Week 3 Reading-Comprehensive Study Notes on Measuring the Sky and Lunar Cycles

Scientific Models, Theories, and Laws

• Fundamental Rules of Scientific Inquiry:

  • First Rule of Science: All scientific knowledge is open to revision based on new experiments or observations.

  • Second Rule of Science: Scientists utilize models as tools to aid understanding, make predictions, conduct experiments, and simplify complex physical phenomena.

• Types of Scientific Models:

  • Scale Models: Physical representations that are scaled-down or scaled-up versions of an actual object (e.g., placing a small model airplane in a wind tunnel to test wing aerodynamics and verify performance predictions).

  • Conceptual Models: Mental abstractions designed to capture the essential features of a system without representing every detail of its underlying reality.

  • Structural Steel Example: When calculating structural forces on a bridge girder, it is modeled as a continuous, solid piece of metal. This conceptual simplification is completely valid for macroscopic force calculations, even though at atomic scales on the order of $10^{-10}\,\text{m}$, the girder is actually a lattice of discrete atoms.

  • Astronomical System Example: Although the modern physical reality is represented by a heliocentric model (Sun at the center with Earth in orbit), a geocentric model (stationary Earth with celestial bodies moving around it) remains a highly effective conceptual tool for organizing daily observations of the rising and setting Sun.

• Theories and Laws:

  • Scientific Theory: A formal framework consisting of both a conceptual model and an explicit statement regarding the model's relative physical truth. For example, the heliocentric theory proposed a mathematical model of planetary orbits along with the physical claim that the Sun actually resides at the center of the Solar System.

  • Scientific Law: A theory that has been repeatedly confirmed by experimental testing and serves as a necessary component across a broad range of other scientific theories.

  • Revision of Laws: Scientific laws are not absolute truths and remain subject to revision. Sir Isaac Newton's law of universal gravitation was shown to be incomplete under extreme gravitational conditions and was superseded by Albert Einstein's theory of general relativity for very massive bodies. However, Newton's law retains its status as a fundamental law because of its wide domain of applicability to everyday physical phenomena.

Historical Foundations of Astronomy

• Practical Motivations Across Early Civilizations:

  • Early societies independently developed sophisticated astronomical systems primarily to construct accurate agricultural calendars (determining precise times for planting and harvesting) and for religious purposes.

  • Mayan Astronomy: The Mayans viewed eclipses as omens of evil; surviving codices contain detailed tables that accurately recorded past eclipses and predicted future solar and lunar eclipses.

  • Babylonian Astronomy: Situated along the Euphrates River roughly $70\,\text{miles}$ from present-day Baghdad, the Babylonians developed advanced sexagesimal (base-60) arithmetic. This mathematics established the modern division of $1\,\text{hour}$ into $60\,\text{minutes}$ and $1\,\text{minute}$ into $60\,\text{seconds}$. They maintained detailed observational logs of stars and planets, believing celestial movements foretold terrestrial events.

  • East Asian Astronomy: Astronomers in China, Japan, and Korea compiled continuous observational records spanning thousands of years, including early documentations of supernovas.

• The Greek Scientific Paradigm:

  • Beginning in the $7\text{th century BC}$, Greek astronomer-philosophers shifted astronomy from record-keeping and omen interpretation (the focus of earlier astronomer-priests) toward developing geometric and physical models to explain celestial motions, establishing astronomy as a formal science.

Ancient Understanding of the Earth's Shape and Size

• Ancient Knowledge of Earth's Geometry:

  • By $1492\,\text{AD}$, scholars thoroughly understood that the Earth was spherical. Greek philosopher-astronomers had already established its spherical shape and accurately measured its dimensions over $1,500\,\text{years}$ earlier.

• Observational Proofs for a Spherical Earth:

  1. Hulled-Down Ships: A receding vessel drops below the horizon well before reaching its visual distance limit. Climbing up a ship's mast restores the distant vessel to view, demonstrating surface curvature.

  2. Lunar Eclipse Shadow: During a lunar eclipse, the shadow cast by Earth onto the Moon is always circular. Aristotle ($384\,\text{BC} - 322\,\text{BC}$) emphasized that a sphere is the only solid shape that consistently produces a circular shadow regardless of orientation.

  3. Latitudinal Stellar Shifts: Traveling north or south reveals new constellations along the horizon while familiar stars drop out of sight, proving curvature along the meridian.

• Eratosthenes' Measurement of Earth's Circumference:

  • Eratosthenes ($276\,\text{BC} - 195\,\text{BC}$) calculated Earth's size using simultaneous solar observations on the summer solstice.

  • In Syene (modern-day Aswan), at noon on the summer solstice, the Sun was directly overhead, casting no shadow and shining straight down a deep well.

  • At the exact same time in Alexandria, a vertical pole cast a shadow showing the Sun was tilted at an angle of $7^\circ$ from vertical.

  • Assuming the Sun is extremely distant, its rays arrive parallel, meaning the shadow angle of $7^\circ$ at Alexandria equals the central angle subtended between Alexandria and Syene from Earth's center.

  • Because $7^\circ$ represents $\frac{7}{360}$ of a full $360^\circ$ circle, the ground distance between the two cities is $\frac{7}{360}$ of Earth's total circumference.

  • Eratosthenes hired a surveyor to pace the distance between Alexandria and Syene, measuring it at approximately $500\,\text{miles}$.

  • Calculation:

$\text{Circumference} = 500\,\text{miles} \times \frac{360}{7} \approx 25,000\,\text{miles}$

• This result is remarkably close to the actual modern value.

Relative Motions and Distances of the Earth, Moon, and Sun

• Ancient Evidence for Moon Proximity:

  • Greek astronomers determined that the Moon is closer to Earth than the Sun based on two physical observations:

  1. Solar Eclipses: The Moon passes directly in front of the Sun, casting its shadow on Earth's surface.

  2. Apparent Angular Velocity: The Moon completes a full circuit relative to background stars in under a month (roughly $28\,\text{days}$), whereas the Sun takes $1\,\text{year}$. Farther objects appear to move more slowly across the field of view (analogous to comparing a car passing nearby on the street to an aircraft flying at $35,000\,\text{feet}$).

• Aristarchus' Heliocentric Model and Geometric Measurements:

  • Aristarchus of Samos ($\text{ca. } 310\,\text{BC} - 230\,\text{BC}$) conducted pioneering astronomical measurements approximately $75\,\text{years}$ before Eratosthenes.

  • He was the first scientist to propose a heliocentric model (a discovery often misattributed to Nicolaus Copernicus $1,700\,\text{years}$ later).

  • Earth-to-Moon Size Ratio: By measuring the size of Earth's shadow during a lunar eclipse, Aristarchus deduced that Earth's radius was roughly $3\,\text{times}$ that of the Moon (the modern value is $3.7$).

  • Relative Sun-Moon Distance Determination: At the exact moment of a first quarter Moon, the Earth, Moon, and Sun form a right-angled triangle with the $90^\circ$ angle centered at the Moon. By measuring the observer angle $\theta$ between the Moon and Sun, relative distance $D_{\text{Sun}}$ can be determined via trigonometry:

$\cos(\theta) = \frac{D_{\text{Moon}}}{D_{\text{Sun}}}$

• Aristarchus measured $\theta = 87^\circ$, concluding that the Sun was $20\,\text{times}$ farther from Earth than the Moon ($D_{\text{Sun}} = 20 \times D_{\text{Moon}}$).

• Analysis of Measurement Error: The actual angle is $\theta \approx 89.8^\circ$, making the Sun over $400\,\text{times}$ farther away than the Moon. Because $\cos(\theta)$ changes rapidly near $90^\circ$, a small measurement error of less than $3^\circ$ resulted in a 20-fold underestimate.

• Relative Sun-Moon Size Determination: Because the Sun and Moon have virtually identical apparent angular sizes in the sky, physical size is directly proportional to distance.

• Since Aristarchus calculated the Sun to be $20\,\text{times}$ farther than the Moon, he concluded the Sun must be $20\,\text{times}$ larger in physical diameter than the Moon ($R_{\text{Sun}} = 20 \times R_{\text{Moon}}$).

• Combining this with Earth's radius being $3\,\text{times}$ the Moon's radius, he derived that the Sun was approximately $\frac{20}{3} \approx 7\,\text{times}$ larger in diameter than Earth.

• Philosophical Conclusion: Finding that the Sun was vastly larger than Earth made it difficult for Aristarchus to conceive that the larger Sun could orbit a smaller Earth, leading directly to his heliocentric hypothesis.

Lunar Phases and Visual Observations

• Apparent Daily Motion vs. Monthly Phase Progression:

  • Driven by Earth's rapid counter-clockwise rotation on its axis, both the Sun and Moon rise in the east and set in the west daily.

  • Because the Moon orbits Earth counter-clockwise in a little more than $28\,\text{days}$, it drifts steadily eastward relative to the Sun, setting roughly $1\,\text{hour}$ later each consecutive evening.

  • Synchronous Rotation: The Moon rotates on its axis in the exact same duration it takes to complete one orbit around Earth. Consequently, observers on Earth perpetually see the exact same face of the Moon.

  • Illumination Reality: Viewed from above the Solar System, exactly one-half of the Moon and Earth are always illuminated by sunlight, while the opposite halves are dark. Phases observed from Earth are light-and-shadow perspectives created by changing orbital positions.

• Detailed 28-Day Lunar Phase Cycle (Observer Facing South):

  • Day 1 (New Moon): The Moon and Sun rise and set simultaneously. The illuminated half faces entirely away from Earth, rendering the Moon completely invisible.

  • Day 2: The Moon sets roughly $1\,\text{hour}$ after the Sun, appearing as a thin, slender crescent near the western horizon.

  • Day 3: The Moon sets roughly $2\,\text{hours}$ after the Sun, appearing as a slightly thicker crescent in the southwest sky.

  • Day 4: The Moon sets over $3\,\text{hours}$ after the Sun, continuing to thicken.

  • Waxing Period: Days 1 through 14. The 2-week period where the visible lit fraction increases while the Moon shifts progressively eastward relative to the Sun.

  • Day 7 (First Quarter): The Moon completes $\frac{1}{4}$ of its 28-day cycle. At sunset, it sits high in the southern sky. It appears half-lit (round on the right, flat on the left) and sets approximately $6\,\text{hours}$ ($\frac{1}{4}\,\text{day}$) after the Sun.

  • Days 8–13 (Waxing Gibbous): The Moon appears "hump-backed" (round on the right, oblong on the left). It sets between $6\,\text{hours}$ and $12\,\text{hours}$ after sunset, appearing farther east each evening at sunset.

  • Day 14 (Full Moon): Halfway through the 28-day cycle. Earth sits directly between the Sun and Moon. The full Moon rises in the east as the Sun sets in the west, reaches high overhead at midnight, and sets in the west at sunrise.

  • Waning Period: Days 15 through 28. The 2-week period where the lit surface progressively shrinks, with the left side remaining round and the right side withering away. The Moon rises progressively later in the night.

  • Days 15–20 (Waning Gibbous): Appears round on the left and oblong on the right. Rises late in the evening after sunset.

  • Day 21 (Third Quarter): The Moon completes $\frac{3}{4}$ of its cycle ($21\,\text{days}$). It rises in the east at midnight, appears half-lit (round on the left, flat on the right), and sets at noon the following day.

  • Days 22–27 (Waning Crescent): Rises in the early morning hours shortly before sunrise, visible primarily near the eastern horizon in dawn light.

  • Day 28 (New Moon): The Moon realigns with the Sun, rising and setting concurrently to restart the cycle.

Lunar Folklore, Superstitions, and Terminology

• Definition of Superstition: A belief, conception, act, or practice resulting from ignorance, unreasoning fear of the unknown or mysterious morbid scrupulosity, trust in magic or chance, or a false conception of causation.

• Traditional Agricultural and Human Superstitions:

  • Standard Crops: Plant during waxing phases (growing Moon); harvest during waning phases (shrinking Moon).

  • Underground Crops (e.g., Carrots): Exception to standard farming rules—harvest during waxing phases and plant during waning phases.

  • Timber and Fruit Harvesting: Pick fruit and cut firewood during waning phases to prevent early rotting.

  • Hair Care: Hair should be cut during waning phases to prevent it from growing back too fast.

  • Mental Health: Reading by moonlight was historically purported to cause lunacy.

• Traditional Names of Full Moons:

  • January: Old Moon

  • February: Snow Moon

  • March: Sap Moon

  • April: Egg Moon

  • May: Milk Moon

  • June: Honey Moon

  • July: Hay Moon

  • August: Grain Moon

  • September: Fruit Moon

  • October: Hunter’s Moon

  • November: Beaver Moon

  • December: Yule Moon

  • Blue Moon: The second full Moon occurring within a single calendar month.

  • Grass Moon: The full Moon occurring closest to the first day of Spring.

  • Harvest Moon: The full Moon occurring closest to the first day of Autumn.