Asteroseismology A brief Introduction

What is asteroseismology?

Asteroseismology Well, asteroseismology refers to the study of the internal structure of stars through the interpretation of their oscillation frequency spectra. Essentially, asteroseismolists try to make use of the oscillation to probe the stellar interior, which is not directly observable. Helioseismology is the comparable study but involving the Sun. In other words, the internal structure of a star can be inferred by studying the pulsational variability that is observed at the surface of a star.

Goal in asteroseismology The ultimate goal of asteroseismology is to improve the evolutionary models of the stars. The theory of stellar evolution is reasonably well established in a global sense, but asteroseismology can enrich it even more.

H-R diagram (On stellar evolution)

They are produced by hot gas churning in the convection zone, Sound waves inside the Sun cause the visible solar disk to move in and out. This heaving motion can be described as the superposition of literally millions of oscillations, including the one shown here for regions pulsing in (red spots) and out (blue spots). The sound waves, represented here by black lines inside the cutaway section, resonate through the Sun. They are produced by hot gas churning in the convection zone, which lies above the radiative zone and the Sun’s core.

Major types of waves in stellar structures Solar oscillation modes are essentially divided up into 2 categories, based on the restoring force that drives them: acoustic, or gravity; these are described by pressure or p-modes and gravity or g-modes  p-mode or acoustic waves have pressure as their restoring force, hence the name "p-mode". Their dynamics are determined by the variation of the speed of sound inside the sun. P-mode oscillations have frequencies > 1 mHz and are very strong in the 2-4 mHz range P-modes at the solar surface have amplitudes of hundreds of kilometers and are readily detectable with Doppler imaging (video) Internal gravity modes, known as g-modes, have gravity (and buoyancy) as the force acting to restore equilibrium. These modes occur where a parcel of fluid vertically displaced from a level in which it is in equilibrium with the surrounding media will show a density diferential, and so will exhibit positive or negative buoyancy in a manner that acts to return the fluid to equilibrium. This will occur in regions in which the primary form of energy transportation is by radiative means. In convective regions a displaced parcel of gas will continueto rise or fall, so standing gravity waves are not able to propagate, and become evanescent within these regions (video)

Types of oscillations The simplest oscillation a star can undergo is a radial one; the star expands and contracts radially and spherical symmetry is preserved during oscillation cycle and these characterized by the radial wave number n (number of nodes) this can be viewed as a sphere with shells. Non-radial oscillations are transverse motion and are characterized by non-radial wavenumbers ℓ (number of surface nodal lines) and m (number of lines that pass through the rotation axis of the star).

Example of Radial oscillation

Different examples of non-radial oscillations (mass elements on white lines do not move during oscillation) (video) 13

Sound rays are bent inside the Sun, like light within the lens of an eye, and circle the solar interior in spherical shells or resonant cavities. Each shell is bounded at the top by a large density drop near the photosphere and bounded at the bottom by an increase in sound speed with depth that refracts a downward propagating wave back toward the surface. The bottom turning points occur along the dotted circles shown here. How deep a wave penetrates and how far around the Sun it goes before it hits the surface depends on the harmonic degree, l. The white curve is for l = 0, the blue one for l = 2, green for l = 20, yellow for l = 25 and red for l = 75

The curving trajectory of pressure waves can be explained in a way by Snell’s Law, where wave speed increase with depth and eventually waves get reflected back to the surface

Frequency and spacings can be described with following equations ν 𝑛,𝑙 =∆ν 𝑛+ 𝑙 2 +ε −𝑙 𝑙+1 𝐷𝑜 ∆ν 𝑛,𝑙 = ν 𝑛,𝑙 − ν 𝑛−1,𝑙 𝑙𝑎𝑟𝑔𝑒 𝑠𝑝𝑎𝑐𝑖𝑛𝑔 = ∆ν ≈135μ𝐻𝑧 𝑓𝑜𝑟 𝑆𝑢𝑛 δ 𝑛,𝑙 = ν 𝑛,𝑙 − ν 𝑛−1,𝑙+2 (𝑠𝑚𝑎𝑙𝑙 𝑠𝑝𝑎𝑐𝑖𝑛𝑔)

Now the question. Can we find relations between frequency modes in stars and their physical properties like mass and radius?

𝑟 0 = δ(𝑛,0) δ(𝑛,1) , ∆ν=𝑙𝑎𝑟𝑔𝑒 𝑠𝑝𝑎𝑐𝑖𝑛𝑔, horizontal curves are age in G and

Now by assuming a constant density ρ, we get 𝑡 𝑑𝑦𝑛 = 𝑟=0 𝑅 𝑑𝑟 𝐶 ,this is called the 𝐷𝑦𝑛𝑎𝑚𝑖𝑐𝑎𝑙 𝑡𝑖𝑚𝑒 𝑠𝑐𝑎𝑙𝑒, it is the time that an acoustic wave takes to travel a stellar medium at sound speed C. Now, the frequency of the fundamental radial mode, 𝑙=0, is proportional to the spacing between adjacent radial modes (∆ν), so ultimately large frequency spacing will be proportional to the dynamical time scale (𝑡 𝑑𝑦𝑛 ). Now, by using basic mechanics and a simplistic method of calculating the time a particle travels in free fall by disregarding pressure forces Now, a particle of mass 𝑚 sitting at a radius 𝑟 within the body of a star; so then we get 𝑚 𝑑 2 𝑟 𝑑𝑡 2 =− 𝑚𝐺𝑀 𝑟 𝑟 2 Now by assuming a constant density ρ, we get ρ= 𝑑 𝑀 (𝑟) 𝑑𝑉 = 𝑑 𝑀 (𝑟) 𝐴ρ𝑑𝑟 , 0 𝑟 𝑑𝑀 (𝑟) = 0 𝑟 4 𝑟 2 πρ𝑑𝑟=𝑀 𝑟 ≈ 4πρ 𝑟 3 3 − 𝑚𝐺𝑀 𝑟 𝑟 2 = −𝑚𝐺4πρ 𝑟 3 3𝑟 2 =𝑚 𝑑 2 𝑟 𝑑𝑡 2 𝑑 2 𝑟 𝑑𝑡 2 + 𝐺4πρ 3𝑟 2 𝑟=0= 𝑑 2 𝑟 𝑑𝑡 2 + λ𝑟 (the sturn Louiville equation again!) Solution is 𝑟 (𝑡) =𝐴𝑐𝑜𝑠 λ 𝑡 +𝐵𝑠𝑖𝑛 λ 𝑡 λ= 4πρ𝐺 3𝑟 2 = ω 2 = (𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦) 2 ≈ 1 𝑡 𝑑𝑦𝑛 2 𝑡 𝑑𝑦𝑛 =𝐷𝑦𝑛𝑎𝑚𝑖𝑐 𝑡𝑖𝑚𝑒 𝑠𝑐𝑎𝑙𝑒 𝑜𝑟 𝐹𝑟𝑒𝑒 𝑓𝑎𝑙𝑙 𝑡𝑟𝑎𝑣𝑒𝑙𝑖𝑛𝑔 𝑡𝑖𝑚𝑒= 3 4πρ𝐺

Notice that 𝑡 𝑑𝑦𝑛 ∝ 1 ρ As mentioned before, the large-frequency spacing will be proportional to the dynamical timescale, so then ∆ν∝ 1 𝑡 𝑑𝑦𝑛 ∝ ρ ∝ 𝑀 𝑅 3 1 2 ∆ν∝ 𝑀 𝑅 3 1 2 (for an arbitrary solar-like star) ∆ν 𝑠 ∝ 𝑀 𝑠 𝑅 3 𝑠 1 2 (for Sun) By combining both and setting ∆ν 𝑠 =135𝐻𝑧 ∆ν= ( 𝑀/ 𝑀 𝑠 ) 1/2 (𝑅/ 𝑅 𝑠 ) 3/2 135𝐻𝑧 or ∆ν= ρ ρ 𝑠 135𝐻𝑧 If only 1 variable is unknown, we have then a way to know either 𝑀 𝑜𝑟 𝑅 for an arbitrary solar-like star in the main sequence; (∆ν) can be obtained from a stellar spectrum.

More on ∆ν= ( 𝑀/ 𝑀 𝑠 ) 1/2 (𝑅/ 𝑅 𝑠 ) 3/2 135𝐻𝑧 and ∆ν= ρ ρ 𝑠 135𝐻𝑧 It is worth noting that if a star during the time it is in the main sequence has a mass that is relatively constant=M; so then you’ll see the evolution mainly between ∆ν and 𝑅, for which we saw from a previous graph that ∆ν decreases with time, therefore 𝑅 should increase as expected in stellar evolution. In other words, if the large frequency spacing ∆ν decreases, the overall density ρ of the star should decreases as well.

Questions?