Directions

1. Obtain a large piece of graph paper from Mrs. Colwell. Carefully, determining the center of the paper. Place a large pencil dot in its position.

2. On this graph paper, draw a 10 cm radius circle as close to the center of the grid as possible using your compass. Label the center of this circle with a large "S" for the Sun. This circle represents the orbit of the Earth around the Sun.

3. From "S" draw a line to the right until it intersects with the circle. This represents the positive x-axis. Neatly label this intersection point as 0^o - Sept. 23. Now draw a line from "S" to the left. This will represent the negative x-axis. Label this intersection point as 180^o - March 21.

4. Since the Earth travels once (360^o) around the Sun in 365 days, use the rough estimate that the Earth moves approximating 1 degree/day, to locate and neatly label each of the positions listed as A through P below on the circle drawn in Step #3.

5. After obtaining a booklet of star photographs and transparencies, locate Mars on each picture and use a short ruler to interpolate the planet's position. Record your results in the chart provided below. After checking with your instructor, share them with the rest of the class by placing them in the appropriate blanks on the board. Notice that the dates on which each of the pictures was taken coincides with the dates already placed on your Earth Orbit.

6. Through each Earth position, lightly sketch in a new "O^o axis" parallel to your original x-axis which passed through the Sun. Then using a protractor, locate the "line of sight" for Mars. Each set of overlays (AB), (CD), etc. will triangulate one Mars position --- you will have a total of "8 spikes". Place a neat circle around each intersection and label them appropriately as M_AB, M_CD, M_EF, M_GH, M_IJ, M_KL, M_MN, and M_OP.

7. Use a ruler to draw in a straight line between two adjacent Mars positions. Then use a compass to bisect the line and a ruler to draw in the perpendicular bisector. Perform this operation up to four times for more accurate results. Extend the bisectors as long as necessary to insure that they intersect.

According to the theorem from geometry, the perpendicular bisectors of any two chords of a circle will intersect in the center of the circle.  You should now be able to draw a circle that represents Mars' orbit. Note that the center of this circle will not pass through the Sun.

Calculations and Conclusions

1. Use two difference sets of Mars' dates and calculate the number of Earth days in one Mars' orbital period. State which sets you analyzed and the number of days for each. Finally, average your two results and convert the result into years (1 Earth year = 365 days)

Combination: ______             _________ days
Combination: ______             _________ days
T = Average = ______ days = ________ years

2. Draw in the line that connects the "center" of Mars' orbit with the Sun. Extend this line all the way across Mars' orbit. Measure this entire length. Then measure the length of the following distances from the Sun to Mars:

            R_P = Perihelion radius = ______ cm
            R_A = Aphelion radius = _______ cm

3. Earth's radius in centimeters = _____________ = 1 AU (Astronomical Unit).  An AU is the average distance from the Earth to the Sun

4. Convert these two radii, R_P and R_A, into Astronomical Units (AU).

R_P = Perihelion radius = ______ AU
R_A = Aphelion radius = _______ AU

5. Your plot of Mars is an ellipse with respect to the Sun. To calculate the degree of its eccentricity, e = c/a, divide the following two measurements:

c = distance from the "center" of Mars' orbit and the Sun = ______ cm = ________ AU
a = one-half the length of the major axis = R_AV = (R_P + R_A) / 2 = ___________ AU

e = c / a = ______ /______ = ________

6. Verification of Kepler's 1st Law.  According to astronomers, the eccentricity of Mars' orbit equals 0.093 Calculate a percent error on your calculation in step 5.

7. Verification of Kepler's 3rd Law.  Calculate Kepler's constant for Mars from your graph by using the formula: k = T² / (R_AV)³.

This formula is derived by setting the gravitational force equal to the centripetal force,
F_G = F_C
where F_G = G M_MM_S / r²
and F_C = M_M v² / r 
when  v = 2pr / T.

Use the values for T (period) from step #1 and R_AV (average radii) from step #5

(#1) T = _______ years
(#5) R_AV = _____ AU
k = T² / (R_AV)³ = _______ / ________ = _________

According to astronomers, Kepler's constant should be 1 for all satellites of the Sun. Calculate a percent error to determine the overall accuracy of your plot.

8. Verification of Kepler's 2nd Law.

By using carbon paper. trace onto a piece of cardboard the sector of the orbit defined by the arc M_GH and M_IJ by first marking the points M_IJ, Sun, and M_GH as well as the arc connecting M_GH and M_IJ. Then remove the carbon paper. By using a ruler, draw in the radii between M_GH and the Sun and M_IJ and the Sun. Cut out this sector and obtain its mass in grams. Place your names on your cardboard, label your sector and record its mass clearly.

Now repeat the above process but with the sector of the orbit defined by M_EF and M_OP by marking the points M_EF, Sun, and M_OP as well as the arc connecting M_EF and M_OP. Then remove the carbon paper. By using a ruler, draw in the radii between M_EF and the Sun and M_OP and the Sun. Cut out this sector and obtain its mass in grams. Place your names on your cardboard, label your sector and record its mass clearly.

Using a piece of string or tape, measure the arc length of each sector.  Next carefully fold the arc length in half and measure its average, center, radius. Record your information in the chart provided below and on each cardboard sector.

9.  What is the relationship between the relative masses of these sectors and their relative areas?

Kepler's 2nd Law states that the ratio of v₁ / v₂ = R₂ / R₁ between any two points 1 and 2 on the same orbital path. This relationship can be rewritten as s₁ / s₂ = R₂ / R₁ if the time period for each of the sectors measured above is equal.

Calculate the ratio of these arc lengths

and then the ratio of these radii.

10.  Finally, calculate the percentage difference between these values. Staple your cardboard sectors to the back of this page of your lab report.