Lab
Kepler's 1st and 2nd Laws
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In this lab we will examine the properties of satellites which travel in elliptical orbits. These satellites have variable speeds since their distances from the sun are constantly changing.
Three empirical laws governing the behavior of satellites in elliptical orbits were published by Johannes Kepler (1571-1630). His research was based on the extensive data compiled on Mars by
Tycho Brahe
(1546-1601).
An ellipse is a conic section formed by slicing a right circular cone with a slanted plane. Every ellipse has two foci and obeys the property that the sum of the distances from each focus to any point on the ellipse is always a constant:
F
_{1}
P + PF
_{2}
= constant
.
Kepler's First Law is called the
Law of Ellipital Orbits
and states that every planet travels in an elliptical orbit about the Sun, which is located at one of the two foci of the ellipse.
The following diagram shows us three important positions on the ellipse in addition to its foci. The perihelion, P, which is the closest position to the Sun; the center, C, which is the middle of the ellipse; and the aphelion, A, which is the most distant position from the Sun.
Step 1:
(Maxwell's method for constructing ellipses)
Obtain a piece of string whose length is 55 cm so that you can tie a loop which is EXACTLY 44 cm in circumference. Fold your sheet of white cardboard into four equal quadrants and carefully mark the center. Next, measure carefully and place two straight pins into the cardboard at (-9,0) and (+9,0) along the x-axis. Loop the string over these pins. Bring the string loop into tension by placing a pencil inside the loop and stretching the string so that the pins and the pencil form a "triangle". Gently move the pencil around the paper and trace out an ellipse. Keep the string tight. This should form an ellipse with a major axis close to 27 cm and a minor axis close to 20 cm.
Step 2:
Kepler's First Law
states that a planet travels in an elliptical orbit with the sun at one foci. Therefore, label the left foci
S
for sun, the left endpoint of the major axis
P
for perihelion, and the right endpoint
A
for aphelion. Ignore the presence of the right focus.
Step 3:
Mark off an
arc
at
A
along the curve of the ellipse that extends 3 cm below
A
[call this point
N
] and the 3 cm above
A
[call this point
M
]. Be careful! These arcs are located along the perimeter, or trace" of the ellipse. Do NOT measure along the vertical line
MxN
. After points M and N are located, draw in two line segments: (1) one connecting
S
to
M
and a (2) second segment connecting
S
to
N
.
Step 4:
Carefully cut the wedge segment created in Step 3 from your ellipse. Make sure that you ONLY cut along the radii
SM
and
SN
and the arc
MN
keeping the point at
S
sharp.
Measure and record the mass of this wedge in grams.
Re-measure its arc length, it should be close to 6 cm.
Step 5:
Kepler's Second Law
states that in equal intervals of time, a line from the planet to the sun sweeps out equal intervals of space. In
Step 4
we delineated one of our two areas of the ellipse.
To verify Kepler's 2nd Law, CAREFULLY begin to cut another wedge shaped segment that starts at
S
, contains point
P
, and is
symmetric
about the radius
PS
. Stop trimming this wedge when it reaches the same mass as the wedge in
Step 4
. CAUTION - start with a large wedge which you can make smaller and NEVER trim away the Sun!
When you have finally produced a wedge that equals as close as possible the same mass as
SMN
, measure and record its mass in grams.
Now measure and record its total arc length in centimeters.
Conclusions
1. One way to classify an ellipse is based on its eccentricity, or how "flattened/oval" it is. Eccentricity is calculated by the ratio of c/a: where "c" in this formula is the distance from the Sun to the Center of the ellipse; and "a" is the length of half of the ellipse's major axis. An eccentricity of 0 means that the ellipse is actually a circle. An eccentricity of 1 means that the ellipse is hyperbolic.
By referencing the right image in the above diagram, you can use your first wedge, cut out in
Step 5
, to measure the values of "a" and "c."
You will now calculate the eccentricity of your ellipse. Give all measurements and calculations to 3 signficant figures.
c =
a =
e =
2. Does the wedge in
Step 5
that is symmetric to the radius
AP
have the same area as the wedge
SMN
produced in
Step 4
? Defend your answer.
3. If your ellipse represented planet X's trip around the Sun, and each "wedge" represents the same amount of orbital time, does planet X have its greatest orbital speed at
A
or
P
? Support your answer.
4. You will now calculate two ratios to determine how the ratio of the planet's speed at the aphelion compares to its speed at the perihelion. Remember that the time,
t
, taken by planet X to travel along each arc length is the same.
arc length
_{aphelion}
=
R
_{A}
=
arc length
_{perihelion}
=
R
_{P}
=
=
=
5. How well does the ratio of arc lengths compare to the ratio of orbital radii? Based on the measuring procedures in the lab, which ratio do you feel is the more "accurate" relationship between our hypothetical planet's aphelion velocity to its perihelion velocity?
Remember to put your names on each sector along with each sector's mass and arc length. Staple them along with your string to the back of your written law report (calculations) when you turn it in.
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