Level : Some mathematical aibility is required.
Objectives
This very rare event will allow a rather precise determination of the distance towards Venus, and towards our Sun.
Our objectives are to apply a simplified 2 Dimensional method, similar to what the EAAE has applied on earlier Eclipse Events. - see Teachers Notice.
Background: The students need to know:
1. Mathematical content : basic trigonometry in 2D - in particular the Sine relation.
2. Astronomical content
Longitude, Latitude, Parallax.
However, before entering the chapter below, you may find it useful to start with this easy to perform Lab Experiment Galileo 1610 Venus Orbit Investigation.
Materials needed : the calculations below may be performed without astronomical equipment.
However, in case you have access to an observatory and thus may contribute with real life images of the transit event, this would help us establishing the first student based measurement of the solar distance. Volunteers, click here
Let us start with some basic astronomy.
You have (probably) all noticed that during summer time the shadows are short (left):
and during winter time the shadows are long (right).
So, the `height' of the Sun (above the horizon) varies from season to season. This height is referred to as the altitude by astronomers.
Astronomers usually describe this variation by considering the conditions at the Earth's Equator:
Note that the solar rays hit our planet at a certain angle with respect to the Equator. Astronomers call this angle the solar declination.
During the Venus transit - the Sun will be positioned above equator, so that we experience Summer in the Northern hemisphere. Mathematically speaking, at this time our Sun has a positive declination.
The solar declination reaches a maximum at +23.44 degrees around June 22.
During wintertime in the Northern hemisphere, the Sun reaches its minimum declination, -23.44 degrees, around December 22. Halfway between these two dates, that is, around March 22 and September 22, the Sun's declination is near 0 deg. These topics were discussed in details during the EAAE-ESA-ESO Sea and Space Project Navigational Chapters.
The exact Solar and Venus' declination values have been measured by astronomers for centuries - knowing these values has been a critical must for navigating the oceans by means of sextant methods.
Accurate values may be found at this Institut de Mecanique Celeste webpage kindly provided by Philippe Duhamel.
Applying these webpages, we find that during the end of the Venus Transit, on June 08 , at 11 UT = 1300 Central European Summer Time,
the Solar declination is "+22 degrees 53 arcminutes 22.9771 arcseconds " ("Geocentric" coordinates - as seen from the Earths center)
1 arcminute is equal to 1/60 Degree, and 1 arcsecond is equal to 1/3600 Degree.
So, converting into ordinary degree units, this becomes
Exercise..
Now show - this is equal to 22,6788 Degrees.
Make a schematic drawing of the Earth - including the directions to the Sun and Venus.
Will you observe the transit happen at the "upper" (Northern) half, or the "lower" (Southern) half of the Sun ?
From our daily life, we are familiar with the geometrical effect that is called stereoscopy. In astronomy this effect is known as parallax:
If an object is at close distance, your left eye and your right eye will give two different views. Nature knows this effect, in fact this is how we are able to estimate distances. If you hold a pencil at a short distance in front of your eyes, you may observe an effect like this:
Note that if you move the close-by pencil out to large distances, this parallax-effect will become smaller and smaller and at some time it will approach zero and vanish. The church is too far away to give any noticeable parallax.
The same type of geometry is apparent during the Venus Transit on June 08, 2004. When viewed from the northern part of our globe, the comparatively (in astronomical terms!) close-by Venus is observed in a slightly different perspective.
Now imagine two observers, one at position 2 and the other at position 1, further to the North.
In general - observers have a globe position defined by their Longitudes and Latitudes . You may find your own geographical latitude in any geographical atlas, or by means of this collection of WWW interactive maps.
During the following pages, we will compare to the situation, where both observers are leaving North/South of each other, approximately at the same longitude. In addition, we will restrict us to the end of the transit, where our Sun is approximately straight South - allowing a simple 2D approximation.
Observer 1 has a geographical latitude relative to Equator, marked with yellow colour. From the drawing below, it is obvious that observer 1 sees the Venus at a lower angle than does observer 2.
So, if we move from position 1 towards more northern latitudes, we will experience that the Venus declination appears to decrease. You have probably been in a similar situation: As you climb a tree, you will `look more and more down' on all your neighbours. This is the already mentioned parallax effect.
You may perhaps know that astronomers frequently apply this geometrical effect by combining pictures of star fields taken half a year apart, when the Earth is at opposite parts of its orbit around the Sun. On such parallax pictures, the images of relatively nearby stars are seen to be shifted with respect to the background stars which are much farther away. From such observations, it has been possible to estimate distances to many nearby stars.
The next drawing shows in more detail by how much the Venus declination is decreasing, when we move from position 2 to position 1:
Note, how the small angle β beta (indicated by the greek letter in the figure) is present in two places. The drawing below shows how we may find beta:
The shaded triangle below will be essential for all our subsequent calculations:
Any High School Mathematical text book will tell us about the famous sine relation:
In our astronomical triangle, the length of "a" will be approximately equal to the length of "c".
If we thus apply the formula on the shaded triangle above, we find:
During the Venus transit 2004 - it would be nice to get data from both the northern and the southern hemisphere.
An observer at a position 3 - say South Africa - will get a parallax relative to position 2
Applying the approximation (valid, if beta is measured in radians units, and if alfa is small) :
We do get this approximation :
or
which now gives the Venus parallax βVenus as
- or , in simplified form :
If we could measure the parallax of Venus relative to the fixed stars, this would be the formula to find the Venus distance.
We are however not able to see the stars during daytime, so we have to measure the parallax relative to the Sun.
The solar distance is not infinite, so the Sun has a parallax too.
This parallax βSun is ( same formula as before ) :
which may again be simplified into :
Observing the parallax of Venus relative to the Sun, will only give us the difference between these two parallaxes (click for details) :
or - simply :
From our previous Lab-exercise, we do know that
- or
Which makes the observed parallax Δβ equal to
In case of bad weather - and as a preparation of what to expect - we now take a look at the real numbers involved.
We have choosen two random positions,
and one in South Africa - position 3 : Geographical LatitudeSouth = Minus 30 Degrees.
Previously, we have calculated the Solar declination, as seen from the Earth Center(or position 2) as being equal to + 22,8897 Degrees
In addition, we did find the declination of Venus - (geocentric) may be found equal to 22,6788 Degrees.
The Earth Radius is now set equal to 6378 km .
Entering all the values above into
we are able to find both KVenus and KSun
Please do this (remember, your pocket calculator should be set to "degrees")...
KVenus = 8482.2 km
KSun = 8476.5 km
Now, let us take a look at the parallax to be expected.
Each position was entered into the Institut de Mecanique Celeste webpage
At +55 Latitude North, we do get a these declination values, corresponding to the end of the transit, approx local noon (1100 UT).
Venus Declination equal to +22 Deg 40 Arcminutes 27.5448 Arcseconds
In South Africa - at minus 30 Latitude South, we do get the following values, also local noon (1100 UT).
Venus Declination equal to +22 Deg 41 arcminutes 7.8303 Arcseconds
Applying 1 arcminute = 60 arcseconds - now show the North South Parallax βVenus (diffence in declinations) will become :
Applying the solar values, now show the Solar Parallax βSun becomes
( +22 Deg 53 arcminutes 29.8560 Arcseconds ) minus (+22 Deg 53 arcminutes 18.3999 Arcseconds) = 11,4561 Arcseconds
We do remember - the Parallax observed : Δβ was equal to βVenus minus βSun
So, as a result, the observed North- South parallax will be in the order of (40,2855 - 11.4561) arcsec = 28,8294 Arcsec,
approximately half an arcminute.
For comparison, the full angular size of the Sun is approx. 30 arcminutes.
Converting the parallax to degrees - we do get (28,8294 / 3600)Degrees = 0,0080 Degrees.
In our theoretical approximations above, we did assume the parallax was measured in Radian units.
As most High School Math books will tell - degrees are converted into radians, simply by multiplying with (pi/180) .
So, the expected parallax in radians will become, show this : 1,39769 10-4
Entering all values into :
We now get
The official value is, for comparison : 43,217 mio Km.
Again, applying the results from our simple previous Lab-exercise, we do already know that
which of course gives
and becomes
Thus, our estimate of the Earth-Sun distance gets equal to 3.33 times 43 mio Km = 143 mio km.
Official value : 149,6 mio Km - so our simplified 2 Dimensional approach gives a deviation less than 6 % .
The drawings from this chapter may be reproduced, in case the EAAE are mentioned.
Teachers Notice... Yes, this method is only an approximation - we do indirectly assume that all measurements are performed close to local noon (Sun being placed in South) - AND that the involved observers live approximately on the same longitude - this means - north-south of each other.
In addition, the original lab exercise involved also assumes that the data for measuring Venus' orbit are made with a declination equal to zero.
These approximations are made both because they are reasonable, and because we want to reach as large an audience as possible.
If you want to go for a 3 Dimensional method - you may find it valuable to visit our prize awarded project Comet Hyakutake .
Otherwise, in case you find the applied 2D method interesting, here is a prize awarded project on how to estimate the distance towards Saturn.
This simple method allowing an estimate of the Lunar Distance during Lunar Eclipses may be useful too.
If you have the chance of contributing with photographic images of the oncoming rare Venus transit, images taken in particular at 1100 UT = 1300 Central European Summer Time on June 08, 2004, please email me.
Hands on Astronomy - a few advices - POS-2002
