Astronomy: Solar System Formation, Kepler's Laws, Tides, and the Moon

Introduction to Forces and Mathematical Models

Newton's Second Law: The phrase "may the force be with you" is scientifically inaccurate; it should be "may the net force be with you," reflecting that $\textbf{F} = m\textbf{a}$ (force equals mass times acceleration), and it's the net force that determines acceleration.
Ptolemy's Epicycles and Fourier Analysis:
- Ptolemy's geocentric model used epicycles (circles moving on larger circles called deferents) to explain planetary motion, including retrograde motion.
- More precise observations would have shown inaccuracies in simple epicyclic motion.
- Ptolemy could have added "epicycles on top of epicycles" to achieve arbitrarily good precision.
- This mathematical technique, developed over a millennium later, is known as Fourier Series or Fourier Analysis.
  - Principle: Any periodic function can be represented as a sum of sines and cosines with different sizes (amplitudes), periods, and phases.
  - Analogy: Representing a shape by summing circles, similar to how epicycles worked.
  - Example: Drawing Homer Simpson with approximately 1,000 epicycles demonstrates how complex shapes (or periodic functions, by drawing it repeatedly) can be accurately reproduced using this principle.

Administrative Announcements

Office Hours: Regular office hours today (no pickle/infrared camera demos). Additional office hours on Friday, 1:10-2:00 P.M. in 121 Campbell, to accommodate students with Monday/Wednesday/Friday 10-11 A.M. classes that are not recorded.
Star Party: Tonight, 7:45-9:30 P.M., on the sixth floor (take the stairs). Weather permitting.
Quiz and Discussion Section: Nearly half of the class has completed it. Remaining students still have time.
Homework Four: Due this Friday. "Talc" sessions (peer help) available today and tomorrow.

Newton's Version of Kepler's Third Law

Unification of Phenomena: Newton's version of Kepler's Third Law unifies seemingly diverse phenomena, such as Jupiter's moons orbiting Jupiter and Earth's moon orbiting Earth, by applying the same physics to any celestial body.
Kepler's Original Law: The square of the orbital period ( $P^2$ ) is proportional to the cube of the semi-major axis ( $a^3$ ), i.e., $P^2 = Ka^3$ . The constant $K$ was thought to be universal for all planets orbiting the Sun.
Newton's Derivation of the Constant: Newton derived the mathematical expression for the constant K: $P^2 = \frac{4\pi^2}{G(m1 + m2)} a^3$
- Where:
 - $P$ is the orbital period.
 - $G$ is the gravitational constant.
 - $m_1$ is the mass of the central body.
 - $m_2$ is the mass of the orbiting body.
 - $a$ is the semi-major axis of the orbit.
Approximation for Disparate Masses:
- If one mass ( $m1$ ) is much greater than the other mass ( $m2$ ), i.e., $m1 \gg m2$ .
- Then the sum $(m1 + m2)$ is approximately equal to $m1$ . ( $m1 + m2 \approx m1$ ).
- In this common scenario, Newton's Third Law simplifies to:
 $P^2 = \frac{4\pi^2}{Gm_1} a^3$
- This approximation is justified because $m_2$ is comparatively tiny.
Applications: This formula allows us to determine the mass of the central body by measuring the orbital period and semi-major axis of an orbiting object:
- Earth orbiting the Sun $\implies$ Sun's mass.
- Moon orbiting Earth $\implies$ Earth's mass.
- Human-made satellites orbiting the Moon $\implies$ Moon's mass.
- Jupiter's moons orbiting Jupiter $\implies$ Jupiter's mass.
- This again highlights the unification of seemingly disparate phenomena under one physical law.
Think-Pair-Share: The constant K for Jupiter vs. Earth:
- Question: How does the constant $K$ in Newton's version of Kepler's Third Law for Jupiter orbiting the Sun compare with that for Earth orbiting the Sun?
- Options: A) much smaller, B) slightly smaller, C) exactly equal, D) slightly greater.
- Correct Answer: B) slightly smaller.
- Explanation:
 - $K = \frac{4\pi^2}{G(m1 + m2)}$ .
 - $m_1$ is the mass of the Sun in both cases.
 - $m2$ for Jupiter is much larger than $m2$ for Earth (Jupiter is about 320 times Earth's mass).
 - Therefore, $(m{\text{Sun}} + m{\text{Jupiter}})$ is a slightly larger number than $(m{\text{Sun}} + m{\text{Earth}})$ .
 - The reciprocal of a slightly larger number is a slightly smaller number. Thus, $K{\text{Jupiter}}$ is slightly smaller than $K{\text{Earth}}$ .
Triumph of Science: This prediction by Newton, later confirmed by increasingly precise observations, exemplifies a triumph of science: making specific quantitative predictions about future observations, not just explaining past ones.
Approximations in Science:
- An approximation is never an exact equality unless the ignored factor is zero.
- Example 1: Newtonian vs. Einsteinian Physics: For airplanes on Earth, Newtonian physics is sufficient and accurate enough for pilots; Einstein's General Theory of Relativity effects are too small to be practically relevant. Similarly, for medieval cannonballs, general relativity was irrelevant compared to air resistance.
- Example 2: Moon's mass in Earth-Moon system: Ignoring the Moon's mass (approx. 1/80th of Earth's mass, or 1/100th for round numbers) in calculations for Earth's mass would lead to a ~1% error. This might be acceptable depending on the required precision of the application, as it greatly simplifies the calculation.
- Importance of Understanding Approximations: Scientists must understand the level at which approximations affect their results. In this course, subtle details that rely on extreme precision will not typically be central to multiple-choice questions unless the point of the question is specifically to identify that subtlety (like the Jupiter-Earth K comparison).
- Deeper Weeds example (Heisenberg Uncertainty Principle): Electron energy levels are often presented as exact, but due to the Heisenberg uncertainty principle, electrons do not reside infinitely long in specific levels, meaning their energy levels are not precisely known, although they are highly precise for practical purposes in this course.

Formation of the Solar System: The Nebular Hypothesis

Observed Facts about Our Solar System:
- All planets orbit the Sun in the same direction (counter-clockwise when viewed from the North).
- Most planets rotate in the same direction as their revolution.
  - Venus is an exception (upside-down rotation).
  - Uranus is on its side.
- Most planets lie in nearly the same plane (within a few degrees).
  - Pluto (a dwarf planet) is a significant exception.
Early Explanations: Kant-Laplace Nebular Hypothesis:
- Origin: Proposed by Immanuel Kant and Pierre-Simon Laplace several centuries ago.
- Core Idea: The solar system initially formed from a cloud of gas and small dust particles.
- Process:
  1. Gravitational Contraction: The cloud began contracting due to Newton's law of gravity.
  2. Slow Initial Spin: The cloud possessed a slow initial spin (e.g., perhaps due to a nearby passing star).
  3. Conservation of Angular Momentum: As the cloud contracted, its spin rate dramatically increased.
    - Law: In the absence of external forces (torques), the product of an object's mass ( $m$ ), its radial size ( $r$ ), and its rotation speed ( $v$ ) remains constant (or more rigorously, $L = I\omega$ where $I$ is moment of inertia and $\omega$ is angular velocity).
    - Demonstration: An ice skater drawing her arms inward reduces her radial size, causing her to spin much faster.
  4. Formation of a Disk: As the spinning cloud contracted and sped up:
    - Centrifugal Force: Particles in the rotating cloud experience an outward centrifugal force (an apparent force in a rotating frame of reference, like feeling plastered against the wall in a spinning amusement park ride).
    - Equatorial Settling: Along the plane of rotation (equatorial plane), the outward centrifugal force counteracts gravity, resisting inward motion. Perpendicular to this plane, there is no significant centrifugal force, allowing gravity to pull particles inward more easily.
    - Result: This natural process leads to the formation of a thin, flat disk of material.
  5. Star and Planet Formation:
    - The Sun forms in the dense center of this disk.
    - Planets form in the outer parts of the disk as material clumps up into larger particles (planet-forming details are still an active area of research).
  6. Clearing of Residual Gas and Dust: Once the Sun and planets have formed, young stars have an enhanced stellar wind (like our Sun's solar wind).
    - This intense early wind blows away the remaining gas and dust, leaving the planetary system relatively devoid of gas between the planets.
Evidence from Observations (Testing the Hypothesis):
- Orion Nebula: An example of a region actively forming stars for millions of years, containing numerous gravitationally contracting and spinning clouds of gas and dust.
- Hubble Space Telescope Images: Observations show protoplanetary disks around young stars, featuring rings and bright, dense spots where planets are believed to be currently coalescing.

Terrestrial Planets: Characteristics and Earth's Structure

Definition: Terrestrial planets (Terra = Earth) are small, rocky, and relatively close to the Sun, in contrast to the larger, gaseous/liquid/icy giants farther out.
General Trends (Don't memorize exact numbers, focus on patterns):
- Proximity to Sun: They are close to the Sun.
- Orbital Periods: They have short orbital periods (a direct consequence of being close to the Sun, according to Kepler's Third Law).
- Size: They are small in radius and mass.
  - Earth is the largest.
  - Venus: ~5% less radius, ~20% less massive than Earth.
  - Mercury: ~1/20th Earth's mass.
  - Mars: ~1/10th Earth's mass.
- Density: They are dense (e.g., Earth's density is much higher than water's $1\text{ g/cm}^3$ ).
- Rotation Rate: Highly variable.
  - Earth and Mars are similar.
  - Mercury and Venus are quite different (Venus is retrograde, rotation period much longer).
- Axial Tilt: Variable.
  - Earth ( $23.5^\circ$ ) and Mars are similar.
  - Mercury has $0^\circ$ tilt.
  - Venus has $177^\circ$ tilt (effectively upside down).
- Moons: Few moons.
  - Mercury and Venus have none.
  - Earth has one (the Moon).
  - Mars has two tiny, diminutive moons.
- Internal Structure: Characterized by rocky outer parts and iron cores (often partly molten).
Earth-Specific Details:
- Internal Structure: Rocky and oceany outer crust, with increasing temperature inward, leading to a partly solid and partly molten iron core.
  - Molten Core Significance: A partly molten core is essential for Earth's healthy magnetic field.
- Plate Tectonics (Continental Drift): The upper mantle and crust float on a soft, churning (convecting) layer (the lower mantle).
  - Consequences: Where these tectonic plates intersect, great geological activity occurs: mountains, volcanoes, and earthquakes.
  - Examples: The Hayward Fault (under Cal Memorial Stadium), the San Andreas Fault, and the "Ring of Fire" around the Pacific Plate (many earthquakes, subduction zones forming volcanoes like those in the Pacific Northwest).
  - Mid-Atlantic Ridge: Plates are moving apart, with magma rising to form a mountain chain. Iceland is a hotspot on this ridge, utilizing geothermal energy for much of its power.
- Atmosphere: Primarily nitrogen, with oxygen being a byproduct of ancient life (photosynthesis by organisms like stromatolites). The atmosphere is very thin.

Tides on Earth

Observation: High and low tides (e.g., extreme tides in the Bay of Fundy), typically two cycles per day.
Primary Cause: The Moon's gravitational pull.
Newton's Initial Simplification: To a first approximation, the Moon-Earth gravitational interaction can be calculated as if their entire masses are concentrated at single points (a concept for which Newton developed integral calculus).
Differential Forces and Two Tides:
- Distance Dependence: Gravity follows an inverse-square law ( $F = G \frac{m1 m2}{D^2}$ ).
- The Moon exerts a stronger gravitational force on the near side of Earth (smaller $D$ ) than on Earth's center, and a stronger force on Earth's center than on the far side of Earth (larger $D$ ).
- Conceptualizing Differential Force: If we consider the force relative to Earth's center (by subtracting the force on the center from all points):
 - On the side nearest the Moon, there's a net force pulling outward toward the Moon.
 - On the side farthest from the Moon, there's a net force pulling outward (away from the Moon).
- Result: Earth is gravitationally "stretched" along the Earth-Moon line, causing two bulges in the water: one facing the Moon and one opposite the Moon.
- Tidal Cycle: As Earth rotates on its axis, any given location experiences these two bulges (high tides) and the two regions of lower water level (low tides) that occur between the bulges, resulting in two high tides and two low tides approximately every 24 hours.
- Reality vs. Idealization: Local geography (bays, continents) and latitude can significantly alter the timing and height of tides, making them more complex than the idealized model.

The Moon's Synchronous Rotation

Observation: We always see the same face of the Moon from Earth, regardless of its phase.
Implication: The Moon is synchronously rotating with Earth, meaning its rotation period around its own axis is exactly equal to its orbital period around Earth (approximately one month).
- Demonstration: If the Moon did not rotate on its axis as it orbited Earth, we would see all sides of it over the course of its orbit. But because it rotates exactly once per orbit, the same side consistently faces Earth.
Cause (Not a Coincidence): This is a fundamental physical phenomenon, not a coincidence (unlike the apparent size of the Sun and Moon during eclipses).
- Early History: The Moon formed from a debris disk around Earth after a Mars-sized object impacted Earth early in its history. The newly formed Moon was very close to Earth and spinning very rapidly.
- Earth's Tidal Forces on the Moon: Earth raised enormous tides on the early Moon. Although the Moon lacks oceans, even its rocky material was significantly elongated (stretched) by Earth's gravity.
- Tidal Braking (Friction): As the rapidly spinning, elongated Moon orbited Earth, different parts of its elongated shape kept facing Earth, causing constant deformation within the Moon's rocks. This internal friction generated heat, which was radiated away.
- Energy Transfer: This dissipated energy had to come from somewhere: the Moon's rotational kinetic energy. This process, known as tidal braking, caused the Moon's rotation to slow down over time.
- Stable Configuration: The Moon continued to slow its rotation until its rotation period matched its orbital period. At this point, the elongated shape of the Moon is permanently aligned with Earth, eliminating the differential stress and, therefore, the internal friction and energy loss. This creates a stable, tidally locked configuration.

Centrifugal Force Demonstration

Illustrating Disk Formation: The concept of centrifugal force causing material to spread outwards in a rotating system was to be demonstrated with a hose taking on a radial configuration when spun, relevant to the formation of the Sun and planets from a spinning disk.