|1. The geographical position
|2. The position of the Sun
|3. The angle of projection w
you need firstly to determine, among other quantities, the distance Δ between the two observers or, to be more precise, the relation Δ/RE 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:
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 (earthradius)!
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.
The simplest method is to measure the elevation of the Sun at noon. The best way to do that is described in Radius of the earth where not the maximal elevation but the time of culmination is of interest.
maximal elevation for northern sites
maximal elevation for southern sites
Measurements of longitude are measurements of time! (A possible device is pictured in Measuring the angular radius of the sun.)
If you know the local sideral time you can derive the longitude by the following algorithm:
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 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.
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 Radius of earth:
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:
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!)
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.
Pooh, a lot of work have been done. But now it is easy, at least in principle, to calculate the angle of projection:
The rectangular positions are then given by:
Finally, we get the projection angle from the following equation:
|last update: 11.04.2012