Astronomy through History

Background

• The puzzle of astronomers in the past
  • Difficulty in predicting and explaining the motions of five planets: Mercury, Venus, Mars, Jupiter, and Saturn.
  • Specific puzzles include:
  1. Retrograde motion of planets.
  2. Motion of Mercury and Venus.

Phenomenon of Retrograde Motion

• Retrograde motion: Despite the general eastward motion of planets on the celestial sphere, they can speed up, slow down, or reverse direction temporarily.
  • Duration of retrograde motion can range from weeks to months.
• Prograde motion: Refers to the planet moving eastward (forward).
• Retrograde motion: Refers to the planet moving westward (backward).

Motion of Mercury and Venus

• Observations from Earth show that:
  • Mercury and Venus are always close to the Sun.
  • They can appear as:
  • Evening stars: Visible in the west just after sunset.
  • Morning stars: Visible in the east just before sunrise.

Models to Explain Motion

• Astronomers developed several models to address the puzzles:
  1. Geocentric model (proposed by Aristotle and other ancient Greeks).
  2. Heliocentric model (proposed by Nicolaus Copernicus).

Ptolemaic Geocentric Model

• Properties of the model:
  1. The Earth is at the center of the Universe.
  2. Celestial bodies move in uniform circular motions.
  3. Each planet moves along a small circle (epicycle) that uniformity revolves around Earth in a larger circle (deferent).
  4. Viewed from above the North Pole, motions are anticlockwise (eastward).
  5. Mercury and Venus’ deferents are attached to the Sun.

Explanation of Ptolemaic Model

• The model explains the puzzles by:
  1. Retrograde motion: This occurs due to the combination of the circular motions of epicycles and deferents.
  2. Motion of Mercury and Venus: With their deferents attached to the Sun, they appear to move alongside it from the perspective of Earth.

Copernican Heliocentric Model

• Properties of the model:
  1. The Sun is near the center of the Universe.
  2. Celestial bodies move in uniform circular motions.
  3. Earth self-rotates on its own axis.
  4. Only the Moon orbits Earth.
  5. Earth and other planets orbit the Sun.
  6. Viewed from above the North Pole, the circular motions are anticlockwise (eastward).

Explanation of Copernican Model

• The model simplifies explanations as follows:
  1. Retrograde motion:
  • Earth moves faster than superior planets and slower than inferior planets. When overtaking a superior planet, it appears to undergo retrograde motion; when overtaken by an inferior planet, it again appears retrograde.
  • The retrograde motion is apparent relative to background stars.
  1. Motion of Mercury and Venus:
  • The orbits of Mercury and Venus lie within Earth’s orbit (inferior planets):
    • Morning star: Before sunrise, as Earth rotates anti-clockwise, Mercury/Venus rises before the Sun.
    • Evening star: After sunset, as Earth continues to rotate anti-clockwise, Mercury/Venus appears after the Sun sets.

Problems with the Copernican Model

1. Copernicus still incorporated over 40 epicycles to align his model with observations.
2. Some predictions lacked accuracy compared to the Ptolemaic model; most phenomena could still be explained by it.
3. Apparent separation between fixed stars should change as Earth orbits the Sun, but this is never observed.
   • Explanation found later: Fixed stars are too far for noticeable changes.
4. The belief that if Earth moves, objects would not fall vertically was prevalent.
   • This misunderstanding was resolved through later developments in classical mechanics.
5. Religious opposition due to the model downgrading Earth’s significance.

Discoveries by Galileo

• Galileo used a telescope to observe celestial bodies, making significant discoveries that supported the Copernican heliocentric model:
  1. The Moon's surface is not smooth; it has mountains and valleys.
  • This suggested celestial bodies were not perfect.
  1. Sunspots appeared on the Sun, indicating that the Sun rotates roughly once a month.
  • This suggests that Earth may also rotate.
  1. Four small moons were discovered orbiting Jupiter, showing Earth is not the center of all celestial movements.
  2. Venus undergoes a full cycle of phases observable from Earth:
  • The Ptolemaic model could not accommodate this observation.
  • The heliocentric model provided strong evidence for Venus orbiting the Sun.

Kepler’s Laws of Planetary Motion

• Kepler discovered planets travel in elliptical orbits rather than perfect circles and formulated three laws:

Kepler's First Law

• Elliptical orbits: Planets orbit the Sun in ellipses, with the Sun at one focus.
  • Distances from the Sun vary:
  • Perihelion: Closest point to the Sun.
  • Aphelion: Farthest point from the Sun.

Kepler's Second Law

• Equal areas in equal times: The line connecting the Sun and a planet sweeps out equal areas during equal time intervals.
  • Implication:
  • A planet moves faster when closer to the Sun due to conservation of angular momentum.

Kepler's Third Law

• Harmonic Law: The square of a planet's orbital period T is directly proportional to the cube of its semimajor axis a of its orbit. Expressed as:
  • T^{2} eta a^{3}
  • Special case for planets in the same system: T^{2} = a^{3}.

Remarks on the Third Law

1. The equation is valid for planets in our Solar System, and for any celestial bodies orbiting a star of similar mass to the Sun.
2. Requires the central massive body to have significantly larger mass than the orbiting body.

Worked Examples

Example 1: Finding Orbital Period

• Given: Semimajor axis of Jupiter = 7.79 imes 10^{8} ext{ km}.
  • Conversion to AU:
  • ext{semimajor axis in AU} = rac{7.79 imes 10^{11}}{1.5 imes 10^{11}} = 5.19 ext{ AU}
  • Applying Kepler’s third law:
  • T^{2} = a^{3}
  • Thus, T = ext{sqrt}((5.19)^{3}) = 11.8 ext{ years}.

Example 2: Verification of Kepler’s Third Law

1. Collect data of orbital semi-major axis and period of planets in SI units.
2. Convert into AU and Earth years:
3. Use T^{2} = a^{3} and logarithmic transformation for graphing.
4. Slope calculated from the graph relates back to Kepler's findings.