INTRODUCTION TO HELIOSEISMOLOGY AND ASTEROSEISMOLOGYsappho.eps.mcgill.ca/...2016/...paperEPSC330.pdf · Introduction to helioseismology and asteroseismology • April 2016 • Term

Introduction to helioseismology and asteroseismology • April 2016 • Term Paper EPSC 320

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INTRODUCTION TO HELIOSEISMOLOGY AND ASTEROSEISMOLOGY

Meryem Berrada

McGill University

[email protected]

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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REFERENCES

Breger, M. "Uncertainties in the Calculated Pulsation Constant Q." DSSN 2. Delta Scuti Star

Newsletter, 2 Mar. 1990. Web. 13 Mar. 2016.

<https://www.univie.ac.at/tops/CoAst/archive/DSSN2/QConstant.html>.

Brown, Jeff. "Re: What Is a Variable Star Observed minus Calculated (O-C) Diagram?" MadSci.

Faculty of Astronomy, 9 Feb. 2000. Web. 7 Mar. 2016. <http://www.madsci.org/posts/archives/2000-

02/950129794.As.r.html>.

Guenther, D. B. "Age of the Sun." Age of the Sun. Astrophysical Journal, Part 1 (ISSN 0004-

637X), Vol. 339, April 15, 1989:1156-1159, Apr. 1989. Web. 7 Mar. 2016.

<http://adsabs.harvard.edu/doi/10.1086/167370>.

Harvey, John. "Helioseismology." Physics Today. N.p., 1 Oct. 1995. Web. 7 Mar. 2016: 32-38.

<https://www.deepdyve.com/lp/aip/helioseismology-UQ0SUm3iCD?key=AIP>.

Jørgen Christensen-Dalsgaard, Jørgen. "Problems of Solar and Stellar Oscillations." Stellar

Oscillations 5 (1983). Phys.au.dk. Institut for Fysik Og Astronomi, Aarhus Universitet Teoretisk

Astrofysik Center, Danmarks Grundforskningsfond, Jan. 2015. Web. 7 Mar. 2016. <http://users-

phys.au.dk/jcd/oscilnotes/print-chap-full.pdf>.

"Kepler: Graphics for 2010 Oct 26 Webcast." Kepler: Graphics for 2010 Oct 26 Webcast. Ames

Research Center, 26 Oct. 2010. Web. 7 Mar. 2016.

<http://kepler.nasa.gov/news/nasakeplernews/20101026webcast/>.

"Main Sequence Stars." Atnf.csiro. Australia Telescope National Facility, n.d. Web. 7 Mar. 2016.

<http://www.atnf.csiro.au/outreach/education/senior/astrophysics/stellarevolution_mainsequence.html>.

Minard, Anne. "The Case of the Missing Sunspots: Solved? - Universe Today." Universe Today -

Space and Asronomy News. Universe Today, 17 June 2009. Web. 7 Mar. 2016.

<http://www.universetoday.com/32642/the-case-of-the-missing-sunspots-solved>.

Nave. "Composition of the Sun." HyperPhysics. Georgia State University, Departement of Physics

and Astronomy, n.d. Web. 7 Mar. 2016. <http://hyperphysics.phy-

astr.gsu.edu/hbase/tables/suncomp.html>.

Nave. "Red Shift." HyperPhysics. Georgia State University, Departement of Physics and

Astronomy, 2000. Web. 7 Mar. 2016. <http://hyperphysics.phy-astr.gsu.edu/hbase/astro/redshf.html#c1>.

"Power Spectrum." Wolfram MathWorld. Wolfram Research, Inc., 2016. Web. 7 Mar. 2016.

<http://mathworld.wolfram.com/PowerSpectrum.html>.

"Spherical Harmonics Symmetries." Ambisonics. Plone Foundation, 2016. Web. 13 Mar. 2016.

<http://ambisonics.iem.at/xchange/fileformat/docs/spherical-harmonics-symmetries>.

"Spherical Harmonic." Wolfram MathWorld. Wolfram Research, Inc., 2016. Web. 13 Mar. 2016.

<http://mathworld.wolfram.com/SphericalHarmonic.html>.

"Stars - Stellar Evolution." Astronomyonline. N.p., 2013. Web. 13 Mar. 2016.

<http://astronomyonline.org/Stars/Evolution.asp>.

Tong, Vincent C.H, and Rafael A. Garcia, eds. Extraterrestrial Seismology. N.p.: Cambridge UP,

2015. Ebooks.cambridge. McGill Livrary, July 2015. Web. 7 Mar. 2016: 11-122.

<http://ebooks.cambridge.org.proxy3.library.mcgill.ca/ebook.jsf?bid=CBO9781107300668>.

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INTRODUCTION TO HELIOSEISMOLOGY AND ASTEROSEISMOLOGYsappho.eps.mcgill.ca/...2016/...paperEPSC330.pdf · Introduction to helioseismology and asteroseismology • April 2016 • Term