Part of our lab will function as Discovery Lab,

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```					        Part of our lab will function as Discovery Lab, which future students might use to
learn about Kepler’s laws and directly work his third law. The following details below
will function as a “mini-lab” within the larger framework of our entire lab. Still, we will
use the information, formulas and methods described below in our entire project. All
right, here it is:

First, the objective is to calculate the mass of Jupiter by observing the orbit of Callisto,
one of its four major moons, with Kepler’s Third Law. We will use only Callisto for two
important reasons. First, for the practical issue of time, one moon is plenty. One moon’s
data is no better than another moon’s data. Moreover, Callisto is the easiest moon to
find, for it is the second largest of the four major moons and is, on average, the farthest
away from Jupiter. These qualities should make the task of finding it the easiest for our
participants.

Second, the formulas.

Kepler’s Third Law is:        T^2=kR^3        where: T= Sidereal Period of moon
R= Semi-Major Axis of moon
k= Gravitational constant between
Jupiter and a specific moon

The sidereal period is the time a moon takes to complete a full orbit of Jupiter. This
value will be measured in days.
The semi-major axis is half the major axis, which is the longest diameter created in a
moon’s orbit of Jupiter. Another way of calculating the semi-major is to find the
average distance of the moon from Jupiter.
The “k” gravitational constant between each moon and Jupiter is different. It is the total
mass of the system observed, in other words the sum of the masses of the moon
and Jupiter.

Now, to find the mass of Jupiter, we do not need to know the mass of Callisto, though we
may use it if we want to. Below are more formulas used to calculate the mass of Jupiter.

Newton’s famous third law is F=ma, with “F” being force measured in Newtons, “m”
being mass measured in kilograms, and “a” being acceleration measured in meters per
second squared.

There is another situation in which force arises: circular motion. The formula for this
case is: F=mv^2/R, with “F” being force measured in Newtons, “m” being mass measured
in kilograms, “v” being velocity measured in meters per second, and “R” being the radius
of the circular motion measured in meters. An equivalent statement of this formula is
F=4mR(Pi)^2/T^2, with “F” being force measured in Newtons, “m” being mass
measured in kilograms, “R” being the radius of the circular motion measured in meters,
Pi being the famous constant that is often estimated at 3.14, and “T” being the period of
revolution measured in seconds.

Now, hold on, because there is yet ANOTHER situation that has its own unique formula
for force. This is the law of universal gravitation. The formula for it is F=GMm/R^2,
with “F” being the force measured in Newtons, “G” being the universal gravitational
constant measured at 6.67*10^-11 Newton meters squared divided by kilograms squared,
“M” being the mass of one object measured in kilograms, “m” being the mass of the
second object measured in kilograms, and “R” being the average distance between the
two objects.

Now we get tricky. In this experiment, “m” will be the mass of Callisto measured in
kilograms, “M” will be the mass of Jupiter measured in kilograms, “G” will be the
universal gravitational constant, “R” will be the average distance between Callisto and
Jupiter measured in meters, and “T” will be the period of Callisto’s orbit around Jupiter
measured in seconds. Next, we will synthesize the aforementioned equations. There is
an inherent flaw in this, for the formula for circular motion is specific to circular motion.
Still, we will use it in our formula manipulations and expect some error in our final
results, which hopefully will not be too egregious.

First, we take the second form of the equation for the force in circular motion,
F=4mR(Pi)^2/T^2. Next, we set next to it the equation for universal gravitational force,
F=GMm/R^2. The right side of each equation equals “F,” so we may set the right side of
each equation equal to each other. Therefore, 4mR(Pi)^2/T^2=GMm/R^2. By
manipulating this equation, we create M=4R^3/GT^2. Notice that in this equation we do
not need to know the mass of Callisto.

With the mass of Jupiter calculated (M), we may now test the validity of Kepler’s third
law, T^2=kR^3. We know the period, T, we have measured the mean distance of Callisto
from Jupiter, R, and we may now calculate k by adding together the mass of Callisto,
which is listed in this report, and the mass of Jupiter that we calculated. Hopefully, there
will be a small margin of error between the two values.

The specific timetable of this discovery lab will encompass eight consecutive nights. The
orbital period of Callisto is 16.689 days. Therefore, in about eight days Callisto will have
reached its farthest distance from Jupiter on both the left side and the right side. We will
then average these eight distances to estimate the average distance between Callisto and
Jupiter. Each night two of our team members will take one or two classmates out to
observe Callisto and Jupiter, at the same time each of the eight nights. This way, each
student will gain experience with the telescope and contribute to collecting the data.

Beyond this discovery lab, we four experimenters will apply the same procedure to each
of the other three moons. We will adjust in each case for the different periods and the
different masses. We will compare our measurements of average distances with those of
the 1998 Keplerites to see whose is more accurate. Furthermore, we will compare our
implicit measurements of Jupiter’s mass with the actual mass to see how much error there
was.

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