Download MATH 112 SPECIAL PROBLEM

MATH 112 SPECIAL PLANETARY PROBLEM
In this problem you will find the parametric equations of Neptune and Pluto.
We will only consider the two dimensional orbits viewed from the north
pole of the Sun way above the solar system. The Sun is at the origin. All
the numbers in the equations will be in astronomical units (AU). An
astronomical unit is the average distance from the Sun to the Earth. The
orbits are ellipses or approximately circles. Your final equations will have
the form:
 x(t ) = a cos( Bt + C ) + h
,

 y (t ) = b sin( Bt + C ) + k
where a is ½ the axis of the ellipse along the x-axis, b is ½ the axis of the
ellipse along the y-axis, and h and k are horizontal or vertical shifts
respectively. Note that C must be the same constant in the equation for x
and y. The same is true for B.
If the planet’s orbit is approximately a circle with radius r, and the Sun at the
center, then
a = b = r , and the shifts are zero, and the above equations become:
 x(t ) = r cos( Bt + C )
.

 y (t ) = r sin( Bt + C )
B is determined by the period, and C is determined by the initial position of
the planets at
t = 0.
Neptune orbits the Sun every 163.7 years and its orbit is approximately a
circle with a radius of 30.05 astronomical units (AU). Pluto orbits the Sun
every 248.54 years and its orbit is an ellipse. The coordinate system has
been chosen so that the x-axis lines up with the major axis of Pluto to make
this problem easier. The major axis of this ellipse is 78.48 AU, while its
minor axis is 76.1 AU. The center of this ellipse is shifted right by 10 AU
from the origin where the sun is located. The following figure shows the
orbits of Neptune and Pluto.
We will let t = 0, be Jan. 1st, 2000. At t = 0 Neptune was at
( x, y ) = (−5.4,−29.6) AU, and Pluto was at ( x, y ) = (−28.0,−9.5) AU.
The Problem:
I) Find the complete parametric equations for Neptune and Pluto with all the
constants a, b, B, C, h, and k determined. You must do all calculation and
solve for all the constants algebraically and show all the steps. Pay
particular attention to choosing the constants that will place the planets in
the correct quadrants.
II) Find the location of the two planets (both the (t, x, y) values in a table and
the locations on the graphs) on Jan 1, 2000, 2050, 2100, 2150, 2200, and
2250. Label these clearly on your graph.
Your report:
Your report should include:
I) A cover sheet.
II) A complete statement of the problem, something like what I have written
above, but in your own words. Do not attach this paper that you are now
holding in your hand to your report.
III) All your calculations.
IV) All the final equations and graphs and tables. The location of the planets
should be indicated on the graphs of the orbits and in a table of (t, x, y)
values. You can present the graphs neatly by hand if you don’t have the
means of transferring the graphs from your calculator to a computer and then
to paper. I can also print very nice graphs for you with MAPLE if you bring
your equations to me during my office hours. Do not use my graph above in
your report.
V) A short conclusion of what this project has contributed to your cosmic
consciousness.
Due Date:
This problem is due Dec. 6th, 2004. At that time Neptune will be at (x,
y)=(0.3, -30.0) AU, and Pluto will be at (-26.5, -14.0) AU. The morning sky
just before sunrise will feature Venus and Mars only a few degrees apart in
the eastern sky about 30 degrees from the horizon, Jupiter higher up in the
southeastern sky, and Saturn in the southwestern sky. Saturn is the last
planet visible without a telescope.
Points:
This problem will be worth 30 points, and will count toward your homework
assignment. I always like to give extra points for especially good work and
presentations for those who will like to make up low points on their first
exam.