Oscillation amplitudes

The mode amplitudes and the corresponding mode heights are parameterized by an enve-lope function according to Equations (2.14)-(2.16). This enveenve-lope allows one to describe the p-mode amplitudes with angular degree` ≤ 2 in the full oscillation power spectrum using only six free parameters. Three parameters describe the center,ν0, and the width,eσi

(i= 1,2), of the envelope, the other three parameters,A<sub>`</sub> (with` = 0,1,2), are the max-ima of the amplitude envelope for the corresponding angular degree`. Figure 2.17 shows the distributions of the 100 initial guesses and the corresponding results of the global fit of these parameters for one block of a 120 day VIRGO time series. Comparing the uniform distribution of the initial guesses with the fit results, it is obvious that the global fit allows a reliable estimation of the parameters of the amplitude envelopes. The distributions of the fit results show a well defined single peak. The best fit is close the maximum of any of the distributions.

The maxima of the amplitude envelope,A_`, and thus the mode amplitudes in the indi-vidual blocks of VIRGO data show a significant variation during the solar cycle, i.e. mode amplitudes are smaller during the solar maximum and higher during the solar minimum.

In order to derive an uncertainty estimate for the amplitudes according to a 4 month time

Table 2.4: Mode parameters of the radial modes (` =0) of the Sun obtained in this work.

The parameters are the result of a global fits of 35 blocks of 120 day VIRGO time series.

The error intervals correspond to a 4 month observation. The fit ranges over 14 consec-utive radial orders (column 1). The mode frequenciesν_n0 (column 2) correspond to Ta-ble 2.2 (see also Section 2.3.1).Column 3lists the mode linewidth,Γn0. The values for the linewidth correspond to the median of the results of the 35 blocks. The 1σerror intervals are defined such that 68% of all fits fall within the bounds set by the subscripts/exponents (i.e. 34% of the fits below/above the median). Column 4 shows the mode amplitudes, An0 = √

πH_n0Γn0. The values are the mean amplitudes of all 35 blocks of VIRGO data.

To take amplitude variations over a solar cycle into account, the 1σerror corresponds to the mean of yearly uncertainty estimates, i.e the standard deviation of the amplitudes of 2-3 blocks of VIRGO data is averaged over the full 14 years of data. Column 5lists the signal-to-noise ratio of the oscillation modes. It is defined as the ratio of the mode height, Hn0, and the background noise at the frequency of the corresponding mode,B(νn0).

n ν_n0[µHz] Γn0[µHz] A_n0 [ppm] H_n0/B(ν_n0) 15 2228.94±0.11 0.95⁺_−0.27^0.23 1.44±0.08 2.7 16 2362.45±0.08 1.06⁺_−0.14^0.17 1.81±0.09 4.2 17 2496.20±0.07 1.09⁺_−0.13^0.19 2.29±0.09 7.7 18 2630.20±0.06 1.04⁺_−0.11^0.10 2.90±0.09 14.9 19 2764.45±0.06 0.97⁺_−0.10^0.12 3.60±0.11 28.2 20 2898.95±0.06 0.99⁺_−0.13^0.10 4.27±0.12 45.9 21 3033.70±0.07 1.07^+0.14_−0.10 4.62±0.13 56.0 22 3168.70±0.07 1.35⁺_−0.12^0.16 4.54±0.13 49.2 23 3303.94±0.08 1.85⁺_−0.11^0.19 4.13±0.12 34.1 24 3439.43±0.11 2.64^+0.18_−0.08 3.56±0.10 20.2 25 3575.17±0.14 3.84⁺_−0.21^0.15 2.97±0.08 11.2 26 3711.15±0.18 5.42^+0.24_−0.31 2.43±0.07 6.1 27 3847.39±0.22 7.39^+0.48_−0.47 2.00±0.06 3.4 28 3983.87±0.28 9.96^+0.73_−0.73 1.64±0.05 1.9

Figure2.17:Distributionofthe100initialguesses(thick,grey)andthecorrespondingresultsoftheglobalfit(fine,black)oftheamplitudeparametersforoneparticularblockofa120dayVIRGOtimeseries.Theindividualpanelscorrespondtothecentralfrequencyν0(topleft),thewidtheσi(i=1,2;topmiddle/right)andthemaximumoftheamplitudeenvelopeA`(`=0,1,2,bottomrow)accordingtoEquations(2.14)-(2.15).Theverticaldashedlineineachpaneldenotesthebestfit.

Figure 2.18: Mode amplitudes,A0, of the radial modes (` = 0) as a function of the mode frequency,ν_n0. The amplitudes are determined from global fits of 35 blocks of 120 day VIRGO time series. The amplitudes are computed according to Equations (2.14)-(2.16).

Thesolid line represents the mean amplitudes, thedashed lines correspond to 1σ error bars.

series, I adopt the same approach as for the mode frequencies. The amplitude parameters are averaged over the 35 blocks of VIRGO data, i.e. over 14 years. The 1σerror estimates are defined as the mean of yearly uncertainties. That implies the standard deviations of the individual parameters from within 1 year, i.e. 2-3 blocks of VIRGO data, are averaged over the full time span. Table 2.3 shows the mean and the 1σ error estimates on the in-dividual parameters of the amplitude envelope. The center of the amplitude envelope is atν₀ = 3067±26 µHz confirming the maximum power of low-degree solar p modes at

∼3 mHz. For the radial modes, I derive a maximum amplitude ofA₀ =4.64±0.13 ppm.

This result matches former measurements of the luminosity amplitude of low-degree so-lar p modes reasonably well. For instance, Kjeldsen and Bedding (1995) measured a luminosity variation of (δL/L)550 =4.7±0.3 ppm where the index refers to a luminosity variation at a wavelength ofλ = 550 nm. This value is determined from velocity mea-surements of low-degree solar p modes using a scaling relation derived in the same paper.

Kjeldsen and Bedding (1995) relate the bolometric luminosity variation and the variation at a particular wavelengthλvia

δL L

bol =δL L

λ

λ 623nm

Teff

5777K. (2.20)

Using this scaling relation, the amplitudes of the radial modes determined in this analysis at λ = 500 nm can be converted to a reference wavelength of λ = 550 µHz. Thus, I obtain a luminosity variation of (δL/L)₅₅₀= 4.22±0.12 ppm. Kjeldsen and Bedding also rescaled former measurements from Woodard and Hudson (1983), Jimenez et al. (1990), and Toutain and Froehlich (1992) and converted them to a reference ofλ=550 nm. Those studies were based on solar intensity measurements and led to a luminosity variation of 3.6 ppm ≤ (δL/L)₅₅₀ ≤ 6.5 ppm which matches the results obtained in this analysis very well.

The estimates of the amplitude parameters allow us to calculate the mode heights, Hn`m, and the mode amplitudes, An`m, according to Equations (2.14)-(2.16). The mode amplitudes and their corresponding 1σerror bar is calculated in the same way as for the amplitude parameters above. I note that the mode amplitudes for modes with` = 1,2 follow the same functional form and are only shifted by a factor of A₁/A₀ ≈ 1.3 and A₂/A₀ ≈0.8. The amplitudesA_n0of the radial modes and their corresponding uncertainty are shown in Figure 2.18 and listed in Table 2.4. The table also shows the signal-to-noise ratio Hn0/B(ν_n0) for the radial oscillations. It is defined as the ratio of the mode height H_n0to the background noiseB(ν_n0) at the frequency of the corresponding mode,ν_n0. The modes included in this analysis have a signal-to-noise ratio of 2. Hn0/B(νn0). 60.