Lecture 14: The Sun and energy transport in stars

Astronomy 111

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Energy transport in stars

•  What is a star? •  What is a star composed of? •  Why does a star shine? •  What is the source of a star’s energy?

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Laws of Stellar Structure I: The Gas Law

•  Most stars obey the Perfect Gas Law: •  Pressure = Density × Temperature

•  In words: Compressing a gas results in higher P & T Expanding a gas results in lower P & T

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Laws of Stellar Structure II: The Law of Gravity

•  Stars are very massive & bound together by their Self-Gravity.

•  Gravitational binding increases as 1/R2

•  In words: Compress a star, gravitational binding gets stronger. Expand a star, gravitational binding gets weaker.

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Hydrostatic Equilibrium

•  Gravity wants to make a star contract. •  Pressure wants to make a star expand. •  Counteract each other:

– Gravity confines the gas against Pressure. – Pressure supports the star against Gravity.

•  Exact Balance = Hydrostatic Equilibrium The star neither expands nor contracts.

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Hydrostatic Equilibrium

Gas Pressure

Gravity

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Core-Envelope Structure

•  Outer layers press down on the inner layers. The deeper you go into a star, the greater the

pressure. •  The Gas Law says:

“More pressure= hotter, denser gas” •  Consequences:

–  hot, dense, compact CORE –  cooler, lower density, extended ENVELOPE

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Core-Envelope Structure

Compact

Core

Extended

Envelope

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Example: The Sun

•  Core: Radius = 0.25 Rsun T = 15 Million K Density = 150 g/cc

•  Envelope: Radius = Rsun = 700,000 km T = 5800 K Density = 10-7 g/cc

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The Essential Tension

•  The Life of a star is a constant tug-of-war between Gravity & Pressure.

•  Tip the internal balance either way, and it will change the star’s outward appearance.

•  Internal Changes have External Consequences

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Why do stars shine?

•  Stars shine because they are hot. – emit thermal (~blackbody) radiation – heat “leaks” out of their photospheres.

•  Luminosity = rate of energy loss.

•  To stay hot, stars must make up for the lost energy, otherwise they would go out.

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Case Study: The Sun

•  Question: How long can the Sun shine?

•  Answer: Consider the internal heat content of the Sun. Luminosity = rate of heat loss.

Lifetime = Internal Heat ÷ Luminosity

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What if no source of energy?

•  The Sun’s Luminosity losses would not be balanced by input of new internal heat. •  The Sun would steadily cool off & fade out.

•  18th Century: – Assumed a solid Sun (iron & rock) – Found Lifetime ~ 10 Million Years

•  No Problem: Earth was thought to be thousands of years old.

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The Age Crisis: Part I

•  Late 1800s: – Geologists: Earth was millions of years old. •  How can Earth be older than the Sun? – Astronomers: The Sun is a big ball of gas in

Hydrostatic Equilibrium.

•  Kelvin & Helmholtz proposed Gravitational Contraction as a source of energy.

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Kelvin-Helmholtz Mechanism

•  Luminosity radiates away heat – Outer layers of the Sun cool at little,

lowering the gas pressure. – Lower Pressure means Gravity gets the

upper hand and the star contracts a little. – Contraction compresses the core, heating

it up a little, adding heat to the Sun. •  Sun could shine for ~100 Million years

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G P

Gravity & Pressure in Equilibrium

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P G

Luminosity radiates away heat & Pressure

Drops

P

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Balance tips in favor of gravity, Sun shrinks.

G P

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Contraction makes core heat up,

increasing the internal Pressure.

G P

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G P

Balance restored, but with higher gravity,

pressure & temperature than before...

Starts the cycle all over again...

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The Age Crisis: Part II

•  Early 1900s: •  Geologists show that the Earth is >2

Billion years old. •  Kelvin Says:

The Geologists are wrong. •  Nature Says:

Kelvin is wrong... There is new physics.

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Nuclear Fusion

•  1905: Einstein demonstrates that Mass and Energy

are equivalent: E=mc2

•  1920s: Eddington noted that 4 protons have 0.7%

more mass than 1 Helium nucleus (2p+2n). If 4 protons fuse into 1 Helium nucleus, the

remaining 0.7% of mass is converted to energy.

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Fusion Energy

•  Fuse 1 gram of Hydrogen into 0.993 grams of Helium.

•  Leftover 0.007 grams converted into Energy:

•  E = mc2 = 6.3x1018 ergs

•  Enough energy to lift 64,000 Tons of rock to a height of 1 km.

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Hydrogen Fusion

•  Question: How do you fuse 4 1H (p) into 4He (2p+2n)?

•  Issues: Four protons colliding at once is unlikely. Must turn 2 of the protons into neutrons. Must be hot: >10 Million K to get protons

close enough to fuse together.

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Proton-Proton Chain

3-step Fusion Chain

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positron

neutrino

2H

positron

neutrino

2H photon

3He

4He

photon

3He

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The Bottom Line

•  Fuse 4 protons (1H) into one 4He nucleus.

•  Release energy in the form of: – 2 Gamma-ray photons – 2 neutrinos that leave the Sun – 2 positrons that hit nearby electrons,

creating two more Gamma-ray photons – Motions (heat) of final 4He and 2 protons.

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The Age Crisis: Averted

•  Luminosity of the Sun is ~4x1033 erg/sec – Must fuse ~600 Million Tons of H into He

every second. – ~4 Million tons converted to energy per

second. – Sun contains ~1021 Million Tons of Hydrogen – Only ~10% is hot enough for fusion to occur.

•  Fusion Lifetime is ~10 Billion Years.

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Test: Solar Neutrinos

•  Question: How do we know that fusion is occurring in

the core of the Sun?

•  Answer: Look for the neutrinos created in Step 1 of

the Proton-Proton chain.

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What are Neutrinos?

•  Massless, weakly interacting neutral particles. •  Travel at the speed of light. •  Can pass through a block of lead 1 parsec

thick!

•  Any neutrinos created by nuclear fusion in the Sun’s core would stream out of the Sun at the speed of light.

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Solar Neutrinos: Observed!

•  Detection of neutrinos is very hard: – Need massive amounts of detector materials – Work underground to shield out other radiation

•  Answer: – We detect neutrinos from the P-P chain in all

the experiments performed to date!

•  Success! But...

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The Solar Neutrino Problem

•  We detect only ~1/3 of the number expected...

•  Why? –  Is our solar structure model wrong? –  Is the subatomic particle theory of neutrinos

wrong? •  The last seems most plausible

–  recent evidence neutrinos have mass –  question for the physics of 21st century!

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Putting Stars Together

•  Physics needed to describe how stars work: •  Law of Gravity •  Equation of State (“gas law”) •  Principle of Hydrostatic Equilibrium •  Source of Energy (e.g., Nuclear Fusion) •  Movement of Energy through star

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Hydrostatic Equilibrium

•  Balance between Pressure & Gravity. –  If Pressure dominates, the star expands. –  If Gravity dominates, the star contracts.

•  Sets up a Core-Envelope Structure: – Hot, dense, compact core. – Cooler, low-density, extended envelope.

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Hydrostatic Equilibrium

Gas Pressure

Gravity

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Energy Generation

•  Stars shine because they are hot. •  To stay hot stars must make up for the

energy lost by shining. •  Two Energy sources available:

– Gravitational Contraction (Kelvin-Helmholtz) •  Only available if star is NOT in equilibrium

– Nuclear Fusion in the hot core.

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Main-Sequence Stars

•  Generate energy by fusion of 4 1H into 1 4He.

•  Proton-Proton Chain: – Relies on proton-proton reactions – Efficient at low core Temperature (TC<18M K)

•  CNO Cycle: – Carbon acts as a catalyst – Efficient at high core Temperature (TC>18M K)

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Proton-Proton Chain

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CNO Cycle:

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•  Fusion reactions are Temperature sensitive: – Higher Core Temperature = More Fusion – ε(PP) ~ T4

– ε(CNO) ~ T18 •  BUT,

– More fusion makes the core hotter, – Hotter core leads to even more fusion, …

•  Why doesn’t it blow up like an H-Bomb?

Controlled Nuclear Fusion

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Hydrostatic Thermostat

•  If fusion reactions run too fast: – core heats up, leading to higher pressure – pressure increase makes the core expand. – expansion cools core, slowing fusion.

•  If fusion reactions run too slow: – core cools down, leading to lower pressure – pressure drop makes the core contract. – contraction heats core, increasing fusion.

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Thermal Equilibrium

•  Heat always flows from hotter regions into cooler regions.

•  In a star, heat must flow: –  from the hot core, – out through the cooler envelope, –  to the surface where it is radiated away as

light.

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Energy Transport

•  There are 3 ways to transport energy: •  1) Radiation:

Energy is carried by photons •  2) Convection:

Energy carried by bulk motions of gas •  3) Conduction:

Energy carried by particle motions

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Radiation •  Energy is carried by

photons. – Photons leave the core – Hit an atom or electron

within ~1cm and get scattered.

– Slowly stagger to the surface (“random walk”)

– Break into many low-energy photons.

•  Takes ~1 Million years to reach the surface.

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Convection

•  Energy carried by bulk motions of the gas.

•  Analogy is water boiling:

Hot blob rises cooler water sinks

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Conduction

•  Heat is passed from atom-to-atom in a dense material from hot to cool regions.

•  Analogy: Holding a spoon in a candle flame, the handle eventually gets hot.

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The Sun

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Energy Transport in Stars

•  Normal Stars: – A mix of Radiation & Convection transports

energy from the core to the surface. – Conduction is inefficient (density is too

low). •  White Dwarfs:

– Ultra-dense stars – Conduction dominates energy transport.

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Summary:

•  Stars shine because they are hot. – need an energy source to stay hot.

•  Kelvin-Helmholtz Mechanism – Energy from slow Gravitational Contraction – Cannot work to power the present-day Sun

•  Nuclear Fusion Energy – Energy from Fusion of 4 1H into 1 4He – Dominant process in the present-day Sun

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Summary:

•  Energy generation in stars: – Nuclear Fusion in the core. – Controlled by a Hydrostatic “thermostat”.

•  Energy is transported to the surface by: – Radiation & Convection in normal stars – Conduction in white dwarf stars

•  With Hydrostatic Equilibrium, these determine the detailed structure of a star.

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Summary:

•  Stars are held together by their self-gravity

•  Hydrostatic Equilibrium – Balance between Gravity & Pressure

•  Core-Envelope Structure of Stars – Hot, dense, compact core – cooler, low-density, extended envelope

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Lecture 8: Telescopes

•  Refracting vs. reflecting •  Angular resolution of a telescope •  Properties of different types of

telescopes •  What is “seeing”? •  What are CCDs?

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Lecture 9: Formation of the Solar System

•  How did solar system form? •  Characteristics of inner and outer Solar

System •  How did angular momentum play a role? •  Define: planetesimals, radioactive age

dating, terrestrial, jovian •  Name the four techniques astronomers

use to find extrasolar planets

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Lecture 10: Terrestrial planets

•  Characteristics of Earth, the Moon, and terrestrial planets

•  How do we know the age of the Earth? •  Where did the Moon come from?

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Lecture 11: Jovian planets

•  Characteristics of Jovian planets, dwarf planets, and other solar system objects

•  Define dwarf planet, asteroid, Kuiper belt, comet

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Lecture 12: Distances to stars

•  Calculate distance to a star given its parallax

•  Define: apparent brightness, luminosity, true binary, visual binary, spectroscopic binary, eclipsing binary

•  What quantities can we measure using binary stars?

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Lecture 13: Colors of stars

•  Why are stars different colors? Why do stars have absorption lines in their spectra?

•  Give the order of the spectral sequence •  Be able to calculate luminosity and know

its units (energy/sec) •  Be able to use the Stefan-Boltzman law

and know its units (energy/sec/area) •  Know the main components and axes of

an H-R diagram

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Lecture 14: Energy transport in the Sun

•  Define: hydrostatic equilibrium, P-P chain, CNO cycle, neutrinos

•  Know what the Perfect Gas Law means •  Describe how the P-P chain makes

energy in the Sun through fusion •  List the three modes of energy transport