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.

Planetary Nebula