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 |  planets  8 planets of the solar system (Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune) and their satellites. |  sun The Sun is in 2/3 of the galactic center towards the edge, at a distance of 30 000 light years of the center. The Sun moves in a speed of 230 km/s around this galactic center during its revolution which it makes in 250 million years. |  galaxies The Galaxy, it is the name datum in our galaxy, it is an enormous spiral wheel of stars, a diameter of 100 000 light years. What appears of the Earth, it is a white continuous band called the Milky Way. |  bigbang There is 15 billion years a tremendous explosion of light gives birth to the space, in time, in the matter, a chaos burning with an inconceivable heat, a formless porridge which is going to swell, to extend in all the directions and to cool quite slowly. |  quotations Some quotations of big men (Aristote, Galilee, Newton, Platon, Laplace, Einstein). |  links Some external links concerning the astronomy.

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|  |  dwarf planets According to the definition which was approved on August 24th, 2006, during the 26th General assembly of the UAI ( International Astronomical Union), three bodies reach the status of dwarfish planet: Pluto, eris, and Ceres. Others objets should soon join this nomenclature. |  solar system The solar system is really much more complex if we take into account all the objects being a part of the system. A considerable number of objects is in the belt of Kuiper and still beyond in the cloud of Oort. |  galaxy groups Heap of galaxies are the biggest structures of the Universe. They are constituted by hundreds of galaxies connected together by their own gravitational attraction. Between the galaxies we find some material constituted by warm gas there, forming a plasma, the temperature of which reaches 10 to 100 million degrees. |  comets Besides planets, satellites and asteroids, the solar system contains comets (hair in Greek). The roaming comets originate in the depths of the space in several lights years. The number of periodic comets is of the order of 2000. |  biographies Some biographies, Einstein, Baade, Lyot, Hale, Hubble, Shapley, Laplace, Maxwell, Newton, Herschel, Kepler, Galilee, Aristotle... |  glossary Definition of certain words or expressions used on this site in link with the astronomy.

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| articles Christian's articles in link with the astronomy (history of the Earth, the electron, the neutrons stars, the rings of Saturn, the particles, the threat of asteroids, journey in the universe up to the atom)... |  exoplanets Exoplanets are situated by definition outside our solar system. Seen the large number of existing galaxies and the large number of stars which they contain, the search for these exo-planets or extra solar planets, is unlimited. |  constellations A constellation is a group of stars of the sky which enough close relations looked the appearance imagination of a figure onto the sky. |  nebulas Of the Latin 'nebula' which wants to say cloud, clouds of gas and dusts in the middle of stars, nebulas are at the same moment active crèches and cemeteries of stars. These magnificence of the sky are lit by the stars which they contain or by stars situated behind them. |  telescopes A telescope is constituted by a mirror which concentrates the light resulting from the observed celestial body and from the objective which supplies an enlarged image with it. The increase and the luminosity are proportional on the surface of the mirror. The presence of the ground atmosphere limits the performances of a telescope. It is to mitigate this inconvenience that spatial telescopes were placed in the space. |  bibliographies Some interesting books in link with the astronomy. |  faq Some simple answers concerning the universe which surrounds us.

 |

| asteroids Numerous small rocky called bodies asteroids are present in the solar system, an important part of them circulate in a ring, between the orbit of Mars and that of Jupiter in 2 to 4 UA.

 |  rings The rings of Saturn are one of more beautiful and of the most surprising spectacles of the solar system. That is why on July 1st, 2004, the spaceship Cassini-Huygens reduced its speed to be captured by the gravity of Saturn to enter in orbit with this one. |  stars A star is a celestial body similar to the Sun, which shines thanks to nuclear reactions which occur in its center. |  multiverse The universe is an expanding cosmic bubble. This bubble creates a new bubble which produces it the others etc.... Quantities of the other universes which give birth to the other universes, it is an interesting idea but which remain at the moment very speculative. |  space probes These instruments that are space probes, perceive in the detail the forms and the composition of objects of the distant regions and offer us exceptional sights of an unequalled precision. |  elements The most usual and practical presentation of the diverse chemical elements is the periodic board of elements also called Table of Mendeleyev. The atoms which have the same number of protons but a number different from neutrons are called isotopes. |  aurora A polar aurora called aurora borealis in the north hemisphere and the aurora australis in the southern hemisphere, is a brilliant phenomenon characterized by sorts of veils extremely colored in the night-sky.

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 The universe is all worked up Imprimer cette page


Mass of Jupiter

      

         

Mercury Mercury the first planet of the solar system in 57,9 million km of the sun is, its diameter is 4880 km.  Venus Venus the second planet of the solar system in 108,2 million km of the sun is, its diameter is 13000 km.  Earth Earth the third planet of the solar system in 149,6 million km of the sun is, its diameter is 12756 km.  Mars Mars the fourth planet of the solar system in 227,9 million km of the sun is, its diameter is 6800 km.  Jupiter Jupiter the fifth planet of the solar system in 778 million km of the sun is, its diameter is 143000 km.  Saturn Saturn the sixth planet of the solar system in 1427 million km of the sun is, its diameter is 120500 km.  Uranus Uranus the seventh planet of the solar system in 2870 million km of the sun is, its diameter is 51120 km.   Neptune Neptune the eighth planet of the solar system in 4496 million km of the sun is, its diameter is 49530 km. 
Moon Earth possess that a single natural satellite, the Moon which orbits in 160000 km of the earth.   satellites of Mars Mars possess 2 known natural satellites, Phobos and Deimos. They orbit near the planet, in some thousand kilometres of this one.   satellites of Jupiter Jupiter possess more than 60 known natural satellites, among which Ganymede, Io, Callisto, Europe and the others... 
  satellites of Saturn Saturn possess 59 known natural satellites, of which Titan, Rhea, Japet, Dioné, Thetys, Enceladus, Mimas, Hyperion, Phoebe, Janus, Epimethea, Promethea, Pandore and the others...  satellites of Uranus Uranus possess at least 27 known natural satellites, among which Titania, Oberon, Umbriel, Ariel...  satellites of Neptune Neptune possess tens of known natural satellites, of which Triton, Proteus, Larissa, Galatea, Despina, Nereid, Thalassa, Naiad...
 

automatic translator

        
Calculation of the mass of Jupiter        
       

We suggest determining the mass of Jupiter by studying the movement of its main satellites: Io, Europe, Ganymede and Callisto.

  1. 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.

  2. The periods of revolution and beams of the orbits of four main satellites of Jupiter were determined and have the following values :

     
    Io
    Europe
    Ganymede
    Callisto
    T(in hours)
    42,5
    85,2
    171,7
    400,5
    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.
         
Correction        
       

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

 
Io
Europe
Ganymede
Callisto
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).
         
Result        
       
M = 4p² / ( 3,15 10-16 *6,67 10-11) = 1,87 1027 kg.
 

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 |  planets  8 planets of the solar system (Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune) and their satellites. |  sun The Sun is in 2/3 of the galactic center towards the edge, at a distance of 30 000 light years of the center. The Sun moves in a speed of 230 km/s around this galactic center during its revolution which it makes in 250 million years. |  galaxies The Galaxy, it is the name datum in our galaxy, it is an enormous spiral wheel of stars, a diameter of 100 000 light years. What appears of the Earth, it is a white continuous band called the Milky Way. |  bigbang There is 15 billion years a tremendous explosion of light gives birth to the space, in time, in the matter, a chaos burning with an inconceivable heat, a formless porridge which is going to swell, to extend in all the directions and to cool quite slowly. |  quotations Some quotations of big men (Aristote, Galilee, Newton, Platon, Laplace, Einstein). |  links Some external links concerning the astronomy.

|
|  |  dwarf planets According to the definition which was approved on August 24th, 2006, during the 26th General assembly of the UAI ( International Astronomical Union), three bodies reach the status of dwarfish planet: Pluto, eris, and Ceres. Others objets should soon join this nomenclature. |  solar system The solar system is really much more complex if we take into account all the objects being a part of the system. A considerable number of objects is in the belt of Kuiper and still beyond in the cloud of Oort. |  galaxy groups Heap of galaxies are the biggest structures of the Universe. They are constituted by hundreds of galaxies connected together by their own gravitational attraction. Between the galaxies we find some material constituted by warm gas there, forming a plasma, the temperature of which reaches 10 to 100 million degrees. |  comets Besides planets, satellites and asteroids, the solar system contains comets (hair in Greek). The roaming comets originate in the depths of the space in several lights years. The number of periodic comets is of the order of 2000. |  biographies Some biographies, Einstein, Baade, Lyot, Hale, Hubble, Shapley, Laplace, Maxwell, Newton, Herschel, Kepler, Galilee, Aristotle... |  glossary Definition of certain words or expressions used on this site in link with the astronomy.

 |
| articles Christian's articles in link with the astronomy (history of the Earth, the electron, the neutrons stars, the rings of Saturn, the particles, the threat of asteroids, journey in the universe up to the atom)... |  exoplanets Exoplanets are situated by definition outside our solar system. Seen the large number of existing galaxies and the large number of stars which they contain, the search for these exo-planets or extra solar planets, is unlimited. |  constellations A constellation is a group of stars of the sky which enough close relations looked the appearance imagination of a figure onto the sky. |  nebulas Of the Latin 'nebula' which wants to say cloud, clouds of gas and dusts in the middle of stars, nebulas are at the same moment active crèches and cemeteries of stars. These magnificence of the sky are lit by the stars which they contain or by stars situated behind them. |  telescopes A telescope is constituted by a mirror which concentrates the light resulting from the observed celestial body and from the objective which supplies an enlarged image with it. The increase and the luminosity are proportional on the surface of the mirror. The presence of the ground atmosphere limits the performances of a telescope. It is to mitigate this inconvenience that spatial telescopes were placed in the space. |  bibliographies Some interesting books in link with the astronomy. |  faq Some simple answers concerning the universe which surrounds us.

 |

| asteroids Numerous small rocky called bodies asteroids are present in the solar system, an important part of them circulate in a ring, between the orbit of Mars and that of Jupiter in 2 to 4 UA.

 |  rings The rings of Saturn are one of more beautiful and of the most surprising spectacles of the solar system. That is why on July 1st, 2004, the spaceship Cassini-Huygens reduced its speed to be captured by the gravity of Saturn to enter in orbit with this one. |  stars A star is a celestial body similar to the Sun, which shines thanks to nuclear reactions which occur in its center. |  multiverse The universe is an expanding cosmic bubble. This bubble creates a new bubble which produces it the others etc.... Quantities of the other universes which give birth to the other universes, it is an interesting idea but which remain at the moment very speculative. |  space probes These instruments that are space probes, perceive in the detail the forms and the composition of objects of the distant regions and offer us exceptional sights of an unequalled precision. |  elements The most usual and practical presentation of the diverse chemical elements is the periodic board of elements also called Table of Mendeleyev. The atoms which have the same number of protons but a number different from neutrons are called isotopes. |  aurora A polar aurora called aurora borealis in the north hemisphere and the aurora australis in the southern hemisphere, is a brilliant phenomenon characterized by sorts of veils extremely colored in the night-sky.

|
 The universe is all worked up Imprimer cette page


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Astronomy - october 15th 2007