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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://ircamera.as.arizona.edu/NatSci102/NatSci102/lectures/suninterior.htm

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 an Astrophysicist"

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