω the angular velocity. Index e stands for
elliptical orbits.
With equation (35) the sidereal angular velocity of Venus around the Sun becomes:
µ
ω
~v =
and that of the Earth:
µ
ω
~e =
dvs
dvse
des
dese
¶2
·
2π
· p~
86400 · Tv
£ −1 ¤
s
(36)
·
2π
· ~e
86400 · Te
£ −1 ¤
s
(37)
¶2
The observer in P moves on his latitude from P2 to P3 during the transit. Thus his distance from
the ecliptic plane varies, and the transit trajectory becomes slightly curvilinear on the Sun's disc.
In our calculation method this trajectory is replaced by a chord, which is passed by Venus with
a constant mean velocity. We should project the motion of our observer in the same way. Thus
we replace the way from P2 to P3 by a chord. This is easy to calculate and could be a good
approximation. Thus the mean velocity of the observer relativ to the centre of the Earth is:
~vrp =
P~3 − P~2
t3 − t2
9
[km/s]
(38)
P~2 , P~3 are the geocentric vectors of the observers location P at the times of the 2. and 3. contacts
respectively. As seen from the Sun the angular velocity of the rotation of the Earth is:
³
ω
~ rp =
´
P~3 − P~2 × ~s
£ −1 ¤
s
~vrp × ~s
=
dese
(t3 − t2 ) · dese
(39)
As seen heliocentric the appearent angular velocity of Venus relativ to the observer P is:
(40)
ω
~ =ω
~v − ω
~e − ω
~ rp
Geocentric, as seen from the observer in P, the orientation of the angular velocity changes and
according to the distance relations we get:
ω
~p = −
dvse
dvse
ω
~ =
(~
ωe − ω
~v + ω
~ rp )
deve
deve
£ −1 ¤
s
or written in detail:
³
´


"µ
#
¶2
¶2
µ
~3 − P~2 × ~s 

P
dvse
2π
des
~e
dvs
p~
[”/s]
ω
~ p = k1 ·
·
−
·
+
deve  86400
dese
Te
dvse
Tv
(tp3 − tp2 ) · dese 
(41)
(42)
For circular orbits dvse = dvs , and dese = des . Thus equation (42) reduces to:

ω
~ p = k1 ·
dvs  2π
dev 86400
µ
~e
p~
−
Te
Tv
¶
³
´

P~3 − P~2 × ~s

+
(tp3 − tp2 ) · dese
(43)
The vector ω
~ q for the observer in Q is calculated in the same way.
9.4 Astronomical Unit
Abbildung 5: Measuring the parallaxes during a transit. Relations of the angles.
Figure 5 shows the geometry of an observation of Venus on the Sun. To symplify we assume that
the gure shows a projection upon the plane through the centre of the Sun and perpendicular to
the vector ~s of the line of sight in the Sun oriented coordinate system.
As mentioned in paragraph 6 Heinz Blatter and Reny Montandon discussed the special problems
and the geometry of the transit in Orion [1] .
10
For the observer in P Venus "draws" her transit trajectory through the point P' on the surface of
the Sun. If we would replace (as a good approximation) the Sun by a projection screen through
the centre of the Sun S, we had the trajectory through the point P.
The observer P sees the distance Venus - centre of the Sun SP 00 under the angle βp .
The observer Q sees the distance Venus - centre of the Sun SQ00 under the angle βq .
The angle αv is the dierence of the angles γp − γq , but not the dierence of the angles βp − βq ,
because SP 00 and SQ00 are seen from dierent perspektives!
The angles γ serve for illustration of this fact but do not have any meaning for our calculations.
We just kwow the angles β .
We may calculate the Astronomical Unit as a good approximation from the relation:
αs =
B
dese
(44)
From the triangles
triangle PQV:
αv + ²p − βp + ²q + βq = 180
triangle PQS:
αs + ²p + ²q = 180
the dierence of the two equations yields:
αv − αs − βp + βq = 0,
=⇒
αv − αs = βp − βq = ∆β
(45)
The parallaxe αv is in good approximation:
αv =
B
deve
By that follows with equation 44:
αv = αs
αv inserted in equation 45 yields:
µ
¶
dese
dvse
αs
− 1 = αs
= ∆β
deve
deve
and with dese =
B
αs
(46)
dese
deve
(47)
=⇒
αs = ∆β
deve
dvse
(48)
from equation 44:
dese =
Thus the astronomical unit:
AE = des =
B dvse
∆β deve
(49)
B dvse des
∆β deve dese
(50)
11
Thank.
My calculations, as well as this paper, are based essentially on the works of Prof. Dr. Heinz Blatter
[1] [2] . He also allowed me to use his gures 1 and 2 . The further development of the method
and the improvement of the approximations are the result of interesting discussions with him. Also
during his stay in Japan we used to have intense discussions by internet. I want to thank him for
all these, as well as for the many useful hints in handling LATEX .
Tabelle 1: Venustransit: Werte und Herkunft der benutzten Parameter
Sonne
Siderische Umlaufszeit
Rektaszension
Deklination
Schiefe der Ekliptik
Ekliptische Länge
Wahrer Mittag
Mittlerer scheinbarer Radius
Massgebender Radius
− für elliptische Bahnen
365.256363
5.116944
22.88
23:26:26
77.8756
11:59:05
959.64
929.48
916.49
d
h
rs
rs − rv
rse − rve
SH 2002, S. 40
CalSKY, für 8.6.2004, 08:22 UT
CalSKY, für 8.6.2004, 08:22 UT
CalSKY, für 8.6.2004, 08:22 UT
CalSKY, für 8.6.2004, 08:22 UT
CalSKY, für 8.6.2004
Herder, S. 243
rSonne − rV enus für innere Kontakte
rs · (es/ese) − rve
rv
rve
Herder, S. 250
CalSKY, für 8.6.2004, 08:22 UT
CalSKY, für 8.6.2004, 08:22 UT
k1 · 12104/(2(ev/es) · 149597870)
rv (es/ese)(1 − vs/es)(1 − vse/ese)/
k1
Herder, S. 250
3600 · 180/π
◦
◦
'"
◦
h:m:s
"
"
"
Venus
Siderische Umlaufszeit
Inklination
Knotenlänge
Mittlerer scheinbarer Radius
− für elliptische Bahnen
224.701
3.3947
76.6949
30.1606
28.8947
d
Erdradius (Aequator)
Bogensekunden pro Radian
6'378.14
206'264.81
km
"/rad
◦
◦
"
Distanzverhältnisse
− für mittlere Entfernungen
Venus-Sonne : Erde-Sonne
Venus-Sonne : Erde-Venus
− für elliptische Bahnen
Ellipse : Kreis, Erde-Sonne
Ellipse : Kreis, Venus-Sonne
Venus-Sonne : Erde-Sonne
Venus-Sonne : Erde-Venus
0.7233297
2.614410
vs/es
vs/ev
1.0150865
1.00398
0.715476
2.514647
ese/es
vse/vs
vse/ese
vse/eve
Herder, S. 250
(vs/es)/(1 − vs/es)
CalSKY, für 8.6.2004, 08:22 UT
Arbeitspapier 3.7.2002
(vs/es) · (vse/vs)/(ese/es)
(vse/ese)/(1 − vse/ese)
CalSKY: Astronomische Online Software, Arnold Barmettler, 2002, http://www.CalSKY.com/
SH: Sternenhimmel, Hans Roth
Herder: Lexikon der Astronomie 1990, Band 2
12
Tabelle 2: Koordinaten und Kontaktzeiten der Beobachtungsorte
Ort
Riad
Antananarivo
Kottamia
Pretoria
Zimmerwald
Pretoria
Murmansk
Pretoria
Longyearbyen
Pretoria
Zimmerwald
Yaounde
Tiksi
Pretoria
Nuuk
Pretoria
Breite
Länge
24:39:00 N
18:52:00 S
29:55:54 N
25:45:00 S
46:52:42 N
25:45:00 S
68:59:00 N
25:45:00 S
78:12:00 N
25:45:00 S
46:52:42 N
3:51:00 N
71:40:00 N
25:45:00 S
64:15:00 N
25:45:00 S
46:46:00 E
47:30:00 E
31:49:30 E
28:12:00 E
7:28:00 E
28:12:00 E
33:08:00 E
28:12:00 E
15:40:00 E
28:12:00 E
7:28:00 E
11:31:00 E
128:45:00 E
28:12:00 E
51:35:00 W
28:12:00 E
2.
Kontakt
5:37:47.3
5:35:26.2
5:38:58.9
5:36:15.8
5:39:45.0
5:36:15.8
5:37:53.1
5:36:15.8
5:37:29.2
5:36:15.8
5:39:45.0
5:39:17.8
5:34:35.8
5:36:15.8
5:38:02.8
5:36:15.8
3.
Kontakt
11:04:00.7
11:08:01.4
11:04:36.2
11:10:02.9
11:04:18.9
11:10:02.9
11:01:30.5
11:10:02.9
11:01:20.5
11:10:02.9
11:04:18.9
11:08:39.6
10:59:33.0
11:10:02.9
11:03:21.7
11:10:02.9
Die Kontaktzeiten wurden mit CalSKY gerechnet
Tabelle 3: Resultate der berechneten Beispiele
Ort
Riad
Antananarivo
Kottamia
Pretoria
Zimmerwald
Pretoria
Murmansk
Pretoria
Longyearbyen
Pretoria
Zimmerwald
Yaounde
Tiksi
Pretoria
Nuuk
Pretoria
Mittelwert
Heutiger Wert
β
∆β
(' )
10:30.2
10:15.7
10:33.2
10:15.4
10:40.4
10:15.4
10:43.8
10:15.4
10:45.7
10:15.4
10:40.5
10:26.7
10:43.9
10:15.4
10:47.0
10:15.5
( )
14.6
Basis
projizierte Winkel zur
Länge
Erdachse
(km)
(◦ )
4'264.371
1.76
17.9
5'233.983
2.53
149'630'456
0.02%
25.0
7'347.351
12.85
149'900'748
0.20%
28.4
8'343.036
16.38
150'241'372
0.43%
30.3
8'921.705
20.36
150'391'830
0.53%
13.8
4'068.695
19.70
150'743'197
0.77%
28.4
8'380.393
28.52
150'578'240
0.55%
31.5
9'317.436
27.72
150'887'974
0.66%
10:28.7
23.7
6'984.675
150'244'396
149'597'870
0.43%
13
Astronomische
Einheit
(km)
149'581'352
zum
heutigen
Wert
-0.01%
Literatur
[1] Blatter, H. und Montandon, R.O., Venustransit 2004, Orion 307, 2001, S.4, Errata in Orion
311, 2002, S.11, sowie Internet:
http://www.astronomie.info/projectvenus
[2] Blatter, H., Messmer, F., Hauswirth, R., Bestimmung der Sonnenparallaxe,
Oktober 2002, Internet: http://www.astronomie.info/projectvenus
[3] Roth, Hans: Der Sternenhimmel, Birkhäuser, 2000, Kosmos, 2002
[4] Lexikon der Astronomie, Herder, 1989
[5] Internetauftritt in astroinfo zum Thema Venus 2004, http://www.astronomie.info/projectvenus
[6] Andreas Verdun: Beispiele aus der Geschichte der Positions-Astronomie,
Orion 310, 3/2002, S. 10
[7] Ferguson James: Astronomy explained on Sir Isaac Newton's principles, ...,to which are added,
a plain method of nding the distances of all the planets ...
London printed for A. Millar, 1764
[8] Roland Brodbeck: Bestimmung der Astronomischen Einheit AE anhand des Venustransits
Orion 312, 5/2002, S. 4, sowie Internet: http://www.astronomie.info/projectvenus
[9] Barmettler Arnold: Astronomische Online Software CalSKY, 2002
Internet: http://www.CalSKY.com/
[10] Roth, Günter Dietmar: Handbuch für Sternfreunde
Springer-Verlag, 3. Auage 1981
[11] Halley, Edmond: A new Method of determining the Parallax of the Sun
Philosophical Transactions of the Royal Society, Vol XXIX (1716)
[12] T. E. R. Phillips , F.R.A.S. and W.H. Steavenson, F.R.A.S.: Splendour of the Heavens
Robert M. McBride & Company New York 1925
14