Kepler's Laws and Newton's Universal Gravitation
Lecture Connections & Historical Context
Carly Simon's "You're So Vain": The lyric "Then you flew your Learjet up to Nova Scotia, to see the total eclipse of the sun" was played due to its relevance to the day's lecture, as the music often links to course material.
Giordano Bruno:
A philosopher burned naked at the stake in Rome in 1600.
His beliefs included: stars are other suns, other planets exist, and the general acceptance of the heliocentric hypothesis.
A monument in his honor is located in Rome in what was called the "Field of Flowers" (previously misidentified as the "Garden of Roses").
Autumnal Equinox: The lecture was given on the autumnal equinox, a time when day and night are of nearly equal length.
Common Misconceptions: The Seasons
Frosted Shredded Wheat Box (An Example): A box of Frosted Shredded Wheat attempted to explain the seasons:
"The different ways Earth tilts toward the sun is what makes our seasons." (Correct and well-stated).
"The tilt decides how much light and heat Earth gets from the sun." (Basically correct).
"It takes one year for Earth to revolve or take its trip around the sun." (Correct).
Incorrect Statement: "While it's revolving, different parts of Earth are closer to the sun than others."
While technically true that the distance varies throughout the orbit, this is not the reason for the seasons.
Differences in Earth-Sun distance have virtually no effect on seasons.
This incorrect idea fails to explain why the Northern and Southern Hemispheres experience opposite seasons.
Actual Reason for Seasons: The tilt of Earth's axis relative to its orbital plane.
Course Logistics
Star Parties:
Attendees can arrive anytime the doors are open.
Might need to wait a few minutes for entry if not there at the start.
Takes about half an hour to view three objects with the Unsalur telescope.
Homework 4: Due this Friday.
Quiz: In discussion section this week.
Kepler's Laws of Planetary Motion
Johannes Kepler: A superb mathematician hired by Tycho Brahe to analyze his astronomical data.
Key Figures in Astronomy/Physics: Students should know Copernicus, Galileo, Kepler, Tycho Brahe, and Newton. Dates are less important than knowing what they did.
Kepler's First Law: Orbits are Ellipses
Statement: Planetary orbits are ellipses, not circles, with the Sun at one focus.
Ellipse Definition: The set of all points such that the sum of the distances from two fixed points (called foci) is a constant (A+B = ext{constant}).
Drawing an Ellipse: Can be done with two tacks, a loop of string, and a pen, where the fixed length of the string ensures the sum of distances from the foci remains constant.
Eccentricity (e): A measure of how "squashed" an ellipse is.
Formula: e = rac{ ext{distance between foci}}{ ext{major axis}} (where the major axis is the longest diameter of the ellipse).
Circularity: If e=0, the foci coincide, and the ellipse is a perfect circle.
Shapes by Eccentricity:
Ellipse: 0 < e < 1 (e.g., e=0.5 or e=0.99 for very squashed ellipses).
Circle: e=0.
Parabola: e=1 (an open curve, essentially an infinitely elongated ellipse).
Hyperbola: e>1 (an even more open curve than a parabola, with two branches).
Semi-major axis: Half of the major axis, representing the average distance of a planet from the Sun in an elliptical orbit.
The Other Focus: For a planet orbiting the Sun, the Sun is at one focus, and nothing (Kepler's ghost, as a humorous aside) is at the other focus.
Conic Sections: Circles, ellipses, parabolas, and hyperbolas are collectively known as conic sections. They can be visualized by intersecting a plane with a double cone at different angles:
Circle: Plane perpendicular to the cone's axis.
Ellipse: Plane tilted relative to the axis, cutting entirely through one cone.
Parabola: Plane parallel to one side of the cone.
Hyperbola: Plane tilted even more, cutting through both cones.
Interstellar Objects: The hyperbolic orbit of an interstellar comet indicates it is not gravitationally bound to our Sun but is merely passing through the solar system.
Kepler's Second Law: Equal Areas in Equal Times
Statement: A line segment joining a planet and the Sun sweeps out equal areas in equal amounts of time (T1 = T2
ightarrow ext{Area}1 = ext{Area}2).Implication: Planets move faster when they are closer to the Sun (shorter lever arm, so they must travel a greater distance to sweep out the same area) and slower when they are farther away from the Sun (longer lever arm, so they travel less distance).
Explanation by Newton: Newton later explained this effect through gravity. When a planet is closer to the Sun, the Sun's gravitational pull is stronger, "whipping it around" faster.
"Slingshot" Effect: This principle is applied in space travel. Spacecraft (like Voyager) can use the gravity of massive planets (e.g., Jupiter) to gain kinetic energy and change trajectory, effectively "slingshotting" towards other destinations (like Saturn).
Observability:
Planets' orbits are nearly circular (low eccentricity), making the speed variation hard to discern directly.
Comets, however, often have very eccentric orbits, spending most of their time far from the Sun (moving slowly) and zipping past the Sun quickly (e.g., Halley's Comet, which has a 76-year orbital period and moves very rapidly when near the Sun).
Copernican vs. Keplerian: Copernicus used perfect circles for orbits, but even his model sometimes needed an offset Sun and epicycles to match observations. The true revolution of Copernicus was shifting the center of attraction from Earth to the Sun, not the perfection of orbital geometry.
Kepler's Third Law: The Harmonic Law
Statement: The square of a planet's orbital period (P) is directly proportional to the cube of its average distance from the Sun (R).
Formulas: P^2 imes R^3 or P^2 = K R^3, where K is a constant.
R (Distance): Technically the semi-major axis, but can be interchangeably used with average distance for most purposes.
Convenient Units (Astronomical Units - AU, Years):
By dividing the equation for an arbitrary planet by the same equation for Earth, the constant K cancels out:
rac{P{planet}^2}{P{Earth}^2} = rac{R{planet}^3}{R{Earth}^3}
ightarrow rac{P{planet}}{P{Earth}}^2 = rac{R{planet}}{R{Earth}}^3If Earth's orbital period (P{Earth}) is defined as 1 year, and its average distance (R{Earth}) as 1 Astronomical Unit (AU), the equation simplifies for calculations within our solar system:
P{planet}^2 = R{planet}^3 (where P is in years and R is in AU).
Example (Mars):
Orbital period of Mars (P_{Mars}) is 1.88 years.
To find its distance (R{Mars}): R{Mars}^3 = (1.88 ext{ years})^2 = 3.5344
R_{Mars} = ext{cube root of } 3.5344 imes 1.52 ext{ AU}This means Mars is about 50% farther from the Sun than Earth.
Exam Calculations: For exams, numbers will be "clean" (e.g., powers of 2, 3, or 10) to allow calculation without a calculator.
Kepler's Motivation: Kepler sought harmony and "music of the spheres," exploring mathematical relationships. He was fascinated by the P^2 = R^3 relationship due to its mathematical and musical properties.
Isaac Newton and the Unification of Physics
Newton's Goal: To explain a range of seemingly disparate phenomena under a common, unified set of rules and assumptions. This is a core purpose of science.
Genius: Newton is considered a unique genius, operating on a different plane than "ordinary geniuses." Like Michelangelo or Mozart, he had an extraordinary impact.
Personality and Life:
Reported to be unfriendly and reclusive.
Became Warden of the Mint later in life, known for giving severe punishments to counterfeiters (for which he was knighted).
Allegedly remained celibate his entire life and was proud of it.
Publication of Principia Mathematica (1686/1687):
Newton was slow to publish his work without "publish or perish" pressure.
Edmund Halley (of Halley's Comet fame), a friend with financial means, encouraged and funded the publication of Newton's masterpiece.
The Principia is a dense and obscure read, even for contemporary English speakers, requiring translation into modern, digestible forms.
Newton's Three Laws of Motion
Law of Inertia: If no net forces act on a body, its speed and direction of motion remain constant.
Contrary to Aristotle: Aristotle believed rest was the natural state of objects. Newton (and Galileo before him) understood that objects stop due to external forces like friction.
Acceleration: A change in speed or direction.
Example: Earth orbiting the Sun at a constant speed is still accelerating because its direction of motion is continuously changing.
Force and Acceleration: A net force acting on a body causes it to accelerate in the direction of the force. The acceleration (a) is directly proportional to the net force (F) and inversely proportional to the mass (m) of the object.
Formula: F = ma (or more precisely, oldsymbol{F} = m oldsymbol{a} for vector quantities).
Example: Kicking a tennis ball causes a large acceleration due to its small mass, while kicking a massive auditorium would cause an imperceptibly small acceleration due to its large mass.
Action-Reaction: For every action, there is an equal and opposite reaction. When two bodies interact, they exert equal and opposite forces on each other.
Example: When pushing on a table, the table pushes back with an equal force; otherwise, one would pass through it.
Apparent Discrepancy: When a person jumps off a chair, they accelerate downwards significantly, while Earth appears not to move. However, Earth does accelerate upwards, but its enormous mass means the acceleration (a = F/m) is infinitesimally small.
Newton's Law of Universal Gravitation
Context: While avoiding the plague, Newton supposedly observed an apple fall (likely apocryphal) and wondered if the same force pulling the apple was also keeping the Moon in orbit around Earth. This was a revolutionary idea, uniting terrestrial and celestial phenomena.
Statement: The attractive force of gravity (F) between any two objects with masses M1 and M2 is directly proportional to the product of their masses and inversely proportional to the square of the distance (D) between their centers.
Formula: F = G rac{M1 M2}{D^2}
Gravitational Constant (G): Newton did not know its value. It is notoriously difficult to measure and is the least precisely known fundamental physical constant.
Consistency with Newton's Third Law: The formula explicitly shows the force is attractive and equal in magnitude for both interacting objects because it's a product of both masses, thus satisfying the action-reaction principle.
Universality: Newton believed this law applied everywhere:
Moon orbiting Earth: Quantitative calculations showed the 1/D^2 relationship correctly predicted the Moon's acceleration.
Jupiter's Moons: Galileo's observations of Jupiter's moons orbiting it fit the same gravitational law.
Double Stars and Star Clusters: Gravity explains why they are bound together.
Galaxies and Galaxy Clusters: Confirmed by later observations, all governed by this law.
Deriving 1/D^2: Newton deduced the inverse square relationship by comparing the acceleration of an apple with that of the Moon. The Moon is about 60 Earth radii away; the acceleration ratio (1/60^2 = 1/3600) matched his predictions.
Newton's Thought Process: The Moon "Falling"
Steps of an Orbit: Newton conceived an orbit as a continuous process of an object "falling" while simultaneously moving with a perpendicular (tangential) velocity.
No Gravity: An object moves in a straight line at constant speed (Newton's 1st Law).
Gravity, No Initial Motion: An object falls directly downwards, accelerating.
Gravity + Perpendicular Motion: By combining these, Newton imagined infinitesimally small steps where an object moves slightly tangentially and falls slightly towards the gravitating body. The continuous sum of these tiny movements creates an orbit.
Invention of Calculus: To deal with these "infinitesimal" changes in time and distance, Newton had to invent calculus (both differentiation and integration).
Doughnut Demonstration: Illustrates that if gravity (represented by a string) is suddenly removed, an orbiting object (doughnut) continues to move tangentially in the direction it was instantaneously moving, not radially outward or spiraling.
Moon Constantly Falling: The Moon is constantly falling towards Earth, but due to its tangential velocity (instilled by early events like a Mars-sized object impacting Earth and forming the Moon), it continuously "misses" Earth and thus stays in orbit.
Cannonball Thought Experiment (from Principia):
Imagine a cannon firing a cannonball from a high mountain.
Low speed: Cannonball falls directly to Earth.
Increasing speed: Cannonball travels further before hitting the curved Earth.
"Orbital Velocity": If fired at a precise speed, the cannonball falls towards Earth at the same rate that Earth's surface curves away from it, resulting in a continuous orbit (a circular orbit in this specific ideal case).
Mass-Independent Acceleration:
Combining F = ma and F = G rac{M1 M2}{D^2}, for an object of mass M2 falling towards a larger body of mass M1:
M2 a = G rac{M1 M2}{D^2} a = G rac{M1}{D^2}This shows that the acceleration due to gravity (a) depends only on the mass of the central body (M1) and the distance (D), not on the mass of the falling object (M2).
Refutation of Aristotle: This mathematically explains why a heavy lead ball and a light tennis ball accelerate and fall at the same rate in a vacuum, a phenomenon Galileo observed experimentally (e.g., astronauts on the Moon dropping a feather and a hammer).
Aristotle's error stemmed from observations in viscous media (water) or with significant air resistance (feathers in air).
Newton's Generalization of Kepler's Third Law
Derivation: Newton derived and generalized Kepler's three empirical laws, showing they were not mere observations but consequences of universal gravitation and his laws of motion.
Generalized Constant: Newton derived the true mathematical expression for Kepler's constant (K): P^2 = rac{4 ext{ } ext{π}^2}{G(M1 + M2)} R^3
M_1 is the mass of the central body (e.g., Sun).
M_2 is the mass of the orbiting body (e.g., planet).
Implications of Constant K:
Kepler could not detect the M2 term because the mass of the Sun (M1) is vastly greater than any planet's mass (M2) (e.g., M{Sun} is 1,000 Jupiters, 320,000 Earths). Thus, (M1 + M2) \approx M_1, making K appear constant.
Modern, precise measurements reveal that K is not exactly constant and does depend slightly on the mass of the orbiting planet.
Measuring Masses using Kepler's Generalized Third Law:
By precisely measuring the orbital period (P) and average distance (R) of an orbiting object, and knowing G and constants like 4 ext{π}^2, we can determine the mass of the central body (M_1).
Mass of the Sun: Determined by observing Earth's orbital period and distance.
Mass of Earth: Determined by observing the Moon's orbital period and distance.
Mass of the Moon: Determined by observing the orbital period and distance of a satellite sent to orbit the Moon.
Significance: This generalization provides a universal method for measuring the masses of celestial bodies, representing a monumental advance in scientific understanding. It is applicable across vast scales from planets and moons to stars and galaxies.
Kepler's Third Law: The Harmonic Law
Statement: The square of a planet's orbital period (P) is directly proportional to the cube of its average distance from the Sun (R).
Formulas: P^2 \propto R^3 or P^2 = K R^3, where K is a constant.
R (Distance): Technically the semi-major axis, but can be interchangeably used with average distance for most purposes.
Convenient Units (Astronomical Units - AU, Years):
For calculations within our solar system, if Earth's orbital period (P{\text{Earth}}) is 1 year and its average distance (R{\text{Earth}}) is 1 Astronomical Unit (AU), the equation simplifies to:
P^2 = R^3 (where P is in years and R is in AU).
1. Calculating the Semi-Major Axis of a Comet:
Given: Orbital period (P) = 66 years
Formula: P^2 = R^3
Calculation:
(66 \text{ years})^2 = R^3
4356 = R^3
R = \sqrt[3]{4356}
R \approx 16.33 \text{ AU}Result: The semimajor axis of the comet's orbit is approximately 16.33 AU.
2. Calculating the Period of a Hypothetical Planet:
Given: Average distance from the Sun (R) = 36 AU
Formula: P^2 = R^3
Calculation:
P^2 = (36 \text{ AU})^3
P^2 = 36 \times 36 \times 36
P^2 = 1296 \times 36
P^2 = 46656
P = \sqrt{46656}
P = 216 \text{ years}Result: The period of the hypothetical planet would be 216 years.
No, according to Newton's First Law of Motion (the Law of Inertia), an external force is not needed to keep an object in motion at a constant speed and constant direction if no net forces (like friction) are acting on it. The notes state: "If no net forces act on a body, its speed and direction of motion remain constant." This directly contradicts the Aristotelian idea that rest is the natural state of objects and that a constant force is required to maintain motion.
According to Newton's Law of Universal Gravitation, the gravitational force (F) is directly proportional to the product of the masses (M1 and M2) and inversely proportional to the square of the distance (D) between their centers: F = G \frac{M1 M2}{D^2}.
If the Moon were three times as far away (3D) and six times as massive (6M2), the new force (F{new}) would be:
F{new} = G \frac{M1 (6M2)}{(3D)^2} = G \frac{6M1 M2}{9D^2} = \frac{6}{9} \left( G \frac{M1 M2}{D^2} \right) = \frac{2}{3} F{old}
Therefore, the gravitational force of the Earth-Moon system would change by a factor of \frac{2}{3} .
According to Newton's Second Law of Motion (F=ma), acceleration (a) is directly proportional to the force (F) and inversely proportional to the mass (m) (a = F/m). If the same force (F) is applied to both asteroids, then the ratio of their accelerations will be the inverse ratio of their masses.
Given:
Mass of Asteroid Jacob (m_J) = 8 \times 10^{21} kg
Mass of Asteroid Peter (m_P) = 2 \times 10^{20} kg
Applied Force (F) is the same for both.
Acceleration of Jacob: aJ = \frac{F}{mJ}
Acceleration of Peter: aP = \frac{F}{mP}
Ratio of Jacob's acceleration to Peter's acceleration:
\frac{aJ}{aP} = \frac{F/mJ}{F/mP} = \frac{mP}{mJ}
Substitute the given masses:
\frac{aJ}{aP} = \frac{2 \times 10^{20} \text{ kg}}{8 \times 10^{21} \text{ kg}} = \frac{2}{8 \times 10} = \frac{2}{80} = \frac{1}{40}
Therefore, the ratio of Jacob's acceleration to Peter'
According to Newton's generalized version of Kepler's Third Law, you would not expect Kepler's original formulation (P^2 \propto R^3) to be exactly correct because the constant of proportionality (K) is not truly universal for all orbiting bodies. Newton derived the true mathematical expression for this constant as P^2 = \frac{4 \pi^2}{G(M1 + M2)} R^3, where M1 is the mass of the central body (e.g., the Sun) and M2 is the mass of the orbiting body (e.g., a planet).
Kepler's original law implicitly assumed K was a constant for all planets. However, Newton's work showed that K depends on the sum of the masses of both the central body and the orbiting body (M1 + M2). While M2 (the planet's mass) is typically much smaller than M1 (the Sun's mass), making (M1 + M2) \approx M1 and K appear nearly constant, the slight variation in the mass of different planets ( M2) means that K is not exactly the same for every planet. Therefore, with highly precise measurements, the original P^2 \propto R^3 would show slight deviations due to the differing masses of the orbiting planets, as captured by Newton's more comprehensive formula.