74 Encyclopedia of the Solar System
(helioseismology); and (3) by the neutrino flux, which now
constrains for the first time elemental abundances in the
solar interior, since the neutrino problem has been solved
in the year 2001.
2.1 Standard Models
There are two types of models of the solar interior: (1) hy-
drostatic equilibrium models and (2) time-dependent nu-
merical simulations of the evolution of the Sun, starting
from an initial gas cloud to its present state today, after
∼8% of the hydrogen has been burned into helium.
The standard hydrostatic model essentially calculates
the radial run of temperature, pressure, and density that
fulfill the conservation of mass, momentum, and energy in
all internal spherical layers of the Sun, constrained by the
boundary conditions of radius, temperature, and radiation
output (luminosity) at the solar surface, the total mass, and
the chemical composition. Furthermore, the ideal gas law
and thermal equilibrium is assumed, and thus the radia-
tion is close to that of an ideal black body. The solar radius
has been measured by triangulation inside the solar system
(e.g., during a Venus transit) and by radar echo measure-
ments. The mass of the Sun has been deduced from the
orbital motions of the planets (Kepler’s laws) and from pre-
cise laboratory measurements of the gravitational constant.
The solar luminosity is measured by the heat flux received
at Earth. From these standard models, a central tempera-
ture of∼15 million K, a central density of∼150 g cm−^3 ,
and a central pressure of 2.3× 1017 dyne cm−^2 have been
inferred. Fine-tuning of the standard model is obtained by
including convective transport and by varying the (inaccu-
rately known) helium abundance.
2.2 Thermonuclear Energy Source
The source of solar energy was understood in the
1920s, when Hans Bethe, George Gamow, and Carl Von
Weizs ̈acker identified the relevant nuclear chain reactions
that generate solar energy. The main nuclear reaction is the
transformation of hydrogen into helium, where 0.7% of the
mass is converted into radiation (according to Einstein’s en-
ergy equivalence,E=mc^2 ), the so-calledp-pchain, which
starts with the fusion of two protons into a nucleus of deu-
terium (^2 He), and, after chain reactions involving^3 He,^7 Be,
and^7 Li, produces helium (^4 He),
p+p→^2 He+e++νe(^2) He+p→ (^3) He+γ
(^3) He+ (^3) He→ (^4) He+p+p
or
(^3) He+ (^4) He→ (^7) Be+γ
(^7) Be+e−→ (^7) Li+νe
(^7) Li+p→ (^8) Be+γ→ (^4) He+ (^4) He
One can estimate the Sun’s lifetime by dividing the avail-
able mass energy by the luminosity,
t◦≈ 0. 1 × 0. 007 m◦c^2 /L◦≈ 1010 years
where we assumed that only about a fraction of 0.1 of
the total solar mass is transformed because only the inner-
most core of the Sun is sufficiently hot to sustain nuclear
reactions.
An alternative nuclear chain reaction occurring in the
Sun and stars is the carbon–nitrogen–oxygen (CNO) cycle,
(^12) C+p→ (^13) N+γ
(^13) N→ (^13) C+e++νe
(^13) C+p→ (^14) N+γ
(^14) N+p→ (^15) O+γ
(^15) O→ (^15) N+e++νe
(^15) N+p→ (^12) C+ (^4) He
Thep-pchain produces 98.5% of the solar energy, and
the CNO cycle produces the remainder, but the CNO cycle
is faster in stars that are more massive than the Sun.
2.3 Neutrinos
Neutrinos interact very little with matter, unlike photons,
and thus most of the electronic neutrinos (νe), emitted by
the fusion of hydrogen to helium in the central core, escape
the Sun without interactions and a very small amount is de-
tected at Earth. Solar neutrinos have been detected since
1967, pioneered by Raymond Davis, Jr., using a chlorine
tank in the Homestake Gold Mine in South Dakota, but
the observed count rate was about a third of the theoreti-
cally expected value, causing the puzzling neutrino problem
that persisted for the next 35 years. However, Pontecorvo
and Gribov predicted already in 1969 that low-energy so-
lar neutrinos undergo a “personality disorder” on their
travel to Earth and oscillate into other atomic flavors of
muonic neutrinos (νμ) (from a process involving a muon
particle) and tauonic neutrinos (νt) (from a process in-
volving a tauon particle), which turned out to be the so-
lution of the missing neutrino problem for detectors that
are only sensitive to the highest-energy (electronic) neu-
trinos, such as the Homestake chlorine tank and the gal-
lium detectors GALLEX in Italy and SAGE in Russia. Only
the Kamiokande and Super-Kamiokande-I pure-water ex-
periments and the Sudbury Neutrino Observatory (SNO;
Ontario, Canada) heavy-water experiments are somewhat
sensitive to the muonic and tauonic neutrinos. It was the
SNO that measured in 2001 for the first time all three lep-
ton flavors and, in this way, brilliantly confirmed the theory
of neutrino (flavor) oscillations. Today, after the successful
solution of the neutrino problem, the measured neutrino
fluxes are sufficiently accurate to constrain the helium abun-
dance and heavy element abundances in the solar interior.