Untitled Notes

Key Figures: At the time of Galileo, significant advancements in the understanding of planetary motion were being made by Tycho Brahe and Johannes Kepler.
- Tycho Brahe (1546 - 1601): An observer who meticulously recorded the positions of celestial bodies.
- Johannes Kepler (1571 - 1630): A mathematician who built upon Brahe's data to formulate his laws of planetary motion.
Copernican Model: Brahe and Kepler placed Copernicus' heliocentric model on a sound mathematical basis, contributing to advances which later influenced Isaac Newton.

Establishment of the Observatory: Brahe established a highly regarded astronomical observatory on the island of Hven at age 30.
- His observatory was significant for its state-of-the-art instruments, which were the best before the invention of the telescope.
Observational Records: Over nearly 20 years, Brahe made continuous observations of the Sun, Moon, and planets, leading to extensive records of their positions.
- Notably observed a supernova, providing crucial data on celestial events, marking a shift in understanding stellar phenomena.
Challenges with Existing Models: Brahe found discrepancies between his detailed observations and the predictions made by Ptolemaic models but lacked the mathematical ability to improve upon these models himself.
Kepler’s Recruitment: Before Brahe's death in 1601, he employed Kepler to analyze his astronomical data, which laid the groundwork for further discoveries in planetary motion.

Overview: Kepler derived his laws from Brahe’s detailed observations, leading to a foundational shift in astronomy.
First Law (Law of Ellipses):
- Each planet orbits the Sun in an ellipse, with the Sun located at one of the focal points.
- This marked a significant departure from the long-held belief that orbits must be circular.
Second Law (Law of Equal Areas):
- The line segment joining a planet to the Sun sweeps out equal areas during equal intervals of time.
- This means planets move faster when closer to the Sun and slower when further away, leading to a variable orbital speed.
- This relationship reflects the conservation of angular momentum: as a planet approaches the Sun, it speeds up; as it recedes, it slows down.
Third Law (Law of Harmonies):
- The square of a planet’s orbital period (P) is directly proportional to the cube of the semi-major axis (a) of its orbit: P^2 ext{ proportional to } a^3 .
- This law allowed for the calculation of distances and periods of planets relative to one another, reinforcing the predictive power of heliocentric theory.

First Law: [ \text{Orbits are ellipses with the Sun at one focus.} ]
Second Law: [ \frac{A1}{A2} = \frac{t1}{t2} \text{ (areas swept out in equal times)} ]
- Where areas A1 and A2 are swept out in time intervals t1 and t2 respectively.
Third Law:
- For any planet orbiting the Sun, the relationship is expressed as: [ P^2 = a^3 ]
  where (P) is the orbital period in Earth years and (a) is the semi-major axis in astronomical units (AU).

Impact on Astronomy:
- Kepler’s laws provided a quantifiable way to understand planetary motion, thus allowing for predictions of planetary positions with improved precision.
- Kepler's laws represented a critical step toward modern celestial mechanics, although they remained descriptive without explaining the underlying forces.

Newton’s Framework: In 1687, Isaac Newton published Philosophiæ Naturalis Principia Mathematica, which provided a theoretical foundation for the observations and laws established by Kepler and others.
Newton's Laws of Motion:
- First Law: An object remains in a state of rest or uniform motion unless acted upon by an external force.
- Second Law: The change in motion of an object is proportional to the force acting upon it: [ F = m a ] where (F) is the force, (m) is mass, and (a) is acceleration.
- Third Law: For every action, there is an equal and opposite reaction.
Role of Gravity: Newton proposed gravity as a universal force acting on all bodies with mass, not just those on Earth, thus providing a systemic explanation for how planets follow elliptical orbits as per Kepler’s laws.

Gravitational Force Equation: The force of gravity between two masses is given by: [ F{gravity} = G\frac{M1 M2}{R^2} ] where (G) is the gravitational constant, (M1) and (M_2) are the masses of the two bodies, and (R) is the distance between their centers.
Implications:
- This law allows the calculation of the gravitational effect on satellites and planets, reinforcing the connection between gravitational forces and orbital mechanics.

Calculating Orbital Periods:
- Example: A planet with a semi-major axis of 50 AU would have an orbital period calculated via Kepler's third law: [ P^2 = a^3 ]. Result: (P ext{ is approximately } 350 ext{ years} ).
Validation of Kepler’s Law: Using known values of other planets (e.g., Venus and Earth) can confirm compliance with Kepler's Third Law:
- For Venus: (P^2 = 0.62^2 = 0.38), and for Earth: (P^2 = 1^2 = 1.00). Both adhere closely to the proportional relationship.

Escape Velocity: To leave Earth’s gravitational pull, a spacecraft must reach an escape velocity of approximately 11 km/s.
- Comparison: A bullet fired from a gun (~0.5 km/s) versus a spacecraft requiring much greater velocity.
Gravity-Assisted Maneuvers: Spacecraft can use the gravitational pull of planets to alter their trajectory, as seen in Voyager 2's journey through the outer solar system.
- This technique allows spacecraft to gain speed or redirect toward additional targets without expending substantial fuel.
Motions of Satellites:
- Satellites and spacecraft in orbit follow the principles established by Newton and Kepler, demonstrating practical applications of gravitational physics.