Download Lecture 2/10 The Sun Ulf Torkelsson 1 The internal structure of the

Lecture 2/10
The Sun
Ulf Torkelsson
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The internal structure of the Sun
The Sun is a main sequence star of spectral class G2. It has existed for 4.5 billion years, and is
slightly less than halfway through its life as a main sequence star. Originally it consisted of 73 %
by mass of hydrogen, 25% helium and 2% metals. As all main-sequence stars it gets its energy
from the fusion of hydrogen to helium in the stellar core, and the dominating fusion process is the
proton-proton-chain like in all low-mass main sequence stars. The ongoing fusion has decreased
the hydrogen fraction at the centre of the star to 33 % and increased the helium fraction to 65 %.
A peculiarity with the abundances in the sun is that the abundance of 3 He has a peak at 0.3R .
The reason for this is that although most of the fusion takes place within 0.2R , some reactions
occur as far out as 0.3R , where 3 He survives for a long time because the transformation to 4 He
progresses slowly at the low temperature.
The energy that is released in the nuclear reactions are transported out from the core through
radiation, and the Sun is radiative out to 0.7R , where the opacity becomes so high that the Sun
becomes convective. The high opacity is the result of that the temperature has dropped down to
2 × 106 K. The Sun then continues to be convective all the way up to the visible surface. The
density drops off with radius very fast in the Sun, and 90% of the mass is concentrated inside of
0.6R .
During its main sequence life time the solar luminosity has increased by 40% and the solar
radius by 10 %. This is a result of that the core is contracting in response to that the number
of particles that can contribute to the pressure is decreasing as hydrogen is converted to helium.
While the core is contracting it is also heating up, which accelerates the nuclear reactions.
This description of the Sun is mostly the result of theoretical modelling, but there are two ways
in which we can test these models. Firstly, helioseismology studies waves on the solar surface.
These waves carry with them information about the interior of the Sun, and by studying the
amplitudes of the wavesp
as a function
p of their wave numbers and frequencies, we can reconstruct
the speed of sound cs = γp/ρ = γkT /(µmH ) in the solar interior. These studies have shown a
remarkable agreement with the theoretical models of the Sun.
Another way to investigate the interior of the Sun is by observing the neutrinos that are
produced during the fusion reactions. These neutrinos hardly interact at all with normal matter
and for that reason they will pass straight through the Sun in 2.3 seconds, while the photons take
105 years to diffuse through the Sun. A few neutrinos can be captured in large experiments on the
Earth. Such experiments have been carried out since the 1960s, but the results did not appear to
agree with the theoretical models of the Sun. The measured neutrino flux was merely a third of the
expected flux. We now understand that this is the result of neutrino oscillations. The neutrinos
that are produced in the Sun, as well as the neutrinos that are detected in most experiments on the
Earth are electron neutrinos, however there are another two flavours of neutrinos, muon and tau
neutrinos, and the neutrinos are oscillating between these three flavours. The neutrino oscillations
also show that the neutrinos cannot be massless as physicists used to believe, but must have some
small masses corresponding to energies of a fraction of an electron volt.
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The solar atmosphere
The Sun has a well-defined surface because it goes from being optically thick to optically thin over
a distance of 500 km in height, which should be compared to the solar radius 700 000 km. This
visible part of the solar atmosphere is known as the photosphere. It is possible to see a fine pattern
of bright cells surrounded by dark walls in the photosphere. This pattern, the granulation, is the
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top of the solar convection zone. In the centre of a granule there is hot gas rising upwards, and
cool gas is sinking downwards in the walls.
Most of the visible solar spectrum is formed in the photosphere. This process is described by
Kirchhoffs laws:
1. A hot, dense gas or hot solid object produces a continuous spectrum with no dark spectral
lines.
2. A hot, diffuse gas produces bright spectral lines (emission lines).
3. A cool, diffuse gas in front of a source of a continuous spectrum produces dark spectral lines
(absorption lines) in the continuous spectrum.
We see that the continuum is formed deep in the photosphere (at an optical depth of 2/3), where
the temperature is 5 770 K. Above this layer there is cooler gas in which most of the absorption
lines are formed.
On top of the photosphere we find the chromosphere in which the temperature increases from
4 400 K at the bottom to 25 000 K at the top. Some spectral lines are formed in the chromosphere
rather than in the photosphere. These are the hydrogen Balmer lines and spectral lines from some
ions. These lines are usually seen as absorption lines against the brighter photosphere, but at the
start and end of the total phase of a solar eclipse it is possible to see the chromosphere with the
naked eye as a thin red ring around the eclipsed Sun. The red light is emission in the Balmer-α line,
and if one would obtain a spectrum of the chromosphere at this moment one would see isolated
emission lines.
There is a thin transition zone on top of the chromosphere. In this transition zone the temperature increases to more than 106 K. Outside of this transition zone there is the corona, which has
a temperature of a few million kelvin. The reason that the corona is this hot is because it is being
heated by processes in the solar magnetic field. The corona can be observed during solar eclipses.
One can distinguish several components in the corona:
• K corona: The K corona is located between 1R and 2.3R from the Sun. This is the
source of continuum radiation. This continuum is generated as light from the photosphere is
scattered by electrons in the corona. It is not possible see any spectral lines in the K corona,
because the absorption lines are smeared out by the thermal motion of the electrons.
• F corona: The F corona starts at 2.3R . The light from the F corona is also scattered light
from the photosphere, but in this case the light is scattered by slowly moving dust particles,
so that the usual absorption lines (Fraunhofer lines) can be observed in the spectrum. The
F corona continues outwards and merges with the zodiacal light, which is light scattered by
dust in the orbital plane of the major planets.
• E corona: The E corona overlaps with both the K and F coronae. The E corona is seen in
spectral lines from highly ionised atoms.
The chromosphere and the corona are also sources of radio emission, and the corona is even
hot enough to emit X-rays. Most of the radio emission is bremsstrahlung that is generated as free
electrons pass close to ions. The X-ray spectrum contains many emission lines from highly ionised
atoms.
It is possible to show that the corona cannot be in hydrostatic equilibrium. We can solve the
equation for hydrostatic equilibrium
GM ρ
dP
=−
,
dr
r2
(1)
P = 2nkT,
(2)
where we can write the pressure as
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if the plasma consists of fully ionised hydrogen and n is the number density of protons n = ρ/mp .
We will assume that the corona is isothermal so that T = 1.5 × 106 K. We then have that
d
GM nmp
(2nkT ) = −
.
dr
r2
This differential equation is separable, so we have
Z n
Z r
n
GM mp 1
1
r0 dn
GM mp dr
= log
=
−
=
−
= −λ 1 −
,
2
n0
2kT r
2kT
r
r0
r
r0
n0 n
which we can write as
r0
n = n0 e−λ(1− r ) ,
where
λ=
GM mp
.
2kT r0
(3)
(4)
(5)
(6)
We can now see that the pressure varies as
r0
P = 2nkT = P0 e−λ(1− r ) ,
(7)
P0 = 2n0 kT.
(8)
where
−λ
That means that we need a pressure of P0 e at infinity to confine the corona. Assuming that the
number density of protons at the bottom of the corona is 3 × 1013 m−3 , we find that the corona
must be confined by a pressure of 5 × 10−4 Pa, which is much higher than the real pressure in the
surrounding interstellar medium. Therefore the solar corona cannot be in hydrostatic equilibrium,
rather is going over into the solar wind, which is blowing out from the Sun at supersonic speeds.
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The magnetic Sun
So far we have neglected magnetic forces, though they are responsible for many of the phenomena
that we observe on the Sun. In general we can write the the force acting on a charge q as it is
moving at a speed v through a magnetic field B as
F = qv × B.
(9)
The effect of this force on an electric charge is that the charge will describe a spiralling motion
along a magnetic field line. On the sun we are not studying individual charges but rather we are
studying the motion of a plasma. In order to understand how the magnetic force affects the plasma
we can notice that nev, where n is the electron density, is equal to the electric current density, J.
However according to Maxwell’s equations we have that the electric current density is
J=
1
∇ × B.
µ0
(10)
It then follows that the magnetic force per volume unit is
J×B=
(∇ × B) × B
µ0
(11)
This force can be split up into two terms, one which is due to a magnetic pressure B 2 /2µ0 , and
one which can be described as a tension in the magnetic field lines. These magnetic forces then
modify the ordinary dynamics of the fluid.
In ordinary hydrodynamics we have acoustic waves that are driven by the pressure variations in
a fluid. In magnetohydrodynamics we get new kinds of waves. The simplest of these waves is the
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Alfvén wave, which is generated by the magnetic tension in the field lines. Alfvén waves propagate
along the field lines at the speed
B
.
(12)
vA = √
µ0 ρ
There are also magnetoacoustic waves, which are affected by both the gas pressure and the magnetic
field.
There are many phenomena on the Sun that are clearly the result of the magnetic fields. The
most obvious of these are the sun spots. Sun spots are dark regions with a temperature that is a
1000 K lower than the rest of the photosphere. These sunspots have strong magnetic fields on the
order of 0.1 T. The reason that the sun spots are cooler appears to be that the magnetic field is
suppressing the convection beneath the sun spot, so that it receives less heat than other parts of
the solar photosphere.
Sunspots typically occur in groups. The western and eastern ends of the sunspot group usually
have the opposite magnetic polarities. At any time one will find that all sunspot group on one
hemisphere, say the northern hemisphere, have their magnetic fields oriented in the same direction,
while groups on the other hemisphere have their magnetic fields arranged in the opposite direction.
The number of sunspots vary cyclically with a period of eleven years. At minimum the Sun
will be practically free of sunspots. Then the first groups of the next solar cycle will appear at
around 40 degrees latitude north and south. Each sun spot will last for about a month, and over
time new sun spots will appear, but as the cycle progresses they will appear at lower and lower
latitudes, so that the last sunspots that are observed will appear at about five degrees latitude. At
solar minimum the sunspot groups shift polarity, so that if during one cycle, the sunspot groups in
the northern hemisphere have leading sunspots with a magnetic north pole, then during the next
cycle the leading sunspots in that hemisphere will have magnetic south poles. This is called Hale’s
polarity law. Apparently the solar magnetic field is changing its direction at solar minimum, and
we should rather speak of a magnetic cycle of 22 years.
The solar magnetic field is generated by a dynamo. The details of how this dynamo generates
the magnetic field are still unknown, but most current models go back to the scenario that was
suggested by Babcock. The Babcock dynamo starts from the observational fact that the Sun
rotates differentially, at the equator it takes the Sun about 25 days to complete one rotation, while
at the poles it takes 28 to 30 days. The result of this is that if we start out with a purely poloidal
magnetic field in the Sun, then the differential rotation will start to wind up and amplify this field.
The resulting toroidal field is what is manifested in the sunspots, and it neatly explains Hale’s
polarity law. However the surface layers on the Sun are convective so the convection will also
transport the magnetic field up and down. As a gas element moves upwards it will expand, but
due to the Coriolis force the expanding gas element will also start to rotate, and thus the magnetic
field becomes twisted. This rotation produces a new poloidal field from the toroidal magnetic
field. However the new poloidal field will have the opposite direction to the old poloidal field, so
at first they will reconnect and annihilate each other, but in the end there will remain only the
new poloidal field, which then serves as the source of a new toroidal field and so on.
The strong magnetic field of a sunspot group extends into the corona above the sunspot group.
Enormous amounts of energy can be stored in these magnetic fields. During the evolution of the
sunspot group these magnetic fields can become tangled, and there may develop sharp gradients
in the magnetic field, for instance in regions where oppositely directed magnetic fields lie close to
each other. These gradients correspond to strong electrical currents according to Eq. (10). These
currents can heat the surrounding medium through Joule dissipation
J2
,
σ
(13)
where σ is the electrical conductivity of the medium (in practice a strong current may even generate plasma instabilities, that lead to a dramatical decrease in the conductivity of the plasma).
Furthermore the electric fields that are associated with the currents can accelerate charged particles to high energies. The most dramatic example of this is magnetic reconnection, in which
oppositely directed field lines can annihilate each other within a small region so that the topology
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of the magnetic field changes. Large amounts of energy can be released in these reconnection
events, and they seem to be the mechanism behind flares and coronal mass ejections on the Sun.
Milder forms of magnetic activity seem to be responsible for maintaining the high temperature of
the solar corona.
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