PHY2019 Observing the Universe 2009-2010
M 2 F G = GM 1M 2 R 2 M 1 R q 2 F E = q 1q 2 4πɛ 0 R 2 q 1 R two protons : F G F E = 4πɛ 0Gm 2 p e 2 = 8 10 37
Ptolemy (83-161 AD) Claudius Ptolemaeus: Roman citizen, possibly Hellenized Egyptian, living in Alexandria Wrote the Almagest, key compendium of all previous Western astronomical knowledge Geocentric model of the Solar System Mark McCaughrean
What Ptolemy s model had to match 129 Geocentric simulation of Solar System in 150AD Pogge
Ptolemy s cosmology 130 Animation courtesy Dennis Duke, Florida State
Nicolaus Copernicus (1473-1543) Argued that geocentric Ptolemaic system was too complicated Heliocentric model more elegant Also explains key features such as eclipses Decades of work yielded De revolutionibus orbium coelestium (On the revolutions of the celestial spheres) (1543, Nuremberg) Orbits still circular though The Copernican principle The Earth is not at the centre of the Universe (although Copernicus thought the Sun was) Mark McCaughrean
Tycho Brahe (1546-1601) Rich Danish nobleman: Tyge Ottesen Brahe Astronomer, astrologer, alchemist Had metal nose: bridge of original lost in duel Built three observatories Uraniborg on island of Hven Reputed to cost 1% of Danish GDP during construction Compare with Apollo project, 0.4% of US GDP Many large, tower-mounted instruments Stjerneborg on Hven Ground-level, instruments underground, to avoid windshake Benátky nad Jizerou, near Prague Funded by Rudolf II, Holy Roman Emperor Tycho advocated geoheliocentrism Sun orbits Earth, but other planets orbit Sun; solved problems with broken Ptolemaic system (phases of Venus, observed by Galileo) Adopted by Catholic Church for many years Uraniborg Observatory Mark McCaughrean
Tycho s cosmology 133 Animation courtesy Dennis Duke, Florida State
Johannes Kepler (1571-1630) Born in Weil der Stadt, Germany Made very significant contributions in optics (explained pinhole camera, refraction in eye, depth perception, formulated eyeglasses, total internal reflection, telescope design) Published in Astronomiae Pars Optica Work with Tycho Forced to move from Graz to Prague due to counter Reformation and his religious beliefs Was Tycho s assistant at first; inherited Tycho s position and data upon his death Detailed analysis of Mars observations showed it to have an elliptical orbit, with Sun at one focus of ellipse Applied successfully to other planets Did not understand why they did this though Led to three laws of planetary motion Mark McCaughrean
Kepler published a number of influential books in astronomy Mysterium Cosmographicum (1596) Geometric polyhedral interpretation of solar system De Stella Nova (1606) Treatise on supernova of 1604 Astronomia Nova (1609) First two laws of planetary motion Harmonices Mundi (1619) Third law of planetary motion Epitome astronomia Copernicanae (1617-1621) Discussion of heliocentrism, all three laws together Rudolphine Tables (1627) Catalogue of Tycho s stellar positions and planetary motions Somnium (1634) Written over long period, hypothesises how sky would look perspective of another planet (Moon): first science fiction Mark McCaughrean
Kepler s cosmology 136 Animation courtesy Dennis Duke, Florida State
Kepler s Laws of planetary motion (I) Perihelion Kepler s First Law: The orbit of a planet around the Sun is an ellipse with the Sun at one focus Actual orbital eccentricities: rp Sun Planet A circle is an ellipse with eccentricity = 0 ra Other focus Aphelion Mercury 0.2056 Venus 0.0068 Earth 0.0167 Mars 0.0934 Jupiter 0.0483 Saturn 0.0560 Uranus 0.0461 Neptune 0.0097 Eccentricity e = r a r p r a + r p e ~ 0.7 here Semi-major axis a = r p + r a 2 137 Semi-major axis Semi-major axis Mark McCaughrean
Kepler s Laws of planetary motion (I) Kepler s First Law - requires solution of equations of motion in a gravitational field 138 Mark McCaughrean
Kepler s Laws of planetary motion (II) Kepler s Second Law: A line joining a planet and the Sun sweeps out equal area in equal time intervals Perihelion Aphelion Planet moves fastest near perihelion and slowest near aphelion, but area of two swept out sectors is equal This is a consequence of the conservation of angular momentum 139 Mark McCaughrean
Derivation of Kepler s Second Law (I) Q v F Δθ r ΔA P vt vr Mass m moves along elliptical orbit with velocity v (vt tangential, vr radial) at distance r from focus F During short interval Δt, body moves from P to Q and radius vector sweeps out angle Δθ Then Δθ vtδt/r During this time, radius vector sweeps out triangle FPQ Area of triangle is ΔA rvtδt/2 Mark McCaughrean
Derivation of Kepler s Second Law (II) Q v F Δθ r ΔA P vt vr L = m ( r x v ) out of plane of figure 141 In limit Δt tends to zero, we have Scalar magnitude of angular momentum Thus rate of sweeping out area As L & m constant, then da/dt is also constant, verifying Kepler s 2nd Law da dt da dt = rv t 2 = r2 2 dθ dt L = mv t r = mr 2 dθ dt = r2 dθ = L 2 dt 2m Mark McCaughrean
Kepler s Laws of planetary motion (III) Kepler s Third Law: The square of a planet s sidereal period around the Sun is proportional to the cube of length of its orbit s semimajor axis P 2 a 3 For circular orbit, semi-major axis is simply radius Mars a = 1.52 AU P = 1.88 yrs Earth a = 1 AU P = 1 yr Jupiter a = 5.2 AU P = 11.86 yrs 142 Mark McCaughrean
Period vs semi-major axis for Solar System Eris 300 100 Uranus Pluto Neptune Period (years) 30 10 3 1 Venus Earth Mars Ceres Jupiter Saturn Straight line is P 2 a 3 0.3 Mercury 0.3 1 3 10 30 Semi-major axis (AU) 143 Mark McCaughrean
The elegance of the heliocentric model 144 Geocentric and heliocentric simulations of Solar System in 1543-1550AD Pogge
Galileo Galilei (1564-1642) Born in Pisa, Tuscany; died Arcetri Physicist, mathematician, astronomer, philosopher, heretic, prisoner Key work on accelerated motion (gravity) Used newly-invented telescope to study solar system and beyond; observations published in Sidereus Nuncius (The Starry Messenger) Support for heliocentricity Observed full set of phases of Venus Suggested that it must orbit Sun rather than Earth Discovery of moons orbiting Jupiter Then why couldn t the Earth orbit the Sun? Observed spots on Sun, mountains on Moon Sun and Moon were not as perfect as philosophy held However, did not agree with Kepler that orbits were elliptical and that Moon caused tides Circle perfect shape for orbits; non-constant Earth rotation Kepler was right in both cases Mark McCaughrean
Isaac Newton (1642-1727) Born in Lincolnshire, England Physicist, mathematician, astronomer, philosopher, alchemist, MP, JP, Master of the Royal Mint, president of Royal Society Mathematics Developed calculus independently of Leibniz Optics Spectral decomposition of light; reflecting telescope Mechanics Three universal laws of motion Gravitation 1/r2 law leads directly to Kepler s Laws of Planetary Motion Uses occult ideas of action at a distance, across a vacuum Must be instantaneous to conserve angular momentum, as seen in the planets Philosophiae Naturalis Principia Mathematica (1687) Origin of modern, physics-based astronomy Mark McCaughrean
M 2 F G = GM 1M 2 R 2 M 1 R Gravity is always attractive (need vector representation to account for this) Newton s gravity provides explanation for Kepler s Laws Kepler s Laws as stated take the Sun as stationary They have broader relevance and applicability (with suitable modifications...)
M2 GM1 M2 FG = R2 M1 R Simplistic derivation of Kepler! s Third Law : ve circular orbit : ME R M! centripetal force provided by gravity 2 GmE M! me v E = R R2 2 ve = 2πR orbital period : PE = ve 2 : substitute for ve PE2 = GM! R square it :! 4π 2 GM! " R3 P 2 R3 (Kepler III) PE2 4π 2 R2 = 2 ve
ve ME PE2 = R M!! 4π 2 GM! " R3 P 2 R3 (Kepler III) Derived for a circular orbit True also for an elliptical orbit......if radius R is replaced by semi major axis a 2a 2 P =! 2 4π GM! " a3 However, considers Sun to be stationary......which is not true Effectively, assumes that ME << M!
ve ME R M! 2 P =! 2 4π GM! " a3 To apply generally, replace M! with mass of central body P2 =! 4π 2 GM " a3 Accounting for more equal masses...! Should be : P 2 = 2 4π G(M1 + M2 ) " a3
Broader application of Kepler s Laws Apply wherever objects are in orbit Binary stars Exoplanet systems Black holes (with modification) Moons and other satellites Example: geostationary satellites Satellites with orbital period equal to Earth rotation period Calculate orbital radius: P 2 = 4π2 a 3 GM e a = 3 GM e P 2 4π 2 a = 3 6.67 10 11 6 10 24 (86 400) 2 4π 2 This is altitude above Earth s centre: altitude above surface is 42 000-6400 = 35 600 km Similarly, can work out period for satellite in low Earth orbit with a 6700 km; P = 5450 s = 91 min = 4.2 10 7 m = 42 000 km Mark McCaughrean
M 2 F G = GM 1M 2 R 2 M 1 R More complete representation : m 2 r 12 F 12 = Gm 1m 2 r 12 2 ˆr 12 m 1 r 12 is a vector that points from body 1 to body 2 F 12 is the force on body 2 caused by body 1 acceleration of body 2 is a 2 where : F 12 = m 2 a 2 Gravity is always attractive
m 2 r 12 F 12 = Gm 1m 2 r 12 2 ˆr 12 m 1 r 12 is a vector that points from body 1 to body 2 F 12 is the force on body 2 caused by body 1 acceleration of body 2 is a 2 where : F 12 = m 2 a 2 m 1 r 21 m 2 Newton s Third Law : F 21 = F 12 F 21 = + Gm 1m 2 r 12 2 ˆr 12 F 21 = Gm 1m 2 r 21 2 ˆr 21 r 21 = r 12 r 21 = r 12 acceleration of body 1 is a 1 where : F 21 = m 1 a 1
acceleration of body 2 is a 2 where : F 12 = m 2 a 2 acceleration of body 1 is a 1 where : F 21 = m 1 a 1 Newton s Third Law : F 21 = F 12 More massive body accelerated less a 2 a 1 = m 1 m 2 Often, larger mass can be considered stationary to a good approximation Body? Considering point masses here... But : a spherical body acts as point mass located at its centre However... doesn t respond as a point mass ( tides )