Mass of Jupiter
Calculation of the mass of Jupiter
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Updated June 01, 2013
We suggest determining the mass of Jupiter by studying the movement of its main satellites: Io, Europe, Ganymede and Callisto.
The movement of a satellite, a mass m is studied in repository one considered as Galilean, having his origin in the center of Jupiter and its axes steered towards distant stars, considered as fixed .On will suppose that Jupiter and its satellites have a distribution of mass in a spherical symmetry. The satellite moves on a circular orbit, at the distance R of the center of Jupiter:
- Determine the nature of the movement of a satellite around Jupiter.
- Determine the speed v of a satellite according to R, of M, mass of Jupiter and of G, constant of universal gravitation.
- Deduct the expression from it of the period of revolution T from the satellite.
- Show that the report T²/ R3 is constant.
The periods of revolution and beams of the orbits of four main satellites of Jupiter were determined and have the following values:
|R(in km)||4,22 105||6,71 105||1,07 106||1,883 106|
- Represent on graduated paper the graph giving the variations of T² According to R3. Conclude.
- By connecting these results to those obtained above, to determine the mass M of Jupiter.
data: G = 6,67 times 10-11N.m2.kg-2.
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The satellite is subjected to the only strength of centripetal gravitation
In the base of Frenet: according to the axis n = GMm /R² = mv²/ (R+h) where from the speed: v² =GM / R
According to the axis t: dv/dt = 0 where from standard of the constant speed and uniform movement.
The circumference of the circle 2pR is crossed in the speed v during the duration T (period 2pR = vT)
Raise to the square and replace v² par son expression 4p²R²= GM / R T²
or T² / R3 = 4p²/(GM)(3rd law of Kepler).
T² According to (R+h) in the cube give a right the guiding coefficient of which is 4pi²/GM
|T(in seconds)||1,53 105||3,07 105||6,18 105||1,44 106|
|R(in m) ||4,22 108||6,71 108||1,07 109||1,883 109|
|T² en s²||2,34 1010||9,42 1010||3,81 1011||2,07 1012|
|R3 en m3||7,51 1025||3 1026||1,22 1027||6,64 1027|
|T² / R3||3,1 10-16||3,14 10-16||3,12 10-16||3,2 10-16|
T² / R3 Is about constant and equal in: 3,15 10-16.
The 3rd law of Kepler is verified well.
Mass of Jupiter: T² / R3 = 4p²/(GM) (3rd law of Kepler).
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|M = 4p² / ( 3,15 10-16 *6,67 10-11) = 1,87 1027 kg. |