Lecture 8: The Internal Structure of the Sun

The Sun

The Sun can be split into two regions: The interior is a sphere with radius R = 7x108m The atmosphere lies on top and has the following layers (from innermost to outermost): The photosphere is about 300 km thick. Most of the Sun's visible light that we see originates from this region. The chromosphere is about 2000 km thick. We only see this layer and the other outer layers during an eclipse. The corona extends outwards for more than a solar radius.

Surface temperature of the Sun is the temperature in photosphere - 5800 K

The Sun Interior is a Spherical Ball of Hot Gas

Main Q: How can a Spherical Ball of Gas radiate energy over long period of time remaining visibly unchanged ? Four ingredients to understanding the Sun (and a star) interior: Balance of forces between Gravity and Pressure - hydrostatic equilibrium Balance of energy produced and radiated away - thermal equilibrium Nuclear fusion in a stellar core as a source of energy. How energy is transported from the core to the surface ?

Basic Physics: Gravity and Gas laws

• Gravity causes all massive objects to be attracted to each other.
  • For two objects, the force causes one object to move towards the other according to Newton's law of gravitation:
  • Micorscopically, there are of order 10⁵⁷ Hydrogen ions (and electrons) in the Sun, all attracted to each other by gravity.
  • But if an ion is outside, most of other ions that attract it are towards the center.

• The Sun is a ball of hot gas (mainly Hydrogen and Helium). If one compresses gas, it heats up and its pressure increases

• Attraction of outer layers by the inner ones towards the center can be balanced by pressure of the inner layers. Then we achieve the hydrostatic equilibrium

• The spherical shape is the equilibrium shape that has least energy (in comparison with irregular shapes). Like, for example, water droplets where the surface tension is balanced by the pressure.

Hydrostatic Equilibrium The particles in a hot gas are in motion. The motion of the gas particles creates gas pressure P A gas in a bottle exerts an outward force on the wall of the bottle: Force = Pressure x Area where Area is the surface area of the bottle. In a star two forces act on the star's gas: Gravity: pulls gas inwards Gas Pressure from below: pushes gas outwards Gas Pressure from above: pushes gas inwards The balance between gravity and net gas pressure difference is called hydrostatic equilibrium.

• When hydrostatic equilibrium is reached, the star will be stable, and its size will stay constant.
• If hydrostatic equilibrium is broken, it is restored very quickly, on a dynamical timescale ~ 15 min for the Sun

Structure of the Sun's Interior

• At deeper layers inside the Sun, the weight of the outer layers is larger. The Sun must have a higher temperature and pressure deep inside in order to balance the larger weight.

• The Sun's temperature and pressure and the gas desnity is largest in the centre and decreases outwards.

Central Temperature Tc = 15 million K Central Density &rhoc = 150 tons/m3 Central Pressure Pc = 1011 atm

Source of Solar energy

• How much energy a hot ball of gas it temperature T~10⁶ contain ?
  • one atom has energy, on average, approx
    
    k_B T = 1.3806503 x 10^-23 J K^-1 x 10⁶ K = 1.4 x 10^-17 J
    
  • We have 10⁵⁷ atoms, so the total energy stored
    
    E ~ 10⁴⁰ J
  
  
  
• Sun luminosity is 4 x 10²⁶ Watt, i.e that much Joules per second
• I.e amount of stored energy in hot gas is enough for time t
  
  t = 10⁴⁰ J / 4 x 10²⁶ W = 2 x 10¹³ sec = 10⁶ years
  Thermal timescale is too short ! Need another source of energy !

Nuclear Fusion: The Source of the Sun's Energy

• 1905: Albert Einstein showed that Mass can be converted to Energy. Equation describing the Equivalence of mass and energy is E = m c².

• 1920: Frederick Aston carefully measured the masses of Hydrogen and Helium. He found: (u = 1.66 x 10^-27kg)
  • m_H = 1.007825 u
  • 4 x m_H = 4.031280 u
  • m_He = 4.002677 u
• 4 Hydrogen atoms are a tiny bit more massive than one Helium atom.

• 1920: Sir Arthur Eddington combined the discoveries of Aston and Einstein with the following proposal:
  
  • In the Sun the nuclear reaction 4 H -> He occurs.
  
  
  • Every time this reaction occurs mass is lost and converted to energy.
  
  
  • Mass difference = 4.8 x 10^-29kg
  
  
  • Fraction of mass lost = (4 x m_H - m_He)/(4 x m_H) = 0.007
  
  
  • Energy produced = 4.8 x 10^-29kg x (3 x 10⁸)² = 4.3 x 10^-12 J

• 1930: Hans Bethe proposed the following reaction steps called the proton-proton chain: (Hans Bethe was awarded the Nobel Prize in physics in 1967 for this work and other discoveries in stellar nuclear astrophysics.)
  
  
  

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  1. ₁¹H + ₁¹H &rarr ₁²H + e⁺ + &nu
     • ₁¹H = nucleus of regular Hydrogen = 1 proton
     • ₁²H = nucleus of Deuterium = Heavy Hydrogen = 1 proton + 1 neutron
     • e^- = electron
     • e⁺ = positron = anti-matter partner to the electron (same mass as electron, but positive charge)
     • &nu = neutrino = neutral particle with very tiny mass
     
     

     This is an extremenly low probability reaction which defines slowness of hydrogen buring in stars and stars long life

     • A proton at T=10⁷ K and &rho=10⁵kg/m has 10⁸ head-on collisions per second with other protons
     • However, most of the collision are fruitless. Either proton is repelled by Coulomb electric force, or proton does not convert into neutron. Only one in 10²⁵ collisions succeed !
     • That means, that on average a given proton will get fused into deuterium after 10²⁵ / 10⁸ s = 10¹⁷ = 10 billion years !!
     • But there are 10³² protons in 1 m³. So we still have 10³² /10¹⁷ s = 10¹⁵ reactions per second in each quibic meter.

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  2. ₁²H + ₁¹H &rarr ₂³He + &gamma
     • ₂³He = light isotope of Helium = 2 protons + 1 neutron
     
     

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  3. ₂³He + ₂³He &rarr ₂⁴He + ₁¹H + ₁¹H
     • ₂⁴He = regular Helium = 2 protons + 2 neutrons
     
     

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  4. There is a competition to the last step. ³He can meet ⁴He rather than ³He. ₂³He + ₂⁴He &rarr add one more ₁¹H and get two ₂⁴He + &nu + e⁺
     

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Where is the released energy ?

Where does released energy go ?

• Photons scatter efficiently and their energy heat the gas

• Neutrinos practically do not interact with anything, and fly through the Sun in just over 2 seconds taking their energy away