The Stars as physical objects

1
The Stars - as physical objects

• Observations
• stars emit energy into space
• we know this because we can see them and light
  is a form of EM radiation, and EM waves carry
  energy

We also know that some stars are brighter than
others (as seen on any clear night)
Question Are brightness differences due to
variations in stellar luminosity or distance (or
both) ?
2
The Magnitude Scale

• Hipparchus (2nd century BC)
• divided stars into six categories of brightness
• Brightest 1st magnitude
• Faintest 6th magnitude

Note the magnitude refers to the apparent
brightness of a star and is not a fundamental
property of the star Apparent brightness is
related to the measured energy flux
3
Radcliff Observatory Oxford

• Norman Pogson (1856) observed
• The energy flux from a 1st magnitude star is
  about 100 times that from a 6th magnitude star
• flux(1st mag. Star) ? 100 x flux(6th mag. Star)
• Recall flux measured energy/m2/sec at detector

4
Pogsons magnitude contract

• Agree (1)
• a difference of 5 magnitudes corresponds to a
  flux ratio of exactly 100 to 1
• Agree (2)
• the stars Aldebran and Altair are magnitude 1
  stars (these will be our standard stars)

5
The point of the contract

• By relating magnitudes to flux ratios, and
  listing standard stars, Pogson set up
• an instrumental system for magnitude
  determination - based upon flux measurements
• a standardized system - all observers will obtain
  the same magnitude for the same given star

6
Where less is more
Sirius
10-cm telescope
Full Moon
Venus
Naked-eye
-6
6
12
0
-12
-9
9
apparent magnitude
Bright
Faint
Sun magnitude -26.75 limit of Hubble Space
Telescope magnitude 25
7
Definitions

• Apparent magnitude (m)
• the magnitude as measured from Earth
• Absolute magnitude (M)
• the apparent magnitude from a fixed distance of
  10 parsecs (about 32.6 light years)

Note Absolute magnitude is a fundamental
measure related to luminosity because the
distance is fixed (we shall define the parsec
soon)
8
Recall

• We saw earlier that
• So, when d 10 pc, the flux varies directly with
  the stars luminosity
• Hence
• Given magnitudes are related to the received flux
• Absolute magnitude must relate to the stars
  intrinsic energy output (luminosity)

? OOTETK
9
A five star result

• With definitions for apparent (m) and absolute
  magnitude (M) in place an important 5 Star
  General result can be derived
• m - M 5 logd / 10pc ()

Where d is the distance to the star in parsecs
10
Why a 5 star result ?

• can determine m (apparent magnitude) from flux
  measurements (and standard star system)
• can calibrate stars according to M (absolute
  magnitude)
• then use formula to find distances
• we literally know the scale of the Universe
  because of the 5 star formula

a difficult task more on this later
NB need to know the distance to at least a few
stars to calibrate M ? stellar parallax (see
later)
11
Absolute magnitude of the Sun

• From Earth, Sun has m -26.75 (from flux
  measurement)
• Earth-Sun distance 1 AU 4.85 x 10-6 pc
• From 5 star formula
• M? -26.75 - 5 log4.85 x 10-6 / 10
• Result Suns absolute magnitude M? 4.82

12
Another BIG result cool!

• We can now in principle find the distance to any
  sun-like star in the entire Universe by measuring
  its apparent magnitude

The no free lunch bit we need to be able to
recognize sun-like stars when we see them
13
Calibrating the Sun

• From Earth, Sun has m -26.75 (from flux
  measurement)
• Earth-Sun distance 1 AU 4.85 x 10-6 pc
• From 5 star formula
• M? -26.75 - 5 log4.85 x 10-6 / 10
• Result Suns absolute magnitude M? 4.82
• Result We can now find the distance to any
  Sun-like star in the Universe

14
Alpha Centauri A

• Alpha Centauri A is a Sun-like star
• Observations ? m -0.01
• 5 star formula with M? 4.82 gives
• -0.01 - 4.82 5 log(d / 10pc), or
• log(d / 10pc) -0.97, so
• d / 10 pc 10-0.97, and finally,
• d(a Cen. A) ? 1.1 parsec

15
Come and get it.

• Getting around the No free lunch problem

We need to be able to recognize like
stars (that is stars with the same physical
properties)
Hence We need to find a distance
independent method of classifying stars
And our meal ticket lies within the study of
stellar spectra
16
Radiation Laws

• The amount of EM energy that an object radiates
  into space is related to its temperature
• PHYSICS hot objects radiate more energy into
  space than cold ones
• The actual amount emitted at a specific
  wavelength, however, is calculated in terms of an
  idealized object

Enter Blackbody radiation theory
17
Blackbody radiators

• Absorb all EM radiation incident upon them (i.e.,
  no reflection hence black)

Have a constant temperature T
Re-radiate EM radiation at all wavelengths
With these idealized quantities a complete
theoretical model can be developed
18
Max Karl Ernst Ludwig Planck (1858 - 1947)
Initiated study of quantum mechanics (QM) in 1900
The ultraviolet catastrophe
the whole procedure was an act of
despair because a theoretical interpretation had
to be found at any price.
19

• What Max Planck did
• Developed an equation that described the way in
  which BBs radiate energy into space
• his equation describes the energy flux of a BB at
  a specific wavelength for a specified temperature

The Planck equation
20
Line connecting peaks of BBs of different
temperature
Energy flux
Planck curve for BB of temperature T1
T1
T2 where T2
Wavelength
21
Temperature

• Temperature is a measure of heat energy
• heat energy is related to the motion of the
  molecules and atoms within a solid / liquid / gas
• hot gas ? molecules move around rapidly
• cool gas ? molecules move around slowly

• Absolute zero temperature at -273 oC
• objects can get no colder - minimum energy state

22
The Kelvin Scale
Uses absolute zero as its starting point.
Introduced by William Thomson (? Lord Kelvin
1824 - 1907)
OOTETK

• T(Kelvin) T(Centigrade) 273
• Water boils at 373 K and freezes at 273 K

23
The astronomers grail - a link between BBs and T
BB of temperature T
Energy flux
Wiens law tells us how lmax varies with
temperature T
lmax
Wavelength
24
Wiens Law
Wilhelm Wien (1864 1928)

• The wavelength (lmax) at which a BB emits the
  maximum amount of energy decreases with
  increasing temperature
• lmax T 2.8977 x 10-3
• units lmax in meters, T in Kelvin

? OOTETK
25
The Suns Spectrum lab3
Peak at l 5000 Å
BB continuum
Infrared
UV
Visible light
26

• Observations
• Maximum energy flux from Sun at lmax 5000 Å
• Note units 1 Å 10-10 meters

Wiens law allows us to calculate the Suns
surface temperature
T (2.8977 x 10-3) / lmax (2.8977 x 10-3 /
(5000 x 10-10)
T? ? 5800 K
? another fundamental result
27
The Stefan - Boltzmann law

• The energy emitted by a BB per second per m2 over
  all wavelengths (i.e., the total energy flux F)
  varies with temperature
• F s T4
• Units Flux in Watts/m2, T in Kelvin

? OOTETK
Constant 5.6704 x 10-8
28
BB of temperature T
The Stefan- Boltzmann law relates to the area
under the Planck curve Area under curve total
flux
Energy flux
Wavelength
29
Applying this to stars

• Stefan-Boltzmann law for a BB
• Flux s T4 (Watts/m2)
• for a spherical BB (e.g., a star) of radius R,
  the surface area is 4pR2 and the luminosity is
• L surface area x Flux (Watts)
• L 4pR2 s T4

? OOTETK
F
Surface area 4pR2
30

• So, now we have an equation linking
• Luminosity, Radius and Temperature
• Idea for observations
• Wiens law gives us the temperature
• Distance and angular size give us the radius
• Stefan-Boltzmann law then gives us the luminosity

An Example The supergiant star Betelegeuse in
Orion
31
looking even more closely at starlight
Continuous spectrum
Blackbody spectrum
l
stars
Absorption line spectrum
BB spectrum with lines
l
Emission line spectrum
l
32
The absorption line spectrum of Vega
T(BB) 7100 K
Absorption lines
l (nm)
33
Spectral analysis is one of the most important
topics in modern astronomy
Theoretical models
Observations
Spectral lines
This solves the no free lunch problem
34

• So, the million dollar question is
• Just what are spectral lines ?
• Before we can answer this question we have to
  first consider a little atomic physics

35
The structure of atoms

• All matter is composed of atoms
• Atom nucleus electrons

Protons (P)
Neutrons (N)

• Atoms distinguished by number of P and N in
  nucleus
• In general, number of electrons number of
  protons

36
Discovering the atom

• Ernest Rutherford (1871 - 1937)
• 1911 scattering experiments reveal nucleus
  electron cloud structure

• Niels Bohr
• Danish physicist (1885 - 1962)
• Developed quantum mechanical model of electron
  orbitals (1913)
• idea is electrons must occupy discrete quantum
  states

37
The Bohr atom imagine electrons to be in orbit
about nucleus
proton
neutron
Electron orbit
electron
1 Å
2 Å
Hydrogen atom
1 Angstrom 10-10m
Helium atom
38
The Hydrogen atom
Allowed electron orbits
First excited state
Proton
E1
E2
Other allowed orbits
E3
increasing orbital energy ?
Ground state (lowest energy state)
39
Quantum mechanics all or nothing

• Electrons can only occupy allowed orbits
• Each electron orbit has a specific energy
• To move between orbits an electron must either
  lose or gain energy
• (by the absorption or emission of EM radiation)
• (or by collisions with other atoms)
• Electrons prefer to be in the lowest energy
  orbital (ground state)

40
Absorption transition - electron gains energy
Photon packet of EM radiation
Proton
E1
E2
Photon
E3
Energy of photon Ephoton E2 - E1
41
Emission transition - electron looses energy
Energy of photon Ephoton E3 - E1
Photon
Proton
E1
E2
E3
42
The energy game

• Energy gained or lost by an electron must be
  exactly equal to the energy difference between
  the final and initial orbitals
• The QM mantra
• Its all or nothing energy wise if an electron is
  to move between allowed orbitals

43

• Photon packet of EM radiation
• Recall
• Photon energy h x frequency
• h Plancks constant
• 6.6 x 10-34 JS

Trivial Pursuits term photon coined by
Albert Einstein in 1905
Nobel Prize in Physics in 1921 (photoelectric
effect)
44
And so, the million dollar pay-off

• Absorption lines
• formed when electrons jump from a low energy
  orbital to a high energy orbital by absorbing EM
  radiation (i.e., a photon)
• Emission lines
• formed when electrons drop from a high energy
  orbital to a low energy orbital by emitting a
  photon

45
Emission transition - electron looses energy
The Hydrogen atom
Photon
Proton
E1
E1
E2
Photon
E3
Absorption transition - electron gains energy
46
Emission lines

• In general, emission lines are associated with
  hot gases
• Two processes are at work
• (1) collisions between atoms place electrons in
  high energy orbitals
• (2) electron transitions to lower energy orbitals
  produces emission lines at discreet wavelengths

47
For Fridays class

• What is the wavelength of the photon emitted when
  an electron in a hydrogen atom jumps from the n
  5 orbital to the n 3 orbital
• What type of telescope would be required to
  detect such photons?
• Answer to show all your calculations and a
  diagram illustrating the transition

48
The absorption line spectrum of Vega
T(BB) 7100 K
Absorption lines
l (nm)
49
Stellar absorption lines

• Absorption line spectra produced by a
• blackbody radiator surrounded by a cool gas
  envelope

Hot, dense interior (blackbody spectrum)
Star
Cool, low density envelope (absorption lines)
50
The basics stellar spectra

• Hot, dense interior
• produces underlying continuum - that is a
  blackbody spectrum
• Cooler, lower density envelope
• electrons absorb specific wavelength continuum
  photons (the ones with just the right energy to
  allow transitions) and produce absorption lines

51
Shape and strength of lines related to
temperature
Flux
Continuum (from hot, dense interior)
l
Photons at these wavelengths are absorbed by
electrons in various specific atoms in the
stars cooler outer envelope
52
Spectral analysis

• Each species of atom has a unique set of allowed
  electron orbits - hence
• unique set of absorption / emission lines for
  each atomic species
• So can identify what stars are made of by
  looking at what spectral lines are present
• Stars are 75 hydrogen, 24 helium, and 1
  everything else

53
Spectral classification

• Classify stars according to their spectral lines
• key idea is to use strength of hydrogen lines
• Cecilia Payne-Gaposhkin (1900 - 1979)
• 1925 Ph.D thesis shows hydrogen line strengths
  are related to temperature

Annie Jump Cannon ? (1863 - 1941) classified
over 250,000 spectra as part of the Harvard
Observatory program to classify stars
54
Stellar Spectra
Ca (HK)
Ha
F
Hb
Hydrogen lines
Ha
G
O
(Sun)
B
G band
K
A
Molecular lines
UV
IR
Optical
55
The spectral classification scheme
Sp type Temperature Comments
O 30,000 hottest and most B 10 -
30,000 luminous stars A 7,500 -
10,000 strongest hydrogen F 6,000 -
7,500 lines G 5,000 - 6000 Sun G2
star K 3,500 - 5000 coolest and least M
3,500 luminous stars
56
Recall - Why classify ?

• Spectral types O B A F G K M
• Method is independent of distance
• Can recognize stars with similar intrinsic
  properties (temperature)
• e.g., sun-like stars spectral type G2
• Can now calibrate spectral type against absolute
  magnitude
• So, back to the 5 star formula

57
The Calibration of absolute magnitude (M)
Spectra
like stars (Same T)
5 star formula (M)
Apparent magnitude (m)
Distance (d)
58
The Fly in the Ointment (FITO)

• We need to know the distances to at least a few
  stars to calibrate M against spectral type

m - M 5 log(d / 10pc)