Stellar Characteristics and the Sun

The Sun as a Star - Stellar Characteristics

Today’s Topics

  • Our Sun is a Star
  • What makes the Sun shine?
  • Historical views
  • Nuclear fusion (E=mc2E = mc^2)
  • Age of the Sun
  • Structure of the Sun
  • Stellar similarities and differences
  • Measuring Stellar Characteristics
  • Stellar luminosity
  • Stellar distances

Stars: The Building Blocks of the Universe

  • Our Sun is one of billions of billions of stars in the Universe (400 billion in the Milky Way alone).
  • The vast majority of the visible matter in the Universe is made of stars.
  • Almost all the light from a galaxy is from the stars it contains.
  • Topics to be covered regarding stars:
    • How do they shine?
    • What gives them their colors?
    • What are they made of?
    • What are they like inside?
    • How are they born, how do they evolve, and how do they die?

The Sun: Basic Facts

Known facts about the Sun that models and theories have to explain:

  • Radius of the Sun: R<em>⊙=7×105R<em>\odot = 7 \times 10^5 km (about 110 × R</em>ER</em>E)
  • Mass of the Sun: M<em>⊙=2×1030M<em>\odot = 2 \times 10^{30} kg (about 300,000 × M</em>EM</em>E)
  • Luminosity of the Sun: L⊙=4×1026L_\odot = 4 \times 10^{26} W
  • Surface temperature of the Sun: T=5800T = 5800 K
  • Composition of the Sun: 70% H, 28% He, 2% other

Earth-Moon Demo

  • Lecture Tutorial on the size of various objects and systems in the solar system (Earth, Moon, Sun, Earth-Moon system, and Earth’s Orbit)
  • Lecture Tutorial: Sun Size, pp. 105-107

What Makes the Sun Shine?

  • Historically, the source of the Sun's energy was a mystery.
  • Ancient thinkers suggested the Sun was a glowing ember of fire.
  • Chemical burning was ruled out because it could only sustain the Sun for a few thousand years, and a burning ember would cool over time.

Kelvin & Helmholtz

  • Suggested that the Sun’s energy could come from slow, steady gravitational contraction in the late 1800s.
  • As gas falls to the center, it loses gravitational potential energy, which converts to heat.
  • Calculations showed this could keep the Sun shining for about 25,000,000 years.
  • Geologists found evidence that the Earth was hundreds of millions or billions of years old (modern estimates are about 4.6 billion years old).

Einstein and Nuclear Fusion

  • Einstein's equation, E=mc2E = mc^2, provided the answer.
  • When 4 protons merge to form a 4He^4He nucleus (alpha particle), mass is converted to energy; this is nuclear fusion.
  • This mechanism allows the Sun to shine at its current rate for 10 billion years.

Nuclear Fusion Process

  • Particles in the Sun’s core collide two at a time.
  • The reaction 4p→4He+energy4p \rightarrow ^4He + energy requires multiple steps:
    1. Two protons combine to form deuterium (2H^2H), a proton and a neutron. A positron (e+e^+) and a neutrino (ν\nu) are emitted to conserve electric charge and lepton number.
    2. Each deuterium nucleus gets an additional proton to make 3He^3He.
    3. Two 3He^3He nuclei fuse to form 4He^4He plus two protons.

Nuclear Fusion Details

  • Positively charged protons repel each other unless they are close enough (~10−1510^{-15} m) for the strong nuclear force to bind them.
  • Nuclear fusion requires very high speeds, implying very high temperatures.
  • Without quantum tunneling, no fusion would occur in the Sun at all.
  • The temperature at the center of the Sun is about 15 million K, hot enough for protons to fuse together to make Helium nuclei.

Energy Released in Nuclear Fusion

  • The energy released can be calculated from E=mc2E = mc^2:
  • Typical chemical reaction releases a few eV.
  • mp=1.673×10−27m_p = 1.673 \times 10^{-27} kg
  • 4mp=6.690×10−274m_p = 6.690 \times 10^{-27} kg
  • mHe=6.643×10−27m_{He} = 6.643 \times 10^{-27} kg
  • Δm=0.047×10−27\Delta m = 0.047 \times 10^{-27} kg
  • E=Δmc2=(0.047×10−27kg)(3×108m/s)2=4.23×10−12J=26.4MeVE = \Delta mc^2 = (0.047 \times 10^{-27} kg)(3 \times 10^8 m/s)^2 = 4.23 \times 10^{-12} J = 26.4 MeV

Mass Conversion and Energy Release

  • Fraction of mass converted to energy:
    • Δmm=0.047×10−27kg6.690×10−24kg=0.007\frac{\Delta m}{m} = \frac{0.047 \times 10^{-27} kg}{6.690 \times 10^{-24} kg} = 0.007 (0.7%)
    • 1 kg H → 993 g He + 7 g converted to energy
  • 4.23×10−124.23 \times 10^{-12} J sounds small, but the reaction happens ~103810^{38} times/second
    • 4×10(−12+38)J=4×1026J4 \times 10^{(-12+38)} J = 4 \times 10^{26} J is released each second
    • L⊙=4×1026W=400,000,000,000,000,000,000,000,000WL_\odot = 4 \times 10^{26} W = 400,000,000,000,000,000,000,000,000 W
  • 600,000,000 tons of H is converted to 596,000,000 tons of He + energy each second!

Age of the Sun

  • To determine the approximate age of the Sun, divide the total mass of the Sun by the amount of Hydrogen converted to Helium each second.
  • Only about 10% of the Hydrogen in the Sun (in the core) will be converted to Helium, so the true lifetime of the Sun is 10 × shorter, or ~ 10 billion years.
  • Since tsolarsyst_{solar sys} ~ 4.6-5 billion years, the Sun is about halfway through its lifetime.
  • t⊙≈2×1030kg6×1011kg/s≈3×1018s≈100billionyearst_\odot ≈ \frac{2 \times 10^{30} kg}{6 \times 10^{11} kg/s} ≈ 3 \times 10^{18} s ≈ 100 billion years

Structure of the Sun

  • For fusion to occur in the center of the Sun, the temperature must be about 15,000,000 K, whereas the surface is only 5800 K.
  • The center of the Sun is so hot because of pressure.
  • The pressure is so high at the center of the Sun because of gravity!
  • The person on the top has no weight to support, the person in the middle supports one other person, and the person on the bottom has to support the two people above him.

Pressure and Energy Transport

  • At every layer of the Sun, outward pressure must equal the pressure caused by the weight of the overlying material.
  • Thus, the pressure in the interior must rise towards the center.
  • At high pressures, the gas is pressed closer together (higher density) and becomes hotter (more frequent collisions lead to higher speeds ⇒ higher internal KE ⇒ higher temp).
  • Two mechanisms of energy transport:
    1. Random walk (radiative diffusion)
    2. Convection (like boiling)
  • It takes more than 100,000 years for a photon released in the fusion reaction to make its way to the surface.

Stellar Interior Quiz

  • If fusion in the solar core ceased today, worldwide panic would not break out tomorrow as the Sun would begin to grow dimmer.
  • d) No, it takes thousands of years for photons created in nuclear reactions at the solar core to reach the surface
  • The Solar Thermostat

Stellar Similarities and Differences

  • People: commonalities and differences
    • People have much in common
      • Same basic structure (2 arms, 2 legs, 2 eyes, 1 nose, etc.)
      • Made up of same stuff (bone, tissue, blood, etc.)
      • Get energy the same way (eating food)
    • People also differ in detail
      • Short, tall, fat, thin
      • Eye, hair, skin color
      • Some live a long time, some don’t
      • Different lifestyles

Stars: Commonalities and Differences

  • Stars are like people in this way.
  • Stars have much in common:
    • Balls of hot gas
    • Made of 70% H, 28% He, 2% everything else
    • Get energy from nuclear fusion
  • Stars also differ:
    • Luminosity
    • Mass
    • Temperature (color)
    • Size (radius)
    • Lifetime
    • End their lives differently

Stellar Luminosity

  • Apparent brightness is a measure of how bright a star appears on Earth.
  • Luminosity is a measure of how much energy per second (W) a star emits.
  • The apparent brightness of an object declines with distance (inverse square).
  • If we measure apparent brightness (energy/sec/m2^2) and we know distance, we can get the luminosity of the star.
  • For Sun, apparent brightness = 1400 W/m2^2 and d = 150 million km = 1.5×10111.5 \times 10^{11} m and
  • Apparentbrightness=Luminosity4π×(distance)2Apparent brightness = \frac{Luminosity}{4\pi \times (distance)^2}
  • L=4π(1400W/m2)(1.5×1011m)2=4×1026WL = 4\pi (1400 W/m^2) (1.5 \times 10^{11} m)^2 = 4 \times 10^{26} W

Distance and Parallax

  • It is relatively easy to measure the apparent brightness of a star.
  • Distance is much harder to measure.
  • For nearby stars (d≤3000d \le 3000 ly) we can use the technique of parallax.
  • Parallax can be understood by putting your finger in front of your face, then alternating closing your two eyes - note how your finger appears to move relative to the more distant objects in the room.

Parallax and Distance Measurement

  • As the Earth orbits the Sun, relatively nearby stars appear to move relative to more distant stars.
  • Because even the nearest stars are so distant, there is a simple relationship between distance and apparent angle a star moves.
  • 1 parsec ≈ 3.26 light years
  • d(inparsecs)=1p(inarcseconds)d (in parsecs) = \frac{1}{p (in arcseconds)}

Lecture Tutorial: The Parsec, pp. 35-37

Stellar Luminosities

  • Stellar luminosities vary from 0.0001L<em>⊙0.0001 L<em>\odot – 1,000,000L</em>⊙1,000,000 L</em>\odot, ten orders of magnitude.
  • Note that most of the stars in this image are at the same distance, so their relative apparent brightness is the same as their relative luminosities.
  • Note that there are many more faint stars than bright stars, suggesting that less luminous stars are far more common.