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lab_7__ex._18____determining_the_mass_of_the_moon.ppt

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Lab 7 Exercise18 – Determining
The Mass of the Moon
This lab is about verifying Kepler’s 3 Laws of
Planetary Motion and using the updated
version of Kepler’s 3rd law which is called now
Newton’s law to calculate the mass of the
Moon.
Kepler’s First Law of Planetary Motion
states that the orbits of the planets are
ellipses with the Sun at one focus.
But the law also applies to any object in orbit
around any other object, like a spacecraft
orbiting around the Moon.
The positional data of the spacecraft around the
Moon is shown in Table 18-1. The time interval
between points is 15 minutes.
To verify Kepler’s First Law use the positional
data (X and Y) and the graph paper provided to
plot the location of the spacecraft over time as it
orbited the Moon.
Did you get an ellipse? Is the Moon at one
focus?
Use a ruler to measure the distance (in mm)
between the foci, major and semimajor axis,
minor and semiminor axis. Calculate the
eccentricity of the ellipse, the orbital period P in
hours (include the remaining time as a fraction of
an hour (e.g. 5 hr, 30 mins = 5.5 hours).
See Fig 18-1 for definitions.
Kepler’s Second Law
States that a line segment connecting the
Sun and a given planet sweeps out equal
areas in equal time intervals.
Prove this law by
calculating the area of
2 triangles in your ellipse.
The areas have to be equal.
Use a ruler to do the measurements for your
triangles. Area = ½ × b × h
b = base
h = vertical height
Kepler’s Third Law
Kepler’s Third Law states that the period of
the orbiting planet (in years) is related to the
semimajor axis of its orbit (in astronomical
units) through the equation P2 = a3.
Kepler failed to take into account the masses
of the objects involved.
Isaac Newton modified Kepler’s Third Law so
that masses were factored into the equation
(m+M)P2 = (42/G) a3
G is the universal gravitational constant and
is equal to 6.668 * 10-8 cm3/gm-sec2
(m+M)P2 = (42/G) a3
We will determine the mass of the Moon M
using this equation and data from your graph.
(42/G) is calculated for you and is given in
the lab as 4.568 * 1013 kg-hr2/km3.
m is the mass of the spacecraft, almost zero
compared to M.
The formula becomes MP2 = (42/G) a3
M = (42/G) a3
P2
You know P in hours and a in mm (from the
ellipse).
We need to convert a from mm in real km.
Determining a in km
You used a ruler to measure the length (in
millimeters) of the major axis, 2a, through the
ellipse you just drew and divided the major axis
by two to find the semimajor axis, a.
Then measure this distance, a, along one of the
axes x or y of your graph to determine how
many lunar radii the semimajor axis is equal to.
Use the fact that the radius of the Moon is 1738
kilometers to convert your semimajor axis into
kilometers. (multiply the value of lunar radii you
got for a with 1738 km)
Plug all the values in the equation
M = (42/G) a3
P2
Your answer will be in kilograms.
To receive full credit for this lab, you must
submit your graph of the positional data of the
lunar orbiter showing the Moon at one focus, a
sheet of loose leaf paper detailing your
calculations for Kepler’s 1st, 2nd and 3rd law,
and with the mass of the Moon clearly marked
in kilograms.
…
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