Science Fair 2015

Detection and Comparison of Solar P-

mode Travel Times at Distinct Solar

Altitudes of Sodium and Potassium Spectra

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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

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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.

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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.

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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

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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.

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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

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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

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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.

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Data

The following graph shows the radius of the fit circles with respect to the

temporal frequency of the cross-section.

2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.515171921232527293133

Radius vs. Frequency

KNa

Frequency (mHz)

Ave

rage

Rad

ius

(Pix

els)

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.

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2.6 2.7 2.8 2.9 3 3.1 3.20

5

10

15

20

25

30f(x) = 23.9399999999999 x − 46.7999999999998f(x) = 22.68 x − 44.1999999999999

Radius vs. Frequency (2.666 mHz - 3.166mHz)

KLinear (K)NaLinear (Na)

Frequency (mHz)

Ave

rage

Rad

ius

(Pix

els)

3.3 3.4 3.5 3.6 3.7 3.8 3.90

5

10

15

20

25

30

35

f(x) = 15.93 x − 30.4449999999999f(x) = 18.9 x − 42.8499999999999

Radius vs. Frequency (3.333 mHz - 3.833 mHz)

KLinear (K)NaLinear (Na)

Frequency (mHz)

Ave

rage

Rad

ius

(Pix

els)

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3.95 4 4.05 4.1 4.15 4.2 4.25 4.3 4.35 4.40

5

10

15

20

25

30

35

f(x) = 18.3000000000002 x − 48.3833333333343f(x) = 16.6500000000001 x − 42.3583333333338

Radius vs. Frequency (4.000 mHz - 4.333 mHz)

KLinear (K)NaLinear (Na)

Frequency (mHz)

Ave

rage

Rad

ius

(Pix

els)

For these three plots, best fit lines were fit to each curve; the equations for each

can be seen on the plots.

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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).

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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.

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Dr. JD Armstrong

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Keith Imada

Dr. Stuart Jefferies

University of Hawaii Institute for Astronomy

Maikalani Advanced Technology and Research Center

All graphs student-generated

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