Warwick PX420 Solar MHD 2015-2016: Helioseismology
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Helioseismology
“At first sight it would seem that the deep interior of the Sun and stars is less accessible to scientific
investigation than any other region of the universe. Our telescopes may probe farther and farther into the
depths of space; but how can we ever obtain certain knowledge of that which is hidden behind substantial
barriers? What appliance can pierce through the outer layers of a star and test the conditions within?”
Sir Arthur Stanley Eddington, 1926: The Internal Constitution of the Stars.
Helioseismology uses the Sun’s natural oscillations to observe beneath the surface of the Sun.
Main oscillations:
• g modes, internal gravity waves where buoyancy is the restoring force. These modes are trapped
beneath the convection zone → small amplitude at surface → predicted amplitudes at surface of
mm s−1 → difficult to observe. Periods > 40 mins.
• f modes, or fundamental modes, which are essentially surface gravity waves (n = 0). Some (high-`)
observed.
• p modes, where the main restoring force is a pressure differential. These modes travel throughout
the solar interior. Periods around 5 minutes, surface amplitudes of approx. 1−10 cm s−1 . Dominant
modes of Sun.
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p modes
A p mode is a standing acoustic (sound) wave. In the adiabatic case the speed of sound is
Cs2 =
γp
,
ρ
(1)
where γ is the first adiabatic exponent, p is pressure, and ρ is density. For an ideal gas (the equation of
state)
kB
p=
ρT,
(2)
µmp
where kB is Boltzmann’s constant, µ is the mean molecular weight, and mp is the mass of a proton.
Therefore
γkB T
Cs2 =
.
(3)
µmp
The speed of sound therefore depends on the temperature and chemical composition of the gas.
In reality we observe frequencies, amplitudes and phases of solar oscillations at the surface. These
properties are compared to those predicted by a Standard Solar Model (SSM) and the tunable properties
of a helioseismic solar model are altered so as to give the best fit to the data. Fig. 1 shows sound speed
and temperature as a function radius for a typical solar model.
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Spherical harmonic description
Since the Sun is a 3D sphere the natural oscillation modes are described by three coordinates: r is the
distance to the centre, θ is co-latitude, φ is longitude. Displacements are given by
ξr (r, θ, φ, t)
= a(r)Y`m (θ, φ) exp(−i2πνt),
(4)
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Figure 1: Sound speed (left) and temperature (right) as a function of radius predicted by Model S of
Christensen-Dalsgaard et al. (1996, Science, 272, 1286).
∂Y`m (θ, φ)
exp(−i2πνt),
∂θ
b(r) ∂Y`m (θ, φ)
exp(−i2πνt),
sin θ
∂φ
ξθ (r, θ, φ, t)
= b(r)
(5)
ξφ (r, θ, φ, t)
=
(6)
where a(r) and b(r) are amplitudes and ν is the cyclic frequency. Y`m (θ, φ) are spherical harmonics given
by
Y`m (θ, φ) = (−1)m c`m P`m (cos θ) exp(imφ),
(7)
where Plm are Legendre polynomials and clm is a normalization constant. The modes are, therefore,
described by three quantum numbers:
• n is the radial order and describes the number of radial nodes.
• l is the harmonic degree, which specifies the total number of surface nodes that are present.
• m is the azimuthal degree, where |m| specifies the number of nodes along the equator.
Figure 2: Spherical harmonic description of modes. The right hand figure shows an ` = 20, m = 16,
n = 14 mode.
2.1
Dependence of frequency on n
Analogous to waves on a string:
Warwick PX420 Solar MHD 2015-2016: Helioseismology
f1
=
f2
=
f3
=
3
v
,
2L
v
,
L
3v
...
2L
(8)
i.e. harmonics are separated evenly in frequency and frequency increases with n. Same is approximately
true for the Sun, but Cs is not uniform in solar interior and so modes are only approximately even
separated.
Figure 3: Waves on a string
2.2
Upper and lower turning points
A p modes is excited in near-surface regions by turbulent convection. As it travels inwards temperature
and therefore sound speed increase → the waves are refracted on a curved trajectory.
Therefore, the radius of the lower turning point depends the angle of the trajectory → radius of lower
turning point depends upon `, with low-` modes traveling deeper in the solar interior than high-`
modes (see fig. 4).
Modes are reflected by the sharp drop in density at the Suns surface. Modes are traveling approximately
radially close to the surface and so the radius of the upper turning point is dependent only on
frequency with the upper turning point of low-ν modes deeper than the upper turning point of high-ν
modes.
Key point: Since the modes sample different regions and therefore the properties of the
modes can be used to infer conditions in the solar interior.
2.2.1
Acoustic cut-off frequency
If the scale height, Λ, is less than the length scale of the mode the pressure changes required to make a
wave cannot be maintained over the length of time that matches the period of the mode. The compressions
are smoothed as the gas can readjust on a timescale that is shorter than the wave period and the wave
is reflected i.e.
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Figure 4: Ray paths taken through solar interior by modes: low-` modes travel deeper in the solar interior
than high-` modes.
• if length scale of mode < scale height: wave transmitted.
• if length scale of mode > scale height: wave reflected.
To determine the scale height, Λ, use the equation of hydrostatic support:
dp
Gmρ
=− 2 .
dr
r
Therefore
GM ρ
d ln p
=
2 .
d ln r
pR
Λ−1 =
(9)
(10)
A standing wave is formed when length scale of mode = scale height. Based on analogy with a string
reflection occurs when wavelength = 2×Λ.
We know that speed=frequency×wavelength and so
ωa =
Cs
.
2Λ
(11)
REMEMBER: ω = 2πν. But equation of state says that p/ρ ∝ T → Λ ∝ T .
T decreases with r until it reaches a minimum T = Tmin at the photosphere.
Here Λ = Λmin , and ωa = ωa,max , the maximum acoustic cut-off frequency.
As waves with frequencies higher than the acoustic cut-off are never reflected they cannot become standing
waves and so are not observed.
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Observing the oscillations
The Fourier transform of a sine wave observed for an infinite amount of time is a delta function at the
frequency of the wave. Real observations can only be made for a finite time. Consider a simple harmonic
oscillator
v(t) = a0 cos(ω0 t − δ0 ),
(12)
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Figure 5: Left: Lower turning points as a function of `. Right: Upper turning point of modes as a
function of frequency.
which is observed between times t = 0 and t = T . The Fourier transform of eq. 12 is
Z T
ṽ(ω) =
v(t)eiωt dt
0
Z Th
i
1
a0
ei(ωt−δ0 ) + e−i(ω0 t−δ0 ) eiωt dt,
2
0
i
h
i
e−iδ0 h i(ω+ω0 )T
eiδ0
1
i(ω−ω0 )T
a0
e
−1 +
e
−1 ,
=
2
i(ω + ω0 )
i(ω − ω0 )
sin[T /2(ω − ω0 )]
sin[T /2ω + ω0 )]
= a0 ei[T /2(ω+ω0 )−δ0 ]
+ ei[T /2(ω−ω0 )+δ0 ]
ω + ω0
ω − ω0
T
T
T
=
a0 ei[T /2(ω+ω0 )−δ0 ] sinc
(ω + ω0 ) + ei[T /2(ω−ω0 )+δ0 ] sinc
(ω − ω0 )
2
2
2
=
Typically when we perform a Fourier transform we create a power spectrum and consider only the positive
axis:
P (ω)
= |ṽ(ω)|2
1 2 2
2 T
'
T a0 sinc
(ω − ω0 ) .
4
2
(13)
In reality solar acoustic oscillations are damped harmonic oscillators:
v(t) = a0 cos (ω0 t − δ0 ) exp (−ηt) ,
(14)
where η is the damping rate. If observed for an infinitely long time the power spectrum is given by a
Lorentzian profile:
1
a0
P (ω) =
,
(15)
4 (ω − ω0 )2 + η 2
where η is the half width at half maximum. If observed for a finite time, T , the power spectrum is a
combination of a Lorentzian and a sinc2 where the shape
• tends towards a sinc2 for ηT << 1,
• tends towards a Lorentzian for ηT >> 1.
Damping rates range from months at low frequencies to days at high frequencies.
Other considerations when deciding how best to make helioseismic observations:
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Figure 6: Sinc squared function
• The resolution of a power spectrum is 1/T and so for two undamped harmonic modes of oscillation
to be resolved the must be separated in frequency by more than 1/T .
• The maximum frequency of power spectrum, Nyquist frequency, is 1/2δt, where δt is the time
between consecutive observations. Then δt < 1/(2ν) i.e. we need at least 2 data points per period
of oscillation to observe the mode.
Since there are lots of closely spaced oscillations (the frequency separation between modes of the same `
and consecutive n is of the order of 140 µHz and consecutive m are separated by less than 1 µHz) we need
observations lasting at least several days and data points to be recorded at least every 100s. Typically
observations last months to years and data points are recorded every minute.
3.1
Ground vs Space based
From a single ground-based observing site the Sun can be observed for no more than 12hrs in any 24hr
period (except at the poles). As discussed above this is not long enough to resolve the individual modes
and so observations over several days are required. However gaps complicate the power spectrum further.
Consider the undamped harmonic oscillator observed for 2 stretches of time, the first between t = 0 and
t = T , and the second between t = t1 and t = t1 + T :
Z
ṽ(ω)
=
0
'
=
T
v(t)eiωt dt +
Z
t1 +T
v(t)eiωt dt
t0
o
T n i[T /2(ω−ω0 )+δ0 ]
T
a0 e
+ ei[(t1 +T /2)(ω−ω0 )+δ0 ] sinc
(ω − ω0 ) ,
2
2
t
T
1
T a0 ei[0.5(t1 +T )(ω−ω0 )+δ0 ] cos
(ω − ω0 ) sinc
(ω − ω0 ) .
2
2
(16)
Warwick PX420 Solar MHD 2015-2016: Helioseismology
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The power is then
P (ω) = T 2 a20 cos2
T
t1
(ω − ω0 ) sinc2
(ω − ω0 ) .
2
2
(17)
Figure 7: Power spectrum of undamped harmonic oscillator that was observed 12hr in every 24hr day.
It is therefore important to observe the Sun as continually as possible: ground-based networks or spacebased observations.
3.1.1
Ground-based networks
Examples are Birmingham Solar Oscillations Network (BiSON, since 1976) and Global Oscillations Network Group (since 1995).
Disadvantages: Still susceptible to weather and instrument failure. Also have atmospheric noise in data.
Advantages: Relatively easy to fix problems and upgrade instruments. Relatively cheap.
3.1.2
Space-based observations
Examples are Global Oscillations at Low Frequencies (GOLF), Variability of solar IRradiance and Gravity Oscillations (VIRGO), and the Michelson Doppler Imager (MDI) on the SOlar and Heliosepheric
Observatory (SOHO) spacecraft, which was launched in 1995. Helioseismic and Magnetic Imager (HMI)
on the Solar Dynamics Observatory (SDO), which was launched in 2010.
Disadvantages: Expensive and impossible to fix problems. Subject to degradation of time.
Advantages: Very few gaps in observations. No noise from Earth’s atmosphere.
3.2
3.2.1
Doppler velocity vs Intensity
Doppler velocity
Line-of-sight velocity is measured from the Doppler shift of lines in the solar spectrum. The velocity
amplitude for each mode is at most 10 cm s−1 . Examples are BiSON, GONG, GOLF, MDI, HMI
3.2.2
Intensity
The amplitude in intensity is a few parts per million (ppm). Difficult to measure intensity variations
from the ground because of fluctuations in Earth’s atmosphere. Highly successful from space. Also much
Warwick PX420 Solar MHD 2015-2016: Helioseismology
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space-based asteroseismology relies on intensity measurements e.g. Kepler. Main example is VIRGO.
3.3
3.3.1
Sun-as-a-star vs resolved
Sun-as-a-star/unresolved
Examples include BiSON, GOLF, VIRGO.
• Get 1 time series that contains all low-` modes (typically ` ≤ 3).
• Higher-` modes are not observed because forward and backward motions described by spherical
harmonics cancel each other out.
• Solar equivalent to asteroseismic observations of other stars.
Figure 8: Left: Frequency-power spectrum of Sun-as-a-star Doppler velocity data collected by BiSON.
Right: Close up of BiSON spectrum.
3.3.2
Resolved
Examples include GONG, MDI, HMI.
• With resolved observations one can use a spatial filter to observe modes of a particular spatial
structure.
• Get time series for different modes.
• Can study global modes or do local helioseismology, that observes oscillations in small tiles on solar
surface.
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Figure 9: Left: Frequency-power spectrum of resolved Doppler velocity data collected by MDI for l = 200,
from Nigam et al., 1998, ApJ, 495, L115. Right: `-ν diagram observed by MDI.
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Successes of helioseismology
Neutrino Problem: Huge discrepancy between number of neutrinos measured on Earth and number
predicted from theoretical models of solar interior.
Original explanation was that there were problems with temperature and pressure in solar interior in
standard solar models. Helioseismology showed this wasn’t the case.
Radius of base of solar convection zone is 0.713 ± 0.001R
Rotation of the solar interior
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