Internal Structure

The Sun is a typical main sequence star. Its principal properties are:

mass

m = Mq

= 1.989 x 1030 kg

radius

R = Rq

= 6.960 x 108 m

mean density

p

= 1409 kg/m3

central density

Pc

= 1.6 x 105 kg/m3

luminosity

L = Lq

= 3.9 x 1026W

effective temperature

Te

= 5785K

central temperature

Tc

= 1.5 x 107 K

absolute bolometric magnitude

Mboi

= 4.72

absolute visual magnitude

MV

= 4.79

spectral class

G2V

colour indices

B - V

= 0.62

U - B

= 0.10

surface chemical composition

X

= 0.71

Y

= 0.27

Z

= 0.02

rotational period

at the equator

25 d

at latitude 60o

29 d

On the basis of these data, the solar model shown in Fig. 12.1 has been calculated. The energy is produced by the pp chain in a small central region. 99% of the solar energy is produced within a quarter of the solar radius.

The Sun produces energy at the rate of 4 x 1026 W, which is equivalent to changing about four million tonnes of mass into energy every second. The mass of the Sun is so large, about 330,000 times that of the Earth, that during the whole main sequence lifetime of the Sun less than 0.1% of its mass is turned into energy.

When the Sun formed about 5000 million years ago, its composition was the same everywhere as its present surface composition. Since energy production is concentrated at the very centre, hydrogen is consumed most

Fig. 12.1. The distribution of temperature, pressure, energy production and mass as functions of radius in the Sun

rapidly there. At about a quarter of the radius the hydrogen abundance is still the same as in the surface layers, but going inwards from that point it rapidly decreases. In the central core only 40% of the material is hydrogen. About 5% of the hydrogen in the Sun has been turned into helium.

The radiative central part of the Sun extends to about 70% of the radius. At that radius the temperature has dropped so much that the gas is no longer completely ionized. The opacity of the solar material then strongly increases, inhibiting the propagation of radiation. In consequence, convection becomes a more efficient means of energy transport. Thus the Sun has a convective envelope (Fig. 12.2).

In: Hannu Karttunen et al. (Eds.), Fundamental Astronomy, 5th Edition. pp. 263-277 (2007) DOI: 11685739_12 © Springer-Verlag Berlin Heidelberg 2007

Fig. 12.2. (a) The rotation rate of the Sun inferred from helio-seismological observations. The equator is at the horizontal axis and the pole is at the vertical axis, both axes being labelled by fractional radius. Some contours are labelled in nHz, and, for clarity, selected contours are shown as bold. (430 nHz is about 26.9 days.) The dashed circle is at the base of the convection zone and the tick marks at the edge of the outer circle are at latitudes 15°, 30°, 45°, 60°, 75°. The shaded area indicates the region in the Sun where no reliable inference can be made with present data. The slanted dotted lines are at an angle of 27° with the rotation axis. (Adapted from Schou et al. 1998). (J. Christensen-Dalsgaard 2007, astro-ph/0610942, Fig. 2) (b) The interior and surface of the Sun. The various kinds of solar phenomena are schematically indicated. (Based on Van Zandt, R.P. (1977): Astronomy for the Amateur, Planetary Astronomy, Vol. 1, 3rd ed. (published by the author, Peoria, III.))

The Solar Neutrino Problem. The central nuclear reactions produce neutrinos at several of the steps in the pp chain (see Fig. 10.5). These neutrinos can propagate freely through the outer layers, and thus give direct information about conditions near the centre of the Sun. When neutrinos from the Sun were first observed in the 1970's, their number was found to be only about a third of what was predicted. This disagreement is called the solar neutrino problem.

In the first experiments only neutrinos from the ppll and pplll branches were observed (Sect. 10.3). Since only a small fraction of the solar luminosity is produced in these reactions, it was not clear what were the consequences of these results for solar models. In the 1990's neutrinos produced in the ppI branch, the main branch of the pp chain, were observed. Although the disagreement with the standard models was slightly smaller in these observations (about 60% of the predicted flux was observed), the neutrino problem still remained.

Perhaps the most popular explanation for the solar neutrino problem was based on neutrino oscillations. According to this explanation, if neutrinos have a small mass (about 10-2 eV), an electron neutrino could change into a ^ or a t neutrino as it passed through the outer parts of the Sun. In the early experiments only electron neutrinos were observed, representing only part of the total number of neutrinos produced.

In 2001 results were announced from experiments in Canada and Japan that measured both the number of electron neutrinos and the total number of neutrinos arriving from the Sun. The total flux agreed with the predictions of the standard solar model, whereas the flux of electron neutrinos were in agreement with the lower values measured in the earlier experiments. This result proved the existence of neutrino oscillations turning some of the electron neutrinos produced in the centre of the sun into other kinds.

The solar neutrino problem can now be considered to be solved. The solution is a great success for the standard solar model. But it has also revealed the existence of neutrino oscillations, proving that neutrinos have a small but non-zero rest mass. This shows that the standard model of particle physics needs to be revised in some respects.

The Solar Rotation. As soon as telescopes were introduced, it was observed from the motions of sunspots

that the Sun is rotating with a rotational period of about 27 days. As early as 1630 Christoph Schemer showed that there was differential rotation: the rotational period near the poles was more than 30 days, while it was only 25 days at the equator. The rotational axis of the Sun is inclined at 7° with respect to the plane of the ecliptic, so that the North Pole of the Sun is best visible from the Earth in September.

The motions of sunspots still give the best information on the rotation near the surface of the Sun. Other surface features also have been used for this purpose. The rotational velocity has also been measured directly from the Doppler effect. The angular velocity is usually written

where f is the latitude with respect to the equator. The measured values of the coefficients are A = 14.5 and B = 2.9 degrees/day.

The rotational velocity deeper down in the Sun cannot be directly observed. In the 1980's a method to estimate the rotation in the interior became available, when it became possible to measure the frequencies of solar oscillations from the variations in spectral lines. These oscillations are essentially sound waves produced by turbulent gas motions in the convection zone. These sound waves have calculable oscillation periods (about 3-12 minutes), which depend on the conditions in the solar interior. By comparing the observed and theoretical values one can get information about the conditions deep inside the Sun. The idea of the method is the same as that used when studying the interior of the Earth by means of waves from earthquakes, and it is therefore called helioseismology.

Using helioseismology, models for the solar rotation throughout the convection zone have been deduced. It appears that the angular velocity in the whole convection zone is almost the same as at the surface, although it decreases slightly with radius near the equator, and increases near the poles. The angular velocity of the radiative core is still uncertain, but there are indications that the core is rotating as a solid body with approximately the average surface angular velocity. At the bottom of the convection zone there is a thin layer known as the tachocline, where the angular velocity changes rapidly with radius. The internal solar rotation

according to the helioseismological studies is shown in Fig. 12.2a.

The solar differential rotation is maintained by gas motions in the convection zone. Explaining the observed behaviour is a difficult problem that is not yet completely understood.

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