M8-S4: Energy Source of Stars

  • analyse and apply Einstein’s description of the equivalence of energy and mass and relate this to the nuclear reactions that occur in stars (ACSPH031)

  • investigate the types of nucleosynthesis reactions involved in Main Sequence and Post-Main Sequence stars, including but not limited to:

– proton–proton chain
– CNO (carbon-nitrogen-oxygen) cycle
  
  • Energy and mass conservation

In addition to the special theory of relativity, Einstein also proposed that the two quantities (mass and energy) are in fact interchangeable. Mass can be converted into energy and energy can be converted into mass. As a result, each quantity does not need to be conserved in its own form as they can inter-convert.

 

The mass and energy conversion or equivalence is governed by the equation:

 

  • Where E is energy in Joules (J)
  • m is the rest mass of a particle or object in kilograms (kg)
  • c is the speed of light in ms-2

 

This form of mass and energy conservation and conversion is commonly observed in nuclear reactions and situations where high-speed particles collide.

Production of energy in stars

  • Stars, including our Sun, produce energy via nuclear fusion. Nuclear fusion is the process in which atomic nuclei physically combine to form one or more different atomic nucleus and additional sub-atomic particles such as neutrons. Nuclear fusion is also sometimes referred to as nucleosynthesis.
  • Nuclear fusion obeys energy-mass conservation but requires the application of energy-mass equivalence.

 

  1. Proton-proton chain reaction

  • Protons normally avoid collision due to electrostatic repulsion but when the temperature is great enough, the rapidly moving protons can overcome this repulsion. As a result, collision between hydrogen-1 nuclei can occur.

 

  • Overview - The simplest form of nuclear fusion occurs between hydrogen nuclei to form helium (diagram on right). The atomic mass of one hydrogen nucleus is roughly 1.008 amu. When four hydrogen nuclei combine, the total mass of resultant products should add up to 4.032 amu due to mass conservation.

 

  • However, the observed mass of a helium nucleus is only 4.003 amu. The lost mass has been converted into energy.

1 atomic mass unit (amu) = 1.661 × 10-27 kg

 

Using Einstein’s mass and energy equivalence:

 

 

 

  • What actually happens?
  1. During the collision between two protons, one of them releases a positron and neutrino to become a neutron which forms deuterium (hydrogen-2).

 

  1. Next, the newly formed deuterium collides with another proton to form helium-3 while emitting a high energy gamma photon.
  2. Finally, two helium-3 nuclei collide to form a single helium-4 nucleus while emitting two protons (hydrogen-1 nucleus).

 

  1. Carbon-nitrogen-oxygen (CNO) cycle

  • Along with proton-proton chain reaction, the CNO cycle is the only other known nuclear fusion reaction that generates helium nuclei from hydrogen nuclei and therefore energy.
  • There are more stages within a CNO cycle which can be better understood when broken down. Hydrogen nuclei are interchangeably referred to as protons as each hydrogen-1 nucleus contains only a single proton.

  1. Carbon-12 nucleus collides with a proton to form nitrogen-13, emitting energy in the process.

 

  1. Nitrogen-13 releases a positron and neutrino to convert one of its protons into a neutron. This forms carbon-13.

 

  1. Carbon-13 collides with a proton to form nitrogen-14, emitting energy in the process. 

 

  1. Nitrogen-14 collides with a proton to form oxygen-15, emitting energy in the process.

  

  1. Oxygen-15 releases a positron and neutrino to convert one of its protons into a neutron. This forms nitrogen-15.

 

  1. Nitrogen-15 collides with a proton to form carbon-12, emitting a helium-4 nucleus in the process.

 

 

  • Although the cycle is broken into numerical stages, it can start at any stage as long as the reactants are present.
  • Similar to proton-proton chain, the CNO cycle allows four protons to be converted into a helium nucleus. While this is happening, two positrons and two neutrinos are released.

 

  1. Triple alpha process

  • The triple alpha process involves three helium nuclei per cycle and creates heavier elements such as carbon-12 and oxygen-16.
  • Triple alpha process can only occur when the star’s temperature is at least 100 MK. This tremendous amount of energy overcomes the electrostatic repulsion between heavier nuclei.
  • What actually happens?

  1. Two helium nuclei collide to form a beryllium-8 nucleus.
  2. Beryllium nucleus can collide with another helium nucleus to form carbon-12, emitting energy (in the form of a photon) in the process.
  3. Carbon-12 can collide with another helium nucleus to form O-16, emitting energy (in the form of a photon) in the process.

 

Energy production varies among stars

  • Main sequence stars
    • Proton-proton chain reaction is the prominent method of producing helium nuclei in stars with mass similar to the Sun.
    • The type of nucleosynthesis reaction depends on the temperature of the star.
      • CNO cycle requires 15 MK (mega-Kelvin) to start and only becomes the prominent source of helium nuclei when the temperature reaches 17-18 MK. For example, our Sun’s core is about 15.7 MK and only 1.7% of helium nuclei are made from CNO cycle.
      • Proton-proton chain’s energy production rate is proportional to the temperature to the power of four (T4) whereas it is to the power of 17 (T17) for CNO cycles.
    • A star remains as a main sequence star as long as there are available hydrogen nuclei to be ‘burned’ to form helium.

 

 

Post-main sequence stars

Red giants (~100 MK)

  • As the amount of hydrogen in the core of the star runs out, the pressure formerly exerted by outward radiation decreases. As a result, the immense gravitational force of the star takes over and pulls outer layers of the star inwards. During this process, their gravitational potential energy is transformed into thermal energy which further heats up the core of the star.
  • Newly derived thermal energy heats up outer layers of hydrogen and continues helium formation. This energy also heats up nearby gases and expands the star, increasing its size (more negative absolute magnitude on HR diagram).
  • Newly formed helium nuclei from burning hydrogen in outer shells are attracted to the core of the star by gravitational force. This increases the mass of the star which in turn increases its temperature.
  • Eventually, the temperature of a red giant becomes high enough to overcome electrostatic repulsion between heavier nuclei such as helium. This allows the occurrence of the triple alpha process.


Supergiants (high mass post-main sequence stars)

As the temperature continues to increase to 500 MK, collision between carbon nuclei become possible to form either neon, sodium or magnesium.

  • Two carbon-12 nuclei can combine to form a neon nucleus, emitting a helium nucleus in the process.

     

    • Two carbon-12 nuclei can combine to form a sodium nucleus, emitting a proton (hydrogen-1) in the process.

     

    • Two carbon-12 nuclei can combine to form a magnesium nucleus, emitting a photon in the process.

     

    • Helium nuclei can collide with existing oxygen-16 nuclei (created from triple alpha process) to form more neon, emitting a photon in the reaction.

     

    • Neon can fuse with helium nuclei to form magnesium

     

    • The ‘burning’ of neon to generate magnesium and heavier elements further increases supergiants’ temperature to the magnitude of 109 K or 1000 MK. At this temperature, oxygen-16 nuclei can fuse to form silicon. Silicon will fuse to form sulfur which in turn form heavier elements such as argon and calcium.
    • Silicon-burning in supergiants eventually stops with the formation of iron which forms the core of these stars.

     

    Low mass post-main sequence stars

      • These stars have weaker gravitational force and thus lower temperature. This means the rate of hydrogen burning is lower.
      • Low temperature conditions in these stars prevent collision between helium nuclei to form larger elements. As a result, the star will eventually cool down after hydrogen in shell layers are depleted.

     

    Type of Star

    Core Temperature Range

    Energy Production Process

    Main Sequence

    ·       Proton-proton chain is prominent between 5 and 18 MK

    ·       CNO cycle becomes more prominent when T > 18 MK

    ·       Proton-proton chain

    ·       CNO cycle

    Post Main Sequence

    Red giant

    ·       100 MK

    ·       Triple alpha process

    Supergiant (high mass)

    ·       High mass stars have enough energy to star triple-alpha process and hence become super giants.

     

    ·       500 – 2000 MK

     

     

     

    ·       Fusion between carbon nuclei to form heavier elements such as Ne, Na and Mg

    ·       Fusion between helium and oxygen or neon to form Mg and Si respectively.

    Low mass

    ·       Between 5 and 100

    ·       Becomes a white dwarf after hydrogen is depleted.

    ·       Proton-proton chain

    ·       CNO cycle

    White Dwarf

    ·       100,000 K to few thousands K

    ·       No energy source to exert pressure against gravitational force.

    ·       Gravitational potential energy.



    Practice Question 3
    (a) Describe two main ways our Sun produces helium. Explain how these processes help the Sun produce its energy. (4 marks)
    (b) Explain why nucleosynthesis of heavier elements such as carbon does not occur in the Sun. (2 marks)
    (c) How do supergiants commonly produce elements heavier than helium? (2 marks)

    Previous section: Hertzsprung-Russell Diagram and Stars' Life Cycle

    Next section: The Electron & Thomson and Millikan's experiments