Using data from the Global Oscillation Network Group (GONG) covering the
period from 1995 to 1998, we study the change with solar activity in solar
f-mode frequencies. The results are compared with similar changes detected from
the Michelson Doppler Imager (MDI) data. We find variations in f-mode
frequencies which are correlated with solar activity indices. If these changes
are due to variation in solar radius then the implications are that the solar
radius decreases by about 5 km from minimum to maximum activity.Comment: To appear in Solar Physic

AD Fiala

AD Wittmann

CA Iglesias

E Anderson

F Laclare

F Noeel

FJ Rogers

HM Antia

J Schou

P Delache

RL Kurucz

SC Tripathy

TL Duvall Jr.

TM Brown

VM Canuto

WA Dziembowski

WR Campbell

English

arXiv

H. M. Antia

Sarbani Basu

J. Pintar

B. Pohl

Crossref

Solar Cycle Variation in Solar F-Mode Frequencies and Radius

H.M. Antia

Using data from the Global Oscillation Network Group (GONG) covering the period from 1995 to 1998, we study the change with solar activity in solar f-mode frequencies. The results are compared with similar changes detected from the Michelson Doppler Imager (MDI) data. We find variations in f-mode frequencies which are correlated with solar activity indices. If these changes are due to variation in solar radius then the implications are that the solar radius decreases by about 5 km from minimum to maximum activity

Antia, H. M.

Basu, Sarbani

Pintar, J.

Pohl, B.

Publications of the IAS Fellows

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SOLAR CYCLE VARIATION IN SOLAR F-MODE
FREQUENCIES AND RADIUS
H. M. Antia
Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005,
India
Sarbani Basu
Institute for Advanced Study, Olden Lane, Princeton NJ 08540, U. S. A., and
Astronomy Deparment, Yale University, P.O. Box 208101 New Haven, CT
06520-8101 USA
J. Pintar and B. Pohl∗
National Solar Observatory, PO Box 26732, Tucson AZ 85726-6732, U. S. A.
February 1, 2008
Abstract. Using data from the Global Oscillation Network Group (GONG) cov-
ering the period from 1995 to 1998, we study the change with solar activity in solar
f-mode frequencies. The results are compared with similar changes detected from
the Michelson Doppler Imager (MDI) data. We find variations in f-mode frequencies
which are correlated with solar activity indices. If these changes are due to variation
in solar radius then the implications are that the solar radius decreases by about 5
km from minimum to maximum activity.
Keywords: Sun: general – Sun: Oscillations
1. Introduction
The fundamental mode or f-mode of solar oscillations are believed to
be surface gravity modes whose frequencies are essentially independent
of the stratification in the solar interior. A large part of the difference
between the observed f-mode frequencies at intermediate degree and
those in a standard solar model has been interpreted as being caused
by the incorrect radius of the solar models used (Schou et al. 1997; An-
tia 1998; Brown and Christensen-Dalsgaard 1998; Tripathy and Antia
1999). The frequencies of these modes can thus be used to measure the
solar radius.
There have been many reports about possible variation of solar ra-
dius with time (Delache, Laclare and Sadsaoud 1985; Wittmann, Alge
and Bianda 1993; Fiala, Dunham and Sofia 1994; Laclare et al. 1996;
Noeel 1997). The reported change in measured angular semi-diameter
varies from 0.1′′ to 1′′, which implies a change of 70 to 700 km in
∗ present address: US Naval Observatories, Washington, DC 20392-5420, U. S. A.
fmodpap.tex; 1/02/2008; 12:40; p.1
2 H. M. Antia et al.
radius. However, there is no agreement among observers about these
variations. It would thus be interesting to look for corresponding varia-
tions in the f-mode frequencies. The reported variations in solar radius
should change the f-mode frequencies by 0.1 to 1 µHz, which is much
larger than the error estimates in these frequencies. Considering how
small the estimated errors in f-mode frequencies are, it is possible to
measure changes in the solar radius as small as a few kilometers over the
solar cycle. Some variation in f-mode frequencies have been reported
in the MDI data by Dziembowski et al. (1998) who find that the solar
radius reached a minimum around the minimum activity period and
was larger by about 5 km, 6 months before and after the minimum.
If this variation is correlated with solar activity then we would expect
much larger change between minimum and maximum activity.
In this work we look for possible variations in the solar radius using
GONG data — which extend over longer time period than the MDI
data used in previous studies. We also look for possible correlations
between change in radius and solar activity. In Section 2 we outline
the technique used in calculating the frequencies and examine possible
systematic errors in computed frequencies. Section 3 gives the results
while in Section 4 we summarize the conclusions from our study.
2. The technique
Since the f-modes have very low power they are barely visible in the
spectra for modes of a given degree ℓ and azimuthal order m that
the GONG project fits to determine solar oscillation frequencies. As a
result, it is difficult to determine the frequencies of these modes reliably
from these spectra. To improve statistics we use the rotation-corrected,
m-averaged, power spectra from GONG data (Pohl and Anderson 1998)
to calculate the mean f-mode frequencies. Because summing over the
2ℓ+1 spectra for a mode of degree ℓ makes the peaks better defined, it
is possible to fit them reliably. Each of these spectra was obtained from
3 GONG months (108 days) of data and has a frequency resolution
of 0.107 µHz. These spectra extend up to ℓ = 200 and frequency ν =
2083 µHz. The f-mode peaks are clearly visible for ℓ >∼ 100. We have
also used two spectra covering only 1 GONG month, which gives a
lower frequency resolution of 0.321 µHz. For these spectra the errors
in computed frequency would be larger as compared to the 3 month
spectra.
We fit each mode separately using a maximum likelihood technique
(Anderson, Duvall and Jefferies 1991) with a symmetric Lorentzian
profile for the peaks. The fit is performed over a region extending to
fmodpap.tex; 1/02/2008; 12:40; p.2
Solar cycle variation in f-mode frequencies 3
Figure 1. Fit to the ℓ = 158 f-mode spectra for months 33–35. Power is in arbitrary
units.
about 20 µHz on either side of the peak and includes leaks from modes
of degree ℓ − 3 to ℓ + 3, ℓ being the degree of the target mode. It is
found that peaks arising from the ℓ+ 1 leak are split into two because
of leaks from peaks of m± 1. Similarly, the ℓ+2 leak is split into three
because of leaks from peaks of m,m ± 2. Although leaks from ℓ − 1
and ℓ− 2 are also split in a similar manner, the power in these peaks
is much smaller. Thus we have explicitly fitted the m-leaks only from
the ℓ+ 1 and ℓ+ 2 peaks. An example of the fit is shown in Fig. 1.
For ν < 1170 µHz, the modes are difficult to fit reliably since the
width is smaller than the resolution limit of the spectra and the power is
relatively low. As a result, in this work we only include higher frequency
modes.
The use of m-averaged spectra may introduce some systematic er-
rors in determining the frequencies. To check for these we compare
the frequencies of p-modes fitted from m-averaged spectra with those
obtained by the GONG project using the individual m spectra. The
fmodpap.tex; 1/02/2008; 12:40; p.3
4 H. M. Antia et al.
Figure 2. Difference in frequencies for months 33–35 between the values obtained
from the m-averaged spectra and those computed from the individual m-spectra.
For clarity only a few error-bars are shown.
GONG project has computed the mean frequency for each n, ℓmultiplet
by fitting the frequencies for individual values of m to polynomials in
m. This mean frequency can be compared with the frequency computed
from the m-averaged spectra. The results are shown in Fig. 2. It may
be noted that f-mode has not been fitted in the individual m spectra
and hence these are not included. It can be seen that the frequency
difference is quite small being of the order of the estimated errors in
the fitted frequencies. Systematic difference between these two sets of
frequencies is <∼ 10 nHz.
Another possible source of systematic errors is the use of incor-
rect even-order splitting coefficients while constructing the m-averaged
spectra. The GONG spectra were obtained by setting these coefficients
to zero. A non-zero value of these even-order coefficients would intro-
duce some systematic shift in frequencies. To estimate the effect of this
we constructed some m-averaged spectra with either a2 = −0.2 nHz or
a4 = 0.1 nHz. The resulting frequency shifts with a2 = −0.2 nHz are
shown in Fig. 3. Similar shifts are found for non-zero a4. Once again
the frequency shifts due to non-zero a2 or a4 are quite small, being of
the order of 10 nHz. Furthermore, the frequency shift for the f-mode
is smaller than those for p-mode with the same ℓ. Although these even
coefficients are known to change with solar cycle, the variation is not
fmodpap.tex; 1/02/2008; 12:40; p.4
Solar cycle variation in f-mode frequencies 5
Figure 3. Shift in frequencies for months 21–23 arising due to non-zero a2
very large at the low frequencies which we are considering in this work.
Thus the shifts shown in Fig. 3 can be considered to be conservative
upper limits to errors that may be expected due to non-zero even
coefficients. From Fig. 2 it can be seen that during a period where
even coefficients are fairly large the difference in frequencies computed
from individual m-spectra and the m-averaged spectra is quite small
— much smaller than the differences seen in Fig. 3.
Considering possible systematic errors, it would be desirable to re-
strict ourselves to low frequency range where these errors are not signifi-
cant. In the frequency range (ν < 1440 µHz) considered in this work the
systematic errors from these effects are expected to be about 10 nHz,
and any time variation in these systematic effects would be smaller.
Having established the reliability of frequencies from m-averaged spec-
tra we calculate the frequencies from 7 different spectra during 1995 to
1998. These spectra are listed in Table 1. This table also lists the solar
activity as measured by the mean daily sunspot number, RI from the
Solar Geophysical Data web page of the US National Geophysical Data
Center (http://www.ngdc.noaa.gov/stp/stp.html).
fmodpap.tex; 1/02/2008; 12:40; p.5
6 H. M. Antia et al.
Table I. List of GONG spectra used
GONG months Starting date Ending date Mean daily sunspot no.
3 – 5 1995-07-18 1995-11-02 14.5 ± 1.3
8 – 10 1996-01-14 1996-04-30 6.0± 0.6
21 – 23 1997-04-26 1997-08-11 14.7 ± 1.1
27 – 29 1997-11-28 1998-03-15 39.5 ± 1.7
33 – 35 1998-07-02 1998-10-17 79.5 ± 2.5
36 1998-10-18 1998-11-22 62.4 ± 4.1
37 1998-11-23 1998-12-28 82.6 ± 4.1
3. Results
We have fitted each of the 7 spectra listed in Table 1 to calculate the
mean frequencies of the f modes. Fig. 4 shows the relative difference
between the observed frequencies for Months 33–35 and Months 8–10.
It can be seen that to a first approximation the relative frequency differ-
ence is independent of frequency and hence this frequency difference can
be interpreted as arising from a change in the solar radius. Of course,
suitable combination of other effects like magnetic field, convection etc.
may also yield frequency differences which are independent of frequency
but that will almost certainly require some fine-tuning of parameters,
while change in solar radius would be a simpler explanation.
To estimate the possible change in radius we take an average over the
relative frequency difference for all 56 f-modes that were successfully
fitted in all spectra:
〈δν〉 =
∑
ℓ
δνℓ
σ2
ℓ∑
ℓ
1
σ2
ℓ
(1)
The differences were taken with respect to calculated frequencies for a
standard solar model, which is a static model, constructed with OPAL
opacities (Iglesias & Rogers 1996) and low temperature opacities from
Kurucz (1991). The OPAL equation of state (Rogers, Swenson & Igle-
sias 1996) was used to construct the model and convective flux was
calculated using the the formulation of Canuto and Mazzitelli (1991).
The radius of the model is 695.78 Mm (Antia 1998). The results are
shown in Fig. 5, which also shows the mean daily sunspot number,
RI , for each of these periods. The change in frequencies appears to be
correlated with the sunspot number.
fmodpap.tex; 1/02/2008; 12:40; p.6
Solar cycle variation in f-mode frequencies 7
Figure 4. Relative frequency differences between f-mode frequencies from the spectra
for months 33–35 and months 8–10.
Figure 5. The averaged frequency difference for f-modes between the Sun and a
standard solar model with radius R = 695.78 Mm as a function of time is shown
by the continuous line. The mean daily sunspot number RI is also shown by points
with scale shown on the right side. For clarity these points have been slightly shifted
along the x-axis.
fmodpap.tex; 1/02/2008; 12:40; p.7
8 H. M. Antia et al.
If the change in frequencies is entirely due to change in radius then
the average frequency difference is related to change in radius (Antia
1998) by
〈
δν
ν
〉 = −
3
2
δR
R
(2)
Thus a relative shift in frequencies by 10−5 corresponds to a change in
radius by 0.67 × 10−5R⊙ ≈ 4.7 km. It may be argued that the change
of frequency may be due to some other effect, e.g., magnetic field. Since
the relative frequency difference between the frequencies at two differ-
ent epochs is almost independent of frequency in the range of degrees
considered here, a change in radius is the simplest explanation for the
frequency shift. Of course, the magnetic field is directly correlated to
activity indices, but the direct effect of magnetic field, is likely to yield
frequency differences that will increase rapidly with frequency since the
higher frequency modes are located closer to solar surface where these
effects are likely to be more important (Campbell and Roberts 1989).
It may be noted that a reduction in solar radius by 4.7 km in 2 years
will generate about 100 times the solar luminosity through release of
gravitational potential energy, if the contraction in homologous. This
indicates that the change in radius is a superficial effect confined to
outer layers where the f-modes are located.
However, the radius appears to decrease with increasing activity.
This is not what is seen by Dziembowski et al. (1998). The magnitude
of change found in GONG data during the period covered by their
study is much smaller than what they have reported. The apparent
discrepancy is most likely due to a different range of degrees used
in their study. They have probably used modes up to ℓ = 300 and
these high wavenumber modes are known to deviate significantly from
the simple dispersion relation (Duvall, Kosovichev and Murawski 1998;
Antia and Basu 1999). As a result, one can expect misleading results if
average is taken over all modes, since it is possible that the deviation
of the modes from the normal dispersion relation has some solar cycle
dependence. A part of the deviation is known to be due to the use
of symmetric profiles in calculating the f-mode frequencies (Antia and
Basu 1999) and it is quite possible that the asymmetry changes with
activity. Thus it would be better to confine our attention to modes with
ℓ < 200, where these effects are minimal. The results shown in Fig. 2
of Dziembowski et al. (1998) imply a decrease in frequency by 10−5
between 1996.8 and 1997.2. If this change is correlated with sunspot
number then we would expect a decrease by at least 5× 10−5 between
1996 and 1998.6 in our results. Certainly no such shift is seen in our
results. This shift is also much larger than the expected systematic
errors in our results.
fmodpap.tex; 1/02/2008; 12:40; p.8
Solar cycle variation in f-mode frequencies 9
Figure 6. The averaged frequency difference for f-modes between the Sun and a
standard solar model with radius R = 695.78 Mm are plotted against the mean
sunspot number during the period of observation. The average over three different
frequency ranges as marked in the figure are compared with each other. The line
marks the best linear fit for the full frequency range. The triangles and squares are
slightly shifted along x-axis for clarity.
In order to check the stability of our results we have repeated the
analysis by restricting the frequencies to 1.17 < ν < 1.32 mHz and
1.32 < ν < 1.44 mHz and the results are shown in Fig. 6. It is clear
that the results are not particularly sensitive to the range of frequencies
used. However, the correlation with solar activity is markedly weaker
in the higher frequency range. From the relative frequency differences
between the observed frequencies and those in a standard solar model
it is known (Tripathy & Antia 1999) that there is a distinct tendency of
relative frequency differences decreasing with frequency in this range.
It is likely that this trend is due to some other effect like magnetic
field which varies with solar cycle. In that case it will interfere with the
signal due to change in radius. It is quite possible that at even higher
frequencies used in MDI study, the correlation is further weakened
because there is an even stronger decrease in the relative frequency
differences.
fmodpap.tex; 1/02/2008; 12:40; p.9
10 H. M. Antia et al.
4. Conclusions
We have calculated the mean f-mode frequencies from the m-averaged
spectra from GONG data covering 108 days. The use of m-averaged
spectra enable us to determine the frequencies reliably. The relative
frequency difference between observed frequencies and those calculated
for a standard solar model are roughly independent of ℓ in the range
that was used in this study (100 < ℓ < 200). Hence it is likely that this
difference is due to difference in solar radius, as other agents are likely
to introduce a steep frequency dependence in relative frequency differ-
ences. The mean shift in f-mode frequencies appears to be correlated
with solar activity indices like the mean daily sunspot number RI .
If this frequency shift is due to change in solar radius then the radius
decreases as activity increases. Further, the change in the solar radius
during the solar cycle is approximately 5 km. This is one or two orders of
magnitude smaller than the changes reported earlier (Delache, Laclare
and Sadsaoud 1985; Wittmann, Alge and Bianda 1993; Fiala, Dunham
and Sofia 1994; Laclare et al. 1996; Noeel 1997).
These results on the change in solar radius with solar cycle are much
less than what has been deduced using MDI data (Dziembowski et
al. 1998) and we believe the discrepancy is due to their using higher
degree modes whose frequencies can be affected by a variety of other
agents, since these modes are localized closer to the solar surface.
Acknowledgements
This work utilizes data obtained by the Global Oscillation Network
Group (GONG) project, managed by the National Solar Observatory,
a Division of the National Optical Astronomy Observatories, which is
operated by AURA, Inc. under a cooperative agreement with the Na-
tional Science Foundation. The data were acquired by instruments op-
erated by the Big Bear Solar Observatory, High Altitude Observatory,
Learmonth Solar Observatory, Udaipur Solar Observatory, Instituto de
Astrofisico de Canarias, and Cerro Tololo Inter-American Observatory.
References
Anderson, E., Duvall, T., and Jefferies, S.: 1990, Astrophys. J. 364, 699.
Antia, H. M.: 1998, Astron. Astrophys. 330, 336.
Antia, H. M., and Basu, S.: 1999, Astrophys. J. 519, 400.
Brown, T. M., and Christensen-Dalsgaard, J.: 1998, Astrophys. J. 500, L195.
Campbell, W. R., and Roberts, B.: 1989, Astrophys. J. 338, 533.
fmodpap.tex; 1/02/2008; 12:40; p.10
Solar cycle variation in f-mode frequencies 11
Canuto, V. M., and Mazzitelli, I.: 1991, Astrophys. J. 370, 295
Delache, P., Laclare, F., and Sadsaoud, H.: 1985, Nature 317, 416.
Duvall, T. L., Jr., Kosovichev, A. G., and Murawski, K.: 1998, Astrophys. J. 505,
L55.
Dziembowski, W. A., Goode, P. R., DiMauro, M. P., Kosovichev, A. G., and Schou,
J.: 1998, Astrophys. J. 509, 456.
Fiala, A. D., Dunham, D. W., and Sofia, S.: 1994, Solar Phys. 152, 97.
Iglesias, C. A., and Rogers, F. J.: 1996, Astrophys. J. 464, 943.
Kurucz R. L.: 1991, in Stellar Atmospheres: Beyond Classical Models. Eds., L.,
Crivellari, T. Hubeny, D. G. Hummer, NATO ASI Series, Kluwer, Dordrecht,
p.441
Laclare, F., Delmas, C., Coin, J. P., and Irbah, A.: 1996, Solar Phys. 166, 211.
Noeel, F.: 1997, Astron. Astrophys. 325, 825.
Pohl, B., and Anderson, E.: 1998, in Structure and Dynamics of the Interior of the
Sun and Sun-like Stars, Eds., S. Korzennik, and A. Wilson, ESA SP418, p297.
Rogers, F. J., Swenson, F. J., and Iglesias, C. A.: 1996, Astrophys. J. 456, 902
Schou, J., Kosovichev, A. G., Goode, P. R., and Dziembowski, W. A.: 1997,
Astrophys. J. 489, L197.
Tripathy, S. C., and Antia, H. M.: 1999, Solar Phys. 186, 1.
Wittmann, A. D., Alge, E., and Bianda, M.: 1993, Solar Phys. 145, 205.
fmodpap.tex; 1/02/2008; 12:40; p.11
fmodpap.tex; 1/02/2008; 12:40; p.12


Solar cycle variation in solar f-mode frequencies and radius

Indian Academy of Sciences

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SOLAR CYCLE VARIATION IN SOLAR F-MODE
FREQUENCIES AND RADIUS
H. M. Antia
Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005,
India
Sarbani Basu
Institute for Advanced Study, Olden Lane, Princeton NJ 08540, U. S. A., and
Astronomy Deparment, Yale University, P.O. Box 208101 New Haven, CT
06520-8101 USA
J. Pintar and B. Pohl∗
National Solar Observatory, PO Box 26732, Tucson AZ 85726-6732, U. S. A.
February 1, 2008
Abstract. Using data from the Global Oscillation Network Group (GONG) cov-
ering the period from 1995 to 1998, we study the change with solar activity in solar
f-mode frequencies. The results are compared with similar changes detected from
the Michelson Doppler Imager (MDI) data. We find variations in f-mode frequencies
which are correlated with solar activity indices. If these changes are due to variation
in solar radius then the implications are that the solar radius decreases by about 5
km from minimum to maximum activity.
Keywords: Sun: general – Sun: Oscillations
1. Introduction
The fundamental mode or f-mode of solar oscillations are believed to
be surface gravity modes whose frequencies are essentially independent
of the stratification in the solar interior. A large part of the difference
between the observed f-mode frequencies at intermediate degree and
those in a standard solar model has been interpreted as being caused
by the incorrect radius of the solar models used (Schou et al. 1997; An-
tia 1998; Brown and Christensen-Dalsgaard 1998; Tripathy and Antia
1999). The frequencies of these modes can thus be used to measure the
solar radius.
There have been many reports about possible variation of solar ra-
dius with time (Delache, Laclare and Sadsaoud 1985; Wittmann, Alge
and Bianda 1993; Fiala, Dunham and Sofia 1994; Laclare et al. 1996;
Noeel 1997). The reported change in measured angular semi-diameter
varies from 0.1′′ to 1′′, which implies a change of 70 to 700 km in
∗ present address: US Naval Observatories, Washington, DC 20392-5420, U. S. A.
fmodpap.tex; 1/02/2008; 12:40; p.1
2 H. M. Antia et al.
radius. However, there is no agreement among observers about these
variations. It would thus be interesting to look for corresponding varia-
tions in the f-mode frequencies. The reported variations in solar radius
should change the f-mode frequencies by 0.1 to 1 µHz, which is much
larger than the error estimates in these frequencies. Considering how
small the estimated errors in f-mode frequencies are, it is possible to
measure changes in the solar radius as small as a few kilometers over the
solar cycle. Some variation in f-mode frequencies have been reported
in the MDI data by Dziembowski et al. (1998) who find that the solar
radius reached a minimum around the minimum activity period and
was larger by about 5 km, 6 months before and after the minimum.
If this variation is correlated with solar activity then we would expect
much larger change between minimum and maximum activity.
In this work we look for possible variations in the solar radius using
GONG data — which extend over longer time period than the MDI
data used in previous studies. We also look for possible correlations
between change in radius and solar activity. In Section 2 we outline
the technique used in calculating the frequencies and examine possible
systematic errors in computed frequencies. Section 3 gives the results
while in Section 4 we summarize the conclusions from our study.
2. The technique
Since the f-modes have very low power they are barely visible in the
spectra for modes of a given degree ℓ and azimuthal order m that
the GONG project fits to determine solar oscillation frequencies. As a
result, it is difficult to determine the frequencies of these modes reliably
from these spectra. To improve statistics we use the rotation-corrected,
m-averaged, power spectra from GONG data (Pohl and Anderson 1998)
to calculate the mean f-mode frequencies. Because summing over the
2ℓ+1 spectra for a mode of degree ℓ makes the peaks better defined, it
is possible to fit them reliably. Each of these spectra was obtained from
3 GONG months (108 days) of data and has a frequency resolution
of 0.107 µHz. These spectra extend up to ℓ = 200 and frequency ν =
2083 µHz. The f-mode peaks are clearly visible for ℓ >∼ 100. We have
also used two spectra covering only 1 GONG month, which gives a
lower frequency resolution of 0.321 µHz. For these spectra the errors
in computed frequency would be larger as compared to the 3 month
spectra.
We fit each mode separately using a maximum likelihood technique
(Anderson, Duvall and Jefferies 1991) with a symmetric Lorentzian
profile for the peaks. The fit is performed over a region extending to
fmodpap.tex; 1/02/2008; 12:40; p.2
Solar cycle variation in f-mode frequencies 3
Figure 1. Fit to the ℓ = 158 f-mode spectra for months 33–35. Power is in arbitrary
units.
about 20 µHz on either side of the peak and includes leaks from modes
of degree ℓ − 3 to ℓ + 3, ℓ being the degree of the target mode. It is
found that peaks arising from the ℓ + 1 leak are split into two because
of leaks from peaks of m± 1. Similarly, the ℓ+ 2 leak is split into three
because of leaks from peaks of m,m ± 2. Although leaks from ℓ − 1
and ℓ − 2 are also split in a similar manner, the power in these peaks
is much smaller. Thus we have explicitly fitted the m-leaks only from
the ℓ + 1 and ℓ + 2 peaks. An example of the fit is shown in Fig. 1.
For ν < 1170 µHz, the modes are difficult to fit reliably since the
width is smaller than the resolution limit of the spectra and the power is
relatively low. As a result, in this work we only include higher frequency
modes.
The use of m-averaged spectra may introduce some systematic er-
rors in determining the frequencies. To check for these we compare
the frequencies of p-modes fitted from m-averaged spectra with those
obtained by the GONG project using the individual m spectra. The
fmodpap.tex; 1/02/2008; 12:40; p.3
4 H. M. Antia et al.
Figure 2. Difference in frequencies for months 33–35 between the values obtained
from the m-averaged spectra and those computed from the individual m-spectra.
For clarity only a few error-bars are shown.
GONG project has computed the mean frequency for each n, ℓ multiplet
by fitting the frequencies for individual values of m to polynomials in
m. This mean frequency can be compared with the frequency computed
from the m-averaged spectra. The results are shown in Fig. 2. It may
be noted that f-mode has not been fitted in the individual m spectra
and hence these are not included. It can be seen that the frequency
difference is quite small being of the order of the estimated errors in
the fitted frequencies. Systematic difference between these two sets of
frequencies is <∼ 10 nHz.
Another possible source of systematic errors is the use of incor-
rect even-order splitting coefficients while constructing the m-averaged
spectra. The GONG spectra were obtained by setting these coefficients
to zero. A non-zero value of these even-order coefficients would intro-
duce some systematic shift in frequencies. To estimate the effect of this
we constructed some m-averaged spectra with either a2 = −0.2 nHz or
a4 = 0.1 nHz. The resulting frequency shifts with a2 = −0.2 nHz are
shown in Fig. 3. Similar shifts are found for non-zero a4. Once again
the frequency shifts due to non-zero a2 or a4 are quite small, being of
the order of 10 nHz. Furthermore, the frequency shift for the f-mode
is smaller than those for p-mode with the same ℓ. Although these even
coefficients are known to change with solar cycle, the variation is not
fmodpap.tex; 1/02/2008; 12:40; p.4
Solar cycle variation in f-mode frequencies 5
Figure 3. Shift in frequencies for months 21–23 arising due to non-zero a2
very large at the low frequencies which we are considering in this work.
Thus the shifts shown in Fig. 3 can be considered to be conservative
upper limits to errors that may be expected due to non-zero even
coefficients. From Fig. 2 it can be seen that during a period where
even coefficients are fairly large the difference in frequencies computed
from individual m-spectra and the m-averaged spectra is quite small
— much smaller than the differences seen in Fig. 3.
Considering possible systematic errors, it would be desirable to re-
strict ourselves to low frequency range where these errors are not signifi-
cant. In the frequency range (ν < 1440 µHz) considered in this work the
systematic errors from these effects are expected to be about 10 nHz,
and any time variation in these systematic effects would be smaller.
Having established the reliability of frequencies from m-averaged spec-
tra we calculate the frequencies from 7 different spectra during 1995 to
1998. These spectra are listed in Table 1. This table also lists the solar
activity as measured by the mean daily sunspot number, RI from the
Solar Geophysical Data web page of the US National Geophysical Data
Center (http://www.ngdc.noaa.gov/stp/stp.html).
fmodpap.tex; 1/02/2008; 12:40; p.5
6 H. M. Antia et al.
Table I. List of GONG spectra used
GONG months Starting date Ending date Mean daily sunspot no.
3 – 5 1995-07-18 1995-11-02 14.5 ± 1.3
8 – 10 1996-01-14 1996-04-30 6.0± 0.6
21 – 23 1997-04-26 1997-08-11 14.7 ± 1.1
27 – 29 1997-11-28 1998-03-15 39.5 ± 1.7
33 – 35 1998-07-02 1998-10-17 79.5 ± 2.5
36 1998-10-18 1998-11-22 62.4 ± 4.1
37 1998-11-23 1998-12-28 82.6 ± 4.1
3. Results
We have fitted each of the 7 spectra listed in Table 1 to calculate the
mean frequencies of the f modes. Fig. 4 shows the relative difference
between the observed frequencies for Months 33–35 and Months 8–10.
It can be seen that to a first approximation the relative frequency differ-
ence is independent of frequency and hence this frequency difference can
be interpreted as arising from a change in the solar radius. Of course,
suitable combination of other effects like magnetic field, convection etc.
may also yield frequency differences which are independent of frequency
but that will almost certainly require some fine-tuning of parameters,
while change in solar radius would be a simpler explanation.
To estimate the possible change in radius we take an average over the
relative frequency difference for all 56 f-modes that were successfully
fitted in all spectra:
〈δν〉 =
∑
ℓ
δνℓ
σ2
ℓ∑
ℓ
1
σ2
ℓ
(1)
The differences were taken with respect to calculated frequencies for a
standard solar model, which is a static model, constructed with OPAL
opacities (Iglesias & Rogers 1996) and low temperature opacities from
Kurucz (1991). The OPAL equation of state (Rogers, Swenson & Igle-
sias 1996) was used to construct the model and convective flux was
calculated using the the formulation of Canuto and Mazzitelli (1991).
The radius of the model is 695.78 Mm (Antia 1998). The results are
shown in Fig. 5, which also shows the mean daily sunspot number,
RI , for each of these periods. The change in frequencies appears to be
correlated with the sunspot number.
fmodpap.tex; 1/02/2008; 12:40; p.6
Solar cycle variation in f-mode frequencies 7
Figure 4. Relative frequency differences between f-mode frequencies from the spectra
for months 33–35 and months 8–10.
Figure 5. The averaged frequency difference for f-modes between the Sun and a
standard solar model with radius R = 695.78 Mm as a function of time is shown
by the continuous line. The mean daily sunspot number RI is also shown by points
with scale shown on the right side. For clarity these points have been slightly shifted
along the x-axis.
fmodpap.tex; 1/02/2008; 12:40; p.7
8 H. M. Antia et al.
If the change in frequencies is entirely due to change in radius then
the average frequency difference is related to change in radius (Antia
1998) by
〈
δν
ν
〉 = −
3
2
δR
R
(2)
Thus a relative shift in frequencies by 10−5 corresponds to a change in
radius by 0.67 × 10−5R⊙ ≈ 4.7 km. It may be argued that the change
of frequency may be due to some other effect, e.g., magnetic field. Since
the relative frequency difference between the frequencies at two differ-
ent epochs is almost independent of frequency in the range of degrees
considered here, a change in radius is the simplest explanation for the
frequency shift. Of course, the magnetic field is directly correlated to
activity indices, but the direct effect of magnetic field, is likely to yield
frequency differences that will increase rapidly with frequency since the
higher frequency modes are located closer to solar surface where these
effects are likely to be more important (Campbell and Roberts 1989).
It may be noted that a reduction in solar radius by 4.7 km in 2 years
will generate about 100 times the solar luminosity through release of
gravitational potential energy, if the contraction in homologous. This
indicates that the change in radius is a superficial effect confined to
outer layers where the f-modes are located.
However, the radius appears to decrease with increasing activity.
This is not what is seen by Dziembowski et al. (1998). The magnitude
of change found in GONG data during the period covered by their
study is much smaller than what they have reported. The apparent
discrepancy is most likely due to a different range of degrees used
in their study. They have probably used modes up to ℓ = 300 and
these high wavenumber modes are known to deviate significantly from
the simple dispersion relation (Duvall, Kosovichev and Murawski 1998;
Antia and Basu 1999). As a result, one can expect misleading results if
average is taken over all modes, since it is possible that the deviation
of the modes from the normal dispersion relation has some solar cycle
dependence. A part of the deviation is known to be due to the use
of symmetric profiles in calculating the f-mode frequencies (Antia and
Basu 1999) and it is quite possible that the asymmetry changes with
activity. Thus it would be better to confine our attention to modes with
ℓ < 200, where these effects are minimal. The results shown in Fig. 2
of Dziembowski et al. (1998) imply a decrease in frequency by 10−5
between 1996.8 and 1997.2. If this change is correlated with sunspot
number then we would expect a decrease by at least 5× 10−5 between
1996 and 1998.6 in our results. Certainly no such shift is seen in our
results. This shift is also much larger than the expected systematic
errors in our results.
fmodpap.tex; 1/02/2008; 12:40; p.8
Solar cycle variation in f-mode frequencies 9
Figure 6. The averaged frequency difference for f-modes between the Sun and a
standard solar model with radius R = 695.78 Mm are plotted against the mean
sunspot number during the period of observation. The average over three different
frequency ranges as marked in the figure are compared with each other. The line
marks the best linear fit for the full frequency range. The triangles and squares are
slightly shifted along x-axis for clarity.
In order to check the stability of our results we have repeated the
analysis by restricting the frequencies to 1.17 < ν < 1.32 mHz and
1.32 < ν < 1.44 mHz and the results are shown in Fig. 6. It is clear
that the results are not particularly sensitive to the range of frequencies
used. However, the correlation with solar activity is markedly weaker
in the higher frequency range. From the relative frequency differences
between the observed frequencies and those in a standard solar model
it is known (Tripathy & Antia 1999) that there is a distinct tendency of
relative frequency differences decreasing with frequency in this range.
It is likely that this trend is due to some other effect like magnetic
field which varies with solar cycle. In that case it will interfere with the
signal due to change in radius. It is quite possible that at even higher
frequencies used in MDI study, the correlation is further weakened
because there is an even stronger decrease in the relative frequency
differences.
fmodpap.tex; 1/02/2008; 12:40; p.9
10 H. M. Antia et al.
4. Conclusions
We have calculated the mean f-mode frequencies from the m-averaged
spectra from GONG data covering 108 days. The use of m-averaged
spectra enable us to determine the frequencies reliably. The relative
frequency difference between observed frequencies and those calculated
for a standard solar model are roughly independent of ℓ in the range
that was used in this study (100 < ℓ < 200). Hence it is likely that this
difference is due to difference in solar radius, as other agents are likely
to introduce a steep frequency dependence in relative frequency differ-
ences. The mean shift in f-mode frequencies appears to be correlated
with solar activity indices like the mean daily sunspot number RI .
If this frequency shift is due to change in solar radius then the radius
decreases as activity increases. Further, the change in the solar radius
during the solar cycle is approximately 5 km. This is one or two orders of
magnitude smaller than the changes reported earlier (Delache, Laclare
and Sadsaoud 1985; Wittmann, Alge and Bianda 1993; Fiala, Dunham
and Sofia 1994; Laclare et al. 1996; Noeel 1997).
These results on the change in solar radius with solar cycle are much
less than what has been deduced using MDI data (Dziembowski et
al. 1998) and we believe the discrepancy is due to their using higher
degree modes whose frequencies can be affected by a variety of other
agents, since these modes are localized closer to the solar surface.
Acknowledgements
This work utilizes data obtained by the Global Oscillation Network
Group (GONG) project, managed by the National Solar Observatory,
a Division of the National Optical Astronomy Observatories, which is
operated by AURA, Inc. under a cooperative agreement with the Na-
tional Science Foundation. The data were acquired by instruments op-
erated by the Big Bear Solar Observatory, High Altitude Observatory,
Learmonth Solar Observatory, Udaipur Solar Observatory, Instituto de
Astrofisico de Canarias, and Cerro Tololo Inter-American Observatory.
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Solar cycle variation in solar f-mode frequencies and radius

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