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------------------------- 2695 - MERCURY - closest planet to the Sun?
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- 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:
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------------- 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 :
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---------------------- ( 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
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---------------- R - 10,085 kilometers
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---------------- T = 11 hours , 58 minutes
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---------------- m = 3.27 * 10^23 kilograms
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- A third orbit on May 25, 2012, yesterday:
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---------------- R - 7,715 kilometers
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---------------- T = 28,800 seconds
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---------------- 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.
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- Mercury smallest, densest, fastest planet. Here are some more messages from Mercury. The Messenger Spacecraft has been in orbit around Mercury since March 18, 2011. Mercury is the closest planet to the Sun and it is the smallest and densest planet.
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- You would think it was the hottest (806F), but, Venus gets that honor due to its greenhouse gases (854F).
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- Mercury is 3,000 miles in diameter and slightly larger than our Moon which is 2,159 miles diameter. There are two other moons in the Solar System that are larger, Ganymede and Titan.
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- Mercury is mostly rock and metal. The purpose of this calculation is to try to learn the proportions of each that make up the core and the crust of the planet.
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- Messenger spacecraft’s orbits have allowed astronomers to calculate the mass of the planet. The mass of Mercury is 3.31 * 10^23 kilograms. Earth is 60 *10^23 kilograms (18 times more massive).
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- This is counter-intuitive, however , even though Mercury’s temperature rises to 806F it is the fastest cooling planet. This is because of the ratio of its surface area to its volume. By “cool” we mean the internal heat escapes into space from the surface until both are at the same temperature.
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- Mercury cooled much faster than Earth because it is a much smaller planet. The total amount of heat contained in a planet depends on its volume. However, the amount that escapes into space only happens at the surface. The total time it takes a planet to lose its internal heat is directly proportional to the surface area to volume ratio.
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--------------------------- Ratio = surface / volume
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--------------------------- Surface area = 4 * pi* r^2
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------------------------- Volume = 4/3 * pi * r^3
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------------------------- Ratio = (4 * pi* r^2 ) / (4/3 * pi * r^3) = 3 / r
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- Because the radius is in the denominator larger objects have a smaller surface area to volume ratio. Smaller objects cool faster. Crushed ice will cool your drink faster than larger ice cubes.
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---------------------------- Radius of Mercury = 1,506 miles
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---------------------------- Mercury Ratio = 3 / 1506
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---------------------------- Radius of Earth = 3,963 miles
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---------------------------- Earth Ratio = 3 / 3963
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- Comparing the two ratios:
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--------------------- Mercury Ratio / Earth Ratio = (3 / 1506) / (3 / 3963) = 2.6
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- This comparison tells us that mathematically Mercury has cooled 2 ½ times faster than the Earth.
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- Gravity separates material by density, the heavier material sinks to the bottom. The result is layers of different material spread through the planet like onion skins. The denser material like iron sank to the center of the planet when everything was molten.
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- Because Mercury cooled rapidly the tectonic and volcanic activity was able to cease after only 1 billion years. Earth is 4.5 billion years and it still has some of this activity. In addition to their cooling ratio, bigger planets simply contain more heat.
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- But, in the case of Mercury ancient craters remain intact. There is little out gassing and low gravity means gas escapes into space easily leaving almost no atmosphere to cause erosion. ( A 100 pound high school student on Mercury would weigh only 38 pounds).
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- Extreme heat and no atmosphere also means there is no rain or snow to cause erosion. As a consequence the geological activity on Mercury’s surface looks very much like the Moon’s.
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- The mass of Mercury is 3.31*10^23 kilograms. To calculate the density we take the ratio of mass to volume. The radius is 2,425 kilometers average. Volume = 4*pi*r^3. Therefore the average density is 5,542 kilograms / meter^3.
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- If we assume an iron rich core has a density of 7,800 kg/m^3. And, we assume the crust being mostly rock has a density of 3,000 kg/m^3, how big would the core be to have the average density we have measured?
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------------------ Outer radius of Mercury = 2,425 kilometers
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-------------------- Inner radius of the core = Rc
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------------------- Core mass + Crust mass = Planet mass
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--------- Core Density * Volume + Crust Density * Volume = 3.31 * 10^23 kilograms
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------------------- 7800 kg/m^3 * (4/3 *pi*Rc^3) + 3000 kg/m^3 ( 4/3 * pi * (2.425*10^6)^3 - (4/3*pi*Rc^3) = 3.31*10^23 kg
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------------------------ Rc^3 = 7.55*10^18 m^3
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------------------------ Rc = 1,960 kilometers
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----------------- The total radius is 2,425 kilometers, so the core is 81% of the planet.
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----------------- The crust is 289 miles thick which is 19% of the planet.
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- Mercury should have a solid iron core, yet, it has a small magnetic field ( 10% that of Earth’s) which means there is a liquid iron core to cause electric currents to flow and a magnetic field to be generated.
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- To explain this astronomers conclude that Mercury’s extreme elliptical orbit and the extreme tidal forces being so close to the Sun heat up the planet through stretching and friction. There have been many surprises as we learn more about Mercury.
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- April 3, 2020 1478 1479 2695
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--------------------- Friday, April 3, 2020 -------------------------
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