Astronomy 217I n s i d e t h e S u n

The story of the internal structure of the Sun is the story of energy transportation and energy generation.

But how to we know what’s happening at optical depth of millions or more?

The vast majority of our observations come from the surface, not the interior.

Examining the Interior

We have 2 clues from deep inside, helioseimology and neutrinos.

Numerical ModelingSince it’s very difficult to observe the Sun’s interior, we rely on numerical models, based on physical principles and tested against observation, to provide information about the Sun’s interior.

The equations of stellar structure are too complex to solve analytically in all but the simplest toy cases. Instead they must be solved by computer.

The result, the standard solar model is among the first triumphs of computational physics.

The most important physical concept included in solar models in Hydrostatic Equilibrium.

Numerically this takes the form

Physically, this says the inward gravitational force must be balanced by the outward pressure.

Hydrostatic Equilibrium

We can use this approximation if all changes are gradual.

The result is the standard solar model.

Standard Solar ModelIn addition to hydrostatic equilibrium there are 3 other vital physical concepts.

Energy GenerationMass Continuity

Energy Transport

Plus boundary conditions like M(0) = 0, L(0) = 0, & M(R☉) = M☉, L(R☉) = L☉, etc.

The form of this transport equation depends on the

dominant energy transport process.

The high temperature of the solar interior fully ionizes the gas, however the decline in Temperature with increasing radius allows atoms to exist in the outer regions.

This divides the solar interior into 2 zones, an inner radiative zone and an outer zone where atoms can exist, increasing the opacity, making convection the most rapid form of energy transport.

Convective Transport

Three-dimensional models of the convective zone can produce results quite similar to the observed granulation of the photosphere.

Convective Confirmation

Vögler, Shelyag, Schüssler … (2005)

These comparisons provide support for the solar model.

Additional support for the solar model comes from Helioseismology.

Doppler shifts of solar spectral lines indicate a complex pattern of vibrations on the surface of the Sun.

These result from standing sound waves in the solar interior.

Different sound waves probe different depths in the Sun.

As the speed of sound depends on the temperature, density and composition of matter, one can deduce these quantities as a function of radius in the Sun.

Helioseimology

Given the Sun’s mass (M☉ = 2 × 1030 kg) and energy production (L☉ = 4 × 1026 W), we find that an average kilogram of the sun produces about 0.2 milliwatts of energy.

This is not much energy, but Sun has been emitting energy at this rate through the billions of years the Earth has existed.

The total lifetime energy output is about 3 × 1013 J/kg.

Chemical sources can only provide ~ 106-7 J/kg.

Gravitational contraction has released the current gravitational binding energy over the Sun’s lifetime.

Gravitational contraction (the Kelvin–Helmholtz mechanism) can only power the Sun for ~ 107 years.

Energy Budget

~ 1011 J/kg⇒

The age estimates of Kelvin were at odds with geological evidence.

In 1904, Ernest Rutherford (1871-1937) proposed radioactivity as the source of solar energy.

Following Rutherford’s discovery of the atomic nucleus in 1909 (with Geiger & Marsden), and his pioneering of nuclear transmutation by fusion in 1919, Arthur Eddington (1882-1944) proposed nuclear fusion as the energy source of the Sun in 1920.

Nuclear fusion releases energy by building bigger nuclei

nucleus 1 + nucleus 2 → nucleus 3 + energy

But where does the energy come from?

Nuclear Fusion

Nuclear Mass!

The relationship between mass and energy comes from Einstein’s famous equation,

E = mc2, where c is the speed of light.

A small amount of mass is the equivalent of a large amount of energy. 1 kg becomes ~ 1017 J.

With temperatures in the center of the Sun of 1.4 × 107 K, the mean particle energy is ~ 1 keV which is insufficient for fusion, but particles in the Maxwell-Boltzmann tail do have enough energy to fuse.

The Sun can therefore tap into the mass-energy by building heavier nuclei, but the nuclei are slightly less massive than the nuclei which fuse to form them.

Nuclear Energy

Nuclear StructureStars are powered by nuclear fusion, building larger, more tightly bound from nuclei smaller nuclei.

Terrestrial nuclear power comes from nuclear fission, breaking apart very heavy nuclei.

Nuclear physics prefers certain configurations of neutrons and protons, which are reflected in binding energy.

Pairing and shells are prime examples.

Full Electron Shell

Proton-Proton Chain

The Sun is powered by the conversion of 4 1H into 1 4He

There are several reaction sequences which “burn” H, but in the Sun the proton-proton chain dominates.

ProtonNeutron

PositronNeutrinoγ-ray photon

1H

1H

1H

1H

1H

1H

1H

1H

2H

2H

3He

3He

4He

PP Chain

4 Fundamental ForcesNuclear reactions in the Sun call on 2 more forces.

Strong nuclear force is responsible for binding nuclei together. It is short range (10−15 m), but the strongest.

Weak nuclear force is responsible for beta decay, converting protons to neutrons or vice versa. It is shorter range (10−18 m) and 1013 weaker that the strong force.

In comparison,

Electromagnetic force is ~1% of the strength of the strong force but infinite in range. It can be attractive or repulsive depending on relative charge.

Gravitational force is very weak (10-38 of the strong force), but always attractive and infinite in range.

Several of the nuclear reactions in the Sun are β decays, involving the conversion of a proton into a neutron. These weak reactions result in the emission of positrons and neutrinos, p → n + e+ + ν.

The positrons rapidly annihilate with a nearby electron, but the neutrinos are stream from the core of the Sun and escape, interacting with virtually nothing.

The flux of solar neutrinos at the Earth is 6 × 1014 neutrinos m-2 s-1, 6 trillion through your hand each sec.

Being able to observe even a small fraction of these neutrinos would give us a direct picture of what is happening in the core of the Sun.

Solar Neutrinos

Solar Neutrino Observations

Observing neutrinos is very challenging because neutrinos are no more likely to interact with terrestrial detectors than they are in the Sun.

Huge detector volumes and the ability to observe single interaction events is required.

KamiokandeChlorineGallium

Neutrino SpectraMore important than imaging the Sun’s thermonuclear core is the information provided by the neutrino spectra.

Detection of solar neutrinos has been ongoing for more than 30 years now. Davis & Koshiba won the 2002 Nobel Prize for their detection.

However, there has always been a deficit in the number of electron neutrinos expected to be emitted by the Sun.

This Solar Neutrino problem was ultimately shown to be the result neutrino oscillations, which interchange the 3 flavors of neutrinos during their passage from the Sun’s core to the Earth, indicating neutrinos have mass. Discovery of oscillations won McDonald & Kajita to 2015 Nobel Prize.

With oscillations accounted for, we now measure the Sun’s neutrino emission to match what the standard solar model predicts.

Neutrino Oscillations

The binding energy released in the PP Chain can be calculated by the difference in mass between the initial particles and the final ones.

Mass of four protons = 6.6943 × 10−27 kg

Mass of helium nucleus = 6.6466 × 10−27 kg

Mass transformed to energy = 0.0477 × 10−27 kg

Energy release per 4He produced = 4.3 × 10–12 J.

This translates to 6.4 × 1014 J per kg of hydrogen, so the Sun converts 6.1 billion kg of H into He every second.

At this rate, it would take 100 billion years to convert 1 solar mass of H into He.

Energy Release

(about 0.71%)

Solar Composition

The conversion of H ⇒ He (and C & O to N) are restricted to the central 10-20% of the Sun.

0.0 0.2 0.4 0.6 0.8 1.010–4

10–3

10–2

10–1

100

R/R

Mass

Frac

tion

Bahcall et al 2005

1H4He3He12C14N16O

Settling of heavier elements

Of Dwarves & GiantsHertzprung-Russell Diagram plots Brightness verses Temperature.

Luminosity is proportional to Temperature and Radius (L ∝ T4R2)

Radius increases to the upper right, so Giants are at the top, Dwarves at the bottom.

Timmes/ASU

Stars like our sun burn hydrogen for billions of years.

When H is exhausted in core, hydrogen burning ignites in shell around the core.

Once hot enough, He burning begins in the core, until He is exhausted.

H & He burning shells around the C+O core drive off the envelope as a planetary nebulae, leaving a white dwarf.

The Future of The Sun

ATNF/CSIRO

Little Ghost Nebula with HST [B: OIII, G: HII, R: NII] (STSCI/NASA)

Envelope of star ejected into space

New white dwarf

Planetary Nebula

Next TimeThe Solar System.

Turn in Homework #7

Read Chapter 8.