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Earl Patrick Bellinger

postdoctoral research fellow

stellar astrophysics centre

department of physics and astronomy

aarhus university

Inverse analysis of asteroseismic data:

a review

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"

"

— Gough & Thompson, The Inversion Problem (1991)

Inverse problems in helioseismology

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"

"

— Gough & Thompson, The Inversion Problem (1991)

a star's

stellar

stellar

Inverse problems in asteroseismology

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“The cause is hidden, but the result is known.”

— Ovid, Metamorphoses (8 AD)

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“The cause is hidden, but the result is known.”

— Ovid, Metamorphoses (8 AD)

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Well-posed problems

  1. existence: a solution exists
  2. uniqueness: the solution is unique
  3. stability: the solution changes continuously with changes to the input

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f(x) = 2x + 1

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Finish the sequence:

​

1, 2, 3, 4, _

6? [Ulam numbers]

7? [numbers n such that (68·10n+7)/3 is prime]

10? [counting in base 5]

17? [n such that n times φ(n) is a palindrome]

42? [an arbitrary sequence I just devised]

5? [natural numbers]

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Consider:

​

x = a + b

y = a + b · (1 + ε)

​

...where ε is arbitrarily small.

If

x = 2

y = 2

then

a = 2

b = 0

...but if

x = 2

y = 2 + ε

then

a = 1

b = 1

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  1. Evolution inversions
  2. Structure inversions
  3. Rotation inversions [e.g., Deheuvels et al. 2012, 2014, Di Mauro et al. 2016, Triana et al. 2017, Benomar et al. 2018, Bazot et al. 2019]
  4. Integrated quantity inversions: mean density, acoustic radius, and core-conditions indicators [Reese et al. 2012, Buldgen et al. 2015a,b, 2016a,b, 2018, 2019]
  5. Glitch analyses [Gough & Thompson 1988, Vorontsov 1988, Gough 1990, Mazumdar et al. 2012, Verma et al. 2014a,b, 2017, 2019]

Inverse problems in asteroseismology

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The evolution inversion problem

Given observations of a star, determine its age, mass, etc.

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The evolution inversion problem

Given observations of a star, determine its age, mass, etc.

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The evolution inversion problem

Given observations of a star, determine its age, mass, etc.

Approaches:

  • Optimize y - M(x, τ) [e.g., Brown et al. 1994, Metcalfe & Charbonneau 2003, Guenther & Brown 2004, Charpinet et al. 2005, Basu et al. 2010, Bazot et al. 2012, Lebreton & Groupil 2014, Metcalfe et al. 2014, Silva-Aguirre et al. 2015, 2017, Lund & Reese 2018, Rendle et al. 2019, Bellinger & Christensen- Dalsgaard 2019, and many more]
  • Model M-1 with machine learning [Pulone & Scaramella 1997, Bellinger et al. 2016, 2019, Verma et al. 2016, Angelou et al. 2017]

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The evolution inversion problem

Bellinger, Hekker, Angelou, Stokholm & Basu (2019), A&A

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Predicted Internal Structure of KIC 6225718

Bellinger, Basu, Hekker & Christensen-Dalsgaard, ApJ submitted

u = P/ϱ

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Comparing Mode Frequencies

Bellinger, Basu, Hekker & Christensen-Dalsgaard, ApJ submitted

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The structure inversion problem

Thompson & Christensen-Dalsgaard (2002), "On Inverting Asteroseismic Data", ESA SP

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The structure inversion problem

Given seismic measurements of a star, determine its internal structure

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The structure inversion problem

Approaches:

  • Regularized least squares [e.g., Tikhonov 1943, Basu & Thompson 1996, etc.]
  • Optimally localized averages [e.g., Backus & Gilbert 1968, 1970, Pijpers & Thompson 1992, 1994, Basu et al. 2002, Bellinger et al. 2017, and many, many more]
  • Differential response [Roxburgh & Vorontsov 2002a,b]

Given seismic measurements of a star, determine its internal structure

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Initial asteroseismic inversions

Gough & Kosovichev (1993), IAU 137

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Basu 2003, Ap&SS

Some Complications

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Inversion results for 16 Cyg A & B

Measuring differences within stellar interiors between stars and stellar models

in the isothermal speed of sound u = P/ρ where P is pressure and ρ is density

stellar center

outer layers

A

B

no difference

Bellinger, Basu, Hekker & Ball (2017), ApJ

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Inversion Results for KIC 6225718

Bellinger, Basu, Hekker & Christensen-Dalsgaard, ApJ submitted

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Alternate Models

Bellinger, Basu, Hekker & Christensen-Dalsgaard, ApJ submitted

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Mixed Mode Averaging Kernels

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Subgiant Structure Inversion

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  • Inverse problems are hard to solve
    • But that's OK: we like hard problems, and other people already did most of the hard work!
  • Asteroseismic inversions have unique difficulties over helioseismic ones
  • Models with convective cores on the main sequence may have issues
  • Models of subgiant cores may be consistent with observations

Inverse Analysis of Asteroseismic Data: Summary

Earl Patrick Bellinger

postdoctoral research fellow

stellar astrophysics centre

department of physics and astronomy

aarhus university

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fin.

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Mixed Mode Kernels

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A seismic scaling relation for stellar age

Bellinger (2019), Monthly Notices of the Royal Astronomical Society (accepted).

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Testing the seismic scaling relations

Bellinger, E. P. (2019), Monthly Notices of the Royal Astronomical Society (accepted).

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Models provided by T. Li, W. Ball, A. Stokholm, J. Ong, M. Yildiz & S. Basu

☉

☉

Evolutionary Modelling of delta Eridani

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delta Eridani: Echelle Diagram

TESS Frequencies by A.G.S.O. de Montellano, R. Garcia & D. Buzasi; Model by J. Ong

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Predicting Mode Frequencies

SONG Frequencies by Torben Arentoft

alias width

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Predicting Mode Frequencies

SONG Frequencies by Torben Arentoft

alias width

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New results: testing the constancy of the physical constants

In collaboration with Jørgen Christensen-Dalsgaard

G’ = 0

G’ < 0

G’ > 0

G’ = 0

G’ < 0

G’ > 0

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New results: testing the constancy of the physical constants

In collaboration with Jørgen Christensen-Dalsgaard

|G/G₀| < 0.43

|G’/G| < 3.5 · 10-12 yr-1

String theory

General relativity

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Initial asteroseismic inversions

Gough & Kosovichev (1993), IAU 137