3. 3
Table of Contents
I. Abstract – Page 4
II. Hypothesis – Page 5
III. Background Information – Page 6-7
IV. Procedure – Page 8-10
V. Data – Page 11-13
a. Radius vs. Frequency Graph
b. Radius vs. Frequency (2.666 mHz – 3.166 mHz) Graph
c. Radius vs. Frequency (3.333 mHz – 3.833 mHz) Graph
d. Radius vs. Frequency (4.000 mHz – 4.333 mHz) Graph
VI. Data Analysis – Page 14
VII. Conclusion – Page 15
VIII. References/Acknowledgements – Page 16-18
4. 4
Abstract
The purpose of this research was to determine whether the difference in travel
times of solar p-waves really is zero below the acoustic cutoff of 4 mHz when
measurements are taken at two different solar altitudes. To do this, full disk doppler
velocity images were taken of the Sun and compiled into two stacks, one for images
taken using a sodium spectra and one for images taken using a potassium spectra. These
two spectra correspond to two different solar altitudes. To determine how similar or
dissimilar the travel times were at these two altitudes, power spectra were generated for
each. The trumpet structure of these power spectra were compared by taking many cross-
sections of the trumpets, fitting circles to the rings generated, and comparing the radii of
these circles with respect to the temporal frequency of the cross-sections. When best fit
lines were fit to the plots of radius vs. frequency for each of the two curves, the quadratic
mean of the error between these lines and their data were far less than the mean error
between the sodium and potassium curves themselves. Such results imply that the
difference between the trumpet structure for the sodium and potassium power spectra
were significantly different and therefore that the travel times were different. In this way,
these results offer evidence against the hypothesis but are not conclusive as more data
and a more detailed analysis would be required to make such a claim.
5. 5
Hypothesis
If the travel times of solar p-waves are measured at distinct solar altitudes, then,
when the power spectra at the two altitudes are compared, no difference in wave travel
time will be detected.
6. 6
Background Information
P-waves, also known as acoustic waves, have pressure as their primary restoring
force, and the variations in the speed of sound within the Sun largely govern the
mechanisms by which these waves behave. Most of these oscillations have frequencies
primarily in the range of 2-4 mHz and are detectable at the solar surface, having
amplitudes of hundreds of kilometers. The large majority of observations of solar p-
waves are conducted through Doppler imaging or spectral intensity imaging, with the
former being used specifically for this research. The study of these p-waves is quite
important, because the analysis of the distribution of p-wave frequencies as a function of
space has allowed for many of the discoveries pertaining to the solar atmosphere and the
Sun’s convection zone within the field of helioseismology.
It is currently believed that solar p-waves exist as evanescent waves in the solar
atmosphere, meaning that the travel times of these waves would be the same regardless of
the height at which they were measured. When imaging the Solar atmosphere, it is
common to use the spectra of many different elements present in the Sun such as sodium
or potassium. Instruments can simply analyze the spectral fingerprint of a specific
element and track the red and blue shifts in these spectral lines, thus measuring the line of
sight velocity. Instruments such as these are utilized to study p-waves. Because the point
of last scattering for the sodium spectra is different than that of the potassium spectra,
instruments that detect acoustic oscillations via one of these elemental spectra would
actually be analyzing different altitudes in the solar atmosphere. According to the current
understanding of the Sun within the heliophysics community, because p-waves should be
7. 7
evanescent at these frequencies, such a difference in solar altitude should have no effect
on the measure of p-wave travel times.
The purpose of the research is to determine whether, when measurements are
taken at two different solar altitudes, the difference in travel times really is zero and
whether or not there is any significant difference below the acoustic cutoff of 4 mHz.
Such research is imperative in the future study of the Sun as such research hinges on the
comparison of data from several instruments that all use different spectra to observe the
Sun.
8. 8
Procedure
Before any analysis of data could take place, images of the Sun had to first be
collected. For this project, the images used were taken by Dr. Stuart Jefferies at the
geographic South Pole with the “MOTH” (Magneto Optical Filters at Two Heights)
instrument he designed with his colleague. This instrument is what made it possible to
obtain images taken in two different spectra, sodium and potassium, and thus at two
different solar altitudes. Additionally, the geographic South Pole was the ideal location to
take these images as the sun is visible for months at a time and continuous viewing is
possible.
Two sets of images were used for this project: a set of three thousand taken in the
sodium spectra and a set of three thousand taken in the potassium spectra. Both sets were
taken at the same time with a ten second cadence over the span of eight-and-a-half hours.
All of these images were full disk images of the Sun (the entire Sun was visible). Once
these images were obtained, two stacks were created for the sodium and potassium
spectra (thee dimensional arrays created from the three thousand images). Only with the
images in stacks would the mathematical analysis be possible.
With the stacks created, the next step was to decide upon what region of the Sun
would be analyzed. To do this, using a program in IDL, a time travel map was created
from the stacks. Time travel maps are maps that display the travel times between two
regions after calculating the Fourier Transform for each stack. On the map, low travel
time is marked by light colors and high travel time is marked by dark colors. In general,
the dark regions indicated areas of magnetic contamination (possibly due to sunspots)
where the p waves were no longer evanescent. Because this project is concerned with
9. 9
only regions where the p waves would be evanescent and the travel time therefore zero
between two altitudes, a region on the Sun was chosen that showed little magnetic
contamination on the time travel maps. It is this region of the Sun that would be analyzed
for the rest of the project.
With the stacks created and the region of interest determined, the analysis of the
images could progress. First, the Fourier Transforms were taken of each stack and
multiplied by their complex conjugates. This produced three-dimensional power spectra
plotting the two spatial frequencies with respect to temporal frequency. For frequencies
under the acoustic cutoff (4 mHz) these power spectra produced “trumpet” shaped
formations with the symmetrical axis of the trumpets aligned with the axis of temporal
frequency. The goal from this point on was to analyze how similar or dissimilar the
trumpet structures were for the sodium and potassium spectra.
To compare the trumpet structures, cross-sections of the power spectra were
examined at specific temporal frequencies. These cross-sections of the trumpet structures
appeared as rings when analyzed; as cross-sections were taken at higher and higher
temporal frequencies the rings would grow and new rings would appear to form near the
center. The eleven cross-sections that were taken for each stack started at 2.667 mHz and
ran to 4.333 mHz at consistent intervals.
To compare how similar the rings at a certain temporal frequency were from the
sodium to potassium spectra, a program was written in IDL that would fit a circle to a
certain ring. The program was written to find the radius that would maximize the average
power present in the width of the circle that was being analyzed. Because the rings of the
trumpet structures are not perfect circles (though quite close), but rather ellipses, the
10. 10
width of the circle was chosen such that the entire ellipse would be contained within its
range. The same width was used for every ring. This program was run for every cross-
section in both the sodium and potassium spectra. As for the logistics of choosing what
ring to fit the circle to, strong rings were chosen that were unambiguously separate from
those adjacent and that were small on the first frame. As the rings grew as higher and
higher temporal frequencies were analyzed, a ring could only be followed for four cross-
sections before it grew too large, and a new smaller ring was followed for the next four
frames. After the program was run for each of the twenty-two cross-sections, the radius
of each ring could be determined and plotted for both the sodium and potassium stacks.
11. 11
Data
The following graph shows the radius of the fit circles with respect to the
temporal frequency of the cross-section.
In this as well as all of the following plots, the red data are representative of the
sodium analysis and the blue data of the potassium analysis. The zigzag pattern is due to
the fact that a certain ring could only by followed for four cross-sections before it got too
large and a new, smaller ring had to be followed. The following three graphs eliminated
this zigzag pattern by breaking this plot into three smaller graphs by ring.
15
17
19
21
23
25
27
29
31
33
2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5
AverageRadius(Pixels)
Frequency (mHz)
Radius vs. Frequency
K
Na
12. 12
y = 22.68x - 44.2
y = 23.94x - 46.8
0
5
10
15
20
25
30
35
2.6 2.8 3 3.2
AverageRadius(Pixels)
Frequency (mHz)
Radius vs. Frequency (2.666 mHz -
3.166mHz)
K
Na
Linear (K)
Linear (Na)
y = 18.9x - 42.85
y = 15.93x - 30.445
0
5
10
15
20
25
30
35
3.2 3.4 3.6 3.8 4
AverageRadius(Pixels)
Frequency (mHz)
Radius vs. Frequency (3.333 mHz - 3.833
mHz)
K
Na
Linear (K)
Linear (Na)
13. 13
For these three plots, best fit lines were fit to each curve; the equations for each
can be seen on the plots.
y = 16.65x - 42.358
y = 18.3x - 48.383
0
5
10
15
20
25
30
35
3.9 4 4.1 4.2 4.3 4.4
AverageRadius(Pixels)
Frequency (mHz)
Radius vs. Frequency (4.000 mHz - 4.333
mHz)
K
Na
Linear (K)
Linear (Na)
14. 14
Data Analysis
Preliminary analysis of the four plots reveals that the circles fit to the rings of the
cross-sections taken from the Fourier Transform of the sodium spectra tend to have large
average radii than that of their potassium counterparts. To find a more quantitative
procedure to this end, the best fit lines had to be used. Once the best fit lines were fit to
each plot, the error between the actual values and the predicted values were found for
each of the twenty-two data points. For both the sodium and potassium data, the
quadratic mean was found. For the eleven measures of error for the sodium data, the
quadratic mean was calculated to be 1.338. For the eleven measures of error for the
potassium data, the quadratic mean was calculated to be 1.031. Next, the difference
between the data values of the sodium radii and potassium radii were found. The
quadratic mean for these eleven measures of difference was found to be 4.980, about four
times the magnitude of the quadratic mean of the two measures of error. In each case, the
quadratic mean was chosen over a traditional mean, because a measure of the magnitude
of the variation was more desirable than a measure of the true mean (which would have
been approximately zero for the two measures of error).
15. 15
Conclusion
After observing the trumpet structures and analyzing the ring cross sections
generated from the power spectra for both sodium and potassium data, evidence was
found against the hypothesis. When looking at the error between the best fit lines and the
plots of radius vs. frequency, the magnitude of the quadratic mean was found to be 1.338
for the sodium curves and 1.031 for the potassium curves. Because the relation between
radius and frequency is not expected to be linear (thus a “trumpet” shape as opposed to a
“cone” shape) these lines of best fit over-approximate the error. In contrast, the quadratic
mean of the error between the measured radii for sodium and the measured radii for
potassium was found to be 4.980. As this figure is approximately four times the
magnitude of the other two measures of error that were themselves over-approximations,
this difference can be considered significant. Such significance provides evidence that p-
waves may not be completely evanescent as expected and that the travel times of these
waves may not be the same at two different heights in the Sun’s atmosphere.
While the conclusion generated from the research is intriguing, it is not yet
definitive. Future research is needed that could collect much more data and perform a
more detailed analysis. With the analysis of many regions on the Sun and a more dense
comparison of the cross sections generated from the separate power spectra for each
region and spectra, a more definitive conclusion can be drawn about the hypothesis.
In any case, this research serves as a point of interest in moving forward as the
implications of conclusive findings against the hypothesis would cause for much of what
the heliophysics community currently knows to be carefully reexamined.
16. 16
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Dr. JD Armstrong
18. 18
Keith Imada
Dr. Stuart Jefferies
University of Hawaii Institute for Astronomy
Maikalani Advanced Technology and Research Center
All graphs student-generated
