Kepler's Law

First of all I hope you all got the idea of what the drawing on the 
last page of the lab was intended to portray.  That solid circle is 
the Moon and the dots are measured positions of the spacecraft at 
EQUAL TIME INTERVALS.

In drawing in the major axis, take some time and care.  The accuracy 
that you can achieve depends on making a good start.  Clearly (?) the
center of the Moon is one focus of the orbit, and that must be on the
major axis.  Most people try to find periluna or apoluna as the other 
point by directly measuring distances from the center of the Moon.
Periluna is the point of closest approach, apoluna the point most distant.
Two points are all that is required to draw a line, so now you can
draw the major axis.  Note:periluna or apoluna are not necessarily going
to be one on the data points.  That would be unbelievable luck.  Lightly
sketch in a section of the orbit around periluna, say, and pull out
your ruler.  Remember that satellites move on smooth paths.  Don't 
play CONNECT-THE-DOTS.

Here's another trick to find the major axis.  Print yourself another copy
of the orbital diagram.  This trick takes advantage of the fact that the
ellipse is a symmetrical figure.  First, you may find it helpful to cut
away the excess paper, leaving just the orbital points (and everything
inside the points of course).  Now fold the page where you think the
major axis might be.  You will have found the major axis when one side
of the data points superimposes over the over side.  Simply transfer
your fold to the orbital chart and you have a very accurate major axis.

The minor axis runs through the center of the ellipse and is perpendicular
to the major axis.

The other focus is the same distance from the center of the orbit as the
center of the Moon is from the center of the orbit.

Questions:

1.  The first law says that the orbit is an ellipse.  Now the definition 
of an ellispe (as I point out in the lecture) is that the distance from
one focus to any point on the ellispe and back to the other focus is a
constant for all points on the ellipse.  Lightly sketch in two sections 
of the orbit.  Again, don't play CONNECT-THE-DOTS (nature doesn't).  Pick
at least two places on the orbit (maybe at quite different places on the
orbit) and test the construction.  That is, measure from one focus to a
point and back to the other focus.  Record the result.  Now try another
point on the orbit.  Did you get the same distance?  If it's an ellipse, you
should.  Of course, this depends on finding the foci correctly, so allow
for a small amount of measurement error.

The eccentricity is defined as f/c or, equivalently, the distance from
the center of the orbit to one focus divided by the length of the semi-major 
axis.

2.  Again pick at least two regions of the orbit.  For this you want to
use the data points since we know these are points equally spaced in 
time.  Don't use consecutive data points, but skipping a point works well.
So you want two pairs of points at very different places in the orbit.
Draw lines from your chosen points to the center of the Moon.  Construct 
an arc between your points as close to the orbital path as you can.  Each
pair should produce an area similar to Area 2 on the explanation under
the Second Law.  I have included that drawing as an attachment.  Notice that
I have split Area 2 into two parts by drawing in a perpendicular from
point A to the line to point B.  The yellow area is clearly a right triangle
whose area is (1/2)(base)(height).  I have labeled the base and height for
you.  The magenta area can also be estimated as a triangle without too
much error.

Now pick your other two data points and repeat this procedure.  Are the areas
equal?

3.  This one is pretty easy - look at the Second Law.

4.  Think of this as a tax form because that's what I had in mind when I wrote
it.  The first steps create a scale for the drawing.  We will be doing this sort
of thing quite a bit in various labs.  Also notice that in this section I show
the calculation in parentheses after each part.

a.  measure the size of the Moon on the drawing in centimeters
b.  the actual lunar diameter is 3476 km from your textbook or 3,476,000 m
c.  here we get a scale for the drawing just like on a map, a correlation
    between actual sizes and drawing sizes.
d.  measured value of the semi-major axis in centimeters
e.  use the scale factor in part c. to convert to meters.
f.  cube the result in e.
g.  multiple the result from f by 8.02 X 10^(-12).  This is the Third Law
    the cube of the semi-major axis times a constant is square of the 
    sidereal period.  So we have just computed P^2.
h.  Take the square root of part g. to give the orbital period in seconds.
i.  Convert part h. to hours.

5.  Here is the actual form of the Third Law.  Do you see P^2 = (constant)*a^3?
    The constant is 8.02 X 10(-12) as above, but now we break the constant 
    into two parts.  One part is 5.92 X 10^(11).  The other part is the
    mass of the Moon plus the mass of the satellite.  I think we can safely
    ignore the mass of the satellite compared to the Moon, so we have
    just calculated the masss of the Moon.

Good Luck.