### Phys 728, Spring 2012            Homework Problem Set #6

6.1  In problem 5.3 of the previous homework, you wrote a program to calculate the u,v coordinates for 12 hours of observation with the EOVSA antennas. Now use them to obtain the synthesized beam (point spread function) of size 512 x 512 pixels by determining an appropriate scale Du, Dv for your u,v plane, gridding the u,v points to fall in the appropriate cell (pixel). Make two beams by following these two procedures: a) In each cell that has a u,v point, add the complex value complex(1,0). For cells that happen to have more than one u,v point (say n points in a cell), the value when you are done gridding will be complex(n,0). This is called "natural weighting." Then do an FFT, and make a print-out of the inner 128x128 cells of the synthesized beam (after shifting by 256,256) as a contour map with contours ranging from -50% to +100% of the peak value (which should be in the center, after shifting). Draw the +50% contour using a different color, or with a thicker line, to make it stand out. b) Do the same procedure, but now using "uniform weighting" where each cell with u,v points in it has a value complex(1,0), no matter how many u,v points may fall into it. Submit your calculation of the Du, Dv scale, the corresponding Dq angular scale of the beam (arcmin/cell), your code, and the two contour plots with x,y axes labeled in arcmin. Measure somehow the major and minor axis of the 50% contour and write down the sizes (full-width-half-maximum) in arcsec. Which one has the best resolution (smallest beam), uniform or natural weighting?

6.2 Download the file uv_model.sav (IDL) or uv_model.mat (MATLAB), which is the fourier transform of a model sky brightness distribution, and apply your two (uniform and natural weighted) gridded u,v point arrays as a sampling function, then do the inverse FFT on the sampled data to obtain dirty maps for the two cases. (a) Display the two images side-by-side, with appropriate annotation, and turn in your code and the images. Note, the zero for the uv data and for the images is the bottom left corner, so to display these items with the zero in the center you will have to shift by 256 pixels in each dimension first. (b) For the natural weighting case, make a six-panel figure like that of Fig. 10 in lecture 6. Label the panels appropriately. Turn in your code for this 6-panel figure as well.