# Astronomy 217

## Prof. Andrew W. Steiner

Oct. 15, 2021

TA James Ternullo

## Last Time

• Solar Angular Momentum
• Properties of the Corona and Chromosphere
• Solar Wind
• Magnetic Braking

## Today

• Models of the Solar Interior

## The Solar Interior

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

## Numerical Modeling

• Since 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.

## Hydrostatic Equilibrium

• The most important physical concept included in solar models in hydrostatic equilibrium.
• Numerically this takes the form $$\frac{dP}{dr} = - \frac{G M(r) \rho(r)}{r^2}$$
• Physically, this says the inward gravitational force must be balanced by the outward pressure.
• We can use this approximation if all changes are gradual.

## Standard Solar Model

• In addition to hydrostatic equilibrium there are 3 other vital physical concepts.
• Mass continuity: $$\frac{dM}{dr} = 4 \pi r^2 \rho(r)$$
• Energy generation: $$\frac{dL}{dr} = 4 \pi r^2 \rho(r) \varepsilon(r)$$
• Energy transport: $$\frac{dT}{dr} = \ldots$$ The form of transport depends on the star and its composition
• Plus boundary conditions

## Convective Transport

• 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 Confirmation

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

## Helioseismology

• 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.

## Energy Budget

• Given the Sun’s mass, $M_{\odot} \approx 2 \times 10^{30}~\mathrm{kg}$ and energy production $L_{\odot} = 4 \times 10^{26}~\mathrm{W}$ 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 $3 \times 10^{13}~\mathrm{J}/\mathrm{kg}$ Chemical sources can only provide about $10^{6-7}~\mathrm{J}/\mathrm{kg}$
• Gravitational contraction has released the current gravitational binding energy over the Sun’s lifetime. $$U = -G \int_0^{R} \frac{m(r) 4 \pi r^2 \rho}{r}~dr \Rightarrow \frac{U}{M} = - \frac{3 M G}{5 R} \approx 10^{11}~\mathrm{J}/\mathrm{kg}$$
• Gravitational contraction (the Kelvin–Helmholtz mechanism) can only power the Sun for $\approx 10^7$ years.

## Nuclear Fusion

• 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: $$\mathrm{nucleus}~1 + \mathrm{nucleus}~2 \rightarrow \mathrm{nucleus}~3 + \mathrm{energy}$$
• But where does the energy come from? Nuclear mass!

## Nuclear Energy

• The relationship between mass and energy comes from Einstein’s famous equation, $E = mc^2$, where $c$ is the speed of light
• A small amount of mass is the equivalent of a large amount of energy. 1 kg becomes about $10^{17}~\mathrm{J}$.
• With temperatures in the center of the Sun of $1.4 \times 10^7$ K, the mean particle energy is about 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 Structure

• Stars are powered by nuclear fusion, building larger, more tightly bound from nuclei smaller nuclei
• Terrestrial nuclear power comes from nuclear fission, breaking apart heavy nuclei.
• Nuclear physics prefers certain configurations of neutrons and protons, which are reflected in binding energy.
• Pairing and shells are prime examples.

## Proton-Proton Chain

• The Sun is powered by the conversion of 4 $^{1}~\mathrm{H}$ into 1 $^{4}~\mathrm{He}$
• There are several reaction sequences which “burn” H, but in the Sun the proton-proton chain dominates.

## Four Fundamental Forces

• Nuclear reactions in the Sun call on 2 more forces.
• Strong nuclear force is responsible for binding nuclei together. It is short range ($10^{−15}~\mathrm{m}$), but the strongest.
• Weak nuclear force is responsible for beta decay, converting protons to neutrons or vice versa. It is shorter range ($10^{−18}~\mathrm{m}$) and $10^{13}$ 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.

## Solar Neutrinos

• 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, $p \rightarrow n + e^{+} + \nu_e$ .
• 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 $6 \times 10^{14}$ neutrinos per $\mathrm{m}^2~\mathrm{s}^{-1}$, 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 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.

## Neutrino Spectra

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

## Neutrino Oscillations

• 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.

## Energy Release

• 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 = $6.6943 \times 10^{−27}$ kg
• Mass of helium nucleus = $6.6466 \times 10^{−27}$ kg
• Mass transformed to energy = $0.0477 \times 10^{−27}$ kg
• Energy release per $^{4}\mathrm{He}$ produced = $4.3 \times 10^{–12}~\mathrm{J}$.
• This translates to $6.4 \times 10^{14}$ 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.

## Solar Composition

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

## Dwarfs and Giants

• Hertzprung-Russell Diagram plots Brightness verses Temperature.
• Luminosity is proportional to Temperature and Radius ($L \propto T^4 R^2$)
• Radius increases to the upper right, so Giants are at the top, dwarfs at the bottom.

## The Future of the Sun

• 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.