About the rotation of the solar radiative interior

ABOUT THE ROTATION OF THE SOLAR RADIATIVE INTERIOR

R.A. GARCÍA1,∗, T. CORBARD2, W.J. CHAPLIN3, S. COUVIDAT4,A. EFF-DARWICH5,6, S.J. JIMÉNEZ-REYES6, S.G. KORZENNIK7, J. BALLOT1,

P. BOUMIER8, E. FOSSAT9, C.J. HENNEY10, R. HOWE10, M. LAZREK2,9,11,J. LOCHARD8,12, P.L. PALLÉ6 and S. TURCK-CHIÈZE1

1Service d’Astrophysique, CEA/DSM/DAPNIA, CE Saclay, 91191 Gif sur Yvette, France2Département Cassini, UMR CNRS 6529, Observatoire de la Côte d’Azur, BP 4229,

06304 Nice Cedex 4, France3School of Physics and Astronomy, University of Birmingham, Edgbaston,

Birmingham B15 2TT, U.K.4HEPL, Stanford University, Stanford, CA 94305-4085, U.S.A.

5THEMIS S.L., 38205, La Laguna, Tenerife, Spain6Instituto de Astrofísica de Canarias, 38205, La Laguna, Tenerife, Spain

7Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, U.S.A.8Institut d’Astrophysique Spatiale, Université Paris XI, 91405 Orsay, France

9Département d’Astrophysique, UMR 6525, Université de Nice-Sophia Antipolis,06108 Nice Cedex 2, France

10National Solar Observatory, P.O. Box 26732 Tucson, AZ 85726-6732, U.S.A.11Laboratoire de Physique des Hautes Energies et Astrophysique (LPHEA), Universite Cady Ayyad

de Marrakech, BP 2390, Marrakech, Morocco.12LESIA, Observatoire de Paris-Meudon, 92195 Meudon Principal Cedex, France

(Received 14 November 2003; accepted 21 January 2004)

Abstract. In the modern era of helioseismology we have a wealth of high-quality data available,e.g., more than 6 years of data collected by the various instruments on board the SOHO mission,and an even more extensive ground-based set of observations covering a full solar cycle. Thanks tothis effort a detailed picture of the internal rotation of the Sun has been constructed. In this paperwe present some of the actions that should be done to improve our knowledge of the inner rotationprofile discussed during the workshop organized at Saclay on June 2003 on this topic. In particular wewill concentrate on the extraction of the rotational frequency splittings of low- and medium-degreemodes and their influence on the rotation of deeper layers. Furthermore, for the first time a full setof individual |m|-component rotational splittings is computed for modes � ≤ 4 and 1 < ν < 2 mHz,opening new studies on the latitudinal dependence of the rotation rate in the radiative interior. It willalso be shown that these splittings have the footprints of the differential rotation of the convectivezone which can be extremely useful to study the differential rotation of other stars where only theselow-degree modes will be available.

∗[email protected]

Solar Physics 220: 269–285, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

270 R.A. GARCIA ET AL.

1. Introduction

The extremely high quality of modern helioseismic data, and the large number ofdata available from a range of observations, is forcing us to revisit and reassessthose techniques of analysis that are employed to both extract the basic parametersof the modes, and then use this information to infer the interior structure of the Sun(e.g., Thompson et al., 2003).

In the case of the former, better-quality power spectra are now allowing us toextract reliable estimates of certain parameters that hitherto we could not constrain.As an illustration, one example of critical relevance to studies of rotation is one’sability to fit the asymmetric arrangement in frequency of individual m at � = 2,in spectra made from ‘unresolved’ observations (Chaplin et al., 2003a, b) above∼ 2 mHz. ‘Sun-as-a-star’ observations of this type (e.g., GOLF1, VIRGO2 andBiSON3) are a crucial source of low-�, core-penetrating splittings. Only three � =2 components – those with m = −2, 0 and 2 (i.e., even (� + m) components) –are identifiable in unresolved data. It has for a long time been standard practiceto fit these modes to a model that assumes a symmetric pattern in frequency, onethat clearly fails to account for any asymmetry (which arises from the influenceof the non-homogeneous distribution of activity over the solar surface). Incorrectmodeling of the underlying resonant spectrum will of course have consequencesfor extracted estimates of the component splittings.

At high frequencies analyses must also contend with increasing resonant-peaklinewidth. This leads to substantial blending of adjacent structure, both between theindividual m of each multiplet and at very high frequencies between neighbouring�. An incorrect estimate of the m-component height ratios within a given multiplet– a parameter that is usually held fixed in the fitting – can also have a deleteriouseffect on the accuracy of the extracted splitting estimate. An in-depth discussionof these issues can be found in Chaplin et al., (2001) and Fletcher, Chaplin, andElsworth (2003).

In the case of resolved observations (MDI4, GONG, etc...), the correct estima-tion of the leakage matrix (which arises from the impossible isolation of individualmodes) is crucial in the determination of the splitting coefficients. Indeed, an errorin the values assumed for the leakage matrix elements will lead to systematic biasof the estimated splitting coefficients. This effect will be particularly important forvery low degree.

In this paper we touch upon the improvements on the data analysis process inorder to better constrain the solar rotation profile toward the core. We will begincomparing the rotational frequency splittings (hereafter splittings) of low-degreepressure modes produced by 5 different helioseismic instruments. This will allow1Global Oscillations at Low Frequency (Gabriel et al. 1995)2Variability of solar IRradiance and Gravity Oscillations (Fröhlich et al., 1995)3Birmingham Solar Oscillation Network (Chaplin et al., 1996)4Michelson Doppler Imager (Scherrer et al., 1995)

ABOUT THE ROTATION OF THE SOLAR RADIATIVE INTERIOR 271

us to combine some of the available data sets in order to reduce the instrumentalnoise and to improve the accuracy of the extracted low-degree low-order p-modesplittings. It will be shown that these splittings are very important, although theyare less deeply penetrating, because they are determined with a very high accuracy(for the reasons formerly mentioned we consider only modes below ∼ 2 mHz).Furthermore we will use, for the first time, some special MDI-masks to infer thesplitting of the odd (� + m) components of the low-degree low-order modes. Butthe splittings of the low-degree low-order modes are only part of the problem. Eff-Darwich, Korzennik, and Jiménez-Reyes (2002) showed that the influence of themedium-� splittings (i.e., 4 < � < 25) on the solar rotation profile below ∼ 0.3R�is very important. An error on their determination cannot be easily compensatedby the low-degree low-order splittings. To address this issue we show here the firstresults of a new fitting procedure (Korzennik, 2003) that will give much bettersplittings of these modes. Finally we discuss the extraction, for the first time, of thecoefficients of the polynomial expansion coming from the low-degree low-orderrotational frequency splittings.

2. Comparison of Splittings Extracted from Different Instruments

To extract the internal rotation of the Sun, we have to combine data from differentinstruments. In fact, those instruments that resolve the solar disk like MDI, GONG5

or LOWL6 are better adapted to measure modes of � ≥ 3 while those that observethe ‘Sun-as-a-star’ are only sensitive to modes of � ≤ 3. Several comparisonshave already been done (e.g., Toutain et al., 1997; Bertello et al., 2000; Howe etal., 2003) to show their differences and similarities but using in general pairs ofinstruments. Here, we wanted to make a full comparison of the splittings extrac-ted from the instruments that are usually combined to make the inversions of thesolar rotation and using the same fitting code to avoid any bias from the extractionmethod. Thus, 760 days-long time series – starting on May 25 1996, during solarcycle minimum – of 5 instruments looking to the ‘Sun-as-a-star’ have been used:GOLF, GONG, MDI, VIRGO and BiSON. For the first four instruments, we haveused a common window function which gives a 83% duty cycle while BiSON hasalready ∼ 80% alone. We have decided not to use the common window for allthe instruments because the duty cycle is reduced to 62%. Unfortunately, we havenot included the LOWL series in this comparison as its duty cycle is too small (∼20%).

Once the periodogram of each instrument was computed, each multiplet wasfitted to a set of Lorentzian profiles using a maximum-likelihood method (see fordetails for example: Appourchaux, Gizon and Rabello-Soares, 1998). The fittedparameters are: the amplitude of the modes (with a fixed theoretical amplitude5Global Oscillation Network Group (Harvey et al., 1996)6Low degree L (Tomczyk et al., 1995)


Figure 1. Synodic rotational frequency splittings of modes � = 1 (upper panel) and 2 (lower panel)of 5 different helioseismic instruments.


ratio between the visible m-components (see for example Chaplin et al., 2001)),its central frequency, the splittings (constrained between 250 and 550 nHz), thelinewidth and asymmetry (same for all the m-components) and the backgroundnoise. Below 2 mHz we have fitted each mode alone while above this frequencywe have fitted the pairs � = 0, 2 and � = 1, 3 together. The resultant splittings forthe modes � = 1 and 2 are plotted on Figure 1. Most of the results agree in 1σ errorbetween all the instruments without any particular systematics on the results (apartfrom the size of the error bars which depends on the instrument as a consequenceof the different signal-to-noise ratio). The average value is also around the expected∼ 400 nHz. While some of the splittings (e.g. � = 1, n = 15 or � = 2, n = 17) aremarginally out of the average, the splittings obtained by different instruments arecompatible between themselves and they can be mixed in order to have a completeset that can be used to infer the solar rotation profile.

3. The Splittings of Low-Degree Low-Order Modes

Compared to other pressure modes, low-order low-degree modes are especiallyinteresting to derive the rotation of the radiative region (below 0.5 R�). Moreoverlow-order pressure modes with a mixed character – i.e., whose frequencies arelower than the maximum Brunt-Väissälä frequency � 470 nHz and thus partlybehave like gravity modes – are much more sensitive to the core rotation than anyother p modes. The two previous assertions can be checked using the so-calledrotational kernels. These functions are denoted Kn�m(r, θ), with n the radial orderof the mode, � its degree, and m its azimuthal order. They appear in the integralthat relates the rotational splittings to the rotation profile:

δνn�m =

∫ R�

0

∫ π

0Kn�m(r, θ)�(r, θ)dθdr , (1)

where δνn�m = νn�

m −νn�0

mis the splitting due to the solar rotation �(r, θ), with θ

the colatitude. Actually this integral is only approximate since the P2-distortionof the solar surface due to the centrifugal forces has been neglected (this is a validapproximation due to the low-rotation speed of the Sun). Further perturbations, likethe magnetic fields, also introduce some corrections to the frequency shift δνn�

m .Figure 2 shows rotational kernels for a few solar oscillation modes. These ker-

nels are derived from a solar model detailed in Turck-Chièze et al. (2001) usingthe following equation (Kosovichev, 1999):

Kn�m(r, θ) = 12In�

ρr2

((ξ 2

r − ξrξh)P2�m +

+ ξ 2h

[dP�m

dθ− 2P�m

dP�m

dθcosθsinθ

+ m2

sin2θP 2

�m

]).

(2)


l=2, n=−2, m=2 l=2, n=10, m=1

l=2, n=1, m=2 l=3, n=1, m=3

Figure 2. Rotational kernels (in arbitrary units) as a function of the colatitude θ (x-axis) and theradial distance to the solar center (y-axis) for the � = 2, n = −2, m = 2 g mode (upper left-handpanel), the � = 2, n = 10, m = 1 p mode (upper right-hand panel), the � = 2, n = 1, m = 2 mixedp mode (lower left-hand panel), and the � = 3, n = 1, m = 3 mixed p mode (lower right-handpanel). The kernels are symmetrical about the solar equator.

P�m is a Legendre polynomial, ρ(r) is the density profile inside the Sun, andIn� is the mode inertia. The rotational kernels of pressure modes show large sens-itivity to the surface layers, in opposition to the kernels of gravity modes (whosesensitivity is maximum in the solar center as shown on the upper left-hand panel ofFigure 2). For pressure modes with a given degree �, the sensitivity to deeper layersincreases with an increase of n. Also, for a given n, the sensitivity to deeper layersincreases with a decrease of �. Therefore low-degree high-order modes seem utterlyuseful to derive the rotation profile in the solar core. However, when n increases the


outer turning point of the oscillation mode is closer to the solar surface. Turbulenceeffects near this surface dampen the modes and widen their spectral lines in powerspectra. It turns out that the splittings corresponding to high-n modes are poorlydetermined: the error bar is much larger than for low-n modes. Figure 3 (left-panel)shows the sensitivity of low-degree modes to the rotation of the layers ≤ 0.2 R�.Figure 3 (right-panel) shows the ratio of this sensitivity to the uncertainty of thesplitting. When this ratio is larger than one, the precision of the splitting is goodenough for the corresponding mode to be used to derive the core rotation profile.For instance, consider the mode � = 1 n = 17 at ν = 2559 µHz. 7% of its

Figure 3. Left: part of the rotational splitting of some pressure modes that is due to the rotation below0.2 R�, as a function of the frequency of these modes. The solid curve is for � = 1, the dashed curvefor � = 2, the dot-dashed curve for � = 3, and the dot-dot-dot-dashed curve for � = 4. The modesbelow � 470 µHz are called mixed modes and are very sensitive to the rotation of the solar core.Right: ratio of the splitting due to the rotation below 0.2 R�, to the uncertainty of the total splittingfor � = 1 (solid line), � = 2 (dashed line), and � = 3 (dash-dotted line).

sectoral splitting (m = �), i.e. � 28 nHz, is due to the layers below 0.2 R�. Yet theuncertainty of its splitting reaches 41 nHz: this mode is hardly informative aboutthe rotation of the solar core. On the other hand, the uncertainty of the sectoralsplitting for the mode � = 1, n = 9 at ν = 1473 is only 5 nHz, whereas � 15 nHzof this splitting comes from the core rotation. Thus this mode is more informativeabout the central solar rotation than the mode � = 1, n = 17 (see also the evolutionof the error bars in Tables I and II as the order of the modes increases). It appearsthat for the � = 1 modes only those with ν ≤ 2200 µHz are informative about thecore rotation. For the � = 2 and 3 modes, two regions are interesting: ν ≤ 1800µHz and 2750 ≤ ν ≤ 3150 µHz. No gravity modes have been detected withcertainty so far from the solar data. In the absence of such detections, the p modeswith the best trade-off between sensitivity to the core rotation and uncertainty of thesplitting are the low-order low-degree modes (and especially the mixed p modeswhen they can be detected).

As discussed before, to infer the solar rotation profile of the inner radiativeregions we have to obtain the lowest-order p modes with the highest accuracy.Therefore, we can try to combine some instruments to improve the signal-to-noiseratio of the otherwise observed power spectrum. This is not a new idea. For ex-


ample, the IRIS network has already used several instruments to improve their dutycycle (Salabert et al., 2002). Here we have used a different approach by averagingthe GOLF time series with the MDI-Gauss and MDI-Gauss-Zero-Mean masks timeseries in order to reduce the non-coherent instrumental noise and thus improvethe signal-to-noise ratio of the modes at low frequency. We have used a 2243day-long series (starting on May 25, 1995) of the space-based GOLF and MDIdisk-integrated velocity time series as they have the highest duty cycle available.The GOLF time series are the standard ones derived from a combination of theblue and red wings of the sodium solar doublet (García et al., in preparation) andintegrated into 60-s cadence to match the MDI rate. This integration is done in thebest way to synchronize both instruments without any interpolation or phase shift.The MDI time series used here are created by applying two different spatial masks,described in Henney et al. (1999), to MDI LOI-proxy Doppler images. The MDI-Gauss mask is defined by Gaussian weights centered on the image disk center. TheGaussian is defined such that the 1/e drop-off is at the image radius defining theinner 20% of the image area. The MDI-Gauss-Zero-Mean mask is the same as theMDI-Gauss mask except it has a mean of zero. Due to geometrical considerations,these series are only sensitive to (� + m) even multiplet components. In addition,two other MDI masks, the north-south and Gauss-plus-north-south, are used toaccess the odd (� + m) components. Both masks are made by dividing the imageinto two equal parts with equal weight but opposite sign.

The MDI time series are more sensitive to higher-degree modes (mainly theGaussian weighted ones) while the effect of adding GOLF cleans up the leakageof modes � > 4. However, for some of the � = 4 modes an averaging of MDI-onlymasks provide better results.

These series cover the rising phase of solar cycle 23 from the minimum to itsmaximum. To avoid any bias in the extracted splitting, due to the solar magneticcycle, we have restricted our analysis to modes below 2 mHz. Indeed, above thisfrequency, the modes undergo substantial frequency shifts during this period (Gellyet al., 2002; Jiménez-Reyes, Jiménez, and García, 2003) and some asymmetriesappear in the arrangements of the components for � ≥ 2 modes (Chaplin et al.,2003a).

Once the periodogram was computed, each multiplet was fitted using the samemaximum-likelihood method described in Section 2. However, we have fitted inthis case symmetrical Lorentzian profiles. Indeed, in this frequency range the peaksare very narrow and no improvement was visible when an asymmetrical profile wasused. Therefore we have favored the fitting with fewer free parameters.

Because of the particular data sets used in this paper, all the odd or even � + m

components of the multiplet are visible in the power spectrum. Thus, for the modes� = 3 and 4, two splittings have been fitted to take into account the differencebetween the sectoral and non-sectoral multiplet components which have a differ-ent sensitivity to the equatorial rotation (see Figure 2). Examples of some fitted� = 1, 2, 3 and 4 modes are shown in Figure 2 of García et al. (2003). The


TABLE I

Synodic splittings of the even (� + m) components obtained from GOLFand MDI linear combination of 2243 day-long velocity time series. Onlythe sectoral components of the first 3 � = 4 modes were fitted; the amp-litudes of the m = 2 components were too small to have a reliable splitting.

� n ν (µHz) δν|m|=� (nHz) δν|m|=(�−2) (nHz)

1.... 6 1039.465 ± 0.001 398.5 ± 1.5

7 1185.596 ± 0.004 401.0 ± 3.6

8 1329.630 ± 0.001 398.7 ± 1.5

9 1472.844 ± 0.004 404.0 ± 3.4

10 1612.719 ± 0.004 412.4 ± 4.2

11 1749.274 ± 0.005 402.0 ± 4.6

12 1885.086 ± 0.006 397.4 ± 6.2

2.... 4 811.632 ± 0.003 404.9 ± 1.9

5 - -

6 1105.141 ± 0.004 399.8 ± 2.2

7 1250.553 ± 0.008 401.4 ± 4.1

8 1394.674 ± 0.005 399.1 ± 2.8

9 1535.860 ± 0.004 406.4 ± 2.4

10 1674.546 ± 0.009 401.0 ± 5.1

11 1810.299 ± 0.012 401.3 ± 6.5

12 1945.752 ± 0.008 380.3 ± 4.4

3.... 6 1161.764 ± 0.003 403.7 ± 3.6 376.7 ± 4.7

7 1306.714 ± 0.004 403.8 ± 4.4 375.8 ± 8.6

8 1450.994 ± 0.005 405.2 ± 6.4 376.4 ± 9.1

9 1591.544 ± 0.007 403.9 ± 9.2 374.3 ± 9.5

10 1729.066 ± 0.012 404.8 ± 14.3 357.1 ± 17.6

11 1865.253 ± 0.014 400.6 ± 15.3 359.8 ± 22.3

12 2001.171 ± 0.017 402.6 ± 18.3 363.6 ± 25.3

4.... 4 912.018 ± 0.0045 402.2 ± 2.8 -

5 1060.900 ± 0.006 380.8 ± 3.0 -

6 1208.995 ± 0.007 401.6 ± 3.7 -

7 1356.379 ± 0.005 407.6 ± 6.2 390.4 ± 4.7

8 1500.287 ± 0.004 407.4 ± 4.9 375.3 ± 4.2

9 1640.932 ± 0.010 413.6 ± 13.5 377.7 ± 8.3

10 1778.044 ± 0.011 410.9 ± 13.9 398.4 ± 10.8

11 1914.718 ± 0.017 413.4 ± 20.8 372.1 ± 13.1

12 2051.773 ± 0.025 405.8 ± 25.6 447.4 ± 24.9


TABLE II

Synodic splittings of the odd (�+m) components obtained from 2243 day-longMDI (north-south and Gauss-plus-north-south) velocity time series. The m = 1component of the mode � = 4, n = 6 could not be fitted due to its smallamplitude. The mode � = 3, n = 6 is at the limit of the S/N and it is probablya bad determination.

� n ν (µHz) δν|m|=(�−1) (nHz) δν|m|=(�−3) (nHz)

1.... 6 1039.473 ± 0.005

7 -

8 1329.645 ± 0.004

9 1472.832 ± 0.014

10 1612.730 ± 0.012

11 1749.286 ± 0.015

12 1885.074 ± 0.017

2.... 6 1105.011 ± 0.109 399.8 ± 2.2

7 1250.561 ± 0.005 387.5 ± 5.1

8 1394.671 ± 0.008 402.8 ± 8.2

9 1535.854 ± 0.006 383.7 ± 6.4

10 1674.536 ± 0.012 386.6 ± 12.1

11 1810.309 ± 0.015 381.1 ± 15.0

12 1945.794 ± 0.024 371.3 ± 23.0

3.... 6 1161.879 ± 0.004 337.2 ± 3.2

7 1306.714 ± 0.006 387.6 ± 4.1

8 1450.994 ± 0.006 388.0 ± 3.9

9 1591.572 ± 0.008 396.2 ± 5.6

10 1729.053 ± 0.013 398.7 ± 8.5

11 1865.263 ± 0.024 388.8 ± 15.6

12 2001.193 ± 0.024 398.5 ± 14.0

4.... 6 1209.005 ± 0.005 406.5 ± 5.4 -

7 1356.397 ± 0.008 403.7 ± 13.61 392.0 ± 8.7

8 1500.286 ± 0.006 396.4 ± 10.01 368.3 ± 7.7

9 1640.931 ± 0.013 391.8 ± 23.08 374.2 ± 15.1

10 1778.069 ± 0.014 399.7 ± 20.66 374.2 ± 16.0

11 1914.747 ± 0.025 385.4 ± 34.07 369.5 ± 25.8

12 2051.651 ± 0.035 398.3 ± 43.16 313.6 ± 32.4


resulting central frequencies and synodic splittings are given in Table I, for the even(�+m) components, and Table II for the odd ones. These results constitute the mostcomplete set of low-degree low-order splittings ever computed.

4. The Splittings of Medium-�� Modes.

Most inversions of the solar rotation rate in the radiative interior are based on datafrom resolved-disk instruments, complemented by data from instruments withoutany spatial resolution. The first type of instrument provides reliable estimates ofrotational splittings for medium- and high-degree modes with degrees and frequen-cies as low as � = 4 and ν � 1.8 mHz, respectively. Until now, it was not possibleto calculate these splittings for modes with frequencies lower than 1.8 mHz, sincetime series used by MDI and GONG to obtain eigenfrequencies are only 72 and108 days long, respectively. In this sense, it is our purpose to use longer time series(� 2100 days) from GONG and MDI instruments and fill the existing gap of datain the low-frequency domain. In particular, we are specially interested in modeswith degrees 4 ≤ � ≤ 25, which are those sensitive to the rotation rate of the solarradiative interior.

These low-frequency modes should not add new independent information tothe inversion problem and hence, it is not expected that the error distribution of theinverted solution will be significantly reduced. However, the lower the frequencyof the modes is, the longer their lifetime is and therefore, the lower the effect ofstochastic excitation in the power spectra. In this sense, it should be much easierto identify peaks with low azimuthal order m in the low-frequency domain andtherefore, the rotational splittings will be more reliable. The addition of these datain the inversion of the rotation rate should give more robustness and reliability toour inferences on the dynamics of the radiative interior.

The current stage of our work is illustrated in Figure 4, where a comparison ispresented between rotational splittings in different frequency intervals for modeswith � < 25. In this sense, Figure 4 (left panels) correspond to sectoral splittings inthe frequency ranges 1.1 to 1.8 mHz, 1.8 to 2.8 mHz and 2.8 to 3.8 mHz, respect-ively. In the same way, Figure 4 (right panels) correspond to rotational splittingsfor modes with |m| = 3 in the frequency intervals afore-mentioned. These datahave been obtained by Korzennik (2003) using 2088 days of MDI observations,being the first time that rotational splittings for frequencies smaller than 1.8 mHzare obtained for modes with � < 25. It is clear that the lower the frequency intervalis, the lower the dispersion and uncertainties of the data are. Although very-low-frequency modes do not penetrate deeper layers of the solar core (�/n < 10−3),they will indeed help to better constrain the rotation in the outer layers of thesolar radiative interior, in particular its latitudinal dependence, as the result of theexcellent data obtained at low m.


Figure 4. Rotational frequency splittings of modes 4 ≤ � ≤ 25 for 3 different frequency ranges: 1.1to 1.8 mHz, 1.8 to 2.8 mHz and 2.8 to 3.8 mHz, from top to bottom respectively. The left panels showthe sectoral splittings (|m| = �). The right panels show the splittings of the |m| = 3 components.

5. Extracting the Solar Rotation Profile

The 2D solar internal rotation profile �(r, θ) is related to the frequency splittingsthrough the 2D integral Equation (1). The observation of individual m componentsfor low-degree splittings potentially opens the new possibility of inferring the lat-itudinal dependence of the rotation rate in the radiative interior. The number of m


components remains however low for low-degree modes and therefore we cannotexpect to obtain good resolution in latitude. However, there is another immediateadvantage in having a measure of the individual m components of the splittingwhich is that we can also obtain a more consistent estimate of a latitudinallyaveraged rotation downwards to the core. This is because, for degrees � > 4 thefrequency splittings are usually obtained by fitting the spectra using the polynomialexpansion (Schou, Christensen-Dalsgaard, and Thompson, 1994)

δνn�m = m

jmax∑j=1

an�j P l

j (m), (3)

where jmax < 2� is chosen in order to stabilize the fit. If we assume that the spectra-fitting procedure, described in Section 2, can be assimilated to a least-squares fitthrough the 2� m components (m �= 0) of the multiplet, then it can be shown thatthe first coefficient of an expansion of Equation (3) is such that:

an�1 =

∫ R�

0Kn�(r)�1(r) dr, (4)

where

�1(r) = 3

2

∫ π

0�(r, θ) sin3(θ)dθ, (5)

and

Knl(r) = 2

In�

ρr2

(ξ 2r − 2ξrξh + (�2 + � − 1)ξ 2

h

). (6)

In previous analyses, for integrated data such as GOLF, only the sectoral splittings( νn��−νn�−�

2�= ∑�

j=1 a2j−1) were measured, and therefore we needed an estimate ofa3 for � = 2 and a3, a5 for � = 3 in order to use Equation (3). These were eitherextrapolated from higher-degree modes or estimated assuming a known rotationprofile in the convection zone (Corbard et al., 1998; Couvidat et al., 2003). Withthe measurement of all m-components for � ≤ 4 we can now get these estimatesdirectly from the data by inverting Equation (3)

an11 = δνn1

1 , (7)

an21 = ( δνn2

1 + 4 δνn22 )/5

an23 = (−δνn2

1 + δνn22 )/5,

}(8)

an31 = ( δνn3

1 + 4 δνn32 + 9 δνn3

3 )/14an3

3 = (− δνn31 − 2 δνn3

2 + 3 δνn33 )/9

an35 = (5 δνn3

1 − 8 δνn32 + 3 δνn3

3 )/126.

(9)


The a coefficients obtained from GOLF + MDI data using these equationsare plotted in Figure 5 for � = 1 (top) to � = 3 (bottom). The lines show thevalue expected for these coefficients for a rotation corresponding to the model bof Corbard et al., (2002), Figure 2. This model has a constant rotation rate in thewhole radiative interior (435 nHz) and the typically observed radial and latitudinalgradients in the convection zone (tachocline, surface shear and differential rota-tion). The general good agreement between the observations and predictions usingthis model indicates that these data are compatible with a rigid radiative interiorrotating at about 435 nHz. It is quite remarkable that the a3 and a5 coefficientsobtained in our analysis correspond in amplitude to the one produced mainly bythe differential rotation of the convection zone. However, two points are slightly offthe general behaviour, the � = 2, n = 12 (where the blending of the components ofthe modes begins to be important in biasing the computed splittings) and the � = 3,n = 6 which is at the level of the detectable signal-to-noise ratio and will be betterdetermined when longer time series are analyzed. The frequency dependence of a1

is also in very good agreement (especially for � = 3) with the one that would beobtained with a rigid rotation and reflects only the expected frequency dependenceof low-degree splittings induced by the Coriolis force (see for example Figure 8 inChristensen-Dalsgaard, 2003).

In order to complete this work and obtain more quantitative results in terms ofuncertainties, radial and latitudinal resolution, it will be necessary to carry out 1Dor 2D inversions of Equation (4) or Equation (1) using also the intermediate-degreemodes obtained as explained in Section 4.

6. Final remarks

We have shown here that we can distinguish between the different components ofthe splitting multiplet and that they differ from a mean splitting in a way that isconsistent with our knowledge of the rotation in the outer layers of the Sun. Thisis a new and interesting result that is potentially of great interest in the perspectiveof future asteroseismic measurements (see also Gizon and Solanki, 2003). It showsthat if we can separate rotational splittings of low-degree modes for stars we willalso be able to reach some information about the differential rotation of the outerlayers of these stars. The precision required for slow rotating solar-like stars isprobably out of reach for now but the effect might become visible for faster rotatingstars.

Acknowledgements

The GOLF experiment is based upon a consortium of institutes (IAS, CEA/Saclay,Nice and Bordeaux Observatories from France, and IAC from Spain) involving a


Figure 5. a coefficient as inferred from MDI + GOLF data for � = 1 (top) to � = 3 (bottom). Thecrosses, triangles and diamonds correspond respectively to a1 − 435, a3 and a5. The dotted, dashedand dot-dashed lines give the expected values for a1 − 435, a3, a5 respectively as calculated fromthe rotation model of Corbard et al. (2002).

large number of scientists and engineers, as enumerated in Gabriel et al., (1995).SOHO is a mission of international cooperation between ESA and NASA. BiSONis funded by the UK Particle Physics & Astronomy Research Council (PPARC).This work utilizes data obtained by the Global Oscillation Network Group (GONG)Program, managed by the National Solar Observatory, which is operated by AURA,Inc. under a cooperative agreement with the National Science Foundation. The datawere acquired by instruments operated by the Big Bear Solar Observatory, HighAltitude Observatory, Learmonth Solar Observatory, Udaipur Solar Observatory,Instituto de Astrofísico de Canarias, and Cerro Tololo Interamerican Observatory.T. Corbard and R.A. García want to thank the CEA/DAPNIA and the CNES fortheir funding and support to the organization of the Saclay workshop.

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