Download Math 256 Project

Halley’s Comet Project for
Differential Equations (Math 256)
Comet Halley from Mount Wilson
Halley's Comet in 2024
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 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 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 and 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 and his law of gravitation ( F = m ⋅ a , and F = −
G⋅M ⋅m
). The 23-year old was
d2
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 law.
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.
y
x
Aphelion
Perihelion
major axis = 2a
minor
axis =
2b
Halley’s Comet
Comets, like all planets, orbit the Sun in elliptic orbits, but their orbits are very eccentric
(the major axis is much larger than the miner 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 figures above).
Astronomers use Astronomical Units (AU) for distance. An Astronomical Unit is the
average distance between the Sun and the Earth. This is the same as the semi-major axis
for the Earth (a in figure above). There are 149.598 × 10 9 m AU . They also use years
instead of seconds. There are 3.15569 × 10 7 s y .
At perihelion on February 9, 1986, Halley’s Comet was only 0.587 AU from the Sun,
well inside the orbit of Venus. The comet was moving at − 11.53205 AU/y at that point.
This is a negative number since the comet moves in retrograde motion (backwards).
The average period of the comet's orbit is 76 years. Comet Halley was visible in 1910 and
again in 1986. Its next perihelion passage will be in early 2062.
We will let t = 0 designate February 9, 1986. With this convention t = 20 is February 9,
2006, for example, and t = 76 is February 9, 2062 when the comet will return to
perihelion again.
We will place the xy-plane on the plane of the comet’s orbit with perihelion on the
positive x-axis as the figure above.
The Mathematics of the Project
In this project we will use differential equations to find the orbit and locations of Halley’s
Comet using Newton’s two laws mentioned above.
Newton’s second law of motion in vector form is:
→
→
F = ma
(1)
→
→
where F is the force vector in N (Newton’s), and a is the acceleration vector in m s 2 ,
and m is the mass in kg.
Newton’s law of gravitation in vector form is:
→
GMm r
F =−
⋅
2
r
r
→
(2)
m3
is the universal gravitational constant, M = 2 × 10 30 kg is
2
s ⋅ kg
r
the mass of the Sun, and m is the mass of a planet or a comet orbiting the Sun, and r is
where G = 6.67 × 10 −11
the vector connecting the Sun to orbiting comet. Note that in (1) and (2) the variables
r r
r
F , a and r are all functions of time.
1) In order to use more conventional units that astronomers use, convert the units of
m3
AU 3
to 2
using the information above.
universal gravitational constant from 2
s ⋅ kg
y ⋅ kg
2) The motion force in (1) and the gravitational force in (2) are equal. Equate the right
hand sides of equations (1) and (2), and cancel the common term.
r
r
r
r
r
3) Using a (t ) = r ′′(t ) , and r (t ) = x(t )i + y (t ) j , convert the equation in 2) to an equation
involving x(t ) , y (t ) and their derivatives. Note that these are the x-y coordinates of the
comet.
4) When two vectors are equal, their components are equal. Equate the x and the y
components on each side of the equation you got in 3). This should give you two secondorder differential equations, one involving x ′′(t ) , x(t ) , and y (t ) , and the other involving
y ′′(t ) , x(t ) , and y (t ) .
5) Let v x = x ′ , and v y = y ′ , and turn each equation into a system of two first-order
differential equations. Collect these into a system of four first-order differential
equations.
6) To come up with four initial conditions for this system, note the position and velocity
of the comet mentioned above under Halley’s Comet. This will actually give you the
four initial conditions that you need for the system in 5).
7) Use MAPLE, or your calculator, to solve this system numerically.
a) With MAPLE use the DEplot( … ) command and the dsolve( … type=numeric)
command. The first one will give you the graph of the orbit, while the second one will
give you the positions at different times after 1986. Make stepsize .005, and use
scene=[x,y].
b)If you use your calculator, you need to set the following setting:
MODE→Graph→Diff Eq
y=→type your four equations and four initial conditions in y1′, yi1,.....
y=→F1→Format→Coord→RECT
→Solution→RK
→Fields→FLDOFF
y=→F7→Axes→CUSTOM
x Axis y?
y Axis y?
8) Present the graph of the orbit of Halley’s Comet and find the xy-coordinates of the
comet for years in table blow and fill the table. Use only three digits after the decimal
point for the xy-coordinates.
t
0
5
10
15
20
25
30
35
38
76
date
Feb. 9, 1986
x-coordinate
y-coordinate
9) Find the lengths of the major and the minor axes of Halley’s Comet (2a and 2b in the
figure), perihelion and aphelion distances to the Sun, and the orbital period for the comet
you get from the solution and fill in the table below.
Halley’s Comet parameters
major axis (AU)
minor axis (AU)
Perihelion distance (to the Sun) (AU)
Aphelion distance (to the Sun) (AU)
Orbital period (y)
Your Report
Your report should be complete and easy to understand by a mathematician who
has not seen this paper and has not been to our class.
Your report should include:
I) A cover sheet. (do not attach any of this paper in your hand to your report).
II) A short and complete statement of the problem in your own words. Do not attach
any part of this sheet to your report.
III) All your math and calculations.
IV) All the graphs and tables (include the two tables in 8) and 9) above).
V) All the MAPLE worksheet or calculator work (attached as Appendix to back of
report)
VI) A short conclusion of what this project has contributed to your cosmic
consciousness.