Download PARALLAX – IT`S SIMPLE! Abstract

PARALLAX – IT’S SIMPLE!
Ilgonis Vilks and Alexandre Costa
EAAE Summer School Working Group (Latvia, Portugal)
Abstract
Parallax is one of the basic methods of stellar distance measurement. It will gain a new
life with a launch of the ESA astrometry satellite GAIA in 2011. To get acquainted with
this method, students will make a device to measure parallax effect. This cardboard
device can be used to calculate distances to the objects inside the classroom and around
the school. Application of this method to measure distances of stars is also discussed.
Students will understand that accuracy of measurements is crucial and that knowledge
of stellar distance is necessary to derive other physical properties of the stars. This
practical activity is suitable for students of age 12 – 18.
INTRODUCTION
Take a pencil and look at it with only one eye open. Then with the other eye. And back
again. The pencil is shifting, jumping against the background (the wall). Now move the
pencil farther away from your eyes. The shift is smaller. This shift is called parallax.
From this simple observation we can derive the basic equation of distance measurement
using parallax. If the parallax angle p is smaller, the distance D to the object is larger.
Using mathematical language we can say that the distance is inversely proportional to
the parallax angle: D ~ 1 / p.
Figure 1. Observing the parallax angle of a pencil [1]
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Figure 2. Annual parallax of a star [2]
The same principle is used in astronomy. Instead of a pencil we take a relatively nearby
star of unknown distance. The far away stars form the background. The main difference
between both measurements is that the baseline b is much larger. In our observation the
baseline was the distance between our eyes. In astronomy the baseline is equal to the
diameter of the Earth’s orbit. Measurements are made when the Earth is at the opposite
points of its orbit (the second measurement is made 6 months after the first
measurement).
Actually astronomers use half of this angle, called the annual parallax (the baseline is
radius of the Earth’s orbit, one end of the baseline is at the centre of the Earth, other end
is at the centre of the Sun). From the right-angled triangle in Figure 2 we can derive the
exact equation for distance calculation: D = b / tg p.
PARALLAX MEASURER
Make a photocopy of Figure 5 onto cardboard. Cut it out along continuous lines and
fold up two triangles along the dashed lines (these will be pointing arrows). The
parallax measurer is ready for work. Now choose the baseline. In our first example it
will be the longer side of A4 paper sheet (297 mm).
Figure 3. Orientation of the parallax
measurer towards the object
Figure 4. Measurement of the parallax angle
Put the sheet of paper on the table and put the parallax measurer at the left side of the
sheet (Figure 3). Choose the object with unknown distance inside the room and rotate
the parallax measurer together with the sheet of paper until both arrows are pointed
exactly to the centre of the object. Now move the parallax measurer to the right lower
corner of the sheet (do not move the sheet itself!). Rotate the parallax measurer (not the
sheet) counter clockwise until arrows point to the object again (Figure 4). Read the
parallax angle from the scale. Try to be as accurate as you can. If you want, you can
repeat measurement several times and use the average value.
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Figure 5. Cardboard device to measure parallax
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Then it is possible to calculate the distance to the object using the previously given
trigonometric equation. Also it is possible to read the corresponding distance straight
from the table printed on the measurer (Distance column 1). It is a good idea also to
measure directly the distance to the object with a ruler and compare both values. The
accuracy of distance measurement using the parallax measurer will be about 5%.
Second Example. Now let’s try to measure greater distances around the school, for
example, distance to a flagpole or other vertical object. This time the baseline must be
larger. We shall use a baseline 10 times larger (297 cm). Instead of a sheet of paper we
take a 3 metre ruler. Two people hold the extended ruler and the third person makes the
measurement with the parallax measurer in the same way as before. The distance to the
flagpole can be calculated or read from the table on the measurer (Distance column 2).
If possible, also check the distance directly with the ruler!
The parallax method is especially good if you cannot measure the distance to the object
directly, for example, if flagpole is on the other side of the river. The same with stars.
You cannot travel to the stars and measure their distance directly using some kind of
“ruler”.
PARALLAX MEASUREMENTS IN ASTRONOMY
The first stellar parallax was measured by German astronomer F.W. Bessel in 1838. The
parallax angle of star Cygnus 61 was very small, some tenths of arcsecond and hard to
measure. Even for the nearest star, Alpha Centauri the annual parallax is less than 1
arcsecond (less than 1/3600 of one arc degree). Until recent time the annual parallaxes
of only a few hundreds of stars were known.
The accuracy of parallax measurements was vastly improved when the European Space
Agency HIPPARCOS satellite went into orbit. Its catalogue, published in 1997, gave
parallaxes of 120 thousand of stars with accuracy of 0.002 arcsecond and therefore it
was possible to determine distances up to 500 parsecs (1 pc = 3,086×1016 m).
Why astronomers need to know distances to the stars? Let’s consider an example. Alpha
Centauri is one of the brightest stars in the sky but its energy output is not very high
(only 1.5 times the energy output of the Sun). Alpha Centauri is a nearby star (1.35 pc),
that’s why it looks so bright. Betelgeuse in Orion is not so bright because it is much
more distant (131 pc). In fact Betelgeuse radiates 9400 times more energy than the Sun!
Apart from the understanding of spatial distribution of stars, the distance to the star is
needed to calculate the real energy output of the star (luminosity). Other physical
properties (core temperature, radius, mass) are also connected to the star’s luminosity.
How far can parallax measurements reach? The diameter of our Galaxy is about 30
thousands parsecs, so it is clear that the HIPPARCOS satellite was able to measure
distances only in the vicinity of the Sun.
The accuracy of parallax measurements is crucial. In 2011 the European Space Agency
plans to launch a new astrometry satellite GAIA that will measure the positions of about
15 million of stars with an accuracy of 10 microarcseconds. This means that distances
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up to 100 thousand parsecs can be measured. GAIA will give information about spatial
distribution of objects in the whole visible part of our Galaxy and will greatly improve
our knowledge of the physical properties of stars.
For larger distances other methods of distance measurements are used. They will be
discussed in the workshop presentation.
References
[1] M. Zeilik, J. Gaustad. Astronomy. The Cosmic Perspective. John Wiley & Sons,
1990, pp. 1 – 822.
[2] H. Karttunen, P. Kroger, H. Oja, M. Poutanen, K. J. Donner (Editors).
Fundamental Astronomy, Springer - Verlag, 1996, pp. 1 – 522.
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