Astronomy Topic 3

• Ancient Greeks were able to apply their geometrical skills to determine the scale and size of the Earth-Moon-Sun system


  Over 2200 years ago
• The ancient Greeks used crude measurements of lengths, angles and time intervals to determine the scale and size of the Earth-Moon-Sun system
• The early calculations of the circumference of the Earth and the sizes of, and distances to, the Moon and Sun used incorrect data
• The methods employed by the ancient Greeks were extremely sound
• Cosmic distance ladder


  The first rungs were established by Eratosthenes and Aristarchus of Samos
• Aristarchus of Samos proposed a Sun-centred (heliocentric) Universe more than 1700 years before Copernicus
• Eratosthenes


  • Used shadows and perhaps an army of men to determine the circumference of the Earth to amazing accuracy (probably)
• Aristarchus of Samos


  • Timed lunar eclipses, used crude protractors (probably) and the width of his thumb at arm's length to calculate the distances to, and diameters of, two of our closest neighbours
• Eratosthenes was in charge of the Great Library of Alexandria in Egypt
• Eratosthenes' method


  1. Noticed the Sun was directly overhead in Syene on the summer solstice

  2. Measured the angle between the Sun and the vertical in Alexandria

  3. Used the proportion between the circumference of the Earth and the distance from Syene to Alexandria to calculate the circumference
• Eratosthenes obtained a value of 5000 stadia for the distance between Alexandria and Syene
• One stadium (or stodion) was the unit of length equivalent to the length of an athletics stadium, but they were not all the same length
• Different sources report an accuracy of between 2% and 20% in Eratosthenes' calculation of the circumference of the Earth
• Aristarchus


  Calculated the Moon's diameter to be between 0.32 and 0.40 times that of the Earth, the correct value is 0.27 times
• The assumption that the rays of light from the Sun are parallel (making the shape of the umbra a cylinder rather than a cone) is not a bad one
• The Sun subtends an angle of only 0.5 degrees at the naked eye, and so the actual rays of light are only inclined at 0.25 degrees at the most to their 'parallel' direction
• The assumption that the Moon moves across the whole of the umbra's diameter (and through the exact centre of the umbra) is more difficult to justify
• Once the relative sizes of the Earth and Moon were determined, it only required a knowledge of the value of the Earth's diameter (as supplied by Eratosthenes) to be able to calculate the diameter of the Moon
• Aristarchus' method for calculating the distance to the Moon

  1. Determining the apparent size of the Moon in the sky

  2. Measuring the width of his thumb and the length of his outstretched arm to obtain a value for the angular size of his thumb and hence that of the Moon

  3. Calculating the Earth-Moon distance using trigonometry
• Aristarchus obtained a value of 2 degrees for the angular size of the Moon, which is very inaccurate (the Moon actually subtends an angle of 0.5 degrees at the naked eye)
• Aristarchus' method for calculating the Earth-Moon distance

  1. Using trigonometry with the incorrect angle of 2 degrees

  2. Using the correct angle of 0.5 degrees
• Using the correct value of 0.5 degrees for the angle gives a much more accurate Earth-Moon distance of 403,000 km
• It is difficult to believe that Aristarchus could be out by a factor of 4 when determining the apparent size of the Moon
• A further ingenious method allowed Aristarchus to determine the relative distances of the Sun and Moon
• Knowing the actual distance of the Moon then made it easy for Aristarchus and others to calculate the distance to the Sun
• This method involved observing the Moon at precisely its quarter or half-full phase, which was and is quite difficult to judge, and somehow obtaining a value for the angle between the Sun and the Moon