Inferring Solar Internal
Structure and Rotation from
p-mode Frequencies and
University of Sheffield, UK
Affiliate scientist, High Altitude Observatory
Singular values for 1-D rotation problem
with 834 p-modes and 100-point radial mesh
2-D rotation inversion
The 1-D rotation example developed in the
last lecture is straightforwardy generalized
to the case of Ω = Ω (r, θ).
e.g. Schou et al. (1994) ApJ 433, 389
Commonly the results of the analysis of the
observations are not individual nlm frequencies.
Rather, the frequencies in each nl multiplet are
fitted as a polynomial in m:
where the P’s are even or odd polynomials in m
of degree j.
The odd coefficients aj can be used as the
data for rotation inversions.
RLS OLA 2-D Rotational Averaging Kernels
(1 s.d. uncertainties on inversion are
indicated in nHz, for a typical MDI dataset)
Inferred rotation inside the Sun
from MDI data (Schou et al. 1998, ApJ 505, 390)
Radial cuts through inferred rotation profile
of the solar interior (at latitudes indicated)
Location and width of tachocline
Tachocline shape in
fitted by a smooth
step like this, with
centre location rc
and width w.
the tachocline: this must be compensated for.
At equator, rc/R = 0.692 +/- 0.002; w = 0.03 – 0.04.
Rotation in the deeper interior
from LOWL + BiSON data (Chaplin et al. 1999, MNRAS 308, 405)
Possibly slightly slow core rotation - but inversions
consistent with uniform rotation in deep interior.
Differences between inversions of data taken at
successive times reveal the “torsional oscillation”
0.99 R 0.95 R 0.90 R 0.84 R
Here we difference rotation inversions relative to
solar minimum at successive 1-year epochs. The
evolution of the rotation rate in the whole
convection zone can be seen.
The rotation rate appears to show quasi-periodic
oscillations near the base of the convection zone
Black: GONG, red: MDI. Filled: RLS, open: OLA
Linearized inversion for solar
The dependence of the frequencies on solar
structure is inherently nonlinear.
But we can use linear inversion techniques if we
assume that the Sun’s structure and frequencies
are small perturbations to those of a known
You have seen in JCD’s lectures that the
frequencies can be written in terms of a
This can describe differences between the Sun
and the reference model.
δF depends on differences in quantities p, ρ and Γ1.
Primary seismic variables
The model quantities that appear in the
equations governing the adiabatic oscillations
are p, ρ, Γ1 (and combinations such as c).
These are therefore the
primary seismic variables.
Except for additional physics (e.g. rotation)
they are the ONLY quantities that can be
inferred from the frequencies unless we
introduce additional assumptions.
For full details see Gough & Thompson (1991), in Solar Interior and
Atmosphere, eds Cox, Livingston & Matthews, p. 519-561 (Univ. of
Invoking hydrostatic equilibrium
There appear to be three independent
unknown functions: δp/p, δρ/ρ, and δΓ1/Γ1.
But the oscillations are presumed to
take place about an equilibrium background
in hydrostatic equilibrium:
Perturbing this gives
Likewise, using the mass equation, δm can be
written in terms of δρ . Hence δp/p can finally
be expressed in terms of δρ/ρ, and the number
of unknown functions reduced from 3 to 2.
Kernels for sound speed and
Kernel for c2
Kernel for ρ
Mass conservation - an additional
Density perturbations cannot be
chosen arbitrarily, as the mass of the
Sun is known, i.e.
Transforming between variable pairs
A key to computing kernels
for other variable pairs is
how to use hydrostatic
equilibrium to transform
between δρ/ρ and δu/u,
where u=p/δ .
with ψ=0 at r=0 and r=R.
from 0 to R logarithmic
E.g. assume the equation of state Γ1=
Γ1(p,ρ,Y) known, where Y is the helium
abundance. So one can express δΓ in
terms of other perturbations
and hence (after some work) derive
kernels for e.g. u and Y.
Kernels for density and helium
Kernel for ρ
Kernel for Y
Formulation of structure inversion
For each observed mean-multiplet frequency we
have a datum (or constraint) of the form e.g.
• Knlc,(r), Knl,c(r) are known functions
• Gsurf(nl) is a term from near-surface errors in the mode
nl are errors in the observations.
Also have in this case the mass-conservation constraint, which can be
written in the same form as the data constraints.
Problem: use these constraints to make inferences
about e.g. δc2 and δρ between Sun and model.
RLS inversion for structure
Perhaps the most obvious approach.
Adjust the unknown functions δc2/c2, δρ/ρ
and Gsurf to get the best fit to the data.
As for rotation, need to regularize, so include
terms in the minimization to penalize solutions
δc2/c2, δρ/ρ that have e.g. large second
derivative. Typically choose Gsurf(ω) to be a
OLA inversion for structure
Try to choose inversion coefficients ci(r0) so that e.g.
is localized near r=r0. If successful, then
is a localized estimate of the relative difference in
sound-speed squared between Sun and model near
OLA inversion for structure
Choose inversion coefficients ci to minimize
where averaging kernel
subject to the constraints
Results of OLA Sound speed
between Sun and a
model, in sense
(Sun minus model)
from BiSON + LOWL data
(Basu et al. 1997, MNRAS 291, 243)
Depth of convection zone
From an inversion for sound speed, can calculate W,
which in the convection zone takes the
constant value -(Γ1-1)
(except in regions of
partial ionization). inversion
location of base of
convection zone is
rcz/R = 0.713 +/- 0.004
From inversions using u and Y, Richard et al.
(1999) determined helium abundance in the
solar convection zone to be 0.248 +/- 0.002
Can also (try to) use W
the HeII bump in W
either by fitting
or from its signature
as a sharp feature
Overshoot at base of c.z.
extent of any overshoot is small –
less than 0.1 HP (0.007R)
Introduction to the hands-on
• Use an IDL-based workbench for 1-D
• Try using both RLS and SOLA methods
• Experiment with trade-off parameters
(λ2 for RLS, θ and δ for SOLA)
• See how solution and error bars, averaging
kernels and inversion coefficients behave
as you vary the trade-off parameters
• Four datasets to try:
- Artificial rotation profile
- Artificial rotation profile with 10nHz s.d. noise
- Uniform rotation profile
- Uniform rotation profile with 10nHz s.d. noise
• There is also a fifth dataset for a mystery
rotation profile, with 2nHz s.d. noise
You are asked to hand in written assessments
of what you can deduce (qualitatively and
quantitatively) about the mystery rotation
profile. There will be a PRIZE (on Friday) for
the best overall deduction.
Kosovichev, A. G. (1999).
Inversion methods in helioseismology and
J. Comp. Applied Math. 109, 1-39.
Thompson, M. J. et al. (2003).
The internal rotation of the Sun
Ann. Rev. Astron. Astrophys. 41, 599-643.