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Measuring the distances to stars

For many centuries it was assumed that the stars resided in a heavenly sphere surrounding the Earth, which was thought to lay at the centre of the Universe. When Galileo found evidence that not all heavenly bodies orbited the Earth, he gave his support to the Copernicus model that placed the Sun, not the Earth, at the centre. But, amongst the objections to this heliocentric model was the one that the stars did not appear to move when viewed from different parts of Earth’s suggested orbit.

Parallax

When a foreground object is viewed from two different positions it appears to move against the background. For example, in the diagram below the foreground tree at C is viewed from two different positions, A and B. From position A the tree at C appears to lie in front of a gap between two background trees. But, from position B the tree at C appears to lie in front of a hut. This effect is known as parallax.

If we observe a foreground object from two different positions, it appears to move against the background. Here the tree at C appears to be in front of the hut as seen from point B, but between two background trees as seen from point A.

The effect of parallax can even be seen by looking through each eye alternatively. In addition, you can verify that the more distant the object the smaller the parallactic effect.

The parallax of stars

As the Earth orbits the Sun, we should notice nearer stars appearing to move against the background of more distant stars. The average distance from the Earth to the Sun (called the Astronomical Unit (AU), was determined in the mid 1700s using the Transit of Venus. I will blog about this history in the near future, but I went to Mongolia in June 2012 to see the last Transit until December 2117.

If we can measure the angle $2p$ between the position of the star in, say, January and its position in, say, July (6 months apart), then a simple bit of trigonometry will give us the distance $d$ of the star.

$\tan (p) = \frac{ 1 AU }{ d } \text{ so } d = \frac{ 1 AU }{ \tan (p) }$

A nearby star will appear to move against the background stars if we view it 6 months apart at opposite sides of our orbit about the Sun.

The definition of the parsec

The unit astronomers use for measuring distances within the Solar System is the Astronomical Unit (AU), as it is a convenient size compared to using kilometres or miles. The value of the AU is 149.6 million km, or $1.496 \times 10^{11} \text{m}$. For example, Jupiter is about 5 AU from the Sun, Saturn about 10 AU and Uranus about 20 AU.

But, the distances of stars is so vast that even the Astronomical Unit is too small. Instead, astronomers use a unit called the parsec. The word is actually a contraction of parallax and second (par + sec), and comes from its definition. A star would be at a distance of 1 parsec if the parallax it exhibited were 1 second of arc (remember there are 60 seconds of arc in an arc minute, and 60 arc minutes in a degree). So, using our definition of stellar parallax we can write

$\boxed{ d \text{ (in parsecs) } = \frac{ 1 }{ p \text{ (measured in arc seconds) } } }$

From this definition we can easily see that, if a star exhibits a parallax $p \text{ of e.g. } 2^{\prime \prime}$ then the distance would be

$d = \frac{ 1 }{ 2 } = 0.5 \text{ pc }$

and if the parallax $p \text{ were } 0.5^{\prime \prime}$ then the distance would be

$d = \frac{ 1 }{ 0.5 } = 2 \text{ pc }$.

To calculate the value of 1 pc in metres we can write

$1 \text{pc} = \frac{ 1.49 \times 10^{11} }{ \tan(1^{\prime \prime}) } = \frac{ 1.49 \times 10^{11} }{ 4.848 \times 10^{-6} } = 3.0857 \times 10^{16} \text{ m }$

The relationship between a parsec and a light year

Measuring stellar parallaxes is the only direct method we have of measuring the distances to the stars. All other methods are based on this initial method as our “yard stick”.

The reason astronomers work in parsecs is because that is the unit most easily calculated when we measure stellar parallaxes. If we measure a parallax angle of $p = 0.5^{\prime \prime}$ then we know the star is at 2 parsecs, if we measure an angle of $p = 0.1^{\prime \prime}$ then the star is at 10 parsecs. Notice in my blog on the supernova in Messier 82, I quoted the distance to M82 in Mega parsecs (Mpc), and in my blog on the most distant galaxy yet discovered, I quoted Hubble’s constant in km/s per Mpc.

Because of the unfamiliarity of the concept of stellar parallax, when astronomers talk to the public we tend to use light years, the distance light travels in a year. This is a much easier concept to understand. We can readily determine the relationship between the two, as to calculate a light year we use the speed of light $c = 3 \times 10^{8} \text{ m/s }$, and the number of seconds in a year, which is $60 \times 60 \times 24 \times 365.25 = 3.15576 \times 10^{7} \text{ s }$, and so the distance light travels in one year is

$1 \text{ ly } = (3 \times 10^{8})(3.15576 \times 10^{7}) = 9.467 \times 10^{15} \text{ m }$

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We see that this is less than the size of a parsec, in fact

$\boxed{ 1 \text{ pc } = 3.26 \text{ ly } }$

The measurement of the first stellar parallax

As I mentioned above, the lack of observed stellar parallax was used as an objection to the idea that the Earth orbited the Sun. The first astronomer to try to measure stellar parallax was James Bradley, in 1729, but he was unsuccessful. The first successful attempt was not until 1838, when Friedrich Bessel measured the parallax of the star 61 Cygni. He found the parallax to be $p = 0.3136^{\prime \prime}$, less than a third of an arcsecond. This is why people had not been able to measure stellar parallax before, the angles for even nearby stars are tiny. Even the closest star to us, Proxima Centauri, has a parallax of only $p = 0.769^{\prime \prime}$, placing it at more than 1 pc away. An angle of $0.769^{\prime \prime}$ corresponds approximately to the angle subtended by an object 2cm in diameter as seen from 5.3km!

The blurring effects of the Earth’s atmosphere (“seeing” as astronomers call it) limits our ability to measure these tiny angles. As a consequence, only a few hundred stars have parallax angles large enough to be able to measure from terrestrial telescopes. By the end of the 19th Century, about 60 stellar parallaxes had been obtained, and the smallest angles measured corresponded to distances considerably less than a kiloparsec.

This made the technique useless for measuring distances on the Galactic scale. Fortunately, in 1914, Henrietta Leavitt discovered the period-luminosity-relationship for Cepheid variables. I mentioned this relationship in passing in my blog on how Edwin Hubble was able to show that the Andromeda Nebula was in fact a galaxy beyond our Milky Way galaxy. I will blog about Leavitt’s discovery in more detail in the future, as it formed the next step in what is often referred to as “the cosmological distance ladder”.

Hipparcos and Gaia

In 1989 the European Space Agency (ESA) launched a satellite called Hipparcos, the main purpose of which was to measure stellar motions (due to the rotation of the disk of the Milky Way), and also to measure parallaxes from space, beyond the blurring effects of the Earth’s atmosphere. Hipparcos orbited the Earth at an altitude of between 507 and 35,888 km (an eccentric elliptical orbit with the centre of the Earth at one of the foci). Hipparcos was able to measure parallaxes down to $p=0.002^{\prime \prime}$, or 2 milliarcseconds (or 2,000 microarcseconds). This meant it could measure the parallax of stars out to a distance of 500pc, and by the end of its 3.5 year mission it had measured the parallax of over 100,000 stars.

In December 2013 ESA launched Gaia, which will go into an orbit about the second Lagrangian point (L2) rather than orbiting the Earth. It has a planned mission length of 5 years, and will be able to measure parallaxes down to 0.0001 arcseconds (10 microarcseconds), which corresponds to a distance of 100,000 pc (100 kpc) or about 320,000 light years. In comparison, the Milky Way galaxy is about 35 kpc in diameter and our Sun lies about 8 kpc from the centre. It is hoped that Gaia will measure the stellar parallax of over 1 billion stars!