1. Introduction to Helioseismology and Asteroseismology
Meryem Berrada

2. Introduction Helioseismology is the study of the sun’s structure
Asteroseismology is the study of other stellar bodies
Identify the internal structure
Variations of brightness  oscillations within the structure
Convection occurring in the deep interior of a star
Observe Doppler shifts in spectrum lines
Standing sound waves  Boundaries : beginning of the convection zone to the surface of the sun

3. Three different types of sound waves
1) Acoustic waves
p-modes
adiabatic process
2) Internal gravity waves
g-modes
Complex: imply a pressure gradient
adiabatic as one of the possible solution
3) Surface gravity waves
f-modes
Incompressible liquid
Constant density

4. Technique
Earthquake is recorded by different stations all around the earth’s surface
In Helioseismology, the source is estimated to any point of the surface
Assume that this point aligns on some great circle
Gives displacement function of the oscillations on that particular great circle.

5. Relationship between Luminosity and
Temperature; 𝐿= 𝑅 2 𝑇 (1)
Mass-Luminosity relationship ; L L ⨀ = M M ⨀ (2)
Time wave at speed c takes to travel
stellar medium; 𝑡 𝑑𝑦𝑛 = 𝑅 3 𝐺𝑀 (3)

6. Spherical harmonics Model the oscillations
Two types of modes : radial & non radial
Sun modes are non-radial: the shape of the star is not preserved during oscillation
Defined by three wavenumbers.
n: radial order, the number of nodes in the radial direction
l: angular degree, the number of nodal lines
m: angular order, the number of nodal lines that cross the equator

7. Doppler shift
Many combinations of wavenumbers that can lead to a similar oscillation
From the Doppler shift analysis, approximate the complexity of the oscillations

8. 𝐹 𝑣 =𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡∗ 𝑛=1 𝑁 𝑦 𝑛 ( 𝑡 𝑛 ) 𝑒 −𝑖𝑡𝑛(2𝜋𝑣)
Fourier Transform
Oscillations are detected as functions of position on the solar disk.
Fourier Transform in time will filter the corresponding frequencies.
𝐹 𝑣 =𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡∗ 𝑛=1 𝑁 𝑦 𝑛 ( 𝑡 𝑛 ) 𝑒 −𝑖𝑡𝑛(2𝜋𝑣)
Fundamental frequency: period-1h  n=0
Oscillation frequencies: period- 5 min  n= 20-30

9. Power Spectrum
Plot of power (energy per unit time ) relative to frequency
Filter noise by setting limits of energy per unit time that can be received from the observed object:
1) Analyze the signal to noise ratio
2) Maximum energy per unit time for a frequency 𝐴 𝐴 0 = 𝐿 𝑀 ( 𝑇 𝑇 𝑒𝑓𝑓 ) 𝑠

10. Assumptions Adiabatic process is occurring
Pressure caused by the dynamics of convection is neglected
No transition zone between the convection zone and the interior
Acoustic waves have specific boundary conditions
Effects of magnetic field are also neglected

11. Simulation of spherical harmonics

12. Quality factor Q
↑𝑄 :Low energy loss = almost no damping ↓𝑄 : Great energy loss = damped oscillation 𝑄=Π 𝑀 𝑀 ⊙ 1 2 𝑅 𝑅 ⊙ − 3 2 𝜌 ~ 𝑀 𝑅 3 𝑔 ~ 𝑀 𝑅 2 𝜌 𝜌 ⨀ = 𝑔 𝑔 ⨀ 𝑅 𝑅 ⊙ = 𝑔 𝑔 ⨀ 𝐿 ⨀ 1/2 𝑇 2 𝐿 1/2 𝑇 ⨀ 2 Gives direct relationship between the observed values and the stellar density.

13. Asteroseismology O-C diagram (Tool to check model )
Observed parameters [O] - Calculated parameters [C] vs time
Curvature= period is changing with time
Quadratic O-C curve = constant rate of change in the period
Upward parabolic curve = steadily increasing period
Increasing slope = real period > that used in model
The calculated part comes from forward and reverse modelling

14. Example of model analysis: variable star Tau V1370
Constellation of Taurus (401 ly)
Decreasing slope = real period ( 𝑑𝑎𝑦𝑠)< that used in model

15. Conclusion
Data: luminosity, temperature, intensity, period and velocity of the sun’s oscillations.
Fourier time transforms : the wavenumbers’ corresponding frequencies
Power spectrum : Oscillation modes
Forward modelling : spherical harmonics, parameters estimation
Inverse modelling : more precise values for parameters
Perspective: diffusion, angular momentum, magnetic fields, transition zone, perturbations

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