Johannes Kepler (1571-1630) German Mathematician and Astronomer Passionately convinced of the rightness of the Copernican view. Set out to prove it!
Kepler s Life Work Kepler sought a unifying principle to explain the motion of the planets without the need for epicycles. Wanted to work with the best observational Astronomer: Tycho Brahe. Obtained Brahe s data after his death. Eventually discovered that ellipses would dramatically simplify the mathematics! No more circles!
Kepler s First Law The orbital paths of planets are elliptical (not circular) with the Sun at one focus. Properties of conic sections known since Euclid
Locus of points produced by this practical geometric construction. Definition on an Ellipse
Property of an Ellipse Major and Minor axes Two Foci. The Sun is at one focus, the other is not physically significant.
Eccentricity The eccentricity of an ellipse is the ratio: e = Distance from the center to a focus Length of the semi-major axis If e = 0 a circle, e = 1 a line The semi-major axis, a, is the average distance between the planet and the Sun. Perihelion = a(1- e) = closest approach Aphelion = a(1+ e) = greatest distance.
Elliptical Terms Visualised
Comments on Elliptical Motion Elliptical motion: No small achievement! Challenged the Authority of Aristotle. Except for Mercury (and Pluto) eccentricity is so small cannot easily distinguish it from circular motion. Hence Ptolemaic and Copernican models did a pretty descent job (for those days). Galileo did not like ellipses!
Kepler s Second Law An imaginary line connecting the Sun to any planet sweeps out equal areas in equal intervals of time. Planets therefore have different speeds at perihelion and aphelion. Challenges Aristotle s insistence that planets have a constant or uniform speed.
Kepler s Second Law Visualised Red arcs all take the same time for equal areas A, B, C.
Kepler s Third Law Laws (1) and (2) published in 1609, based on a long study of the motion of Mars. An appeal to simplicity in mathematics. During next 10 years extended to all known planets and devised 3 rd law. The square of the planet s orbital period is proportional to the cube of its semimajor axis. or: P 2 /a 3 = Constant
Orbital Properties of the Planets Planet Semi-Major Axis, a Period, P Eccentricity, e P 2 /a 3 Mercury 0.387 0.241 0.206 1.002 Venus 0.723 0.615 0.007 1.001 Earth 1.000 1.000 0.017 1.000 Mars 1.524 1.881 0.093 1.000 Jupiter 5.203 11.86 0.048 0.999 Saturn 9.537 29.42 0.054 0.998 Uranus 19.19 83.75 0.047 0.993 Neptune 30.07 163.7 0.009 0.986 Pluto 39.48 248.0 0.249 0.999
Further Notes Period is Sidereal period Astronomical Unit = semi-major axis of the Earth s orbit (or average Sun-Earth distance) Deviations in P 2 /a 3 for Uranus and Neptune is mutual gravitational effect. Predictive Law.applies to all planets.
Incidentally: Kepler and The Star of Bethlehem In 1604 Kepler observed a brilliant conjunction (a close apparent mutual approach)of Jupiter, Saturn and Mars. He calculated that this type of conjunction takes place every 805 years. Hence previously occurred in AD 799 and in (February) 6 BC. Now regarded as the Star of Bethlehem.
The Size of the Solar System Kepler s 3 rd Law (P 2 /a 3 = Constant) only gives the relative scale of the solar system: Orbital periods known in terms of Earth years. Semi-major axis known in terms of that of the Earth. Need to determine the actual length of the Astronomical unit (A.U.)..how? Need reliable measure of the parallax when the planet is closest to us (hence biggest parallax)
Reminder: Parallax Consider a planet as seen against the background stars (very far away). View from A and B are different the planet moves with respect to the background stars Apparent angular displacement is Parallax.
Early Attempts Mars in direct opposition Earth directly between Mars and the Sun. Tycho Brahe tried in 1582 Twice per day for Earth diameter baseline (before Dawn and at Sunset) He claimed Success! However. 0.012 o was too small for his instruments!
Try again! In 1672 Cassini measured Mars in opposition at Paris, while his assistant, Richer, did the same in South America. Concluded that Mars was ~4000 Earth diameters away (at opposition). Hence AU ~87 million miles (Modern day value : 93 million miles) Fluke - as large experimental uncertainty!
Halley s Solar Transits November 7 1677 observed the transit of Mercury across the face of the Sun. Realised these rare events could be used to determine the A.U.
Halley s Predictions In 1716 Halley predicted the next transits of Venus would occur on: 6 th June 1761 and 3 rd June 1769 (He never lived to see it.) Next possible opportunity would be: 9 th December 1874 and 6 th Dec 1882 Next chance was that: 8 th June 2004! Why so rare? Venus orbital plane is slightly (3.4 o ) inclined to that of the ecliptic.
Halley s Quest. Halley realised that the transit of Venus would give much better accuracy for A.U. Observers need to measure time when Venus enters and exits the Sun s disc. Need accurate clock and telescope (quadrant) Needed observers all over the world. (unprecedented, non-political collaboration) Correlate results to determine the A.U.
The Calculation in principle Assume orbit is circular (for simplicity) Earth-Sun average distance is 1 AU. Venus s orbit has radius of ~ 0.7AU. Therefore, at closest approach, Venus is only 0.3AU from Earth. From parallax measurement, with known baseline, can determine distance to Venus. Hence find AU.
Halley s Legacy 150 observations of Venus s transits worldwide. Even so, experimental problems persisted Optical distortions, atmospheric turbulence A.U. found to be 91 million miles. Modern determination using radar methods gives A.U. = 149,597,870km!
Images of the Transit of Venus: 8 th June 2004
The Transit of Venus The Atmosphere of Venus illuminated by the Sun.
Venus Transit: Close up views
Quest for Simplicity Kepler s Laws discovered empirically. Based upon observational evidence. What is the underlying reason the laws to work? What forces are involved? This question was addressed by Isaac Newton.
Isaac Newton (1642-1727) By the time he was 25, he had discovered the laws of motion including gravity. Only published them 20 years later at the prompting of Halley!
The Title Page of Newton s Principia, 1686 The Mathematical Principles of Natural Philosophy One of the most influential books in Physics
Newton s First Law: Inertia Every body continues in a state of rest, or in a state of uniform motion in a straight line ( inertia ), unless it is compelled to change state by an external force acing on it. Aristotle thought (wrongly) the natural state was at rest, and to move required a force. Newton: Uniform motion requires no force! Observation needs a frictionless environment.
Newton s Second Law: Mass Changing speed or direction implies an acceleration. The resultant acceleration (a) is directly proportional to the applied force (F). The constant of proportionality being the mass (m) of the body. F = ma
Newton s Second Law - continued The greater the force, the greater the acceleration of the body. For a constant force, the smaller the mass of an object, the larger its acceleration. SI unit of force is the Newton. 1 Newton of force is needed to make a 1 kilogram object accelerate at the rate of 1 meter per second, every second.
Newton s Third Law To every action there is an equal and opposite reaction. Forces do not occur in isolation. If A pulls on B, then B necessarily exerts a force on A too. (eg: jet planes, bullet recoil, hammer )
Gravity Newton hypothesised that any object with mass exerted an attractive gravitational force on other massive objects. To Newton this force was action at a distance. (no physical link between bodies) Now regarded as a property of space a force field that influences massive objects.
Newton s Law of Gravity Every particle of matter in the universe attracts every other particle of matter with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them. F = Gm m r 1 2 2 G = Gravitational constant = 6.67x10-11 Nm 2 /kg 2
Inverse-Square Law Inverse square force rapidly weakens with distance from the source but never quite reaches zero!
Gravity and Circular Motion Direction of force is along the line between the two bodies. If motion was circular, speed would be constant.
Equations for Planetary Motion Mutual attraction between Sun and planets. F = GM Force for circular motion can be shown to be: Assume circular motion: forces must be equal. r S 2 M M Pv 2 F = r P
Algebra! Hence: Simplifies to: GM r S 2 M P = M Pv r 2 GM S = v r (1) 2 Planet s speed v is : v = circumference orbital period = 2πr P (2) Substitute (2) in (1): GM S = 4π P 2 2 r 3
Kepler s Third law! Rearrange equation: P 2 4 = 2 π r GM S 3 or P r 2 3 = GM S = 4π 2 GM S P 2 4π r 2 3 = constant If the G value is known (Cavendish - 1793) then the mass of the Sun can be found! Mass of Sun ~2.0x10 30 kg, Earth ~6.0x10 24 kg
Principia: From Ballistics to Satellites A Vertical motion B, C Parabolic motion E Circular motion D, F Elliptical Motion
Conclusions Newton s laws of motion combined with that of gravity explains motion of all objects. All earth-bound objects All heavenly bodies Without an exact inverse square law the universe would be very different!
Are Newton s Laws Always Valid? NO! Two exceptions: When the speed of the motion approaches the speed of light: Relativity When the dimensions become very small - the world of atoms: Quantum Mechanics The motions of everyday objects follow Newtonian Mechanics.