Laboratory Exercise
THE ORBIT OF MERCURY
Introduction and History
The planets orbit the sun in elliptical orbits with the sun at one focus, as first discovered by Johannes Kepler, using extensive observations of Mars provided by Tycho Brahe. Mercury has the shortest orbital period (P = 0.241 years) of all the planets and next to Pluto, it also has the most eccentric orbit (e = 0.2056).
As Mercury is observed to move, it never gets very far from the sun, as seen from the earth. It moves out a certain distance from the sun and then moves back, passing either in front of or behind the sun, before moving out on the other side. When Mercury reaches its greatest distance from the sun as seen from the earth, it is also at its greatest angle from the sun. This angle is called its greatest elongation and is further described as an angle east or west of the sun. The greatest elongations are show below in Figure 1.
Figure 1. Elongations of Mercury
Standing on the earth facing south and looking at Mercury and the Sun in the sky, if Mercury is seen east of the sun, the elongation is an eastern angle (ex. ÐGEE = 20^{o}E). For Mercury west of the sun, the elongation is a western angle (ex. ÐGWE = 29^{o}W).
Notice that when Mercury is at either greatest elongation, the line from earth to Mercury forms a right angle with the line from Mercury to the sun. If the line from the earth to the sun is drawn in, it is seen that the earth, Mercury and the sun are at the vertices of a right triangle. Hence, no matter where the earth is in its orbit, if
Mercury is observed at greatest elongation, the sun, earth and Mercury will always form a right triangle, with Mercury located at the right angle.
Thus, if Mercury is observed at many greatest elongations, its orbit can be determined by repeatedly using the sun-earth-Mercury right triangles to calculate the distance of Mercury from the sun and plotting these distances versus some angle, like longitude.
We will assume in this analysis that the earth’s orbit is circular (e = 0.0), with a constant radius of 1.0 astronomical units (AU) and centered on the sun. This is not a bad approximation since the eccentricity of the earth’s orbit is e = 0.0167, the second smallest of all the planets except for Neptune (e = 0.0086).
Laboratory Procedure – Kepler’s 1^{st} Law (Law of Ellipses)
We will use the observations of Mercury’s greatest elongations (both eastern and western) listed in Table 1, which were made during the years 1967 to 1969.
Table 1. Mercury’s Greatest Elongations for 1967 – 1969
Date |
Greatest Elongation Ð (^{o}) (E = East, W = West) |
1967 February 16 |
18E |
March 31 |
28W |
June 12 |
25E |
July 30 |
20W |
October 9 |
25E |
November 18 |
19W |
1968 January 31 |
18E |
March 13 |
27W |
May 24 |
23E |
July 11 |
21W |
September 20 |
26E |
October 31 |
18W |
1969 January 13 |
18E |
February 23 |
26W |
May 6 |
21E |
June 23 |
23W |
September 3 |
27E |
October 15 |
18W |
December 28 |
19E |
Start Excel and create a new workbook with two worksheets. Select worksheet 1 and rename it "Mercury's Orbit". Rename worksheet 2 "Circle". Save the workbook on your lab disk. Now, study the earth-Mercury-sun right triangle below in Figure 2, which shows the relative position of Mercury at the first elongation in Table 1, where ÐGEE = 18^{o}E = -18^{o}.
Figure 2. Mercury at ÐGEE = 18^{o}E = -18^{o}
Clearly, the distance of Mercury from the Sun, d, is given by the formula d = 1 AU x sin(ÐGEE) = 1 AU x sin(18^{o}) = 0.31 AU
Month |
Day |
Year |
Year Day Number |
Leap Years (1901) |
Julian Day Number |
Greatest Elongation (^{o}) |
Distance of Mercury from Sun (AU) |
Mercury's Longitude(^{o}) |
x (AU) |
y (AU) |
|||||||||
2 |
16 |
67 |
47 |
16 |
2439538 |
-18 |
0.31 |
-72.00 |
0.10 |
-0.29 |
|||||||||
3 |
31 |
67 |
90 |
16 |
2439581 |
28 |
0.47 |
104.38 |
-0.12 |
0.45 |
|||||||||
6 |
12 |
67 |
163 |
16 |
2439654 |
-25 |
0.42 |
49.33 |
0.28 |
0.32 |
|||||||||
7 |
30 |
67 |
211 |
16 |
2439702 |
20 |
0.34 |
231.64 |
-0.21 |
-0.27 |
|||||||||
10 |
9 |
67 |
282 |
16 |
2439773 |
-25 |
0.42 |
166.62 |
-0.41 |
0.10 |
|||||||||
11 |
18 |
67 |
322 |
16 |
2439813 |
19 |
0.33 |
342.05 |
0.31 |
-0.10 |
|||||||||
1 |
31 |
68 |
31 |
16 |
2439887 |
-18 |
0.31 |
271.98 |
0.01 |
-0.31 |
|||||||||
3 |
13 |
68 |
72 |
16 |
2439928 |
27 |
0.45 |
447.39 |
0.02 |
0.45 |
|||||||||
5 |
24 |
68 |
144 |
16 |
2440000 |
-23 |
0.39 |
388.36 |
0.34 |
0.19 |
|||||||||
7 |
11 |
68 |
192 |
16 |
2440048 |
21 |
0.36 |
571.67 |
-0.31 |
-0.19 |
|||||||||
9 |
20 |
68 |
263 |
16 |
2440119 |
-26 |
0.44 |
508.65 |
-0.37 |
0.23 |
|||||||||
10 |
31 |
68 |
304 |
16 |
2440160 |
18 |
0.31 |
685.06 |
0.25 |
-0.18 |
|||||||||
1 |
31 |
69 |
31 |
17 |
2440253 |
-18 |
0.31 |
632.72 |
0.01 |
-0.31 |
|||||||||
2 |
23 |
69 |
54 |
17 |
2440276 |
26 |
0.44 |
791.39 |
0.14 |
0.42 |
|||||||||
5 |
6 |
69 |
126 |
17 |
2440348 |
-21 |
0.36 |
729.36 |
0.35 |
0.06 |
|||||||||
6 |
23 |
69 |
174 |
17 |
2440396 |
23 |
0.39 |
912.67 |
-0.38 |
-0.09 |
|||||||||
9 |
3 |
69 |
246 |
17 |
2440468 |
-27 |
0.45 |
853.63 |
-0.31 |
0.33 |
|||||||||
10 |
15 |
69 |
288 |
17 |
2440510 |
18 |
0.31 |
1030.03 |
0.20 |
-0.24 |
|||||||||
12 |
28 |
69 |
362 |
17 |
2440584 |
-19 |
0.33 |
959.97 |
-0.16 |
-0.28 |
|||||||||
Sun |
0.00 |
0.00 |
Figure 3. The completed 'Mercury's Orbit' worksheet.
We will now build an Excel table in the 'Mercury's Orbit' worksheet which displays the calculated positions of Mercury around the sun during 1967 - 1969. Follow the cell instructions below to complete the worksheet seen in Figure 3.
Cell(s) Address Cell(s) Content: Number, Text, or =Formula or (Instruction)
(Do NOT enter text after ";" as these are comment statements)
A1 Month B1 Day C1 Year
D1 (Wrap Text) Year Day Number
E1 (Wrap Text) Leap Years (1901)
F1 (Wrap Text) Julian Day Number
G1 (Wrap Text) Greatest Elongation (^{o})
H1 (Wrap Text) Distance of Mercury from Sun (AU)
I1 (Wrap Text) Mercury's Longitude (^{o})
J1 x (AU) K1 y (AU) I21 Sun
A2:A20 (Fill in months from Table 1)
B2:B20 (Fill in day of month from Table 1)
C2:C20 (Fill in year as two digit number from Table 1)
D2 =INT(275*(A2/9)) - 2*INT((A2+9)/12) + B2 - 30 ; Day of year
D2:D20 (Fill Down)
E2 =INT((C2-1)/4) ; Leap years since 1901
E2:E20 (Fill Down)
F2 =2415020 + 365*(C2) + D2 + E2 - 0.5 ; Julian date
F2:F20 (Fill Down)
G2:G20 (Fill in elongation angles from Table 1. GEE = negative angle, GWE = positive angle)
H2 =SIN(ABS(G2*PI()/180))
H2:H20 (Fill down) ; Mercury's heliocentric distances
I2 =IF(G20, (((F2-$F$2)/365.25)*360) + (90-ABS(G2)),(((F2-$F$2)/365.25)*360) - (90-ABS(G2))) ; Mercury's heliocentric longitudes
I2:I20 (Fill down)
J2 =H2*COS(I2*PI()/180)
J2:J20 (Fill down)
K2 =H2*SIN(I2*PI()/180)
K2:K20 (Fill down)
J21 0.00 K21 0.00 ; Sun's position
We can now plot Mercury’s orbit, using the Chart Wizard tool in Excel. Select the Chart Wizard tool and from the window that pops up select the following:
Chart Type: XY (Scatter) (No point connection)
Chart Source Data: Use the Add command to plot two data series.
Series 1: Name: Mercury
X Values: =Mercury's Orbit!$J2:$J$20
Y Values: =Mercury's Orbit!$K2:$K$20
Series 2: Name: Sun
X Values: =Mercury's Orbit!$J21
Y Values: =Mercury's Orbit!$K21
Chart Options: Title: Mercury
x-axis: x(AU)
y-axis: y(AU)
Axes: Value (x) axis
Value (y) axis
Gridlines: None
Legend: None
Chart Location: As object in Mercury's Orbit worksheet.
Figure 4. Mercury’s Orbit with sun's (_{*}) position
After your chart appears, double click in the area of the plot itself and change the fill effects to "none" to get rid of the gray background. Your chart should appear similar to Figure 4. We can now attempt to fit various closed figures to Mercury's orbit, like circles and ellipses, as did Kepler studying the orbit of Mars. We will first approximate the shape of Mercury’s orbit by an off-centered circle of radius a and center at (x, y) = (h, k), which has cartesian (x, y) and polar (r,q)expressions respectively of:
Select the second worksheet called 'Circle'. Now follow the cell instructions below to complete the second worksheet seen in Figure 5.
Cell(s) Address Cell(s) Content: Number, Text, or =Formula or (Instruction)
(Do NOT enter text after ";" as these are comment statements)
A1 Model Longitude q (^{o}) B1 Model Distance r (AU)
C1 x (AU) D1 y (AU)
E1:F1 (Merge Cells) (x - h)^{2} + (y - k)^{2} = a^{2}
G1 Units
A2 0.00
A3 =A2+10
A3:A38 (Fill Down)
B2=($F$2*COS(A2*PI()/180) + $F$3*SIN(A2*PI()/180)) + SQRT(($F$2*COS(A2*PI()/180) + $F$3*SIN(A2*PI()/180))^2 - $F$2^2-$F$3^2+$F$4^2)
B2:B38 (Fill Down)
C2 =B2*COS(A2*PI()/180)
C2:C38 (Fill Down)
D2 =B2*SIN(A2*PI()/180)
D2:D38 (Fill Down)
E2 h = E3 k = E4 a =
E5 Position E6 Center
F2 0.00 F3 0.00 F4 0.00
F5 x F6 0.00
G2 AU G3 AU G4 AU
G5 y G6 0.00
To plot the orbit of Mercury with the off centered circle go to the 'Mercury's Orbit' worksheet and click on the chart. Copy it and paste it into the 'Circle' worksheet. From the Chart menu pick the Source Data … option. In the Source Data window use the Add command to add two more data series.
Series 3: Name: Model
X Values: =Circle!$C$2:$C$38
Y Values: =Circle!$D$2:$D$38
(Double click on one of the model data points in the chart to bring up the Format Data Series window. From the Patterns tab menu, set the following attributes.)
Line: Automatic Color: Black
Marker: None
Series 4: Name: Center
X Values: =Circle!$F$6
Y Values: =Circle!$G$6
(Double click on the center data point in the chart to bring up the Format Data Series window. From the Patterns tab menu, set the following attributes.)
Line: None
Marker: Style + Foreground: Black Background: White
Now, you have to pick some values of h, k, and a and enter them in cells F2, F3, and F4 respectively to try fitting different circles to Mercury’s orbit. When you have discovered the best values of h, k, and a, record your values below:
h = __________________
k = __________________
a = __________________
Hint: to help find values of h, k, and a, use the facts that -1.0 < h < 0.0, 0.0 < k < 1.0 and 0.0 < a < 1.0. Good values will produce an orbit close to the one seen in Figure 5.
Figure 5. The completed Circle worksheet (only shown through rows 1 - 31). Notice the calculated orbit (solid line), the center of the circle (+), and the position of the sun (_{*}) now appear separately.
Now make a copy of the 'Circle' worksheet and put it after 'Circle'. Rename it 'Ellipse'. We will fit Mercury's orbit with an ellipse, to compare it with our off centered circle fit. An ellipse with a semi major axis a greater than its semi minor axis b and center at (x, y) = (h, k) has cartesian (x, y) and polar (r, q) expressions respectively of:
So follow the instructions below to modify specific cell instructions and produce the completed worksheet seen in Figure 6.
Cell(s) Address Cell(s) Content: Number, Text, or =Formula or (Instruction)
(Do NOT enter text after ";" as these are comment statements)
E1:F1 (Merge Cells) r = a(1-e^{2})/(1 + ecosq)
A2 0.00
A3 =A2+1
A3:A362 (Fill Down)
B2 =$F$2*(1 - $F$4^2)/(1 + $F$4*COS(A3*PI()/180))
B2:B362 (Fill Down)
C2 =$F$5+B2*COS(A2*PI()/180)
C2:C362 (Fill Down)
D2 = $F$6+B2*SIN(A2*PI()/180)
D2:D362 (Fill Down)
E2 a = E3 b = E4 e =
E5 x_{o} = E6 y_{o} = E7 Position
E8 Center E9 Sun (Focus 1) E10 Focus 2
F2 0.00 F3 0.00 F4 =SQRT(F2^2-F3^2)/F2
F5 0.04 F6 0.08 F7 x
F8 =F5 - SQRT(F2^2-F3^2) F9 =F5
F10 =F5 - 2*SQRT(F2^2-F3^2)
G2 AU G3 AU G5 AU G6 AU
G7 y G8 =F6 G9 =F6 G10 =F6
To plot the orbit of Mercury with the ellipse, click on the chart and from the Chart menu pick the Source Data … option. In the Source Data window use the Add command to modify and add the following data series.
Series 1: Name: Mercury
X Values: =Mercury's Orbit!$J$2:$J$20
Y Values: =Mercury's Orbit!$K$2:$K$20
Series 2: Name: Model
X Values: =Ellipse!$C$2:$C$362
Y Values: =Ellipse!$D$2:$D$362
Series 3: Name: Sun
X Values: =Ellipse!$F$9
Y Values: =Ellipse!$G$9
Series 4: Name: Focus 2
X Values: =Ellipse!$F$10
Y Values: =Ellipse!$G$10
Series 5: Name: Center
X Values: =Ellipse!$F$8
Y Values: =Ellipse!$G$8
Figure 6. The completed 'Ellipse' worksheet (only shown from rows 1 - 31). Notice the calculated orbit (solid line), the center of the ellipse (x), and the two foci (one is now the position of the sun and the other just marks an empty point in space).
Now, you have to pick some values of the semi major axis a and the semi minor axis b and enter them in cells F2 and F3 respectively, to try fitting different ellipses to Mercury’s orbit. When you have discovered the best values of a and b, record your values below (and the eccentricity, e):
a = __________________
b = __________________
e = __________________
Hint: to help find values of a and b use the facts that 0.00 < a < 1.00 and 0.00 < b < 1.00 as well as a b. Good values will produce an orbit close to the one seen in Figure 6.
Questions and Exercises
1.) How would you compare the fit of Mercury’s orbit by an off-centered circle as compared to a true ellipse? Which is better or worse?