Introduction to helioseismology and asteroseismology • April 2016 • Term Paper EPSC 320
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INTRODUCTION TO HELIOSEISMOLOGY AND ASTEROSEISMOLOGY
Meryem Berrada
McGill University
The field of Helioseismology aims to understand the inner structure and the oscillations of the sun.
Similarly, Asteroseismology is the study of other stellar bodies. These fields aim to create models
of the stars’ oscillations. A star’s internal structure cannot be observed directly, as a result, these
fields use seismology to establish a density profile. Indeed, there are enough stars in the universe
to have a variety of data, but the further a star is from earth, the greater the background noise is.
Thus, it is easier to study the sun first and then use it as a guide to study other stellar bodies. In
addition, the study of the sun’s internal structure aims to improve the knowledge of nuclear energy
generation, energy flows, interaction of magnetic fields with matter, and particle acceleration to
high energies. (Harvey) This paper introduces the physics occurring in the stellar interior, the
background information necessary to study stars, the process of data filtering, and the theory
behind modelling.
INTRODUCTION
Stellar objects have variations of brightness, and these variations are interpreted as vibrations or
oscillations within the structure. (Kepler) This source of agitation is caused by convection
occurring in the deep interior of a star. (Harvey) In fact, there is a pattern of outward and inward
oscillations of the gases that is observed on the surface of a star. Knowing that our sun is mainly
composed of helium, these oscillations lead to the analysis of the Doppler shifts in spectrum lines,
in the idea to evaluate the interior structure of the sun (Nave, Composition) Helioseismology also
uses the physics of wave propagation, more precisely standing waves, to understand the variations
of density and velocity in the interior structure. In fact, observations led to the understanding that
the boundary near the surface of the sun has a large density drop, while the lower boundary of the
convection zone demonstrates an increase in speed. (Harvey) These observations infer on the
frequencies of oscillation, the sun’s internal structure, and the state of evolution. In the same idea,
the internal structure of a star infers on the characteristics of the core, the convection zone and the
radiative zone.
BACKGROUND
The standing waves that are received from the sun belong to three different types. First of all, the
primarily observed waves are acoustic waves, which generate p-modes. (Harvey) The effects of
gravity on the wave propagation and the changes in gravitational potential are negligible. Also,
this equilibrium situation assumes an adiabatic process in the convection region. It is considered
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to be a reasonable approximation as long as the equilibrium structure, in this case the stellar body,
varies poorly compared with the oscillations. (Jørgen) This oscillation mode is the one that is
mainly considered due to its simplicity. Secondly, another type of wave that standing waves belong
to is internal gravity waves, which generate g-modes. (Harvey) This situation is more complex.
The g-modes consider a layer of gas stratified under gravity, implying a pressure gradient. It is
also assumed that the internal gravity waves occur in an equilibrium situation. This assumption
simplifies the calculations; a small variation in equilibrium quantities means that the gradient of
those quantities can be neglected, compared with the gradient of perturbations of gravitational
potential. (Jørgen) Lastly, there is another type of waves which is surface gravity waves, which
generate f-modes. (Harvey) This mode of oscillation considers an incompressible liquid at constant
density on a free surface. The free surface assumption implies that the surface boundary has a
constant pressure. In this assumption, a constant density implies that there are no perturbations in
the gravitational potential. (Jørgen) These descriptions emphasize the possible variations in
equilibrium and in gravitational potential as there are primordial to the modelling.
In seismology, the location of an earthquake can be estimated from the travel time rays between
the epicenter and a receiver. This method is efficient when the earthquake is recorded by different
stations all around the earth’s surface. The technique is similar in helioseismology. Although there
are no geophones located on the sun’s surface, the source of the oscillations can be first estimated
to any point of the surface. The technique requires the assumption that this picked point for a
source aligns on some great circle, by which a wave may have travelled. The frequencies observed
on that path are then correlated in order to get the displacement function of the oscillations on that
particular great circle. (Harvey) This time-distance method requires the analysis of the data on
various directions.
The pattern of oscillations of another stellar object can only be approximated after the estimations
of the age of the star. This estimation is necessary as it infers on the amount of gas and the general
composition of a star. (Guenther) Numerical models of stellar evolution use the object’s luminosity
and estimated surface temperature in order to approximate its helium composition, as shown in
figure (a).
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Figure (a): Hertzsprung-Russell diagram.
Describing the evolution of stellar bodies relative to
their surface temperature (Kelvin) and luminosity
(solar units). (Kepler)
The numerical models use a basic relationship between luminosity and temperature for the stars
in the main sequence. (Stars) This category of stars is illustrated in figure (a).
𝐿 = 𝑅2𝑇4 Equation (1)
The observations lead to estimations on the luminosity of the observed stellar object. The
luminosity is then compared with that of other known stellar objects, along with the Hertzsprung-
Russell diagram to be inferred on the surface temperature. The luminosity and temperature are
then used in the computation of the radius of that particular stellar body, using equation (1).
Considering that equation (1) is the primary guide to modelling, Helioseismology and
Asteroseismology are actually limited to the modelling of main sequence stars.
DATA PROCESSING
Spherical harmonics must be considered in order to evaluate the wave propagation pattern for a
rotating sphere. This method can determine the number of nodes on a surface, which will lead to
modeling the oscillations of a stellar object. Two types of modes can be analyzed: radial and non-
radial. Most solar modes are non-radial, meaning that the shape of the star is not preserved during
oscillation. The non-radial mode is defined by three wavenumbers. First, the radial order n
corresponds to the number of nodes in the radial direction. Second, the angular degree l,
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corresponds to the number of nodal lines. Finally, the angular order m indicates the number of
nodal lines that cross the equator. (Jørgen)
Figure (b): Models of spherical harmonics according to
different combinations of m, n variable values. Different
combinations describe oscillations of different directions and
amplitudes. (Spherical, Ambisonics)
As illustrated in figure (b), there exist a large number of possible combinations of wavenumbers
that can lead to a similar oscillation. In order to filter the possible combinations, the Doppler shift
analysis needs to be included. The theory states that as a source is approaching a receiver, the wave
frequencies increase, and are then blue shifted. Inversely, as a source is receding, the wave
frequencies decrease, and are then red shifted. These red and blue shifts are analogous to what is
illustrated in figure (b). (Nave, Red Shift)
Considering that the source of oscillation is along a great circle, the oscillations can be detected as
functions of position on that particular solar disk. (Jørgen) On the same hand, the angular diameter
of a star is hard to obverse, and is much easier to calculate based on the stellar radial velocity. The
stellar radial velocity can be computed by integrating over the solar disk by the means of Fourier
Transform in position. This technique has proven to filter all modes above l=4. (Tong) Then, a
Fourier Transform in time will filter the corresponding frequencies. (Jørgen) The Fourier transform
of the time series 𝑦𝑛(𝑡𝑛) is displayed in equation (2). (Tong)
𝐹(𝑣) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ∗ ∑ 𝑦𝑛(𝑡𝑛)𝑒−𝑖𝑡𝑛(2𝜋𝑣)𝑁𝑛=1 Equation (2)
The Fourier time transform depends on the target frequency v, and the sum of the time
series 𝑦𝑛(𝑡𝑛), where 𝑡𝑛 is the time at which the frequency is observed. From a Fourier Transform
plot, the real amplitude of the plot represents the amplitude of the oscillation about the mean value.
Plus, the phase of the target frequency is equal to the ratio of the real and imaginary parts of
equation (2). Equation (2) is evaluated at frequencies νk = k/T, where t is the time corresponding
to the observation of that particular frequency and k varies from 1 to half the number of data
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samples. The range of frequencies varies from the time T to 1/T. However, if the signal-to-noise
ratio is high, longer periods might be necessary to analyze the target frequency. (Tong) The
frequencies that are evaluated are not completely random; there exists a recurrent pattern on stars.
So far, the observations of pulsating stars and other main sequence stars lead to a total of known
frequencies for as many as 106 modes. In fact, more than half this number has been observed on
the sun, leading to a great understanding of the sun’s oscillation modes.
Further on, the sun’s fundamental frequency is used to filter the background noise. The
fundamental period of oscillation is about an hour, while the other oscillations have a period of
about five minutes. Comparing both categories of frequencies allow for a better filtering of the
frequencies belonging to the acoustic waves that are occurring in the convection zone. In fact,
while the fundamental frequency has no nodes, the other oscillation frequencies have from 20 to
30 nodes, which need to be considered during the spherical harmonic approximations. (Jørgen) In
other words, from the fundamental frequency it is possible to model the main oscillation mode of
the sun, and then from the other frequencies it is possible to create a density profile of the
convection zone.
From a power spectrum, the oscillation modes corresponding to each observed frequency can be
identified. A power spectrum is a plot of power relative to frequency. For a given signal, the power
is the energy per unit time that is recorded by the observer. (Power Spectrum, Wolfram) As
mentioned earlier, the background noise might come from the instruments or from the other stellar
objects. The noise from the other stellar objects cannot be controlled, but a satisfying way to filter
it is by setting limits of energy per unit time that can be received from the observed object. In fact,
limits to the background noise are set such that they are contained between three and ten times
lower that the target mode height on the power spectrum plot. After filtering the data, researchers
analyze the signal to noise ratio. It is calculated that, for an observation time longer than two
months, a ratio greater than 10 leads to a precision on the target frequency of less than 0.2μHz.
Similarly, for an observation period longer than four months, a ratio greater than 3 leads to a
precision on the target frequency of less than 0.2μHz. These values of the signal-to-noise ratio can
be used as boundaries for a second data filtering. The maximum energy per unit time that can be
received from the observed stellar object can be estimated using equation (3). (Tong)
𝐴
𝐴0=
𝐿
𝑀(
𝑇
𝑇𝑒𝑓𝑓)𝑠 Equation (3)
Here, the maximum energy per unit time for a frequency νmax, as observed on the power spectrum,
is the mode amplitude A [cms−1 or ppm], the maximum solar mode amplitude is A0, the stellar
luminosity L, the stellar temperature T, the effective temperature of the sun Teff, and the stellar
mass M are in terms of the solar units. The exponent s depends on the signal received. It is set to
0 for a radial velocity assumption and to 2 for the intensity of fluctuations in the power spectrum.
(Tong) The minimum energy per unit time that can be received from the observed stellar object is
estimated from stars less massive than the sun; as this mass-luminosity relationship will provide
the lowest ratio. (Main)
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𝐿
𝐿⨀= (
𝑀
𝑀⨀ )
4
Equation (4)
After identifying the oscillation frequencies and modes coming from a given stellar object, the
physics of spherical harmonics are used in order to model the position of the oscillation.
THEORY
The modeling of a stellar object’s interior is based on various assumptions. First, the
parameterization of the convection process is set on the surface layers of the convection zone (the
outer layers of the sun’s structure). As mentioned earlier, this leads to the assumption that an
adiabatic process is occurring. One of the assumptions made during stellar modelling is that the
turbulent pressure caused by the dynamics of convection is negligible. It is also assumed that there
is no transition zone between the convection zone and the interior. This implies that the acoustic
waves have specific boundary conditions, leading to the analysis of standing waves. Also, in order
to ease the models, the effects of magnetic fields are also neglected. The microphysics and the
previous assumptions are tested while observing frequencies from a specific stellar object. If the
data is not compromised by any assumptions, then the theory holds and the modeling can begin.
(Jørgen)
The models need to agree with the observed sets of frequencies, amplitudes and phases computed
from the Fourier Transforms. As illustrated in figure (c), the relative intensity of the oscillations is
compared to stellar objects of other size in order to make a correlation between the size of a star
and the intensity of the waves observed.
Figure (c): A Kepler “concert” of Red Giant Stars.
Data collected from the Kepler Mission from
NASA. Describing the relative intensity
(amplitudes) with respect to time, relative to the
size of the stellar bodies. (Kepler)
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The correlation in figure (c) implies that smaller stars will induce standing sound waves of lower
intensities and smaller pulse width. (Kepler) However, there is no numerical relation for this
correlation that can be used in stellar models. This figure is only a guide to understand the data.
This figure also shows that a correlation between the pulse width and the size can be made.
Similarly, figure (d) illustrates the correlation between the pulses of stars and their category in the
Hertzsprung-Russel diagram.
Figure (d): Taking the pulse of stars. Describing the
variation in amplitude and pulse width relative to the
size of stellar objects. (Kepler)
The illustration (c) also indicates that there is a limit in the observable intensity amplitudes of
pulses. On the other hand, both figures emphasize on a correlation between the body size and pulse
width. Thus, when it is necessary to differentiate between a binary composition and the general
secular stability of a star, models use the length of phases as a guide, rather than the relative
intensity. (Tong)
After considering the physics of the stellar interior and the correlations between the observed data,
spherical harmonics provide with a 3D model of the oscillations. The spherical harmonics
𝑌𝑙𝑚(𝜃, 𝜙) describe the angular position of the solution to Laplace’s equation in spherical
coordinates. The solution varies with 𝜃, the polar coordinate with 𝜃 ∈ [0 , 𝜋], and with 𝜙, the
azimuthal coordinate with 𝜙 [0,2𝜋). Also, the angular order m varies with respect to the angular
degree l, such that m = -l, - (l-1),…, 0,… (l-1), l. (Spherical, Wolfram)
𝑌𝑙𝑚(𝜃, 𝜙) = √
(2𝑙+1)(𝑙−𝑚)!
(4𝜋)(𝑙+𝑚)!𝑃𝑙
𝑚(𝑐𝑜𝑠𝜃)𝑒𝑖𝑚𝜙 Equation (5)
In equation (5), 𝑃𝑙𝑚(𝑐𝑜𝑠𝜃) is Legendre polynomial as a function of m and l. (Spherical, Wolfram)
This is set such that the integral of | 𝑌𝑙𝑚 |2 over the unit sphere is 1, constraining the stellar models
to a unit volume. Basically, varying the combinations of wavenumbers m and l is the only way to
obtain different direction, amplitude and position of oscillation. As illustrated in figure (e), higher
orders of wavenumbers lead to more complex oscillation models. In fact, the solution 𝑌00(𝜃, 𝜙)
describes a simple sphere, while the solution 𝑌30(𝜃, 𝜙) describes a sum of three vertically
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propagating waves. However, comparing with the solution 𝑌33(𝜃, 𝜙), which is similar to the
solution of lower order 𝑌00(𝜃, 𝜙), it is clear that a combination of wavenumbers of the same value
doesn’t result in a complex oscillation solution. This is an indicator that the wavenumbers m and
l must not be equal to each other.
Figure (e): Spherical harmonic combinations.
Describing the various possible oscillation
modes that can be obtained with these particular
combinations of m, l variables values. (Spherical,
Wolfram)
The observations on the sun show that the oscillations are complex and are varying from degrees
0 to 1500, in which the range is too broad to create a stellar model. One solution is to diminish the
telescopes’ sensitivity, so that they can only detect a few degrees. Another solution is to limit the
observations to a single solar disk. This will isolate the modes that belong to that particular disk.
In which case, the fluctuations in intensity in the solar disk can be approximated by the equation
(6). (Jørgen)
𝐼(𝜃, 𝜑, 𝑡) = √4𝜋 ∗ 𝑅𝐸{𝐼0𝑌𝑚𝑙 (𝜃, 𝜑)𝑒−𝑖𝜔0𝑡} Equation (6)
Where the function 𝐼(𝜃, 𝜑 , 𝑡) denotes the intensity with respect to the polar coordinate 𝜃, the
azimuthal coordinate 𝜑 and time t, 𝜔0 is the angular frequency, and 𝐼0 is the initial intensity. This
is derived by taking the real part of the spherical harmonic solution. As it is difficult to determine
the exact position of a fluctuation is space-time, equation (7) denotes how the fluctuation can be
located with respect to the whole-disk observation. (Jørgen)
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𝐼(𝑡) = 1
𝐴∬ 𝐼(𝜃, 𝜑, 𝑡)𝑑𝐴 Equation (7)
Here, the intensity is evaluated on A, the area of the disk. In order to generate models for other
stellar objects, the system of coordinates that is used in the integration must be convenient. In fact,
it appears that setting the polar axis towards the observer ease the calculations. The final expression
that is used to describe the amplitudes of oscillation is:
𝐼(𝑡) = 𝑆𝑙(𝐼)
𝐼0cos (𝜔0𝑡) Equation (8)
Where, 𝑆𝑙(𝐼)
can be written as;
𝑆𝑙(𝐼)
= 2√(2𝑙 + 1) ∫ 𝑃𝑙(𝑐𝑜𝑠𝜃)𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜃𝑑𝜃𝜋/2
0 Equation (9)
(Jørgen) Similarly, the velocity of the oscillations is modeled from the same method. In this case,
since the velocity of oscillation is derived from the Doppler shift of spectral lines, the values
obtained correspond to only the line-of-sight component of velocity. This component may be
written as:
𝑉(𝜃, 𝜑, 𝑡) = √4𝜋 ∗ 𝑅𝐸{𝑎𝑟𝑉0𝑌𝑚𝑙 (𝜃, 𝜑)𝑒−𝑖𝜔0𝑡} Equation (10)
(Jørgen) Where the function 𝑉(𝜃, 𝜑, 𝑡) denotes the velocity with respect to the polar coordinate,
the azimuthal coordinate and time. The variable 𝑎𝑟 is the unit vector in the radial direction, and 𝑉0
is the initial velocity. As for the intensity, the velocity is derived from the real component of the
spherical harmonic solution. As observed from the whole-disk, the velocity of propagation of the
standing sound waves can be located using equation (11). (Jørgen)
𝑉(𝑡) = 𝑆𝑙(𝑉)
𝑉0cos (𝜔0𝑡) Equation (11)
Where, 𝑆𝑙(𝑉)
can be written as;
𝑆𝑙(𝑉)
= 2√(2𝑙 + 1) ∫ 𝑃𝑙(𝑐𝑜𝑠𝜃) cos2 𝜃 𝑠𝑖𝑛𝜃𝑑𝜃𝜋/2
0 Equation (12)
The intensity of fluctuations and the velocity of propagation infer on the surface oscillations.
Now, the properties of the stellar interior, such as the density profile, can be approximated using
the quality factor Q. (Jørgen)
𝑄 = Π (𝑀
𝑀⊙)
1
2(
𝑅
𝑅⊙)
− 3
2 Equation (13)
In this case, the period of oscillation is defined by Π = 2𝜋
𝜔0 with units of (MHz)-1, the stellar mass
M in solar units 𝑀⊙, and the approximated radius R in solar units 𝑅⊙. As mentioned in the Data
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Processing section, it is observed that the period of oscillation varies around five minutes, which
limits the expected values of the quality factor. (Jørgen) From the relationship between mass and
density, it is possible to make estimations on the density of the outermost layers of the convection
zone. The mean density can be expressed as:
𝜌 ̅~𝑀
𝑅3 Equation (14)
(Breger) There also exists a relationship between mass and gravity:
𝑔 ~𝑀
𝑅2 Equation (15)
(Breger) Bringing equations (14), (15) equation (13) can be rewritten as;
�̅�
�̅�⨀=
𝑔
𝑔⨀
𝑅
𝑅⊙ Equation (16)
Here, �̅�⨀ and 𝑔⨀ are the solar units of density and gravity respectively. (Breger) Now, using
equation (1), equation (16) can be rewritten as to have a direct relationship between the observed
variables and the density of the layers. (Jørgen)
�̅�
�̅�⨀=
𝑔
𝑔⨀
𝐿⨀1/2
𝑇2
𝐿1/2𝑇⨀2 Equation (17)
All variables are in terms of solar units, which leads to great simplifications. However, these
equations must be solved numerically since the relative luminosity and surface temperature are
only approximations relative to a specific set of values. From helioseismology, the principal
components of the sun’s inner structure are uncovered. It is found that about half of the mass and
98% of the energy generation is focused in a core of radius a quarter its total radius 𝑅⊙. Plus, the
core is surrounded by a radiative zone, where energy is transported by radiation up to 0.713𝑅⊙.
Then follows the convection zone, where energy is mainly transported by convection up to the
surface. (Harvey) In addition to evaluating the models’ accuracy, the quality factor is still
considered, in Asteroseismology, in order to create a density profile of the stellar bodies.
Unfortunately, models cannot consider the location of sunspots, gaps, or other uncommon solar
activity. For example, researchers have found that missing sunspots could be due to stream jets. In
fact, these are not well understood and need to be studied in more depth in order to be properly
included in the solar models. (Minard) In addition, stars are evolving and by doing so, their radius,
composition and brightness changes. The models that are studied today aim to integrate the
changes caused by the evolving solar object and the changes emerging from incorrect assumptions.
One of the tools used to guide models for variable stars is the O-C diagram. These include binary
companions, pulsars, eclipsing stars and much more. The O-C diagram uses the set of predictions
obtained from forward and inverse modeling (calculated parameters, C) to compare with the
observations (observed parameters, O). The difference between the observed and the calculated
values is plotted on the vertical axis with respect to time on the horizontal axis. (Brown) During
stellar evolution, secular changes can be observed in the time scale. A curvature in the O-C diagram
implies that the period is changing with time. For example, a constant rate of change in the period
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produces a quadratic O-C curve, and a steadily increasing period produces an upward parabolic
curve. From these evaluations, forward and inverse modelling are once again used in combination
to ameliorate the accuracy of the parameters that constitute the models. (Harvey)
CONCLUSION
In summary, Helioseismology and Asteroseismology intend to model the interior structure of
stellar bodies by constructing a density profile. This is mainly done by using the physics of
standing acoustic waves and the adiabatic process. Considering that the wave pattern depends on
the medium it is travelling through, the stellar evolution diagram, along with the stellar radius and
mass, can be used to estimate the composition of the stellar body. This infers on the density of the
convection zone. Then, this parameter is used along with spherical harmonics to model the
oscillations of the stellar body. From a Fourier transform in time, it is possible to find the
wavenumbers that correspond to the observed frequencies. Given an expected range of frequencies
and amplitudes, a power spectrum plot will filter the unwanted frequencies. Finally, from a
combination of forward and inverse modelling, more accurate values of the parameters that
constitute the models can be found. In helioseismology, the modelling process resumes to the
approximate width of the core (center to 0.25𝑅⊙), the radiation zone (0.25𝑅⊙ ≤ 𝑅 ≤ 0.713𝑅⊙)
and the convection zone (0.713𝑅⊙ to the surface). To summarize the outcomes of spherical
harmonics, the possible degrees of oscillations of the sun are limited to n=20 to 30, with 𝑙 ≥ 4 and
approximately 106 possible combinations. In Asteroseismology, the models do not conclude to a
particular structure, but they are also limited to about 106 possible combinations. On the other
hand, the models used today are not yet accurate. Further research is done in order to include the
processes of diffusion, angular momentum, magnetic fields, a transition zone and the perturbations
caused by convection. In any case, the science of helioseismology has led to a great leap in
understanding solar convection.
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