Download Halley`s Comet Project Calculus III

Halley’s Comet Project
Calculus III
Comet Halley from Mount Wilson, 1986
"The diversity of the phenomena of nature is so great, and the treasures hidden in the
heavens so rich, precisely in order that the human mind shall never be lacking in fresh
nourishment."
Johannes Kepler
This term we will study Halley’s Comet, its position as a function of time, and Kepler’s
Second Law of planetary motion. I will hand you weekly problems, which I call H
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.
Halley’s Comet Project
Calculus III
This term we will study the orbit of Halley’s Comet and its position as a function of time.
I will hand you weekly problems I will call H problems. We will use power series to
estimate the locations of the comet at various times during the 76 years it takes to orbit
the Sun. You will summarize the results of these problems at the end of the term in a
project report.
Edmond Halley's Comet
In 1705 Edmnnd Halley predicted, using Newton’s newly formulated laws of motion, that
the comets seen in 1531, 1607, and 1682 are all the same comet and would return in
1758 (which was, alas, after his death). The comet did indeed return as predicted and was
later named in his honor. The average period of Halley's orbit is 76 years. Comet Halley
was visible in 1910 and again in 1986. Its next passage will be in early 2062.
Comets, like all planets, orbit the Sun in elliptic orbits, but their orbits are very eccentric
(the major axis is much larger than the minor axis). The point where the comet is closest
to the Sun is called perihelion, and the point where it is the farthest is called aphelion
(see the figure in the refresher sheet attached).
At aphelion in 1948, the comet was 35.25 AU from the Sun, while at perihelion on
February 9, 1986, it was only 0.5871 AU from the Sun. An astronomical unit (AU) is the
semi-major axis for Earth, which is about 93 million miles.
The ellipse’s semi-major axis is a, while its 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 .
We will let t = 0 designate February 1986. With this convention t = 20 is February
2006, and t = 76 is February 2062 when the comet will return to perihelion again.
The orbits of the Earth, Uranus, Neptune and Halley’s
Comet
Close up view of the orbit of Earth and Halley’s Comet
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( B ) + h

2π t
 y (t ) = a sin(
)+k
B

As above, but change period to B
2π t

 x(t ) = a cos( B ) + h

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
Problem H1
a) Write the parametric equation of a circular orbit with radius a centered at the origin
with parameter t , and an orbital period of op . The planet is at (a , 0) at t = 0 .Your
answer will involve sine and cosine functions.






b) Write the parametric equation of an elliptic orbit with major axis 2a along the xaxis, 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)






c) Shift the ellipse in b) 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
(see Refresher ). Write the equation for this orbit. Your equations should be in terms of
a, b, e and op :






d) The orbit of Halley’s Comet has the following values:
a = 19.34
AU ,
b = a 1− e2
AU
e = 0.97
op = 76
years
AU is an “astronomical unit” which is the average distance from the Sun to the Earth ( a
for Earth).
Kepler’s Law states that the line connecting the Sun to the planets or comets sweeps
equal areas in equal time. The equation in c) ignores this law and will, therefore, give the
correct orbit, but incorrect locations for Halley’s Comet. We will see in Problem H2
how to find the correct positions. If Halley’s Comet is at perihelion at t = 0 (Feb. 1986),
find the incorrect location of this planet using the equation in c) at the given times below.
Put your answer in ordered pairs ( x, y ) and use three decimal places. Perihelion is when
the planet is closest to the Sun (for our problem this is (a − a ⋅ e , 0) )
time in years
t = 0 (Feb 1986) :
Incorrect locations
..................................................................
t = 0 .5 :
..................................................................
t = 1:
..................................................................
t = 5:
..................................................................
t = 10 :
..................................................................
t = 20 (Feb 2006) :
..................................................................
t = 30 :
..................................................................
t = 40 :
..................................................................
t = 50 :
..................................................................
t = 60 :
..................................................................
t = 70 :
..................................................................
t = 75 :
t = 76 (Feb 2062) :
..................................................................
..................................................................
e) Graph the elliptic orbit and locate the above locations on your graph. Use MAPLE,
and attach your graph. 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.
> with(plots):
> f:=t->a*cos(2*Pi*t/op1)-a*e; g:=t->b*sin(2*Pi*t/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});
Problem H2
Johann Kepler in 1609 discovered that planets and comets 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. Since the line joining the Sun to the planet is
shorter when the planet is closer, the length of the orbit traveled by the planet in a given
interval of time would be larger to make the areas swept equal.
y
Planet moves faster
Planet moves slower
x
2b
a.e
Aphelion
Perihelion
2a
For a circular orbit the eccentricity e is zero, but as the orbit gets more eccentric
(elongated), e approaches 1. 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 (0 , − a ⋅ e) . 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( op ) − a ⋅ e


 y (t ) = b ⋅ sin( 2π t )

op
or:
2πt

 x(t ) = a ⋅ cos( op ) − a ⋅ e

2πt
 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.
Equation (1) does not account for Kepler’s Second Law (It assumes an almost constant
velocity). To account for that Kepler developed the following equation called Kepler’s
Equation:
2π t
= E − e ⋅ sin( E )
op
(2)
For a given time t , you first solve for E from (2) and then plug E in equation (1) instead
2π t
:
of
op
 x(t ) = a ⋅ cos( E ) − a ⋅ e

2
 y (t ) = a 1 − e ⋅ sin( E )
(3)
The variable E is called eccentric anomaly, while the expression 2π t
is called mean
op
anomaly. Note that for a circular orbit when e = 0 , these two are the same, but as e gets
closer to 1, these two will be different.
Equation (3) will give the correct position of the comet at a given time t . The only
problem with this is that because equation (2) is an implicit equation in E, and cannot be
solved for E, you must solve for E using a numerical technique. Fortunately your TI
calculator and MAPLE have SOLVE commands to do this for us ( solve (equation in x ,
x) for TI and MAPLE)
We will study techniques to approximate E as a function of t in explicit form in
problems H3 and H4. This will give us E (t ) = an expression in t which we will then
plug into (3) for E as an expression.
a) For problem H2 let t equal the values in the table below, solve for E from (2) using
the solver command on your calculator or MAPLE (make sure your calculator is in radian
mode). Now use (3) to find the correct locations for Halley’s Comet. Write the location
in ordered pairs ( x, y ) , and carry your results to three decimal places.
Time in years
Value of E
Correct locations of the comet
t = 0 (Feb 1986) :
....................
..................................................................
t = 0 .5 :
....................
..................................................................
t = 1:
....................
..................................................................
t = 5:
....................
..................................................................
t = 10 :
....................
..................................................................
t = 20 (Feb 2006) :
...................
..................................................................
t = 30 :
....................
..................................................................
t = 40 :
....................
..................................................................
t = 50 :
....................
..................................................................
t = 60 :
...................
..................................................................
t = 70 :
....................
..................................................................
t = 75 :
....................
..................................................................
t = 76 (Feb 2062) :
.................
..................................................................
b) Plot the orbit and locate these locations as you did in Problem H1.
c) Observe the difference in these locations and that in Problem H1 and summarize with a
short statement.
Problem H3
We saw in problem H2 that to find the correct locations of Halley’s Comet we had to
solve the following implicit equation for E (eccentric anomaly):
2π t
= E − e ⋅ sin( E ) ,
op
(1)
and then plug the value of E into the orbital equation for Halley’s Comet given by:
 x(t ) = a ⋅ cos( E ) − a ⋅ e

2
 y (t ) = a 1 − e ⋅ sin( E )
(2)
Implicit equations are not very convenient when scientist want to predict the location of
planets and comets in the sky, or want to design spacecraft to land on or fly by these
celestial objects. It is important to find an explicit expression for E as a function of time,
E (t ) = some expression i n t , that we can plug directly in the arguments of the cosine
and sine functions in (2).
In this H problem and the next we will study power series that will approximate E as an
explicit function of t . First, we need to study Bessel functions before we can proceed.
Bessel functions, like sin( x ), cos( x ), and ln( x ) functions, are called transcendental
functions and can be presented explicitly only by power series. They are written as
J 0 ( x), J 1 ( x), J 2 ( x), J 3 ( x), .....
The subscript gives the order of the function (the above are Bessel functions of order 0,
order 1, order 2, order 3, …. ). Bessel function of order k is the solution to the following
differential equation:
x 2 y ′′ + xy ′ + ( x 2 − k 2 ) y = 0,
k = 0, 1, 2, 3, ... .
(3)
For example, J 2 ( x) is the solution to x 2 y ′′ + xy ′ + ( x 2 − 2 2 ) y = 0 . In Chapter 7 we will
study differential equations, and in section 8.10 and a later H problem we will learn
techniques to solve these differential equations. The solutions to these differential
equations are given by the power series:
(−1) i x 2i
J 0 ( x) = ∑
2 2i
i = 0 (i! ) 2
∞
∞
J 1 ( x) = ∑
i =0
(−1) i x 2i +1
i! (i + 1)!2 2i +1
∞
J 2 ( x) = ∑
i =0
(4)
(−1) i x 2i + 2
i! (i + 2)!2 2i + 2
.
.
.
1) Write a general power series for a Bessel function of order k .
J k ( x) = ...................................................................................... ……
2) Write the first four terms of the power series of each Bessel functions in (4), in
exact form, and end each with + ⋅ ⋅ ⋅ to indicate infinite series. Leave the
5
denominators in factorial and power form like 2! 3! 2 to show the patterns (DO NOT
EXPAND THESE INTO LARGE NUMBERS)
J 0 ( x) =
J 1 ( x) =
J 2 ( x) =
J 3 ( x) =
3) Turn the summations in equation (4) above to partial sums, and choose n for the
n
upper limit of the sums
∑
such that the partial sums will give Taylor polynomials
i =0
T20 , T21 , T22 , and T23 for J 0 ( x), J 1 ( x), J 2 ( x), and J 3 ( x) , respectively:
n = ………………..
4) Enter the Taylor polynomials T20 ( x), T21 ( x), T22 ( x), and T23 ( x) approximations
for J 0 ( x), J 1 ( x), J 2 ( x), and J 3 ( x) , respectively, into a MAPLE worksheet or your
calculator [the command is: sum ( … ,i = 0 .. n); for MAPLE and
∑ (...., i, 0, n) for
TI ]. Plot these four functions on the same set of axes on the window x ∈ [0,10] ,
y ∈ [−1,1] and attach your graphs.
5) MAPLE knows these functions as BesselJ (k , x) , where k is the order and x the
independent variable. For example BesselJ (2, x ) is J 2 ( x) . Your calculators
unfortunately don’t have Bessel functions in their catalogue. Use MAPLE to graph
J 0 ( x) through J 3 ( x) on the same set of axes and on the same windows as in 4) and
attach the graphs.
6) Write a short statement as to how the partial sum of the series form of Bessel
functions and MAPLE’s Bessel functions compare. Where are they similar, where
are they different.
Problem H4
We saw in problems H2 and H3 that to find the correct locations of Halley’s Comet we
had to numerically solve the following implicit equation for E (eccentric anomaly):
2π t
= E − e ⋅ sin( E ) ,
(1)
op
and then plug the value of E into the orbital equation for Halley’s Comet given by:
 x(t ) = a ⋅ cos( E ) − a ⋅ e

2
 y (t ) = a 1 − e ⋅ sin( E )
.
(2)
In order to avoid having to solve the implicit equation (1) numerically, astronomers and
mathematicians have developed a solution for the eccentric anomaly E (t) as an explicit
function of t , which is a power series form given by:
E (t ) =
∞
J ( k ⋅ e)
2π t
2π t ⋅ k
+ 2∑ k
⋅ sin(
)
op
k
op
k =1
.
(3)
In (3), J k (k ⋅ e) is the Bessel function of order k that we studied in H3 with arguments
e, 2e, 3e,.... . Note that J k (k ⋅ e) itself is a transcendental function and has a power series
expansion.
You will use MAPLE to do this problem. See the note below if you would like to use
your calculator. You can enter this power series as written in (3) into MAPLE using
BesselJ (k , x) syntax in MAPLE for J k (k ⋅ e) . Note that k is the order, and x is the
argument, which is k ⋅ e here.
1)With e = 0.97 for Halley’s Comet, use MAPLE to find the approximate (decimal)
values for the terms J 1 (e), J 2 (2e) , J 3 (3e), J 4 (4e) , and write 2πt / 76 plus the first four
terms of the power series for E(t) in (3), then end with + ⋅ ⋅ ⋅ to indicate infinite series.
Leave the term 2π t op as 2π t 76 , but turn all the coefficients of the sine functions into
decimals.
E (t ) = .........................................................................................................................................
.....................................................................................................................................................
2) Enter equation (3) in MAPLE using the first 50 terms (k=0..50), using the function
notation for E (t ) [this will look like E := t->sum (….) ]. Enter the following values of t
in the table below to find the values of E (t ) . Plug these values of E (t ) into equation (2)
to find the ( x, y ) locations of Halley’s Comet and fill the table below:
Time in years
Value of E
Approximate location of the comet
t = 0 (Feb 1986) :
....................
..................................................................
t = 0 .5 :
....................
..................................................................
t = 1:
....................
..................................................................
t = 5:
....................
..................................................................
t = 10 :
....................
..................................................................
t = 20 (Feb 2006) :
...................
..................................................................
t = 30 :
....................
..................................................................
t = 40 :
....................
..................................................................
t = 50 :
....................
..................................................................
t = 60 :
...................
..................................................................
t = 70 :
....................
..................................................................
t = 75 :
....................
..................................................................
t = 76 (Feb 2062) :
.................
..................................................................
3) Write a short statement as to how this compares with your correct locations you got in
problem H2.
*Note: You can use your calculator to do this problem, but since your calculator does not
know Bessel functions, you need to use the power series for J k (k ⋅ e) :
∞
J k ( k ⋅ e) = ∑
i =0
(−1) i (k ⋅ e) 2i + k
,
i! (i + k )!2 2i + k
and plug that in (3) to get:
(−1) i (k ⋅ e) 2i + k
2i + k
∞ ∑
2πt
2πt ⋅ k
i = 0 i! (i + k )!2
E (t ) =
+ 2∑
⋅ sin(
).
op
k
op
k =1
∞
The calculator will give similar answers to MAPLE if you use the first fifty terms for
both of the series (partial sums). The calculator is, however, excruciatingly slow. If you
do this, it is best to store the numbers in op, and e and enter this equation with op and e
symbols and not numbers. Store the function as f(t), and then enter f(0), f(0.5) f(1), ….
to get values for E(t).
Problem H5
In previous H problems we used Bessel functions to model the orbit and the location of
Halley’s Comet. In this last H problem we will actually solve a differential equation to
find the power series of one of these Bessel functions as an example of how the power
series for Bessel functions are derived. I will hand you a guideline to ask you to
summarize the results of problems H1 through H5 into a project report next week
Bessel function of order k, J k (x) as we have seen before, is the solution to the following
differential equation:
x 2 y ′′ + xy ′ + ( x 2 − k 2 ) y = 0,
k = 0, 1, 2, 3, ... .
For example, J 2 ( x) is the solution to x 2 y ′′ + xy ′ + ( x 2 − 2 2 ) y = 0 .
If we let k=0, and choose proper initial conditions, the solution to the initial value
problem :
x 2 y ′′ + xy ′ + x 2 y = 0,
y (0) = 1,
y ′(0) = 0
(1)
is the Bessel function of order 0 ( that is J 0 ( x) ).
a) Solve the above initial value problem (1) using the power series technique. Make sure
you show all your steps and put the final answer in ∑ form.
b) Find the interval of convergence of J 0 ( x) .
c) Assuming that you can solve the differential equation for any Bessel function J k (x) ,
find the interval of convergence of the general Bessel function J k (x) . The
∑
form for
J k (x) was found in H3.
d) Graph several Taylor polynomials for J 0 ( x) until you reach one that looks like a
good approximation to J 0 ( x) over the interval [-5, 5]. Present the graphs and the Taylor
polynomial that does this approximation
This problem will be graded on the use of good mathematical notation and complete
write up of your work.
Writing Your Project Report
You are now ready to present your scientific work on Kepler’s Laws as applied to
Halley’s Comet. Here is a guideline for your presentation for the results of problems H1
through H5.
a) Please do not attach or refer to any of the H problems in your report. You can cut and
paste results it they are neatly done, but use your own words. Write your report as if
someone who does not know anything about the H problems, but knows math is
reading your report. Your report should be word processed.
b) You will summarize all the information that you have learned in the H problems in
your report.
Your report:
1) Introduction: Summarize Kepler’s Laws and what the project will present ( the exact
implicit and the approximate explicit equations for E ) just in words ( No equations in
Introduction).
2) Main Presentation : Organize the information in a way that will make sense to an
outside reader. First present Kepler’s first two laws. Next present Halley’s Comet and
its orbital elements (a, e, op, b). Then the implicit equation that Kepler derived, and
finally the approximate explicit form that were later developed, and compare the results.
You also need to present Bessel functions in there as well. Emphasize Power Series for
E(t) and Bessel Functions that we have learned in this class. Include all the tables and
the formulas and the graphs that we have developed in the H problems. The solution to
J 0 ( x) you found in H5 can be presented here or as an appendix in the back. If you
present it as an appendix in the back, mention here that as an example we will present the
solution for the first Bessel function in Appendix A.
3) Summary: Summarize the results of this project and all that you have learned just in
words. The summary will be just in words with no equations or graphs or tables.