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----------------- - 1727 - Pluto and Charon - Do the Math?
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- The New Horizon space craft arrives Pluto July 2015. Its 7 instruments are already taking data starting this month. Pluto has 5 moons, Charon, which is half its size, Nix and Hydra and 2 other moons without names that are only 60 to 100 miles diameter.
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-------------------- Pluto’s diameter is 1,485 miles ---------- 18.7% Earths
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-------------------- Pluto’s mass is 0.2% that of Earth’s. This Review teaches how the mass of Pluto and its moon Charon can be calculated.
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- ----------------- Pluto density = 1,750 kilograms / meter^3 This is 31.7% the average density of Earth. Pure water is 1,000 kilograms / meter^3
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--------------------- Pluto and Charon orbit the Sun in 248 days.
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-------------------- The orbit radius is 3,670,000,000 miles. Earth is 93,000,000 miles or one AU, astronomical unit. Therefore, Pluto is 39 AU or 39 times the Earth-Sun distance. It has an elliptical orbit that varies from 29.5 AU to 49.5 AU.
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--------------------- One day on Pluto takes 153.3 hours. 6 times that of Earth’s.
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- Pluto is a Dwarf Planet part of the Kuiper Belt Objects that orbit the Sun at 30 AU to 50 AU. To date astronomers have discovered more than 1,000 Kuiper Belt Objects. Pluto was just the first one because it is highly reflective. There are many more KBO’s yet to be discovered, estimated to be over 100,000 of them that are over 60 miles diameter. Pluto is 1,485 miles in diameter.
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- Pluto has a large moon, Charon, that is half its size and 4 smaller moons. Pluto reflects 55% of its sunlight because its surface is ice, nitrogen ice, with small amounts of methane ice and carbon monoxide ice. There is a small nitrogen atmosphere surrounding the planet.
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- Charon’s orbit is 12,160 miles from Pluto. Our Moon is 238,855 miles from Earth. Charon’s diameter is 756 miles and its mass is 10% that of Pluto Charon reflects 45% of its sunlight. Charon’s surface is water ice with small amounts of ammonia.
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- Pluto’s density is 1,750 kilograms/meter^3 which would indicate that it is 70% rock. Charon’s density indicates it is 50% rock and 50% water ice.
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- At least 20% of the KBO’s have their own moons If a KBO has a moon , their orbits allow astronomers to calculate their mass, volume, density and some estimate of their composition. Here is how it is done. I do not have the data on Pluto and Charon so I will substitute an eclipsing binary pair of stars that I do have data on.
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- First observation is a compete orbit of the pair takes 2.63 years. p = 2.36 years.
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- Using the Doppler shift of spectrum light the orbital velocities are measured. One star has a velocity of +6.8 kilometers per second and - 6.8 kilometers per second. These are symmetrical so the orbits must be circular rather than too elliptical. The other star has orbital velocity of +20.4 km/sec and -20.4 km/sec. The “+” velocity is coming towards us and is blue shifted in its spectrum. The “- “ velocity is going away from is and is redshift, longer wavelengths, in its spectrum.
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- Since we know the orbital velocity and the period of one orbit we can calculate the orbital circumference.
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------------------- distance = velocity * time
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------------------ d = 20.4 km/sec * 2.63 years * 3.156*10^7 sec / year.
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------------------- d = 1.69 * 10^9 kilometers.
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------------------ v = orbital velocity of 45,634 miles per hour.
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- The “d” is the circumference of orbit, the radius is found by dividing by 2*pi.
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----------------- r = d / 2*pi
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----------------- r = 2.7*10^8 kilometers.
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----------------- r = 167,770,000 miles
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- There are 1.5*10^8 kilometers in 1 AU, astronomical unit, the distance from Earth to Sun, 93,000,000 miles.
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----------------- r = 2.7 *10^8 / 1.5 * 10^8 = 1.8 AU
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- The radius for the second of the pair is calculated:
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------------------ d = 6.8 km/sec * 2.63 years * 3.156*10^7 sec / year.
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------------------ d = 56.44 *10^7 km
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------------------ r = 56.44*10^7 / 2 * pi = 8.98*10^7 km * 1 AU / 1.5 * 10^8 km
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----------------------- r = 0.6 AU
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- Johannes Kepler ( 1571 - 1630) discovered 3 laws of planetary motion: (1) Most orbits are an ellipse with the larger planet at one focus. Circular is a special type of ellipse for some unique orbits. (2) As a planet moves around its orbit it sweeps out equal areas in equal times. (3) More distance the orbit has the slower the average orbital speed. The relationship is the square of the period of orbit equals the cube of the radius.
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----------------------- p^2 = r^3
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- This equations works with no constants of unit conversion when the period is in years, the radius is in AU’s and the mass is in Solar Mass. When other units become involved Isaac Newton’s version for this equation becomes:
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----------------- p^2 = 4*pi^2 / G * 1 / ( M+m) * r^2
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---------------- (M+m) = 4*pi^2 * r^3 / p^2
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- For an ellipse the “r” becomes the semi- major axis, radius of a circle is a special case. Since the pair are always on opposite sides of the center of mass their separation is:
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----------------- a = 1.8 AU + 0.6 AU = 2.4 AU
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----------------- p = 2.63 years
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------------------(M+m) = a^3 / p^2 = 2.4^3 / 2.63 ^2 = 2 Solar Mass
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--------------- To find the mass of each body we realize that the orbital velocities are inversely proportional to mass:
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---------------- M / m = vm / vM = 20.4 km/sec / 6.8 km / sec = 3
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--------------- M = 3m
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-------------- M + m = 2
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--------------- m = 0.5 Solar Mass
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--------------- M = 1.5 Solar Mass
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- Therefore, we have calculated the masses of the binary pair by observing their period of orbit and measuring their orbital velocities. Next we could measure their diameters, calculate their volumes, calculate their densities and estimate their composition by knowing the densities of various materials, water = 1,000 kilograms / meter^3
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- Request other Reviews about Pluto and Kuiper Belt Objects:
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- #1724 Visiting Pluto, New Horizon spacecraft
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- #1632 Kuiper Belt Objects
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- #1364 Pluto and Charon
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- #1129, #1279 Pluto demoted to a Dwarf Planet.
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- #1660 Kuiper Belt Dwarf Planets
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- # 33 Kepler’s laws, Issac Newton biography
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- #543, #685 KBOs discovered 8-14,-2005 we have learned a lot since then.
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--- Some reviews are at: -------------- http://jdetrick.blogspot.com -----
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--- 707-536-3272 ---------------------- Monday, January 19, 2015 ---
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