Essen, horizontal view

Internet Project
Observing, Photographing and Evaluating the
Transit of Venus, June 8th, 2004

Essen, equatorial view

Project 2: Determing of the own geographical coordinates and the projected distance of different observers
To be able to derive an own measure of the Astronomical Unit from the
self determined value of the Sun's parallax by the following equation (see the
related paper (in German or in English))
you need firstly to determine, among other quantities, the distance
Δ between the two observers or, to be more precise, the
relation Δ/R_{E} between the distance and the
radius of the Earth. Secondly, you must determine the angle of projection
w because you must calculate the the projection of the
distance parallel to the direction to the Sun.
The distance Δ can be calculated when the
geographical coordinates of the both observers are known. For the
calculation of w it is necessary to determine the local
sideral time of the observers and the position of the Sun, that
means its geocentric equatorial coordinates.
There are two tasks for measurements:
 Determination of the own geographical position. This may be done in
advance.
 Measurement of the Sun's position on June 8th which can be done best at
noon.
Of course, very exact measures of both quantities are easily to be obtain,
the geographical position, for instance, by GPS (Global Positioning System),
the position of the Sun from every astronomical almanach or computer program.
Nevertheless, we will try to get own measures in order to understand better
the foundations of the astronomical distance ladder!
Because the geographical positions influence the result for the parallax of
the Sun only weakly quite rough methods of "astronomical navigation"
are sufficient.
For these measurements, it is suitable to have determined the exact
direction to south in advance (project
3)!
1.1 Geographical latitude
1.1.1 at night
The geographical latitude equals the elevation of the northern celestial pole
and, with sufficient accuracy, that of Polar Star:
Unfortunately, there is no as simple method for observers on the southern
hemisphere! But it is possible to apply the method which is described for the
Sun, below, to a star with known declination.
1.1.2 during the daytime
The simplest method is to measure the elevation of the Sun at noon. The best
way to do that is described in project 3
where not the maximal elevation but the time of culmination is of interest.



maximal elevation for northern sites
 maximal elevation for southern sites



1.2 Geographical longitude
Measurements of longitude are measurements of time! (A possible device is
pictured in project 4.)
1.2.1 at night
If you know the local sideral time you can derive the longitude by the
following algorithm:
 Your actual sideral time differ from that of Greenwich by 4min/degree
times your longitude (which we count positive if your site is east of
Greenwich):
 The sideral time at the time t(UT) can be derived from that at midnight:
The factor 1.0027379 is due to the fact that, during 24 hours, the Earth
rotates more than once and, therefore, more than 24 h sideral time pass.
 The Greenwich sideral time at midnight of the day of interest may be
taken from an astronomical almanach.
Combining these relations we find the following equation which makes it
possible to derive the longitude from the local sideral time at t:
The simplest way to determine local sideral time is to use the "celestial
clock":
A more precise way is to measure the time of culmination of a star with known
right ascension:
To be able to recognize the time of culmination you should know the exact
direction to south.
1.2.2 during the daytime
In this case, you should measure the time of local midday, that is the time at
which the Sun culminates, by the method described in
project 3:
If you live east of Greenwich the Sun will culminate before 12.00 UT! For
exactness, this equation had to be corrected by the equation of time. But
because in June this correction would be small it has been omitted.
Your longitude, therefore, can be calculated in the following way:
2. The position of the Sun
The position of the Sun ca be determined by the same procedures as described
above (Caution: There is a certain danger to get into a tautology!)
 The declination can be determined by measuring the maximal elevation.
 The right ascension can be determined by observing the exact time of
culmination and calculating the associative sideral time.
3. The angle of projection
To be able to calculate the angle of projection w you must know the
vector connecting two observers and the vector directing to the Sun in the
same coordinate system. As we already know the position of the Sun in
geocentric equatorial coordinates we want to transform the positions of the
observers in that system, too.
 As the picture above shows the declination of an observers equals,
obviously, his geographical latitude;
 When the observer is located in the xyplane (as in the picture above)
his right ascension is, obviously, zero. Additionally, the vernal equinox is
just culminating for him, that is his local sideral time is 0.00h. Then, due
to the rotation of the Earth his right ascension is growing with the same rate
as his sideral time. Therefore, his right ascension always equals his sideral
time:
Pooh, a lot of work have been done. But now it is easy, at least in principle,
to calculate the angle of projection:
 Transform the equatorial positions of the observers from polar into
rectangular coordinates:
The rectangular positions are then given by:
 The distance, therefore, is the length of the vector connecting the both
observers:
 Transform, in the same manner, the position of the Sun to rectangular
coordinates.
 The angle of projection can then be calculated with the scalar product:
Finally, we get the projection angle from the following equation:
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Prof. Dr. Udo Backhaus
last modification: March 28^{th}, 2008