Ellipses Homelab: Name: ____________Class: _______
Materials needed: string, cardboard, two stick pins, paper, protractor,
and metric ruler
Background: Kepler discovered that the orbit of Mars is
elliptical. He plotted the orbits of
the other planets and determined that they are also elliptical. Kepler then
formulated his Law of Elliptical Orbits:
Each planet moves around the sun in an orbit whose shape is an ellipse,
with the sun at one focal point. From any point on this ellipse, the sum of the
distances from that point to the two foci is constant in length. In each planet-sun system, the sun is at one
focus while the other focus is a point in space not occupied by any
astronomical body. Kepler
came up with this theory by plotting the orbit of Mars, his assignment when he
was invited to become one of Tycho Brahe’s assistants at his new observatory in
circumference = 2 π
√ ( (a2
+ b2) / 2) area = π *
a * b a2 =
b2 + c2
Procedure:
1. Center a sheet of paper over the cardboard and tape it
down.
2. Press 2 stick pins, exactly 5.00 cm apart and
centered side to side and top to bottom, through the paper into the cardboard.
3. Cut a piece of string (dental floss works
great) a length about 3 to 4 times the distance between the pins, plus enough
for a knot.
4. Tie a knot to form a loop and lay the loop on
the paper so that is encloses the two pins.
5. Place a sharp pencil inside the loop and
stretch the loop taut against the pins (keeping the string flat on the paper
surface). Draw an ellipse around the two
pins by keeping the pencil point as far from the pins as the string will allow.
6. Remove the string and pins.
7. Draw the major axis through the two
pinholes. This will be the longest
distance across the ellipse.
8. Draw the minor axis through the shortest
distance across the ellipse. It will be
perpendicular to the major axis at its midpoint. Use a ruler and protractor.
9. Label the two foci. These are the holes left by the pins. By default the left is the sun and is labeled
by placing a small circle around the pin hole.
The other pin hole is labeled F.
10. Label the perihelion on
your drawing. This is the point on the
orbit closest to the sun (one of the foci).
It is at one end of the major axis.
11. Label the aphelion on
your drawing. This is the point farthest
from the sun in the orbit of the planet.
It also is at one end of the major axis.
Questions:
1. In cm and to the correct number of
significant figures, record the:
length of major axis: __________cm length of minor axis: _____________cm
2. The semimajor axis
is measured from the midpoint of the major axis to the aphelion.
length of semimajor axis:
__________cm
3. Eccentricity equals the distance between the
foci divided by the length of the major axis.
eccentricity = distance between foci =
___________ (no units here)
length
of major axis
4. The eccentricity of a circle is zero. Look up the eccentricity of the orbit of the
earth around the sun.
List it: ____________________
(for bonus)
5. Calculate the circumference of your ellipse:
_______________
6. Calculate the area of
your ellipse: __________________
7. Confirm the ellipse by solving: a2
= b2
+ c2 Was your ellipse drawn correctly? (Show
all work here).
Part 2:
From any point on this
curve, the sum of the distances from that point to the two foci is constant in
length. In each planet-sun system, the
sun is at one foci.
The other foci is a point in space, not
occupied by any astronomical body.
1. Draw and label the major axis as AB. Label the center of the ellipse as C.
2. Draw and label the minor axis as GH.
3. Indicate the position of a planet in its
orbit at two different times (any two places on the ellipse). Label one position P1 and the
other P2. Draw in the lines:
P1Q, P1F, P2Q, and P2F
4. Measure the following lengths:
length of P1Q = __________cm
length
of P1F = __________cm
length
of P2Q =
__________cm
length
of P2F = __________cm
length of P1Q + P1F =
________cm (remember: when
adding, it’s decimal places)
length
of P2Q + P2F =
________cm
Are
the sums P1Q + P1F and P2Q + P2F equal or unequal? If the difference between the two is greater
than 0.1 cm you need to locate new points and try again or do another ellipse.
Part 3 – Construction of
another ellipse
Construct an ellipse whose
major axis is 16 000 000 km and whose eccentricity is
0.5 (hint: let
1 cm = 1 000 000 km)
a) If this ellipse were the
orbit of a planet, what would be the aphelion distance of the planet?
___________________ km
b) What is the length of
the minor axis of this ellipse?
___________________ km