The Mercury Project
Calculus II
Einstein’s theory of general relativity showed why Mercury’s perihelion shifts very
slowly around the sun. This was a powerful factor motivating the adoption of general
relativity.
This term we will study the orbit of Mercury, its position as a function of time, and
Kepler’s Second Law of planetary motion. I will hand you weekly problems, which I call
m problems. You will hand these problems back to me, they will be graded, and handed
back to you. You will collect these problems and will summarize the results at the end of
the term in a project report.
23
Problem m0
Johann Kepler in 1609 discovered that planets orbit the Sun in elliptic orbits, and that
their orbital velocity is not constant but varies. The following summarizes Kepler’s first
two laws (See Figure):
1) The planets orbit the Sun in Elliptic orbits with the Sun at one of the focal points.
2) The line joining the Sun to a planet sweeps out equal areas in equal time.
His second law, simply said, means that planets slow down when they are farther from
the Sun, and speed up when they are closer.
y
Planet moves faster
Planet moves slower
x
2b
a.e
Aphelion
Perihelion
2a
The ellipse’s semi-major axis is a, while the semi-minor axis is b. The eccentricity, the
measure of its elongation, is e and is given by e = 1 − b 2 a 2 , which can be solved for b
to give b = a 1 − e 2 . Eccentricity is between 0 and 1. For a circular orbit e = 0 , and for
a very elongated orbit e is close to 1. The distance from the center of the ellipse to either
focal point is a ⋅ e .
Orbital values for the planet Mercury are:
a = 0.387
e = 0.206,
AU ,
b = a 1− e2
op = 0.241
AU ,
years
AU is astronomical unit, which is the average distance from the Sun the Earth.
The area of an Ellipse is given by
A = π ab .
24
In this, and all the subsequent m problems, please round your answers to four decimal
places, unless otherwise mentioned, and include units for the results, where applicable.
Find the area of the orbital ellipse of Mercury:
A =……………………………………………
Every 1/20th of the orbital period (op/20), the line from the Sun to Mercury sweeps
exactly 1/20th of the area A you found above. This is true regardless of where Mercury is.
Fill in the table the areas swept by the line from the Sun to Mercury. These should be all
the same numbers, and equal to 1/20th of the area you found above.
Exact Area swept in 1/20th of op
Time interval
t = 0 to t =
op
.
20
………………………
t=
4op
5 op
to t =
20
20
……………………….
t=
10 op
11 op
to t =
20
20
………………………
Note: In order to make calculations in the m problems easier with your calculator or
MAPLE program, it is essential to store the formulas with variable names, and then store
all the numerical values into variable names, before you attempt to evaluate the formulas
in these problems.
 4πβ 
ab  4π ( β − α )

Here is an example you will see in m5. Calculate A1 =
− sin 

4 
op
 op 
for α = 0, β = op / 20 and also for α = 4op / 20, β = 5op / 20 :
> A1:=abs(a*b/4*(4*Pi*(bet-alp)/op1-sin(4*Pi*bet/op1)));
1
π ( bet − alp )
π bet  
A1 := a b  4
− sin 4
 
4
op1

 op1  
> a:=0.387; e:=0.206; b:=a*sqrt(1-e^2); op1:=0.241; alp:=0; bet:=op1/20;
area:=evalf(A1);
a := 0.3870 e := 0.2060 b := 0.3787 op1 := 0.2410 alp := 0 bet := 0.0121
area := 0.0015
> alp:=4*op1/20;bet:=5*op1/20;
area:=evalf(A1);
alp := 0.0482 bet := 0.0603
area := 0.0230
25
Problem m1
Refer to the back of this m1 handout for a refresher on parametric equations of conic
sections.
a) Write the implicit equation of a circle with radius a centered at the origin.
…………………………………………………….
b) Write the parametric equation of a circle with radius a centered at the origin with
parameter t , and a period of 2π . Your answer will involve sine and cosine functions.






c) Write the parametric equation of a circular orbit with radius a centered at (h, k ) with
parameter t , and a period of 2π . The planet’s position at t = 0 should be at (h + a, k )






d) Find the location of this planet, in exact form, at:
t=0 :
..................................................................
t =π /4 :
..................................................................
t =π /2 :
..................................................................
t =π :
..................................................................
t = 3π / 2 :
..................................................................
t = 2π :
..................................................................
26
e) Write the parametric equation of a circular orbit with radius a centered at the origin
with parameter t , and an orbital period of op . Your answer will involve sine and
cosine functions.






f) Write the parametric equation of an elliptic orbit with major axis 2a along the x-axis,
minor axis 2b along the y-axis. The ellipse is centered at (0 , 0) with parameter t , and
an orbital period op . The planet’s position at t = 0 should be at (a , 0)






g) Shift the ellipse in f) left so that the origin is at the right focal point. Note that the
distance from center to each focal point is a ⋅ e , where e is the eccentricity of the ellipse.
Write the equation for this orbit. Your equations should be in terms of a, b, e and op :






h) The orbit of Mercury has the following values:
a = 0.387
AU ,
b = a 1− e2
AU
e = 0.206
op = 0.241
years
AU is an “astronomical unit” which is the average distance from the Sun to the Earth ( a
for Earth). If Mercury is at perihelion at t = 0 , find the location of this planet at the
given times below. Put your answer in ordered pairs. Perihelion is when the planet is
closest to the Sun (for our problem this is (a − a ⋅ e , 0) )
27
t=0 :
..................................................................
op
:
..................................................................
20
4op
=
:
..................................................................
20
5 op
=
:
..................................................................
20
10 op
=
: ..................................................................
20
11 op
=
:
..................................................................
20
t=
t
t
t
t
i) Graph the elliptic orbit and locate the above locations on your graph, and attach your
graph. Use a graphing software such as GRAPH, WINPLOT, or MAPLE, with a window
of − 0.5 AU to 0.5 AU in both directions, and a scale of 0.1 AU . Connect the origin
op
to the above points and shade the three slices, one from t = 0 to t =
, one from
20
4op
5op
10op
11op
t=
to t =
, and one from t =
to t =
.
20
20
20
20
This is an example of how you can plot the orbit of a planet and place the planet's positions on the orbit using MAPLE.
For this example a=1.5 AU, b=1.2 AU, op=3 years, e=0.6. The two locations were calculated for t = 0.15 year and
t = 0.25 year.
> with(plots):
> f:=t->a*cos(2*Pi*t/op1)-a*e; g:=t->b*sin(2*Pi*t/op1);
πt 
f := t → a cos 2
 − a e
 op1 
πt 
g := t → b sin 2

op1


> a:=1.5: b:=1.2: e:=0.6: op1:=3:
> p1:=plot([f(t),g(t),t=0..3],x=-3..3,y=-2..2,scaling=CONSTRAINED,
xtickmarks=[-1,1],ytickmarks=[-1,1]):
p2:=pointplot({[f(.15),g(.15)],[f(.25),g(.25)]},symbol=CIRCLE,
color=black,scaling=CONSTRAINED):
display({p1,p2});
28
Refresher on Parametric Equations of Conic Sections:
Parametric equation of a circle r = a ,
center at (0,0), period 2π :
 x(t ) = a cos(t )

 y (t ) = a sin(t )
Parametric equation of an ellipse, major
axis 2a, minor axis 2b, center at (0,0),
period 2π :
 x(t ) = a cos(t )

 y (t ) = b sin(t )
As above, but shift center to (h, k ) :
 x(t ) = a cos(t ) + h

 y (t ) = b sin(t ) + k
As above, but shift center to (h, k ) :
 x(t ) = a cos(t ) + h

 y (t ) = a sin(t ) + k
As above, but change period to B
2π t

x
(
t
)
=
a
cos(
)+h

B

2π t
 y (t ) = a sin(
)+k
B

As above, but change period to B
2π t

x
(
t
)
=
a
cos(
)+h

B

2π t
 y (t ) = b sin(
)+k
B

Parametric equation of an ellipse, major
axis 2a , minor axis 2b , eccentricity e ,
center at (−a ⋅ e, 0)
2π t

 x(t ) = a cos( B ) − a ⋅ e

2π t
 y (t ) = a 1 − e 2 sin(
)
B

y
Planet moves faster
Planet moves slower
x
2b
a.e
Aphelion
Perihelion
2a
29
Problem m2
The following figure shows the orbit of a planet around the Sun. The point of the orbit
closest to the Sun is called perihelion, and the point farthest is called aphelion. To
simplify the calculations for this problem, without loss of generality, we will place the
origin at the focal point where the Sun resides, the x-axis along the major axis. The length
of the major axis is 2a , and that of the minor axis is 2b . The center of the ellipse is then
at (− a ⋅ e, 0) . We will also let time t equal zero when and the planet is at perihelion.
With these assumptions, the parametric equations of the orbit of a planet are:
2π t

 x(t ) = a cos( B ) − a ⋅ e

2π t
 y (t ) = b sin(
)
B

(1)
Where op is the orbital period in Earth years, and b = a 1 − e 2 .
y
Planet moves faster
Planet moves slower
x
2b
a.e
Aphelion
Perihelion
2a
Note that when e = 0 , then b = a , and the above equations turns into the parametric
equations of a circle with center at the origin.
Equation (1) does not account for Kepler’s second law (In fact it has total disregard for
orbital velocity). To account for that, we can add a term to the arguments of the cosine
and sine functions. This is an approximation to an otherwise difficult problem, but is a
very good one for e ≤ 0.21 :
30

 2π t
2π t 
+ e ⋅ sin(
) − a⋅e
 x(t ) = a ⋅ cos 
op 

 op

 y (t ) = b ⋅ sin  2π t + e ⋅ sin( 2π t )
 op

op 


(2)
Equations (1) and (2) (Models 1 and 2) give the position of a planet as a function of time
in years. The values of a, b, e, and op for Mercury are given in problem m1.
Find the locations for Mercury for the following times for the two models above. You
calculated the first model’s locations in problem m1, and can just copy them here. Refer
to the Note in m0 to make your calculations easier.
Model 1
t=0 :
Model 2
t=0 :
...............................................
op
:
.................................................
20
4op
=
:
................................................
20
5 op
=
:
................................................
20
10 op
=
: ................................................
20
11 op
=
:
................................................
20
op
:
.................................................
20
4op
=
:
................................................
20
5 op
=
:
................................................
20
10 op
=
: ................................................
20
11 op
=
:
................................................
20
t=
t=
t
t
t
t
t
t
t
t
...............................................
Graph the elliptic orbit and locate the planet locations for model 2, as you did for model 1
in problem m1, with the same viewing window and scales. Connect the Sun to the above
op
points and shade the three slices, one from t = 0 to t =
, one from
20
4op
5op
10op
11op
t=
to t =
, and one from t =
to t =
.
20
20
20
20
31
Problem m3
In problems m3 through m6 we will work on finding the area swept by a line connecting
the Sun to a planet using geometry and integral calculus. The graph in Fig. 1 is given by
the parametric equation:
 x = f (t )

 y = g (t )
Fig. 1
Fig 2
A
O
Fig 3
C
Fig 4
t=β
B
B
A
O
O
D
D
C
a) Find the area OCA in Fig 2 in terms of f , g and α only.
Area OCA = …………………………………………....
32
t =α
b) Find the area of the triangle ODB in Fig. 3 in terms of f , g and β .
Area ODB = …..…………………………………………
c) If the area DCAB in Fig. 4 is A1, find the area A of the slice OAB in terms of
f , g , α , β and A1 (think of adding and subtracting areas of triangles to A1).
Area OAB = …………………………………………….………………………. Eq. (1)
The above equation gives the area swept by a line connecting the Sun to a planet, if the
functions f (t ) and g (t ) are the parametric equations for the orbit of that planet. We will
program this equation for Mercury to find areas swept in problem m4.
33
Problem m4
A planet’s elliptic orbit has major axis 2a along the x-axis, minor axis 2b along the yaxis, eccentricity e , orbital period op , and the Sun at the right focal point and the planet
at perihelion at t = 0 . There are two models that predict the position of this planet.
Model 1:
2π t

 x(t ) = a ⋅ cos( op ) − a ⋅ e


 y (t ) = b ⋅ sin( 2π t )

op
(1)
Model 2:

 2π t
2π t 
+ e ⋅ sin(
) − a⋅e
 x(t ) = a ⋅ cos 
op 

 op

 y (t ) = b ⋅ sin  2π t + e ⋅ sin( 2π t )
 op

op 


(2)
Write the formula for the area A swept by the line connecting the Sun to the planet from
t = α to t = β you found in m3 (Eq. (1) in m3). We will call it AS for area swept.
AS = ………………………………………………………………………..
Program this equation in MAPLE, or your calculator, to find the area swept for the two
models, as follows. First note that MAPLE is case-sensitive, while your TI calculator
may not be. Following the note in m0, let f 1(t ) and g1(t ) equal to x(t ) and y (t )
functions for model 1, and f 2(t ) and g 2(t ) be equal to x(t ) and y (t ) functions for
model 2. Use function notation for these functions.
Let the area swept be called ASM1 and ASM2 (for area swept model 1, and area swept
model 2). Let the area DCAB in m3, which we called A1 , be called A1M1 and A1M2
(for A1 model 1, and A1 model 2). These are expressions, not functions. Use alp and bet
for alpha and beta.
Do not declare any numeric values for any constants or variables at this stage.
34
Your program in MAPLE will look like this (note that in some versions of MAPLE op is
reserved, so call it op1):
> restart; interface(displayprecision = 4): Digits := 20:
> f1:=t->a*cos(2*Pi*t/op1)-a*e;
g1:=t->b*sin(2*Pi*t/op1);
f2:=t->a*cos(2*Pi*t/op1+e*sin(2*Pi*t/op1))-a*e;
g2:=t->b*sin(2*Pi*t/op1+e*sin(2*Pi*t/op1));
> ASM1:=A1M1+ .......;
ASM2:=A1M2+ .......;
Your calculator functions and expressions will look like this ( sto → is the store key):
a ∗ cos(2π t / op ) − a ∗ e sto → f 1(t )
b ∗ sin(2π t / op ) sto → g1(t )
a1m1 + sto → asm1
a1m2 + sto → asm2
Save this MAPLE program, or functions and expressions in your calculator. We will find
formulas for A1 (A1M1, and A1M2) for the two models in m5, and will input values for
the variables including alpha and beta in m6.
35
Problem m5
In problem m3 you wrote the formula for the areas swept by a line from the Sun to a
planet. Model 1 is simple but inaccurate, model 2 is more complicated but more accurate.
The only missing part of the equations is area A1 , which we will find in this problem .
Given a parametric curve
B
t=β
t =α
 x(t ) = f (t )
 y (t ) = g (t )

A
O
D
C
Area A1 between this curve and the x-axis from t = α to t = β (area DCAB in the figure
above), as we will see in class, is given by:
β
A1 =
∫ ydx ,
α
y = g (t ),
dx = f ′(t )dt
.
β
=
∫α g (t ) f ′(t )dt
Eq. (1)
The absolute value sign above is to insure positive areas.
36
Write the integral formulas for A1 for model 1 (that is starting with equation (1) in m4,
find f ′(t ) and then plug in f ′(t ) and g (t ) in equation (1) above, but do not integrate
here). Use chain rule to find the derivative of f (t ) , and show your steps. Pull all the
multiplicative constants out of the integral and simplify the integrand. We will integrate
this on next page.
A1 (model 1)
=……………………………………………………………………………………Eq. (2)
37
Your next task is to integrate the integral equation for A1 for model 1 (Eq. (2) you found
above) by hand. Start with equation (2) above, use the double angle identity to convert
the sine squared to a square-less cosine, and integrate using substitution. Show all your
work here. The result for A1 should have no integral sign and should be in terms of
a, b, α , β , and op and should be simplified.
A1 (Model 1):
=………………………………………………………………………………
Eq. (3)
It is not easy to find A1 for model 2 as we did for model 1. We will leave the integral
formula for A1 for model 2 as is in Eq. (1) above, but will replace f (t ) and g (t ) with
f 2(t ) and g 2(t ) .
β
A1 (Model 2)
=
∫α g
2
(t ) f 2′ (t )dt
Eq. (4)
38
Problem m6
In problem m5 you found formulas for area A1 for model 1 (Eq. (3) in m5) and for model
2 (Eq. (4) in m5). We will now find numerical values for A1 and, finally, the areas swept
by the line connecting the Sun to Mercury.
Add to your MAPLE program, or calculator functions and expressions you wrote in m4,
new lines to define A1M1 and A1M2, using equations (3) and (4) in m5. These are
expressions, not functions.
Now you can declare numerical values for a, b, e, op and α and β . You can now
change alpha and beta to change the intervals and get corresponding values for the areas.
Find numerical values for the three time intervals given in problem m0 for the
expressions for A1 for model 1 and model 2, and list the areas in the following table. The
values of the orbit of Mercury and the intervals are given in m0 and repeated here.
a = 0.387
b = a 1− e2
AU ,
AU
e = 0.206
op = 0.241
α , β = 0,
years
op
4op 5op
or
,
20
20 20
Model 1 area A1 (A1M1)
t = 0 to t =
op
:
20
t=
4op
5 op
to t =
:
20
20
t=
10 op
11 op
to t =
:
20
20
or
10op 11op
,
20
20
Model 2 area A1 (A1M2)
.........................................
t = 0 to t =
op
:
20
.....................................
t=
4op
5 op
to t =
:
20
20
....................................
t=
10 op
11 op
to t =
:
20
20
.........................................
.....................................
....................................
And finally, find the areas swept for model 1 and model 2 in the following table:
39
Model 1 area swept by line connecting the Sun
to Mercury (ASM1)
t = 0 to t =
op
:
20
t=
4op
5 op
to t =
:
20
20
t=
10 op
11 op
to t =
:
20
20
.........................................
Model 2 area swept by line connecting the Sun
to Mercury (ASM2)
t = 0 to t =
op
:
20
.....................................
t=
4op
5 op
to t =
:
20
20
....................................
t=
10 op
11 op
to t =
:
20
20
.........................................
.....................................
....................................
According to Kepler’s Second Law, the areas above must be the same, but neither of the
above models is exact. Model 2, however, should be better than model 1.
40
Problem m7
In problem m6 you found the approximate areas swept by a line from the Sun to Mercury
for two models. Model 1 is simple but inaccurate, model 2 is more complicated but more
accurate. You found the exact areas swept during these intervals (1/20th of the area of the
orbital ellipse) for Mercury in problem m0. Fill in the areas for both models from the
second table in m6 here, compare to the values in problem m0 and find the percent errors
for each interval and fill in the error columns. Note that percent error is:
% error =
approximate − exact
× 100
exact
Exact area swept in 1/20th of an orbital period (op/20) from m0 :…………………………
Model 1 area swept by line
connecting the Sun to Mercury
for time intervals :
op 

0 , 20  :


..................
 4op 5 op 
 20 , 20  :


10 op 11 op 
 20 , 20  :


...................
% error
Model 2 area swept by line
connecting the Sun to Mercury
for time intervals:
………..
op 

0 , 20  :


………..
 4op 5 op 
 20 , 20  :


..................
10 op 11 op 
 20 , 20  :


.................. ………..
41
...................
% error
……..…
……..…
.................. ……..….
Writing Your Project Report
You are now ready to present your scientific work on Kepler’s Second Law for Mercury.
Here is a guideline for your presentation for the results of problems m0 through m7.
a) Please do not attach or refer to any of the m problems in your report. Write your
report as if someone who does not know anything about the m problems, and has
never been to our class, but knows math, is reading your report. You are writing
your report for an OUTSIDER.
b) You do not need to present all the preliminary steps in m1. Present the main ideas of
the two models, planet locations for the intervals we have worked with, the area formulas,
numerical values for the areas, and the differences in the accuracy of the two models.
Present all the tables, and graphs that are relevant to understanding these main ideas.
Your report:
1) Introduction: Summarize Kepler’s Laws (m0) and the two models that we have been
working with (m2). Present the orbital values (a, b, e, op) for Mercury (m0). Summarize
what you will be doing in this project.
2) Project Report: Present the two models and planet locations you found for each
model in m1 and m2, with tables and graphs. Present the equations for the areas swept
by a line from the Sun to Mercury by starting with a figure similar to Fig. 4 in m3, and
starting with Area OAB in m3. You can then derive and present the area equations for
A1 in m5 for each model (Eqs. (3) and (4) in m5).
Present the exact area that should be swept in 1/20th of an orbital period (m0). Present
 op   4op 5op 
10op 11op 
for model 1 and
the areas swept for the periods 0, , 
,
, and 
,

20 
 20   20 20 
 20
model 2 and the errors in a table (m7).
3) Summary: Summarize the results of this project and all that you have learned.
42