The Standard Solar Model

One of the most successful models in modern astrophysics, the standard solar model has four basic assumptions:

  1. The sun evolves in hydrostatic equilibrium.
  2. Energy is transferred in the Sun by radiative, convective, and neutrino transport.
  3. Thermonuclear fusion of hydrogen into helium is the sole energy source inside the Sun.
  4. The Sun was initially homogeneous.

Hydrostatic equilibrium implies a local balance between pressure and gravity and to describe this condition in detail, one must specify the temperature, density, and composition of the stellar material. This combination is known as an equation of state (the simplest of these equations of state is the familiar ideal gas law studied in introductory physical science and chemistry classes). By far, the dominant processes of energy transport inside the Sun are radiative and convective flow. The efficiency of each process at every point inside the Sun is determined by the local temperature gradient (how rapidly the temperature changes over distance), the ability of the stellar material to absorb energy (its opacity) and the ratio of specific heats.

In our sun, the fusion of hydrogen into helium via the proton-proton chain is the dominant process. Hydrogen fusion via the CNO (carbon-nitrogen-oxygen) cycle becomes the dominant energy generation process at core temperatures greater than about 20 million Kelvin.

The final assumption made about the Sun is that it was highly homogeneous at its arrival on the zero-age main sequence (the point at which hydrogen fusion was initiated). At that time, it is believed, the Sun was almost completely convective. The figure below depicts the values of the standard solar model:

The standard solar model

The standard method of plotting data is to use the fractional mass interior to a specific point on the x-axis and the percentage of the dependent value on the y-axis:

Solar model as mass fraction

Note that the luminosity (red circles) reaches 100% (indicated by a y-value of 1.00) at a fractional mass of approximately 070. This means that all of the Sun's energy is generated by a thermonuclear core that contains 70% of the Sun's mass. The curve describing the radius (purple squares) as a function of fractional mass, however, shows that 70% of the Sun's mass is within an area that occupies only the inner 30% (i.e. y-value = 0.30) of the Sun's radius!

It may be easier to visualize the dependent values when plotted against the fractional radius of the Sun, however. In the graph show below, the thermonuclear core is seen to extend over the inner 30% (0.30 Solar Radii) of the Sun's interior, the radiative zone extend up to 70% (0.70 Solar Radii) of the distance to the surface, and the convective zone represents the upper 30% of the solar interior (From 0.70 Solar Radii to the photosphere at 1.0 Solar Radii). It can also be seen that the core contains approximately 60% of the Sun's mass and only the hydrogen in the core is being depleted by the proton-proton cycle.

Standard solar model as radius fraction

The Proton-Proton Cycle

In our sun, the fusion of hydrogen into helium via the proton-proton chain is the dominant process. Hydrogen fusion via the CNO (carbon-nitrogen-oxygen) cycle becomes the dominant energy generation process at core temperatures greater than about 20 million Kelvin. Our sun's core is believed to be approximately 15.6 million K.

The Sun's initial chemical composition was believed to be 73% hydrogen, 25% helium and 2% metals. The core's current composition is thought to be 35% hydrogen, indicating a main sequence age of about 5 billion years.

The Thermonuclear Reaction

The proton-proton cycle

The nuclear reactions may be expressed as equations:

1H + 1H —> 2H + e+ + ν (must occur twice)

2H + 1H —> 3He + γ (must occur twice)

3He + 3He —> 4He + 1H + 1H

Dramatis Personae
1H = hydrogen nucleus (proton) sometimes written as p
2H = deuteron (isotope of hydrogen containing one proton and one neutron)
3He = isotope of helium with only a single neutron
4He = ordinary helium nucleus (the alpha particle)
e+ = anti-electron, or positron (sometimes written as β+)
ν = electron neutrino
γ = gamma ray (high energy electromagnetic radiation)

Post-Main Sequence Evolution