KEPLER'S 3RD LAW - ADVANCED CALCULATOR

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Kepler's 3rd Law - Advanced G · m · t² = 4 · π² · r³

The above equation might seem intimidating but you may notice the familiar t² = r³ embedded in the formula. The 'G' is the Universal Constant of Gravitation and the 'm' represents the sum of the masses of 2 orbital bodies. 't' is the time of one orbital period and 'r' is the average orbital radius. By taking the value of 4 π² as 39.730 and using the C.G.S. value of 'G' = 6.6739 x 10^-8, the formula becomes:

m · t² = 5.9153 · 10⁸ · r³
where 'm' is stated in grams, 't' in seconds and 'r' in centimeters. The formula might still seem unwieldly but don't worry - this calculator does all the work for you !!!

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1) Satellites that are in geosynchronous orbit circle the Earth once per day. This eliminates the need for continuous repositioning of satellite receiving dishes because even though the satellite is moving, it stays in the same position relative to the Earth. Given that the Earth's mass is 5.979x10²⁷ grams (5.979x10²⁴ kilograms) and that the satellite's orbital period must be 86,400 seconds (one day), what altitude is required for a geosynchronous orbit?
Click on the 'RADIUS' button, enter the time and mass, click on 'CALCULATE' and the answer is 4.226 x10⁹ centimeters or 4.226 x10⁷ meters or 42,260 kilometers or 26,260 miles. (This is the distance as measured from the Earth's center).
2) The Moon orbits the Earth at a center-to-center distance of 3.86 x10¹⁰ centimeters (3.86 x10⁸ meters). As for determining the mass to use, scroll to the top of the page. You will see that the 'M' in the formula stands for the mass of both orbital bodies. Usually, the mass of one is insignificant compared to the other. However, since the Moon's mass is about 1/81 that of the Earth's, it is importatnt to use the sum of their masses. Since the Moon's mass= .0735 X10²⁷ grams and the Earth's mass = 5.979 x10²⁷ grams, then their sum = 6.0525 x10²⁷ grams (6.0525 x10²⁴ kilograms). Now that you have this information, how long does it take the Moon to make one revolution around the Earth?
Click on the 'TIME' button. Enter the radius and mass data. Click on 'CALCULATE' and the answer is 2,371,000 seconds or 27.44 days.
3) Every 151,200 seconds, Io orbits Jupiter at an orbital radius of 4.218 x 10¹⁰ centimeters (4.218 x 10⁸ meters). What is Jupiter's Mass?
Click on the 'MASS' button. Enter the radius and time data.
Click on 'CALCULATE' and the answer is 1.94 x 10³⁰ grams (1.94 x 10²⁷ kilograms).

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