Order

Apollo and Dionysus

• Friedrich Nietzsche (1872) argues that the world is fundamentally meaningless, yet classical Athenian tragedy transcends pessimism through a tension between Dionysian (chaos) and Apollonian (order) influences.

• The study of aesthetics benefits from the duality of the Apollonian and the Dionysian, just as reproduction depends on the duality of the sexes and their periodic reconciliation. Nietzsche emphasizes that the future of art is bound up with this duality.

• Quote (from Nietzsche): the further development of art is bound up with the duality of the Apollonian and the Dionysian; reproduction depends upon the duality of the sexes, their continuing strife and reconciliation.

Apollo and Dionysus (Generalized)

• Apollonian influence: Order, form, structure; ↓ entropy.

• Dionysian influence: Chaos, disorder, randomness; ↑ entropy.

• Life, culture (including classical Athenian tragedy), and physical systems emerge from the interplay between Apollonian and Dionysian.

Apollo and Dionysus (Our cast of characters)

• Ordered and differentiated structures in the universe (Apollonian).

• The history of the tragic form mirrors the evolutionary history of the universe as a tension between Dionysian and Apollonian influences.

• The 2nd law of thermodynamics: The entropy of an isolated system tends to increase over time. Reality becomes more disordered and undifferentiated over time (Dionysian).

Pop stars and observational cosmology (recall from LECTURE 2)

• The Hubble-inspired universe is populated by many stars and many galaxies.

• Hooker telescope: Completed in 1917; 2.5-m (100-inch) telescope at Mount Wilson Observatory (California, USA).

• Edwin Hubble used the Hooker telescope to establish:

  • CLAIM 1: The universe is larger than the Milky Way galaxy.

  • CLAIM 2: The universe is expanding.

• Source reference: Hubble’s observational program; Great Debate context from LECTURE 2.

Walter Baade and stellar populations

• Walter Baade (1942) is regarded as a pivotal observational astronomer after Hubble.

• Baade discovered two distinct stellar populations: Pop I and Pop II stars.

• Baade used the same Hooker telescope as Hubble but with more red-sensitive photographic emulsions, revealing Pop II stars that Hubble missed.

• Pop I stars: Younger, hotter (bluer), metal-rich, concentrated in the arms of galaxies.

• Pop II stars: Older, cooler (redder), metal-poor, found in galactic bulges and halos.

The 200-inch Hale telescope and Pop III extension

• 1949: Hale telescope (5.1 metres) at Palomar Observatory.

• Baade later used the Hale telescope to test his stellar-population concept.

• Later extensions include Pop III stars: the first luminous stars formed from primordial gas.

• Stellar population descriptions:

  • Pop I: Youngest stars formed < 10 Gyr ago; typically metal-rich; masses up to ~100 M⊙; located mainly in spiral arms.

  • Pop II: Among the longest-lived stars; formed after Pop III enrichment; metal-poor; masses up to ~100 M⊙.

  • Pop III: First luminous stars; formed ~100–250 Myr after the Big Bang; metal-free; massive up to ~1000 M⊙; short-lived.

Simulation imagery and examples

• Simulation image: Formation of a Pop III star (Abel et al., 2002).

• Pop III stars have largely been hypothesized or simulated within the CDM framework (Abel et al., 2002; Bromm et al., 1999, 2002; Bromm & Loeb, 2004).

• The Methuselah star (HD 140283) is an example of a Pop II star; its age is estimated via stellar models.

• The Sun is an example of a Pop I star.

Chronology of cosmic events

• Birth of the Methuselah star: approximately 14.46 billion years ago.

  • (Bond et al., 2013)

• Big Bang and birth of the observable universe: ~13.8 billion years ago.

  • (Planck/Aghanim et al., 2020)

• Birth of the Milky Way: ~13 billion years ago.

  • (Xiang & Rix, 2022)

• Birth of our solar system: ~4.6 billion years ago.

• Birth of Earth: ~4.54 billion years ago.

  • (Dalrymple, 2001; Tera, 1981)

Problems in cosmic dating

• PROBLEM 1: Age paradox

  • Methuselah star: ~14.46 Gyr vs Universe: ~13.8 Gyr.

  • It seems impossible for any constituent to predate the universe itself.

• PROBLEM 2: An 8.4 Gyr gap between Milky Way birth (13 Gyr) and the solar system birth (4.6 Gyr).

The 8.4 Gyr gap and solar siblings

• The gap was filled by the lives and deaths of the Sun’s stellar ancestors (Pop II and Pop III).

• Evidence suggests the Sun was born in a populous star-forming region; thousands of the Sun’s solar siblings are now scattered across the Milky Way.

• Gaia mission (ESA) aims to map Milky Way stars and identify the Sun’s solar siblings.

• PROBLEM 1 is resolved by recognizing tension between stellar astrophysics (stellar models) and cosmology; uncertainties in measurement and modeling at the oldest regime.

• PROBLEM 2 is resolved by recognizing that the Sun is a Pop I star (3rd generation).

The formation of our solar system

• The solar system formed about 4.6 billion years ago via two hypotheses:

  • H1: Kant–Laplace nebular hypothesis (Kant, 1755; Laplace, 1796).

  • Central collapse of a giant cloud raised temperature and pressure enough for nuclear fusion to begin, forming a star (the Sun).

  • The collapsing nebula formed a disk; most mass migrated to the center, with the rest flattening into a disk.

  • H2: (Alternative or complementary idea; implied in note) Disk fragmentation and planet formation within the solar nebula.

Kant, early natural science, and tidal locking

• Kant’s contributions to natural science include:

  • A paper on the Moon showing it keeps the same face to Earth (tidal locking).

  • Three papers around the 1755 Lisbon earthquake anticipating seismology (Kant, 1756a, 1756b, 1756c).

• Tidal locking result: Time for Moon to rotate on its axis equals time to orbit Earth = 27.3 days.

The solar system: planet formation and types

• The planetesimal hypothesis (Safronov, 1969): after sun birth, dust clumps into planetesimals; via gravity, clumps attract more clumps, building planets.

• Planet classification by distance from the Sun:

  • Terrestrial planets: Mercury, Venus, Earth, Mars (inner rocky bodies).

  • Frost line (snow line) marks boundary beyond which volatile ices condense; separates terrestrial and gas/ice giants.

  • Gas giants: Jupiter, Saturn (large gaseous planets).

  • Ice giants: Uranus, Neptune (ices and volatile compounds).

  • Trans-Neptunian objects (e.g., Pluto) orbit beyond Neptune; reclassified as a dwarf planet in 2006.

The Sun: fusion energy and lifetime

• Nuclear fusion in the Sun’s core via hydrogen burning to helium.

• Energy release via mass–energy conversion: $E = mc^2$.

• Approximate solar parameters:

  • Mass of the Sun: M_igodot \,\approx\, 2\times 10^{30} \,\text{kg}.

  • Core comprises about 10% of the Sun’s mass, i.e., m{core} \approx 0.10 Migodot.

  • Mass converted to energy: about 0.7% of the mass of the resulting helium photons, yielding total energy $E_{fusion} = m c^2$ with $m \approx 1.4\times 10^{27} \,\text{kg}$ and $E \approx 1.26\times 10^{44} \text{ J}$.

• Current solar luminosity: L_igodot \approx 3.8\times 10^{26} \text{ W}.

• Estimated solar lifetime on the main sequence: $\text{Lifetime} \approx 10.5\ \text{billion years}.$

• The Sun is currently about halfway through its main-sequence life.

• End-of-life scenario: red giant phase will engulf Mercury, Venus, and likely Earth (as the Sun exhausts core hydrogen).

Planets: distances and temperatures

• The five brightest planets visible to the naked eye: Mercury, Venus, Mars, Jupiter, Saturn.

• Disk of planetary temperatures (approximate average surface temperatures):

  • Terrestrial planets: Mercury ≈ -167°C, Venus ≈ 464°C, Earth ≈ 15°C, Mars ≈ -65°C.

  • Gas giants: Jupiter ≈ -110°C, Saturn ≈ -140°C.

  • Ice giants: Uranus ≈ -195°C, Neptune ≈ -200°C.

• Trans-Neptunian objects beyond Neptune (e.g., Pluto).

• The eight planets and their $AU$ distances:

  • Mercury: 0.39 AU; Venus: 0.72 AU; Earth: 1.00 AU; Mars: 1.52 AU;

  • Jupiter: 5.20 AU; Saturn: 9.54 AU; Uranus: 19.20 AU; Neptune: 30.06 AU.

• Definition: 1 AU ≈ 149,597,870,700 m ≈ 1.496×10^{11} m; average Earth–Sun distance.

Goldilocks zone

• The Goldilocks zone (habitable zone) is the region around a star where conditions allow liquid water on a planet’s surface.

• Analogy: Goldilocks tale—Earth is at just the right distance to maintain water in liquid form: not too hot, not too cold.

• Portrayed across the solar system diagram: Mercury (too hot), Venus (hot), Earth (just right), Mars (cold).

Distances, light, and time: 1 AU and light travel time

• 1 AU ≈ 1.496×10^11 m.

• If cycling at 25 km/h, it would take ≈ 682.6 years to cover 1 AU.

• Light travel time: at speed c ≈ 3.0×10^8 m/s, 1 AU corresponds to about 499 seconds ≈ 8 minutes 19 seconds.

• Thus, the Sun as we see it is the Sun as it was 8 minutes 19 seconds ago.

• Visual: Sun’s photons as Earth’s energy currency; photons travel from Sun to Earth and drive most Earth processes.

Solar energy receipt by Earth

• Solar constant: $GSC = 1361 \text{ W m}^{-2}.$ (energy per unit area at 1 AU)

• Geometry: Earth intercepts sunlight with cross-sectional area A = \pi Rigoplus^2 where Rigoplus = 6.37\times 10^6 \,\text{m}.

• Power intercepted by Earth: P_igoplus = GSC \times A = 1361 \times \pi (6.37\times 10^6)^2 \approx 1.73\times 10^{17} \,\text{W}.

• Energy per photon: $E<em>{photon} = \frac{h c}{\lambda}$ with $\lambda \approx 550\ \text{nm}$, $h = 6.626\times 10^{-34} \ text{ J s}, \ c = 3.0\times 10^8 \ text{ m/s}.$ For $\lambda = 550 \text{ nm}$, $E</em>{photon} \approx 3.61\times 10^{-19} \text{ J}.$

• Average wavelength of visible light: between 400 nm and 700 nm; midpoint ~ 550 nm.

• Number of photons hitting Earth per second:

  • \dot{N}{photons} = \frac{Pigoplus}{E_{photon}} \approx \frac{1.73\times 10^{17}}{3.61\times 10^{-19}} \approx 4.79\times 10^{35} \text{ s}^{-1}.

• The text notes that this corresponds to almost five hundred decillion photons per second (short scale; decillion ≈ $10^{33}$ in the short scale).

The energy transfer metaphor

• The sun–Earth energy transfer is likened to a bicycle wheel: Sun as hub; photons as spokes; Earth as wheel.

• Various biological and physical processes harness and convert solar energy to power life.

Theory of heat and energy transfer

• Two competing theories historically:

  • Caloric theory of heat (Lavoisier, 1789): Heat as a weightless caloric fluid flowing from hot to cold.

  • Mechanical theory of heat (Thomson, 1798; Clausius, 1857): Heat as transfer of kinetic energy of atoms/molecules; hot bodies have faster particles.

• Paddle wheel experiment (Joule, 1849): A paddle wheel immersed in fluid with weights rotating it.

  • Result: Temperature of the fluid increases due to stirring; the amount of temperature rise is proportional to the work done.

  • This supports the mechanical theory and links mechanical work with heat energy; foundational for the First Law (energy conservation).

• Predictions:

  • Caloric theory predicted no heat generation from stirring (incorrect).

  • Mechanical theory predicted heat generation from stirring (correct).

• Conclusion: Heat is not a material substance but a form of energy transfer; the paddle-wheel experiment demonstrates energy conservation and connects mechanical work with thermal energy.

Entropy and microscopic foundations

• Thermodynamics studies heat, work, temperature, energy, entropy, and matter.

• Temperature: average kinetic energy of particles; hotter bodies have faster-moving particles; colder bodies have slower-moving particles.

• Entropy: the number of microstates consistent with a macrostate; Boltzmann’s formulation: $S = k<em>B \ln \Omega$ where $\Omega$ is the number of microstates; $k</em>B$ is Boltzmann constant.

Entropy: examples and the second law

• Example 1: A pan of hot water in contact with a cold metal spoon.

  • Heat flows from hot water to cold spoon due to higher average kinetic energy of hot-water molecules.

  • Temperature of spoon rises; water cools slightly; the two systems move toward a common temperature.

  • The number of accessible microstates increases; entropy increases.

• The second law states: The entropy of an isolated system tends to increase over time, i.e., systems evolve toward greater disorder.

• The universe as an isolated system tends toward greater entropy and thermal equilibrium.

• Open systems (like living organisms) can maintain local order by exporting energy and entropy to the environment.

• Recall: Prigogine’s theory of dissipative structures (Lecture 1).

The 0th to 3rd laws and caveats

• The four laws of thermodynamics (summary):

  • 0th law: If two systems are each in thermal equilibrium with a third system, they are in thermal equilibrium with each other. Establishes the concept of temperature.

  • 1st law: Energy can neither be created nor destroyed, only transformed (conservation of energy).

  • 2nd law: The total entropy of an isolated system tends to increase over time.

  • 3rd law: As temperature approaches absolute zero, the entropy of a perfect crystal approaches a minimum value (zero for a perfectly ordered crystal).

• Caveats about the 2nd law (Uffink, 2001):

  • Some view it as statistical rather than a strict law.

  • It may be incomplete or a derivation from statistical mechanics.

  • There are many formulations and interpretations; no single universally settled wording.

  • Historical debates led to committees (British Association, 1891) without definitive closure.

Schrödinger’s paradox (What is Life?)

• In 1943, Schrödinger asked how living organisms can maintain order (life) despite the 2nd law.

• Questions (Q1–Q2) guiding Schrödinger’s inquiry:

  • Q1: How can events in space and time within living organisms be accounted for by physics and chemistry?

  • Q2: What makes living organisms distinct from non-living matter?

• Schrödinger’s paradox: If life exists, the 2nd law seems violated, yet life clearly exists; thus the 2nd law appears violated though it cannot be violated.

• This is termed Schrödinger’s paradox.

Schrödinger’s paradox: examples and reasoning

• Example 1: Paramagnetism

  • Oxygen in a magnetic field tends to align with the field, but thermal motion keeps molecules oriented randomly.

  • Net alignment emerges as a macroscopic order due to large numbers of particles.

• Example 2: Brownian motion

  • Fog droplets move randomly due to collisions with air molecules; gravity causes a slow downward drift.

  • On average, the upper boundary of the fog sinks, showing macroscopic order from microscopic randomness.

• Example 3: Diffusion

  • Potassium permanganate dissolved in water diffuses from higher to lower concentration due to random molecular motion.

  • Macroscopic uniformity emerges from microscopic randomness.

• These examples illustrate how microscopic chaos can yield macroscopic order; the paradox is resolved by recognizing that life relies on large numbers and complex organization.

Schrödinger’s proposed resolution and responses

• RESPONSE 1 (Life needs gross structure): Life requires many cooperating particles to achieve statistical stability, order, and predictability.

• RESPONSE 2 (Need for information storage): Life also requires complex information storage (notably in aperiodic crystals) to maintain structure.

• The most essential living-cell component must be an aperiodic crystal capable of stable yet non-repetitive information storage.

  • Example 1: NaCl displays a periodic crystal structure.

  • Example 2: Quasicrystals are aperiodic yet ordered; they can store complex information.

• Conclusion: Schrödinger argues that life combines gross statistical structure with complex information storage (aperiodic crystals).

The diffusion simulation (Appendix)

• Appendix provides Python code to simulate diffusion in a 40×20 box:

  • 800 particles; 200 steps; random-walk updates; reflecting boundaries.

  • Visualized via Matplotlib; animation saved as an mp4 using ffmpeg.

• Purpose: illustrate diffusion as a practical demonstration of microscopic randomness leading to macroscopic order.

Connections and real-world relevance

• The Apollonian–Dionysian duality maps to how complex systems (cosmic, biological, cultural) arise from balancing order and chaos.

• The 2nd law constrains all closed systems, but open systems (like Earth and life) sustain local order by exporting entropy to the environment.

• Observational cosmology (Hubble, Baade, Hale) reshaped our view of the universe’s size, structure, and age, linking astronomy to foundational physics.

• Kant and Schrödinger bridge early modern philosophy with physics, showing how questions about life, structure, and knowledge connect to thermodynamics and cosmology.

• The diffusion model and quasicrystal concept illustrate how information and order can emerge from simple, local rules and interactions.

Key formulas to memorize

• Energy–mass equivalence: $E = m c^2$

• Boltzmann entropy: $S = k_B \ln \Omega$

• Planetary distances and units:

  • 1 AU ≈ $1.496 \times 10^{11} \text{ m}$

• Visible-light photon energy (example with 550 nm):

  • $E_{photon} = \dfrac{h c}{\lambda}$ with $\lambda = 550\ \text{nm},\ h = 6.626 \times 10^{-34} \ \text{J s},\ c = 3.0 \times 10^{8} \ \text{m/s}$

  • For 550 nm, $E_{photon} \approx 3.61 \times 10^{-19} \text{ J}$

• Solar constant: $GSC = 1361 \text{ W m}^{-2}$

• Cross-sectional area of Earth: $A = \pi R<em>\oplus^2,\quad R</em>\oplus \approx 6.37 \times 10^6 \text{ m}$

• Intercepted power by Earth: $P_\oplus = GSC \times A \approx 1.73 \times 10^{17} \text{ W}$

• Photon flux to Earth: $\dot{N}<em>{photons} = \frac{P</em>\oplus}{E_{photon}} \approx 4.79 \times 10^{35} \text{ s}^{-1}$

• Sun’s lifetime on main sequence: $\approx 10.5\ \text{billion years}$

• Estimated Sun lifetime using current luminosity and fusion energy: $\text{Lifetime} \approx \frac{L<em>\odot}{E</em>{fusion}} \approx 10.5\ \text{Gyr}$

• Universe age (Planck/ToI values): $t_{univ} \approx 13.8\ \text{Gyr}$