The Sun swings
For a long time there were no possibilities in solar physics research to determine directly information about the solar structure. All knowledge was based on theoretical considerations and model calculations. Not until the 1970s helioseismology developed to the branch of solar physics that allowed to revise the theoretical results on the observations of low-frequency sound waves that show up on the solar surface.
Roger K. Ulrich and – independent from him – John W. Leibacher with Robert F. Stein elaborated the necessary theoretical foundations. They calculated that the Sun is possibly subject to oscillations and swings like a musical instrument. These oscillations hum through the whole solar body. In the same way as the tones characterize a musical instrument and its design the sound waves that travel through the Sun are characteristic for the internal properties of the Sun. In turn it should be possible to determine these internal solar properties by investigating the sound waves probing the Sun. This is in deed where helioseismology is very successful.
The first evidence of solar oscillations goes back to Robert Leighton in 1962. Together with Robert W. Noyes and George W. Simon he investigated flow velocities at the solar surface by the Doppler effect and found indications of a wave motion with a period of 5 minutes. At that time these waves were considered to be an atmospheric phenomenon. The theoretical ideas, that these sound waves penetrate through the whole Sun were proofen by Franz-Ludwig Deubner in 1975.
Since then solar oscillations, and based on them the solar interior, were studied with great precision. The details that were derived about the solar interior exceed by far the usual accuracy of any other investigation of an astronomic object.
The Sun as a Resonance Cavity
Three types of oscillations are descriminated on the Sun. The main source of information are acoustic waves. The pressure gradient is their driving force. These waves are called p modes (p: pressure).
Waves where boyancy drives the oscillation are called g modes (g: gravity). Theoretically they might exist in the Sun, but until now they were not detected.
A special type of waves are surface waves. They correspond to waves, that are known from the surfaces of deep water. They are called f modes (f: fundamental).
The discrete spectrum of waves as it is observed on the Sun is only possible if the waves penetrate through large areas in the solar interior and finally interfere constructively. This comes abouth as following: sound waves are refelected inwards at the solar surface because of the rapidly decreasing gas pressure. Towards the solar center the temperature increases continously. Connected with this temperature increase the sound speed increases as well. The lower part of a wave front that is reflected at the surface with a non-vanishing inclination angle moves faster than the upper part. The wave turns away from its originally almost vertical orientation towards the center and migrates back towards the surface. Waves migrating inwards and waves migrating outwards interfer and for certain frequencies this interference leads to formation of standing waves with characteristic patterns. A single standing wave – or a oscillation mode – has a surface amplitude in the order of cm/sec. The simultaneous occurrence of approximately 10 millions of standing waves in the Sun results at the surface in the well-known phenomenon of the “Five-Minute Oscillations” with velocity amplitudes of some kilometers per second.
Following a sound wave on its way from the surface, the wave is directed first almost vertically towards the solar center. As the sound speed increases the wave path is more and more bended and the waves misses the center of the Sun. Therefore, the path of a wave through the solar interior depends particularly on the course of the sound speed in the solar interior. The point of the closes encounter with the solar center is called turning point of the modus. Having passed the turning point the wave migrates back outwards until it reaches the surface again. There the wave is reflected as by a mirror turning is orientation back towards the center.
These kind of sound waves are the origin for standing waves that are generated if inwards and outwards migrating waves are superimposed. In the Sun we have to speak of standing waves in a three-dimensional sense.
The oscillation pattern at the surface exhibits node lines, where the motion is at rest. The total number of node lines at the surface is called harmonic degree l. Some modes show node lines that connect the poles. These number m of node lines is alway lower than or equal to the harmonic degree l. In the solar interior the oscillation modes do also exhibit the nature of standing waves by having a certain number n of nodes along the radius. This number of radial nodes is called radial order. Any mode can be definetively identified by the three numbers n, l, and m.
The seismic exploration of the Sun
There are two ways to measure solar oscillations. The first possibility is to capture a dopplergram. Spectral lines from lifting regions at the solar surface are blueshifted due to the Doppler effect. Light from lowering regions is redshifted. The degree of a spectral line shift is directly connected with the velocity component in the direction of the observer. A dopplergramm, i.e. the determination of the line shifts at each point on the solar surface, displays these velocity components.
A second possibility to observe oscillations are intensity variations of a spectral line or the total intensity of the Sun. This is caused by pressure variations connected with the sound waves and those lead to variations in temperature. The atomic processes that contribute to the generation of spectral lines are perturbed by these temperature variations, hence the intensity of a spectral lines varies in common mode with the oscillation. This phenomenon can also be observed in the total solar intensity, i.e. in integrated white light recordings of the Sun. In that sense, the Sun is a pulsating or a variable star. But the intensity variations of 10^-5 magnitudes are much smaller then in regular variable stars, where they can be as high as 8 magnitudes in Mira variable stars. Independently from the observation technique, the signal contains simultaneously contributions from a large number of oscillations which are excited on the Sun. Helioseismology has the goal to derive information about the solar interior from such a signal that depends on location and time on the solar disc. The desired information are sound speed, course of density and temperature, internal dynamics, or the excitation of the oscillations. In principle, this is similar to finding a general model for the Sun that reflects all the observed properties of the oscillations. In practice, the analysis consists of several steps.
In a first step the solar rotation and large-scale flow components of supergranulation are removed from the single exposures. The rotation and supergranulation signal are obtained by averaging many exposures. Due to the 5-minute period the oscillations are extincted as long as enough exposures are averaged. Correcting for differential rotation and supergranulation the solar oscillations come forward clearly.
Second, spatial filters are used to split the single series of exposures of the oscillation velocity signal into a multitude series that are ordered according to the harmonic degree and the azimuthal order of the oscillations.
The third step consists of a Fourier analysis, in order to seperate the single modes, that vary in the harmonic degree. The result is either displayed in a diagnostic diagram or in a power spectrum for a fixed pair of l and m values. The spectral power density, i.e. a measure for the quota of a single frequency to the total signal, is displayed in both cases. The diagnostic diagram displays the spectral power density as function of frequency and harmonic degree l, whereas a spectrum depends only on frequency. A discrete spectrum of oscillations can be seen in either cases. The narrowband ridges in the diagnostic diagram connect oscillations with identical radial order. The f-modes are on the lowest ridge. It becomes clear, that the spectral power density is not uniformly distributed, but it is concentrated on certain ridges.
The frequency resolution is given by the length of the recorded time series. In order to be able to resolve single oscillations a recording duration of at least 1 day is required. The upper diagnostic diagram and spectrum were created from 36 days time series obtained with the GONG network. Uninterrupted long time series are challanging because of the natural dayly interruption of earth-bound observations at night. Recordings in the australian summer from sites close to the south-pole are one possible way out. There, the recordings of the Sun are uninterrupted for several months.
Even longer uninterrupted data sets are obtained by the aid of a network of solar observatories that are distrubuted around the globe. The “Global Oscillation Network Group” (GONG) maintains such a network since 1995. The observatories are located on various geographic lenghts on the earth.
Therby, caused only by the weather, the Sun can be observed 87% of a day on average. The fundig of this network is provided by the U.S. National Science Foundation, but many international collaborations are involved. Currently, there are worldwide 130 GONG members from 67 different institutions and 20. They are all organized in several teams to investigate specific scientific questions. Each of the six telescopes consists of two mirrors, that follow the Sun on their daily path on the Sky. The mirrors direct the light horizontically into a cargo container, that serves as observatory building. The whole instrumental equipment is stored in this container. The optical system is sealed by a filter window and has an effective aperture of 2.8 cm.
Several optical instruments are contained in a box close to the focus of the objective that has a focal length of 1 m. These instruments can be moved into the ray path and allow a calibration of the telescope. A variable polarization retarder can also be moved into the ray path, in order to map the line-of-sight component of the solar magnetic field. All these mechanisms are computer controlled and work in general automatically. Actually, the capture of a surface dopplergram is made possible by a hybrid filter with a bandpass of 1 Å. This bandpass isolates the Ni I line at 6768 Å. The filter consists of a 5 Å interference filter and three following bifringent elements. Two of these bifringent elements are made of combination of calcit and ammonium-dihydrogen-phosphate. The size is selected in a way to enable a thermal compensated bandpass. All elements are kept in oven at constant temperature that is stabilized to 0.000001 K.
The core of the instrument is a polarizing michelson inferometer with an optical retardation of 30000 wavelengths. The interferometer was designed to have a wide field of view and to be thermally stable. The interference pattern of the interferometer takes the form a squared cosine and is sampled with a rotating disc along the filtered solar spectrum. Due to the presence of the Frauenhofer line a modulation is generated. The phase of the modulation is a good measure for the doppler shift.
In the first years of operation, a modified conventional CCD camera was used for the image capturing. The resolution of this camera was 8 arcseconds per pixel. In the meantime these cameras were replaced by new ones with an resolution of 2.5 arcseconds for each of the 1024×1024 pixels. Three images are taken in every modulation cycle in order to determine the phase, the amplitude and the brightness at each pixel. The captures are intergrated over 60 seconds and the result is digitally stored on tape.
The container harbors the complete electronic assembly, a computer for data recording and all the instruments. Hence the observatory can be operated autonomously at remote locations by two computers and a accurate clock. Usually, there is a snapshot of doppler velocity, line strength, continuum intensity and Zeeman splitting in every minute – day and night. In the nightime the instrumental parameters and the weather conditions are recorded only. As soon as the program detects that the Sun has risen, the instrument coverage is removed automatically. Then the instrument is pointed close to the Sun and a guiding sensor gives precise positioning orders. In the case of a cloudy sky the computer estimates the Sun’s position and adjusts the positioning of the instrument. As soon as there is in daytime enough sunlight, a calibration sequence is called. The observations are carried out until sunset. Afterwards the instrument stows itself again autonomously. The instrument can be run for one week without human intervention. Then, the only need is to change the tapes or to do maintenance if necessary.
Since 1996 there is the other opportunity to use the Michelson-Doppler-Interferometer (MDI) aboard the space observatory SOHO to get uninterrupted time series from the Sun. Observations with an observatory in space from a suitable orbit avoid interruptions by the day and night cycle. The SOHO spacecraft which is a joint project of ESA and NASA delivers data of highest quality. SOHO was launchend in 1995. The scientific operation was started in 1996 from an orbit around the first Lagrange point between Earth and Sun. Because of a technical defect in 1998 it seemed that SOHO was lost. But skillful actions brought SOHO back. Time series from SOHO cover on average 96% of a day. SOHO harbors three instruments for helioseismic data recording. The instrument GOLF (Global Oscillations at Low Frequency) and VIRGO (Variability of solar Irradiance and Gravity Oscillations) are aimed especially to detect g modes. But up to now g modes could not be found doubtlessly. The instrument SOI/MDI (Solar Oscillations Investigation / Michelson Doppler Imager) uses a Fourier tachometer based on a design where two spectral bands can be used with two tuneable Michelson interferometers. These are used to record in every minute the observations of the Doppler velocity over the whole solar surface with a spatial resolution of 2 arcseconds. This corresponds to 800000 simultaneous and independent velocity measurements at different places. Especially, studies of solar oscillations and important details about the solar interior became possible with these two instruments MDI and GONG. Valuable data come from the LOWL instrument of the High Altitude Observatory on Mauna Loa, Hawaii, which utilizes a magnetooptical filter. Recently a further instrument was added on Tenerife to form a network of two sites. One of the first networks that consists meanwhile out of six stations is the Birmingham Solar Oscillation Network (BiSON). This network does not record resolved images of the solar surface, but averages the solar oscillations over the whole solar disc. Other ground-based networks are IRIS “Installation d’un Reseau International de Sismologie solaire” and TON ” Taiwan Oscillation Network”.
Conclusions about the structuring of the solar interior are drawn by means of inversions. The principle of inversion methods is simple. The paths of every two scillation modes with different radial degree n or harmonic degree l differ from each other. As these modes have different frequencies, the integrated sound speed on their paths must be different, too. This can knowledge is used in turn for the seismic investigation of the Sun. The relation between frequency and sound speed is know, but the sound speed can not be measured directly, but only the frequencies. In order to determine the sound speed, this integral relation is inverted. This is a well-know mathematical and often difficult problem. Therefore, helioseismologists spend a lot of time on the development of effective mathematical techniques and computer software for the inversion of the oscillation frequencies. Based on the frequency difference between two modes, the sound speed can be determined in that region in the Sun that was passed by only one of the modes.
For a detailed map of the physical properties of the solar interior, many mode frequencies must be measured. Afterwards, a careful analysis of the different wavepaths throught the solar interior must be performed. Aa a result, the sound speed in every region of the Sun is determined.
Modes with m=l are more concentrated to the solar equator, while modes with lower azimuthal order m reach higher latitudes. The investigation of modes with different order m allows studies of properties of the Sun that depend on latitude. Especially, this is important for the determination of the rotation rate in the solar interior.
Global Properties of the Sun
The seismic determination of the internal structure of the Sun shows that the theoretical calculations and simulations are in well agreement with the real Sun. There are only small differences in the sound speed profile of less than 0.5%. This is a remarkable result, as it shows the great possiblities of modern physics that lead to the present substantiated state of knowledge about such a complex object as the Sun.
Starting from the inverted sound speed, the central temperature of the Sun can be determined to be 15.7 ± 0.3 x 10^6 K. This result was available directly after the first observation campaign with SOHO in 1997. Newer experiments at the Sudbury Neutrino Observatory lead to overall approvement of this result in 2002. Up to that date, there was still speculation, that the reason for the too low rate of measured neutrinos, which are set free during under high tempertature during the fusion processes in the solar core, might be a lower central temperature of the Sun. Now, it became clear, that neutrinos are able to change their characteristics which makes them undetectable for the earlier neutrino detectors.
Also starting from the inverted sound speed, the depth of the convection zone can be determined with great precision. Sarbani Basu and H. M. Antia found that the ratio of depth of the convection zone and solar radius is given by 0.287 ± 0.001, i.e. in the outer 28.7 % of the Sun the material is always in motion transporting energy from the solar interior to the surface.
The solar radius is determined from the frequencies of the f-modes. They depend as 1/R^3/2 on the solar radius. Being acoustic waves, the p modes can not be used for the determination of the solar radius as their frequency depend on the sound speed additionally. The analysis results in a value for the solar radius of 695700 ± 100 km.
Observations of the solar surface reveal a non-uniform rotation. The equator needs 25 days whereas regions at the poles need approximately 30 days for one full revolution. This result is in principle not very amazing, because in contrast to the Earth the Sun is a gasball. Hence, there is no reason for assuming that the Sun would rotate as a solid body. However, the origin of the observed solar differential rotation is not completely understood. The question, how the differential rotation profil at the surface continues in the solar body was answered by inverting the p-mode frequencies.
The upper figure shows the solar rotation in a cross section view, determined on the basis of GONG data. The colors represent the rotation period (blue slow, red fast). A welcome sign of this inversion result is the agreement of the rotation rate near the surface with earlier observations of, e.g., sunspots. The rotation rate is almost constant on all latitudes from the surface down to a depth marked by the dotted circle. In this region there are only minor variations in the differential rotation. The dotted circle marks the lower boundary of the convection zone. There, energy is transported by gas motions. The whole convection zone therefore rotates very similar as the surface. The transition to a constant rotation rate at the bottom of the convection zone seems to be very sharp. The details and reason for this transition are still topics of current research. Further observations from GONG and SOHO will be very important for this research topic. Moreover as can be seen from this figure, only less is known about the rotation in the central regions of the Sun. The next years will reveal more details about the central rotation on the basis of new observations and improved data analysis methods.
Consulting data from various years and comparing those, amazing results are detected. The solar differential rotation changes with time.
(Courtesy R. Howe, GONG, Tucson)
From time to time, the rotation is slightly accelareted and decelerated again along a degree of latitude The zones of accelaration and decelaration migrate with the 11-years solar cycle from higher latitudes towards the equator. This results in the impression of a zonal flow directed to the equator. These bands of slightly faster and slower rotation seem to be in close connection with the locations of sunspot appearance that also show a tendency towards the equator during the solar cycle.
Even more amazing is the fact, that the rotation rate varies also with time at the bottom of the convection zone. There the variation occurs with a period of 1.3 years.
(Courtesy R. Howe, GONG, Tucson)
These results give hope, that helioseismology will reveal further details from the bottom of the convection zone in the near future. In those regions a differentially rotating sphere rubs against the solar core, which might be the origin for the solar magnetism. Because, electromagnetic induction is there most effective.
Therefore, helioseismology is now also focusing on investigating the solar dynamo. This includes the probing of the whole convection zone, because the magnetic field is generated deep in the Sun and rises somehow towards the surface, where it can results in visuable effects as, e.g., in sunspots. Moreover, the effect of the internal matter flow, through wich the acoustic waves penetrate, needs to be understood, before information about the magnetic field in the deep solar interior can be derived. In turn, this allows an investigation of the flows in the solar interior.
Flows and Streams in the Solar Interior
The solar surface reveals a flow pattern in form of the granulation. This granulation consists of many single convection cells, which have an average size of 1500 km and a mean life time of 5 min. Dopplergrams reveal an even bigger convective cell structure, the supergranulation. Those have an extension of up to 30000km and outlast several days. For a long time, there is a debate, whether even more bigger convection cells are possible. These “Giant Cells” have not been detected yet, but they should reach through the whole convection zone. Based on the theoretical considerations from Markus Roth and Michael Stix, it was possible for Markus Roth, Rachel Howe and Rudi Komm to derive a detection limit for the state-of-the-art data analysis techniques. They found, that any large-scale flow with a long life time and a velocity down to 10 m/s could be detected within the convection zone. Moreover they determined the signal, that such flows would imprint in the helioseismology data.
As such signals are not observed, the matter in giants cells must stream very slowly, if these cells exist at all. Flow components in the solar interior are variable, i.e., they change with time. Therefore, the amplitude and the course of a flow component vary. The signal caused by such varying flow components is variable itself. Markus Roth and Michael Stix showed, that a changing large-scale flow component in the Sun leads to a restructuring of the sharp peaks in the power spectrum. Side lobes must emerge. This structur can be seen as fingerprint of the flow components in real solar data.
It is now a challange, to develop new helioseismic inversion methods for surveying the convection zone.