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Distance measurement in Astronomy
The measurement of distance is crucial to our understanding of the scale of the Universe.
Radar
The distance from the Earth of objects in the Solar System can be measured using radar. A
pulse is sent out and the time taken for the reflected pulse to be received is recorded.
Knowing the speed of electromagnetic radiation in free space and the time between
transmission and reception the radar pulse enables us to find the distance of the object. For
example the elapsed time would be 2.56 s for the Moon and up to 50 minutes for Jupiter and
around five and a half hours for Pluto. (The last two numbers depend on the relative
positions of the Earth, Jupiter and Pluto in their orbits)
In the example of the distance of the Moon the time for the pulse to reach the Moon is 1.28s.
The pulse travels at the speed of electromagnetic waves in a free space (3x10 8 ms-1) and so
the distance of the Moon from the Earth can be calculated from:
s =vt = 3x108x1.28 = 3.84x108 m = 3.84x105 km = 240 000 miles
Parallax
The difference in direction of a star viewed from the two ends of a line with a length equal to
the radius of the Earth’s orbit is called the PARALLAX of the star.
Stars that are close to the earth clearly have a larger parallax than ones far away. In other
words their direction when viewed from the Earth changes significantly as the Earth orbits the
Sun.
Earth
Distant stars
Parallax 
Centauri
2
Sun
Parallax 
Earth
Distant stars
Figure 1
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By significantly we mean a fraction of a second of arc. In the example shown  Centauri
(distance 1.33 parsecs) has a parallax of 0.75 “ of arc.
Astronomical unit
One Astronomical unit (AU) is defined as the mean distance of the Earth from the Sun
(1.5x1011 m)
The light year
This is the distance that light travels in free space in one year = 9.5x1015 m
The Parsec
The radius of Earth’s orbit = 1.5x1011 m, and therefore the distance is found from:
tan(1”) = 1.5x1011/d so
d = 3.06 x1016 m
1 parsec is the distance at which an object subtends an
angle of one second using the radius of the Earth’s orbit
as the baseline.
Distances between galaxies are usually measured in light
years or Mega parsecs (Mpc).
One second of
arc
One parsec
1 Parsec = 3.06x1016 m = 2.04x105 AU = 3.26 light years
1 Mega parsec (Mpc) = 3.26x106 light years = 3.097x1022 m
Radius of
Earth’s orbit
Earth
Sun
Figure 2
The parallax of a number of stars is shown in the following
table.
Star
Parallax Distance (l.y)
(" of arc)
A Centauri
0.750
4.3
Barnard's Star
0.545
6.0
Sirius
0.377
8.6
Procyon
0.285
11.4
Star
Parallax Distance (l.y)
(" of arc)
Vega
0.133
25
Arcturus
0.097
34
Aldebaran
0.054
60
Castor
0.001
570
At distances much greater than this the parallax method becomes impossibly difficult to
measure. Remember that 1" of arc is the angle subtended by a human head almost ¾ of a
2
kilometer away. Therefore the parallax of Castor is the same as the angle subtended by a
human head at a distance of almost 750 km!
Another method for measuring larger distances had to be found.
Cepheid variables
The solution came early in the twentieth century as a result of studies of a variable star (one
whose brightness changes with time) in the constellation of Cepheus.
Period
Brightness
Radius
Figure 3
The brightness of the star varied in a particular way (see Figure 3) and in 1912 Miss
Henrietta Leavitt of Harvard College observatory discovered an important connection
between the period and brightness. This is now known as the period-luminosity relationship.
Many other stars were found to vary in a similar way and the group of stars was called
Cepheid variables. (There are actually two types of Cepheid variable but we will just consider
one type here).
The period-luminosity relation means that if you can measure the period of a Cepheid
variable you can find its luminosity. Knowing how bright the star really is and then measuring
how bright it appears to be will then give the distance of the star from the Earth. The
discovery of Cepheid variables in the Andromeda nebula (M31) enabled its distance from
Earth (over two million light years) to be found.
Two ways of presenting the period luminosity law are shown by the graphs in Figure 4.
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Luminosity (Sun = 1)
106
104
Population I Cepheid variables
2
10
1.0
-2.0
-1.0
0.0
1.0
2.0
3.0
log period (days)
Absolute magnitude
-5
-4
-3
-2
Population I Cepheid variables
-1
0
+1
0.1
1
10
100
Figure 4
Period (days)
Of course the period of a variable star in distant galaxies is really difficult to measure and so
yet another method was needed to push back the limits of cosmic distance measurement.
The Tully-Fisher relationship
This relationship, named after the two American Astronomers who discovered it, is not yet
widely used because of lack of reliable data. It states that the more luminous a galaxy the
faster it rotates. Therefore measurement of the rotational speed of galaxies using the
Doppler effect gives a way of determining their distance from us.
Astronomers therefore have to turn to the work of Edwin Hubble.
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