Name:
Class
Phys1411/1403 - Goderya
Kepler’s Laws: Moons of Jupiter and Derivation of Mass of Jupiter - Planetarium
Introduction
Johannes Kepler used Tycho Brahe’s data on planetary motion to derive three laws that govern the motion of
planets around the Sun.
1st Law: Law of Ellipses
Planets move around the Sun in an elliptical orbit with Sun at one of the focus
Source: Thinglink.com
Geometry of Ellipse
An ellipse has two axes, a major axis and a minor axis. The Sun is not at the center but at one of the foci as
shown in the diagram. The distance of the planet closest to the Sun is called perihelion and the distance
farthest from the Sun is called aphelion. The distance from the center to the perihelion point is called the semi
major axis a and the distance from the center to the focus is called c. The ratio of c/a is called the
eccentricity e of the ellipses. The short axis is called semi minor axis b. The eccentricity of the ellipse can
also be calculated by other equations for example
1
. However, these equations though useful are
not practical for observational astronomy. If we define Ra as the aphelion distance and Rp as the perihelion
distance from the Sun, then a more useful equation is
1 .
2nd Law: Law of Areas
A line from a planet to the Sun sweeps over equal areas in equal interval of times
Source: Cengage Learning
Second law basically says that the planets moves fastest when it is at perihelion and slowest when it is at
aphelion. This law has its basis in conservation of angular momentum and energy.
3rd Law: Law of Periods
A planet’s orbital period P squared is proportional to its average distance from the Sun a (semi major axis)
cubed.
If we measure the period in earth years and the distance in astronomical units then:
a
Mass of a Planet
We can use Newton’s Universal Law of Gravity, Centripetal force and Kepler’s third law to derive and
expression (formula) to calculate the mass M of the Planet. This expression is;
Using G = 6.67 x 10 -11 m3/s2kg we can write M as
6.0
10
… … … … . 1 Where a is now in meters and P in seconds. This equation is more user friendly and it will give us mass in kg.
Note: Kepler’s 3rd Law assumes that the planet is orbiting the Sun. When we apply the equation to
moon of Jupiter we need to keep in mind that the moons orbit Jupiter and not the Sun.
MJ/Ms = 1.9x1027/1.99 x 1030=9.548x10-4, so the equation;
Needs to be scaled properly for mass
of Jupiter. This scaling factor in our case amounts to multiplying a in AU by 0.09846. In our
calculations below we do that by using 1AU = 1.5 x 10 10 m instead of 1 AU = 1.5 x 10 11m.
2 Part I: Data Collections and Calculations
Your SI will setup the Planetarium to show the orbits of Galilean Moons of Jupiter (Io, Europa, Ganymede, and
Calisto) one at a time but not in any particular order. For convenience in measurements the orbits of the moon
and the meridian will also be visible, but remember that in real situation when viewing with the telescope these
are not there and that the view itself may also be different. For each of the moon measure the time for one
complete orbit and find the orbital period. Then use Kepler’s 3rd law to find the distance of the moon form
Jupiter. Record the data in table below and complete the remaining columns. Note that observations and
experimental measurements are rarely done a single time, usually an average with uncertainty is needed, but
we will not worry about calculating uncertainty for this class we will just do average of two observations.
(Hint: Typical periods may be 2, 4, 6 and 17 days. Your calculations of period in days should be of 3 significant digits).
Moon 1:
Orbit
Number
1
2
Start
Date
Start
Time
End
Date
End
Time
Period P
in days
Period P in
seconds
Period in
years.
Average
Calculate the average distance in AU form Kepler’s 3rd Law (
/
) use period in years: ____________
Convert the distance a into meters (multiply by 1.5 x 1010 m): __________________
Calculate the
Moon 2:
Orbit
Number
1
2
where a is now in meters and P is in seconds: ___________________
Start
Date
Start
Time
End
Date
End
Time
Period P
in days
Period P in
seconds
Period in
years.
Average
Calculate the average distance in AU form Kepler’s 3rd Law (
/
) use period in years: ____________
Convert the distance a into meters (multiply by 1.5 x 1010 m): __________________
Calculate the
Moon 3:
Orbit
Number
1
2
where a is now in meters and P is in seconds: ___________________
Start
Date
Start
Time
End
Date
End
Time
Period P
in days
Period P in
seconds
Period in
years.
Average
Calculate the average distance in AU form Kepler’s 3rd Law (
/
) use period in years: ____________
Convert the distance a into meters (multiply by 1.5 x 1010 m): __________________
Calculate the
where a is now in meters and P is in seconds: ___________________
3 Moon 4:
Orbit
Number
1
2
Start
Date
Start
Time
End
Date
End
Time
Period P
in days
Period P in
seconds
Period in
years.
Average
/
Calculate the average distance in AU form Kepler’s 3rd Law (
) use period in years: ____________
Convert the distance a into meters (multiply by 1.5 x 1010 m): __________________
Calculate the
where a is now in meters and P is in seconds: ___________________
Part II: Data Analysis and Questions
1. Calculate the average from four values (one for each moon) of
: ________________
2. Use equation (1) from introduction to calculate the mass of Jupiter in kg.
3. Mass of Earth is 6 x 1024 kg. Using your answer from equation 1, what is the mass of Jupiter relative to the
mass of Earth.
4. Use the image below and information from part I, to identify and label the four moons.
Moon 1: _________________________
Moon 2: _________________________
Moon 3: _________________________
Moon 4: _________________________
Source: http://pics-about-space.com
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