The Solar Interior

Now that we have studied the physics of stellar interiors, we are ready to concentrate on the star we can study in the most detail, our Sun.  From our discussion of the H-R diagram, we see that the Sun is a completely ordinary star of spectral type G2 V, residing quite near the middle of the main sequence, so much of what we learn in studying the Sun will apply to stars in general, especially dwarf stars (luminosity class V).

Using scaling arguments, we have seen that the average density, core pressure and core temperature are as given in the table below.  These are contrasted with the more exact values determined from computer models of the Sun's interior (the so-called standard model).  We also show the mass fraction of hydrogen (X) and of helium (Y) at the surface and at the core.
 

From Scaling Arguments		From Standard Model
Core Temperature	1.4 x 107 K	Central Temperature	1.58 x 107 K
Core Pressure	2.7 x 1014 N m-2	Central Pressure	2.50 x 1016 N m-2
Average Density	1410 kg m-3	Central Density	1.62 x 105 kg m-3
X at surface	0.73	X at core	0.336
Y at surface	0.26	Y at core	0.643

The Sun is about 4.52 billion years old, and has been converting hydrogen to helium in its core all that time.  More than half of the hydrogen has been used up already, in the core, so the Sun is about middle-aged.  The solar model can be traced in time from an initial composition through to its present age.  As the composition changes due to using up the hydrogen, the other parameters of the Sun such as its radius, temperature, and luminosity must change as well.  According to the figure below, the solar radius has increased by about 12% over its lifetime, the surface temperature has increased by about 20%, and the luminosity (4pR²sT⁴) has increased by 30%.  However, this implies about a 5% increase in the last 500 million years, and it is not clear that such an increase is compatible with known Earth climate history.


Theoretical understanding of how the radius, temperature, and luminosity of the Sun has
changed over the lifetime of the Sun.  The changes may be more extreme than actually
occurred, given what we know of Earth's climate history.  Figure from Carroll and Ostlie,
Introduction to Modern Astrophysics, Addison-Wesley, New York, 1996.

Computer models such as the standard model must fit the observable constraints that we have, e.g. the mass, radius, surface temperature, luminosity, and composition.  We have some other constraints such as the age of the solar system (4.5 billion years), but we would like to have other information to be sure that the model is correct.  Currently, we have two additional methods, both pioneered in the 1970's, that actually let us "see" inside the Sun.  These are

• solar neutrino detections
• helioseismology

One of these, helioseismology, agrees spectacularly with the standard model and can give us information about the interior rotation of the Sun.  The other, solar neutrino measurements, disagrees rather spectacularly!  Recall our reactions that make up the proton-proton chain:

 

PP I Chain (69%)
	11H +		11H =>		21H + e+		+ ne
	21H +		11H =>		32He + g
	32He +		32He =>		42He +	2	11H

 

PP II Chain (31%)
	32He +		42He =>		74Be + g
	74Be +		e- =>		73Li	+ ne
	73Li +		11H =>		2 42He

 

PP III Chain (0.3%)
	74Be +		11H =>		85B + g		+ ne
			85B =>		84Be + e+		+ ne
			84Be =>		2 42He

The electron neutrinos in each reaction are highlighted.  Neutrinos are rather mysterious particles, nearly or completely massless and very unreactive.  They pass through the Sun as if it were not there, so while photons reach us from the core only after 1 million years of tortuous meandering, neutrinos come directly to us.  However, if traveling 350,000 km through a star cannot stop them, how can we detect them?  The neutrino in the yellow cell, above, can be captured by a Chlorine-37 nucleus in the following reaction:
 

Neutrino Capture (Homestake)
3717Cl		+ ne	=>		3718Ar + e-

In the Homestake Gold Mine in South Dakota, a tank of 100,000 gallons of cleaning fluid (C₂Cl₄) acts as a neutrino detector, in a pioneering experiment by Raymond Davis.  Every several months the tank is purged to detect the Ar atoms produced in the reaction.  From the standard solar model, it is expected that only slightly more than 1 atom per day will be produced among the 10³⁰ Chlorine atoms in the tank!  In fact, Davis only finds less than 1 every two or three days.  Other experiments have been set up to detect neutrinos, including the far more numerous neutrinos produced in the PP-I chain (purple cell, above).  Results of all of the experiments indicate that the number of neutrinos detected is not consistent with the standard model.

Helioseismology is the study of "seismic" pressure and gravity waves inside the Sun.  Just as we can study earthquakes to learn about the Earth's interior, we can also study "sunquakes" to learn about the Sun's interior.  Sound waves are generated randomly, probably mostly from small-scale convective turbulence near the surface of the Sun.  They rattle around inside the Sun, but those modes that are at special wavelengths to match spherical harmonics of the Sun can live a long time and resonate inside the Sun.  The wavelength-frequency (k-w) relationship of these resonances tell us the structure of the inside of the Sun.  The results of helioseismology agree very well with the standard model.

So what can we say about the failure of the neutrino results?  Perhaps the problem is due to neutrino physics itself, which is poorly understood.  One possibility is that neutrinos oscillate between two forms, one of which we can detect and the other of which is not detectable.  The question is still unanswered at this time, but if the neutrino can be shown to have mass (it is known to be less than 7.2 eV so far) then oscillations can be calculated and compared with observations.