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Sunday, March 22, 2009

PHYSICS OF THE SUN

this course and apply them to understanding the source of all life on earth - our sun. We
will learn that the sun operates according to principles that we can understand, and on
the basis of this we can even predict the manner in which it will eventually die.
2. Basic solar facts:
a) Mass o 30
9
f sun 2 10 kg = 333,000 Earth's
b) Diameter of sun 1,392,000 km = 10 Earth's
c) Age of sun 4.6 billion years
d) Rotation Period = 25 days at equator, 36 at poles (surface)
e) Tem
= ×
=
=
perature = 15 million 0K at core, 5770 0K at surface
f) Density = 8 gold at the core, average is ~ 1.5 water
g) Composition: 72% H, 25% He, rest is metals
3. The sun puts out a huge amount of en
×
27 24
16
ergy. In quantitative terms we measure this in by
its luminosity, 3.83 10 joules per second. The power output is 3.83 10 kilowatts.
This is equal to 8 10 of the largest power plants on Ear
× ×
×
9
th, meaning those which produce
~5000 MW of power. Another way of expressing this: every second the sun puts out as
much energy as 2.5 ×10 (2 billion) such power plants would put out every year.
4. What powers the sun? The earth is very old (billions of years). If there was a chemical
fuel (say, coal or oil) at most that would last a few million years. But the sun is many
thousands of tons older than that. Only after the discovery of 2 did we know the
real secret. The sun gets its energy from the fusion (the coming together and combining
of atomic nuclei. For this e
E=mc
xtremely high temperatures, density, and pressure is needed.
4. The basic point is that, as you can see in the picture,
the combined mass of 4 protons is higher than than
that of the helium nuclei into which they convert
through the process shown earlier. The difference
then appears in the form of kinetic energy of the
released particles, which in random form is heat.
5. Protons repel protons, but the only way in which they
will participate in a fusion reaction is when they can
come sufficiently close. This requires that they smash
into each other at sufficiently high speeds, and hence
nuclear fusion in the sun require
0
s core temperatures
greater than about 8 million K. He nuclei can also
fuse with each other to release energy, but because
they are heavier much higher temperatures & densities
(about 100 million 0K for helium fusion) are required.
Thus, stars fuse hydrogen first. Each second, the sun
turns 4 million tons of hydrogen into energy.
6. Fusion takes place only in the core of the sun (see diagrams below) because it is only
there that the hydrogen gas is hot enough. From the core, the heat gets out by the
emission of photons (radiation zone). The hot gas then exchanges heat with the sun's
relatively cold exterior through convective currents. Huge columns of hot gas move
from the inside towards the surface. After giving up most of the heat, the "cold" gas
sinks towards the centre and the cycle goes on.
7. For over 4 billion years the sun has been nearly steady in maintaining its size. It is in a
state of equilibrium between two big forces acting oppositely to each other:
a) The hot hydrogen gas seeks to expand outwards because the thermal velocity of H
atoms leads to a pressure directed outwards.
b) Gravity tries to squeeze the star inwards because every piece of matter attracts every
other piece. So the gravitational pressure is inward, increasing toward the core.
As in the diagrams below, imagine a piece of solar material at some distance from the
centre of the sun. The two forces acting upon it must exactly balance in equilibrium. If
for some reason the sun cools down, the pressure of gas will decrease and the sun will
contract to a new equilibrium point. Conversely, the sun will expand if the rate of fusion
were to increase.
Look at the second diagram. Let ( ) be the mass contained within radius . We will not
assume that the density is constant in . First find the inward directed gravitational force
on a shel
M r r
r
2
2
2
l of matter at radius and thickness , ( ) ( )4 . There is a
net pressure as shown, and we will call the difference of pressures. Then obviously
4 . Hence, ( )
r dr dF GM r r r dr
r
dP
dF dP r dP GM r
ρ π
π
×
= −
= × =− 2 2
2
0
( ) or, ( ) ( ). The total mass
upto radius is, ( ) ( )4 . If we knew ( ) then ( ) would also be known.
By solving the differential equation, we would also then k
r
r dr dP GM r r
r dr r
r Mr r r dr r Mr
ρ ρ
ρ π ρ
= −
= ∫ ′ ′ ′
-3
3
now ( ). To give us a better
understanding, suppose for simplicity that the sun is approximately uniform. Then the
density is, 1.4gm cm . Hence the pressure at the centre of the s
(4 /3)
P r
M
R
ρ
π
≈ ≈
9
un can be
computed, 3 10 atmospheres. centre
P GM
R
ρ
≈ ≈ ×
TE Ts
R Rs
8. The equilibrium of forces is a nearly perfect one,
but there are small disturbances and these cause
"earthquakes" (actually sun-quakes) to occur on
the surface. The study of the surface of the sun and
sun-quakes is an area known as "helioseismology"
(helio=sun, seismology is the study of vibrations
and earthquakes). The pulsating motions of the sun
can be seen by measuring the Doppler shifts of
hydrogen lines across the face of the sun. Some
parts are expanding towards the earth while adjacent
regions contract away. This is like the modes of a
ringing bell. Vibrations propagate inside the sun,
and the waves travel through a hot dense gas. They
experience reflection and refraction since the speed
at which a wave travels changes when it goes from
a region of high to low density or vice-versa. All this
is being studied by astrophysicists today as they map
out the sun's interior.
9. Let us calculate the surface temperature of a plane
4
t circulating the sun. We shall use the
Stefan Boltzman law that you studied in the lecture of heat: the power radiated by a black
body per unit surface area at temperature is .
For the
T σT
4
rmal equilibrium, we must have that all the
power absorbed from the sun is re-radiated by
the planet. Let be the sun's flux at its surface,
Then, . Since radiation decreases by the
S
S S
P
P=σT
( )4 2
2
distance squared, flux at earth
This must be multiplied by , which is the
crossectional area of the planet. On the other hand,
for emission from the planet, flux at it
Rs
R S R
E
E
P T
R
P
σ
π
= = ×
=
( )
4 2
2
2 2 4 4
s surface . We must multiply this by 4 ,
the area of planet, to get the total radiated power. Now impose the equilibrium condition:
4 . This gives 4 , and hence
E E
Rs
R E E E S R E
T R
P R P R T T
σ π
π π σ σ
=
× = × × = ( )1/ 2
2
8
11
. This is
true for any planet, so let's see what this gives for the earth using 7 10 metres,
1.5 10 metres, and 5800 . This gives 280 , which is very sensible!
Rs
E R S
S
o
S E
T T
R
R T K T K
=
= ×
= × = =
Of course, we have assumed that the earth absorbs and emits as a black body. True?
2.8 kTS
Planck Radiation Law
2.8 kTE photon energy
earth
sun
Greenhouse gas absorption
energy
Strong adsorption in the
infrared. (rotational
and vibrational
Photon ti )
CO2 or
H2O n
1
)
2
3
2
1
Energy levels of an oscillator:
( n
n
n
ε n ω =
=
=
= + 􀀽
n = 0
10. Actually, the black body assumption is only approximately true. About 30% of the sun's
radiation reflects off our atmosphere! In fact the average surface temperature of the earth
is about 290K, or about 13 0C. The extra warming is mainly due to the "greenhouse
effect". Certain gases trap the sun's radiation and it is not able to get out, thus leading
to greater absorption. 2 4 2 2 2
2
The main greenhouse gases are CO , CH , H O. (N , O and Ar are
transparent to solar and earth radiation.) CO is now 360 ppm (parts per million) of the
atmosphere, up from 227 ppm in 1750 2
0
2 2
, before the industrial revolution. Plants turn H O +
CO into O plus organics. Some estimates predict a 3 to 10 C rise in the earth's surface
temperature over the next 100 years due to t 2 he increased CO greenhouse effect.
In the above, you can see the narrow window in which the greenhouse gases absorb the
sun's radiation. But why are only some gases responsible, and not others?
11. The answer lies in quantum m
2
echanics. In a previous lecture you learned that molecules
can exist only in certain states that have very specific values of energy. For example, a
molecule of CO can oscillate and have equally spaced energy levels as shown below:
The energy at which these molecules can absorb radiation happen to lie in the spectrum
of the re-radiated energy from the earth's surface. Thus, they effectively trap the outgoing
radiation, leading to enhanced earth temperatures. For any molecule, we can both
calculate (using quantum mechanics) the frequencies at which radiation is absorbed or
emitted. We can also experimentally measure the frequencies. Both of these are in very
good agreement, and this is one of the reasons why we have such confidence in the
correctness of quantum mechanics.
Earth Sun
The Sun in Five Billion Years
Core
H burning shell
He
The Sun as a Red Giant
diameter = 1 AU
The Sun as a Main
Sequence Star
diameter = 1/100 AU
Helium Depletion in Core
H burning shell
He burning shell
Carbon
Core
12. Finally, I have some bad news: the sun is going to die because the hydrogen supply is
eventually going to run out. There is, of course, no immediate danger - the Sun can last
another 5 billion years on core hydrogen fusion. During this time it will be consuming
hydrogen and producing helium. At the end of this phase, the core of the sun will have
mostly helium with the little bit of hydrogen left almost entirely outside in a thin layer.
The reaction rate will fall, and the core will no longer be able to balance the pull due to
gravity. This will cause a shrinkage of the core. As a consequence the temperature will
increase. A new phase of the sun is about to start.
13. At this point a fusion reaction in next core zone
begins. This lifts the envelopes and the sun will
brighten up significantly. It is now a Red Giant
star, and as you can see below, it is huge! The
earth will be swallowed up by the expanded sun.
14. As remarked earlier, helium can also "burn" to
produce carbon, but it needs a much higher
temperature. Eventually even the He will be
depleted, and the reaction rate will fall. Again
the small reaction rate means that the core will
shrink, and heat up to the point that it crosses
the carbon fusion threshold of 600 million K.
Low mass stars cannot reach this temperature.
Envelope expands to a supergiant, many times
larger than even a Red Giant.

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