Physics of Astronomyweek 1 Thus. 6 April 2006

Astronomy: Universe Ch.5: Light

If there’s time: Plank mass

Astrophysics: CO 5: Spectra

Seminar: WebX workshop etc. for new students

Looking ahead

Universe Chapter 5: The Nature of Light

Guiding Questions1. How fast does light travel? How can this speed be measured?2. Why do we think light is a wave? What kind of wave is it?3. How is the light from an ordinary light bulb different from the light

emitted by a neon sign?4. How can astronomers measure the temperatures of the Sun and stars?5. What is a photon? How does an understanding of photons help explain

why ultraviolet light causes sunburns?6. How can astronomers tell what distant celestial objects are made of?7. What are atoms made of?8. How does the structure of atoms explain what kind of light those atoms

can emit or absorb?9. How can we tell if a star is approaching us or receding from us?

Galileo unsuccessfully attempted to measure the speed of light by asking an assistant on a distant hilltop to open a lantern the moment Galileo opened his lantern.

Light travels fast.

Light travels through empty space at a speed of 300,000 km/s, called c

In 1676, Danish astronomer Olaus Rømer noted that the exact time of eclipses of Jupiter’s moons varied based on how near or far Jupiter was to Earth.

This occurs because it takes different times for light to travel the different distances between Earth and Jupiter.

Improving measurements of c

In 1850, Frenchmen Fizeau and Foucalt showed that light takes a short, but measurable, time to travel by bouncing it off a rotating mirror. The light returns to its source at a slightly different position because the mirror has moved during the time light was traveling a known distance.

White light is composed of all colors which can be separated into a rainbow, or a spectrum, by

passing the light through a prism.

Visible light has a wavelength ranging from 400 nm (blue) to 700 nm (red).

Light is electromagnetic radiation. It has a wavelength and a frequency

.

Although Isaac Newton suggested that light was made of tiny particles 130 years earlier, Thomas Young demonstrated in 1801 that light has wave-like properties. He passed a beam of light through two narrow slits which resulted in a pattern of bright and dark bands on a stream.

This is the pattern one would expect if light had wave-like properties.

Imagine water passing through two narrow openings as shown below. As the water moves out, the resulting waves alternatively cancel and reinforce each other, much like what was observed in Young’s double slit experiment.

This is the pattern one would expect if light had wave-like properties.

It turns out that light has characteristics of both particles and waves. Light behaves according to the same equations that govern electric and magnetic fields that move at the speed c, as predicted

by Maxwell and verified by Hertz.

Light is a form of electromagnetic radiation,

Electromagnetic radiation consists of oscillating electric and magnetic fields. The distance between two successive wave

crests is the wavelength, .

Stars produce electromagnetic radiation in a wide variety of wavelengths in addition to visible light.

Astronomers sometimes describe EM radiation in terms of frequency, , instead of wavelength, . The relationship is:

Speed = distance/time

c = Where c is the speed of light, 3 x 108 m/s

WIEN’S LAW: The peak wavelength emitted is

inversely proportional to the temperature.

In other words, the higher the temperature,

the shorter the wavelength (bluer) of the

light emitted.

A dense object emits electromagnetic radiation according

to its temperature.

BLACKBODY CURVES: Each of these curves shows the intensity of light emitted at every wavelength for idealized glowing objects (called “blackbodies”) at three different temperatures.

Note that for the hottest blackbody, the maximum intensity is at the shorter wavelengths and the total amount of energy emitted is greatest.

Astronomers most often use the Kelvin or Celsius temperature scales.

In the Kelvin scale, the 0 K point is the temperature at which there would be no atomic motion. This unattainable point is called absolute zero.

In the Celsius scale, absolute zero is –273º C and on the Fahrenheit scale, this point is -460ºF.

The Sun is nearly a blackbody.

Wien’s law relates wavelength of maximum emission for a particular temperature:

max = 3 x 10-3 Tkelvins

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

ENERGY FLUX = T4 = intensity =Power/AreaENERGY FLUX is measured in joules per second per square meter of a

surface, and the constant = 5.67 x 10-8 W m-2 K-4

Wien’s law and the Stefan-Boltzmann let us discover the

temperature and intrinsic brightness of stars from their colors.

Energy of a photon in terms of wavelength:

E = h c / where h = 6.625 X 10-34 J s

or h = 4.135 X 10-15 eV

h = Planck’s constant

Energy of a photon in terms of frequency:

E = h where is the frequency of light

High energy light has short wavelength and high frequency.

Each chemical element produces its own unique set of spectral lines.

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

Continuum = rainbow of light

Law 1 A hot opaque body, such as a perfect blackbody, or a hot, dense gas produces a continuous spectrum -- a complete rainbow of colors with without any specific spectral lines. (This is a black body spectrum.)

Emission lines due to electron relaxation

Law 2 A hot, transparent gas produces an emission line spectrum - a series of bright spectral lines against a dark background.

Absorption lines due to electron excitation

Law 3 A cool, transparent gas in front of a source of a continuous spectrum produces an absorption line spectrum - a series of dark spectral lines among the colors of the continuous spectrum.

Kirchhoff’s Laws

Here is the Sun’s spectrum, viewed with a prism or diffraction grating.

But, where does light actually come from?

Light comes from the movement of electrons

in atoms.

Rutherford’s experiment revealed the nature of atoms

Alpha particles from a radioactive source are channeled through a very thin sheet of gold foil. Most pass through, showing that atoms are mostly empty space, but a few bounce back, showing the tiny nucleus is very massive.

An atom consists of a small, dense

nucleus surrounded by electrons

Nucleus = protons + neutrons

• The nucleus is bound by the strong force.

• All atoms with the same number of protons have the same name (called an element).

• Atoms with varying numbers of neutrons are called isotopes.

• Atoms with a varying numbers of electrons are called ions.

Spectral lines are produced when an electron jumps from one energy level to another within an atom.

Bohr’s formula for hydrogen lines

E = hc/ = E0 [ 1/nlo

2 – 1/nhi2 ]

nlo = number of lower orbit

nhi = number of higher orbit

R = Rydberg constant

= wavelength of emitted or absorbed photon

The wavelength of a spectral line is affected by the relative motion between

the source and the observer.

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

= wavelength shift

o = wavelength if source is not movingv = speed of source

c = speed of light

What can we learn by analyzing starlight?

• A star’s temperature – by peak wavelength

• A star’s chemical composition – by spectral analysis

• A star’s radial velocity – from Doppler shifts

Guiding Questions

1. How fast does light travel? How can this speed be measured?2. Why do we think light is a wave? What kind of wave is it?3. How is the light from an ordinary light bulb different from the light

emitted by a neon sign?4. How can astronomers measure the temperatures of the Sun and stars?5. What is a photon? How does an understanding of photons help explain

why ultraviolet light causes sunburns?6. How can astronomers tell what distant celestial objects are made of?7. What are atoms made of?8. How does the structure of atoms explain what kind of light those atoms

can emit or absorb?9. How can we tell if a star is approaching us or receding from us?

Practice problems

Pick a few to work on together

No homework assignment for Universe Ch.5

Do the Universe Online self-test for Ch.5

BREAK

Then we’ll derive Planck mass from some of these fundamental concepts, if we have time…

Calculating the Planck length and mass:

1. You used energy conservation to find the GRAVITATIONAL size of a black hole, the Schwartzschild radius R.

2. Next, use the energy of light to calculate the QUANTUM MECH. size of a black hole, De Broglie wavelength .

3. Then, equate the QM size with the Gravitational size to find the PLANCK MASS Mp of the smallest sensible black hole.

4. Finally, substitute M into R to find PLANCK LENGTH Lp

5. and then calculate both Mp and Lp.

1. Gravitational size of black hole (BH):R = event horizon

2

2

1

22

Gravitational energy kinetic energy

GmMmv

rGM

You solved for rv

The Schwarzschild radius, inside which not even light (v=c) can escape, describes the GRAVITATIONAL SIZE of BH.

2grav

GMR

c

2. Quantum mechanical size of black hole

in :

____________

Energy of photon wavelengthof particle

hc hE pc p Mc

Solve for wavelength terms of mass M

The deBroglie wavelength, , describes the smallest region of space in which a particle (or a black hole) of mass m can be localized, according to quantum mechanics.

3. Find the Planck mass, Mp

2

2

:

____________

p

p

p

Schwartzschild radius deBroglie wavelength

R

GM h

c M c

Solve for the Planck mass

M

If a black hole had a mass less than the Planck mass Mp, its quantum-mechanical size could be outside its event horizon. This wouldn’t make sense, so M is the smallest possible black hole.

4. Find the Planck length, Lp

These both yield the Planck length, Lp. Any black hole smaller than this could have its singularity outside its event horizon. That wouldn’t make sense, so L is the smallest possible black hole we can describe with both QM and GR, our current theory of gravity.

2

, , :

______________

______________

p

p

p

hcSubstitute your Planck mass M intoeither R or

GGM

Rch

M c

5. Calculate the Planck length and mass

2 3

34 8 112

2

:

6 10 , 3 10 , 7 10

, _____________

_________________

ms

p

pp

Usethese fundamental constants

kg m m mh x c x G x

s s kg s

hctoevaluate the Planck mass M

G

GMand the Planck length L

c

These are smallest scales we can describe with both QM and GR.

Break

Then we’ll continue with Astrophysics…

Astrophysics: CO Ch.5

Light and the interaction of matter:

• Spectral lines, Kirchhoff’s laws, Dopper shift

• Photon energy, Compton scattering

• Bohr model

• Quantum mechanics, deBroglie, Heisenberg

• Zeeman effect, Pauli exclusion principle

Astro. Ch.5: Interaction of light & matter

History of Light quantization:• 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 +

Astro. Ch.5: Interaction of light & matter

History of atomic models:• Thomson discovered electron, invented plum-pudding model• Rutherford observed nuclear scattering, invented orbital atom• Bohr used deBroglie’s matter waves, quantized angular momentum, for better H atom model. En = E/n2

• Bohr model explained observed H spectra, derived phenomenological Rydberg constant • Quantum numbers n, l, ml (Zeeman effect)• Solution to Schrodinger equation showed that En = E/l(l+1)• Pauli proposed spin (ms=1/2), and Dirac derived it

Improved Bohr model

Solution to Schrodinger equation showed that En = E/l(l+1),Where l = orbital angular momentum quantum number (l<n).

Almost the same energy for l=n-1, and for high n.

Quantum Mechanics

Light as waves with wavelength l: classicalLight as particles with discrete energy (Planck) E = hc/ = pc

Electrons as particles with momentum p=mv: classicalElectrons as waves with p=h/: de Broglie wavelength = ___

Heisenberg: 2x p 2E t

Magnetic fields can spin charged particles

Cyclotron frequency: An electron moving with speed v perpendicular to an external magnetic field feels a

Lorentz force (no change in energy here): F=ma

(solve for =v/r)

Compare to Zeeman effectStart HW (see hints): #4, 9, 14, 17

Magnetic fields can interact with intrinsic spin

Zeeman effect: A particle with intrinsic spin m has more or less energy depending on its orientation with an

external magnetic field

Start HW (see hints): #4, 9, 14, 17