Slide. Historical Overview

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1 Slide Historical Overview

Nicolaus Copernicus (1473-1543) First astronomer to offer scientific evidence for Helio-centric Cosmology (Helio=Sun; Sun-centered).

Heavenly motions are uniform, eternal, and circular or compounded of several circles (epicycles). 2. The center of the universe is near the Sun. 3.

2 Nicolaus Copernicus ( ) First astronomer to offer scientific evidence for Helio-centric Cosmology (Helio=Sun; Sun-centered). The major parts of theory (published in De Revolutionibus orbium coelestium - On the Revolutions of the Celestial Spheres) are: 1. Heavenly motions are uniform, eternal, and circular or compounded of several circles (epicycles). 2. The center of the universe is near the Sun. 3. Around the Sun, in order, are Mercury, Venus, Earth and Moon, Mars, Jupiter, Saturn, and the fixed stars. 4. The Earth has three motions: daily rotation, annual revolution, and annual tilting of its axis. 5. Retrograde motion of the planets is explained by the Earth's motion. 6. The distance from the Earth to the sun is small compared to the distance to the stars.

Galileo Galilei (1564-1642) Major role in scientific revolution of the Renaissance. Developed ideas about gravity that influenced Newton and Einstein later.

3 Galileo Galilei ( ) Major role in scientific revolution of the Renaissance. Developed ideas about gravity that influenced Newton and Einstein later. Of many accomplishments, was first to use a telescope to study the sky (observed Jupiter and discovered the socalled Galilean moons). This showed that celestial objects orbited something other than the Earth. Also observed that Venus has phases much like the Moon. Both of these observations lent strong support to Copernican - Heliocentric Cosmology.

4 Problems with Geocentric Model : 2. Phases of Venus Geocentric Model Heliocentric Model

5 Problems with Geocentric Model : 2. Phases of Venus

6 Problems with Geocentric Model : 3. Moons of Jupiter

7 The significance of what he saw: Strong support to the heliocentric model from numerous observations Discoveries of sunspots, moons around other planets, landscape on the moon Observed that gravitational acceleration is independent on mass; formulated the Principle of Relativity - basis of Special and General Relativity Galileo published in Italian, not Latin. Widely read. Language of the people, rather than language of the scholars. Transition from a faith-based science to an observation-based science. Rejected the old view (still alive now!) that the only path to true understanding is through religious faith The Bible tells us how to go to heaven, not how the heavens go. He came under fire from the Catholic Church and was forced to give a public denial of the heliocentric/copernican system, and was placed under house arrest for the last 10 years of his life. Was not pardoned by the Church until Science in Italy was dealt a severe blow. The center of scientific investigation shifted to northern Europe.

Tycho Brahe (1546-1601) Foremost astronomer after the death of Copernicus. King Frederick II of Denmark set him up at Uraniborg, an observatory on the island of Hveen.

8 Tycho Brahe ( ) Foremost astronomer after the death of Copernicus. King Frederick II of Denmark set him up at Uraniborg, an observatory on the island of Hveen. With new instruments (quadrant), Brahe could measure positions of stars and planets to 4 minutes of arc (1/8th the diameter of the full moon). With new instruments (quadrant, similar to present-day sextant), Brahe could measure positions of stars and planets to better than 4 minutes of arc (1/8th the diameter of the full moon). Made the most accurate observations of planet positions ever recorded.

9 Using the Quadrant (similar to a Sextant) to measure a star s altitude. An observer sights a star along the Quadrant while a plumb line measures the angle.

Tycho Brahe (1546-1601) Also Credited with observing the supernova of 1572, demonstrating that the stars do change. He coined the term Nova for new star in a book.

10 Tycho Brahe ( ) Also Credited with observing the supernova of 1572, demonstrating that the stars do change. He coined the term Nova for new star in a book. Term still used today. Brahe failed to find evidence for the Earth s motion and presumed the Copernican model was false. Brahe endorsed the Geocentric cosmology.

Johannes Kepler (1571-1630) Brahe moved to Prague in 1599. Kepler joined him to develop cosmological model consistent with Brahe s observations.

11 Johannes Kepler ( ) Brahe moved to Prague in Kepler joined him to develop cosmological model consistent with Brahe s observations. Kepler initially obtained excellent agreement with Brahe s data with the planets moving in spheres and equants, similar to Ptolemy s model, except for 2 data points which were off by 8 minutes of arc (twice the accuracy of Brahe s measurements). Kepler kept struggling and finally rejected Ptolemaic model and eventually realized that that Brahe s data were only consistent with a model where planets (1) orbit the Sun and (2) their orbits are ellipses.

12 Observed Planet and uncertainty on the measurement Position Predicted by Model

13 y = ax 3 +bx 2 +cx+d y = sin(x) / x y x

Johannes Kepler (1571-1630) Kepler s Laws: published in 1609 in Astronomica Nova (The New Astronomy). 1st Law: A planet orbits the Sun in an ellipse, with the Sun at one focus of the ellipse.

14 Johannes Kepler ( ) Kepler s Laws: published in 1609 in Astronomica Nova (The New Astronomy). 1st Law: A planet orbits the Sun in an ellipse, with the Sun at one focus of the ellipse. 2nd Law: A line connecting a planet to the Sun sweeps out equal areas in equal time intervals. 3rd Law: The square of the orbital period, P, of a planet is equal to the cube of the average distance, a, of the planet from the Sun.

15 Ellipses defines an ellipse Aphelion Sun Perihelion

16 Advances in mechanics and celestial dynamics

17 Dynamics Finally, we talk about what causes the motion! Connection between force and motion The concept of force gives us a quantitative description of the interaction between two bodies or between a body and its environment Johannes Kepler was probably the first who suggested that it is some force which is responsible for the orbital motion of Mars

18 Aristotle: a natural state of an object is at rest; a force is necessary to keep an object in motion. It follows from common sense B.C. Galileo: was able to identify a hidden force of friction behind common-sense experiments

19 Galileo: If no force is applied to a moving object, it will continue to move with constant speed in a straight line Inertial reference frames Galilean principle of relativity: Laws of physics (and everything in the Universe) look the same for all observers who move with a constant velocity with respect to each other.

A legend says that Galileo dropped cannonballs of unequal weights from the Leaning Tower of Pisa to show that both objects reach ground at the same time. This story is probably not true.

23 A legend says that Galileo dropped cannonballs of unequal weights from the Leaning Tower of Pisa to show that both objects reach ground at the same time. This story is probably not true. In fact, Galileo experimented with balls rolling down a ramp. Galileo experimentally proved that objects fall with the same acceleration independently on their masses. He found a correct mathematical formula describing this motion: the distance traveled by a falling body increases as the square of the time that has passed.

25 Sir Isaac Newton ( ) Developed Universal theory of Gravity, and set his three Laws of Motion, which are framework for classical mechanics, including basis for modern engineering. -1st Law: An object at rest, remains at rest unless acted on by an outside force. An object with uniform motion, remains in motion unless acted on by an oustide force. - 2nd Law: An applied force, F, on an object equals the rate of change of its momentum, p, with time. - 3rd Law: For every action there is an equal and opposite reaction.

26 Start with Kepler s 3rd law: For circular orbit: Newton s Theory of Gravity Insert this into Kepler s 3rd law: Multiply both sides by factors: Yields: Force!

27 Newton s Theory of Gravity By Newton s 3rd law, this is the force on mass M from mass m, so the force on mass m should be: Equating these two formula gives: where Defining and we get the usual formula:

28 Robert Hooke Spring force, Hooke s law, spring clocks Biological research with a microscope Discovered plant cells and coined the term cell One of the fathers of paleontology; proponent of evolution (>200 years before Darwin) Built one of the first reflecting telescopes Pioneering work on the theory of light Bitter priority dispute with Newton Proposed inverse square law of gravitation

29 Leibniz-Newton calculus priority dispute

30 Gottfried Leibniz These are Leibniz notations: Integral sign as an elongated S from Summa and d as a differential (infinitely small increment).

Leonhard Euler 1707-1783 Read Euler, read Euler, he is the master of us all

logarithms, power series, calculus of variations, origin of analytic number theory,

Integrated Leibniz and Newton s calculus Three of the top five most beautiful

31 Leonhard Euler Read Euler, read Euler, he is the master of us all Pierre-Simon Laplace f(x), complex numbers, trigonometric and exponential functions, logarithms, power series, calculus of variations, origin of analytic number theory, origin of topology, graph theory, analytical mechanics, 80 volumes of papers! Integrated Leibniz and Newton s calculus Three of the top five most beautiful formulas are Euler s Most beautiful formula ever the beam equation : a cornerstone of mechanical engineering

32 Introduce Angular Momentum: Look at rate-of-change of L as a function of t: = 0, Conservation of Angular Momentum

33 Derive Kepler s Laws from Newton s Laws Define Center of Mass Frame for 2 objects for i=1..n objects Differentiate with time, Differentiate with time, again: by Newton s 3rd Law

34 Derive Kepler s Laws from Newton s Laws Choose Frame with R=0 =0 Define Reduced Mass of the System : μ= m1 m2 / (m1+m2)

35 Derive Kepler s Laws from Newton s Laws Now write out total Energy of the System Insert: And : r = r 2 - r 1

36 Derive Kepler s Laws from Newton s Laws Total Energy Becomes: Grouping Terms : And: Which Gives: Total Energy of the System is the sum of the Kinetic Energy of the reduced mass and the potential energy of the total mass/reduced mass system.

37 Derive Kepler s Laws from Newton s Laws Similarly for the Angular Momentum: Insert: Leads to: The Two-Body problem may be treated as a One-body problem with μ moving about a fixed mass M at a distance r.

38 Derive Kepler s Laws from Newton s Laws The Two-Body problem may be treated as a One-body problem with μ moving about a fixed mass M at a distance r.

39 Derive Kepler s Laws from Newton s Laws Take cross product: A x (B x C) = (A C) B - (A B) C

40 Derive Kepler s Laws from Newton s Laws D is a constant ^ v x L and r lie in the same plane, so must D D is directed toward Perihelion, D determines e (eccentricity) of ellipse. Maximum reached with ^r and D point in same direction.

41 Derive Kepler s Laws from Newton s Laws A (B x C) = (A x B) C Kepler s 1st Law!

42 Derive Kepler s Laws from Newton s Laws Integrate from Focus to r: da = ½ r 2 dθ da dt = ½ r2 dθ dt Let : v = vr + vθ = dr ^ ^ dt r + r dθ dt θ da dt = ½ r vθ r and vθ are perpendicular, therefore : r vθ = r x v = L / μ = L / μ da L = Kepler s 2nd Law! dt 2μ = Constant

43 Derive Kepler s Laws from Newton s Laws Integrate 2nd Law with time: A = ½ L/μ dt A = ½ (L/μ) P, where P = period b P 2 = A = π ab 4π 2 a 2 b 2 μ 2 L 2 a L 2 / μ 2 1st Law: r = L = μ [ GMa (1-e 2 ) ] ½ GM (1 + e cosθ) Kepler s 3rd Law P 2 4π 2 a 3 = G (m1 + m2)

44 Calculate Orbital Speed at Perihelion and Aphelion Use Kepler s 1st Law At Perihelion Aphelion Sun Perihelion At Aphelion Total Energy:

45 Other types of orbits

46 Total Energy in a Bound Orbit Etot = -GMμ 2a = -Gm 1m2 2a = <U> / 2 That is the total energy is 1/2 of the average potential energy. This is the Virial Theorem. More in depth derivation by Rudolf Clausius (reproduced in book). It has many applications in astrophysics, including providing evidence for Dark Matter in Clusters of Galaxies! Rudolf Julius Emanuel Clausius (January 2, 1822 August 24, 1888), was a German physicist. He is one of the central founders of the science of thermodynamics.

47 High Tide in Bay of Bundy Low Tide

49 What Causes Ocean Tides?

50 Saturn as seen by Spacecraft Cassini, October 2004

51 Tidal Forces Force of Moon on m at center of Earth Force of Moon on m on surface of Earth Fc,x = -G Mm / r 2 Fc,y = 0 Fp,x = (G Mm/s 2 )cosϕ ΔF = Fp - Fc = G Mm s 2 = (r - Rcosθ) 2 + (Rsinθ) 2 = r 2 M= mass of Moon; m= test mass Fp,y = -(G Mm/s 2 )sinϕ cosϕ 1 G Mm i - s 2 r 2 ^ s 2 ( - ) ( ) sinϕ ^j ( 1-2Rcosθ ) R << r r

52 ΔF = Fp - Fc = G Mm Tidal Forces s 2 = (r - Rcosθ) 2 + (Rsinθ) 2 = r 2 cosϕ 1 G Mm i - s 2 r 2 ^ s 2 ( - ) ( ) 1-2Rcosθ r ΔF G Mm/r 2 [ cosϕ ( 2Rcosθ 1 + ) -1 ] ^ i r - G Mm/r 2 [ 1 + 2Rcosθ ] sinϕ j^ r ( sinϕ ^j ) R << r For the Earth-Moon, cosϕ 1, r sin ϕ = Rsinθ, sin ϕ = (R/r) sinθ ΔF G MmR/r 3 [ 2cosθ ^i - sinθ ^j ]

53 Tidal Forces ΔF G MmR/r 3 [ 2cosθ ^i - sinθ ^j ] Force of the Moon on the Earth Differential (relative) force on the Earth, relative to the center

54 Tidal Forces ΔF G MmR/r 3 [ 2cosθ ^i - sinθ ^j ] When will the tidal force equal the force of gravity? Consider the Earth-Moon system: Mm, Rm = mass of moon, radius of moon ME, RE = mass of Earth, radius of Earth G Mmm/Rm 2 = ( 2GMEm/r 3 ) Rm Assume constant density: Mm = (4/3)πRm 3 ρm r 3 = (2ME/Mm ) Rm 3 = (2RE 3 ρe) / (Rm 3 ρm) Rm 3

55 Tidal Forces G Mmm/Rm 2 = ( 2GMEm/r 3 ) Rm Assume constant density: Mm = (4/3)πRm 3 ρm r 3 = (2ME/Mm ) Rm 3 = (2RE 3 ρe) / (Rm 3 ρm) Rm 3 r = ( 2ρE ρm ) ⅓ RE or r = fr ( ρe ρm ) ⅓ RE, fr = 2 ⅓ = Roche, using a more sophisticated assumption for the density of the Moon calculated fr = 2.456

58 Comet Shoemaker-Levy 9 after passing through Jupiter s Roche Limit

59 Saturn as seen by Spacecraft Cassini, October 2004

60 Planetary Moons form when a moon crosses the Roche Limit Saturn Janus Epimetheus Rings Mimas Enceladas Tethys Other Moons Rings Dione Rhea Rings Roche Limit Distance from Saturn Center in RS

61 Effects of tides Slow down the rotation of earth Seabed slips under the water bulges Friction slows down the rotation The day was 18 hours long 900 million yr ago Slide

62 The Tidally-Locked Orbit of the Moon The Earth also exerts tidal forces on the moon s rocky interior that slow down its rotation. It is rotating with the same period around its axis as it is orbiting Earth (tidally locked). We always see the same side of the moon facing Earth. Slide

63 Acceleration of the Moon s Orbital Motion Earth s tidal bulges are slightly tilted in the direction of Earth s rotation. Gravitational force pulls the moon slightly forward along its orbit. Slide

64 Effects of tides 1. Synchronization of the rotational and orbital period 2. Tides cause the heating of the interiors of the interacting bodies (Io) 3. If the bodies are too close to each other, they can be disrupted by tides (Roche limit). Slide

Tycho Brahe ( )

Tycho Brahe ( ) Tycho Brahe (1546-1601) Foremost astronomer after the death of Copernicus. King Frederick II of Denmark set him up at Uraniborg, an observatory on the island of Hveen. With new instruments (quadrant),