Abstract: This lab will use the program CLEA

Abstract: This lab will use the program CLEA and
the simulation The Revolutions of the Moons of Jupiter to track the moons of
Jupiter for 12-hour intervals. These points will be plotted, and a sine wave
will be fit to each of the moons location curves. The period and amplitude of
this wave will then be used to calculate the mass of Jupiter. This will be
compared to the actual mass of Jupiter and discussed.

Introduction: The mass of Jupiter can be
calculated by observation of the moons that orbit the planet. The physical
observation of Jupiter is time consuming and almost impossible in Ireland
because of the weather. A telescope is also needed for actual observation. To
overcome these issues a simulation program CLEA was used to simulate the moons
of Jupiter as they orbit. Kepler’s Third Law (M = a³/p²) is used to calculate
the mass of Jupiter.

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Method: The
CLEA program simulation The Revolution of the Moons of Jupiter
was opened. The moons positions were logged 18 times at 12-hour intervals
making note of the cloudy days.

The program then plots the
points for one moon on a graph. The period, amplitude and t-zero (the point
where the curve crosses the Y axis going from negative to positive) were found
and input into the fit sine curve. The curve was then manipulated to best fit
the points. The period and amplitude of this curve was then noted and converted
to years and A.U. This was repeated for all the moons of Jupiter.

Kepler’s Third Law Mass =
amplitude³/period² was then used to calculate the mass of Jupiter according to each
moon in Solar Masses.

 

Results:

Table 1: Location Data for
the 4 moons of Jupiter

 

 

Figure 1: Sine curve fit
to Callisto location points

Period of Callisto =
16.688 days/365 = 0.0457 years

Radius of Callisto = 13.2
Jupiter diameters/1050 = 0.01257 A.U.

Mass of Jupiter From
Callisto = 0.01254³(A.U.)/0.0457²(years) = 0.0009509 M?

 

Figure 2: Sine curve fit
to Ganymede location points

Period of Ganymede = 701days/365
= 0.0195 years

Radius of Ganymede = 7.49
Jupiter diameters/1050 =0.00713 A.U.

Mass of Jupiter From Ganymede
= 0.00713³(A.U.)/ 0.0195²(years) =0.000953 M?

 

 

 

 

 

 

Figure 3: Sine curve fit
to Europa location points

Period of Europa = 3.584 days/365
= 0.00982 years

Radius of Europa = 4.7952
Jupiter diameters/1050 =0.004567 A.U.

Mass of Jupiter From Europa
= 0.004567³(A.U.)/ 0.00982²(years) = 0.0009878 M?

 

Figure 4: Sine curve fit
to Io location points

Period of Io = 1.768 days/365
= 0.004839 years

Radius of Io = Jupiter
diameters/1050 = 0.002868 A.U.

Mass of Jupiter From Io = ³(A.U.)/
²(years) = 0.001007 M?

 

 

 

 

 

Discussion/Conclusion:

 

Average mass of Jupiter =
(0.0009509+0.000953+0.0009878+0.001007)/4 = 0.0009751 M?

Mass of Jupiter in Earth
masses = 0.0009751/3*10?? = 325.058 Earth masses

Actual mass of Jupiter = 0.0009546
M? or 317.83 Earth masses 1

 

The calculated mass of
Jupiter is close to the actual mass of Jupiter. Callisto and Ganymede’s masses
of Jupiter came out as the closest with Europa close behind. The least accurate
of the moons is Io. This may have been due to the small orbit of this moon.
Many times, it was impossible to mark the location as the moon is behind
Jupiter. The period of this moon is also very small resulting in difficulty
when trying to fit the sine curve to the limited points found.

 

Jupiter has moons that are
further away than Callisto. These moons will have larger periods than Callisto
because the period is the time it takes to orbit the planet. The further away
from a planet a moon is the longer the orbit will take.

 

A 10% error in amplitude
would cause a larger error than a 10% error in period because in the formula amplitude
is cubed where as period is only squared. Amplitude ends up as the larger
number.

 

The moons orbit around
Earth: 27.3 days/365 = 0.07479 years

Mass of Earth (question
4): 0.00256³ A.U./0.07479² years = 0.000002999 or 2.999×10?? M?

Actual mass of Earth = 3.003467×10??
M? 2

 

 

 

 

 

 

 

 

 

1 https://ipfs.io/ipfs/QmXoypizjW3WknFiJnKLwHCnL72vedxjQkDDP1mXWo6uco/wiki/Jupiter_mass.html

2 https://www.wolframalpha.com/input/?i=mass+of+earth+in+solar+mass+units

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