The Laws of Planetary Motion and Newtonian Mechanics Study Guide

Contributions of Tycho Brahe and Johannes Kepler

• Historical Context: Around the time Galileo began his experiments with falling bodies, Tycho Brahe and Johannes Kepler significantly advanced the understanding of planetary motions, building a mathematical foundation for the Copernican heliocentric theory and setting the stage for Isaac Newton.

• Tycho Brahe (1546–1601):

  • Brahe was a Danish nobleman and the most significant pre-telescopic observer in Europe.

  • His interest sparked early, notably after studying an exploding star (supernova).

  • The Hven Observatory: With patronage from King Frederick II, he established an observatory on the island of Hven. For nearly 20 years, he maintained a continuous and precise record of the positions of the Sun, Moon, and planets.

  • Data Accuracy: He found that planetary positions differed from the published tables based on Ptolemy’s work. While he was a master observer, Brahe lacked the mathematical ability to create a better model from his data.

  • Transition to Prague: After Frederick II died in 1597, Brahe moved to Prague to serve as the court astronomer for Emperor Rudolf of Bohemia. There, he hired Johannes Kepler as an assistant.

• Johannes Kepler (1571–1630):

  • Kepler was a German mathematician and astronomer from a poor family who studied at the University of Tubingen.

  • He was a proponent of the Copernican heliocentric hypothesis.

  • Research with Brahe: Brahe tasked Kepler with finding a theory of planetary motion that matched the Hven observations. Brahe was initially guarded with his data, but Kepler gained full access after Brahe’s death in 1601.

  • Development of the Laws: Kepler spent over 20 years analyzing the data, eventually publishing his first two laws in The New Astronomy (1609).

Kepler’s Laws of Planetary Motion

• Definition of Orbit: The path an object takes through space.

• The First Law (Law of Ellipses):

  • Every planet moves around the Sun in an orbit that is an ellipse.

  • The Sun is located at one of the two foci of the ellipse.

• The Second Law (Law of Equal Areas):

  • A straight line joining a planet and the Sun sweeps out equal areas in space during equal intervals of time ($t$).

  • Orbital Speed Variation: This law implies that a planet moves faster when it is closer to the Sun and slower when it is farther away.

• The Third Law (Harmonic Law):

  • The square of a planet’s orbital period ($P$) is directly proportional to the cube of the semimajor axis ($a$) of its orbit.

  • Formula (Proportionality): $P^2 \propto a^3$

  • Formula (Standard Units): When $P$ is in Earth years and $a$ is in astronomical units (AU), the equation is $P^2 = a^3$.

  • Application: This law allows for the calculation of relative distances from the Sun based on orbital time.

Geometry and Properties of Ellipses

• Conic Sections: The ellipse is a member of a family of curves including the circle, parabola, and hyperbola, formed by the intersection of a plane and a cone.

• The Foci (Focus): Two special points inside the ellipse. The sum of the distances from these two points to any point on the curve is constant.

• Drawing an Ellipse: An ellipse can be constructed with two tacks (foci) and a loop of string. When the string is kept taut with a pencil, the resulting curve is an ellipse.

• Anatomical Parts of an Ellipse:

  • Major Axis: The widest diameter of the ellipse.

  • Semimajor axis ($a$): Half the major axis; it represents the average distance from the Sun in planetary orbits.

• Eccentricity ($e$):

  • The ratio of the distance between the foci to the length of the major axis.

  • Zero Eccentricity: Foci are in the same location, resulting in a circle.

  • Maximum Eccentricity: Approaching $1.0$, the ellipse appears nearly flat.

  • Example: Mars has an eccentricity of approximately $0.1$, meaning its orbit is nearly circular but the deviation is critical for understanding its motion.

Isaac Newton’s Laws of Motion

• Context: Born in 1643, Newton established a conceptual framework in Philosophiae Naturalis Principia Mathematica (1687) that unified the work of Galileo, Brahe, and Kepler.

• First Law (Law of Inertia):

  • Every object continues in a state of rest or uniform motion in a straight line unless compelled to change by an outside force.

  • Inertia: The tendency of objects to maintain their current state of motion.

• Second Law (Force and Acceleration):

  • The change of motion (momentum) of a body is proportional to and in the direction of the force acting upon it.

  • Acceleration: The rate of change in velocity. Newton showed that acceleration is proportional to the applied force.

• Third Law (Action and Reaction):

  • For every action, there is an equal and opposite reaction.

  • Forces always occur in pairs. If an object exerts a force on another, that object exerts an equal and opposite force back.

Physical Properties: Mass, Volume, and Density

• Mass: A measure of the total amount of material within an object.

• Volume: The measure of the physical space an object occupies, measured in cubic units (e.g., $cm^3$).

• Density: The ratio of mass to volume ($\text{Density} = \text{Mass} / \text{Volume}$). It describes how tightly packed material is.

• Density of Common Materials:

  • Gold: $19.3\,g/cm^3$

  • Lead: $11.3\,g/cm^3$

  • Iron: $7.9\,g/cm^3$

  • Earth (bulk): $5.5\,g/cm^3$

  • Rock (typical): $2.5\,g/cm^3$

  • Water: $1\,g/cm^3$

  • Wood (typical): $0.8\,g/cm^3$

  • Insulating Foam: $0.1\,g/cm^3$

  • Silica Gel: $0.02\,g/cm^3$

• Astronomical Extremes: Densities range from a comet’s tail ($10^{-16}\,g/cm^3$) to a neutron star ($10^{15}\,g/cm^3$).

Conservation Principles: Momentum and Angular Momentum

• Linear Momentum: Defined as the product of an object's mass ($m$) and its velocity ($v$). In the absence of an outside force, momentum is conserved ($p = m \times v$).

• Angular Momentum: A measure of the rotation of a body as it revolves around a fixed point.

  • Definition: The product of an object's mass, its velocity, and its distance from the center of rotation ($L = m \times v \times r$).

  • Conservation of Angular Momentum: If mass and distance change, velocity must adjust to keep the total angular momentum constant.

  • Examples:

    • A figure skater spins faster when pulling their arms inward (decreasing distance leads to increased velocity).

    • A collapsing star increases its rotation speed as its radius shrinks.

    • Kepler’s second law is a direct result of this conservation; as a planet's distance to the Sun decreases, its speed increases.

Newton’s Universal Law of Gravitation

• Hypothesis: Newton proposed that the same force causing an apple to fall on Earth (gravity) also keeps the Moon in orbit.

• Universal Attraction: Gravity is a mutual attraction between all material bodies in the universe.

• Inverse Square Law: The gravitational force ($F_{gravity}$) decreases with increasing distance ($R$) in proportion to the inverse square of the separation ($1/R^2$).

• Mass Proportionality: Force is proportional to the product of the masses ($M_1$ and $M_2$) of the two bodies.

• Gravitational Formula:

  • $F_{gravity} = G \frac{M_1 M_2}{R^2}$, where $G$ is the universal gravitational constant.

• Gravity and Acceleration: Force causes acceleration. At Earth's surface, gravity causes a downward acceleration of $9.8\,m/s^2$.

• Weight vs. Mass: Mass is constant, but weight is the force of local gravity. One would weigh less on the Moon or Mars due to weaker gravitational pull.

• Newton’s Version of Kepler’s Third Law: Newton generalized Kepler's law to include mass:

  • $a^3 = (M_1 + M_2) \times P^2$

  • In the solar system, because the Sun's mass is so dominant, $M_1 + M_2 \approx 1$, which is why Kepler did not notice the mass factor.

Orbital Mechanics in the Solar System

• Important Orbital Points:

  • Perihelion: The point in an orbit closest to the Sun where the object moves fastest.

  • Aphelion: The point farthest from the Sun where the object moves slowest.

  • Perigee: The point closest to Earth for an Earth-orbiting object.

  • Apogee: The point farthest from Earth for an Earth-orbiting object.

• Planetary Data (Average):

  • Mercury: $a = 0.39\text{ AU}$; $P = 0.24\text{ years}$; eccentricity = $0.21$.

  • Neptune: $a = 30.06\text{ AU}$; $P = 164.82\text{ years}$; eccentricity = $0.01$.

• Asteroids and Comets:

  • Asteroids: Mostly located in the asteroid belt between $2.2$ and $3.3\text{ AU}$ (between Mars and Jupiter).

  • Comets: Generally have larger orbits and much higher eccentricities (often $0.8$ or higher). They spend most of their time moving slowly far from the Sun.

Satellite Motion and Escape Velocity

• Circular Satellite Velocity: The speed required for an object to fall "around" Earth in a circle. This is approximately $8\,km/s$ ($17,500\,mph$).

• Launch Requirements: While a satellite stays in orbit due to gravity once positioned, immense energy is required to accelerate it to orbital speed outside the atmosphere.

• Orbit Decay: Satellites in very low Earth orbits may lose energy and fall due to atmospheric friction (drag).

• Escape Speed: The speed necessary to leave Earth forever and not be captured by its gravity. For Earth, this is approximately $11\,km/s$ ($25,000\,mph$).

• Interplanetary Flight: Spacecraft use gravity assists (flybys) from planets to gain or lose energy and redirect their trajectories (e.g., Voyager 2, Galileo).

• Weightlessness: Astronauts feel weightless not because of a lack of gravity, but because they are in a state of continuous free fall around Earth.

Multi-Body Interactions and the Discovery of Neptune

• Perturbations: Small gravitational disturbances on an orbit caused by the influence of other bodies besides the primary star.

• The Case of Uranus: After its discovery in 1781 by William Herschel, Uranus showed discrepancies in its predicted orbit even after accounting for Jupiter and Saturn.

• Discovery of Neptune (1846):

  • John Couch Adams and Urbain Le Verrier independently calculated that an unknown eighth planet was pulling on Uranus.

  • Adams provided coordinates to the British Royal Astronomer, while Le Verrier sent his findings to Johann Galle in Berlin.

  • Galle located Neptune on September 23, 1846, within one degree of Le Verrier’s predicted position.

Cultural and Historical Impact of Scientific Breakthroughs

• Shift in Worldview: The transition from old superstitions to a rational, manageable, and mechanical "clockwork" universe influenced literature and philosophy.

• Poetic Responses:

  • John Donne: Expressed anxiety and doubt regarding the loss of traditional certainties as the heliocentric theory emerged.

  • Alexander Pope: Celebrated the clarity and "light" that Newton's laws brought to the understanding of nature.

  • Arthur Hugh Clough: Voiced concerns that science reduced the universe and humanity to mere chemical and mechanical functions.

  • Robinson Jeffers: Suggested that astronomy serves to humble humanity by highlighting the vastness of the cosmos through "star-swirls."

Questions & Discussion

• Q: What would the orbital period of an asteroid be with a semimajor axis of 3 AU?

  • A: Using $P^2 = a^3$, we find $P^2 = 3^3 = 27$. Therefore, $P = \sqrt{27} \approx 5.2\text{ years}$.

• Q: By what factor would weight change if Earth had the same mass but double the radius?

  • A: Because force is inversely proportional to the square of the distance, weight would reduce by $(1/2)^2 = 1/4$.

• Q: If a star has twice the mass of the Sun, and a planet orbits it in 4 years, what is the distance?

  • A: Using $a^3 = M \times P^2$, we have $a^3 = 2 \times 4^2 = 32$. The distance $a$ is the cube root of 32, which is approximately $3.2\text{ AU}$.