Unit 12: Beginnings of modern astronomy Modern Astronomy

Unit 12: Beginnings of Modern Astronomy

Course logistics recap
- Friday, September 12: no in-person class; a video lecture is provided and should be watched before the in-person class on Monday, September 15.
- A short quiz (Quiz Friday, September 12) will be available in Canvas and remains open until the start of class on Monday, September 15 so students can complete it any time before then.
- Class focus: Units 12 and 13; review of heliocentric vs geocentric models, retrograde motion, and epicycles; then introduction to the beginnings of modern astronomy with Tycho Brahe.
- Assignments: due every Wednesday by 11:59 PM; two types:
- Pre-class assignments: unlimited attempts before due date.
- Homeworks: two attempts before due date; second attempt can improve score; aim for ~100% on the homework.
- Videos and slides are posted in Canvas; will be shown and discussed in class.
Quick recap: geocentric vs heliocentric models and retrograde motion
- Geocentric model: Earth at the center; Earth is the center of the solar system; thought to be the center of everything in the cosmos.
- Heliocentric model: Sun at the center of the solar system; Earth and other planets orbit the Sun; supported by observations and later physics.
- Retrograde motion: planets occasionally move backward (opposite to the usual eastward motion across the sky) before continuing their normal prograde motion.
- Epicycles (geocentric explanation): planets move on small circular paths (epicycles) that themselves travel around the deferent; used to account for retrograde motion within a geocentric framework.
- In heliocentric theory, retrograde motion arises naturally from the relative motion of Earth and outer planets as Earth overtakes or is overtaken in its orbit; epicycles aren’t required.
- Both models can explain eclipses and seasons; historically, both were equally precise for many observations, but the question was which model better represents reality and is simpler (Occam's Razor).
- Occam’s Razor (as discussed): the simpler model tends to be preferred; the heliocentric model is simpler because it requires fewer ad hoc constructs (epicycles).
- Current consensus: the heliocentric model is correct; the geocentric model with epicycles is not the accurate description of reality.
Parallax as a key observational test
- Parallax: apparent shift in the position of nearby stars relative to distant background stars as the Earth orbits the Sun.
- Tycho Brahe (1546–1601) built precise instruments and could not detect stellar parallax with the measurements available at the time.
- Brahe therefore doubted the heliocentric model and favored a geocentric (or hybrid) view due to the lack of observed parallax.
- Parallax concept reminder: if the Earth moves around the Sun, nearby stars should show a tiny annual shift against faraway stars. The lack of detectable parallax at Brahe’s time was a major observational challenge for heliocentrism.
Tycho Brahe and Kepler: data-led shift toward heliocentrism
- Brahe’s data were extraordinarily precise for the era and were later used by his student Kepler.
- Kepler (using Brahe’s data) demonstrated that planetary orbits are not perfect circles; they are ellipses.
- Kepler’s context: though his teacher supported a geocentric view, Kepler’s analysis of Brahe’s data led him to solidly support a heliocentric model with elliptical orbits.
Kepler’s first law: planets move in elliptical orbits with the Sun at a focus
- Ellipse geometry basics:
- Ellipse has two foci; the Sun sits at one focus in a planet’s orbit.
- Semi-major axis: $a$ ; semi-minor axis: $b$ .
- Eccentricity: $e = rac{c}{a} = rac{ ext{distance from center to focus}}{a}, ext{ where } c = ae.</li><li>The distance between the two foci is$ 2c = 2ae $.</li><li>Kepler’s first law: planets orbit the Sun in ellipses with the Sun at one focus, not at the center.</li><li>Implication: the orbit is not a perfect circle; the distance to the Sun changes over the orbit, affecting orbital speed.</li><li>Elliptical parameters are often described by the semi-major axis$ a $and semi-minor axis$ b $; the magnitude of eccentricity$ e $determines how stretched the ellipse is.</li><li>Practical note: for many planets, the ellipses are nearly circular (small$ e $), but the exact deviation is essential for precise orbital dynamics.</li><li>Visual aid: an ellipse with center, major axis length, and a focus offset from the center where the Sun sits.</li></ul></li><li>Kepler’s second law: equal areas in equal times; varying orbital speed<ul><li>Statement: a line from the Sun to the planet sweeps out equal areas in equal time intervals.</li><li>Consequence: orbital speed is not constant:</li><li>Planets move faster when near the Sun (near perihelion).</li><li>Planets move slower when far from the Sun (near aphelion).</li><li>Intuition / diagram description: when the planet is far from the Sun, a given time interval sweeps a tall, narrow sector (large radial distance but small angular sweep); near the Sun, a similar time interval sweeps a larger angular sweep but over a shorter radial distance; the swept area remains constant.</li><li>Metaphor: a planet “coasts” slowly when far and speeds up when close to the Sun due to gravitational influence and angular momentum conservation.</li><li>Related term: areal velocity, which is proportional to the angular momentum of the planet around the Sun and remains constant for a given orbit.</li></ul></li><li>Kepler’s third law: P^2 ∝ A^3 (empirical relation) with units explained<ul><li>Kepler measured orbital periods and the semi-major axes for planets.</li><li>The third law relates the orbital period (P) to the semi-major axis (A) by:$ P^2 = A^3. $</li><li>P is the orbital period in years.</li><li>A is the semi-major axis in astronomical units (AU).</li><li>This law is empirical for planetary orbits observed in the solar system; later Newtonian gravity provides a theoretical basis for it.</li><li>Worked examples (from the lecture):</li><li>Saturn:<ul><li>Semi-major axis:$ A \,\approx\, 10 \,\text{AU} $(approximation; real value ~9.5–10 AU)</li><li>Compute$ A^3 = 10^3 = 1000 $; then$ P^2 = 1000\Rightarrow P = \sqrt{1000