A&A 493, 185-191 (2009)
DOI: 10.1051/0004-6361:200811047
I. W. Roxburgh^{1,2}
1 - Astronomy Unit, Queen Mary, University of London, Mile End Road, London E1 4NS, UK
2 -
LESIA, Observatoire de Paris, Place Jules Janssen, 92195 Meudon, France
Received 27 September 2008 / Accepted 26 October 2008
Abstract
Aims. We investigate the diagnostic potential of
p-modes and the origin of the periodicity in their small separations.
Methods. We used theoretical analysis, phase-shifts, modelling. and data analysis.
Results. The periodicity in the small separations between modes of
is determined by the acoustic radius of the base of the outer convective envelope. The mean variation is determined primarily by the structure of the inner core. The separations are related to the inner phase shifts differences
which we show can be determined directly from the frequencies. The modulation period is shifted slightly by the frequency dependence of the phase shifts and the amplitudes. We present results using data from the BiSON, IRIS, and GOLF experiments, and a solar model, all of which give a modulation period of Hz corresponding to an acoustic radius s.
Key words: stars: oscillations - stars: interiors
The small separations
between p-modes of oscillation have been widely used as diagnostics of the solar and stellar interiors (cf. Gough 1983; Provost 1984; Christensen-Dalsgaard 1984, 1988; Ulrich 1986), as they are primarily determined by the interior structure of the star (cf. Tassoul 1980). Since the predicted amplitudes of modes of degree
are considerably larger than those of ,
for some stars these may be the only modes that can be reliably determined and the only small separations available being those between modes with
defined as (cf. Roxburgh 1993)
(1) |
Figure 1: Top panel: ; middle panel: , bottom panel: , for solar model A. in units of 2500 Hz. | |
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Figure 1a shows the separations
d_{01}, d_{10} multiplied by
Hz) for a solar modelA (a smoothed version of model S, Christensen-Dalsgaard et al. 1996); they
display a periodic modulation about a mean curve which is not present in the d_{02} separations (Fig. 1c).
The periodicity is clearer in Fig. 1b which plots the 5 point differences (Roxburgh & Vorontsov 2003a). (Hereafter R&V = Roxburgh & Vorontsov.)
(2) | |||
(3) |
Figure 2: Variation of small separations with for frequencies obtained by the BiSON, IRIS and GOLF experiments. All data sets show the same periodicity and the same phase. | |
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Figure 2 shows the small separations for 3 solar data sets from the ground based experiments BiSON and IRIS and the GOLF instrument on SOHO. The BiSON data set was for the time span 1993-2003 (Chaplin et al. 2007; Verner 2008); the IRIS data set for 1989-99 (Fossat et al. 2003) and the P3 GOLF data set for 1996-1999 (Gelly et al. 2002). All show the same periodicity and phase. These data are analysed in more detail in Sect. 8 below.
Inspection of Figs. 1a and b gives a modulation period of
Hz. A more quantitative estimate is obtained by fitting a low order curve to remove the trend (as shown in Figs. 1a,b) and taking the Fourier transform of the residuals. The trend was removed by a least squares fit to the function
(4) |
Figure 3: Residuals to a low order fit; a) , b) . | |
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Figure 4: Power spectra of the residuals: a) , b) . | |
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As is well known, simple asymptotic analysis shows that discontinuities and steep gradients in the acoustic structure of a star induce periodic modulations in the frequencies with period
where
is the acoustic depth of the region of rapid variation (Vorontsov 1988; Gough 1990; R&V 1994b; Monteiro et al. 1994).
Likewise such discontinuities induce periodic variations with period 1/(2t_{1})
where t_{1} is the acoustic radius, the two being just alternative representations of each other (R&V 2001).
Further details are given in Sect. 7 below.
The acoustic radius t and acoustic depth
are defined by
(5) |
For model A used in the calculations for Figs. 3a and b, the base of the convective zone is at an acoustic radius of t=1422 s or equivalently an acoustic depth of s; so the equivalent modulation periods are Hz and Hz. Also in model A the location of the region of rapid change in acoustic variables due to HeII ionisation is around s, s, and the corresponding modulation periods are Hz and Hz.
It is clear from Figs. 4a and b that the signal in the separations with a modulation period of Hz corresponds to the acoustic radius of the region at the base of the convective zone, not the acoustic depth. The fact that the peaks are at Hz rather than Hz is due to two factors: a) the acoustic waves are not pure sine waves but are subject to a frequency dependent phase shift; b) the amplitudes of the contributions to the oscillating component are frequency dependent; both contribute to the Fourier transform shifting the peak from 352 to Hz. Details of the various contributions to the oscillating component and the frequency shift are given in Sect. 7 below.
As shown by R&V (2000, 2003a,b), by matching the solution of the oscillation equations
in the inner and outer layers at some intermediate acoustic radius t_{f}, the adiabatic oscillation eigenfrequencies
of a spherical star necessarily satisfy the Eigenfrequency equation
(6) |
(7) |
(8) | |||
(9) |
The inner and outer phase shifts as defined in Eqs. (8) and (9) are continuous functions of frequency. For any frequency they are given by partial wave solutions of the oscillations equations (cf. R&V 2000): for these are solutions which are regular at the centre and satisfy the the surface gravitational boundary conditions; for they are solutions which satisfy the surface boundary conditions. For modes of degree this completely determines since for the equations reduce to second order and the oscillating gravitational potential and its derivative , d/dr do not contribute to the determination of , whereas for modes , d/dr = 0 at the surface as the external gravitational dipole moment is identically zero for modes. The Eigenfrequency Eq. (6) is then given by demanding continuity between the inner and outer solutions at some t=t_{f} which gives the discrete eigenfrequencies .
The partial waves and eigenfrequencies can always be represented as in Eqs. (6)-(9); the value of this representation is that, with the above definition of (Eq. (7)) both are almost independent of t_{f} in the intermediate layers of a star (cf. R&V 1996) and is almost independent of . is therefore determined by the structure of the outer layers () and by the structure of the inner layers ().
Figure 5: Phase shifts and for model A: a) variation with acoustic radius for Hz. cannot be separated in these figures; b) variation with frequency for x_{f}= 0.9, t_{f}/T=0.615; c) variation of and with frequency. | |
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Figure 5a shows the variation of and with acoustic radius for Hz for model A; Fig. 5b shows their variation with frequency evaluated at x=x_{f}=0.9; whilst Fig. 5c shows the variation with frequency of the differences and . As stated above . The difference displays the periodic variation with modulation period Hz, caused by the base of the convective envelope.
Using the eigenfrequency Eq. (6), taking
to be independent of ,
and
as continuous functions of ,
the small separations
d_{01}, d_{10} can be expressed as
(10) | |||
(11) |
The surface phase shift ,
which has within it the modulation from the HeII ionisation zone, contributes to the small separations, this produces the considerable scatter in the residuals plotted in Fig. 3a. However since
(12) |
Since the small separations are governed by the phase shift difference we here show
how to determine this difference directly from the frequencies. We write the Eigenfrequency Eq. (6) in the form
(13) |
(14) |
(15) |
Figure 6: a) as determined from the frequencies and Eigen-frequency Eq. (14) with Hz. b) Phase shift difference determined from the eigenfrequencies by interpolation (points), the model values (continuous periodic line), the mean variation (smooth line). c) Power spectra of residuals to mean variation - the smaller maximum is that obtained using the points determined from the frequencies. | |
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We now draw on theoretical analysis that is essentially independent of (Fig. 5c) and that both and can be considered as continuous functions which vary on a scale sufficiently large for us to be able to interpolate between the points determined by the frequencies. This enables us to subtract the two curves to determine the values of . The points in Fig. 6b show the resulting values of at the eigenfrequencies obtained using cubic spline interpolation. Note that these values are independent of the choice of mean large separation and T since the addition to in Eq. (15) cancels out in the subtraction. The solid curve is the value obtained from solving the oscillation equations for partial waves and calculating the inner phase shifts at x_{f}=0.9 as illustrated in Fig. 5c. I emphasise that the points displayed in Fig. 6b are obtained from the frequencies alone. The small differences between the points and the continuous curve are due to several factors: a) interpolation is not exact; b) is not exactly equal to ; c) the are not exactly constant in the outer layers. Nevertheless the agreement is good. Panel (c) shows the power spectra of the residuals to a low order fit as in Eq. (5). The amplitude derived from the points in Fig. 6b is somewhat less than that derived from the continuous model values, but the location of the maxima are almost the same at Hz.
In Sect. 8 below this analysis is applied to data from the BiSON. IRIS and GOLF experiments.
The equations governing the Eulerian pressure perturbation p', gravitational potential perturbation
and radial displacement
for an oscillation with angular frequency
are (cf. Unno et al. 1979)
(16) |
(17) |
(18) |
(19) |
The equation governing the variable
(Eq. (7)) can then be reduced to one second order equation with acoustic radius
as independent variable namely
(20) |
(21) |
(22) |
(23) |
At the base of the convective zone the potentials U_{0}, Q are discontinuous and vary on a scale short compared with the wavelength of an oscillation mode, Fig. 7 shows the variation of these potentials for model A. Here Hz is a normalisation factor corresponding to Hz. We have superposed curves for Hz in Fig. 7; U_{0} varies very slightly with frequency whilst .
Figure 7: Acoustic potentials at base of the convective zone: (solid lines), (dashed lines), and (fine dotted lines), at a frequencies Hz. | |
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These sharp changes induce a periodic contribution
to the phaseshifts, which can be obtained by integrating Eq. (22) over the discontinuity (cf. R&V 2001). The discontinuity can be approximately represented as
(24) | |||
(25) |
(26) |
(27) |
(28) |
A simple illustration of this is to take the function is
(29) |
Figure 8: Power spectrum (Ampliude^{2}) of simple model Eq. (28). The modulation period corresponding to the base of the convective zone is Hz but the maximum power is shifted to Hz due to the frequency dependency of the phase shifts and of the amplitudes of the two components. | |
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The behaviour of the small separations for data from the BiSoN, IRIS, and the GOLF experiment on SoHO were shown in Fig. 2, they all show a behaviour similar to each other, and similar to that of the theoretical model.
To determine the phase shift differences and resulting power spectra for the modulation period we follow the analysis given in Sect. 6. We use the same frequency range for all data sets confining the analysis to modes of radial order n=9 to 24 where the estimated errors on the frequencies are less than Hz. The resulting values for the phase shift differences are shown in the top 4 panels of Fig. 9. Note that a error of Hz in the frequencies translates into a error of 0.002 in . We then remove the mean trend by a low order fit as in Eq. (4) and take a power spectrum of the residuals to estimate the modulation period. The results for this frequency range are shown in the bottom panel of Fig. 9, giving the peak in the power spectra at Hz (Bison), Hz (IRIS), Hz (GOLF) and Hz (model A). There is reasonable agreement between the model and data sets.
Figure 9: derived from the frequencies as described in Sect. 6 for a) model A, b) a BiSON data set, c) IRIS data set, d) GOLF data set. All data sets were analysed using frequencies in the range 1400-3500 Hz. Bottom panel: power spectra of residuals to a low order fit as in Eq. (4). The modulation period at the peak of the power spectra are as listed. There is good agreement between the values derived from solar data and that of the model. | |
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Next we undertook a Mont-Carlo simulation with 10 000 random realisations of the errors - that is the frequencies were taken as
with r_{k} generated by a random number generator with standard deviation 1 and a Gaussian distribution.
This gave the following results for the mean peak modulation period and 1
standard deviation:
However as shown above the modulation period is not simply 1/(2t_{1}) but is enhanced by 6-7 Hz due to the frequency dependence of the phase shifts and the amplitudes of the oscillating signal. One might try to fit a model function to the data to determine these contributions at the same time as determining the frequency corresponding to the base of the convective zone (cf. Verner et al. 2004), but this depends on having a reliable parameterised model of the region below the base of the convective zone and of the internal phase shift difference. We will return to this matter in a subsequent paper where we also search for solar cycle variations.
We have show that the periodicity in the small separations of p-modes of degree is due to the region of sharp change in acoustic variables at the base of the convective envelope, the modulation period being primarily determined by the acoustic radius t_{1} of this interface. These separations are determined by the difference between the internal phase shifts of modes of degree . This phase shift difference can be directly derived from the frequencies using the Eigenfrequency equation. The modulation period is not precisely 1/(2 t_{1}) as the peak frequency is modified by the frequency dependence of the phase shifts , and by the frequency dependence of the amplitudes of the resulting signal. This periodic signal is not seen in small separations of modes of degree . This is because modes with degree differing by 2 are almost in phase when they reach the base of the convective zone whereas modes differing by 1 in degree are almost out of phase. Other odd separations of modes of degree will display a similar periodicity.
We have not so far considered the diagnostic value of the combination which is directly calculated at eigenfrequencies and is known at once has been calculated by interpolation at simply by adding to . Even though and cannot be separately determined the combination will nevertheless contain the signature of the HeII ionisation which can be extracted by a variety of procedures (cf. Brodskii & Vorontsov 1989; Vorontsov & Zharkov 1989; Gough 1990; Lopes et al. 1997; Perez-Hernandez & Christensen-Dalsgaard 1998; Roxburgh & Vorontsov 2001; Verner 2004) and used to infer the Helium abundance and entropy of the convective envelope.
Since the mean variation of with frequency is determined by the structure of the inner regions of a star it provides a diagnostic of the internal structure; this can be used as the basis of crude inversion procedure to probe the internal density distribution by parameterising the structure in terms of values of the polytropic index at a few interior points (cf. Roxburgh 2000; R&V 2002), even when one only has data on modes. Moreover as shown by R&V (2003b) the requirement that the -dependent inner phase shifts must match on to an -independent surface phase shift can be used to find a best fit interior model out of a set of such models, independent of the structure of the surface layers, even when one only has modes with (R&V 2003b). The mean values of the small separation d_{01} and large separations, and their ratios, can be used with a 0,1 asteroseismic diagram to place constraints on a possible stellar model (Christensen-Dalsgaard 1988; Mazumdar & Roxburgh 1993). Of course one can do better when one also has modes of higher degree, but as remarked at the beginning of this paper, for some stars we may only be able to detect modes of degree .
Acknowledgements
I thank Graham Verner and the BiSON network for providing the 11 year frequency set used in the analysis, and the UK Science and Technology Facilities Council (STFC) which supported this work under grants PPA/G/S/2003/00137 and PP/E001793/1.