ASTR 161: Orbits and Gravity Study Guide

Course Logistics and Performance Review

• Homework Assignments:     - The first homework assignment is scheduled to be posted to Blackboard after class.     - The due date is Tuesday, September 16, 2025 (09/16/2025) at 11:59 PM.     - The content of the assignment covers Lectures 1 through 4.
• Recap of Previous Topics:     - Navigation principles.     - The actual shape of the Earth.     - Apparent motions of the planets as seen on the sky.     - Stellar parallax as a measure of distance.     - The Ptolemaic model, which was fundamentally geocentric.
• Quiz Q1: Celestial Navigation:     - Question: Baltimore is roughly $39^\circ$ north of the equator. What is the altitude of the north celestial pole (NCP) as seen from Baltimore?     - Answer: $39^\circ$.     - Explanation: The altitude of the North Celestial Pole is equal to the observer’s latitude in the Northern Hemisphere.
• Quiz Q2: Stellar Parallax Calculation:     - Question: Wolf 359 is a red dwarf star that has appeared many times in works of fiction. It is one of the nearest stars to the Solar System and the stellar parallax as observed from Earth is $0.415$ arcseconds. What is the distance to Wolf 359 in parsecs?     - Options: A) $0.415\,\text{pc}$, B) $7.86\,\text{pc}$, C) $2.41\,\text{pc}$, D) $0.127\,\text{pc}$.     - Answer: C) $2.41\,\text{pc}$.     - Calculation: Using the formula $d = \frac{1}{p}$:         - $d = \frac{1}{0.415} \approx 2.41\,\text{pc}$.
• Quiz Q3: Telescopic Discoveries:     - Question: Galileo made many important discoveries with his telescopic observations of the sky. What did he discover about the Moon?     - Options: A) The Moon has no atmosphere, B) The Moon is made of cheese, C) The Moon causes the tides in Earth’s oceans, D) The Moon has mountains.     - Answer: D) The Moon has mountains.

Historical Developments in Astronomy and the Middle Ages

• Developments in the Islamic World:     - Significant advancements occurred through new mathematical techniques.     - Adoption of the concept of zero, which was originally imported from India.     - Critiques of the Ptolemaic Model:         - Hasan Ibn al-Haytham (1028): Early critic of the geocentric mechanics.         - Nasir al-Din al-Tusi (1261): Noted scholar depicted in murals at the Maragheh observatory, where he was recorded teaching astronomical principles.     - Regional Variations: Alternative geocentric models were developed in Al-Andalus (Islamic Spain), such as the work of Nur ad-Din al-Bitruji in the 12th century.
• Geo-heliocentric (Hybrid) Models:     - These models proposed that planets orbit the Sun, but the Sun itself orbits the Earth.     - Nilakantha Somayaji: Proposed such a model in Kerala, India (1500).     - Tycho Brahe: Proposed the Tychonic system in Denmark (1587).

The Shift to Heliocentrism: Copernicus and Galileo

• Nicolaus Copernicus:     - Main Work: De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres), published in 1543.     - Core Concept: Heliocentric model where the Earth and other planets orbit the Sun.     - Limitations: The model was not perfect and still relied on epicycles because Copernicus insisted on perfectly circular orbits.
• Galileo Galilei and the Telescope:     - Invention: The telescope was invented in the Netherlands in 1608.     - Improvement: Galileo built his own version with significant improvements and was the first to document systematic sky observations.     - Lunar Observations (1609): Galileo observed mountains and craters on the Moon, proving it was not a perfect, smooth sphere.     - Galilean Moons (1610): Discovered the four largest moons of Jupiter. This proved that not everything revolved around the Earth. The Juno spacecraft provided high-resolution imagery of these moons in June 2016.     - Practical Uses of Moons: Galileo proposed using the Galilean moons as a "celestial clock" for determining longitude. While impractical for use at sea, this method was utilized on land.     - Phases of Venus (1610-1611): Galileo observed that Venus goes through a full cycle of phases (like the Moon), which is only possible in a heliocentric system and impossible in the Ptolemaic geocentric system.

Tycho Brahe and Johannes Kepler

• Tycho Brahe (d. 1601):     - He was the last major astronomer to work without the aid of a telescope.     - Uraniborg (1576-1597): Established the first dedicated research observatory in Europe.     - Legacy: Produced a massive volume of highly accurate planetary observations that exceeded previous records of precision.
• Johannes Kepler (1571-1630):     - Kepler inherited Tycho Brahe's extensive observational data.     - He used this data to improve the heliocentric model, realizing that the orbit of Mars was not a circle, but rather an ellipse.

Kepler’s Laws of Planetary Motion

• Kepler’s First Law:     - Definition: The orbit of a planet is an ellipse with the Sun at one focus.     - Generalization: All orbits are conic sections with the orbited body at one focus.
• Geometry of Elliptical Orbits:     - Axes:         - Major axis: The longest diameter of the ellipse.         - Minor axis: The shortest diameter of the ellipse.         - Semi-major axis ($a$): Half of the major axis.         - Semi-minor axis ($b$): Half of the minor axis.     - Eccentricity ($e$): Measures how "flat" the ellipse is, determined by the distance between the foci.         - $e = \sqrt{1 - \left(\frac{b}{a}\right)^2}$     - Specific Distances:         - Pericentre distance (closest approach): $a \times (1 - e)$.         - Apocentre distance (farthest point): $a \times (1 + e)$.         - For Earth, the eccentricity is low: $e = 0.017$.
• Conic Sections in Orbits:     - Bound/Closed Orbits:         - Circle: $e = 0$.         - Ellipse: $0 < e < 1$.     - Unbound/Open Orbits:         - Parabola: $e = 1$.         - Hyperbola: $e > 1$.
• Kepler’s Second Law:     - Definition: A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.     - Implication: A planet moves more slowly when it is farther from the Sun and faster when it is closer (pericentre).
• Kepler’s Third Law:     - Definition: The square of a planet’s orbital period ($P$) is proportional to the cube of the semi-major axis ($a$) of its orbit.     - Formula: $P^2 \propto a^3$.     - Solar System Units: If using Earth years for $P$ and astronomical units (au) for $a$, the formula simplifies to:         - $P^2 = a^3$.     - Eccentricity Independence: The law depends only on the semi-major axis, not the eccentricity of the orbit.     - Example (Mars): Period ($P$) = $1.88\,\text{years}$; Semi-major axis ($a$) = $1.52\,\text{au}$.         - Check: $(1.88)^2 \approx 3.53$ and $(1.52)^3 \approx 3.51$.     - Lecture Question (Venus):         - Given the semi-major axis of Venus is $0.723\,\text{au}$, what is the orbital period?         - Calculation: $P = \sqrt{(0.723)^3} = \sqrt{0.3779...} \approx 0.615\,\text{years}$.         - Results: $0.615\,\text{years}$, which is approximately $225\,\text{days}$.