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The Question
(Submitted December 16, 1998)
I am a science teacher in a middle school in Istanbul, some
of my students asked me how a scientist can measure the
temperature of the core of the sun? Since I am not sure, and
the archives don't really say, I thought I would give this a
try. The only thing I can think of is that by measuring the
amount and different kinds of radiation that come from the core
scientist can then extrapolate the approximate temperature?
The Answer
That is a really good question, and one which is surprisingly
simple on the surface (or to a first order approximation).
First lets look at a pot of water at room temperature. (I know this
seems strange, but bear with me on this.) If you leave it alone, after
a few days, if nothing starts growing, the water level will go down as
water evaporates off of the top, until finally you are left with an
empty pot. Right?! At room temperature, the water isn't boiling, but
every now and again a water molecule can "jump" into the air. How fast
this happens is determined by hydrostatic equilibrium. (i.e. a balance
between what is pushing one way with what is pushing the other
way). In the case of the pot of water, it is how much pressure (from
gravity pulling down on the atmosphere) is pushing down on the water,
compared with how much the water molecules, which have some energy and
want to move, are pushing up on the air.
The actual definition of boiling point is when the atmospheric
pressure = the vapor pressure of the liquid. This is why at high
elevations (like Mt. Ararat) the boiling point of water is much lower
than at sea level (Istanbul). At higher elevations there is less air
pressure to push on the water, so it can boil (escape) at a lower
temperature. The boiling point of water is about 90 degrees
Celsius at 3,000m instead of 100 degrees at sea level.
So gravity pushes on the atmosphere, and the weight of all that air
pushes on the water to prevent it from boiling, UNLESS you add energy.
Well, the sun works the same way. If the Sun were not burning, gravity
would compress all the gas down into a much smaller space. Since the
sun is bigger than just a ball of gas held down by gravity, we know
(along with other things, like the fact that it is glowing VERY
brightly) that there must be some source of energy in there. (In the
sun's case the only thing that can produce enough energy over such a
long period of time is nuclear fusion.) If we can measure the size
(radius) of the sun, and have a good estimate of its mass, using some
physics we can calculate the temperature in the center needed to hold
the rest of the sun up.
The reason that we can't just look at the radiation (light) that comes
from the center of the sun to measure its temperature is that the sun
is so dense that even the most energetic gamma-rays can only travel a
few centimeters in the center before being absorbed by another atom,
and then re-emitted a little while later. (This is called Compton
scattering).
If you light a fire, the farther away from the fire you get, the
cooler it feels. This doesn't mean that less energy is given off, but
that the same amount of energy that is sent from the fire is spread
out over a larger area, and doesn't feel as hot. (This is called the
Inverse Square Law) The surface of the sun glows at about 5,000
degrees C, not because it has nuclear fusion going on, but because it
absorbs the energy from the center and then sends it on to us. A good
animation showing this can be found at:
http://zebu.uoregon.edu/textbook/inv.html
The X-rays we see from the sun come from the Corona, a region of VERY
hot (about 1,000,000 degrees) gas above the surface of the sun, but
which we can only see during a total eclipse. (There will be a total
Eclipse visible in Turkey this summer.)
The only thing that we can detect that comes directly from the center
of the sun are neutrinos. These small particles are so non-reactive
that over 99.9% of them go right through the earth without touching
(affecting) anything. The small fraction that we do detect gives us
information on the nuclear processes in the center of the sun.
More mathematical details are below:
Here is the full equation which equates the pressure of the gas to the
volume it takes up:
dP/dr = -rho GM(r)/r^2
P is the pressure at radius r, rho is the density, usually denoted
with the Greek letter. G is the gravitational constant. M(r) is the
mass INTERIOR to radius r, since this is the material which pulls down
on a particle at radius r.
If we calculate pressure using the simple estimate
P = nkT
where n is the number density particles, k is the Stefan-Boltzmann
constant, and T is the temperature in degrees Kelvin, we get the
relation:
T_c ~ (G M_sun / R_sun) (m_p/k)
Here k is the Stefan-Boltzmann constant and m_p is a proton mass,
M_sun and R_sun are the solar mass and radius.
This gives T_c ~ (6.67e-8*2e33/7e10) (1.67e-24/1.38e-16) ~ 2.3e7 K
Which is actually quite close to more accurately derived numbers.
Just knowing the solar radius and Mass and that the sun is supported
in hydrostatic equilibrium basically sets the interior temperature.
More detailed modeling uses more complex and accurate microphysics and
fits the observed radius, mass and solar luminosity to derive the
run of temperature in the sun.
Hoscakal
Mike Arida, Tod Strohmayer and Andy Ptak
for "Ask a High-Energy Astronomer"
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