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---------------------------- - 2306 - How Far Away is the Sun?
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- In the 1600 and 1700’s seafaring adventurers determined their position of latitude and longitude at sea using sextants and clocks. Sextants worked well in measuring the angle height of the Sun or certain stars above the horizon. It was high noon when the Sun was at its highest point and the angle above the horizon told you your latitude (north-south).
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- Longitude on the other hand was a time measurement. To determine longitude seafaring navigators needed to know how much time elapsed since high noon in one location and high noon in another in order to tell how far you traveled east-west.
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- If my clock is exactly 1 PM in Greenwich, Greenland and at their current location the Sun is highest at 12:00 PM , noon, then their location is 15 degrees west of Greenwich longitude. One hour is 15 degrees because 24 hours * 15 degrees = 360 degrees, one Earth rotation.
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- However, clocks were not accurate in those days. Pendulum clocks did not work well on a rolling ships. Mechanical clocks were not invented yet.
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- Ole Romer a Danish astronomer in 1676 hit upon using the moons of Jupiter as a clock. The four moons Io, Europe , Callisto, and Ganymede were tracked with precise orbital periods. Io is the largest and it had an orbit of 1.88 days. As Io moved behind Jupiter it was a tick and as it first reappeared on the other side of the Jupiter disk it was a tock. The tick-tock was very repeatable and served as a pendulum.
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- But, when they started using Io for a clock they learned that the actual time kept lagging eclipses time by some amount. And, after six months, when the Earth was the farthest distance form Jupiter in its orbit about the Sun it was a full 16 minutes behind schedule. Six months later when Earth-Jupiter distance was the shortest the Io eclipse tick-tock was back to the precise time again.
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- Romer realized that the time difference of 16 minutes was the additional time it took light from Jupiter-Io to traverse the orbital diameter of Earth around the Sun. If he knew the diameter of the orbit he could calculate the speed of light. To do that he needed to know the distance to the Sun.
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- He could measure this radius distance to the Sun by using parallax. He would measure the Sun’s position relative to the fixed far distant stars in the background. And, 12 hours later measure the angle as the position shifted due to one side of the triangle being the diameter of the Earth. Then simple geometry could calculate the height of the triangle.
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- If you all are thinking alike than someone is not thinking. The Sun is too bright you can not see the stars in the background.
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- In 1685 Giovanni Cassini tried another route. He measured the distance to Mars, then calculated the distance of Mars from the Sun and subtracted. Cassini used a ship in the South Pacific to measure the position of Mars relative to distant stars. At the same time in the Paris Observatory he made the same measurement. When the ship returned home they compared their position measurements and determined a parallax angle of 34 arc seconds. Using the base of the triangle, the diameter of the Earth, 12,720 kilometers divided by the sine of the angle 34 arc seconds = 77,800,000 kilometers from Earth to Mars.
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- Next Cassini used Kepler’s formula that the period of the planet squared is equal to the radius of the orbit cubed :
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-------------------------------------------- P^2 = a^3
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- Where P is the period in Earth years and a is the distance from Mars to the Sun in astronomical, Earth-Sun distance. The period of Mars is 1.88 years.
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------------------------------------------- 1.88^2 = 3.5344 = a^3
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------------------------------------------- a = 1.52 astronomical units
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- Since the Earth-Sun distance is one astronomical unit then the distance Earth-Mars must be .52 astronomical units and Cassini measured that distance to be 77,800,000 kilometers. Setting up the proportional equation and dividing 77,800,000 kilometers by .52 he determined that one astronomical unit, or the Earth-Sun distance, to be 149,600,000 kilometers.
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- Therefore, the diameter of Earth’s orbit is 299,200,000 kilometers. And Romer had determined that it took 16 minutes for light to travel that distance according to the Io-Jupiter tick-tock. The speed of light must be 299,200,000 kilometers / 960 seconds = 311,666 kilometers per second.
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-------------- He was not far off from the current number of 299,292 kilometers/second.
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- Now Romer could construct a table for the delay in the Jupiter moon clock for each month of the year and have an accurate time piece to determine longitude on the open seas.
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- Kepler’s law for planet orbits was actually more complicated than my calculation. He realized that the planetary orbits were ellipses, not circles. His law states that a line joining a planet and the Sun sweeps out equal areas in equal intervals of time and stated that the square of a planet’s sidereal period around the Sun is directly proportional to the cube of the length of its orbit’s semi major axis. Using Newtonian mechanics the equation becomes:
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----------------------------------------------- P^2 = 4*pi*a^3 / G*( m + M )
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- Where G is the gravitational constant and m is the mass of Mars and M the mass of the Sun.
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- With more careful calculations the table to add the speed of light to the tick-tock of the moons of Jupiter being eclipsed can become a very accurate clock as long as there is a clear sky and Jupiter is in view.
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- Those guys in the 1600’s were pretty smart.
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- March 13, 2019. 645
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--------------------- Wednesday, March 13, 2019 -------------------------
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