Assignment 1.  ASTR 407		            Due: Monday Jan 8, 10:00 AM
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   I assume you can write and use simple C/Perl/pyton codes (or
   some other programming environment), and use software to make plots.
   For this homework, each solution must have a written/textual outline
   explaining what you have done (think of it as explaining how you 
   approached the problem to another student in the course, refering
   to numbered figures/tables as though it was an article).  If you 
   wrote computer code for the question, you should provide a printout 
   of the numerical code you used to solve the problem and the output 
   of the code for each case you execute it for; make your code output
   well explained (for example, print out column headings).

1. Find the zero (numerically) of f(x) = x**3 - 6x - 7 by coding a
   Newton's method using x=2 as a starting guess (by inspection,
   the root is clearly between x=2 and 3).  How many iterations are 
   needed to get to 1 part in a billion  accuracy?  (That is, the 
   absolute value of the difference from the previous iteration is
   less that 1.0e-9).  Thus, show a table where iteration 0 is the
   intial guess, and then your columns are  
       iteration        current_estimate_x       diff_with_last_iter
   Then repeat with using x=3 as a starting guess.  By plotting a 
   graph of the function explain why one of these guesses is better
   for this numerical method than the other.

2. Review the midpoint rule for performing an approximate numerical 
   calcuation of a definite integral, and briefly summarize how it
   works.  Then write a piece of code that uses this method to
   estimate the integral of f(x) = 1/x  from x=1 to x=10 numerically. 
   Show that this method converges (to the correct answer easily
   obtainable from the analytical integration you should do), as you
   increase the number of 'bins' from 10 to 100 to 1000.

3. Given the differential equation versus time t for the variable 'a':
	                     da/dt = 2 * a**(-4)
   (that is, 2 times 'a' to the negative fourth power) then, 
   if we consider the case where a(t=0) = 1, 

	i.  Analytically (not numerically), solve the equation and
	    determine a(8).
	ii. roughly sketch (or plot with a computer) a graph of 
            a(t) from t=0 to 8.
            
4. Download the file MPCxyz.txt from the web site.  Your professor
   has obtained the 3-dimensional cartesian position and velocity
   vectors for the first 9999 minor planets, so the file format is
      n  x  y  z vx vy vz
   where n is the object's identification number.  You need not provide
   code for this question as you are just plotting.  The units are in
   au and au/year, and positions and velocities are heliocentric.
   
   i. Adjusting your plotting limits so that you see every minor planet
      within 7 au of the sun, make a plot of the position of all these
      objects projected down onto the (x,y) plane.  Draw circles at the
      semimajor axes a for all that planets that appear on the diagram
      (they are listed in appendix E.1).  You must choose your axes
      so that the circles appear as circles on your plot (if you software
      doesn't force square axes, you may have to fiddle).  Make the symbol
      for the data points small so that you can see them all.
      
   ii.Compute the norm of the two vectors r and v (many plotting programs
      may let you do this inside the program, otherwise you may need to
      read in MPCxyz.txt to a program and write out another file).
      Make a plot of v as a function of r out to r=7 au.