01 Celestial mechanics.pdf

Lecture 1 Celestial mechanics  Motions of objects in outer space Mechanics is the area of physics concerned with the relationships between force, matter, and motion among physical objects. Celestial mechanics deals with the motions of objects in outer space. ...

Lecture 1 Celestial mechanics  Motions of objects in outer space Mechanics is the area of physics concerned with the relationships between force, matter, and motion among physical objects. Celestial mechanics deals with the motions of objects in outer space.  Outline History – Geocentric model Observations Aristotelian physics Efbrazil, CC BY-SA 4.0, via Wikimedia Commons  Geocentric model Historically, the apparent motions of the celestial objects were described by European philosophers using the geocentric model.  Observations Two observations supported the idea that Earth was the center of the universe.  Earth First, Earth seems to be unmoving from the perspective of an earthbound observer; it feels solid, stable, and stationary.  Sun, Moon, & planets Second, the Sun appears to revolve around Earth once per day. The Moon and planets also appear to revolve around Earth about once per day. Stellarium Web Online Star Map https://youtu.be/uI2yZQOKOGk  Stars The stars appeared to be fixed on a celestial sphere rotating about once each day about an axis through the geographic poles of Earth. Geek3, CC BY-SA 3.0, via Wikimedia Commons  Celestial objects orbit Earth at the center The geocentric model describes the universe with Earth at the center. The celestial objects all orbit Earth. https://www.youtube.com/watch?v=jN4xVxKP7t4  Celestial objects have their own motions Alexey Elfimov, CC BY 3.0, via Wikimedia Commons The celestial objects have their own motions. The stars have a daily westward motion. Compared to the stars: – The Sun has a yearly eastward motion. – The Moon has a monthly eastward motion. This Photo by Unknown Author is licensed under CC BY-NC-ND  Aristotelian physics In the 4th century BC, an influential philosopher, Aristotle, wrote works based on the geocentric model. Aristotelian physics is the form of natural science or natural philosophy described in the works of Aristotle.  Celestial object is embedded in a sphere According to Aristotle, the celestial objects are embedded in concentric spheres that rotate at fixed rates. Geek3, CC BY-SA 3.0, via Wikimedia Commons  Uniform circular motion The celestial spheres are composed of the special element aether, the sole capability of which is a uniform circular motion at a given rate relative to the daily motion of the sphere of stars.  Order of spheres from Earth outward The order of spheres from Earth outward follows the decreasing orbital periods of the celestial objects. – Relative to the daily westward motion of the outermost sphere of stars: The Moon’s sphere has an eastward motion in one month. The Sun’s sphere has an eastward motion in one year.  Summary Celestial mechanics – History (scientific inter- disciplinarity) Geocentric model (mathematics) – Observations (astronomy) – Aristotelian physics Lecture 1 – Part 2 Celestial mechanics  Outline History (cont.) – Celestial objects appear to revolve around Earth with own motions (observations) Geocentric model Aristotelian physics – Further observations Adjustments – Eccentric – Equant  Sun’s tilted path around Earth The Sun’s movement over the course of a year traces out a path along the ecliptic against the sphere of stars. The ecliptic is inclined to the celestial equator. – The Sun is north of the celestial equator for about half of the year.  Seasons On Earth, seasons are the result of the Sun’s tilted path around Earth. – The summer solstice is the time when the Sun reaches its most northerly excursion relative to the celestial equator on the celestial sphere. It is seen as the middle of summer. J.hagelüken, CC BY-SA 3.0, via Wikimedia Commons  Differences in the lengths of the seasons Astronomers observed differences in the lengths of the seasons. – Duration of the seasons of the year 2024 This is inconsistent with a premise of the Sun moving around Earth in a circle at uniform speed.  Eccentric The eccentric is generally attributed to Hipparchus (190 – 120 BC). According to Hipparchus, moving the center of the Sun’s path slightly away from Earth would satisfy the observed differences in the lengths of the seasons.  Equant Equant is a mathematical concept developed by Claudius Ptolemy in the 2nd century AD. The Ptolemaic system predicted various celestial motions considerably better than without the equant.  Constant angular speed The equant point is placed so that it is directly opposite to Earth from the eccentric. A celestial object was conceived to move at a constant angular speed with respect to the equant.  Adjustment to Aristotelian physics The equant was an adjustment to Aristotelian physics. The moving object’s speed will vary during its orbit, faster in the bottom half and slower in the top half, but the motion is considered uniform because the object goes through equal angles in equal times from the perspective of the equant point.  Summary History – Observations Celestial objects appear to revolve around Earth with own motions – Geocentric model – Aristotelian physics » Uniform circular motion Differences in the lengths of the seasons etc. – Adjustments » Eccentric & equant » Constant angular speed  Outline History (cont.) – Observations Celestial objects appear to revolve around Earth with own motions (observations) – Geocentric model » Aristotelian physics Planetary retrograde motion – Heliocentric model – Geoheliocentric model  Copernicus’ model In the 16th century, a mathematical model of a heliocentric system was presented by Nicolaus Copernicus.  Heliocentric model In the heliocentric model, the daily-rotating Earth and the planets revolve around the Sun at the center of the universe. Only the Moon revolves around Earth.  Earth’s rotation around its axis The apparent motion of celestial objects around Earth over the course of one day is caused by Earth’s rotation around its axis.  Earth orbits the Sun Earth orbits the Sun. As seen from the orbiting Earth, the Sun appears to move with respect to the stars. Tfr000 (talk) 16:54, 15 March 2012 (UTC), CC BY-SA 3.0, via Wikimedia Commons  Planetary retrograde motion Copernicus’ system resolved the issue of planetary retrograde motion. The planets generally drift slowly eastward relative to the stars. However, a planet periodically appears to stop its eastward drift, and then drift back toward the west. Then, it appears to resume its normal motion west to east. This Photo by Unknown Author is licensed under CC BY-SA Stellarium Web Online Star Map  Earth & the planets revolve around the Sun In the heliocentric model, Earth and the planets revolve around the Sun. The heliocentric model argued that the planetary retrograde motion was apparent: a planet that Earth is passing seems to move backwards against the stars. This Photo by Unknown Author is licensed under CC BY-SA  Summary Geocentric model Heliocentric model – Planetary retrograde motion This Photo by Unknown Author is licensed under CC BY-SA  Tychonic system The Tychonic system is a model of the universe published by Tycho Brahe in 1588.  Geoheliocentric model It is a geoheliocentric model: Earth is at the centre of the universe, the Sun and Moon and the stars revolve around Earth, and the other five planets revolve around the Sun.  Mathematically equivalent to heliocentrism https://www.youtube.com/watch?v=6laRU_BzhvU At the same time, the motions of the planets are mathematically equivalent to the motions in Copernicus’ heliocentric system under a simple coordinate transformation.  Could be explained by the Aristotelian physics The Tychonic system combines what Tycho saw as the mathematical benefits of the Copernican system with the philosophical and “physical” benefits of the geocentric model. The Aristotelian physics of the time offered no physical explanation for the motion of Earth, whereas it could easily explain the motion of celestial bodies.  Summary Heliocentric model Geoheliocentric model – Aristotelian physics Celestial objects are embedded in rotating spheres  Scripture portraying Earth as being at rest Tycho also cited the authority of Scripture in portraying Earth as being at rest. – 1 Chronicles 16:30 “tremble before him, all the earth. The world is firmly established; it shall never be moved.” This Photo by Unknown Author is licensed under CC BY-SA  Summary This Photo by Unknown Author is licensed under CC BY-SA History – Observations Celestial objects appear to revolve around Earth with own motions – Geocentric model » Aristotelian physics Planetary retrograde motion – Heliocentric model – Geoheliocentric model » Aristotelian physics » Scripture  Outline History (cont.) – Tycho’s astronomical observations Kepler’s laws of planetary motion Efbrazil, CC BY-SA 4.0, via Wikimedia Commons  Tycho’s astronomical observations Tycho was also known for his comprehensive and unprecedentedly accurate astronomical observations.  Comprehensive & accurate observations He devoted many of his efforts to improving the accuracy of the existing types of instrument. – For example, a quadrant is an instrument used to measure angles up to 90. He designed larger versions of these instruments, which allowed him to achieve much higher accuracy.  Kepler’s laws of planetary motion Kepler’s laws of planetary motion was published by Johannes Kepler between 1609 and 1619. The laws modified the heliocentric theory of Copernicus, replacing its circular orbits with elliptical trajectories, and explaining how planetary velocities vary.  Heliocentric model Kepler had believed in the Copernican model.  Kepler’s religious view Much of Kepler’s enthusiasm for the Copernican system stemmed from his theological convictions about the connection between the physical and the spiritual. – The universe itself was an image of God, with the Sun corresponding to the Father, the stellar sphere to the Son, and the intervening space between them to the Holy Spirit.  Sun was the source of motive force In Kepler’s religious view of the cosmos, the Sun (a symbol of God the Father) was the source of motive force in the Solar System.  Planetary orbit Kepler introduced the revolutionary concept of planetary orbit, a path of a planet in space resulting from the action of physical causes, distinct from previously held notion of celestial sphere to which planet is attached.  Summary Tycho – Geoheliocentric model Aristotelian physics – The Sun is attached to a rotating sphere. Scripture – Earth is at rest. Kepler – Heliocentric model Kepler’s religious view – The Sun a symbol of God the Father – The Sun’s force causes Earth to move in an orbit.  Analyzed Tycho’s astronomical observations Kepler analyzed Tycho’s observations of the orbit of Mars. He could not reconcile Tycho’s highly precise observations with a circular fit to Mars’ orbit. He set about trying to fit a non-circular orbit to the data.  First law Finding that an elliptical orbit fit the Mars data, Kepler concluded Semimajor axis that all planets move in ellipses, with the Sun at one focus—his first law of planetary motion. Ag2gaeh, CC BY-SA 4.0, via Wikimedia Commons  Ellipse An ellipse is a plane curve surrounding two focal points (𝐹1 and 𝐹2 ), such that for all points on the curve, the sum of the two distances to the focal points is a constant. – It generalizes a circle, which is a shape consisting of all points in a plane that are at a given distance from the centre. The major axis of an ellipse is its longest diameter. The semimajor axis is one half of the major axis.  Second law Based on measurements of Earth and Mars, Kepler created a formula in which planets sweep out equal areas in equal times— his second law of planetary motion. Faster motion as a planet moves closer to the Sun Kepler supposed that the motive power radiated by the Sun weakens with distance, causing faster or slower motion as planets move closer or farther from it. This Photo by Unknown Author is licensed under CC BY-NC-ND  Summary History – Early astronomical observations Eccentric & equant – Geocentric model – Circular orbit – Tycho’s comprehensive & accurate astronomical observations Kepler’s laws of planetary motion – Heliocentric model – Elliptical orbit  Third law With Tycho’s data and his own astronomical theories, Kepler treated relationships between planetary orbital velocity and orbital distance from the Sun much more precisely and attached new physical significance to them. He articulated what came to be known as the third law of planetary motion: The square of a planet’s orbital period is proportional to the cube of the length of the semi-major axis of its orbit.  Further planet has a slower orbital speed The third law expresses that the farther a planet is from the Sun, the slower its orbital speed, and vice versa. https://youtu.be/3PCdNaUaYt4  Summary History – Tycho’s astronomical observations Kepler’s laws of planetary motion – Heliocentric model – Elliptical orbit https://youtu.be/3PCdNaUaYt4  Summary Celestial mechanics – History Early observations – Geocentric model – Aristotelian physics Planetary retrograde motion (observation) – Heliocentric model Tycho’s astronomical observations – Kepler’s laws of planetary motion  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