The Post-Main-Sequence Phase and Electron Degeneracy (Part 3)

When core hydrogen fusion slows down, a main-sequence star with mass greater than 0.4 M⊙M_\odot becomes a giant

This transformation occurs due to several processes:

Core Changes

  • Main Sequence Definition: Fusion of hydrogen into helium in a star’s core.

  • Helium Convection:

    • Stars with mass > 0.4 M⊙ do not transport helium out of their cores by convection.

    • This leads to chemical composition differences between their cores and outer layers throughout their lives.

  • Core Fusion Slows:

    • When core hydrogen is largely converted into helium, fusion in the core drastically slows down.

    • The core's temperature is not high enough to enable its helium to fuse into other elements.

Expansion and Hydrogen Shell Fusion

  • Loss of Support:

    • As the rate of core fusion decreases, a star can no longer support the crushing weight of its outer layers.

    • The hydrogen-rich gas just outside the core is compressed inward and heated.

  • Hydrogen Shell Fusion:

    • This compression causes the hydrogen to begin fusing into helium in a shell a few thousand kilometers thick surrounding the core.

    • The proton-proton chain is the major source of energy in this process.

  • Expansion Mechanism:

    • Photons created in shell fusion are generated closer to the star’s outer layers than core photons.

    • These shell photons have more energy to push the outer layers outward, causing significant expansion.

Physical Characteristics of Giants

  • Surface Cooling:

    • Expanding far from the fusing shell, the surface gases cool.

    • The temperature of the star’s bloated surface decreases between 3000 K and 6000 K, depending on the star’s total mass.

  • Increased Luminosity:

    • Despite cooler surfaces, a giant is more luminous because it has much more surface area, emitting more photons each second.

    • As a full-fledged giant, our Sun will shine 2000 times more brightly than it does today.

  • Solar Example:

    • In about 5 billion years, our Sun will have a helium core and swell into a giant with a radius of about 0.5 AU.

    • This will vaporize Mercury and perhaps cause Venus to spiral into the Sun, scorching Earth to a cinder.

Mass Loss in Giants

  • Continuous Gas Leakage:

    • The outer layers of giant stars constantly leak gases into space; this mass loss is often significant.

    • A typical mass loss rate for a giant is roughly 10−7M⊙10^{-7}M_\odot ​per year.

  • Spectroscopic Evidence:

    • When studied spectroscopically, the escaping gases exhibit narrow absorption lines.

    • Lines from gases coming towards us are slightly blueshifted due to the Doppler effect, corresponding to expansion velocities of around 10 km/s.

  • Comparison to Main-Sequence Stars:

    • For comparison, in a main-sequence star, such as the Sun, mass-loss rates are only around 10−14M⊙10^{-14}M_\odot per year.

Helium fusion begins at the center of a giant

  • Initial Core State:

    • When a star first becomes a giant, it’s hydrogen-fusing shell surrounds a compact core of almost pure helium, initially too cool for helium fusion.

    • Helium produced by the shell settles onto the core.

  • Evolution based on Stellar Mass:

    • For Low-Mass Giants (0.4M⊙0.4 M_\odot to 2M⊙2 M_\odot) - Helium Flash:

      • Electron Degeneracy:

        • Helium atoms in the core are squeezed into a crystal-like solid, ionizing into nuclei and electrons.

        • Electrons are crowded, influenced by the Pauli exclusion principle, forcing faster motion to avoid being identical.

        • This creates electron degeneracy pressure, supporting the core and preventing collapse.

      • Abnormal Pressure-Temperature Relationship:

        • Unlike ordinary gas, pressure in a degenerate core does not increase with temperature.

        • Without this 'safety valve', the core cannot expand and cool when overheated; it just gets hotter.

      • The Helium Flash:

        • Eventually, the core temperature reaches ~100 million K, initiating core helium fusion.

        • Due to degeneracy, the core doesn't expand, leading to a dramatic increase in temperature and fusion rate over a few hours.

      • Degeneracy Lifted:

        • At ~3.5⋅108K3.5\cdot10^8K, the degenerate helium transforms into an ordinary gas.

        • The 'safety valve' operates: increased temperature raises pressure, causing the core to expand and cool, moderating fusion.

      • Post-Flash Consequences:

        • The expanding core cools the hydrogen-fusing shell, decreasing its fusion rate.

        • The outer regions contract, and the star shrinks to a new hydrostatic equilibrium.

        • The helium-fusing giant becomes smaller, dimmer, and hotter than before the flash.

    • For Higher-Mass Giants (>2M_{\odot}) - No Helium Flash:

      • Non-Degenerate Core:

        • Gravitational pressure smoothly compresses and heats the core to helium fusion temperature without becoming degenerate.

      • Normal Pressure-Temperature Relationship:

        • If helium fusion overheats the core, increased pressure causes it to expand, cooling gases and slowing fusion.

        • If too little energy is produced, gravity compresses the core, increasing temperature and fusion rate.

        • This 'safety valve' of changing pressure with temperature prevents collapse or explosion.

      • Evolutionary Path:

        • These stars evolve smoothly across the giant region on the H-R diagram, getting larger until core fusion begins.

        • They then contract slightly, but less than stars undergoing a helium flash.

Life in the giant phase has its ups and downs

  • Helium Fusion Processes:

    • Triple-alpha process (Dominant):

      • Three helium atoms fuse to become a carbon atom (4He+4He+4He→12C+γ4He+4He+4He\to12C+\gamma).

      • Two steps:

        • Two helium nuclei fuse to create beryllium4He+4He→8Be4He+4He→8Be.

        • If a third helium nucleus collides with beryllium within 10−8s10^{-8}s, stable carbon is formed.

        • Otherwise, beryllium decays back into two helium atoms.

      • Releases energy as photons (γ\gamma).

    • Carbon-oxygen fusion:

      • Carbon fuses with another helium nucleus to produce oxygen (12C+4He→16O+γ12C+4He→16O+γ ).

      • Energy released as gamma rays.

    • Duration of helium fusion:

      • About 10% as long as the hydrogen fusion phase on the main sequence (e.g., Sun: 10 billion years on main sequence, 1 billion years as a giant).

      • Core's further gravitational contraction ceases during helium fusion.

  • Variable Stars (Instability Strip):

    • Post-core helium fusion evolution:

      • Stars move partly back towards the main sequence on the H-R diagram (right to left).

      • Pulsations:

        • Pressure-temperature 'safety valves' are imperfect, causing stars to overheat, expand, then cool, contract, and overheat again.

        • This leads to instability and pulsation.

    • Instability Strip: Region on the H-R diagram where these pulsating variable stars are found.

    • Types of Variable Stars:

      • RR Lyrae variables:

        • Lower-mass, post-helium-flash stars passing through the lower end of the instability strip.

        • Periods shorter than 1 day.

      • Type I Cepheid variables (Cepheids):

        • Higher-mass stars passing back and forth through the upper end of the instability strip.

        • Note: Type II Cepheids formed earlier with fewer metals.