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Measuring the Distance to the Stars

This is a very nice example of some very simple mathematics achieving something which  for centuries appeared impossible – measuring the distance to the stars.  Before we start we need a few definitions:

• 1  Astronomical Unit (AU) is the average distance from the Sun to the Earth.  This is around 150,000,000km.
• 1 Light Year is the distance that light travels in one year.  This is around 9,500,000,000,000km.  We have around 63000AU = 1 Light Year.
• 1 arc second is measurement for very small angles and is 1/3600 of one degree.
• Parallax is the angular difference in measurement when viewing an object from different locations.  In astronomy parallax is used to mean the half the angle formed when a star is viewed from opposite sides of the Earth’s solar orbit (marked on the diagram below).

With those definitions it is easy to then find the distance to stars.  The parallax method requires that you take a measurement of the angle to a given star, and then wait until 6 months later and take the same measurement.  The two angles will be slightly different – divide this difference by 2 and you have the parallax.

Let’s take 61 Cyngi – which Friedrick Bessel first used this method on in the early 1800s.  This has a parallax of 287/1000 arc seconds.  This is equivalent to 287/1000 x 1/3600 degree or approximately 0.000080 degrees.  So now we can simply use trigonometry – we have a right angled triangle with opposite side = 1 AU and angle = 0.0000080.  Therefore the distance is given by:

tan(0.000080) = 1/d

d = 1/tan(0.000080)

d = 720000 AU

which is approximately 720000/63000 = 11 light years away.

That’s pretty incredible!  Using this method and armed with nothing more than a telescope and knowledge of the Earth’s orbital diameter,  astronomers were able to judge the distance of stars in faraway parts of the universe – indeed they used this method to prove that other galaxies apart from our own also existed.

Orion’s Belt

The constellation of Orion is one of the most striking in the Northern Hemisphere.  It contains the “belt” of 3 stars in a line, along with the brightly shining Rigel and the red super giant Betelgeuse.  The following 2 graphics are taken from the great student resource from the Royal Observatory Greenwich:

The angles marked in the picture are in arc seconds – so to convert them into degrees we need to multiply by 1/3600.  For example, Betelgeuse the red giant has a parallax of 0.0051 x 1/3600 = 0.0000014 (2sf) degrees.  Therefore the distance to Betelgeuse is:

tan(0.0000014) = 1/d

d = 1/tan(0.0000014)

d = 41,000,000 AU

which is approximately 41,000,000/63000 = 651 light years away.  If we were more accurate with our rounding we would get 643 light years.  That means that when we look into the sky we are seeing Betelgeuse as it was 643 years ago.

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All content on this site has been written by Andrew Chambers (MSc. Mathematics, IB Mathematics Examiner).

### New website for International teachers

I’ve just launched a brand new maths site for international schools – over 2000 pdf pages of resources to support IB teachers.  If you are an IB teacher this could save you 200+ hours of preparation time.

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