PHYS 3380 - Astronomy The first exam will be next Monday, September 29 at the regular class time. It...

PHYS 3380 - Astronomy

The first exam will be next Monday, September 29 at the regular class time. It is closed book but you may bring in one 8 1/2 X 11 inch “cheat sheet” with writing on both sides. It will cover everything I have covered up to and including last Wednesday’s class (9/17/14).


Center of MassNote: the previous calculations assumed that the mass of the orbiting body was much smaller that the central body - center of orbit at center of central body

Newton showed that two objects attracted to each other by gravity actually orbit about their center of mass - the point at which the objects would balance if they were connected.

This idea is used to find planets orbiting other stars - massive planets cause star to move against background stars

Center of Mass - Binary StarAnimation

PHYS 3380 - Astronomy Einstein1905 - “The Year of Physics”

Submitted doctoral thesis "A New Determination of Molecular Dimensions”

Published five pioneering papers in "Annalen der Physik" - revolutionized physics:

"On A Heuristic Point of View Concerning the Production and Transformation of Light"

- electromagnetic radiation must consist of quantums or photons - explained the photoelectric effect- became the foundation of quantum theory - what he received the Nobel Prize for in 1921

"On the Electrodynamics of Moving Bodies" - special relativity - new interpretation of the conception of space and time - observer can never detect their uniform motion except relative to other objects - coordinate systems- speed of light constant - independent of motion relative to light source

"Does the Inertia of a Body Depend upon its Energy Content?" - the equivalence of mass and energy - E = mc2


General Relativity

1916 published "The Foundation of the General Theory of Relativity”

- generalized special theory of relativity- observer cannot distinguish between inertial forces due to acceleration and uniform gravitational forces- gravity is curvature of space-time

- curvature dependent on mass- acceleration of mass dependent on space-time curvature

Numerous implications on astronomy and astrophysics

- orbital motion- black holes- big bang- formation and structure of galaxies


Proof of General Relativity

Einstein convinced that light deflection by the gravitational field of the sun could be observed during a total solar eclipse

- photograph section of sky where eclipse would occur- during eclipse, photograph same section and measure difference in positions- predicted deflection of 1.75 arcseconds for starlight grazing Sun’s surface

Theory predicted the deflection of light in a gravitational field

Several failed observations of total solar eclipses before proof in 1919- observed eclipse in island of Principe in the Gulf of Guinea in western Africa and Sobral, Brazil- found shift in stars outward from Sun

- 1.61±0.30 arcseconds at Principe- 1.98±0.12 arcseconds at Sobral


Einstein’s eclipse

.


Proof of General Relativity

Precession of long axis of Mercury’s orbit

Newton’s formulation predicted precession of 5557.62 arcseconds per century

- actually 43.11 arcseconds more - about 29 km past position predicted by Newton per orbit - accumulative - 12,000 km per century - easily measured

Einstein predicted 43.03 arcseconds per century difference

Effect since observed on Venus, Earth, and asteroid Icarus


Light

PHYS 3380 - AstronomyLight

What is light? - A vibration in an electromagnetic field through which energy is transported.

Light as a wave

Light as a particle E = hf photon

f = c

The dual nature of light or wave-particle duality:

PHYS 3380 - Astronomy Properties of Waves

Period: time to complete one cycle of vibration - from crest to crest or trough to trough

Frequency (f): number of crests passing a fixed pointper second

Frequency= 1/period

Amplitude (a): maximum displacement fromequilibrium

Wave length (l): distance between successive crests

Speed (of a wave) (s)= wave length x frequencys= l x f


Light is a vibration in an electromagnetic field through which energy is transported

- a transverse wave - vibration or oscillation is perpendicular to direction of propagation of wave (vs. longitudinal wave - vibration or oscillation is in the same direction as propagation of wave)

So electrons can be manipulated by light. Electrons wiggle up and down as light passes by.

Light as a Wave


Visible light ranges through 7 major colors from long wavelengths (low frequency - red) to short wavelengths (high frequency - violet) - Red, orange, yellow, green, blue, indigo, violet (Roy G Biv)

•For a wave, its speed: s = x f

•But the speed of light is a constant, c.

•For light: x f = c

•The higher f is, the smaller is, and vice versa.

•Our eyes recognize f (or ) as color.


Light as a Particle (Photon)

•Light propagates as quanta of energy called photons•Photons •move with speed of light•have no mass•are electrically neutral

• Energy of a photon or electromagnetic wave:

E = hf = h c/ where

h = Planck’s constantf = frequency of a light wave - number of crests passing a fixed point in 1 secondc = velocity of light

= wavelength of a light wave•Higher frequency/shorter wavelength - higher energy


The Electromagnetic Spectrum

Most wavelengths of light can not be seen by the human eye.

The visible part of the electromagnetic spectrum lies between ultraviolet and infrared light (between about 400 and 700 nm). The higher the frequency (shorter the wavelength), the higher the photon energy. Radio waves are at the long wavelength end of the spectrum and gamma rays are at the short wavelength end of the spectrum.


Light as Information Bearer

Spectrum of a distant object - a spectrum is the amount of energy or intensity at different wavelengths.

By studying the spectrum of an object, we can learn its:1 Composition2 Temperature3 Velocity

We can separate light into its different wavelengths (spectrum).


Electron Energy LevelsElectrons can not have just any energy while orbiting the nucleus.Only certain energy values are allowed.Electrons may only gain or lose certain specific amounts of energy.

Each element (atom and ion) has its own distinctive set or pattern of energy levels - holds the key to studying of distant objects in the universe.

This diagram depicts the energy levels of Hydrogen. 1 eV (electron volt) = 1.6 X 10-19 J

Electron jumps to higher energy levels can only occur with addition of the particular amounts of energy representing differences between possible energy levels. Energy levels are quantized - study of electron energy levels called quantum mechanics. Atom gains this energy either from KE of another atom colliding with it or from absorption of energy carried by light - falls to lower energy level by emitting light or transfer of energy by collision.


Absorption and Emission. When electrons jump from a low energy shell to a high energy shell, they absorb energy. When electrons jump from a high energy shell to a low energy shell, they emit energy. This energy is either absorbed or emitted at very specific wavelengths, which are different for each atom.

When the electron is in a high energy shell, the atom is in an excited state.

When the electron is in the lowest energy shell, the atom is in the ground state.


The Hydrogen Atom. The hydrogen atom is the simplest of atoms. Its nucleus contains only one proton which is orbited by only one electron. In going from one allowed orbit to another, the electron absorbs or emits light (photons) at very specific wavelengths. Note - wavelength is often written as and the unit used is an angstrom (A) = 10-8 m

PHYS 3380 - AstronomyInteraction of Light with Matter

So each electron is only allowed to have certain energies in an atom.

Electrons can absorb light and gain energy or emit light when they lose energy.

It is easiest to think of light as a photon when discussing its interaction with matter. Only photons whose energies (colors) match the “jump” in electron energy levels can be emitted or absorbed.

Hydrogen

So visible emission spectrum is created when a gas is heated and collisions in gas continually bump electrons to higher energy levels - emit photons of specific wavelength as they fall back to lower levels. Absorption spectrum is produced when white light is passed through cloud of cool gas. Photons of specific wavelengths absorbed as electrons jump to higher energy levels.

Emission Spectrum

Absorption Spectrum

PHYS 3380 - Astronomy Emission Spectra

•The atoms of each element have their own distinctive set of electron energy levels.•Each element emits its own pattern of colors, like fingerprints.•If it is a hot gas, we see only these colors, called an emission line spectrum.

Orion Nebula in Ultraviolet

PHYS 3380 - AstronomyAbsorption Spectra

• If light shines through a cool gas, each element will absorb those photons whose colors match their electron energy levels.

• The resulting absorption line spectrum has all colors minus those that were absorbed.

• We can determine which elements are present in an object by identifying emission and absorption lines.

Hydrogen


Temperature and Thermal Energy

Why does water burn your skin so much quicker than air?Why is falling into a 32º F lake more dangerous than standing outside naked on a 32º F?

Temperature - measure of the average kinetic energy of the particles in a substance - particles in box on right have higher temperature - higher velocity = more KE = higher temperature

Both boxes have same temperature - particles have same average velocity/KE - box on right has more thermal energy - energy contained in a substance - more particles


This diagram compares three common temperature scales. The Fahrenheit scale is used in the United States, but nearly all other countries use the Celsius scale. Scientists prefer the Kelvin scale because O K represents

absolute zero, the coldest possible temperature.

PHYS 3380 - Astronomy Thermal/Blackbody Radiation

Photons are produced whenever charged particles are accelerated- A moving charge gives rise to a magnetic field, and if the motion is changing (accelerated), then the magnetic field varies and in turn produces an electric field - electromagnetic radiation - photons

In an opaque object or dense gas cloud, photons can’t easily escape - they “bounce around” in the object. This randomizes their radiative energies and resulting photon energies depend only on the body’s temperature - produces a continuous spectrum called a thermal radiation or blackbody spectrum.

Blackbody - a hypothetical body that completely absorbs all wavelengths of thermal radiation incident on it

- does not reflect light - appears black if temperature low enough so as not to be self-luminous. - all blackbodies heated to a given temperature emit thermal radiation with the same spectrum - required by thermal equilibrium - distribution of blackbody radiation as a function of wavelength - the Planck law, cannot be predicted using classical physics. - the first motivating force behind the development of quantum mechanics

PHYS 3380 - Astronomy Key Features of a Blackbody Spectrum

- a dense object produces light at all possible wavelengths if the object is above absolute zero.

- since everything in the universe is above 0 K, all dense objects (solids, liquids, thick gases) will produce a thermal spectrum.

- the shape of a continuous spectrum depends on only the temperature of the object not its chemical composition.

- as the temperature of an object increases, more light is produced at all wavelengths -as the temperature of an object increases, the peak of thermal spectrum curve shifts to shorter wavelengths (higher frequencies)

--cool things appear red or orange, hotter things appear yellow or white, and very hot things blue or purple.


Temperature (K) of Black

Body

Wavelength (max) at Which Most Radiation is

Emitted

Type of Radiation

3 0.1 cm Radiowaves

300 0.001 cm "Far" Infrared

3,000 1000 nm "Near" Infrared

4,000 750 nm Red Light

6,000 500 nm Yellow Light

8,000 375 nm Violet Light

10,000 300 nm "Near" Ultraviolet

30,000 100 nm "Far" Ultraviolet

300,000 10 nm "Soft" X-Rays

1.5 million 20 nm "hard" x-rays

3 billion 0.001 nm Gamma rays


Hotter objects emit more total radiation per unit surface area.

E = T4 ( = 5.67 x 10-8 watts/m2 K4) - Stefan-Boltzmann Law

Hotter objects emit photons with a smaller wavelength (higher average energy.)

max (nm) = 2.8983 x 106 nm-°K/T 106 / T(K) [nm] - Wien’s Law


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Φ =π Bλ dλ0

∞

∫Bλ is the Planck Law brightness - the amount of energy per unit

time per unit surface area per unit solid angle per unit wavelength

emitted by a blackbody as function of wavelength. The extra factor

of π converts from a brightness per solid angle to total brightness.

Bλ (T) =2c 2h

λ5(ehc /(λkT ) −1)

c is the speed of light, h is Planck's constant, k is Boltzmann's constant

Φ = π2c 2h

λ5(ehc /(λkT ) −1)dλ

0

∞

∫

Φ = 2πc 2h1


0

∞

∫

Derivation of the Stefan-Boltzmann Law

PHYS 3380 - AstronomySolid Angle

The solid angle subtended by a surface S - the surface area of a unit sphere covered by the surface's projection onto the sphere. This can be written as

where n is a unit vector from the origin, da is the differential area of a surface patch, and r is the distance from the origin to the patch. Written in spherical coordinates with the colatitude (polar angle) and for the longitude (azimuth), this becomes

Solid angle is measured in steradians, and the solid angle corresponding to all of space being subtended is 4π steradians.

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=ˆ n ⋅dar2S

∫∫

€

= sinφdθS

∫∫ dφ


Consider the solid angle subtended by one face of a cube of side length centered at the origin. Since the cube is symmetrical and has six sides, one side obviously subtends 4π/6 steradians.


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Φ=2πc 2h1


0

∞

∫Let

u =hc

λkT

λ =hc

ukT

dλ = −hc

u2kTdu

Then

Φ = 2πc 2h−hc

u2kThc

ukT

⎛

⎝ ⎜

⎞

⎠ ⎟5

(eu −1)0

∞

∫ du

Φ = 2πc 2hkT

hc

⎛

⎝ ⎜

⎞

⎠ ⎟4

u3

eu −10

∞

∫ du

Φ =2πc 2hk 4T 4

h4c 4 ζ (4)Γ(4)


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Φ=2πc 2hk 4T 4

h4c 4ζ (4)Γ(4)

ζ is the Riemann zeta function :

ζ (n) =1

k nk=1

∞

∑

ζ (4) =π 4

90Γ is the gamma function :

Γ(n) = (n −1)!

Φ =2πc 2hk 4T 4

h4c 4

π 4

903!

Φ =2π 5k 4

15h3c 2T 4

Φ =σT 4

σ = 5.67 ×10−8 J/m2s K4 - Stefan - Boltzmann constant


So, the luminosity of a star depends on temperature and size (surface area).

Amount radiated from every square meter equals σT4 (Stefan-Boltzmann Law)

- the hotter the star, the more energy radiated per square meter

Total amount radiated (luminosity):

L = 4πR2σT4

R is star's radius, surface area = 4πR2

Stellar luminosities generally given in number of solar luminosities:

LS=3.826 X 1026 W

If we measure L and T, we can estimate R

T can be determined using Wein’s Law


The Stefan-Boltzmann Law can be used to estimate the Earth’s temperature:

€

Sun's total luminosity is :

LS = 4πRS2σTS

4

The amount of energy the Earth intercepts from the Sun (PE) is the

total emitted energy reduced by the ratio of the solid angle

subtended by the Earth at the Sun divided by the total solid angle 4π .

The Earth is a sphere with radius RE at distance r =1AU so

PE = LS

πRE2 /r2

4πThe luminosity of the Earth is :

LE = 4πRE2σTE

4


€

In steady state, the luminosity of the Earth must balance the radiative

power input from the Sun :

PE = LE

LS

πRE2 /r2

4π= 4πRE

2σTE4

4πRS2σTS

4( )

πRE2 /r2

4π= 4πRE

2σTE4

TE =RS2rTS

TE = 5780 K6.96 ×108m

2(1.496 ×1011m)= 278.8 K


Wein’s Law can be derived by taking the derivative of the Planck brightness with respect to wavelength and determining where this function is zero, i.e. set dB/d=0 and solve for . (I’ll leave that exercise for you, if you want to try it.)

The Stefan-Boltzmann Law, Wein’s Law, and Newton’s Universal Law of Gravitation together allow us to determine a star's type, its mass, its temperature, its rate of energy production, its diameter, its life expectancy and its future fate.

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