Laboratory 7
Objective:
In this laboratory Johannes Kepler’s Laws of Planetary Motion will be studied with emphasis on his third law. The significance of Kepler’s third law will be understood by graphing its relations.
Background:
Explaining the motion of
the planets became a pursuit of many early scientists. Nicholas Copernicus, in 1543 proposed a theory
that the solar system was heliocentric. In other words, the planets orbit the central
Sun in circular orbits. This was a
profound statement during his time since the accepted theory of the solar
system was that it was geocentric,
everything revolved around the Earth, including the Sun.
Later Johannes
Kepler in 1609 proposed the three laws of planetary motion, using the detailed
observations of Tycho Brahe. Kepler’s
motivation was to find a simple explanation of how the planets orbited the Sun,
as Copernicus thought, but soon discovered that the simplicity of circular
orbits was not correct. Instead he found
that the planets orbit the Sun in elliptical orbits. This was a small change that led to an
abundance of information about how planets orbit the Sun.
In 1687 Isaac
Newton published the Principia in which he explained Kepler’s laws this time
including using the force of gravity.
The law of gravity states that the gravitational force between two
celestial bodies is proportional to the product of the two masses and inversely
proportional to the square of the distance between them. If the distance between two celestial bodies
is doubled, the gravitational attraction between them is reduced by 1/4. If you triple the distance the force of
gravity between the two celestial bodies drops to 1/9.
Kepler’s
Laws of Planetary Motion
1. Kepler’s 1st Law states that the orbit of a planet around the Sun is an ellipse, with the Sun’s center of mass at one foci.
The eccentricity, e, of an ellipse can be determined by the following relationship:
Where c is the distance between the two foci
a is the Semi-Major Axis.
The eccentricity is how ‘squashed’ the ellipse is compared to a circle. Notice that if c = 0, the two foci are at the same point and we have a circle. The Semi-Major Axis and the eccentricity is all that is needed to describe the size of a planet’s orbital path as well as its shape.
2. Kepler’s 2nd Law states that a line joining a planet and the Sun sweeps out equal areas in equal intervals of time.
As a planet travels along the ellipse, it can be visualized to “sweep” out an area, A, during a given time interval t. To maintain equal areas the planet has to move faster when closer to the Sun, then when further away.
3. Kepler’s 3rd Law states that the square of the Orbital Period of a Planet orbit is proportional to the cube of the Semi-major axis.
Where P is the Orbital Period
a is the Semi-Major Axis.
The Orbital Period is the time it takes for
the planet to complete one orbit around Sun and must be measured in Years. The Semi-Major Axis represents a distance the
planet will travel and must be measured in measured in Astronomical Units
(A.U.)
Procedure:
Part
1: Observing the Inner Planets
1. Open
the Sky6 software.
2. Set the view to the 3D Solar System view. On the toolbar
click on 
. You should now see an overhead view of the
solar system.
3. Zoom in until the orbit of Mars fills your screen. In addition you should see the orbits Mercury, Venus and Earth.
4. Open the time window and set the time to advance in 1-day increments.
5. Left
click on Mercury to select it. Set the date to
6. Press the single forward arrow to move the planet in its orbit by one day. Press the double forward arrow to speed the motion and watch Mercury revolve around the Sun.
7. Stop the motion of Mercury after it makes one full orbit and reaches the starting point again. If you miss the starting point, just use the single forward or backward arrow to get back to the point.
8. You have just witnessed one “year” as defined on Mercury, or one solar revolution! Record the date on your data sheet under End Day.
9. Repeat steps 5-8 for Venus, Earth and Mars.
Part 2: Observing the Outer Planets
1. Now zoom out until the orbit of Pluto fills your screen.
2. In the time window, set the time skip to 1 Year, using the dropdown menu.
3. Repeat steps 5-8 from Part 1, for all of the outer planets.
Part 3: Calculations
1. Convert all of your Orbital Periods from days
into years by dividing each period by 365.25 days. For example the Orbital Period of Mercury in
years is;
2. Convert all of your distances from kilometers to Astronomical units by dividing the distance by 149600000 km. For example the Distance from the Sun to Mercury is
3. Now calculate the Orbital Period squared,
and the Distance cubed,
.
Part 4:
Graphical representation of the data
1. Using the graph paper provided plot the Orbital Period (P) in years vs. Distance (a) in A.U.
2. On a separate piece of graph paper plot vs. .
3. Answer the remaining question.