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Revision: 1.2, Sun Feb 24 18:34:11 2019 UTC (4 years, 1 month ago) by tplarson Branch: MAIN CVS Tags: Ver_LATEST, Ver_9-5, Ver_9-41, Ver_9-4, HEAD Changes since 1.1: +30 -6 lines bug fixes |
Visualizing the Sun: An Introduction to the SoshPy Visualization Package a more extensive discussion of helioseismology with added graphics can be found at http://solar-center.stanford.edu/SoSH/ . short instructions can be found in quickstart_visual.txt . the study of oscillations inside the sun is called helioseismology. in particular, we are here concerned with acoustic waves for which the sun provides a resonant cavity. the Sonification of Solar Harmonics (SoSH) Project is a collaborative research initiative that seeks to transform helioseismology into a listening experience for the scientist and nonscientist alike. it has developed the SoSH Tool, which is able to sonify solar acoustic data by filtering and transposing it up to the range of human hearing. the SoshPy Visualization Package was created to provide complementary graphical representations of the solar harmonics being sonified. in what follows we provide a brief discussion of helioseismology, which augments the discussion in the documentation provided with the SoSH Tool (instructions_audio.txt). we then move on to a detailed description of the included python module and the various scripts that use it. Section 1: Helioseismology consider that we are here concerned with sound, which is simply the coherent vibration of matter. now imagine a plasma element (a chunk of matter) at the surface of the sun. in response to a given mode, which inhabits the entire sun, this single plasma element will oscillate at the mode frequency. in other words, it moves back and forth; some part of this motion will be in the radial direction, some in the latitudinal direction, and some in the longitudinal direction. a model can tell us how much of the motion lies in each direction. we call the radial motion the vertical displacement, and we put the latitudinal and longitudinal components of the motion together to make the horizontal displacement. now, how are we to measure this displacement? actually, it turns out to be much easier to measure the speed of our plasma element, which is simply the rate at which the displacment is changing. in mathematical terms, we say the velocity of the plasma element is the derivative of its position. since the plasma element is moving in response to one of the sun's harmonics, it can be modeled as a simple harmonic oscillator. in other words, its motion is mathematically equivalent to the motion of a weight attached to a spring. in particular, the model tells us that the magnitude of the the plasma element's velocity is equal to the magnitude of its displacement multiplied by its frequency, which is simply the mode frequency. it is the velocity of the plasma element that we are able to measure. the details of this measurement are beyond the scope of this discussion, but in short we are actually measuring the frequency of spectral lines of atoms at the solar surface. for plasma at rest, these spectral lines have precise frequencies that can be measured in a laboratory. the motion of the plasma causes these frequencies to be shifted slightly. this is called the doppler effect, and it is exactly the same phenomenon that you can observe with sound waves when a siren drives past: when the siren is approaching it sounds higher, and when it is moving away it sounds lower. we use the doppler effect to create velocity images of the sun's surface, and for this reason we call them dopplergrams. each pixel of the image corresponds to one of our plasma elements, and the pixel value tells us how fast that element is moving toward or away from the observer. here we run into a fundamental limitation of our method of observation. we would like to know both the vertical and horizontal components of the velocity. unfortunately, a dopplergram only gives us the velocity along the observer's line of sight. at the center of the solar image, this is exactly the vertical component, and at the very edge, it is the horizontal component. everywhere else we measure a mix of the two, and nowhere can we measure both of them. this is one of the reasons we have to rely on a model. so what quantity should we plot to represent a mode? it turns out that for a large majority of modes, the horizontal component is very small compared to the vertical component at the surface, and for many purposes can be neglected entirely. hence, by default we will show the vertical component of velocity only. the scripts, however, are also able to plot the latitudinal and longitudinal components, as well as these two combined into the total horizontal component. they can also plot the magnitude of the total velocity, as well this magnitude squared. we are also able to plot either a surface view or an interior view. in the latter case, we will show a plane containing the solar rotation axis. also in this case we plot the velocity scaled by the square root of the background density. this is for two reasons. firstly, since the velocities drop off rapidly with depth while the density increases rapidly, the quantity plotted shows visible variations throughout the interior. secondly, the square of this scaled velocity is actually equal to the energy density of the mode. note that for every quantity plotted, the colormap will be scaled to use the full range of the data. therefore, in a surface view, the velocity squared is indistinguishable from the energy density, because the background mass density is constant over the solar surface. similarly, for plots of single modes, we do not scale the displacement amplitudes by frequency to get a velocity, because this would also be invisible. when we begin adding modes together, however, we must properly account for their relative amplitudes, so in this case the frequency scaling is necessary because the frequency will be different for different modes. Section 2: Surface Views - drawharmonics.py and addharmonics.py for instructions on running these scripts, see the quickstart guide. in what follows i give a more detailed account of how they work. we start with drawharmonics.py, which plots surface views of single modes. but first we need a small amount of mathematical background. an oscillation mode on the sun, also known as a harmonic, is given by the product of a function of latitude and longitude and another function of radius only. the first function is called a spherical harmonic, and for these we have explicit analytical expressions. the function of radius is called a radial eigenfunction, and these must be computed numerically by solving a system of different equations. for a surface view, we only need the spherical harmonic, which is given by Ylm = Plm(cos(theta))*exp(im*phi), where the Plm's are associated legendre polynomials. in the coordinate system that we use, theta is the angle from the solar rotation axis, so it different from the latitude by 90 degrees. the angle phi is exactly the longitude. each spherical harmonic is described by two integers, l and m. the spherical harmonic degree l is >= 0, and the azimuthal order m ranges from -l to l. the associated legendre polynomial, however, depends on l and |m|, the absolute value of m. more details can be found in instructions_audio.txt . at the beginning of the script, the arrays containing theta and phi are generated by the function image2sphere(). this function is exactly the same as that used to generate artificial spherical harmonics for scientific analysis. it takes as arguments the number of pixels in the x and y directions (default 1000) as well as three parameters describing the orientation of the solar image. the first of these is bangle, which is defined as the latitude of the subobserver point. in other words, it is the tilt of the solar rotation axis towards the observer. similarly, the parameter pangle is the tilt in the plane of the image. these angels default to 30 degrees and 0 respectively. a third parameter, distobs, gives the distance of the observer in solar radii. it defaults to 220 (the actual distance between the sun and the earth is about 215 solar radii), but this can be set lower or higher to zoom in or out of the solar image. the script then enters an input loop wherein the user is prompted to enter the l, m, and n of the desired mode. this final integer, the radial order n, describes the behavior of the mode along the solar radius. for a surface view, it serves only to determine the ratio of the vertical and horizontal components. next we must calculate the associated legendre polynomial. this is ultimately done by the function setplm(), which again is exactly the same routine used for scientific analysis. to find the Plm for a given l and m, we start by using an explicit expression for the Plm with l=m, and then use a recursion in l to find the Plm with the requested l. hence, a total of l-m Plm's are actually calculated, along with their corresponding derivatives. to save on computing, these are all saved for potential use in subsequent iterations of the loop. next we construction the spherical harmonic, Ylm, as well as its derivative in both the theta and phi directions. the derivative in the theta direction gives the theta component, and dividing the derivative in the phi direction by sin(theta) gives the phi component. the radial component is given directly by the Ylm. in all cases, we must take the real part to find the physically meaningful quantity. next, we read the vertical to horizontal ratio from a table. this table was extracted from a model using the function writesurfacemodel(). the table contains both the magnitudes of the vertical and horizontal components of the displacement at the solar surface as well as a theoretical estimate of their ratio. more details of the model are provided in the next section. the script drawharmonics.py arbitrarily gives the vertical component (rc) a magnitude of 100 and sets the magnitude of the horizontal component (hc) according to the theoretical ratio. finally, the python dictionary varlist is populated with the six plotting options: Vr - radial or vertical component Vt - theta or latitudinal component Vp - phi or longitudinal component Vh - magnitude of horizontal component, =sqrt(Vt^2+Vp^2) Vmag - magnitude of total velocity, =sqrt(Vsq) Vsq - magnitude squared, =(rc*Vr)^2+(hc*Vh)^2 note that Vr, Vt and Vp are signed quantities, while Vh, Vmag and Vsq are strictly positive. once the script terminates, all six quantities will be available in varlist. unless the script was started with animate=0, the plot will be animated. to do so, it uses the two functions defined at the top of the script. the first, calcylmt(i), serves to evolve the mode in time. to do so, it require the arrays ylm, dylmt and dylmp to already exist. the time dependence is given by exp(-iwt), but rather than using the mode's actual frequency, we simply set w=2pi/nframes so that the animation will loop seamlessly. nframes is the total number of frames in the animation and can be set when the script is called. note that calcylmt(i) only calculates the requested quantity for each frame, rather than the full six which were calculated for the first frame. the second function needed for the animation, ylmanimate(i), simply takes the array returned by calcylmt(i) and uses it to set the data for a previously existing image instance. this is the function that must be passed to the matplotlib animation utility, and it is also called to save frames to disk. we now move on to the script addharmonics.py, which as the name suggests plots sums of modes. it is a relatively straightforward extention of the previous script, but the first notable difference is that the input loop does not accept the list of modes to plot. rather, this list is set once at the beginning of the script. that is, the user first enters the number of modes, followed by the l, m and n of the desired modes. the script then enters its input loop, wherein plotting parameters may be changed. in other words, to plot a new combination of modes, the script must be rerun. for each mode, the Ylm and its derivatives are calculated as before and the three resulting components are stored in lists. the same table is read to get the model values, but now the vertical (rc) and horizontal (hc) magnitudes are used directy. we will also scale the displacement amplitudes by frequency (freq) to get the proper relative velocity amplitudes. these three parameters are likewise stored in lists. next the sums are performed and the six plotting variables are stored in varlist as before. now, however, when we animate the plot, we give each mode in the sum its proper relative frequency. in particular, we use w=2pi*freq*freqscale/nframes, where freqscale is a parameter that can be set when the script is called. because the frequencies have already been scaled to be in units of millihertz, the default value of freqscale is 1/3. in general, the animation will no longer loop seamlessly. to see the sum evolve further in time, one may increase freqscale. one may also wish to simultaneously increase nframes. note that the animation functions in this script are named calcsumt(i) and sumanimate(i). here we must insert a short discussion of colormaps. for the signed quantities Vr, Vt and Vp, we would typically like for the value of zero to map to the center of the colormaps. for a single mode, this is almost always accomplished automatically, because the data range will usually be symmetric around zero. depending upon the orientation of the solar image, however, certain modes (often those with m=0) will not satisfy this requirement, perhaps leading to unexpected results. to correct this situation, by default we force the color mapping to be symmetric around zero for the signed quantities. this means that the full range of the colormap will not be used. to revert to automatic scaling, one may specify colorshift=0 when starting the script. the default value is 1. in either case, the color mapping is determined by the data in the first frame. this is adequate for single modes. for sums, however, it may happen that a subsequent frame will exceed the data range of the first frame, in which case the colormap will saturate. this is usually not a problem, but if one wishes to avoid this behavior, one may specify colorshift=2. in this case, the script will step through calcsumt(i) for i in range(nframes) and find the full range of possible data values in advance. we may also add the capability to change the range as the animation proceeds, but this is not yet fully implemented. Section 3: Interior Views - drawradial.py and addradial.py these two scripts operate in close analogy to their surface counterparts. the most important difference is that these require a much more extensive model for input. in particular, the full radial eigenfunctions must be read from a file, along the mesh on which they were calculated. the full eigenfunction actually consists of two functions, one giving the amplitude of the displacement in the vertical direction, and one giving the amplitude of the displacement in the horizontal direction. we call these two functions xir and xih respectively, and they each depend on both spherical harmonic degree l and radial order n. finally, we will also need the background mass density used to construct the model. all of this data has been packaged into an hdf file available from the SoSH website: http://solar-center.stanford.edu/SoSH/#mods . at the beginning of each of these scripts, we call the function loadmodel(), which reads the mesh and mass density used for the model, as well as the l, n and frequency of all the modes in the model, and puts them in global variables. a second function, getradial(l, n), will return the actual displacement eigenfunctions for the modes we wish to plot. after loadmodel(), the scripts call image2rtheta(), which returns the arrays containing the r and theta coordinates for each point in the image. like its surface counterpart, this function can take as arguments the number of pixels in the x and y directions (default 1000). the only other paramter it can take, however, is distobs, the observer distance in solar radii. in other words, the orientation of the r-theta plane is fixed. distobs should in general be set to the same value used for the surface views; in both cases it defaults to 220. note, however, that these scripts can take another parameter, rsurf, which is the fractional image radius at which to truncate the model (default 1.0). in some cases you may wish to remove the surface values if you find they dominate the plots. this will reduce the size of the area occupied by the plot. as mentioned above, the radial eigenfunctions will be scaled by the mass density. for the value of phi in drawradial.py, we have arbitrarily chosen pi/4m for the right half of the image, so that the real and imaginary parts of the spherical harmonic (recall the factor of exp(imphi)) will be the same. this means that all three components will be comparable. of course, the value of phi on the left half of the image will be pi/4m + pi. for m=0, we set phi=0. in addradial.py, we set phi according to the first mode entered.
Karen Tian |
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