- Today, May 26, 2012, the Messenger spacecraft is in orbit around the planet Mercury. the math to put it in orbit was developed in the 16th century. Tycho Brahe used the newly developed telescope to accurately measure the orbits of the planets. He died in 1601 and his assistant Johannes Kepler inherited his data that he had collected. Kepler spent 20 years analyzing the data. He concluded that to match the data planets had to be in elliptical orbits about the Sun and not the perfect circular orbits that the consensus at the time believed.
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- His mass analysis concluded that equal time intervals were swept by the radius vector from the Sun to the planet as the planet orbited the Sun.
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- Kepler’s third law in math was that the time it takes for a planet to make one full revolution around the Sun is proportional to Time^2 / Radius^3 = Constant. R is half the major axis of the Ellipse.
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- If factors are proportional they can be turned into an equality with the right Constant of Proportionality. Isaac Newton did the math to accomplish this for Kepler’s laws. He first assumed that the orbit were ellipses. He knew that a stable orbit was a balance of forces.
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- The force of gravity obeyed the inverse square law. The force varied inversely as the square of the distance between the masses.
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------------------------- Gravity Force = Constant / Radius^2
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-------------------------- F = Constant / R^2
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--------------------- Acceleration of object in circular orbit = velocity^2 / Radius
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------------------------- a = v^2 / R
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--------------------- Newton had is own law of motion = Force = mass * acceleration
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---------------------- F = m*a
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----- Substituting: F = m * v^2 / R
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----- Setting forces equal -------- Constant / R^2 = m* v^2 / R
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------ Solving for v^2 = Constant * R / m * R^2
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---- Calculating the period of an orbit = T = circumference / velocity
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------------------ T = 2 * pi * R / v
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------------------ T^2 = 4 * pi^2 * R^2 / v^2
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-------Substituting v^2 -------- T^2 = ( 4*pi^2 * m / Constant) * R^3
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- This was Kepler’s 3rd law of orbits because with a constant mass the factor , ( 4*pi^2 * m / Constant) , is simply another constant.
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---------------------- T^2 / R^3 = Constant
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- Newton defined the Constant of Proportionality with the Gravitational Constant , G.
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----------------------- T^2 / R^3 = 4 * pi^2 / G * m
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--------------------- G = 6.67*10^-11 meters^3 / (kilograms * seconds^2)
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-------------------- T^2 / R^3 = (5.91*10^11) * m
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- We will use this equation to solve for the mass of Mercury, but, first let’s try it out on the International Space Station that we know is orbiting at 6,738 kilometers radius. And we know the mass of the Earth is 5.9 *10^24 kilograms.
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------------------ T^2 = 5.91 ( 6.738*10^6)^3 / 5.9*20*10^24 = 30,640,000 seconds
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------------------- T = 92.3 minutes.
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- The International Space Station circles the Earth about every 90 minutes. You can see it. The scheduled pass over’s for your area are found on the web.
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- Now we have confidence to use this math on the planet Mercury On April 25, 2011 the Messenger spacecraft completed one orbit in 12 hours and 2 minutes. The radius of the orbit was 10,124 kilometers.
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--------------- T^2 / R^3 = 5.91*10^11 / m
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---------------- m = 5.91*10^11 * ( 10.124 * 10^6 meters )^3 / (43.32*10^3 seconds)^2
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---------------- m = 3.27*10^23 kilograms
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--------------- Another orbit on September 14, 2011
---------------- R - 10,085 kilometers
---------------- T = 11 hours , 58 minutes
---------------- m = 3.27 * 10^23 kilograms
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- A third orbit on May 25, 2012, yesterday
---------------- R - 7,715 kilometers
---------------- T = 28,800 seconds
---------------- m = 3.27 * 10^23 kilograms
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- The elliptical orbits around Mercury are changing but the mass remains constant. The radius, velocity, and period all adjust to keep the equation constant working according to the laws of physics.
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- My astronomy textbook had the mass of Mercury at 3.3 * 10^23 kilograms. Mercury is just 5.5% the mass of the Earth. The old guys had it right. An announcement will be made shortly, stay tuned.
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(1) To learn more : Request review # 1369, #1343, # 1165, #814, #44 five other reviews of things we have learned about this closest planet to the Sun
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707-536-3272, Sunday, May 27, 2012
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