The coherent properties of six oscillations over a two week period in which seven days of equatorial diameter measurements were analyzed, are confirmed by the addition of an extra day of data. The two large 1 (the principal order number in the spherical harmonic expansion of the eigenfunction) g-mode oscillations may be candidates for the slowly rotating mode locked structures. For the four low frequency p-modes, periodic nature is observed in the daily power levels, varying with periods of several days. This is attributed to beating between rotationally split m states for a given 1 value. Nonradial modes are a major contribution to the observed solar oscillations. The nonradial character of the observed modes allows the depth dependence of the internal solar rotation to be investigated

Caudell, T. P.

Hill, H. A.

English

NASA Technical Reports Server

EVIDENCEFORTHEEXISTENCEOFNONRADIALSOLAROSCILLATIONS:
SOLARROTATION*
ThomasP. Caudell and Henry A. Hill
Department of Physics
University of Arizona
I. INTRODUCTION
In the recent work of Hill and Caudell (1978) the solar origin
of the observed long period oscillations in apparent diameter was
demonstrated. One of the strongest pieces of evidence obtained at
SCLERAT for this conclusion was the phase coherency of six oscillations
over a two-week period in which seven days of equatorial diameter mea-
surements in the 1973 solar oblateness data of Hill and Stebbins (1975)
were analysed. This analysis further showed, through examination of
the ratio of oscillatory power at two different scan amplitudes in the
finite Fourier transform edge definition, the FFTD (see Hill, Stebbins,
Oleson 1975 for discussion of diameter measurement technique), that
the four higher frequency modes were likely to be p-modes while the
two lower to be g-modes with large values of _ _ 20-40, where _ is
the principal order number in the spherical harmonic expansion of the
eigenfunction. The discovery that g-modes of large _ value are ex-
cited to an observable level allows one to place new constraints on
solar models (Hill and Caudell 1978).
This paper extends this form of analysis through inclusion of an
additional day of data, which falls between the first four and the
last three days of the previous analysis. Whenplaced on the phase
diagrams, this subsequent addition confirms the original choice of
phase solutions and strengthens the solar normal mode hypothesis. The
power of the oscillations as a function of day for the p-modes are also
analysed for periodicity. Weobserve that the power fluctuates for
each of the peaks in a well defined manner, going nearly to zero at the
minima. This implies the existence of beating between states of approx-
imately equal amplitudes, an interpretation which in the light of the
properties of the FFTD suggests that the beating is not between states
of the different _ values. It further suggests that the beating is
between rotational split states that differ only in m, the azimuthal
order number, and is reminiscent of Wolff's (1977) work on rotation in
DA white dwarfs. With this interpretation, mode identification of
these oscillations is made, allowing inference of the eigenfunction
weighted internal rotation rate of the sun (Hill 1978).
665
https://ntrs.nasa.gov/search.jsp?R=19800016767 2020-03-21T18:35:17+00:00Z
II. PHASECOHERENCY
The phaseas definedby the Fouriertransformfor eachpeak in
the powerspectrumwascomputedand shifted"toa commontimeof day.
Sincethisis observationallydeterminedto withina multipleof 2_,
the solutionwas pickedwhichgavethe bestfit for a constantphase
shiftperday (seeHilland Caudell1978for details). In Figure!
is plottedthe phasesolutionsfor the equatorialobservationsas a
functionof date,withthe new pointon the 17thincluded.Theseare
indeeduniquesolutionsin thatdisplacingthe lastthreedays up or
down multiples_of 2_ increases ignificantlythe X2 variantof the
linearfit. To illustratethisproposition,the squarerootof X2 for
a givenfit to the originalseven-daysobservationsforeachpeak is
givenin Figure2 plottedagainstthedisplacementin phaseof the last
threedaysas a unit. The standarddeviationof an individualobser-
vationhas beenset arbitarilyto unity. Notethe parabolicnatureof
thesecurvesandthat it is unlikelythata phaseshiftof 2_ hasoc-
curredbetweenthefour-and three-daysetsof observations.This in-
dicatesthatthe oscillationsobservedat SCLERAare phasecoherent
overperiodsof two weeks,furtherproofof theirsolarorigin.
III. TEHPORALPOWERVARIATIONS
The phasecoherencyof the oscillatorypowerovera twoweek
periodimpliesa relativelyhighlevelof stability,i.e.lowdamping.
Thisfactallowsone to examinethe powerof the oscillationsas a
functionof day withoutsuchcomplicatingfactorsthatlead to a change
in themode duringthe interim.Variationsin powerwillariseif
beatingoccursbetweendifferentunresolvedeigenfrequencieswithina
singleobservedpeak (Wolff1976). The beatingpairsof modesmay dif-
fer in m valueas wellas _ valuewhen solarrotationisaccountedfor
(Gough1977). Due to the theoreticallypossiblevaluesof _ in the ob-
servedpeaksand the variationin detectionsensitivitywith_ and m of
the FFTDused in thesediametermeasurements,the temporaldependence
of powercan be valuablein distinguishingbetweenthe two cases.
In Figure3 is plottedthe equatorialoscillatorypoweragainst
date forthe fourp-modepeaks. Throughthe datapointshavebeenfit
a functionof the formAZcos2(_t + @) whereamplitudeA, beatfre-
quencyRb and phase@ were adjustedto minimizetheX2 variant. The
resultantfunctionshavebeendrawnin the figure. Here_ represents
the frequencyof themode in milliHertzand _b the bestfitbeat fre-
quencyin radiansper day. Bothquantitiesare listedin the firsttwo
columnsof TableI. Becauseof the samplingin the timeon a daily
basisthereare higheraliasedvaluesof _b whichwill giveequallygoodfits.
666
There are severalmajor conclusionsto bedrawn from the power
plots. First, the cosine squared functionof a single beat frequency
does indeedfit the observationsquite well. This indicatesthat
perhapsonly one pair of states dominatesthe beatingin each peak.
Second,the power at the minimum goes nearly to zero, often by a
factorof lO0 below the maximum value. Only if the amplitudesof the
two constituentstates are equal to within 20% will this be possible.
If the beatingstateswere of differentZ value, this observational
result requiresthat their amplitudesvary with Z just as the vari-
ation of the FFTD sensitivitywith _ (seeHill 1978 for discussionof
dependenceof the sensitivityof the FFTD). This must be true for
each of the observedpeaks. However, such an assumptionregarding
the _ dependenceof the amplitudesis not requiredif the beats occur
betweenrotationallysplit modes. Since the FFTD favors the mode of
highest_ and Im],the beatingin a given peak of the power spectrum
will be dominatedby the two modes with m = ±_. This appearsto be
the most likely phenomenonto occur in light of the observationsto
date.
In the followingwe presentan interpretationbased on the case
where modes of nonzero_ value are excitedand the beatingpattern
is dominatedby the associatedm = ±_ states. Beats betweenstates
with Iml < _ only add lower beat frequenciescommensuratewith those
for the Im] = _ states. This will allow extractionof information
about the internalrotation of the sun.
IV. ROTATIONALSPLITTINGAND INTERNALROTATION
The hydrogenatom, when given an axis ofsymmetrythroughthe
impositionof an externalmagneticfield, splitsthe otherwiseenergy
degeneratem states and producesa multipletof states,2_ + l in
number for each Z value. As in the hydrogenatom, when the sun is
given an axis ofsymmetryby rotation,a degenerateset of states with
a given _ value will split its eigenfrequencyinto a 24 + l multiplet
(Gough1977, Ledouxand Walraven 1958). The eigenfunctionin the
presenceof rotationhas the form,
where
(2)
Bk_ = Repl }2 + _(_+I 2 r2dr_k_, )nk_
_0
667
is the localrotationalfrequency,p themassdensity,r the radius
of the mass element,and theremainingfunctionsare definedas com-
ponentsof the displacementvectorfora normalmode of a nonrotating
starwrittenas
_k£(r)P_(cosO)eim@, nk£(r)d p_(cosO)eim@ ,
The angles 0 and @ are the standardsphericalcoordinates,P_(x) the
associatedLegendrepolynomialand k is the radial order relatedto
the number of radial nodes in the eigenfunction. Clearly Bk£R is the
eigenfunctionweightedmean of the internalrotationalfrequency.
Identificationof the modes contributingto the four p-modeoscil-
lationsas well as the spatialfilteringfor SCLERA type diameter
measurementsmust be clarifiedbefore 8k£_ can be determined.
The theoreticaleigenfrequencyspectrumof the sun is rich in
states of differing£ value, many ofwhich lie close togetherin
frequency(Iben 1976 and Wolff 1978a). Due to the low resolution
of an eight hour data run, severalof these statesmay form the con-
stituentsof a single observedpeak in power (Hill 1978, §2.2.3).
The detectionsensitivityto a particular£ value for the diameter
measurementsmade at SCLERA dependson the dimensionsof the aperture
and the latitudeof the measurement. Hill (1978)calculatedthe
detectionsensitivityfor oscillationsas a functionof £ value at
the equatorusing the Solutionsto the wave equationgiven by Hill,
Rosenwaldand Caudell (1978). It was found in the region between
= 0 and 12 the relativedetectionsensitivityincreaseswith the
value by a factor of nearly 4 in power. Out of this analysisalso
came the result that for a given £ value, the relativesensitivityto
m reachesa pronouncedmaximumwhen m = ±£ and is a minimumfor the
m = 0 case. Combiningthese two theoreticalresultswe concludethat
althoughseveraldifferenteigenmodesmay be contributingto a given
oscillation,the largest£ value will be detectedthe easiest. If
rotationallysplit, the m = ±£ states will producethe strongest
detectableeffect.
Consultingthe eigenfrequencyspectrumassociatedwith a standard
solar model (Iben 1976, Wolff 1978a),a tentativemode identification
can be made for the modes that dominatethe observed power spectrum.
This is accomplishedby pickingthe theoreticalfrequencieswhich fall
within the observedpeaks. The k value and the maximum £ values as-
sociatedwith each observationalpeak are listed in columns3 and 4
of Table I. Applyingthe resultsof the previousparagraph,we
668
conclude that beating must be occurring between the plus and minus m
states for these largest _ values, Column 5 of the table is the re-
sult of dividing the appropriate alias of the beat frequency by these
maximum_ values to give the smoothest variation in rotation rate;
column 6 is the inverse of column 5 in days where the standard de-
viation has been estimated to be one day. There is also another
solution giving a smoother rotation curve. For this solution, the
rotation rate increases more rapidly than the one given in the table.
The last column in Table 1 gives estimates of the depth to which
the p-mode has substantial amplitude (Wolff 1978b, Figure 2). Inside
this point, the amplitude exponentially decreases to the center, thus
weighting the rotational velocity curve exterior to this point. These
numbers illustrate the tenet that the various beat frequencies contain
the requisite information to extract the depth dependence of solar
rotation.
The correlation between columns 6 and 7 in Table 1 appears to be
statistically significant. Based on this analysis the interior of the
sun appears to be rotating at a somewhat larger rate over the surface,
a result not inconsistent with the recent work on the five-minute
oscillation (Ulrich, Rhodes, and Deubner 1978) and Gough's (1977)
interpretation of 12,2 day periodicities reported in the Princeton
solar oblateness observations (Dicke 1976).
V. CONCLUSION
The addition of a new data point on the phase plots of Hill and
Caudell (1978) confirms the coherent properties of the observed oscil-
lations. The two large _ g-mode oscillations identified in the pre-
vious work may be candidates for the slowly rotating mode-locked struc-
tures of Wolff (1976). For the four low frequency p-modes, periodic
nature is observed in the daily power levels, varying with periods of
several days. This is attributed to beating between rotationally
split m states for a given _ value, an effect similar to that sug-
gested for the DA white dwarfs (Wolff 1977)." In any case, nonradial
modes are a major contribution to the observed solar oscillations;
the nonradial character of the observed modes allows the depth
dependence of the internal solar rotation to be investigated.
The authors wish to acknowledge the use of the computing
facilities of Kitt Peak National Observatory and the assistance of
Mr. Ross Rosenwald with the numerical techniques.
*This work was supported by the National Science Foundation and the
Air Force Geophysical Laboratories.
tSCLERA is an acronym for the Santa Catalina Laboratory for
Experimental Relatively by Astrometry and is a research facility
jointly operated by the University of Arizona and Wesleyan University.
669
Table 1
frad_ /rad_
v('mNz) b_day/ k _ Bk_ (days)day/ Trot reff/Re
0.606 0.57 l lO 0.258 24.4_+l.O .77
0.539 1.15 l 8 0.249 25.2_+l.O .75
0.463 1.41 l 6 0.288 21.8_+l.O .72
0.414 1.05 l 4 0.263 23.9_+l.O .64
REFERENCES
Dicke,R. H. 1976,SolarPhys.,3__77,2 1.
Gough,D. G. 1977,M.N.R.A.S.,submitted.
Hill,H. A. 1978,The New SolarPhysics,ed.J. A. Eddy (Westview
Press,Boulder,Colorado).
Hill,H. A. and Caudell,T. P. 1978,M.N.R.A.S.,in press.
Hill,H. A., Rosenwald,R. D., and Caudell,T. P. 1978,Ap. J., in
press.
Hill,H. A. and Stebbins,R. T. 1975,Ap. J., 200,471.
Hill,H. A., Stebbins,R. T., and Oleson,J. R. 1975,Ap. J., 200,484.
Iben,I.,Jr.,and Mahaffy,J..1976,Ap. J. (Letters),20__99,3 .
Ledoux,P., andWalraven,Th. 1958,Handbuchder Physik,Vol.LI.,
ed, S. Flugge(Berlin,Springer-Verlag),353.
Ulrich,R. K.,Rhodes,E. J., and Deubner,F. L. 1978,preprint.
Wolff,C. L. 1976,Ap. J., 20___55,612.
Wolff,C. L. 1977,Ap. J., 21___66,784.
Wolff,C. L. 1978a,see §2.2.3of Hill1978.
Wolff,C. L. 1978b,submittedfor publication.
670
I J i | | e e | | • | e | e
36"rr '. mHz o/°
32"n" __'_/_ °/°
P.STr 0.539/ o/o"
o/O /" -
-" _ .o/
247/" /° / / /° /
,2Tr .o/°/_.366 / j_/o/°"
8Tr "°/,./°/°
47/" /°/° _o "__0"248
_'1 _ I i I I I I I P I i I I
8 9 I0 II 17 19 2021
Dote
(Sept1973)
Fig. 1. The phaseas defined by the Fourier transform of the observed
oscillations as a function of date. The phase for a given
day and predetemined time of day is defined to within an ad-
ditive multipleof 2_ radians. Forthis plotthe multiple
has beenpreviouslychoosen(Hilland Caudell1978)to yield
the bestfit to the datanot includingthe 17thof September.
The resultsof the 17thhave beenaddedmakingno changein
the previousfits. The frequencyfor eachoscillatoryperiod
obtainedfromtheseobservationsis listedon the figurein
mHz,
671
' • ! ! | I I l I I I
I 0.414 mXz .606 mHz.
O •
.366 mHz
I .539mHz"
b 0
2
.248 mHz .463 mHz
0 _ , , I , I I I I I I
-2-1 0 +1 +2 -2-1 0+1 +2
Displocements
Fig.2. The squarerootof theX2 variantof the linearfitsin
Fig.l wherethe lastthreedays havebeendisplacedup
and downmultiplesof 2_. The standarddeviationsfor
the individualobservationshavebeenset to one. The
zeroof the horizontalscaleis takenas the solutions
plottedin Fig.I.
672
I I i ! I I i I I I l m I t I s
5O
0 - _0.606 mltz"
.:_ 50 o
-
0.559 mHz_i ° m,
50 o)= -,
0
0.. 0.463 mtlz0
50
OAI4
0 8 9 I0 II 17 1920 21
Date(Sept.1973)
Fig. 3. The power level in units of milli-arcsecsquaredper fre-
quency bin (0.03- mHz) for the four p-mode oscillations
as a functionof date. The smoothcurve is the least
square fit to the points with the functionA2cos2(Rbt+ €).
673
Discussion
[Questions interjected during the paper]
Stellingwerf: What determines the slopes in Fig. i?
Hill: For instance, if there were exactly 24 periods of oscillation in one
day, the slope can, for example, be either 0, +i, +2, -i, or -2. But if the
period comes out to be, say, 45.5 minutes -- that's 31.65 periods in a day,
then it's the 0.65 that determines the slope.
A. Cox: What is the 0.463 mHz oscillation like in Fig. i?
Hill: This one is quite close to being diurnal -- it's off by about i .
[After Talk]
J. Cox: One question about differential rotation -- if you imagine that the
rotational angular frequency is constant on a cylinder, isn't the interpre-
tation ambiguous because of the variation from latitude to latitude in the
rotation rate, i.e. differential rotation?
Hill: But if the oscillation is coherent over a time long compared to the
period of oscillation, thenwhen one measures rotational splitting, one
f
measures an average over the entire Sun. So it doesn't matter whether it's
a longitudinal slice in cylinders or a slice in spherical surfaces. One
doesn't measure the splitting locally; it's a weighted average. The equation
that was written down by van Horn had 0 outside the integral because he had
uniform rotation, but as soon as you have non-uniform rotation, _ goes inside
the integral and you get a weighted average of _ over the entire volume.
674
Wesselink: Did I see that the rotational period was decreasing as you go
down?
Hill: Yes, the period was decreasing as you went in.
Wesselink: Then it goes up again?
Hill: No, I think that's just scattering. The scatter was around one day
in the period. There must be several interpretations, and What I've pre-
sented is just one interpretation. If you want to use it, you must conclude
there's a trend there. There may also be more structure.
675


Evidence for the existence of nonradial solar oscillations: Solar rotation

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