The Standard Solar Model: Its Content and Meaning

Adam L. Bruce

Abstract

In the 20th century a physical model for the behavior of our sun was developed in order to understand

and better predict its actions, specifically those if its core. Over time it has proved to be one of the most

complete and successful theoretical constructs in modern astronomy. The basic assumptions of the model

are discussed as well as the SSM predictions for composition, energy production and other physical

quantities. Challenges to the model are also discussed as well as their resolutions.

1. Introduction

The Sun is the most prominent body in our sky; its light not only reaches us at the distance of 1 AU,

but also provides enough energy to support nearly all organisms on our planet’s surface, as well as

blocking out all other heavenly bodies that can be seen from earth when it’s present in the sky (with the

exception of the Moon at various phases). For a long portion of our history, scientific knowledge about

the Sun was elusive and disregarded in favor of mysticism and theological beliefs, but with the advent of

modern scientific and empirical studies, certain facts about the Sun were ascertained. The Standard

Solar Model (SSM) was developed in the early 20th century as a theoretical system conceived to

describe our sun’s behavior by means of a scientific and physical model after those of particle

physics (SM) and cosmology (SCM).1 The SSM is constantly evolving due to revisions that take

into account more subtle aspects of certain physical elements, but has never been completely

refuted on experimental grounds. There have been challenges and problems faced by the SSM,

but in all circumstances the evidence that contests its validity has been shown to come from

sources outside the model itself.

2. Assumptions

There are four underlying assumptions which are made use of in the SSM. The first is that the sun

develops and evolves in hydrostatic equilibrium. The principle of hydrostatic equilibrium implies that

there exists a localized gravitational force balanced by a pressure gradient, which in turn creates pressure

gradient force in the opposite direction. This causes the gaseous layers of the Sun to remain in hydrostatic

balance when the sum of these forces is zero.

This can all be expressed as the simple differential:

���� = − ���

�� (1)

Where P is pressure, ρ is density and m is the mass which is contained in any given radius r.4

Equation (1) can be simplified even further if we rewrite m/r2 as a derivative of height, that is to say ∆ ��� =

∆ℎ = dh, thus giving us:

� = −�ρ �ℎ (1a)

It is also to be noted that from this we can derive a direct relation between pressure and the Solar

Radius (height) by integrating both sides such as:2

� � = − � �ρ �ℎ → = −�ρ � �ℎ → = −�ρℎ The existence of this proportion in particular becomes necessary for the SSM as it satisfies the

requirement of being able to directly calculate a value for P given any solar radius, which is needed for

various theoretical predications.

The problem such as it stands now suffers from a lack of definition as it is described in (1). To

describe in detail the hydrostatic equilibrium as it exists, or rather is assumed to exist, in the Sun we must

specify the temperature, density, and composition of its material. This is known as the Equation of State

and can be related by the simple law for an ideal gas:

Pp = ��ℛ

μ (2)

Where T is temperature, µ is the mean molecular weight and ℛ is the ideal gas constant.5 In more

recent advancements of the SSM, (2) has been modified to include other more subtle physical forces such

as the Debye-Hückel correction, which takes into account the coulomb-interactions of solar particles (see

3.1), done by a modification to the “perfect” pressure (Pp) assumed in (2) of the form:

Pg = Pp�1 − ���

� (��)!/�#�/� �!/�

��/� $%�/&' (2a)

Where Pg is the gas pressure, N0 is Avogadro’s number, e is the charge of an electron (not the physical

constant), k is the Boltzmann constant and ζ is given by the summation ∑ �(*+& + *+) -./.'+ where Zi, Xi, Ai

are the charge, mass number and relative abundance of an element i respectively.8

The second assumption of the SSM is that energy in the Sun can be transferred though any one of

the processes of radiation, conduction, convection or neutrino losses.4,5 Each of these accounts for some of

the solar energy produced, that being said however, radiation and convection are the dominant forms of

energy transport (the conditions which cause convection to occur are outlined in 3.3). The measure of the

energy flow through any given shell is shown by the temperature gradient produced by each process, thus

they can be modeled by:

���� = − �0�1

23�45���� (3)

For radiative transport, and

���� = 61 − 2

7 8 6��8 ��

�� (4)

For conduction transport, where κ is the opacity of the stellar material, L is its luminosity and γ is the

ratio given for the specific heats Cp/CV .1

The measure of the energy balance of any shell in the Sun, known as the Thermal Equilibrium10, can

be obtained as a derivative of the shell’s total luminosity equated to the sum of the energy entering the

bottom of the shell and the energy produced by nuclear reactions within the shell, such as:

�1�� = 4πr2ρ6 �<

�=∙�� + ? �@�=8 (5)

Where the term �<

�=∙�� is the nuclear energy generation rate per unit mass, coupled with ? �@�=

known as the “entropy term”, which describes stellar evolution on a time-explicit model. Dropping the

entropy term would only describe a static, non-evolving, time-exclusive model.9 It should also be noted

that the Sun’s luminosity accounts for the amount of energy per second flowing outward from any given

solar shell.9 This makes the conceptual model, mathematically shown in (5), to be one of equating the

change in solar energies exclusively.

The third assumption of the model is that the thermonuclear reactions which take place in the

Sun’s core are its only source of energy. This is required to correctly model the nature of the nuclear

fusion which takes place at the core, and therefore is necessary to the SSM as a whole, since the behavior

of the Sun’s core is the main reason for establishing such a model in the first place. The reaction which

takes place fuses hydrogen nuclei into helium nuclei.1 This is the most prominent at the core since

temperatures can reach > 16 MK, which is more suited to fusion reactions taking place that the

considerably lower temperature on the core.1

The final assumption is that the sun was initially a homogenous structure in its primordial stages.

Physical evidence for this claim is shown by a measure of the initial “heavy” elements present in the

stellar core, since in the reactions which take place in a star such as our sun these elements are neither

created nor destroyed.1 Our best chemical model of stellar evolution relies, for that reason, on the

abundance of hydrogen versus helium in the Sun, both of which are destroyed and produced respectively

in the most common solar reaction (pp-I).

3. Theoretical Predictions of the SSM

From these assumptions we can build a solar theory contained in mathematical and chemical models

that describe and produce numerical values for various physical characteristics such as mass, luminosity,

pressure, and density for any given solar radius. These essential quantities can then be combined and

juxtaposed in various ways so as to describe the physical processes involved in creating the conditions

inside the star.

3.1 Solar Composition

Given the assumptions in section two, we should be able to describe the solar composition by a

simple algebraic manipulation to (2), solving for µ. In particular this shows the overall change in the solar

composition throughout the star, which gives a decline in µ in as r increases. This decline occurs as a

result of the reactions which fuse hydrogen atoms to make helium-4 atoms.10 Since this constitutes

evidence that the Sun should not have a completely homogeneous structure, it also shows the evolution of

it since the primordial times where it would have be homogeneous as noted by the fourth assumption of

the model.

The evolution of the Sun in terms of its chemical composition can also be attributed to the

hydrogen-helium fusion which takes place at its core. The fusion occurs by a general reaction of the form:

4p+ → 4He + 2e+ + 2ve (6)

Where the elemental superscript indicates an isotope of helium, and where two positrons (e+) and

two electron-neutrinos (ve) are produced in accordance with the principle of Lepton conservation.10 This is

accomplished through one of two processes: the CNO (Carbon-Nitrogen-Oxygen) Chain (CNO) and the

proton-proton chain (pp). A third process, the Triple Alpha Chain, can also facilitate the reaction in (6),

but has only been observed to take place in older stars which have accumulated a large amount of helium

from the activity of either the CNO or pp chain.3 The dominant process for (6) to take place by in our star

is a branch of the pp chain called the pp-I chain, modeled chemically by the sequence:

p+ + p+ → &2 A + e+ + ve + 0.42MeV

&2 A + p+ → +�& BC + D EE + 5.49MeV �& BC + �& BC → 2p+ + F& BC + 12.86 MeV (7a-c)

It is responsible for roughly 84.6% of the solar energy generated from our sun, where &2 A is a

deuterium atom, �& BC and F& BC are the isotopes He-3 and He-4 respectively, and p+, e+, ve, D EE are

protons (p+), positrons (e+), electron-neutrinos (ve), and gamma-rays (D EE).1 pp-I is dominant on the

interval (10, 14) MK and is the coolest solar fusion reaction known, since the temperatures on the interval

(0, 10] MK are too low for solar fusion to take place. Another 13.8% of the solar energy is generated by

the pp-II branch, which can occur if:

�& BC + F& BC → GF HC + D EE (8)

And is dominant on the interval [14, 23) MK.12 The CNO chain produces the other 1.6% of the solar

energy output but is not dominant until (20, 25) MK, thus it is a secondary process in relatively temperate

star such as our Sun.1 Over time these reactions will noticeably alter the solar composition from a

hydrogen based core to a helium-4 based core. At this point there are speculations that the triple alpha

chain can take effect, but there are uncertainties.10

3.2 Solar Energy Producing Regions

It was noted in the foregoing section that the thermonuclear reactions which power the Sun can

only exist at its core, since it’s only there that the high temperatures and pressures which are required by a

fusion reaction are known to exist. Therefore, since the assumption has been made that these reactions are

the only source of fuel for a star (assumption 3), any attempt to describe the solar energy productions in

physical and observable terms must start from there.

Since we cannot directly observe the core of the sun we are only left with equating it to some

observable quantity. This can be accomplished by taking a measure of the solar luminosity created by the

pp and (to a lesser extent) the CNO chains together. This is modeled by:

�1�� = 4JK&ρε (9)

Where ε is the energy production rate.1 This is a simplified version of (5), where only the behavior of

a specific shell is described in a static model. It can be inverted to solve for ε, and then successive values

taken to the point where ε depreciates significantly, thus showing the regions of greatest energy

production. By this method it has been found that the drop in ε occurs at M ��N��5+4=+OP ~ 0.21 M⊙,1 where

the first term is the point of depreciation and the latter is the total radius of the Sun. This is approximately

the area of the core, meaning that our initial thoughts were correct.

3.3 Convection in Solar Shells

Convection occurs in a physical body, such as the sun, when local perturbations to a discrete

region causes the region’s density to become noticeably different than those of the regions surrounding it,

thus making it unstable. This produces a buoyant force which acts upon the region and causes it to rise

and fall in a large bulk motion of solar material. This motion is important because it carries energy and

mixes the stellar material between various solar shells (while preserving hydrostatic equilibrium among

them, as noted in assumption 1).1

Given the assumption that any given region of the sun changes adiabatically and that it receives

most of its energy from the outside area by a radiative processes (assumption 2), we can create a

condition for the stability of the region:

V�4� < V4�X (10)

Where the temperature gradients are defined with: V�4� = �Y1�23�45���Z and V4�X = V��[+OP = �\]�

�\]� . This

is called the Schwarzchild Stability Criterion. When the stability criterion is violated, the region itself

becomes instable and the process of convection begins in the solar shell. From this model, Convection

reaches its highest stage of efficiency at the point where the adiabatic temperature is roughly equivalent to

the total gradient.1 Once these gradients are calculated they show where convection is most active in the

star, this occurs in the outer shells of the sun, with radiation being more prevalent near the core.

Convection is seen to be at a maximum at approximately 0.7 R⊙.

4. The Solar Neutrino Problem

The only significant challenge to the SSM came in the late 1960s and was left unsolved for almost 30

years. The so called “Solar Neutrino Problem” (SNP) came into focus when Raymond Davis Jr. (1914-

2006) performed an experiment in the Homestake Gold Mine using large tanks of perchloroethelene,

chosen because of its high amount of chlorine, to facilitate a decomposition reaction between the

neutrinos and the chlorine such as:

37Cl + ve → 37Ar + 5.0Mev (11)

37Ar being a radioisotope of argon. Thus enabling Davis to measure the amount of electron neutrinos

which had passed through the tank. When the final measurements were taken, it was shown that the

measurement was roughly 1/3 the amount of the solar neutrinos predicted.7 This was later confirmed by

other experiments such as SAGE in the former USSR, GALLEX in Italy, and Kamiokande and Super

Kamiokande in Japan. This suggested that the calculation of the neutrino flux by John N. Bacall (1935-

2005) was incorrect, but when it was checked no errors were found.10 The SNP became a focal point of

solar research until 2002.

The now accepted solution to the SNP was first suggested conceptually in 1957 by Bruno Potecorvo

(1913-1993), 11 years before the deficit was first observed at Homestake, and was first mathematically

worked out by Portecorvo in 1967.11 It suggests that a neutrino released from the sun can change flavor

(electron, lepton or tau) on its journey to earth based on the probability of a mismatch between the flavor

and mass eigenstates. The flavor of a neutrino is made up of two mass eigenstates, which in turn are both

made up of two flavor eigenstates, all of which are sinusoidal functions. The relationship between the

various neutrino eigenstates is given by the nonlinear system:

|_`ab_ = c da+∗+

_|`+ _b

|_`+b_ = c da++

4_|`a _b (12)

Where |_`4b_ is a neutrino of a given flavor (e, µ, τ), |_`+b_ is a neutrino of a given mass (1, 2, 3), * is a

complex conjugate to Uai, a non-identity PMNS matrix given by:7

U = fdC1 dC2 dC3d$1 d$2 d$3dh1 dh2 dh3i = fC+42/& 0 00 C+4&/& 00 0 1i

The propagation of the various eigenstates through space is similar to that of a wave. In particular

the propagation of the mass eigenstates, |_`+b_, can be described as a simple planewave solution of the

form:

|_`+(j)b_ = Ck+(<.=kNlmmmn olmmmn )|_`+(0)b_ (13)

Where p+ is the energy of the eigenstate i, t is the time from the start of the propagation, qrmmmn is the

three-space momentum, and srmmmn is the position relative to the starting place.7 p+ is effected by the

untrarelativistic limit, since the particles are traveling close to the speed of light. Therefore, the limit

being |qrmmmn| = q+ ≫ u+, we can find an approximation of p+ such as p+ ≅ q+ + �.�&N. .7 The limit applies to

all neutrinos since the masses of the particles are all less than 1 eV and their energies are all greater than 1

MeV, making the Lorenz factor > 106 in all cases. Using the approximation for p+ as well as letting t = L,

where L equals the distance traversed by the particles (because of the closeness to the speed of light) and

finally dropping the unnecessary phase factors, we obtain:

|_`+(w)b_ = Ck.x.�y�z. |_`+(0)b_ (14)

Which is much easier to find solutions for than (13).7

With these forces in effect, it becomes apparent that eigenstates with different masses tend to

propagate at different speeds. The heavier ones lag behind while the lighter ones accelerate ahead. This

change in the mass eigenstates causes interference between the flavor eigenstates. This is because there is

a significant phase shift in their sinusoids, produced by the mass eigenstates, thus causing the linear

combination of a second eigenstate of a different flavor, contained in the mass eigenstate, to become a

non-zero value, while at the same time decreasing the value of the linear combination of the original

flavor. This causes the neutrino to appear a different flavor than its original. The probability of this

occurrence is represented by:

{ a→| = {}_`|{_`a(j)b__{& = ~c da+∗ d|++

Ck+�.�1&N. ~&

(15) Where α and β are the original and oscillated flavors respectively. A more convenient form of this is:

a→| = �a| − 4 c ℜ(

+��da+∗ d|+da�d|+∗ ) sin& �∆u+�& w

4q+ � + 2 c ℑ(+��

da+∗ d|+da�d|+∗ ) sin �∆u+�& w2q+ � (16)

Where ∆u+�& is defined as u+& − u�&. This rewrites (15) with the sinusoidal functions representing the

eigenstates evaluated at the parameters of oscillation. Apart from being more convenient it is also much

more direct.7

All of these equations counted c and ћ as natural units, that is to say as having the value of 1. The

resolution of the phase is often written with them restored as:7

∆��5�1

FћN. = �����

FћN. ∙ ∆����� 1

�����

N. ≈ 1.267 ∆����� 1

�����

N. (17)

Where the 1.267 is unitless.

The first experiment to detect every flavor of neutrino was the SNO Laboratory in Ontario

Canada. When the measurements were made the amount of neutrinos of any flavor detected were in

agreement with Bacall’s original calculations.6

5. Closing

It has been shown how the SSM is one of the most successful theories of modern astronomy. The four

assumptions it draws on are both reasonable and have yet to be shown incorrect. The predictions

themselves have for the most part been unquestioned, and in the case where the model was brought into

question, the source of the issue was not found within the model itself, but in the nature of what it was

measuring. The equations for the model accurately describe the behavior of all the portions of the sun,

both what we can observe directly and indirectly, and it is expected that this should continue to be true,

since only a fundamental change in the laws of physics as we know them could skew our methods. There

is always this possibility in physics, but for now it is a great improbability.

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