Martian Project
Calculus I
Spirit and Opportunity Mars rovers send pictures home from Mars
On Christmas Day, 1642, the year Galileo died, there was born a male infant so tiny that, as his
mother told him in later years, he might have been put into a quart mug, and so frail that he had
to wear a bolster around his neck to support his head. This unfortunate creature was entered in
the parish register as “Isaac sonne of Isaac and Hanna Newton”. There is no record that the wise
men honored the occasion, yet this child was to alter the thought and habit of the world.
James Newman
Introduction:
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
the Figure at the end of this handout):
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 times.
His second law, simply said, means that planets slow down when they are farther from the Sun,
and speed up when they are closer. Since the line joining the Sun to the planet is shorter when
the planet is closer, the length of the orbit covered by the planet in a given interval of time would
be larger to make the areas swept equal.
Kepler did not have the physics or the mathematical tools to prove his own discovery, and it was
left for the genius of Sir Isaac Newton to do that, in 1665, using his second law of motion ( F =
ma). The 23-year old was a student at the University of Cambridge when an outbreak of the
Plague forced the university to close down for 2 years. Those two years were to be the most
creative in Newton’s life. He conceived the law of gravitation, the laws of motion, differential
calculus, and the proof of Kepler’s Laws.
Mathematics of Orbits:
An ellipse is described by the length of the semi-major axis a , and the length of the semi-minor
axis b ( refer to the Refresher on Parametric Equations sheet at the end of this handout). The
ellipse’s eccentricity, the measure of its elongation, is e and is given by:
e = 1−
b2
.
a2
This relationship can be solved for b to give:
b = a 1 − e2 .
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 .
Note that when a = b , we have e = 0, and the ellipse is a circle. Our planets have eccentricities
of 0.009 (Neptune) to 0.206 (Mercury).
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 project, 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 center of
the ellipse is then at ( − a ⋅ e, 0) . We will also let time t = 0 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( op ) − a ⋅ e


 y (t ) = b ⋅ sin( 2π t )

op
or:
2π t

 x(t ) = a ⋅ cos( op ) − a ⋅ e

2π t
 y (t ) = a 1 − e 2 ⋅ sin(
)

op
(1)
Where op is the orbital period in Earth years.
Note that when e = 0 , the above equations turns into the parametric equations of a circle with
center at the origin and radius equal to a.
Although equation (1) models the shape of the orbit correctly, it 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 < 0.2 :

 2π t
2π t 
+ e ⋅ sin(
) − a⋅e
 x(t ) = a ⋅ cos 
op 

 op

 y (t ) = a 1 − e 2 sin  2π t + e ⋅ sin( 2π t )
 op

op 

(2)
Equations (1) and (2) give the position of a planet as a function of time in years. The x- and ycomponents of orbital velocity are given by:
d x(t )

 v x (t ) = dt

d y (t )
v y (t ) =
dt

(3)
And finally the orbital velocity as a function of time is given by Pythagoras’s Theorem.
v(t ) = v 2 x (t ) + v 2 y (t ) .
(4)
The orbital constants for Mars are given in the following table:
Semi-major axis in (AU)
a
1.524
Eccentricity
e
0.0934
Orbital Period (years)
op
1.88
AU is an Astronomical Unit, which is Earth’s semi-major axis (the mean distance from the Sun
to Earth), and is about 93 million miles.
The Project:
Your task in this project is to calculate the location and the orbital velocity of Mars for the
simple (and inaccurate) model given by equation (1), and the better approximation model given
by equation (2). You will make a table and plot the orbital velocity for 1.88 year (one Martian
year) for the two models and will compare them. Use three decimal places in all your
numerical results. Here are the steps you can take to arrive at the result:
A) The simple model:
0) Calculate the average orbital velocity of Mars by noting that Mars travels the circumference
of its elliptic orbit in 1.88 year. The following is a simple approximate equation for
circumference of an ellipse (there is no simple exact formula):
a2 + b2
C ≅ 2π
2
Average orbital velocity is then this distance C divided by time op for Mars. The units will be
AU/y. All your calculations for the instantaneous velocity in the following steps should orbit
this average velocity.
1) Find the x and y locations of Mars for time increments of 0.188 year from t = 0 to t = 1.88 for
the simple model of equation (1). You should have 11 points. Make a graph of the elliptic orbit
and indicate the locations of Mars for the 11 time calculations with times indicated on each
point.
2) Find v x (t ) and v y (t ) for model (1) in terms of a, e, and op . Do not plug in numerical
constants at this time. You should find the derivatives by hand using the derivative rules we
have learned in this class. Write a statement here for each step describing how you found the
derivative by using the derivative rules. For example:
[
[
]
2
3
d
c ⋅ e kx + d ⋅ e qx
dx
2
3
d
d
=
c ⋅ e kx +
d ⋅ e qx
dx
dx
2
3 d
d
= c ⋅ e kx ⋅
kx 2 + d ⋅ e qx
qx 3
dx
dx
= ⋅⋅⋅
g ( x) =
]
[ ]
[
]
[ ]
derivative of sum rule
multiplicative constant, and chain rules
3) Find v(t ) for model (1) using equation (4) in terms of the constants a, e, and op . You should
be able to simplify this expression greatly using trigonometric identities.
4) Plug in values for the constants a, e, and op in v(t ) and find numerical values for orbital
velocity for the time increments mentioned in step 1). Tabulate and graph this function.
B) The more accurate model:
5) Find the x and y locations of Mars for time increments of 0.188 year from t = 0 to
t = 1.88 for the more accurate model of equation (2). You should have 11 points. Make a graph
of the elliptic orbit and indicate the locations of Mars for the 11 time calculations with times
indicated on each point.
6) Find v x (t ) and v y (t ) for model (2) in terms of a, e, and op . Do not plug in numerical
constants at this time. You should find the derivatives by hand using the derivative rules we
have learned in this class. Write a one line statement here for each step describing how you
found the derivative by using the derivative rules as in step 2) above.
7) The expressions in step 6) will be too complicated to find v(t ) for this model as we did in
step 3). To find v(t ) for model 2), plug the constants a, e, and op into equations for v x (t ) and
v y (t ) , and find numerical values for each velocity component with the same time increments as
in step 1), and then find velocity using equation (4).
8) Calculate the orbital velocity now by using equation 4) for every time data point you have for
the components of velocity in step 6). Tabulate and graph this function.
Your Report
Present all the mathematics and the calculations for both models.
Present the locations and the orbital velocities for each model in separate tables. Each table
should have four columns (for t, x, y, and v).
Make a graph of orbital velocity as a function of time for each model. Choose a scale that will
show the differences in velocities well.
Finally make separate graphs of the elliptic orbit and indicate the locations of Mars for the 11
time calculations with time and orbital velocity indicated for each point. Make sure that the
graphs are large enough to cover one whole graph paper each.
Your report should then have two tables with 4 columns each, two ellipses with location of Mars
and its velocity indicated on these points, and two graphs of velocity vs. time.
Your report should be complete and easy to understand by a mathematician who has not
seen this handout and has not been to our class.
Your report should include:
I) A cover sheet.
II) A short and complete statement of the problem in your own words. Do not attach any
part of this handout to your report.
III) All your calculations.
IV) All the graphs and tables.
V) A short conclusion of what this project has contributed to your cosmic consciousness.
Refresher on Parametric Equations of Conic Sections:
Parametric equation of a circle r = a center Parametric equation of an ellipse, major
axis 2a, minor axis 2b, center at (0,0),
at (0,0), period 2π :
period 2π :
 x(t ) = a cos(t )

 x(t ) = a cos(t )
 y (t ) = a sin(t )

 y (t ) = b sin(t )
As above, but shift center to (h, k ) :
As above, but shift center to (h, k ) :
 x(t ) = a cos(t ) + h
 x(t ) = a cos(t ) + h


 y (t ) = a sin(t ) + k
 y (t ) = b 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