410505 
Topic 1: Analyzing Numerical Data 
Fall 2011 
Application 1: Hubble's Law and Model Fitting
Web Notes
Lecture notes:
Monday August 29
Wednesday August 31
Hubble's article "A relation between distance and radial velocity
among extragalactic nebulae",
Proc. Natl. Acad. Sci. USA 15, 168 (1929),
http://www.pnas.org/content/15/3/168.full.pdf.
K.A. Olive and J.A. Peacock, "Bigbang cosmology", in C. Amsler et al.,
Phys. Lett. B667, 1 (2008),
http://pdg.lbl.gov/2009/reviews/rpp2009revbbangcosmology.pdf.
Some Type Ia supernova data from
http://dark.darkcosmology.dk/~tamarad/SN/.
Algorithms for Fitting Data to a Straight Line from
W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery,
"Numerical Recipes in C" (Cambridge University Press 1992),
http://www.nrbook.com/a/bookcpdf.php,
Section 15.2
Fitting Data to a
Straight Line.
Python programming: bookmark and browse
Homework Assignment 1: Due Sunday September 11 before 11:59 pm
NOTE: PHY 410 do Problems 1 and 2. PHY 505 do all problems 1, 2, 3

Modify the Hubble program to make a leastsquares fit to the 9 groups
(open circles, ?) in Figure 1 of Hubble's
1929 article
and compare the slope of
the fitted straight line to Hubble's value of K. Estimate the
age of the Universe that this value implies.
Note: Hubble states in his article: "Two solutions have been made, one using
the 24 nebulae individually, and the other combining them into 9 groups
according to proximity in direction and in distance." He does not specify
the 9 groups, so you need to figure how he selected them. The distances
are given in Table 1, but not the directions. Presumably he had a galaxy
catalog handy on his desk with the directions of each object listed. You
can look them up at
http://spider.seds.org/ngc/ngc.html.

Find Hubble's constant from the intercept and slope
output of the supernova program and compare with Hubble's value.
Explain any discrepancies you observe.
Note: Hubble's Law in the simple nonrelativistic form v = rK +
const. does not work for supernovae. You should use the relativistic form
given in the
web notes
\begin{equation}
\mu = 25 + 5\log_{10}\left(\frac{cz}{H_0}\right) + 1.086(1q_0)z
+ \ldots
\end{equation}
Assume that q_{0} = 1.

PHY 505 ONLY: Divide the supernova data set into
two subsets, low redshift and high redshift. Compute the slope
separately for each of the two subsets. Can you conclude from your
results that the expansion of the Universe is constant, accelerating,
or decelerating? You can also try to determine q_{0} using
Gnuplot fit
on the full dataset.
Submit homework on UBlearns as one PDF file. Click on the "Assignments" tab,
open the "Homework Assignment 1" folder, click on "Homework 1", and
scroll down to the "Attach File" button.
You must click the "Submit"
button when you are done for me to see your file.

LaTeX is the most widely
used software for scientific documentation, and is strongly recommended for
preparing homework assignments.

If you really prefer to use
Microsoft Word, install PDF support to save as PDF. Optional:
MathType is a
better equation editor, not free but not too expensive.

In an emergency scan
handwritten solutions to PDF.
Application 2: Earthquakes and Other Natural Hazards
Web Notes
Lecture notes:
Wednesday September 7
Friday September 9
Codes:
quake.py
quake.cpp
grid.py
grid.cpp
H. Kanamori and E.E. Brodsky,
The Physics of Earthquakes,
Physics Today 54, 3440 (2001).
How many earthquakes with magnitude between 1 and 10 on the
Richter scale have been recorded?
Check out the data from the USGS
National Earthquake Information Center!
Tunguska Event:
Comet?
Asteroid?
NASA Near Earth Object Program
A Near
Earth Object Fact Sheet
Homework Assignment 2: Due Sunday September 18 before 11:59 pm

Download the NEIC Global dataset of earthquakes with magnitudes greater than
1.0 and use a
quake
code to fit the Gutenberg Richter Law. The
raw data do not provide a reasonable fit. Explain why and fix the code to
obtain a more reliable estimate of the slope constant b. Check your
answer with values given in the references.
PHY 505 ONLY:
Download a different event data set (e.g. from the Southern California
Earthquake Center
http://www.scec.org/resources/data/, or any other source you
can find on the Web), perform a linear fit and compare your results with
the NEIC global set.

The NEIC dataset contains columns labeled
LAT LONG
in addition
to MAGNITUDE
. Make a density map of
quakes on Earth's surface. Compare density maps of low M and high
M events. Does this help explain the bad fit in problem 1?
Optional: superimpose the density map on an outline of tectonic plates and/or
continents.
PHY 505 ONLY:
Construct a 3D bar graph or contour plot (see
3Dim
Plot with a ColorMap (pm3d))
of total seismic moment (energy)
as function of latitude and longitude. You need to construct an equal area
grid on Earth's surface and accumulate the energy released in each grid
element as a 2D histogram. Compare your results with the event density map.

Download the Near Earth Object Fact Sheet
and extract the "Orbital Period" and "Semimajor Axis" data.
Plot the dataset, fit it to a wellknown Law of Physics, and determine any
physical parameters involved in the fit.
Choose another pair of columns that you think might give an interesting
plot and discuss.
Application 3: Global Warming, Sunspots, and the FFT
Web Notes
Lecture notes:
Monday September 12
Codes:
co2.py
co2.cpp
fftdemo.py
fftdemo.cpp
J.L. Sarmiento and N. Gruber,
Sinks for Anthropogenic
Carbon,
Physics Today 55, 30 (2002).
R.F. Keeling, S.C. Piper, A.F. Bollenbacher and J.S. Walker,
"Atmospheric Carbon Dioxide Record from Mauna Loa",
CDIAC
http://cdiac.ornl.gov/trends/co2/siomlo.html
The
Scripps CO_{2} Program
website has a large collection of
current
datasets.
Report of the
Intergovernmental Panel on Climate Change (2007)
http://www.ipcc.ch/.
Algorithms for Fast Fourier Transforms from
W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery,
"Numerical Recipes in C" (Cambridge University Press 1992),
http://www.nrbook.com/a/bookcpdf.php,
Section 12.2
Fast Fourier Transform
(FFT),
Section 12.4
Fast Sine and Cosine
Transforms.
Wikipedia article on
Sunspots
Sun Spot Cycle
Reviews of Solar Physics
Homework Assignment 3: Due Sunday September 25 before 11:59 pm

Run the
fftdemo
code and interpret the power spectrum: can
you measure the parameters of the damped oscillator from the power spectrum?
What is the largest $N$ you
can transform on your computer in a reasonable amount of time with the
DFT and FFT algorithms?
Is there any advantage in this example in using a larger $N$?
PHY 505 ONLY: Measure the CPU time dependence of the DFT and FFT algorithms as functions
of $N$, plot, and compare with theoretical expectations.

The Mauna Loa CO_{2} concentration
http://cdiac.ornl.gov/ftp/trends/co2/maunaloa.co2
increases with
every passing year
in addition to oscillating with the seasons. To Fourier analyze the
seasonal oscillation, the linearly increasing trend needs to be
subtracted from the data.
Fit the trend to a linear function $f(t)=a_0 + a_1t$.
Subtract this linear increase from the data, generate a power spectrum and
interpret your result.
If the current trends continue, when will the concentration reach
toxic levels?
http://en.wikipedia.org/wiki/Carbon_dioxide#Toxicity.
PHY 505 ONLY:
Is the linear increase accelerating or decelerating?
Fit to a quadratic function $f(t)=a_0 + a_1t+a_2t^2$
(e.g. using Gnuplot), subtract and generate a power spectrum.
How does this change your results from the linear fit?

Download sunspot data from NASA
http://solarscience.msfc.nasa.gov/greenwch/spot_num.txt,
generate a power spectrum and
describe any interesting features you observe.