410-505 Topic 1: Analyzing Numerical Data Fall 2011

## Application 1: Hubble's Law and Model Fitting

Lecture notes: Monday August 29    Wednesday August 31

Hubble's article "A relation between distance and radial velocity among extra-galactic 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, "Big-bang cosmology", in C. Amsler et al., Phys. Lett. B667, 1 (2008), http://pdg.lbl.gov/2009/reviews/rpp2009-rev-bbang-cosmology.pdf.

Some Type Ia supernova data from http://dark.dark-cosmology.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

1. Modify the Hubble program to make a least-squares 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.

2. 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 non-relativistic form v = rK + const. does not work for supernovae. You should use the relativistic form given in the web notes $$\mu = 25 + 5\log_{10}\left(\frac{cz}{H_0}\right) + 1.086(1-q_0)z + \ldots$$ Assume that q0 = 1.
3. 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 q0 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

Lecture notes: Wednesday September 7    Friday September 9

H. Kanamori and E.E. Brodsky, The Physics of Earthquakes, Physics Today 54, 34-40 (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!

### Homework Assignment 2: Due Sunday September 18 before 11:59 pm

1. 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.

2. 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 3-D bar graph or contour plot (see 3-Dim Plot with a Color-Map (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 2-D histogram. Compare your results with the event density map.

3. Download the Near Earth Object Fact Sheet and extract the "Orbital Period" and "Semimajor Axis" data. Plot the dataset, fit it to a well-known 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

Lecture notes: Monday September 12

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/sio-mlo.html

The Scripps CO2 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

1. Run the fft-demo 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.

2. The Mauna Loa CO2 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?

3. 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.