12/27/2003

给我一个支点，我能移动地球　――阿基米德

给我一架望远镜，我可以称出天神的质量

Here is the planet Jupiter taken through a Schmidt-Cassegrain 8 inch telescope with f/10 focal ratio.  Besides the band structure on Jupiter, the Great Red Spot (GRS) is clearly visible near the center of the left picture.  A close examination of the photo reveals 3 satellites along the diagonal from upper-left to lower-right, with a faint hint of the fourth one near the upper left corner.  They are the four Galilean satellites, Io, Europa, Ganymede and Callisto, discovered 394 years ago by the famous Italian astronomer.  One hour and 6 minutes later, the second picture, shown on the right, was taken.  The satellites moved with respect to the Jupiter by as much as 10 arcseconds.  By approximating the satellite orbits as perfect circles and ignoring the tilt of the satellite orbit plane with respect to the line of sight, the orbit can be expressed as

where  are the distances from Jupiter, orbit radii, orbit periods and orbit phases respectively.  A Maximum Likelihood (ML) estimator[1] is programmed to solve the unknowns from 3 or more observations of ’s.  Over a course of 3 hours and 57 minutes, 7 photos are taken and the ML solution gives

The data (dots) and ML solution (lines) are shown in the following plot.  It is worth noting that Ganymede moved only 10 pixels in the observing time span and yet produces a far more accurate solution than Callisto, which left a motion of 35 pixels in the images. 

The mass of Jupiter is related to the radius and period by Newtonian gravity

where G is the gravitational constant, 6.673E-11m³.kg^-1.s^-2. Plugging in this equation yields a mass of 1.12, 1.63, 1.55, 1.29 x 10²⁷ kg respectively.

The actual Jupiter mass is 1.899 x 10²⁷ kg[2].  Considering the approximations made and the short observation span, underestimation of 15-40% the weight of the Roman God of the heaven is perhaps justified.

As a side note, from the position of the GRS on Jupiter, the rotation speed of 9 hours per revolution can be estimated.  This fast spinning, in conjunction with the larger size, is the main cause of the 7% fatter waist.

[1] “Fundamentals of Statistical Signal Processing: estimation theory”, Steven M. Kay. 1993, p193-195

[2] “Solar System Dynamics”, C.D. Murray and S.F. Derrmott, Cambridge University Press, 1999.