Physics of Astronomy last lecture Tues.30.May 2006 Observations and explanations … … modern physics and astrophysics… Your research projects Looking ahead Let’s discuss the italicized text…

An incomplete overview of what we’ve done! What do we observe? What sense do we make of observations? How do things move? Why do things move? Forces and momenta Work and energy Oscillations … modern physics and astrophysics …

You’ve learned to find major star groupings …

“Arc to Arcturus, speed on to Spica”

How does the Sun (appear to) move? Why?

Describing celestial motions quantitatively Altitude-AzimuthRight ascension-Declination

What causes the seasons?

Lunar motion: when are eclipses possible?

Ptolemaic vs Copernican models Geocentric vs Heliocentric Strengths and weaknesses of each model?

Retrograde motion: observed & explained

Planetary motion Kepler - Why ellipses? Wait a century … need Newton’s mechanics … coming up … answers to questions outstanding since Aristotle… How do things move? Why do things move? Forces and momenta Work and energy Oscillations … modern physics and astrophysics …

Kinematics: How things move Linear motion: v(t) = dx/dt a(t) = dv/dt = d 2 x/dt 2 Uniform circular motion: a = v 2 /r

Dynamics: Why things move:  F=ma F = F x i + F y j where F x = m a x and F y = m a y

Orbital Dynamics F = ma GmM/r 2 = m v 2 /r Kepler’s laws (for M>>m)Generalization (for M~m) K1: orbits are elliptical - about center of mass K2: equal areas in equal times- conservation of L K3:

Linear and angular momenta Forces change linear momentaTorques change angular momenta  F=dp/dt (= ma) where  =dL/dt = r  F = I  where Linear momentum p = mv Angular momentum L = mv  r = I  is conserved if  F=0 is conserved if   =0 s=R , v=R , a tan =R    mr 2 K lin =p 2 /(2m) = ½ Mv 2 K ang =L 2 /(2I) = ½ I  2 Ex: Rocket propulsion, center of massEx: Kepler’s 2d law m 0 dv/dt = - v 0 dm/dt

Relationship between force and work/energy Work done = force. displacement in the same direction Dot product: A. B = AB cos 

Work done by a varying force Example: Spring obeys Hooke’s law: F = -kx

Potential energy U : F = -dU/dx only conservative forces have potential energy Kinetic energy K = T = Work done = ½ mv 2 When is mechanical energy conserved? Mechanical energy E = K + U

Energy conservation Conservative force: Work done doesn’t depend on path taken (curl x F = 0) Net work done around a closed path = 0 potential energy U depends only on x, and F x = -dU/dx E tot = K + U = constant (conservation of mechanical energy) Gravity and Electrostatics are conservative Friction and Magnetism are not conservative

Ex: Energy conservation in rotation lab Loss of potential energy→work done→increase in Kinetic energy -  U → W → +  K

Ex: Escape velocity and black hole Not even light can escape (v=c) if it is closer than r to a black hole. This is the Schwarzschild radius R (v=c)=_____________ MRMR mv m v→0, r→0

Hamiltonian formulation of equations of motion Easier: 1st order, scalar differential equations instead of  F=ma: 2nd order, vector differential equations. Conserved momenta p i are immediately evident, for position coordinates q i for which dH/dq i = 0

Virial Theorem = /2 where = average value of potential energy over one cycle Example: For gravitationally bound systems in equilibrium, the total energy is always one-half of the potential energy.

Energy diagrams and Power Power = rate of change of Energy: P = dE/dt Minimum energy = stable state (F=0)

Oscillations Systems oscillate about energy minimum Ex: Spring oscillates about equilibrium x 0 Displacement x(t) = A cos (  t +  )

Frequency of oscillation of mass on spring Angular frequency = angular speed =  = 2  f where frequency f = 1/T and T = period. Differentiate: Simplify: Solve for  2 =sqrt(k/m)

Moving on to modern physics… Oscillations: Bohr atom Conservation of energy + quantization of angular momentum Quantum Mechanics Spectra Stellar spectra Stars Parallax Astronomy Flux, luminosity, temperature, etc Light Electromagnetism and Maxwell equations Light, optics, QM, Cosmology

Bohr model for the Hydrogen atom (Same from energy conservation or the virial theorem.) Quantization of orbital angular momentum: L = mvr = nh/2  Eliminate v 2, and solve for

Where did that ‘h’ come from? Planck’s constant Blackbody solution h = smallest unit of angular momentum INVENTION OF THE QUANTUM

Bohr’s synthesis → SPECTRA Bohr combined Rutherford’s model of the orbiting electron with deBroglie’s hypothesis of electron wavelengths: angular momentum would be quantized in electron orbits Derived orbit radii and energy levels for H-like atoms. Despite unanswered questions (such as how could such orbits be stable?), Bohr’s model fit the observed Balmer spectrum and explained spectra beyond the visible range.

Observed spectrum of the Sun and Hydrogen Calculate energies of H lines from their colors: E = hc/  hf = pc Planck constant h = 6.63 x J.s Energy units: 1 eV = x J

Sun’s spectrum and range of electromagnetic spectrum

Spectra can involve diffraction, refraction and/or interference – discuss examples

Resolution limit (Ex: Venus by naked eye?)

Interference bright spots

Refraction: v = c/n; n 1 sin  1 = n 2 sin  2

The brightness of spectral lines depend on conditions in the spectrum’s source.

Wien’s law relates wavelength of maximum emission for a particular temperature: max (m) = 2.9 x T kelvins Stefan-Boltzmann law relates a star’s energy output, called E NERGY F LUX, to its temperature E NERGY F LUX =  T 4 = intensity =Power/Area Boltzmann constant  = 5.67 x W m -2 K -4 Stellar Spectra → Temperature & Flux

Spectra and O B A F G K M

Maxwell-Boltzmann distribution of a hot gas of temperature T The number of gas particles per unit volume with speed between v and dv is:

Our Sun fuses H→He, produces E=  mc 2

Finding the sizes of nearby stars

Hertzsprung-Russell diagram Bigger + Hotter → Brighter stars: L=4  T 4 R 2

Finding the distance and sizes of distant stars

Lives of stars

Deaths of stars

Our Galaxy and local cluster

The observable Universe

Light We know all this from the light of stars But what is light, exactly, and how does it work? Electromagnetism and Quantum mechanics ….

How light rays work: Ex: Telescope

Why: Electromagnetism

Maxwell equations  Light waves E(x,t)=E 0 sin (kx- w t) and B(x,t)=B 0 sin (kx- w t) solve Faraday’s and Ampere’s laws. Electromagnetic waves in vacuum have speed c = 1/  (     ) and energy/volume = 1/2 e 0 E 2 = B 2 /(2 m 0 )

Ex: Doppler Shifts Red Shift: The observer and source are separating, so light waves arrive less frequently. Blue Shift: The observer and source are approaching, so light waves arrive more frequently.  / o = v/c v = speed of source c = speed of light  = wavelength shift o = wavelength if source is not moving

Synthesis: Big Bang Faster recession of distant galaxies: universe is expanding 3K radiation: universe is cooling Primordial abundances of H, He and metals: early universe is understood Inflation: solution of horizon and flatness problems

But light is not just a wave… Stefan-Boltzmann blackbody had UV catastrophe Planck quantized light, and solved blackbody problem Einstein used Planck’s quanta to explain photoelectric effect Compton effect demonstrated quantization of light hc/ = K max + 

Quantum cosmology? Quantum Mechanics explains the very small General Relativity explains the very massive (theory of gravity)

Problem: the early singularity could be outside its own event horizon?! “Laws of physics break down” R  R=

Planck scales Last week’s workshop derived these fundamental sizes: Planck mass ~ 3 x kg ~ 4 x m A black hole smaller than this could be outside its own event horizon, so QM and gravity are not both consistent at this scale. ~ s At earlier times, our familiar laws of physics “break down”.

Outstanding cosmological questions What physics operated before the Planck time? What is gravity? Higgs? Graviton? Other? What is dark matter? Neutrinos? Wimps? What is dark energy? Why does universe’s expansion accelerate? How to unite gravity with QM? Loop quantum gravity? Superstrings? D-branes? Supersymmetric particles?

We need a new theory of “quantum gravity” String theory? Loop quantum gravity? Will one of these resolve the crisis and become our ultimate GUT?

How to choose which model? Criteria: * New model answers old Q * Predictions pass tests * New puzzles solvable * Simplicity, beauty * More? My generation articulated this problem. Your generation will solve it.

Other outstanding questions? Fundamental questions:Technical questions:

Looking ahead Final exam Thursday Peer evals: Friday Online survey Friday Eval conferences next Mon + Tues Summer! Next year…