How is Light Made? How is Light Made? Can be considered as EITHER - - PDF document

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How is Light Made? How is Light Made?

Deducing Temperatures and Deducing Temperatures and Luminosities of Stars Luminosities of Stars (and other objects (and other objects… …) ) Review: Electromagnetic (EM) Radiation Review: Electromagnetic (EM) Radiation

• EM radiation: regularly varying electric & magnetic fields

– can transport energy over vast distances.

• “Wave-Particle Duality” of EM radiation:

– Can be considered as EITHER particles (photons) or as waves

• Depends on how it is measured
• Includes all of “classes” of light

– ONLY distinction between X-rays and radio waves is wavelength λ

G a m m a R a y s U l t r a v i

• l

e t ( U V ) X R a y s V i s i b l e L i g h t I n f r a r e d ( I R ) M i c r

• w

a v e s R a d i

• w

a v e s

10-15 m 10-6 m 103 m 10-2 m 10-9 m 10-4 m

Increasing wavelength Increasing energy

Electromagnetic Fields Electromagnetic Fields

Direction

• f “Travel”

Sinusoidal Fields Sinusoidal Fields

• BOTH the electric field E and the magnetic

field B have “sinusoidal” shape

Wavelength Wavelength λ λ

Distance between two identical points on wave

λ

Frequency Frequency ν ν

number of wave cycles per unit time registered at given point in space

inversely proportional to wavelength

time 1 unit of time (e.g., 1 second)

2 Wavelength and Frequency Wavelength and Frequency

Proportional to Velocity v Inversely proportional to temporal frequency ν

Example: AM radio wave at ν = 1000 kHz = 106 Hz λ = c/ν = 3 × 108 m/s / 106 Hz = 300 m λ for AM radio is long because frequency is small

λ = v/ν = c /ν (in vacuum)

“ “Units Units” ” of Frequency

• f Frequency

meters cycles second second meters cycle cycle 1 1 "Hertz" (Hz) second c ν λ ⎡ ⎤ ⎢ ⎥ ⎡ ⎤ ⎣ ⎦ = ⎢ ⎥ ⎡ ⎤ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦

Light as a Particle: Photons Light as a Particle: Photons

Photons: little “packets” of energy Energy is proportional to frequency

E = hν

Energy = (Planck’s constant) × (frequency of photon) h ≈ 6.625 × 10-34 Joule-seconds = 6.625 × 10-27 Erg-seconds

Generating Light Generating Light

• Light is generated by converting one class
• f energy to electromagnetic energy

– Heat – Explosions

Converting Heat to Light Converting Heat to Light The Planck Function The Planck Function

• Every opaque object (a human, a planet, a star)

radiates a characteristic spectrum of EM radiation

– Spectrum: Distribution of intensity as function of wavelength – Distribution depends only on object’s temperature T

• Blackbody radiation

visible infrared Intensity (W/m2) radio ultraviolet

0.1 1.0 10 100 1000 10000

Planck Planck’ ’s Radiation Law s Radiation Law

• Wavelength of MAXIMUM emission λmax

is characteristic of temperature T

• Wavelength λmax ↓ as T ↑

http://scienceworld.wolfram.com/physics/PlanckLaw.html

λmax

As T ↑, λmax ↓

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Sidebar: The Actual Equation Sidebar: The Actual Equation

• Derived in Solid State Physics
• Complicated!!!! (and you don’t need to know it!)

h = Planck’s constant = 6.63 ×10-34 Joule - seconds k = Boltzmann’s constant = 1.38 ×10-23 Joules per Kelvin c = velocity of light = 3 ×10+8 meter - second-1

( )

2 5

2 1 1

hc kT

hc B T eλ λ = − Temperature dependence Temperature dependence

• f blackbody radiation
• f blackbody radiation
• As object’s temperature T increases:

1. Wavelength of maximum of blackbody spectrum (Planck function) becomes shorter (photons have higher energies) 2. Each unit surface area of object emits more energy (more photons) at all wavelengths

Shape of Planck Curve Shape of Planck Curve

• “Normalized” Planck curve for T = 5700K

– Maximum Intensity set to 1

• Note that maximum intensity occurs in visible region of

spectrum for T = 5700K

http://csep10.phys.utk.edu/guidry/java/planck/planck.html

Planck Curve for T = 7000 Planck Curve for T = 7000-

• K

K

• This graph is also “normalized” to 1 at maximum
• Maximum intensity occurs at shorter wavelength λ

– boundary of ultraviolet (UV) and visible

http://csep10.phys.utk.edu/guidry/java/planck/planck.html

Two Planck Functions Two Planck Functions Displayed on Logarithmic Scale Displayed on Logarithmic Scale

• Graphs for T = 5700K and 7000K displayed on

same logarithmic scale without normalizing

– Note that curve for T = 7000K is “higher” and its peak is farther “to the left”

http://csep10.phys.utk.edu/guidry/java/planck/planck.html

Features of Graph of Planck Law Features of Graph of Planck Law

T T1

1 < T

< T2

2 (e.g., T

(e.g., T1

1 = 5700K, T

= 5700K, T2

2 = 7000K)

= 7000K)

• Maximum of curve for higher temperature
• ccurs at SHORTER wavelength λ:

– λmax(T = T1) > λmax(T = T2) if T1 < T2

• Curve for higher temperature is higher at ALL

WAVELENGTHS λ

⇒ More light emitted at all λ if T is larger – Not apparent from normalized curves, must examine “unnormalized” curves, usually on logarithmic scale

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Wavelength of Maximum Emission Wavelength of Maximum Emission Wien Wien’ ’s s Displacement Law Displacement Law

• Obtained by evaluating derivative of Planck

Law over temperature T

Human vision range 400 nm = 0.4 µm ≤ λ ≤ 700 nm = 0.7 µm (1 µm = 10-6 m)

[ ] [ ]

3 max

2.898 10 meters K T λ

−

× =

Colors of Stars Colors of Stars

• Star “Color” is related to temperature

– If star’s temperature is T = 5000K, the wavelength

• f the maximum of the spectrum is:

(in the visible region of the spectrum, green)

nm m m 579 579 . 5000 10 898 . 2

3 max

= ≅ × =

−

µ λ

Colors of Stars Colors of Stars

• If T << 5000 K (say, 2000 K), the wavelength of

the maximum of the spectrum is:

(in the “near infrared” region of the spectrum)

• The visible light from this star appears “reddish”

– Why?

nm m m 966 966 . 3000 10 898 . 2

3 max

≅ ≅ × =

−

µ λ

Blackbody Curve for T=3000K Blackbody Curve for T=3000K

• In visible region, more light at long λ

⇒ Visible light from star with T=3000K appears “reddish”

Colors of Stars Colors of Stars

• If T << 5000 K (say, 2000 K), the wavelength of

the maximum of the spectrum is:

(peaks in the “near infrared” region of the spectrum)

nm m m 1450 449 . 1 2000 10 898 . 2

3 max

≅ ≅ × =

−

µ λ

Colors of Stars Colors of Stars

• Color of star indicates its temperature

– If star is much cooler than 5,000K, the maximum of its spectrum is in the infrared and the star looks “reddish”

• It gives off more red light than blue light

– If star is much hotter than 15,000K, its spectrum peaks in the UV, and it looks “bluish”

• It gives off more blue light than red light

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Betelguese Betelguese and and Rigel Rigel in Orion in Orion

Betelgeuse: 3,000 K (a red supergiant) Rigel: 30,000 K (a blue supergiant)

Planck Curves for Planck Curves for Rigel Rigel and Betelgeuse and Betelgeuse

Plotted on Log-Log Scale to “compress” range of data

RIgel Betelgeuse

Luminosities of stars Luminosities of stars

• Sum of all light emitted over all wavelengths

is the luminosity

– A measure of “power” (watts) – Measures the intrinsic brightness instead of apparent brightness that we see from Earth

• Hotter stars emit more light at all wavelengths

through each unit area of its surface

– luminosity is proportional to T4 ⇒ small increase in temperature makes a big increase in luminosity

Luminosities of stars Luminosities of stars

• Stefan-Boltzmann Law

L = Power emitted per unit surface area σ = Stefan-Boltzmann Constant ≈ 5.67 × 10-8 Watts / (m2 K4)

• Obtained by integrating Planck’s Law over λ
• Luminosity is proportional to T4

⇒ small increase in temperature produces big increase in luminosity

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T L σ =

Consider 2 stars with same Consider 2 stars with same diameter and different T diameter and different T’ ’s s What about large & small stars What about large & small stars with same temperature T? with same temperature T?

• Surface Area of Sphere ∝ R2

– R is radius of star

• Two stars with same T, different

luminosities

– the more luminous star must be larger to emit more light

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How do we know that Betelgeuse How do we know that Betelgeuse is much, much bigger than is much, much bigger than Rigel Rigel? ?

• Rigel is about 10 times hotter than

Betelgeuse (T = 30,000K vs. 3,000K)

– Rigel gives off 104 (=10,000) times more energy per unit surface area than Betelgeuse

• But these two stars have approximately

equal total luminosity

– therefore diameter of Betelguese must be about 102 = 100 times larger than Rigel

So far we haven So far we haven’ ’t considered t considered stellar stellar distances distances... ...

• Two otherwise identical stars (same

radius, same temperature ⇒ same luminosity) will still appear vastly different in brightness if at different distances from Earth

• Reason: intensity of light inversely

proportional to the square of the distance the light has to travel

– Light wave fronts from point sources are like the surfaces of expanding spheres

Use Stellar Brightness Difference Use Stellar Brightness Difference

• If one can somehow determine that brightnesses
• f 2 stars are identical, then use their relative

brightnesses to find their relative distances

• Example: the Sun and α Cen (alpha Centauri)

– spectra look very similar ⇒ temperatures are almost identical (from Planck function)

• diameters are also almost equal
• deduced by other methods

⇒ luminosities about equal

• difference in apparent magnitudes ⇒ difference

in relative distance

– Check using parallax distance to α Cen

The The Hertzsprung Hertzsprung-

• Russell Diagram

Russell Diagram Hertzsprung Hertzsprung-

• Russell (

Russell (“ “H H-

• R

R” ”) Diagram ) Diagram

• Graphical Plot of Intrinsic

Brightness as function of Surface Temperature

• 1911 by Hertzsprung (Dane)
• 1913 by Henry Norris Russell
• Stars Tend to “Cluster” in Certain

Regions of Plot

– “Main Sequence” – “Red Giants” and “Supergiants” – “White Dwarfs”

• Star “Types” based on

Temperature

Star Types Star Types

O B A F G K M