"EAAE Summerschools" Working Group
Centre for Science and Mathematics Education, Utrecht University (The Netherlands)
How is it possible astronomers know distances in the universe where they can't arrive at all? In this workshop participants will get an insight in the parallax method, that laid the basis for the measurement of the universe. This method is based on the triangulation method that is used on the earth.
We will start outdoor: participants will measure by triangulation the distance to an object they can't reach. They will use simple instruments made by themselves. No calculations are necessary, the distances can be found by a geometrical method. Building on this method and measurement, we develop insight in the parallax method. We will see how it has been applied to find basic distances in the sky, from the moon until the nearby stars.
This series of activities will help you to understand how distances in the universe can be measured by the parallax method. This method, like several other methods to determine distances, is based on the principle that a triangle is completely known with only three elements. Therefore those methods are called triangulation methods. They are used by surveyors to make a map of a region. To understand those methods it is not necessary to make calculations or to know about trigonometry. We have only to measure angles and the length of a basis line that is within our reach, then we make a scale drawing. So we need two instruments for measuring (an angle-measure tool and a tape measure) and a ruler, a protractor, paper and pencil for drawing, that's all!
We will start with an outdoor activity, measuring the distance to an object (tree, house) without arriving there, like surveyors do.
It will become clear that practically there are some problems in using the same method for objects in the sky. Thus, another method is proposed to determine a triangle, using the phenomenon of parallax. Also this method we will use first in our own environment. Then it will be shown how it was used to determine the distance to the moon, some planets and nearby stars.
Determining a distance by triangulation
You need an angle-measuring tool (the professional one is called theodolite and is used by surveyors), one or two sticks, about as high as yourself and a tape measure.
Here's how we might measure the distance of a tree, like in the situation that is sketched above. Imagine you can walk from A until B, but you can't reach the tree, for instance because there is a river between you and the tree. The trick is that you determine the shape and the dimensions of the triangle ABC. When you measure AB (the baseline) and the two angles a (BAC) and b (ABC), the triangle ABC is completely known. As you may see, when the tree is more far away (C'), the angle at C becomes smaller and the shape of the triangle becomes different.
Activity 1: Determining a real distance by triangulation.
Mark two places, A and B, for instance by standing vertical sticks in the ground. The points A and B, together with the tree at point C, form a triangle. Measure the distance AB with the tape measure.
Stand at point A and measure BAC using an angle-measuring tool by looking at the tree and at the stick in B. Then move to the point B and measure ABC. So you know the length of one side of the triangle - our baseline - plus the angles at which the other sides branch off from it. Construct on paper the triangle ABC on scale. Using your ruler and the scale, you can simple calculate the distance from every point of your baseline to the tree.
a. When the distance to an object increases and the baseline stays the same, show that the two angles are becoming very similar each other, they differ less and less. What is the consequence for the measurements and the calculation of the distance?
b. Show that you can overcome this problem in two ways: to make the baseline greater or to use a more accurate angle-measuring tool.
c. Imagine you want to determine the distance to the moon in this way. In this case you need a very long baseline. How long may it be? What will be the practical objection to use in this case the method of triangulation?
Determining a distance by parallax
Even if the method of triangulation is a clever method, and very much used on the earth by surveyors, it is not useful when the object is very far away as celestial objects are. In that case we may use the parallax method, a specific triangulation method. First we introduce the method in our environment.
The principle of parallax
Activity 2: the phenomenon of parallax
When you hold out a finger and view it only with your left eye, then with your right eye, your finger seems to shift relative to the distant background. This is the phenomenon of parallax.
Activity 3: greater and smaller parallax
Raise your finger to your nose and sight a nearby object. Alternately blink your eyes as before and observe the apparent shift of your finger between left and right. Now move your finger a little farther away and blink again. The parallax shift is smaller than it was before. Move your finger farther and the parallax shift becomes even smaller.
We can deduce the following rule: the farther an object, the smaller is parallax, and its converse, the smaller the parallax, the farther the object.
Here is the key to measuring the distances of objects around us, from objects a few inches away to stars in outer space. In fact we have to construct again a triangle, but in a slightly different way as before, using the phenomenon of parallax. We will use now a mark that we see in the same direction as the object from which we want to determine the distance.
How to use parallax to measure a distance
First you'll apply the method of parallax to measure the distance from your eyes to a finger that you hold with stretched arm in front of you. Look with one eye at the finger, and look for a mark on the wall or outside, that is just behind the finger. Remember that mark, now look with your other eye at your finger, behind it there will be another mark. The situation is like in the drawing below. When you look now at the marks without looking at your finger, you can measure the angle ß with your angle-measuring tool.
In this drawing a is somewhat greater than ß, but more the marks are far away, more the angles a and ß become similar. (Show this by yourself.) When you consider a = b, you know the angle in the top of the triangle that is formed by your two eyes and your finger. When you consider this a isosceles triangle, with the distance between your eyes as baseline, you can draw on scale the whole triangle and, with the help of a ruler, measure the distance from your eyes to your finger.
Activity 4: determining the distance of your finger by using parallax
Now we will do the just described experiment.
Hold the arm with the finger stretched. You can use marks on the wall of the classroom, but also marks you can see outside through the window.
Measure the distance between your eyes with a ruler.
Measure b with your angle-measuring tool.
Make a drawing on scale and find from that the distance between your eyes and your finger.
Control the answer by measuring this distance directly with a ruler or a tape measure.
Activity 5: determining a real distance in the outdoors by using parallax
Now we'll do the same activity outside for instance with the tree you used in activity 1 to determine the distance. This time you don't use the distance between your two eyes as baseline, but a greater one, like the distance between A and B. In A as in B an observer has to look for a far away object (at the horizon) that is directly behind (or in line) with their position and the tree. They tell each other the object they notice to appear in line with the tree, then they can both measure directly the angle between those two marks. Measuring the baseline with a tape measure, gives again the triangle that can be drawn on scale from which you can derive the distance to the tree.
Notice that this method works also when the observers in A and B can't see each other! They only need to be able to see both the tree and the same marks at the horizon. So this method is useful in astronomy for measuring great distances, but the baseline has to be very large, for instance the earth diameter.
Parallax in Astronomy
1. Distance to the Moon
The method of parallax has been used first in astronomy to determine the distance to the Moon. A successful measurement was carried out already in the second century B.C. by Hipparchus. He used observations of the solar eclipse of March 14, 189 B.C. Witnesses living near the Hellespont, the narrow strait of North-western Turkey, reported that the eclipse was total. However observers at Alexandria saw only four-fifths of the sun obscured by the lunar disk. Hipparchus assumed that the Sun is sufficiently distant so that during the several minutes of maximum eclipse, she served as a stationary marker against which the Moon's parallax could be gauged. So about 1/5 of the Sun's diameter had been shifted because parallax from the two terrestrial vantage points. We see the Sun under an angle of about 0.5 degree, therefore one-tenth of a degree is the Moon's parallax over a baseline extending between the Hellespont and Alexandria. Combining, Hipparchus deduced from this the Moon's distance: between thirty-five and forty one Earth-diameters. The true value is approximately thirty Earth-diameters. Respectably close, considered that the work was carried out more than 2.000 years ago.
2. Distance to Mars
Between 1671-1673, Cassini and Richer used the parallax method to determine the distance to Mars. They used the moment that Mars was as near as possible to the earth. Cassini stayed in Paris, Richer went to the Cayenne estuary in French Guyana (South America) for doing measurements.
Questions concerning conditions for determining the distance to Mars. Sketch drawings to clarify that the two observers Richer and Cassini
had to be as far as possible from each other
had to use stars as markers
had to do their observations exactly at the same time
had to do their observations when Mars is more near the Earth
Which difficulties can you see in determining the distance to the moon or Mars in this way?
Cassini combined the measurements of Richer with his own measurements in Paris and could calculate the parallax with respect the distance Paris - Cayenne, the baseline. This gave the distance to Mars, which made possible to calculate the distance to the sun, by using Kepler's laws.
Flamsteed, an English astronomer, used in the same period the parallax method to determine Mars distance, but he did it in a different way. He used an old method invented by Tycho Brahe, which allowed him to stay in England. The trick was to measure the planet's position with respect to neighbour stars at the same location four hours before and after its meridian transit, as illustrated in the figure below. The observer at D would measure Mars's position at A and A', whereas without parallax it would have moved from B to B'. (Of course Mars needs to be in a position this is possible in reality.)
There were already previous calculations of the distance Earth-Sun: Copernicus found 3 million kilometre, Tycho Brahe 8 million km, Kepler more than 20 million km. Cassini found 140 million km. Flamsteed found a similar distance to the sun as Cassini.
This enormous distance was a source of astonishment, but even if Cassini's numbers were not so accurate and not everybody agreed with it, it appeared to be the best method until that time.
From two points on Earth, A and B, two astronomers observe simultaneously the position of a planet P on the background of distant objects. Knowing the positions of A and B permits to determine the angle a (diurnal parallax) under which, from the planet P, one sees the radius R of the Earth at the equator. This difficult computation requires a precise knowledge of the shape of the Earth.
3. Distance to the stars.
The diurnal parallax measured and calculated by Cassini i.e. was about 9² " 0.0025°. To measure the parallax of stars, that are farther away than the sun, it was clear that it was necessary to use an even greater basis than the diameter of the earth. The annual parallax can be used. Look the figure.
The motion of a nearby star S, with respect to distant stars, as seen from the Earth E which rotates around the Sun (successive positions E1 and E2, at times separated by about 6 months here). The Earth orbit is quasi-circular and is represented here in perspective.
The annual parallax is the angle v, under which, from the star, one sees the radius of the Earth's orbit (more precisely, the semimajor axis of this orbit, or the Astronomical Unit of distance, the AU).
But Cassini and his colleague astronomers didn't succeed to find a parallax for even one star! This was long considered as a counter-indication for the sun-centred world picture, because it was considered as impossible that the stars could be so far away that the annual parallax couldn't be measured. The next activity will make clear why this was reasonable.
Activity 6: why it lasted so long before parallax of stars was found.
Consider in the following calculations the star S perpendicular above the plane of the sun and the earth's orbit around the sun: SCE = 90°.
a. Show that when the annual parallax v = 45°, the distance from the sun to the star S is 1 AU.
b. Show that, when a star S has v = 1°, the distance from the sun to S is about 57 AU.
c. Tycho Brahe was able, even without a telescope, to measure angles as small as 0,005°, but even no stars with this small annual parallax were found. Calculate how far the stars should be at least from the sun. (in AU) (Use the answer of b.)
d. Cassini was able to measure angles about 0,001°, but also he found no parallax. If the stars were so far away, it would also mean that the space was incredible empty! It stayed a strong argument against a sun-centred world picture.
e. Between 1830 and 1840 three astronomers, Bessel, Struve and Henderson measured the first annual stellar parallax. A star in Cygni appeared to have a parallax of 0.0001° (about 0.35²) thus enabling a first determination of the distance to a star! Calculate this distance (in AU), using the answer of c.
f. They measured also the distance to a Centauri, now known as the closest neighbour to the sun. The annual parallax appeared to be 0.76².
Activity 7: Measuring the parallax of a star by photographing.
To measure the parallax, photos of a small part of the starry sky are made by a telescope. Below there are drawings made from two photos taken from the same part of the starry sky at 6 months of interval.
The whole photographic plate has a dimension of 1/3 arc second (1/3² ), so about 0.0001°. As you may see, some stars stayed at the same place, while others have moved in the time of half year.
a. What has the astronomer to do, to be sure that the stars didn't move in reality, but that the difference is caused by the parallax?
b. Which star on the photo is the nearest to the earth? What is its annual parallax?
c. Which of those stars are the most remote? What is their annual parallax?
d. When star D is at 18 light-years distance from the earth, what are the distances of the stars A, B and E in light-years?
Ferguson, K. (1999), Measuring the Universe.
Fucili, L, Genseberger,R. and Ros, R., "How the transit of Venus can be used to determine the Earth-Sun distance".
Genseberger, R. and Wielinga, R (1996), Ontwikkeling van ideeën over het heelal. Enschede/Utrecht: SLO-CDb
Helden, A. van (1985), Measuring the Universe. Chicago. The University of Chicago Press.
Hirshfeld, A.W. (2001), Parallax, the Race to Measure the Cosmos. New York: W.H Freeman and company
Pecker, J.C. (2001), Understanding the Heavens. Berlin, New York: Springer.
A simple angle-measuring instrument
There are various possibilities to build an angle measure instrument. A very simple one can made according the following instructions:
Fix a 180° scale on a piece of tempex (format A4).
Three pins (for instance small sticks of wood) are then enough to measure angles between objects seen from your eyes. The material makes it possible to fix the pins perpendicular on the board, just by pushing them through the paper into the surface.
You keep the instrument horizontal, well fixed on a horizontal surface (table).
One of the pins is placed in the centre of the small circle, from there you look at the first object and place a second stick into the board where you see the two sticks and the object in one line. Then you look at the second object, keeping the board in the same position. You place the third pin in that direction, into the scale of the angle measurer. Now you can calculate the angle by reading the corresponding numbers and subtract them.