Review Chapters 1-8 1.The Celestial Sphere –History –Positions 2.Celestial Mechanics –Elliptical Orbits –Newtonian Mechanics –Kepler’s Laws –Virial Theorem

Review Chapters 1-8

1. The Celestial Sphere– History– Positions

2. Celestial Mechanics– Elliptical Orbits– Newtonian Mechanics– Kepler’s Laws– Virial Theorem

3. Continuous Spectrum of Light– Stellar Parallax– The Magnitude Scale– The Wave Nature of light– Blackbody Radiation– Quantization of Energy – The Color Index

Review Chapters 1-8

4. Special Relativity– Lorentz Transformations

– Time and Space in Relativity

– Relativistic Momentum

– Redshift

5. Interaction of Light and Matter– Spectral Lines

– Photons

– The Bohr Model of the Atom

– Quantum Mechanics and Wave-Particle Duality

6. Telescopes– Basic Optics

– Optical Telescopes

– Radio Telescopes

– Infrared,UV,X-ray and Gamma-Ray astronomy

Review Chapters 1-8

7. Binary Systems and Stellar Parameters– Classification of Binary Stars

– Mass Determination Using Visual Binaries

– Eclipsing, Spectroscopic Binaries

– The Search for Extrasolar Planets

8. The Classification of Stellar Spectra– The Formation of Spectral Lines

– The Hertzsprung-Russell Diagram

Midterm Exam 1

• The exam– Part I - in-class

• Conceptual questions• “Easy” calculations (bring a calculator)• Three sheets (back and front) of notes (no xerographic reduction)

– Part II - Take-home• Application of the material to Astrophysical Problems• More of a pedagogical leaning exercise

• Preparation– Material from textbook ch 1-8

• Concepts• Equations

– Examples from textbook– Homework problems!!! See solutions on web…

Possible Problem Topics• Motions of heavenly bodies

• Position on celestial sphere

• Elliptical Orbits

• Newtonian Mechanics and Kepler’s Laws

• Escape Velocity

• Stellar Parallax

• Magnitude Scale, Flux, Luminosity,…

• Basic Properties of Light

• Blackbody Radiation– Wien’s and Stefan/Boltzmann– Planck Function (usage…not derivation)– Monochromatic Flux/Luminosity

• Color Index, Bolometric Correction

• Redshift

• Spectral Lines– Kirchoff’s Laws– Spectrographs

• Photons – Photoelectric, Compton effects– Energy in terms of eV-nm

• Model of the Atom– Wavelengths of emitted photons

• Quantum Mechanics– Wave-Particle Duality– Uncertainty Principle– Pauli Exclusion Principle

• Telescopes– Basic (Snell’s Law/Reflection)– Diffraction limit/Resolution/Seeing– Brightness/Focal Ratio/Magnification– Mounts– Diameter/Resolving Power

• Special Relativity– Time Dilation,Length contraction– Redshift– Relativistic Momentum

• Binary Star Systems– Classification– Mass Determination

• Classification of Stellar Spectra– H-R Diagram

• ….

The Celestial Sphere

HistoryThe Greek traditionCopernican Revolution

Positions on the celestial sphereAltitude-Azimuth coordinate system Equatorial coordinate systemDaily,Seasonal changes, Precession

Measurements of time

Positions on the Celestial Sphere

The Altitude-Azimuth Coordinate System • Coordinate system based on observers

local horizon• Zenith - point directly above the

observer • North - direction to north celestial pole

NCP projected onto the plane tangent to the earth at the observer’s location

• h: altitude - angle measured from the horizon to the object along a great circle that passes the object and the zenith

• z: zenith distance - is the angle measured from the zenith to the object z+h=90

• A: azimuth - is the angle measured along the horizon eastward from north to the great circle used for the measure of the altitude

Equatorial Coordinate System

• Coordinate system that results in nearly constant values for the positions of distant celestial objects.

• Based on latitude-longitude coordinate system for the Earth.

• Declination - coordinate on celestial sphere analogous to latitude and is measured in degrees north or south of the celestial equator

• Right Ascension - coordinate on celestial sphere analogous to longitude and is measured eastward along the celestial equator from the vernal equinox to its intersection with the objects hour circle

Hour circle

Positions on the Celestial SphereThe Equatorial Coordinate System

• Hour Angle - The angle between a celestial object’s hour circle and the observer’s meridian, measured in the direction of the object’s motion around the celestial sphere.

• Local Sidereal Time(LST) - the amount of time that has elapsed since the vernal equinox has last traversed the meridian.

• Right Ascension is typically measured in units of hours, minutes and seconds. 24 hours of RA would be equivalent to 360.

• Can tell your LST by using the known RA of an object on observer’s meridian

Hour circle

What is a day?

• Solar day– Is defined as an average interval

of 24 hours between meridian crossings of the Sun.

– The earth actually rotates about its axis by nearly 361 in one solar day.

• Sidereal day– Time between consecutive

meridian crossings of a given star. The earth rotates exactly 360 w.r.t the background stars in one sidereal day = 23h 56m 4s

The period (sidereal) of earth’s revolution about the sun is 365.26 solar days. The earth moves about 1 around its orbit in 24 hours.

Precession of the Equinoxes

• Precession is a slow wobble of the Earth’s rotation axis due to our planet’s nonspherical shape and its gravitational interaction with the Sun, Moon, etc…

• Precession period is 25,770 years, currently NCP is within 1 of Polaris. In 13,000 years it will be about 47 away from Polaris near Vega!!!

• A westward motion of the Vernal equinox of about 50” per year.

Celestial MechanicsFun with Kepler and Newton

•Elliptical Orbits

•Newtonian Mechanics

•Kepler’s Laws Derived

•Virial Theorem

Elliptical Orbits 3Kepler’s Laws of Planetary Motion

Kepler’s First Law: A planet orbits the Sun in an ellipse, with the Sun at one focus of the ellipse.

Kepler’s Second Law: A line connecting a planet to the Sun sweeps out equal areas in equal time intervals

Kepler’s Third Law: The Harmonic Law

P2=a3

Where P is the orbital period of the planet measured in years, and a is the average distance of the planet from the Sun, in astronomical units (1AU = average distance from Earth to Sun)

The Geometry of Elliptical Motion

Description in book (pp25-27. ). Example 2.1.1

€

b2 = a2(1− e2)

r =a(1− e2)

1+ ecosθ

A = πab

Kepler’s First Law

Kepler’s First Law: A planet orbits the Sun in an ellipse, with the Sun at one focus of the ellipse.

• a=semi-major axis• e=eccentricity• r+r’=2a - points on

ellipse satisfy this relation between sum of distance from foci and semimajor axis

Kepler’s Second Law

Kepler’s Second Law: A line connecting a planet to the Sun sweeps out equal areas in equal time intervals

Kepler’s Third Law

Kepler’s Third Law: The Harmonic Law

P2=a3

• Semimajor axis vs Orbital Period on a log-log plot shows harmonic law relationship

Newton’s Laws of Motion

• Newton’s First Law: The Law of Inertia. An object at rest will remain at rest and an object in motion will remain in motion in a straight line at a constant speed unless acted upon by an external force.

• Newton’s First Law: The net force (thesum of all forces) acting on an object is proportional to the object’s mass and its resultant acceleration.

• Newton’s Third Law: For every action there is an equal and opposite reaction

€

rF net =

r F i

i=1

n

∑ = mr a

Newton’s Law of Universal Gravitation

• Using his three laws of motion along with Kepler’s third law, Newton obtained an expression describing the force that holds planets in their orbits…(derivation in book, done on blackboard)

€

F = GMm

r2

Work and Energy

• Energetics of systems • Potential Energy• Kinetic Energy • Total Mechanical Energy• Conservation of Energy• Gravitational Potential

energy

• Escape velocity€

U = −GMm

r

€

K =1

2mv 2

€

vesc = 2GM /r

Derivations on pp37-39

Derivation of Kepler’s First Law

• Consider Effect of Gravitation on the Orbital Angular Momentum

• Central Force

Angular Momentum Conserved

• Consideration of quantity leads to equation of ellipse describing orbit!!!

• Derivation on pp43-45

€

rL = μ

r r ×

r v =

r r ×

r p

€

ra ×

r L

Derivation of Kepler’s Second Law

• Consider area element swept out by line from principal focus to planet.

• Express in terms of angular momentum

• Since Angular Momentum is conserved we obtain the second law

• Derived on pp 45-48

€

dA

dt=

1

2

L

μ

Derivation of Kepler’s Third Law

• Integration of the expression of the 2nd law over one full period

• • Results in

• Derived on pp 48-49

€

dA

dt=

1

2

L

μ

€

A =1

2

L

μP

€

P 2 =4π 2

G(m1 + m2)a3

Virial Theorem

• Virial Theorem: For gravitationally bound systems in equilibrium the Total energy is always one half the time averaged potential energy

• The Virial Theorem can be proven by considering the quantity and its time derivative along with Newton’s laws

and vector identities

• Many applications in Astrophysics…stellar equilibrium, galaxy clusters,….• Derivation on pp 50-53

€

E =U

2

€

Q ≡r p i •

r r i

i

∑

Distance and Brightness

•Stellar Parallax

•The Magnitude Scale

Stellar Parallax

• Trigonometric Parallax: Determine distance from “triangulation”

• Parallax Angle: One-half the maximum angular displacement due to the motion of Earth about the Sun (excluding proper motion)

With p measured in radians

€

tanθ = B /d

d = B /tanθ

€

d =1AU

tan p≈

1

pAU

PARSEC/Light Year

• 1 radian = 57.2957795 = 206264.806”• Using p” in units of arcsec we have:

• Astronomical Unit of distance: PARSEC = Parallax Second = pc 1pc = 2.06264806 x 105 AU

• The distance to a star whose parallax angle p=1” is 1pc. 1pc is the distance at which 1 AU subtends an angle of 1”

• Light year : 1 ly = 9.460730472 x 1015 m • 1 pc = 3.2615638 ly

€

d ≈206,265

p"AU

€

d ≈1

p"pc

•Nearest star proxima centauri has a parallax angle of 0.77”•Not measured until 1838 by Friedrich Wilhelm Bessel•Hipparcos satellite measurement accuracy approaches 0.001” for over 118,000 stars. This corresponds to a a distance of only 1000 pc (only 1/8 of way to centerof our galaxy)•The planned Space Interferometry Mission will be able to determine parallax angles as small as 4 microarcsec = 0.000004”) leading to distance measurements of objects up to 250 kpc.

The Magnitude Scale

• Apparent Magnitude: How bright an object appears. Hipparchus invented a scale to describe how bright a star appeared in the sky. He gave the dimmest stars a magnitude 6 and the brightest magnitude 1. Wonderful … smaller number means “bigger” brightness!!!

• The human eye responds to brightness logarithmically. Turns out that a difference of 5 magnitudes on Hipparchus’ scale corresponds to a factor of 100 in brightness. Therefore a 1 magnitude difference corresponds to a brightness ratio of 1001/5=2.512.

• Nowadays can measure apparent brightness to an accuracy of 0.01 magnitudes and differences to 0.002 magnitudes

• Hipparchus’ scale extended to m=-26.83 for the Sun to approximately m=30 for the faintest object detectable

Flux, Luminosity and the Inverse Square Law

• Radiant flux F is the total amount of light energy of all wavelengths that crosses a unit area oriented perpendicular to the direction of the light’s travel per unit time…Joules/s=Watt

• Depends on the Intrinsic Luminosity (energy emitted per second) as well as the distance to the object

• Inverse Square Law:

€

F =L

4πr2

Absolute Magnitude and Distance Modulus

• Absolute Magnitude, M: Defined to be the apparent magnitude a star would have if it were located at a distance of 10pc.

• Ratio of fluxes for objects of apparent magnitudes m1 and m2 .

• Taking logarithm of each side

€

F2

F1

=100(m1 −m2 ) / 5

€

m1 − m2 = −2.5log10

F1

F2

⎛

⎝ ⎜

⎞

⎠ ⎟

•Distance Modulus: The connection between a star’s apparent magnitude, m , and absolute magnitude, M, and its distance, d, may be found by using the inverse square law and the equation that relates two magnitudes.

Where F10 is the flux that would be received if the star were at a distance of 10 pc and d is the star’s distance measured in pc. Solving for d gives:

The quantity m-M is a measure of the distance to a star and is called the star’s distance modulus

€

100(m−M ) / 5 =F10

F=

d

10pc

⎛

⎝ ⎜

⎞

⎠ ⎟

2

€

d =10(m−M +5)/ 5 pc

€

m − M = 5log10(d) − 5 = 5log10

d

10pc

⎛

⎝ ⎜

⎞

⎠ ⎟

Einstein’s Postulates of Special Relativity

• The Principle of Relativity. The laws of physics are the same in all inertial reference frames.

• The Constancy of the Speed of Light. Light moves through vacuum at a constant speed c that is independent of the motion of the light source.

A reference frame in which a mass point thrown from the same point in three different (non co-planar) directions follows rectilinear paths each time it is thrown, is called an inertial frame.

– L. Lange (1885) as quoted by Max von Laue in his book (1921) Die Relativitätstheorie, p. 34, and translated by Iro).

Proper Time And Time Dilation

€

t2 − t1 =(t2

′ − t1′) + (x2′ − x1

′)u /c 2

1− u2 /c 2

€

(x2′ − x1

′) = 0

€

Δt =Δ ′ t

1− u2 /c 2

€

Δtmoving =Δtrest

1− u2 /c 2= γΔtrest

Proper Length and Length Contraction

• Measure positions at endpoints at same time in frame S’ and in frame S, L’=x2’-x1’

€

x2′ − x1

′ =(x2 − x1) − u(t2 − t1)

1− u2 /c 2

€

L′ =L

1− u2 /c 2

€

Lmoving = Lrest 1− u2 /c 2 = Lrest /γ

Redshift

• Can determine radial velocity of object by measuring shift in spectral lines….

• See example 4.3.2 p99

• For v<<c we have€

z ≡λ obs − λ rest

λ rest

€

obs = ν rest

1− vr /c

1+ vr /c

€

obs = λ rest

1+ vr /c

1− vr /c

€

z =1+ vr /c

1− vr /c−1

€

z +1 =1+ vr /c

1− vr /c

€

vr

c=

(z +1)2 −1

(z +1)2 +1

€

vr

c= z =

λ obs − λ rest

λ rest

(radial motion)

Relativistic Momentum and Energy

€

K = mc 2(γ −1)

€

E = γmc 2

Kinetic Energy

Total Energy

Rest Energy

€

E = mc 2

€

E 2 = p2c 2 + m2c 4

Momentum Energy Relation

€

rp =

mr v

1− u2 /c 2= γm

r v

Momentum

The Continuous Spectrum of Light

•The Nature of Light

•Blackbody Radiation

•The Color Index

Speed of Light

• Ole Roemer(1644-1710) measured the speed of light by observing that the observed time of the eclipses of Jupiter’s moons depended on how distant the Earth was from Jupiter. He estimated that the speed of light was 2.2 x 108 m/s from these observations. The defined value is now c=2.99792458 x 108 m/s (in vacuum). The meter is derived from this value.

• Measurement of speed of light is the same for all inertial reference frames!!!

Special Relativity(will come back to this topic..soon)

Takes an additional 16.5 minutes for light to travel 2AU

The Wave Nature of Light

• Light impinging on double slit

• Exhibits Inerference pattern

€

€

d sinθ =nλ

(n −1

2)λ

⎧ ⎨ ⎪

⎩ ⎪

Interference condition

(n=0,1,2,…for bright fringes)

(n=1,2,…for dark fringes)

INTERFERENCE

WAVEhttp://vsg.quasihome.com/interfer.htm

http://vsg.quasihome.com/interfer.htm









Electromagnetic Waves

Electromagnetic Wave speed

€

c=με

=3 × 8 /m s Light is indeed anElectromagnetic Wave

€

με ≈(8.85 × − sC/m 3 ⋅ )(kg π × −7m⋅ /kg C )

Waves are Transverse

€

rE⊥rB⊥rS

Electromagnetic Spectrum

Region Wavelength

Gamma Ray nm

X-Ray 1 nm<10 nm

Ultraviolet 10 nm<400 nm

Visible 400 nm<700 nm

Infrared 700 nm<1 mm

Microwave 1mm<10 cm

Radio 10 cm<

Radiation Pressureand the Poynting Vector

Radiation Pressure

Radiation Pressure is significant in– extremely luminous objects such as:

• early main-sequence stars • red supergiants• Accreting compact stars

– Interstellar medium dust particles

Poynting Vector

€

rS=

μ

rE×

rB

•The rate at which energy is carried by a light wave is described by the Poynting vector.•Instantaneous flow of energy per unit area per unit time (W/m2) for all wavelengths.•Points in the direction of the electromagnetic wave’s propagation.•Radiant Flux: Time average (over one period) of the Poynting vector

•Because an electromagnetic wave carries momentum it can exert a force on a surface hit by light…

€

S =1

2μ0

E0B0

€

Frad =S A

ccosθ

€

Frad =2 S A

ccosθ

absorption)

reflection)

Particle-like nature of lightPhotons

• Photon = “Particle of Electromagnetic “stuff””

• Blackbody RadiationFailure of Classical Theory

Radiation is “quantized”

• Photo-electric effect (applet)

€

E=hLight is absorbed and emitted in tiny discrete bursts

http://www.lon-capa.org/~mmp/kap28/PhotoEffect/photo.htm

http://en.wikipedia.org/wiki/Photoelectric_effect

http://en.wikipedia.org/wiki/Blackbody_radiation

Blackbody Radiation

• Any object with temperature above absolute zero 0K emits light of all wavelengths with varying degrees of efficiency.

• An Ideal Emitter is an object that absorbs all of the light energy incident upon it and re-radiates this energy with a characteristic spectrum.Because an Ideal Emitter reflects no light it is known as a blackbody.

• Wien’s Law: Relationship between wavelength of Peak Emission max and temperature T.

• Stefan-Boltzmann equation: (Sun example)

Blackbody Radiation Spectrum

€

maxT = 0.002897755mK

€

L = AσT 4

Blackbody

L:Luminosity A:area T:Temperature

Blackbody Radiation

€

maxT = 0.002897755mK

€

L = AσT 4

http://www.mhhe.com/physsci/astronomy/applets/Blackbody/frame.html







Planck’s Law for Blackbody Radiation

•Planck used a mathematical “sleight of

hand” to solve the ultraviolet catastrophe.

•The energy of a charged oscillator of frequency f is limited to discrete values of Energy nhf.•During emission or absorption of light the change in energy of an oscillator is

hf.•The mean energy at high frequencies tends to zero because the first allowed oscillator energy is so large compared to the average thermal energy available kBT that there is almost zero probability that this state is occupied.

•Planck seemed to be an “Unwilling

revolutionary”.He viewed this “quantization” merely as a calculational trick…Einstein viewed it differently…light itself was quantized.

€

u(λ ,T) =8πhc

λ5(ehc / λkBT −1)

Derivation of Stefan-Boltzmann Law

€

etotal =c

4u(λ ,T)dλ =

λ = 0

∞

∫ 2πhc 2

λ5(ehc / λkBT −1)dλ

0

∞

∫

€

x = hc /λkBT

€

etotal =2πkB

4T 4

c 2h3

x 3

ex −1( )dx

0

∞

∫€

x 3

(ex −1)dx

0

∞

∫ =π 4

15

€

etotal =2π 5kB

4

15c 2h3T 4 = σT 4

€

σ = (2)(3.141)5(1.381×10−23 J /K)4

(15)(2.998 ×108 m /s)2(6.64 ×10−34 J ⋅s)3

σ = 5.67 ×10−8W ⋅m−2 ⋅K−4

•Stefan-Boltzmann’s Law can be obtained from Planck’s Law by simply integrating the spectral density function over all wavelengths

•Subsituting and evaluating

•We obtain:

The Stefan-Boltzmann constant is a derived constant depending on kB, h and c !!!!!

Color/Temperature Relation

•Planck’s law and Astrophysics

•Monochromatic Luminosity and Flux

•Bolometric Magnitude

•Filters, measured flux

•The Color Index

Planck’s Law and Astrophysics

Power radiated per unit wavelength per unit area per unit time per steradian: (units are W m -2 m-1 sr -1 )

€

Bλ (T) =c

4πu(λ ,T) =

2hc 2 /λ5

(ehc / λkBT −1)

Power radiated per unit wavelength per unit area per unit time per steradian: (units are W m -2 s sr -1 )

€

Bν (T) =c

4πu(ν ,T) =

2hν 3 /c 2

(ehν / kBT −1)

In spherical coordinates the amount of radiant energy per unit time having wavelengths between anddemitted by a blackbody radiator of temperature T and surface area dA into a solid angle is given by:

€

dΩ ≡ sinθ ⋅dθ ⋅dφ

€

Bλ (T)dλdAcosθdΩ = Bλ (T)dλdAcosθ sinθdθdφ

€

Bν (T)dνdAcosθdΩ = Bν (T)dνdAcosθ sinθdθdφ

In terms of B:

Consider a model star consisting of a spherical blackbody of radius R and temperature T. Assuming that each patch dA emits isotropically over the outward hemisphere, the energy per second having wavelengths between anddemitted by the star is:

€

Lλ dλ = Bλ dλdAcosθ sinθdθdφA

∫θ = 0

π / 2

∫φ= 0

2π

∫Angular integration yields a factor of π , the integration of dA over the surface of the sphere yield 4πR2.

€

Lλ dλ = Bλ dλdAcosθ sinθdθdφA

∫θ = 0

π / 2

∫φ= 0

2π

∫

€

Bλ (T)dλ =σT 4

π0

∞

∫

Note that:

Monochromatic Luminosity and Flux

€

Lλ dλ = 4π 2R2Bλ dλ

€

Lλ dλ =8π 2R2hc 2 /λ5

ehc / λkBT −1dλ

Monochromatic Luminosity

Monochromatic Flux received at a distance r from the model star is:

€

Fλ dλ =Lλ

4π ⋅r2dλ =

2πhc 2 /λ5

ehc / λkBT −1

R

r

⎛

⎝ ⎜

⎞

⎠ ⎟2

dλ

Fd is the number of Joules of starlight energy with wavelengths between anddthat arrive per second per one square meter of detector aimed at the model star, assuming that no light has been absorbed or scattered during its journey from the star to the detector. Earth’s atmosphere absorbs some starlight, but this can be corrected. The values of these quantities usually quoted for stars have been corrected and would correspond to what would be

measured above Earth’s atmosphere.

Why do we keep the wavelength dependence?

Filters!!! Sf(

€

mbol = −2.5log10 Fλ dλ0

∞

∫( ) + Cbol

S

The Color IndexUVB Wavelength Filters

• Bolometric Magnitude: measured over all wavelengths.

• UBV wavelength filters: The color of a star may be precisely determined by using filters that transmit light only through certain narrow wavelength bands:

– U, the star’s ultraviolet magnitude. Measured through filter centered at 365nm and effective bandwidth of 68nm.

– B,the star’s blue magnitude. Measured through filter centered at 440nm and effective bandwidth of 98nm.

– V,the star’s visual magnitude. Measured through filter centered at 550nm and effective bandwidth of 89nm

• U,B,and V are apparent magnitudes

Sensitivity Function S()

Color Indices and Bolometric Correction

Color Indices• A star’s absolute color magnitudes can be

determined from the apprarent color magnitudes by using eqn 3.6 if the distance is known.

• U-B color index:difference between its ultraviolet and blue magnitudes.

• V-B color index:difference between its blue and

visual magnitudes.

Stellar magnitudes decrease with increasing brightness. A star with a smaller B-V index is bluer than a star with a larger value of B-V!!!!

Because a star’s color index is a difference in magnitudes it is independent of the star’s distance

€

U − B = MU − MB

€

B −V = MB − MV

Bolometric Correction•The difference between a star’s bolometric magnitude and its visual magnitude is known as the bolometric correction given by:(example 3.6.1)

BC=mbol-V=Mbol-MV

Color Indices and Bolometric Correction

Color Indices• A star’s absolute color magnitudes can be

determined from the apprarent color magnitudes by using eqn 3.6 if the distance is known.

• U-B color index:difference between its ultraviolet and blue magnitudes.

• V-B color index:difference between its blue and

visual magnitudes.

Stellar magnitudes decrease with increasing brightness. A star with a smaller B-V index is bluer than a star with a larger value of B-V!!!!

Because a star’s color index is a difference in magnitudes it is independent of the star’s distance

€

U − B = MU − MB

€

B −V = MB − MV

Bolometric Correction•The difference between a star’s bolometric magnitude and its visual magnitude is known as the bolometric correction given by:(example 3.6.1)

BC=mbol-V=Mbol-MV

Filter Response

€

U = −2.5log10 Fλ SU dλ0

∞

∫( ) + CU

€

V = −2.5log10 Fλ SV dλ0

∞

∫( ) + CV

€

B = −2.5log10 Fλ SB dλ0

∞

∫( ) + CB

€

U − B = −2.5log10

Fλ SU dλ0

∞

∫Fλ SB dλ

0

∞

∫

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟+ CU −B

€

B −V = −2.5log10

Fλ SB dλ0

∞

∫Fλ SV dλ

0

∞

∫

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟+ CB−V

€

CU −B ≡ CU − CB

€

CB−V ≡ CB − CV

If one assumes that B is slowly varying across the bandwidth of the filter S can be approximated by a step function S=1 inside the filter’s bandwidth and S=0 otherwise.The integrals for U,V and B can be approximated by the value of the Planck function B at the center of the filter bandwidth,multiplied by that bandwidth. Therefore for the filters listed on p 75 of the text, we have

€

U − B = −2.5log10

B365ΔλU

B440Δλ B

⎛

⎝ ⎜

⎞

⎠ ⎟+ CU −B

€

B − V = −2.5log10

B440ΔλB

B550ΔλV

⎛

⎝ ⎜

⎞

⎠ ⎟+CB−V

Look at example 3.6.2 T=42000K U-B=-1.19 and B-V=-0.33

The constants CU,CB and CV differ and are chosen such that the star Vega(T=9600K, use applet) has a magnitude of zero as seen through each filter. This does not imply that Vega would be equally bright when viewed through them.

Filter Response

€

U = −2.5log10 Fλ SU dλ0

∞

∫( ) + CU

€

V = −2.5log10 Fλ SV dλ0

∞

∫( ) + CV

€

B = −2.5log10 Fλ SB dλ0

∞

∫( ) + CB

€

U − B = −2.5log10

Fλ SU dλ0

∞

∫Fλ SB dλ

0

∞

∫

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟+ CU −B

http://astro.unl.edu/naap/blackbody/animations/blackbody.html

€

U − B = −2.5log10

Fλ SU dλ0

∞

∫Fλ SB dλ

0

∞

∫

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟+ CU −B









Spectral Type, Color and Effective Temperaturefor Main-Sequence Stars

Spectral Type B-V Te(K)

O5 -0.45 35,000

B0 -0.31 21,000

B5 -0.17 13,500

A0 0.00 9,700

A5 0.16 8,100

F0 0.30 7,200

F5 0.45 6,500

G0 0.57 6,000

G5 0.70 5,400

K0 0.84 4,700

K5 1.11 4,000

M0 1.24 3,300

M5 1.61 2,600 0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

-1 -0.5 0 0.5 1 1.5 2

B-V

Temperature

From Frank Shu, An Introduction to Astronomy(1982), Adapted from C.W. Allen, Astrophysical QuantitiesNote that this table does not quite agree with our text!!!!

Spectral Type, Color and Effective Temperaturefor Main-Sequence Stars

QuickTime™ and a decompressor

are needed to see this picture.

Spectral Type, Color and Effective Temperaturefor Main-Sequence Stars (continued)



Color-Color Diagram

• Relation between the U-B and B-V color indices for main sequence stars.

• Would be a straight line if stars were true black bodies. Some light is absorbed as it travels through a star’s atmosphere. The absorption being wavelength dependent alters the distribution of radiation from that of a blackbody.

• Best agreement to Blackbody radiation for very hot stars….

Interstellar Reddening

One also needs to correct color indices for interstellar reddening. As the light propagates through interstellar dust, the blue light is scattered preferentially making objects appear to be redder than they actually are…

The Interaction of Light and Matter

• Spectral Lines• Photons• Rutherford-Bohr Model of the

Atom• Quantum Mechanics and

Wave-Particle Duality

Fraunhofer Spectral Lines

• Josef von Fraunhofer(1787-1826) had catalogued 475 dark spectral lines in the solar spectrum.

• Fraunhofer showed that we can learn the chemical composition of the stars. Identified a spectral line of sodium in the spectrum of the Sun

http://en.wikipedia.org/wiki/Fraunhofer_lines

http://en.wikipedia.org/wiki/Fraunhofer_lines

Spectral LinesKirchoff’s Laws of Spectra

• Robert Bunsen (1811-1899) created a burner that produced a “colorless” flame ideally suited for studying the spectra of heated substances.

• Gustav Kirchoff (1824-1887) and Bunsen designed a spectroscope that could analyze the emitted light.

• Kirchoff determined that 70 of the dark lines in the solar spectrum corresponded to the 70 bright lines emitted by iron vapor.

• Chemical Analysis by Spectral Observations “Spectral Fingerprint”.

•A hot dense gas or hot solid object produces a continuous spectrum with no dark spectal lines•A hot, diffuse gas produces bright spectral lines (emission lines)•A cool, diffuse gas in fron of a sources of a continuous spectrum produces dark spectral lines (absorption lines) in the continuous

spectrum.

Spectral LinesApplication of Spectral Measurements

• Stellar Doppler Shift• Galactic Doppler Shifts• Quasar Doppler Shifts



http://sci2.esa.int/interactive/media/flashes/2_2_1.htm

Radial Velocities





Spectral LinesSpectrographs

• Spectroscopy• Diffraction grating

equation

(n=0,1,2,…)

• Resolving Power

€

d sinθ = nλ

€

Δ = nN

http://wps.prenhall.com/wps/media/objects/610/625137/Chaisson/CH.00.002/HTML/CH.00.002.s5.htm

PhotonsPhotoelectric Effect

• Photoelectric Effect• Kinetic energy of ejected

electrons does not depend on intensity of light!

• Increasing intensity will produce more ejected electrons.

• Maximum kinetic energy of ejected electrons depends on frequency of light.

• Frequency must exceed cutoff frequency before any electrons are ejected

Einstein took Planck’s assumption of quantized energy of EM waves seriously. Light consisted of massless photons whose energy was:

Einstein was awarded the Nobel Prize in 1921 for his work on the photo-electric effect

€

E photon = hν =hc

λ

€

Kmax = E photon − Φ = hν − Φ =hc

λ− Φ

http://phet.colorado.edu/simulations/sims.php?sim=Photoelectric_Effect

Photo-electric Effect



PhotonsCompton Scattering

• Compton Scattering• Wikipedia entry• Arthur Holly Compton (1892-

1962) provided convincing evidence that light manifests particle-like properties in its interaction with matter by considering how (x-ray) photons can “collide” with a free electron at rest.

• Conservation of energy and momentum leads to the following:

• Showed that photons are massless yet carry momentum!!!

€

Δ = f − λ i =h

mec(1− cosθ)

€

E photon = hν =hc

λ= pc

Compton Wavelength:is the characteristic change in wavelength in the scattered photon.€

C =h

mec= 0.00243nm

http://www.student.nada.kth.se/~f93-jhu/phys_sim/compton/Compton.htm

http://en.wikipedia.org/wiki/Compton_scattering

http://en.wikipedia.org/wiki/Compton_scattering

Rutherford-Bohr Model of the Atom

• Ernest Rutherford (1871-1937)

• Rutherford Scattering





Observation consistent with scattering from a very small (10,000 times smaller radius than the atom) dense object… The nucleus

http://en.wikipedia.org/wiki/Ernest_Rutherford

Wavelengths of Hydrogen





Bohr’s Semi-classical Model of the Atom• Bohr assumed for the

electron proton system – to be subject to

Coulomb’s Law for electric charges

– Quantization of angular momentum for the electron orbit. L=nh/2π

• Quantization condition prevents electron from continuosly radiating away energy

• Discrete energy levels for electron orbit



Bohr’s Semi-classical Model of the Atom























Bohr Hydrogen atom and spectral lines

Quantum Mechanics and Wave-Particle Duality

• QM Tunneling• DeBroglie Matter Wave• Heisenberg Uncertainty

Principle• Schroedinger Equation and QM

atom• Spin and Pauli Exclusion

Principle

http://phet.colorado.edu/sims/quantum-tunneling/quantum-tunneling.jnlp

DeBroglie Matter Wave





Quantum Mechanics and Wave-Particle Duality

• Fourier • Wave Packets

€





Uncertainty Principle



http://phet.colorado.edu/sims/fourier/fourier.jnlp

http://phet.colorado.edu/sims/quantum-tunneling/quantum-tunneling.jnlp

Schrodinger Wave Equation and Hydrogen Atom





http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html

Additional Quantum Numbers and splitting of spectral lines

• Normal Zeeman Effect







Can measure magnetic fields by examining spectra!!!!

Spin and the Pauli Exclusion Principle



No two electrons can share the same set of four quantum numbers

Chemistry….The periodic table….

Electron Degeneracy Pressure and White Dwarfs

• Exclusion Principle for Fermions (spin-1/2 particles) and uncertainty principle provide electron degeneracy pressure that is the mechanism that prevents the further collapse of white dwarves….

Telescopes

• Basic Optics• Optical Telescopes• Radio Telescopes• Infrared, UV, X-Ray and Gamma Ray Astronomy• All-Sky Surveys and Virtual Observatories

Reflection and Refraction

Reflection

Refraction€

θ1 = θ2

€

n1 sinθ1 = n2 sinθ2

Reflection and Refraction



Lenses

• Focal length determined by radii of curvature , R1and R2, of the lens surfaces and the index of refraction of the lens material

• Convention for sign of radii of curvature is concave --> negativeconvex --> positive

• Wavelength dependent

€

1

f= (nλ −1)(

1

R1

−1

R2

)

Lensmaker formula for thin lenses

Mirrors

• For spherical mirrors

• f=R/2

• Wavelength independent

Focal Plane/Plate Scale

Image Size related to focal length f of telescope

€

y = f tanθ

€

y ≈ fθ

€

dθ

dy=

1

f

Plate ScaleUsually interested in how large an angle is subtended by a pixel in a CCD as well as how large an angle the CCD array can observe.

Frisco Peak Telescope w/STL-6303Plate scale (“/pixel)=206265*s/ff=32”*25.4mm*10=8128mm=8128000μms=9μm 3072x2018 pixels, 27.5 mm x 18.4 mmPlate Scale=0.22”/pixelFOV~11.63’x7.78’

dθ(“)=1radian/206265=dy/f

Resolution and Rayleigh CriteriaDiffraction

• Single Slit diffraction• Consider a point in the slit

D/2 away from another point in slit. The pair of rays from these two points will exhibit destructive interference if they arrive at a point on the screen being 1/2 wavelength

out of phase

• Now divide the slit into fourths

€

D

2sinθ =

λ

2

€

sinθ =λ

D

€

D

4sinθ =

λ

2

€

sinθ = 2λ

D

€

sinθ = mλ

D

Condition for minimum in light intensity on screen

Resolution Single Slit Diffraction



Diffraction from a Circular Aperture Airy Disk







Diffraction limited spatial resolution for a telescope with an aperture of diameter D is given by:

Intensity as a function of angle from center of the image is given by a Bessel function



Rayleigh Criterion for Resolution

• Somewhat arbitrary definition of barely resolvable is that the maximum of one of the Airy disks lies at the first minimum of the other Airy disk

http://astronomy.swin.edu.au/cosmos/R/Resolution








Hubble ST resolution limit



Atmospheric Seeing

• Why do stars twinkle?• Stars are so distant as to

effectively be point sources.• Planets are resolvable.• Plane wavefronts of light

from a distant point source become distorted as the wavefronts passes through a turbulent layer in the atmosphere.

• The index of refraction in the turbulent layer is not uniform

http://en.wikipedia.org/wiki/Image:Seeing_Moon.gif

http://en.wikipedia.org/wiki/Image:Eps_aql_movie_not_2000.gif

Brightness of Image





Refracting Telescope

• Angular Magnification



Reflecting Telescopes



Telescope Mounts

• Altitude Azimuth Equatorial

Radio Telescopes

• Can observe the universe in radio wavelengths from the surface of the earth

Radio Telescopes

Antenna Pattern



Interferometry

Infrared, Ultraviolet, X-ray and Gamma-ray Astronomy

• Can view universe at other wavelenghts as well….

Classification of Binary Stars

• Optical Double Two stars that lie along nearly the same line of sight. Similar RA and DEC. At greatly different distances. Not gravitationally bound.

• Visual Binary Both stars in the binary system can be spatially resolved. Possible to monitor the motion of each member (if the orbital period is not too long..). Can observe angular separation. If distance to system is known linear separation can be determined.

• Astrometric Binary In some cases only one member of a binary system can be seen. Its oscillatory motion betrays the presence of the other member of the binary system. The center of mass of the binary system moves at a constant velocity.

•Eclipsing Binary For binary systems that have orbital planes oriented approximately along the line of sight one star may pass in front of the other, blocking the light of the other. This results in a variation of light received from this system. The light curve indicates the presence of two stars and can provide information about the radii and relative temperatures of the stars.

•Spectrum Binary A spectrum binary is a system with two superimposed, independent, discernible spectra. Periodic shifts in the wavelengths of spectral lines due to the changing motion of the stars.

•Spectroscopic Binary If the orbital period is not too long and the orbital plane has a component along the line of sight, a periodic shift in wavelength will be observable. Both spectra will be available if the luminosities are comparable. In the case of a large difference in luminosity only one set of spectra will be observable. The periodic shift in wavelength in this case still reveals the presence of a binary system.

Mass Determination using visual binaries

• Measure Period• Measure orbital semi-

major axis• Apply Kepler’s 3rd law• Even when distance to

system is not known, angular measurements allow determination of mass ratio…!!!

• If distance and inclination known can determine individual masses

• Derivation from pp 183-185











C.M frame ==> R=0



Angles subtended by semi-major axes can be observed directly. Even if distance d is unknown mass ratio can be determined

Mass Determination using visual binaries







€

a = a1 + a2 Need to know distance

Total mass from Kepler’s law

Individual masses from ratio

Need to account for inclination…



The Mass Function

• For small eccentricity• Do not need to know inclination

angle to determine mass ratio• To obtain sum of masses do need

to know inclination angle• Derivation pp187-188











€

v1 = 2πa1 /P

v2 = 2πa2 /P

€

a1 = Pv1 /2π

a2 = Pv2 /2π

Mass-Luminosity Relation

• Luminosity is highly correlated to stellar mass !!!! Why???

L/L(Sun) ~ [M/M(Sun)]**3.9

Eclipsing Binaries

• Can use eclipses to determine radii and ratio of temperatures from eclipsing binaries!!!

• Radii from First contact time, minimum light time,…

• Derivation pp190-191

• Example 7.3.1


are needed to see this picture.QuickTime™ and a

decompressorare needed to see this picture.

ExoPlanet detection

• Radial Velocity (Wobble)

• Transit

The Classification of Stellar Spectra• The Formation of Spectral Lines

• The Hertzsprung-Russell Diagram

Understanding Spectral Lines

Need to understand…

• The atom

• Statistical Mechanics

Atomic TransitionsBoltzmann Energy DistributionSaha Ionization Equation

First excited state occupancy for hydrogen atom from Boltzmann Equation

Hertzsprung-Russell diagram

• Ejnar Hertzsprung (1873-1967) Danish Engineer and amateur astronomer

• 1905 publication confirming correlation between luminosity and spectral type

• Noticed that type G and later stars could have a range of luminosities…The brighter stars of these classes were GIANTS in order to achieve their brightness at the lower flux at lower temperatures



Spectroscopic Parallax• Can use H-R diagram to

estimate absolute brightness of star given its spectral type

• Use apparent brightness and distance modulus formula

• To obtain distance

• Scatter of +/- 1 magnitude results in factor of 1.6 uncertainty in distance



Documents

Review Chapters 1-8 1.The Celestial Sphere –History –Positions 2.Celestial Mechanics –Elliptical Orbits –Newtonian Mechanics –Kepler’s Laws –Virial Theorem