Celestial Mechanics Lecture 1 - Part 3 PDF

Lecture 1 – Part 3 Celestial mechanics This Photo by Unknown Author is licensed under CC BY-NC-ND  Outline History (cont.) – Observations Celestial objects appear to revolve aro...

Lecture 1 – Part 3 Celestial mechanics This Photo by Unknown Author is licensed under CC BY-NC-ND  Outline History (cont.) – Observations Celestial objects appear to revolve around Earth – Geocentric model Planetary retrograde motion – Heliocentric/geoheliocentric model Galileo’s observations with the telescope – Disproved the geocentric model Efbrazil, CC BY-SA 4.0, via Wikimedia Commons  Galileo’s observations with the telescope With the invention of the telescope in 1609, observations made by Galileo Galilei called into question some of the tenets of geocentrism. Stellarium Web Online Star Map  Moons of Jupiter Galileo could see the moons of Jupiter and stated that they orbited around Jupiter. – Galileo observed small “stars” close to Jupiter. Observations on subsequent nights showed that the positions of these “stars” relative to Jupiter were changing. He concluded that they were orbiting Jupiter: he had discovered Jupiter’s Jan Sandberg, Attribution, via Wikimedia Commons largest moons.  Not everything revolved around Earth This was a significant claim as it would mean not only that not everything revolved around Earth as stated in the geocentric model, but also showed a secondary celestial body could orbit a moving celestial body.  Venus exhibits a full set of phases Galileo observed that Venus exhibits a full set of phases similar to that of the Moon. – Galileo saw Venus at first small and full, and later large and crescent. This Photo by Unknown Author is licensed under CC BY-SA  Venus orbits the Sun This proved that Venus orbits the Sun and not Earth, and disproved the geocentric model.  Summary History – Observations Celestial objects appear to revolve around Earth – Geocentric model Planetary retrograde motion – Heliocentric/geoheliocentric model Galileo’s observations with the telescope – Disproved the geocentric model  Outline History (cont.) – Observations Celestial objects appear to revolve around Earth with own motions – Aristotelian physics Tycho’s comprehensive & accurate astronomical observations – Kepler’s laws of planetary motion Motions of objects – Newton’s laws » Provided an explanation for Kepler’s laws » Aristotelian physics was not correct Efbrazil, CC BY-SA 4.0, via Wikimedia Commons  Newton’s laws In 1687, Isaac Newton formulated the laws of motion and universal gravitation.  Laws of motion Newton’s laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it.  First law A body remains at rest, or in motion at a constant speed in a straight line, unless acted upon by a force.  Second law The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed.  Momentum, mass, & velocity By “motion”, Newton meant the quantity now called momentum, which depends upon the mass of a body and the velocity at which that body is moving. The momentum (𝐩) of a body is the product of its mass (𝑚) and its velocity (𝐯): 𝐩 = 𝑚𝐯  Force, mass, and acceleration Newton’s second law states that the rate of change of the momentum is the force. If the mass (𝑚) does not change with time, then the force (𝐅) equals the product of the mass and the rate of change of the velocity, which is the acceleration (𝐚): 𝐅 = 𝑚𝐚  Third law If two bodies exert forces on each other, these forces have the same magnitude but opposite directions.  Conservation of momentum Newton’s third law relates to a more fundamental principle, the conservation of momentum. Suppose that two bodies interact. The momentum exchanged between them adds to zero, so the total change in momentum is zero. https://www.youtube.com/watch?v=hsDjzBdCTWs  Provided an explanation for Kepler’s laws Newton provided an explanation for Kepler’s laws.  Acceleration of a planet Newton computed the acceleration of a planet moving according to Kepler’s first and second laws. – The direction of the acceleration is towards the Sun. – The magnitude of the acceleration is inversely proportional to the square of the planet’s distance from the Sun (the inverse square law).  Sun is the physical cause This implies that the Sun may be the physical cause of the acceleration of planets.  Force acting on a planet Newton defined the force acting on a planet to be the product of its mass and the acceleration (Newton’s second law of motion). So: – Every planet is attracted towards the Sun. – The force acting on a planet is directly proportional to the mass of the planet and is inversely proportional to the square of its distance from the Sun.  Law of universal gravitation A force is also acting on the Sun (Newton’s third law of motion). So Newton assumed, in his law of universal gravitation: – All bodies in the Solar System attract one another. – The force between two bodies is in direct proportion to the product of their masses and in inverse proportion to the square of the distance between them.  Sun’s orbit is barely perceptible As the planets have small masses compared to that of the Sun, the Sun’s orbit is barely perceptible. Binary simulator  Aristotelian physics was not correct After the work of Newton, it became generally accepted that Aristotelian physics was neither correct nor viable.  Summary History – Observations Celestial objects appear to revolve around Earth with own motions – Aristotelian physics Tycho’s comprehensive & accurate astronomical observations – Kepler’s laws of planetary motion Motions of objects – Newton’s laws » Provided an explanation for Kepler’s laws » Aristotelian physics was not correct