Oct 4 - Galileo to Newton Flashcards
Planetary Motion at the Time of Copernicus
1473 - by that time, tables of planetary motion based on the Ptolemaic model had become noticeably inaccurate. But few people were willing to undertake the difficult calculations required to revise the tables
Copernicus - Finding a better way to predict planetary positions:
Decided to try Aristarchus’s Sun-centered idea
Recognized the much simpler explanation for apparent retrograde motion offered by a Sun-centered system
Through complex mathematical details - able to discover simple geometric relationships which allowed him to discover each planet’s orbital period around the sun and its relative distance from the sun in terms of earth-sun distance
Why was the copernican model was converted many times after publishment?
While Copernicus had been willing to overturn Earth’s central place in the cosmos, he held fast to the ancient belief that heavenly motion must occur in perfect circles.
This incorrect assumption forced him to add numerous complexities to his system (including circles on circles much like those used by Ptolemy) to get it to make decent predictions.
In the end, his complete model was no more accurate and no less complex than the Ptolemaic model
How did Tycho start his discoveries?
Decided to observe a widely anticipated alignment of Jupiter and Saturn. To his surprise, the alignment occurred nearly 2 days later than the date Copernicus had predicted. Resolving to improve the state of astronomical prediction, he set about compiling careful observations of stellar and planetary positions in the sky.
Over a period of three decades, Tycho and his assistants compiled naked-eye observations accurate to within less than 1 arcminute—less than the thickness of a fingernail viewed at arm’s length.
Why was Tycho unsuccessful?
He was convinced that the planets must orbit the Sun, but his inability to detect stellar parallax [Section 2.4] led him to conclude that Earth must remain stationary.
He therefore advocated a model in which the Sun orbits Earth while all other planets orbit the Sun
Kepler’s initial Mars observations
Focused lots on Mars, which posed the greatest difficulties in matching the data to a circular orbit
Kepler found a circular orbit that matched all of Tycho’s observations of Mars’s position along the ecliptic (east–west) to within 2 arcminutes
Kepler’s key discovery was that planetary orbits are not circles but instead are a special type of oval called…
an ellipse
The long axis of the ellipse is called its…and half is…
major axis, each half of which is called a semimajor axis
The short axis of an ellipse is called…
the minor axis
By altering the distance between the two foci, you can draw ellipses of varying…
Eccentricity, a quantity that describes how much an ellipse is stretched out compared to a perfect circle
A circle is an ellipse with zero eccentricity, and greater eccentricity means a more elongated ellipse
Kepler’s 3 laws of Planetary Motion
1: The orbit of each planet about the Sun is an ellipse with the Sun at one focus
2: A planet moves faster in the part of its orbit nearer the Sun and slower when farther from the Sun, sweeping out equal areas in equal times
3: More distant planets orbit the sun at slower average speeds, obeying the precise mathematical relationship
KEPLER - 1: The orbit of each planet about the Sun is an ellipse with the Sun at one focus
Tells us that a planet’s distance from the Sun varies during its orbit
Its closest point is called perihelion (from the Greek for “near the Sun”) and its farthest point is called aphelion (“away from the Sun”)
The average of a planet’s perihelion and aphelion distances is the length of its semimajor axis
We refer to this simply as the planet’s average distance from the Sun.
KEPLER - 2: A planet moves faster in the part of its orbit nearer the Sun and slower when farther from the Sun, sweeping out equal areas in equal times
“Sweeping” refers to an imaginary line connecting the planet to the Sun, and keeping the areas equal means that the planet moves a greater distance (and hence is moving faster) when it is near perihelion than it does in the same amount of time near aphelion.
Perihelion = longer sweep
Aphelion = shorter sweep
KEPLER - 3: More distant planets orbit the sun at slower average speeds, obeying the precise mathematical relationship:
- p(squared) = a(cubed)
The letter p stands for the planet’s orbital period in years and a for its average distance from the Sun in astronomical units (AU)
AVG. speed declines with distance from the sun
RECALL: speed = distance/time
Not a straight line on graphs
3 objections to the Copernican Revolution:
Aristotle had held that Earth could not be moving because, if it were, objects such as birds, falling stones, and clouds would be left behind as Earth moved along its way.
The idea of noncircular orbits contradicted Aristotle’s claim that the heavens—the realm of the Sun, Moon, planets, and stars—must be perfect and unchanging.
No one had detected the stellar parallax that should occur if Earth orbits the Sun
Galileo’s answer to 1: Earth could not be moving because, if it were, objects such as birds, falling stones, and clouds would be left behind as Earth moved along its way.
Galileo defused the first objection with experiments that almost single-handedly overturned the Aristotelian view of physics
In particular, he used experiments with rolling balls to demonstrate that a moving object remains in motion unless a force acts to stop it (an idea now codified in Newton’s first law of motion
Galileo’s answer to 2: The idea of noncircular orbits contradicted Aristotle’s claim that the heavens—the realm of the Sun, Moon, planets, and stars—must be perfect and unchanging.
The second objection had already been challenged by Tycho’s supernova and comet observations, which demonstrated that the heavens could change. Galileo then shattered the idea of heavenly perfection after he built a telescope
Galileo’s answer to 3: No one had detected the stellar parallax that should occur if Earth orbits the Sun
The third objection—the absence of observable stellar parallax—had been of particular concern to Tycho. Based on his estimates of the distances of stars, Tycho believed that his naked-eye observations were sufficiently precise to detect stellar parallax if Earth did in fact orbit the Sun
Tycho’s argument required showing that the stars were more distant than Tycho had thought and therefore too distant for him to have observed stellar parallax. Although Galileo didn’t actually prove this fact, he provided strong evidence in its favor
What determines the strength of gravity?
Newton expressed the force of gravity mathematically with his universal law of gravitation.
3 statements of Newton’s law of Gravitation:
Every mass attracts every other mass through the force called gravity.
The strength of the gravitational force attracting any two objects is directly proportional to the product of their masses. For example, doubling the mass of one object doubles the force of gravity between the two objects.
The strength of gravity between two objects decreases with the square of the distance between their centers. (We therefore say that the gravitational force follows an inverse square law)
Mathematically, all three statements of the Universal Law of Gravitation can be combined into a single equation
Fg is the force of gravitational attraction, M1 and M2 are the masses of the two objects, and d is the distance between their centers
The symbol G is a constant called the gravitational constant, and its numerical value has been measured to be G = 6.67 × 10−11 m3/(kg × s2).
In what 4 ways did Newton extend Kepler’s laws?
Planets Are Not the Only Objects with Elliptical Orbits
Ellipses Are Not the Only Possible Orbital Paths
Objects Orbit Their Common Center of Mass
Orbital Characteristics Tell Us the Masses of Distant Objects
Planets Are Not the Only Objects with Elliptical Orbits
Kepler wrote his first two laws for planets orbiting the Sun, but Newton showed that any object going around another object will obey these laws.
EX: satellites orbit around Earth; orbiting object moves faster at the nearer points in its orbit and slower at the farther points
Ellipses Are Not the Only Possible Orbital Paths
Ellipses (which include circles) are the only possible shapes for bound orbits—orbits in which an object goes around another object over and over again
However, Newton discovered that objects can also follow unbound orbits—paths that bring an object close to another object just once
For example, some comets that enter the inner solar system follow unbound orbits. They come in from afar just once, loop around the Sun, and never return
More specifically, Newton showed that bound orbits are ellipses, while unbound orbits can be either parabolas or hyperbolas Together, these shapes are known in mathematics as the conic sections, because they can be made by slicing through a cone at different angles
Objects Orbit Their Common Center of Mass
We usually think of one object orbiting another object, like a planet orbiting the Sun or the Moon orbiting Earth.
However, Newton showed that two objects attracted by gravity actually both orbit around their common center of mass—the point at which the two objects would balance if they were somehow connecte
Orbital Characteristics Tell Us the Masses of Distant Objects
Recall that Kepler’s third law is written p2=a3, where p is a planet’s orbital period in years and a is the planet’s average distance from the Sun in AU
Newton found that this statement is a special case of a more general equation that we call Newton’s version of Kepler’s third law
This equation allows us to calculate the mass of a distant object if we measure the orbital period and distance of another object orbiting around it; moreover, we can use any units, not just years and AU
Newton’s version of Kepler’s third law also explains another important characteristic of orbital motion. It shows that the orbital period of a small object orbiting a much more massive object depends only on its orbital distance, not on its mass. That is why an astronaut does not need a tether to stay close to the spacecraft during a space walk
How do gravity and energy allow us to understand orbits?
The law of gravitation explains Kepler’s laws of planetary motion, which describe the simple and stable orbits of the planets, and Newton’s extensions of Kepler’s laws explain other stable orbits, such as the orbit of a satellite around Earth or of a moon around a planet
But orbits do not always stay the same. For example, you’ve probably heard of satellites crashing to Earth from orbit, proving that orbits can sometimes change dramatically
Orbital energy:
Orbital energy—the sum of its kinetic and gravitational potential energies—stays the same
This fact is a consequence of the law of conservation of energy
Orbits cannot change spontaneously
Left undisturbed, planets would forever keep the same orbits around the Sun
Energies in an orbit:
A planet orbiting the Sun has both kinetic energy (because it is moving around the Sun) and gravitational potential energy (because it would fall toward the Sun if it stopped orbiting).
The amount of kinetic energy depends on orbital speed, and the amount of gravitational potential energy depends on orbital distance
Because the planet’s distance and speed both vary as it orbits the Sun, its gravitational potential energy and kinetic energy also vary
Gravitational encounters
Although orbits cannot change spontaneously, they can change through exchanges of energy. One way that two objects can exchange orbital energy is through a gravitational encounter, in which they pass near enough that each can feel the effects of the other’s gravity
