The key to understanding the method is the triangulation method.
Triangulation is:
"in navigation, surveying, and civil engineering, a technique for
precise determination of a ship's or aircraft's position, and the
direction of roads, tunnels, or other structures under construction.
It is based on the laws of plane trigonometry, which state that, if
one side and two angles of a triangle are known, the other two sides
and angle can be readily calculated. One side of the selected triangle
is measured; this is the baseline. The two adjacent angles are
measured by means of a surveying device known as a theodolite, and the
entire triangle is established. By constructing a series of such
triangles, each adjacent to at least one other triangle, values can be
obtained for distances and angles not otherwise measurable.
Triangulation was used by the ancient Egyptians, Greeks, and other
peoples at a very early date, with crude sighting devices that were
improved into the diopter, or dioptra (an early theodolite), and were
described in the 1st century AD by Heronof Alexandria."
( from: The Encyclopedia Britannica CD Rom Version 2003 )
The NASA published a fine article on how this method can be applied to
the measurement of the AU using Venus Transits. This is the best
source I could find as it contains basic information (in rather simple
words) on the method:
"Suppose you were trying to measure the width of a canyon from your
side to the distant canyon wall. You couldn't pace the distance. Could
you come up with a way to make this measurement safely?
...
Surveyors and geologists encounter this kind of problem all the time,
and over the course of centuries, they have found a simple way to sole
this problem. They use a method called 'triangulation'.
In ordinary land surveying, imagine a distant mountain peak and two
observers are located at 'A' and 'B' separated by a few miles (the
length 'S'). The base angles at A and B can be measured with an
instrument called a theodolite. By knowing the base distance A to B,
and the baseline distance S, the distance to the peak can be worked
out with a simple scaled drawing or with trigonometry.
( http://image.gsfc.nasa.gov/poetry/venus/triangulation.jpg )
Suppose it was hard for you to measure the two base angles in the
triangulation method. This could easily happen if the object were so
far away that your instrument could not accurately discern that these
angles were different than 90 degrees. For example, if the object is
10 miles away, and your baseline is only 5 feet long, the two base
angles would have a measure of 89.9946 degrees. This angle differs
from 90 degrees by only 0.0044 degrees which equals 16 seconds of arc
(there are 60 minutes or arc/degree x 60 seconds of arc/minute of arc
= 3600 seconds of arc per degree!) This would be a very difficult
angle to measure even with very expensive modern surveying equipment!
Astronomers run into this problem all the time. To solve this problem,
they don't bother measuring the base angles at all. Instead, they
measure the vertex angle in the triangle. It turns out that this angle
is very easily measured using photographic techniques. The method is
called trigonometric parallax or just 'parallax' for short.
Extend your arm in front of you, hold your thumb up, and alternately
open and close your eyes. You will see your thumb's position move
against the more distant background in front of you. Astronomers call
this the parallax shift as the figure below illustrates:
( http://image.gsfc.nasa.gov/poetry/venus/parallax.gif )
By knowing the distance between your eyes (2 x R) and how much this
shift measures in degrees (twice the measure of the parallax angle q
), you can calculate the distance to your thumb (D)! The formula that
you use is:
Tan (q) = R / D
But this same principle applies to measuring distance to objects far
away from you too...like the planets. Today, astronomers use
photographs of stars taken 6 months apart. During that time, Earth has
traveled from one side of its orbit to the other, and the orbit
baseline is twice 93 million miles (150 million kilometers). By
measuring how far the image of a star has shifted relative to the far
more distant stars in the background between, say, January and June,
astronomers can accurately measure angles as small as 0.001 seconds of
arc or 0.0000003 degrees.
Believe it or not, long before the time when photography had been
invented, astronomers were using the parallax technique to measure the
distances to the nearby planets. By the time of Kepler in the early
1600s, astronomers knew exactly how far the planets were from the sun
in terms of the distance from earth to Sun, but they didn't know
exactly how many kilometers this distance equaled.
When you make a scale model of the solar system with its nine planets,
how do you know that in this model, the actual distance from Sun to
Earth is 93 million miles and not, say, 153 million or 23 million? At
the time of Kepler, the best estimates for this distance were as small
as 5 million miles! The answer is that you have to come up with a way
to actually measure this distance, and just like land surveyors, you
can't travel across distance to make this measurement."
( all from: http://image.gsfc.nasa.gov/poetry/venus/TRACEvenus.html )
As for the real calculations I suggest you visit the NASA page as
there are many illustrating charts and images or you chose the
following:
The step by step procedure to do the calulation ist given here:
( http://www.vt-2004.org/usage/vt-forms.html )
As there are many formula hard to transform to the limited formation
that can be used here, please see the procedure on the page I gave.
If you want to visit more pages on the Venus Transit, an extensive
commented link of links can be found at:
( http://www.transitofvenus.org/ )
I hope this helps you to understand the method and solves your problem.
Please post a clarification request if anything should still be unclear.
till-ga |