Physics of Astronomylast 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-Azimuth Right ascension-Declination

What causes the seasons?

Lunar motion: when are eclipses possible?

Ptolemaic vs Copernican modelsGeocentric vs Heliocentric

Strengths and weaknesses of each model?

1 1 1

S P P

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 = d2x/dt2

Uniform circular motion: a = v2/r

Dynamics: Why things move: F=ma F = Fx i + Fy j where Fx = m ax and Fy = m ay

Orbital Dynamics

F = ma

GmM/r2 = m v2/r

Kepler’s laws (for M>>m) Generalization (for M~m)K1: orbits are elliptical - about center of massK2: equal areas in equal times - conservation of L

K3:

2 2

2 21

x y

a b

2shell shell

GmdF dM

r

2 2 34G M m P a 2 2 34GMT R

Linear and angular momenta

Forces change linear momenta Torques change angular momenta

F=dp/dt (= ma) where =dL/dt = r F = I where

Linear momentum p = mv Angular momentum L = mvr = I

is conserved if F=0 is conserved if =0

s=R, v=R, atan=Rmr2

Klin=p2/(2m) = ½ Mv2 Kang=L2/(2I) = ½ I2

Ex: Rocket propulsion, center of mass Ex: Kepler’s 2d law

m0 dv/dt = - v0 dm/dt

Relationship between force and work/energy

Work done = force . displacement in the same direction

Dot product: A.B = AB cos

21

2x

x x x x

dv dxW m dx m dv mv dv mv

dt dt F dx

Work done by a varying force

Example: Spring obeys Hooke’s law: F = -kx

( )W kx dx F dx

Potential energy U : F = -dU/dx

only conservative forces have potential energy

Kinetic energy K = T = Work done = ½ mv2

When is mechanical energy conserved?

Mechanical energy E = K + U

Kinetic and Potential energy vs time

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5

time(s)

Kinetic(J)

Potential(m/s)

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 Fx= -dU/dx

• Etot = 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

Energies vs time

-1.00E-020.00E+001.00E-022.00E-023.00E-024.00E-025.00E-026.00E-027.00E-028.00E-02

0 5 10 15 20 25 30 35

Time (s)

En

erg

y(J)

U=mgh

Kt=1/2 mv̂ 2

Kdisk

Ktop

Kpulley

Ktot

Etot

Ex: Escape velocity and black hole

210 0

2____________

i i f f

Kinetic energy Gravitational energy

K U K U

GmMmv

rEscapevelocity v

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)=_____________

M

Rm v

m

v→0, r→0

Hamiltonian formulation of equations of motion2

( ) ... :2

pH T V V x treat p and x independently

mdH dx

dp dt

dH dp

dx dt

Easier: 1st order, scalar differential equations instead of F=ma: 2nd order, vector differential equations.

Conserved momenta pi are immediately evident, for position coordinates qi for which dH/dqi = 0

Virial Theorem

<E> = <U>/2where <U> = 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 x0

Displacement x(t) = A cos (t + )

Frequency of oscillation of mass on spring

Angular frequency = angular speed = = 2f

where frequency f = 1/T and T = period.

Differentiate:

Simplify:

Solve for 2=sqrt(k/m)

2

2

2

2( cos ) ( cos )

F ma

d xkx m

dt

dk A t m A t

dt

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

2 2 20 0

2 2

20

( )

4 4

4

F ma

kqQ qQ e ZeF

r r r

Ze vm

r r

(Same from energy conservation or the virial theorem.)

Quantization of orbital angular momentum:L = mvr = nh/2Eliminate v2, and solve for

22

1 1 2

2 4

1 12 2 28

on

o

hnr r where r

Z me

Z meE E where E

n h

Where did that ‘h’ come from?

Planck’s constant

Blackbody solution

h = smallest unit of angular momentum

INVENTION OF THE QUANTUM

http://www.mines.edu/Academic/courses/physics/phgn341/Lectures/Lecture37

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 10-34 J.s

Energy units: 1 eV = 1.602 x 10-19 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?)

1.22

D

Interference bright spots

sind m

Refraction: v = c/n; n1sin1 = n2sin2

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 10-3 Tkelvins

Stefan-Boltzmann law relates a star’s energy output, called ENERGY FLUX, to its temperature

ENERGY FLUX = T4 = intensity =Power/Area

Boltzmann constant = 5.67 x 10-8 W m-2 K-4

Stellar Spectra → Temperature & Flux

Spectra and O B A F G K M

http://www.gothard.hu/astronomy/astroteaching/anloos/specclass.html

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:

23

22 24

2

mvkT

v

mn dv n v e dv

kT

Our Sun fuses H→He, produces E=mc2

Finding the sizes of nearby stars

Hertzsprung-Russell diagram

Bigger + Hotter → Brighter stars: L=4T4R2

Finding the distance and sizes of distant stars

Lives of stars

Deaths of stars

Our Galaxy and local cluster

The observable Universe

LightWe 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

1 1 1

object imaged d f

Why: Electromagnetism

Charge E field Current B field

Faraday Changing B E

Ampere Changing E B

0

AdEq I0ldB

lE ddt

d B

lB ddt

d E

00

Maxwell equations Light waves

E(x,t)=E0 sin (kx-t) and B(x,t)=B0 sin (kx-t)solve Faraday’s and Ampere’s laws.

Electromagnetic waves in vacuum have speed c = 1/() and energy/volume = 1/2 0 E2 = B2 /(20 )

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 sourcec = speed of light = wavelength shift

o = wavelength if source is not moving

Synthesis: Big BangFaster 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

1 cose

h

m c

hc/ = Kmax +

Quantum cosmology?

Quantum Mechanics explains the very small

General Relativity explains the very massive (theory of gravity)

http://fusionanomaly.net/quantummechanics.html

http://www.phys.lsu.edu/dept/gifs/quantum.gif

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 10-8 kg

~ 4 x 10-35 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.

~ 10-43 s

At earlier times, our familiar laws of physics “break down”.

p

hcM

G

3p

hGPlanck length L

c

5p

hGPlanck time

c

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?http://www.columbia.edu/cu/record/archives/vol23/vol23_iss18/28c.gif http://www.cpt.univ-mrs.fr/~rovelli/rovelli.html

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: email Friday

Online survey Friday

Eval conferences next Mon + Tues

Summer! Next year…