HOW DID KEPLER DO IT?

Tycho Brahe (1546 - 1601) spent about twenty years measuring the positions of the planets against the background of the stars; especially Mars which could be seen for most of the year, whereas Mercury and Venus were visible for only parts of the year: in the evenings when they were in eastern elongation (east of the Sun) between points 4 and 1 in their orbits (Diagram 1); and in the pre-dawn sky when they were west of the Sun between points 2 and 3. At other times they were lost in the glow of the Sun.

M_w V_w Greatest western elongation of mercury and Venus west of the Sun before sunrise,

M_e V_e greatest eastern elongation of Mercury and Venus east of the Sun in the evenings.

Between points 1 and 2 and between 3 and 4 the two planets are lost in the glow of the Sun, then being less than 10° from the Sun.

In this diagram the sizes of the orbits are drawn to scale. Mercury's sidereal year is only 88 days and its synodicperiod 116 days. It takes Mercury 116 days to move from a certain point against the sky relative to the Earth and Sun to return to the same point after one revolution round the Sun. Venus has a synodic period of 584 days.

Of these days Mercury spends 11 days in moving from point 1 to point 2 and 26 days from point 3 to 4 so that it is invisible for 37 days out of the 116 days of it's synodic period. The remainder, 79 days is equally divided between eastern and western elongation from points 2 to 3 and 4 to 1; about 40 days each. Venus is visible between points 2 and 3 and between points 4 and 1 for periods of 245 days and invisible for periods of 12 days between points 1 and 2 and also for 83 days between points 3 and 4.

The angles of greatest elongation varied for the positions in the orbits of these planets because their orbits are ellipses as Kepler proved. Kepler (1571 - 1630) made use of the average greatest elongation when the angles Earth - planet. - Sun were 90º to work out the distances of these planets from the Sun in terms of the Earth's distance which was not accurately known in those days - so Kepler called it 1 ASTRONOMICAL UNIT. From Brahe's measurements Kepler found the average greatest elongation of Venus to be 46,5° and that of Mercury 22,75°. In the case of Mars, Kepler made use of the position of Mars when it was in QUADRATURE, i.e. when the angle Mars - Earth - Sun (M E S) was 90° (Diag 2).

From Brahe's measurements, Kepler found that, on average 105,5 days elapsed between the opposition of Mars when Sun - Earth - Mars were in a straight line (S - E₁ - M₁ and the quadrature of Mars when the angle Sun - Earth - Mars, was a right angle, 90°.

Now, in 105,5 days the Earth moves through 105,5 ÷ 365,25 x 360 degrees. This equals 104°. In the same period of time. Mars moves from M₁ to M through an angle of 105,5 ÷ 687 x 360 = 55°. (Mars' sidereal period, the time that Mars takes to make one revolution round the Sun is 687 days and it was well known from Tycho's measurements)

Therefore, the angle ESM = 104 - 55 = 49º.

Since the sum of the angles of a plane triangle is 180°, angle EMS = 180 - (90 + 49) = 180 - 139 = 41°. So Kepler was able to make a scale drawing of triangle SEM and measure according to scale the relative distance SM of Mars from the Sun in terms of the Earth's relative distance ES which is 1 astronomical unit. In Diagram 3 the Earth's distance from the Sun is reduced to 200 millimetres. To determine the relative distance of Mars: construct the right angle SEM and construct the angle ESM which equals 90° - 41 = 49° Then measure the distance SM, the relative distance of Mars from the Sun. It is found to be 305 mm, correct to 1 millimetre.

Therefore:

astronomical units..

To determine the distance of Venus construct angles SEV = 46,5° and ESV = 90 - 46,5 = 43,5° on SE. The angle SVE will be 90°

Therefore:

astronomical units

The reader should try and construct these triangles and see how easy it is to get answers close to those obtained by Kepler.

Kepler found the correct values for the relative distances of these planets by making use of trigonometry. In the triangle SEM, right angled at E, the ratio SE divided by SM is the sine of angle SME.

The ratio SM <- SE = 1— ratio of SE over SM this = 1 — sin 41 = 1 — 0,6561 =1,524. astronomical units. In triangle SEV (for Venus) SV — SE = sine of SEV = sine of 46,5° = 0,725 astronomical units. In the case of Mercury: sine of SEMe = sine 22,75° = 0,3867. astronomical units.

Jan Eben van Zyl