Phys/Geog 182 Week 6 (Mon. class) Gravitation and the Motion of the Planets of the Solar System
The Copernican Revolution The model of the Greeks (attributed primarily to Ptolemy) had the Earth at the center of everything. Copernicus proposed a model of the solar system with the Sun at the center (hence a solar system), but the planets had circular orbits. Galileo made a telescope and used it to view the sky,
and saw that the phases of Venus refuted the geocentric model of the Greeks. Tycho Brahe and Johannes Kepler developed a more detailed heliocentric model with elliptical orbits. See OpenStax Astronomy sections 2.2 and 2.4 for the historical material. Retrograde motion of a planet occurs over several weeks,
and involves motion to the west, as compared to prograde (direct) motion, which is to the east (relative to the stars of the celestial sphere). The Geocentric Model of planetary motion (Greek philosophy) The Geocentric Model does explain retrograde motion, using concepts like deferent and epicycle. However,
it does not predict the motion with much accuracy, and does not predict phases of Venus (seen with a telescope). Ptolemys Model of planetary motion used deferents (big circles) and epicycles (little circles centered on a point that moves on the deferent). This involved up to 80 circles to describe 7 objects! Occams
Razor says: Simplify this! Nicholas Copernicus and his Heliocentric model of the Solar System explained this in a simpler way with the Sun at the center. Retrograde motion is seen in this model, using Earth and Mars as the example.
Retrograde Motion of Mars as seen from Earth Galileo Galilei and the Birth of Modern Astronomy Galileo built a telescope in 1609 and looked at the sky. Four objects: The Moon
The Sun Jupiter Venus (and much more) Galileo looked at the Moon and saw mountains, craters, valleys, and topography like you might find on the Earth.
The Moon was perhaps an object like the Earth! By projecting an image of the Sun, he could see imperfections on the Sun. Sunspots could be seen to move from east to west on the Sun and he deduced that the Sun rotated about once a month.
Galilean Moons of Jupiter Small point of light could be seen near Jupiter. By observation during several weeks he deduced that these were moons and that they revolved around Jupiter. Perhaps this planet was like the Earth, with several moons
of its own. It also seemed like a miniature model of the heliocentric solar system. Venus Phases in the Heliocentric model These are consistent with the observations in a telescope. Venus Phases in the Geocentric model are obviously wrong as
soon as you observe with a telescope. This refutes Ptolemy! Geocentric vs. heliocentric theories Both described the positions and movement of the Sun, Moon, and 5 visible planets, as seen without a telescope. The geocentric theory was too complicated (80 circles!). (Occams razor could be invoked to seek a simpler way.) Once the telescope was used to observe Venus, the
geocentric theory could not explain the phases of Venus. The heliocentric theory of Copernicus explained many of Galileos observations, but also used circular orbits. More accurate measurements did not agree with the simple theory of Copernicus (circles had to be replaced by ellipses in the newer theory of planetary motion by Kepler). After Copernicus
and Galileo, two major figures changed the way we come to understand the Universe: Keplers laws of planetary
motion Newtons laws of mechanics Further development of the heliocentric theory More detailed observations were made by Tycho Brahe (commonly called Tycho, 1546 - 1601). He made observations of a supernova in 1572
which convinced him that it was a distant star. He received an island and built an observatory to measure planetary motion to high accuracy over a period of more than 20 years. His observations were inherited by an assistant, Johannes Kepler, when Tycho died in 1601. Tycho Brahe
obtained data over a period of 21 years that were later used by his assistant Johannes Kepler to determine that planetary orbits are NOT circles,
but are ellipses. Johannes Kepler and the Laws of Planetary Motion Kepler used decades of Tychos observations in his mathematical calculations, to determine the shape of the planetary orbits, and the speed of the
planets as they went around the Sun. This massive effort resulted in three major statements about the characteristics of planetary orbits: Keplers three laws of planetary motion. Keplers three laws of planetary motion Orbital paths of the planets are ellipses.
An imaginary line connecting the planet with the Sun sweeps out equal areas of the ellipse in equal intervals of time. The square of a planets orbital period is proportional to the cube of its semi-major axis. Kepler published this in 1609, the same year that Galileo built his first telescope.
Keplers laws of planetary motion Keplers first law: The orbital paths of the planets are elliptical, with the Sun at one focus. Keplers second law: An imaginary line connecting the Sun to any planet sweeps out equal areas of the ellipse in equal intervals of time.
Keplers third law: The square of the planets orbital period is proportional to the cube of its semimajor axis. An Ellipse can be drawn with string and TWO foci For an ellipse, r1 + r2 = 2a
The eccentricity is defined as: e = c/a A circle results when e = 0 Some Properties of Planetary Orbits
Keplers laws of planetary motion Keplers first law: The orbital paths of the planets are elliptical, with the Sun at one focus. Keplers second law: An imaginary line connecting the Sun to any planet sweeps out equal areas of the ellipse in equal intervals of time. Keplers third law: The square of the planets
orbital period is proportional to the cube of its semimajor axis. Keplers Second Law: equal areas in equal time This also means higher speed at closer distances. Another graphic on Keplers Second Law:
The Astronomical Unit is about 150,000,000 km Keplers laws of planetary motion Keplers first law: The orbital paths of the planets are elliptical, with the Sun at one focus. Keplers second law: An imaginary line connecting the Sun to any planet sweeps out equal areas of the ellipse in equal intervals of
time. Keplers third law: The square of the planets orbital period is proportional to the cube of its semimajor axis. Keplers Third Law: P2 (in years) = a3 (in a.u.) Basically, it means that large orbits have long periods.
Real orbits have the center of mass as one focus For the Sun and planets, this is not a large effect. For binary stars, the center of mass
may be near the middle of the line connecting them. Newtons Laws of Physics
First law: inertia Second law: F = ma or acceleration = force / mass Third law: Action and Reaction
means that forces occur in pairs. These can be used to show that orbits should obey Keplers 3 laws. Isaac Newton developed a quantitative and explanatory theory of mechanics, explaining the motion of objects resulting from forces.
Newtons First Law: The law of inertia. An object will continue in its motion without change of velocity unless it is acted on by a net external force. Newtons Second Law: F = ma The acceleration of a mass is proportional to the total force acting upon it, and inversely proportional to the mass of the object.
Newtons Third Law: Action-reaction For every force acting upon an object (action), there is a force acting on another object (reaction) which has the same magnitude (size) but points (acts) in the opposite direction. Newton also developed the universal law of gravity.
Gravitational force varies with the distance between the objects. It depends on the product of the two masses, i.e., m1 x m 2 and on the inverse of the square of the distance between the masses (assuming they are small
compared with the distance). 1/r2 The Suns gravity causes planets to move on a path called an orbit. These orbits obey Keplers Laws. Newtons Laws explain Keplers Laws Newtons Laws account for all three of Keplers Laws.
The orbits of the planets are ellipses, but it is also possible to have orbits which are parabolas or hyperbolas. (conic sections) Edmond Halley predicted a comet would return in 1758 and every 76 years after that. (seen in 1910, 1986, and will return in 2061) Halleys comet has an elliptical orbit extending out past Neptune. William Herschel discovered Uranus in 1781 by accident. After 50 years it was seen to deviate from an elliptical orbit, and a calculation led to the discovery of Neptune in 1846.
To be precise, elliptical orbits would only occur if there were only the Sun and one planet. There are 8 planets and other objects which cause deviations from the perfect elliptical orbit. Reading for this section See the OpenStax book Read Ch. 3 (skipping the Examples) Skip the poetry if you like.
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