Asteroseismology

This is a essay I wrote for my 4th year of undergraduate studies in 2010. The infomation contained in it is now a little out of date I recommend anyone with an interest in the topic reads William Chaplin's 'Music of the Sun' and looks at the results from the last few years of Kepler data, because they are amazing.


Asteroseismology

Stuart J. Mumford

Abstract

Asteroseismology is the study of the oscillation modes of stars, it has developed from Helioseismology which has provided a fascinating amount of information about the internal structure and processes of the Sun. The recent launches of high precision space based photometry instruments has opened the doors to an influx of data suitable for Asteroseismology. This will in turn, have the ability to transform our understanding of the universe’s structure and evolution, just as solar physics stood on the brink of a helioseismic revolution in the late 1960’s astrophysics now stands on the cusp of a period of rapid discovery due to asteroseismology.

Introduction

Humanity has been staring into space probably for as long as our species has existed, with the invention of the telescope, and advancements in technology, we have been able to peer deeper and deeper into the black. However, how can we look below the surface of an object, such as the Sun, which is opaque to our eyes? Until 1960 the internal workings of the Sun could only be theorised, we had no observational mechanism to study it, then in the summer of 1960 observations of the movement of the solar surface were being taken to study the lifetime of convection features, granules. What Bob Leighton found was the surface of the Sun was ringing with a peak oscillation period of about 5 minutes, this discovery was the beginning of a new field of science, Helioseismology.(William J. Chaplin 2006)

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Fig. 1 | Velocty field of the Sun where the dark areas are
receeding patches. (Leighton et al., 1962)

The discovery paper, published in 1962 detailed the observational techniques, and demonstrated by use of doppler difference images, regular movement on the solar surface. (Leighton, Noyes, and Simon 1962)The method used was a scanning ‘spectroheliograph’ (Leighton 1959) which was adapted not to measure magnetic field but doppler shift in the spectral lines. It achieved this by developing the method where the intensity of the light slightly to the blue side, and slightly to the red side of a Fraunhoufer line was recorded onto two superimposed plates, the apparatus had a scanning period of a few minutes meaning that two separate observation runs gives the change in the surface velocity field, a method which is still used for doppler measurements of the Sun. If the velocity field on the surface of the Sun was only due to the granulation and convective noise, a subtraction of two observation runs should result in random noise. Instead what Leighton found was a well structured velocity field which had a period of around 5 minutes, this is shown in Figure 1. This was the beginnings of what is now called ‘Helioseismology’ the oscillations of the Sun.

After the initial publication by Leighton, further analysis techniques were applied to the data and the oscillation spectra was displayed as a power spectra. Initially this analysis was done in just the frequency domain, however soon two dimensional power spectra in both angular frequency and wavenumber (\(\omega\) and \(k\) ) were created. These graphs required a major push in available instrumentation to obtain the oscillation spectra with high enough resolution to form observable features on the plots. The quest for higher quality data of the spectra is a factor in the current asteroseismic observations just as much as at the birth of the field. The results of a 1975 paper (Deubner 1975) gave for the first time the 5 minute oscillation signal in a \(\omega\) and \(k\) power spectra which is shown in Figure 2a, this is compared to a modern power spectra with data from the SOHO (Solar and Heliospheric Observatory) satellite and MDI (Michelson Doppler Imager) instrument in Figure 2b.

The power spectra in Figure 2a and an independent verification from (Rhodes, Ulrich, and Simon 1977) allowed conformation of the theory which described the solar oscillation signal as trapped acoustic modes in the Sun (Ulrich 1970), which theorised the distinctive overtone ridges in the \(\omega\) - \(k\) power spectra. The acoustic wave theory presented in (Ulrich 1970) described the surface oscillation as trapped acoustic waves resonating in a spherical cavity with a size dependant upon the horizontal velocity of the wave. An acoustic wave trapped in the convection zone of the Sun will refract totally at the depth where its horizontal velocity is equal to the local sound speed. The ridges on the \(\omega\) - \(k\) diagram are formed because at each horizontal wavenumber there are overtones of frequencies resonating in the cavity, in the same way a musical instrument has many resonating overtones to form a musical note. (William J. Chaplin 2006)

Helioseismology and Asteroseismology

As asteroseismology is a direct extension of helioseismology, a discussion of the principal results and data types available for helioseismic analysis of the Sun is applicable to asteroseismology. The primary comparisons between Astero and Helioseismology can be drawn from what is termed ‘Global Helioseismology’ which is the study of whole Sun oscillation modes. It is clear that these oscillation modes are roughly equivalent to the observable modes on other stars, where spatial resolution is not available.

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Fig. 2 a | The first published \(\omega\) - \(k\) power spectra.
(Deubner, 1975)
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Fig. 2 b | A modern \(\nu\) - \(l\) power spectra
from SOHO/MDI. ( l is related to k )

Global helioseismology is the study of oscillation modes which propagate through the whole Sun, which are averaged in longitude, and symmetrical in latitude. (William J. Chaplin 2006) These modes can be observed either on small patches of the solar surface as they were in the 1960’s or they can be observed as integral measurements of the whole solar disk as the Birmingham group and others did in the 1970’s. The newer field of local helioseismology studies the propagation of acoustic wave modes under small patches of the solar surface, this allows studies of wave speed under sunspots or bulk plasma flows in the solar convection region. (William J. Chaplin 2006) While the results of local helioseismology have been extensive, the kind of data required for the analysis techniques used requires high spatial resolution and therefore is not applicable to the discussion of asteroseismology.

Helioseismic and Asteroseismic Analysis

Helioseismic data is a linear combination of many frequencies on a photometric or spectroscopic signal. However the required information is the fractional power of the signal against frequency and / or wavenumber, to extract these properties from the raw data requires detailed analysis. There are many different analysis techniques available to use, the basic analysis for time series data is well known but assumes an uninterrupted data set which is evenly spaced. (C. Aerts, J. Christensen-Dalsgaard, and D.W. Kurtz 2010) The helioseismic or asteroseismic data however does not conform to these assumptions, so new analysis techniques were developed for these data. The basic analysis is to separate a frequency spectra from the recorded time series data this is generally done using a number of methods, depending upon the type and quality of the data. Data quality is dependant upon the duty cycle (length and gaps in the data) and signal to noise ratio of the observations which depends upon the source of the data. Ground based observations either of the Sun or of other stars are subject to the day night cycle of the observing location, or other periodic variations, such as, seasonal visibility of stars or non-uniformity of the Earth’s orbit around the Sun. Where as, spaced based observations, while sensitive to other sources of data interruption, generally have far superior duty cycles to their ground based counterparts. There are many detailed analysis methods used to extract frequency spectra, most of which are based on some form of the Fourier series, with consideration for both periodic and non-periodic data gaps, and other artefacts of the data. For a full discussion of the mathematics of these analyses see (C. Aerts, J. Christensen-Dalsgaard, and D.W. Kurtz 2010) or (J. Christensen-Dalsgaard 2003).

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Fig. 3 | Diagram showing mode penetration through the stellar
interior, low angular degree modes penetrate deeper.

While the oscillation spectra from the Sun contains angular degrees from l = 0 to l \(\approx\) 1500 it is not possible to successfully extract all these wave modes out of a data signal at once, as the amplitudes are too small and the frequencies too close. Therefore some form of spatial filtering must be done before Fourier analysis in any form is to be applied to the data, to extract high wavenumber modes from the data spatial resolution is needed, where as for low wavenumber modes, spatial filtering is achieved by ‘Sun as a Star’ observations where the high degree modes average out of the data due to the higher number of nodes visible on the solar surface. (J. Christensen-Dalsgaard 2003; C. Aerts, J. Christensen-Dalsgaard, and D.W. Kurtz 2010) The other limiting factor to the quality of the frequency spectra is the length of the data series and any gaps in it, this further complicates the analysis. The main issue with gaps in the data series, apart from introducing noise around the actual frequencies, is if the gaps are periodic. These periodic gaps lead to ‘alias’ frequencies, frequencies introduced into the data which represent the frequencies of the gaps, for example a sidereal day gap will introduce a frequency at \(11.5741\mu Hz\). (C. Aerts, J. Christensen-Dalsgaard, and D.W. Kurtz 2010)

After the frequency analysis stage is complete the next stage in the analysis is to identify the modes and angular degrees of those modes in the frequency spectra, this is a crucial analysis step in determining physical parameters of the star from the data. This is due to the mode numbers holding the information of the path length through the star and the frequencies which the modes take holds information about the sound speed along the path the mode takes through the star. The spherical harmonic functions that describe the modes of oscillation are given by a three indices \(l,n,m\) where l is the angular degree, which is related to the horizontal wavenumber, n is the radial order, the number of nodes in the radial direction and m is the harmonic order. For a spherically symmetric star there is a frequency degeneracy of the \(m\) order, such that for all m for a given l the frequencies are the same. (C. Aerts, J. Christensen-Dalsgaard, and D.W. Kurtz 2010) However the oscillation modes of a star are not spherically symmetric, the primary deviation from the spherical symmetry being that caused by rotation, this leads to a offset in the frequency of a \(l\) mode such that \(\nu_{l,n,.m}=\nu_{l,n}+m\Omega\) where \(\Omega\) is a factor of the rotational velocity. (J. Christensen-Dalsgaard 2002) This ‘small frequency splitting’ is used to determine properties of stellar rotation, differential rotation and other properties originating in departures from spherical symmetry.

Following the determination of modes and their respective frequencies, various other techniques have been developed to infer more information about the star, the main technique for determining information about the interior properties of the star is ‘Inversion’ which is the process by which parameters describing the interior structure of the star are inversely reconstructed from the frequencies of the oscillation modes. The frequency of a mode gives information about the local sound speed through the portion of the star which the modes propagates. As the sound speed also holds information about the density and other parameters of the star, it is possible to infer a detailed radial structure of many stellar parameters by inverting high wavenumber modes for information on the outer edge. And then subtracting this already known information from the small wavenumber modes, which penetrate deeper into the star, to give a radial profile of the parameter being measured. (See Figure 3) (J. Christensen-Dalsgaard 2002; J. Christensen-Dalsgaard 2004)

Data Types and Sources

As already mentioned there are two principal data types for asteroseismology, spectroscopy and photometry. Spectroscopy is normally used for helioseismic data as it provides better data, largely due to the fact that granulation on the surface of Sun like stars causes higher noise on photometric data. Asteroseismic data has two main sources, space based and ground based observations. The space based observations are currently photometric due to the ability of these instruments to collect data for large numbers of stars and their consequential overlap with planet finding missions. While the important ground based observations are spectroscopic data because of the higher quality instruments and lack of space based instrumentation. Ground based spectroscopy is essential to asteroseismology, it is however not the source of the main oscillation data, it is used to provide fundamental parameters of target stars as to extract detailed information out of asteroseismic data a large amount of stellar modelling is performed. This modelling is dependant upon estimates or values of key stellar parameters which are supplied by ground based spectroscopic observations. However due to the lack of a space based spectrometer or a dedicated large network of ground based telescopes around the globe, it is very hard and expensive to get high quality, long spectroscopic data series to obtain good oscillation spectra. There currently is one notable exception to this, for a 26 day period over the end of 2006 and the beginning of 2007, a coordinated multi-site campaign of spectroscopic observations of Procyon were performed to obtain an oscillation spectra. (Arentoft et al. 2008) The results from these observations are of very high quality, better than is expected from the Kepler photometric data, the frequencies for 55 modes spanning 20 radial orders were obtained, this is comparable with good ‘Sun as a Star’ observations of our Sun, the spectra is shown in Figure 4d. (Bedding et al. 2010)

Space based photometry has provided an influx of good quality photometric data in the last 4 or 5 years starting with the launch of CoRoT (COnvection ROtation and planetary Transits) in 2006, and with some data coming from the MOST (Microvaribilty and Oscillations of STars) mission before that. CoRoT was the first spaced based instrument which was partly designed for photometric asteroseismology, it has two alternating fields of view and does medium length runs of 150 days in a target field.(Boisnard and Auvergne 2006) The larger source of data is the NASA Kepler satellite which was launched in 2008 and ended its first commissioning observation run in October 2009. The Kepler satellite even more than CoRoT has a large focus on exoplanet detection however there is a dedicated asteroseismology group KASC (Kepler Asteroseismic Science Consortium)(Christensen-Dalsgaard et al. 2007). The Kepler satellite has a fixed field of view so will provide spectra up to 5 years in length, with very minimal interruption, and with an increased sensitivity and short cadence data for asteroseismology, Kepler is a fantastic data source.

Results of Helioseismology

The analysis of the global oscillation modes of the Sun has been an active field for around 50 years, helioseismology in this time has deeply enhanced our understanding of the Sun and its interior. Helioseismology has been used to refine our models of the solar interior, both its core and composition, its structure and radial rotation profile, as well as giving the most accurate determination of fundamental parameters such as radius and mass. In the 1980’s and 1990’s when helioseismology was developing as a field, the first measurements of the solar neutrino flux were made. The neutrino flux was found to be substantially smaller than the solar models of the day predicted (William J. Chaplin 2006), this triggered a 35 year trade off between the solar physicists who observed that their solar models predicted the bulk properties of the Sun well and the particle physicists who also had no reason to disbelieve their theories. In the end helioseismology gave the solar physicists the results to confirm their models which put the ‘ball in the court’ of the particle physicists, and in 2002 the solar neutrinos were observed in the correct quantities when all three neutrino flavours were measured, meaning that neutrinos had mass and ‘flavour change’ between the Sun and earth.(Dermott J. Mullan 2010)

As well as this dramatic result for global helioseismology, by using the ‘Inversion’ techniques the radial rotation profile for the Sun was determined, this confirmed the existence of the region of the Sun called the Tachocline, a region below the convection zone where there is a transition between differential rotation and solid body rotation. (Spiegel and Zahn 1992) As well as this, more fundamental parameters are able to be inferred using models with helioseismic data as input or calibration output, the SOHO satellite increased the quality of helioseismic data available and as a result increased our determination of the solar radius. (Schou et al. 1997)

More recently ‘Local Helioseismology’ has given detailed information about specific features of the Sun, by using the available spatial resolution and high degree modes it is possible to study the plasma flows under sunspots, to detect and study zonal flows in the convection zone and to detect and ‘image’ sunspots on the far side of the Sun using their effect on the wave modes that propagate round to the earth side of the Sun. (J. Christensen-Dalsgaard 2002) All these results are dependant upon high spatial resolution and therefore, as already stated, will not be considered in terms of asteroseismology.

Asteroseismology Results

Just as helioseismology has given us a detailed understanding of our star, applying the principals and methods to other stars should increase our understanding of the populations and evolution of our universe, as well as providing detailed information on specific stars. However due to the obvious differences in data sources and types there will be limitations as to what analysis can be performed, even beyond the differences between global and local helioseismology. The primary restriction upon the asteroseismic data, as mentioned in Section [sub:Helioseismic-and-Asteroseismic-Analysis], is for whole star observations, that in angular degree modes higher than \(l=2\) the oscillations partly or wholly cancel out in the signal(J. Christensen-Dalsgaard 2003). This severely restricts the ability to do inverse analysis on asteroseismology data in general(J. Christensen-Dalsgaard 2004). However concentrating on new Kepler data there is plenty of potential for exciting science from asteroseismology, for both solar type stars as well as stars in other stages of their life.

CoRoT Results

The CoRoT spacecraft does not have the data quality of the newer Kepler mission, however its data still generates good asteroseismic results. A 2008 paper, (Appourchaux et al. 2008), observes solar like oscillation on HD 49933, which is Sun-like, it has a radius of 1.34 \(R_{\bigodot}\), using a 60 day observation run they successfully processed an oscillation spectra (shown in Figure 4b) and extracted approximately 10 p modes for each angular degree \(l=0,1,2\) however the quality of the data was such that the analysis was not straight forward and mode identification was challenging. A second 2009 paper (Barban et al. 2009) detected solar like oscillations on the star HD 181420, their analysis was also successful, however they did not find clear signatures of the \(l=2\) degree due to large rotational splitting causing the \(l=2\) degree to blend into the \(l=0\) degree.

Kepler Results

The initial results from the Kepler spacecraft were very successful for the first post-commissioning 33 day run of observations, (Chaplin et al. 2010) published the first solar-like asteroseismic paper using Kepler (which was written 5 days after the publication of the data (Karoff et al. 2010)), an oscillation spectra (shown in Figure 4c) was obtained and 20 modes of oscillation identified over \(l=0,1,2\) this is very positive result for the first published results from a new data source. As well as identifying the oscillation modes the basic properties of the three stars studied in this paper were obtained, such as mass to 6% and radius to 2%.

As well as many results and analysis for solar-like stars asteroseismology can reveal information about stars in other phases of their evolutionary cycle, including what phase they are in. In (Kallinger et al. 2010) the global properties of 1000 red giant type stars are analysed through an automated pipeline, the global parameters are obtained by using the large and small frequency separations, to give fundamental parameters for the stars as well as giving automated mode identification. These data used in these red giant papers is the long cadence data type from the Kepler spacecraft, which is ~28 minutes, as opposed to the data used in (Chaplin et al. 2010) which is the short cadence data, which is ~ 58 seconds, as the detection of modes on solar-like stars demands the high cadence data. These data used in (Kallinger et al. 2010) are also some of the first analyses to be done upon the ‘q2’ Kepler data which is the first extended observing run, when this observing run is combined with the ‘q1’ first scientific run, and the ‘q0’ installation observations, there is currently \(\gtrsim\)130 days of Kepler data available.

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Fig. 4 (a) A Solar spectra from the BiSON network, from 8 years of spectroscopic observations. (Christensen-Dalsgaard, 2004)
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Fig. 4 (b) A oscillation spectra of HD 49933 from the CoRoT mission. (Appourchaux et al., 2008)

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Fig. 4 (c) A oscillation spectra from Kepler of KIC 3656476. (Chaplin et al., 2010)
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Fig. 4 (d) A ground based spectroscopic oscillation spectra of Proycon. (Bedding et al., 2010)

Future Results

Both the current CoRoT and 33 day Kepler observations hold potential for much more asteroseismology work. However Kepler’s next data observing session will be much longer and can be combined with the first 33 day observations (with a 1.8 day gap) to give an extended data series making it easier to extract from the spectra mode identification, as well as rotational splitting’s of the frequencies in both n and l angular degrees, information on stellar activity cycles by their fingerprints upon the oscillation spectra and possibly signatures of differential rotation of the star.(Karoff et al. 2010) On top of the data available from current satellites, NASA is planning the launch of more planet hunting satellites which cross over with asteroseismology, and as can be seen from (Arentoft et al. 2008) multi-site spectroscopic observations hold the promise of even better data, just as multi-site telescope networks gave the first long time series data of the Sun.

Conclusion

From 1962 when Leighton, Noyes, and Simon (1962) discovered the first oscillations on the solar surface Helioseismology has grown to be one of the corner-stones of solar physics, it supplies direct information about the internal structure of the Sun which is not obtainable in any other manner, as well as being a fantastic source of data for the properties of every aspect of the Sun. Asteroseismology promises to use the knowledge collected over the last 45 years of helioseismic analysis and apply it to any other star in the observable universe for which data can be collected. The potential for such a source of information about the properties and life cycle of the universe has the potential to change our understanding of the universe.

The current asteroseismology results demonstrate that with careful analysis the details of low degree modes can be found, along with rotational splitting of the frequencies and signatures of stellar cycles. Even the most optimistic solar physicist in 1975 when helioseismology was in its infancy would not have imagined the amount of information the Sun held about itself let alone the ability to extract similar information about other stars. Today asteroseismology stands on the cusp of a data influx from Kepler and the promise of further similar space based missions, as these data are analysed our understanding of our place in the universe will grow as we simultaneously discover more about other stars and the planets around them.

References

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