Birmingham Solar-Oscillations Network Research Articles

A comparison of the use of genetic and hill-climbing algorithms to fit low-l solar p-mode spectra

We present a comparative study of low-l solar p-mode parameters extracted by genetic-algorithm and ‘standard’ hill-climbing minimisation routines. To effect this we make use of observations made in integrated sunlight by the Birmingham Solar-Oscillations Network (BiSON) and the GOLF instrument on board the ESA/NASA SOHO satellite, in addition to artificial data. We find that over the central part of the p-mode range the two fitting routines return similar results. However, at low frequencies — where the S/N in the modes is low and their resonant peaks narrow — we find that the genetic routine appears to offer more robust estimates of the underlying parameters.

Solar p-mode frequencies at $\mathsf{\ell=2}$: What do analyses of unresolved observations actually measure?

We have studied in detail the extraction of estimates of p-mode frequencies from unresolved observations of the visible disc of the Sun. Examples of data of this type include ground-based observations made by the Birmingham Solar-Oscillations Network (BiSON), and space-borne observations made by the GOLF and VIRGO/SPM instruments on board the ESA/NASA SOHO satellite. The fitting of the modes is complicated in practice by the asymmetric arrangement in frequency of the three components (, 0 and 2) that are prominent in such data. In order to investigate the effect of this we used a series of 10-yr artificial datasets into which varying degrees of asymmetry were introduced. The sets were designed to mimic the characteristics of the BiSON and GOLF data, and were analyzed both with and without the BiSON window function from the period 1992 through 2001. Since reliable estimates of the asymmetry have only recently been extracted from unresolved observations (Chaplin et al. [CITE]a) it has for a long time been standard practice to fit the modes to a model that assumes a symmetrically arranged multiplet. We have tested the impact of this on the accuracy of the extracted frequencies. Furthermore, we demonstrate that asymmetric models can be successfully applied, provided the data are of sufficient length and quality. We also discuss the implications of our simulations for analyses of real solar data.

Use of genetic algorithms to determine low-lrotational p-mode splittings at high frequencies

We present a comparative study of genetic and standard fitting routines applied to the task of extracting reliable estimates of the rotational splitting of full-disc, low-angular-degree (low-l) solar p-mode data at high frequencies. 100 artificial proxies of a 10-yr data base of observations made by the Birmingham Solar-Oscillations Network (BiSON) were used to test the two approaches. All sets were analysed over the frequency range from 3000 to 4000 μHz. Previous work, based on non-linear ‘hill-climbing’ fitting techniques, has demonstrated the unfortunate tendency for full-disc estimates of the splitting to overestimate the true, underlying values at high frequencies. Here, we show that the resulting bias is less severe when a genetic-fitting approach is adopted. This is largely the result of the number of erroneous ‘null-valued’ estimates of the splitting being considerably reduced: these estimates are, in effect, re-introduced into the expected normal distribution of fitted splittings. We also illustrate the diverse control one has when using a genetic algorithm as a fitting routine; this diversity is shown to allow further refinement in the estimate of the rotational splitting. Finally, we address the issue of the reliability of the formal splitting uncertainties returned by the mode fitting, and find that complications arising from the strong anticorrelation between the splittings and their error estimates are not alleviated by the use of the genetic technique.

Studies of the solar mean magnetic field with the Birmingham Solar-Oscillations Network (BiSON)

The first analysis of 10 yr of solar mean magnetic field data from the Birmingham Solar-Oscillations Network (BiSON) instrument at Sutherland, SA is presented. Long-period (>1 d) variations correlate well with those observed by the Stanford group, but our daily mean values have a statistical precision over 10 times better than previous work. Differences between the absolute values determined from Sutherland and Stanford are discussed, as are systematic annual variations of the order of 15 per cent. At high frequencies (>0.5 mHz), the mean noise spectral density for our decade of observations is 0.53 G(2) Hz(-1) , setting a lower limit to the noise level than that obtained from brief observations by GOLF on board the ESA/NASA satellite SOHO in 1996. The high-frequency mean noise spectral density is different in different years. The lowest annual value, obtained for 1997, is 0.22 G(2) Hz(-1) , whilst the highest, for 1995, is 1.46 G(2) Hz(-1) . It is not yet clear whether these variations are related to solar activity.

Observation of, and temporal variations in, solar p-mode multiplet frequency asymmetries at = 2

The predominant contribution to the frequency splitting of low-l solar p modes arises from the rotation of the solar interior and this lifts the frequency degeneracy in l to give a symmetric pattern where the observed (synodic) separation between adjacent m (i.e. for \Deltam\=1) is similar to400 nHz. Magnetic fields can also contribute to the splitting, but they do so in such a way as to introduce asymmetries in the arrangement of the components within each multiplet. In disc-integrated data this effect may become apparent when lgreater than or equal to2. Here, we attempt to extract estimates of the frequency asymmetries at l=2 from the analysis of disc-integrated data collected by the ground-based Birmingham Solar Oscillations Network (BiSON) and the GOLF instrument on board the ESA/NASA SOHO satellite. Our analyses demonstrate that we have evidence for there being non-zero asymmetries present (significance approximate to3-4sigma) during an epoch coincident with high levels of surface activity close to the maximum of solar cycle 23. The asymmetries are indistinguishable from zero at minimum levels of activity near the cycle 22/23 boundary. We also compare the observed asymmetries with those calculated from a model that is based upon the recent predictions of Moreno-Insertis & Solanki. While the level of agreement between the two is found to be reasonable, the observations suggest (though with poor constraints placed upon this) that the influence on the mode frequencies of high-latitude activity may not be as strong as in the model.

Does the Energy Supplied to Low-[CLC][ITAL]l[/ITAL][/CLC] Solar [CLC][ITAL]p[/ITAL][/CLC]-Modes Vary over the Activity Cycle?

We report on the average behavior of the excitation and damping of low angular degree (low l) solar p-mode oscillations over the decade from 1991 to 2000 using both long and short time duration Fourier transforms. The data in question were collected by the ground-based Birmingham Solar Oscillations Network. Throughout most of the period under study, the energy supply rate to the modes remains roughly constant—implying a near-constant level of forcing—while the damping and velocity power show a fairly smooth increase and decrease, respectively, in response to increasing levels of solar activity (in line with previous findings). However, here we uncover evidence of there being a sharp increase in the mode velocity power over a brief period of approximately 100 days centered on 1998 late March. The magnitude and sign of this are contrary to the expectation based on the long-timescale, solar-cycle trend; such unusual behavior is absent in the damping. This implies that the forcing of, or rate of energy supplied to, the modes increased in magnitude over this period.

On the measurement bias of low-$\ell$ solar $\vec{p}$-mode excitation parameters: The impact of a ground-based window function

We present a study of the impact of a ground-based, multi-station window function on estimates of the power and damping of low-l solar p modes extracted from fits to resonant structure in the frequency domain. The window functions come from six-site observations made by the Birmingham Solar-Oscillations Network (BISON) over the 10-yr period beginning 1991 January. Two strategies were adopted. In the first, we used an 800-d time series of continuous observations made by the GOLF instrument on board the ESA/NASA SOHO satellite. These data were modulated by a variety of BISON window functions, with fractional duty cycles ranging from similar to0.4 to similar to0.8, and the resulting series analyzed. In the second we generated artificial 10-yr time series and studied the effect on these of the complete BISON window.

Peak finding at low signal-to-noise ratio: low- solar acoustic eigenmodes at n<= 9 from the analysis of BiSON data

We make use of 9 yr of full-disc helioseismic data - as collected by the ground-based Birmingham Solar-Oscillations Network (BiSON) - to search for low-frequency, low-angular-degree (low-l) acoustic modes. A range of tests are applied to the power spectrum of the observations that search for prominent mode-like structure: strong spikes, structure spanning several bins signifying the presence of width (from damping), and the occurrence of prominent multiplet structure at l greater than or equal to 1 arising principally from the solar rotation and made from several spikes separated suitably in frequency. For each test we present analytical expressions that allow the probability that the uncovered structure is part of the broad-band noise background to be assessed. These make use of the cumulative binomial (Bernoulli) distribution and serve to provide an objective measure of the significance of the detections. This work has to date uncovered nine significant detections of non-broad-band origin that we have identified as low-l modes with radial overtone numbers n less than or equal to 9.

On the measurement precision of solar p-mode eigenfrequencies

We make use of 3456 d of observations of the low-l p-mode oscillations of the Sun in order to study the evolution over time of the measurement precision of the radial eigenfrequencies. These data were collected by the ground-based Birmingham Solar-Oscillations Network (BiSON) between 1991 January and 2000 June. When the power spectrum of the complete time series is fitted. the analysis yields frequency uncertainties that are close to those expected from the returned coherence times of the modes. The slightly elevated levels compared with the prediction appear to be consistent with a degradation of the signal-to-noise ratio in the spectrum that is the result of the influence of the window function of the observations (duty cycle 71 per cent). The fractional frequency precision reaches levels of a several parts in 10(6) for many of the modes. The corresponding errors reported from observations made by the GOLF instrument on board the ESA/NASA SOHO satellite, when extrapolated to the length of the BiSON data set, are shown to be (on average) about similar to25 per cent smaller than their BiSON counterparts owing to the uninterrupted nature of the data from which they were derived. An analysis of the BiSON data in contiguous segments of different lengths, T, demonstrates that the frequency uncertainties scale as T-1/2. This is to be expected in the regime where the coherence (life) times of the modes, tau(nl), are smaller than the observing time T (the 'oversampled' regime). We show that mode detections are only now beginning to encroach on the 'undersampled' regime (where T < τ(nl)).

Rigid rotation of the solar core? On the reliable extraction of low- rotational p-mode splittings from full-disc observations of the Sun

We present low-l rotational p-mode splittings from the analysis of 8 yr of observations made by the Birmingham Solar-Oscillations Network (BiSON) of the full solar disc. These data are presented in the light of a thorough investigation of the fitting techniques used to extract them. Particular attention is paid to both the origin and magnitude of bias present in these estimates. An extensive Monte Carlo strategy has been adopted to facilitate this study-in all, several thousand complete, artificial proxies of the 96-month data set have been generated to test the analysis of real 'full-disc' data. These simulations allow for an assessment of any complications in the analysis which might arise from variations in the properties of the p modes over the 11-yr solar activity cycle. The use of such an extended data set affords greater precision in the splittings, and by implication the rotation rate inferred from these data, and reduces bias inherent in the analysis, thereby giving a more accurate determination of the rotation. The grand, weighted sidereal average of the BiSON set is 434 +/-2 nHz, a value consistent with that expected were the deep radiative interior (r/R<0.5) to rotate at the same frequency, and in the same 'rigid' manner, as the more precisely and accurately studied outer part of the radiative zone.

Analysis of the solar cycle and core rotation using 15 years of Mark-I observations: 1984-1999

High quality observations of the low-degree acoustic modes (p-modes) exist for almost two complete solar cycles using the solar spectrophotometer Mark-I, located at the Observatorio del Teide (Tenerife, Spain) and operating now as part of the Birmingham Solar Oscillations Network (BiSON). We have performed a Fourier analysis of 30 calibrated time-series of one year duration covering a total period of 15 years between 1984 and 1999. Applying different techniques to the resulting power spectra, we study the signature of the solar activity changes on the low-degree p-modes. We show that the variation of the central frequencies and the total velocity power (TVP) changes. A new method of simultaneous fit is developed and a special effort has been made to study the frequency-dependence of the frequency shift. The results confirm a variation of the central frequencies of acoustic modes of about 450 nHz, peak-to-peak, on average for low degree modes between 2.5 and 3.7 mHz. The TVP is anti-correlated with the common activity indices with a decrease of about 20% between the minimum and the maximum of solar cycle 22. The results are compared with those obtained for intermediate degrees, using the LOWL data. The frequency shift is found to increase with the degree with a weak l-dependence similar to that of the inverse mode mass. This verifies earlier suggestions that near surface effects are predominant.

Changes in convective properties over the solar cycle: effect on p-mode damping rates

Measurements of both solar irradiance and p-mode oscillation frequencies indicate that the structure of the Sun changes with the solar cycle. Balmforth, Gough & Merryfield investigated the effect of symmetrical thermal disturbances on the solar structure and the resulting pulsation frequency changes. They concluded that thermal perturbations alone cannot account for the variations in both irradiance and p-mode frequencies, and that the presence of a magnetic field affecting acoustical propagation is the most likely explanation of the frequency change, in the manner suggested earlier by Gough & Thompson and by Goldreich et al. Numerical simulations of Boussinesq convection in a magnetic field have shown that at high Rayleigh number the magnetic field can modify the preferred horizontal length scale of the convective flow. Here, we investigate the effect of changing the horizontal length scale of convective eddies on the linewidths of the acoustic resonant mode peaks observed in helioseismic power spectra. The turbulent fluxes in these model computations are obtained from a time-dependent, non-local generalization of the mixing-length formalism. The modelled variations are compared with p-mode linewidth changes revealed by the analysis of helioseismic data collected by the Birmingham Solar-Oscillations Network (BiSON); these low-degree (low-l) observations cover the complete falling phase of solar activity cycle 22. The results are also discussed in the light of observations of solar-cycle variations of the horizontal size of granules and with results from 2D simulations by Steffen of convective granules.

Changes to low-ℓ solar p-mode frequencies over the solar cycle: correlations on different time-scales

We have studied variations in the frequencies of low-l solar p modes through the analysis of nine years of helioseismic data collected by the Birmingham Solar Oscillations Network (BiSON)dagger. This is the first time that such a long data set has been explored with the extra accuracy afforded by fitting the modes to asymmetric profiles. The epoch covered (1991-99) spans the declining activity phase of solar cycle 22, and a substantial portion of the initial activity increase during cycle 23. The complete time series has been split into contiguous segments of length 27, 54, 108 and 216 d in order to facilitate the study of changes occurring on different time-scales. Further, we have characterized the observed shifts as a function of six well-known indicators of solar activity. These indices reflect changes taking place in the photosphere, chromosphere and corona, but only over the visible hemisphere of the Sun. Since the low-l eigenfrequencies respond to global variations in activity, we discuss the implications of this mismatch for the analyses performed. We demonstrate that, as expected, the low-l modes adjust to changes in the activity measures on time-scales as short as a few months. Our analysis indicates that all six proxies correlate equally well (at the level of precision of the data) with the measured shifts. Further, the sensitivity of the shifts to changes in five of the activity indices is the same, to within similar to 15 per cent (1 sigma) or so, on the falling and rising phases considered. There is, however, a slight suggestion that the sensitivity to changes in the disc-averaged line-of-sight magnetic field component (as determined from daily Kitt Peak magnetograms) may be higher on the rising phase of the cycle.

Helioseismic Constraints on Rotation and Magnetic Fields in the Solar Core

In recent years, more and more precise measurements have been made of solar oscilla- tion frequencies and line widths. From space, the Solar and Heliospheric Observatory/Michelson Doppler Imager (MDI) data has led to much progress. From the ground, networks, like Global Oscillation Network Group (GONG), Taiwanese Oscillation Network (TON), and Birmingham Solar Oscillations Network (BiSON) have also led to much progress. The sharpened and enriched oscil- lation spectrum of data have been critically complemented by advances in the treatments of the opacities and the equation of state. All of this has led to a significantly more precise probing of the solar core. Here we discuss the progress made and suggest how the core may be better probed with seismic data on-hand. In particular, we review our knowledge of the rotation and structure of the core. We further argue that much may be learned about the core by exploiting the line width data from the aforementioned sources. Line-width data can be used to place sharper constraints on core properties, like the degree to which the Sun rotates on a single axis and the upper limit on magnetic fields that may be buried in the core.

Width and Energy of Solarp‐Modes Observed by Global Oscillation Network Group

We present measurements of mode width, Γ, and mean square velocity power, , derived from all currently processed 108 day Global Oscillation Network Group (GONG) time series and discuss their implications for p-mode excitation and damping. Assuming stochastic excitation, we estimate the energy per mode, E, and the energy supply rate, dE/dt. For modes with l = 9-150, the mean square velocity power and the mode energy peak at about 3.15 mHz reaching values of ≈ 1.4 × 103 cm2 s-2 and E ≈ 2.2 × 1028 ergs. The energy supply rate reaches a maximum value of dE/dt ≈ 2.5 × 1023 ergs s-1 near 3.6 mHz. The mode width shows the familiar plateau between 2.5 and 3.1 mHz with a near 2.9 mHz, which is strongest for l ≈ 40. This dip is most prominent during solar cycle minimum and disappears with increasing magnetic activity. The energy supply rate decreases on average by about 2.7% from the previous activity minimum to the currently highest level of activity. The solar cycle variation of dE/dt shows no frequency dependence, which is in marked contrast to the other mode parameters. The mode parameters are adequately represented by power laws in several frequency ranges, for example, dE/dt ~ ν6.89±0.07 for 2.4 mHz ≤ ν < 3.0 mHz and dE/dt ~ ν-5.62±0.27 for 3.75 mHz ≤ ν < 4.5 mHz. The solar cycle variation of these parameters can then be expressed as changes of a few percent in the power-law exponents and multipliers. Our results agree reasonably well with previous studies of Birmingham Solar-Oscillations Network and Big Bear Solar Observatory data.

Observational Upper Limits to Low‐Degree Solarg‐Modes

Observations made by the Michelson Doppler Imager (MDI) and Variability of solar IRradiance and Gravity Oscillations (VIRGO) on the Solar and Heliospheric Observatory (SOHO) and by the ground-based Birmingham Solar Oscillations Network (BiSON) and Global Oscillations Network Group (GONG) have been used in a concerted effort to search for solar gravity oscillations. All spectra are dominated by solar noise in the frequency region from 100 to 1000 μHz, where g-modes are expected to be found. Several methods have been used in an effort to extract any g-mode signal present. These include (1) the correlation of data—both full-disk and imaged (with different spatial-mask properties)—collected over different time intervals from the same instrument, (2) the correlation of near-contemporaneous data from different instruments, and (3) the extraction—through the application of complex filtering techniques—of the coherent part of data collected at different heights in the solar atmosphere. The detection limit is set by the loss of coherence caused by the temporal evolution and the motion (e.g., rotation) of superficial structures. Although we cannot identify any g-mode signature, we have nevertheless set a firm upper limit to the amplitudes of the modes: at 200 μHz, they are below 10 mm s-1 in velocity, and below 0.5 parts per million in intensity. The velocity limit corresponds very approximately to a peak-to-peak vertical displacement of δR/R☉ = 2.3 × 10-8 at the solar surface. These levels which are much lower than prior claims, are consistent with theoretical predictions.

Source of excitation of low- solar p modes: characteristics and solar-cycle variations

We investigate various properties of the excitation source that is responsible for driving the acoustic p-mode oscillations of the Sun. Current prejudice places this in the superadiabatic layer of the convection zone. We consider in detail how the precise nature of the resonant mode spectrum is modified: (i) as a result of the impact of different source-multipole mixtures; and (ii) as a function of the radial extent of the source. To do this, we model the observed resonant spectra with the solutions to a simple, one-dimensional wave equation which is intended to describe the essential elements of the solar resonant acoustic cavity, Further, we also fit these models to the low-l peaks in a high-resolution power spectrum generated from data collected by the Birmingham Solar-Oscillations Network (BiSON), We also use the extensive BiSON data set to search for variations in the source characteristics over the solar cycle.

Variations in the excitation and damping of low- solar p modes over the solar activity cycle

We have searched helioseismic data collected by the Birmingham Solar Oscillations Network (BiSON) for solar-cycle changes to those low-ℓ p-mode parameters that relate to the excitation and damping of the resonances. These data — collected between 1991 and 1997 — cover the complete declining phase of solar activity cycle 22 (up to and including the cycle 22/23 boundary). Over the range 2600≤ν≤3600 μHz, we uncover a mean 24±3 per cent increase in the frequency-domain linewidths; a mean decrease of 46±5 per cent decrease in the mode heights, and a mean decrease of 22±3 per cent in the modal velocity powers. The rate at which energy is supplied to the modes remains constant, at the level of precision of the observations (measured change 0±4 per cent). We use expressions derived from the equation of a damped harmonic oscillator to illustrate the diagnostic properties of the observables: these indicate that both the signs and relative sizes of the extracted variations can arise from changes solely to the net damping; the net forcing of the modes need not change. The results possibly hint at the changes being maximal at frequencies near ∼3100 μHz. They might therefore suggest an origin for the observed variations that is peaked in the superadiabatic layer of the convection zone, which couples most strongly to the eigenfunctions of modes at the centre of the p-mode spectrum.

Depth of excitation and reversal of asymmetry of low- solar p modes: a complementary analysis of BiSON and VIRGO/SPM data

We use the solutions to a simple, one-dimensional wave equation – which is intended to describe the essential elements of the solar resonant acoustic cavity – as formalistic models to which to fit low-l modes in observational helioseismic power spectra. We have analysed data collected in velocity, by the ground-based Birmingham Solar-Oscillations Network (BiSON), and in intensity, by the full-disc VIRGO Sun photometers (SPM) on board the ESA/NASA SOHO satellite. Our analysis extracts, as a function of the modal radial overtone number n, direct estimates of: the location of the excitation source of the acoustic resonances; and the size of the component of the convective granulation that is correlated to the observations of the resonances, and is responsible for the reversal of asymmetry of the mode profiles when different observation techniques are applied (i.e. intensity or Doppler velocity measurements). This information is coded in the complicated forms of the observed, resonant profiles.

An analysis of solar p-mode frequencies extracted from BiSONdata: 1991-1996

We present a comprehensive frequency analysis of Doppler velocity observations of the visible solar disc made by the Birmingham Solar Oscillations Network (BiSON) from 1990-1996, i.e. covering the falling phase of activity cycle 22, up to and including the cycle 22/23 boundary. We have fitted low-degree (low-l) solar p modes in a variety of power spectra of differing lengths generated from these data. The analysis of the extracted frequencies reveals the expected clear solar-cycle dependence; in addition, there is now sufficient accuracy in the data to show that the low-l modal eigenfrequencies are less affected by the solar cycle than their higher l counterparts. The observed low-degree frequency shifts up to approximate to 3900 mu Hz are consistent - at the level of precision of the data - with an inverse mode-mass scaling. At frequencies above this, the blending of modes adjacent in frequency space makes it increasingly difficult to extract reliable frequency estimates. However, our data show indications of a turnover and possibly also an eventual sign change in the solar cycle shifts at frequencies above;approximate to 4000 mu Hz, as seen in higher l data. We have parametrized the observed shifts as a function of the 10.7-cm radio flux, and produced an activity-corrected, average frequency table which incorporates eigenfrequencies from 18 4-month and 9 8-month spectra. We also present the fitted frequencies from a 32-month power spectrum, generated from data collected at or close to solar minimum. In addition, we also searched for frequency asymmetries in the l = 2 mode multiplets. These could result from strong near-surface magnetic activity, or a buried magnetic field. Our fits merely place an upper limit to any mean asymmetry - over the range 10 less than or equal to n less than or equal to 21 - of between approximate to - 80 and approximate to 170 nHz (3 sigma).