How is Light Made? How is Light Made? Deducing Temperatures and - - 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… …) )

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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 radiation is everywhere around us. It is the light that we see, it is the heat that we feel, it is the UV rays that gives us sunburn, and it is the radio waves that transmit signals for radio and TVs. EM radiation can propagate through vacuum since it doesn’t need any medium to travel in, unlike sound. The speed of light through vacuum is constant through out the universe, and is measured at 3x108 meters per second, fast enough to circle around the earth 7.5 times in 1 second. Its properties demonstrate both wave-like nature (like interference) and particle-like nature (like photo-electric effect.)

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Electromagnetic Fields Electromagnetic Fields

Direction

• f “Travel”

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Sinusoidal Fields Sinusoidal Fields

• BOTH the electric field E and the magnetic

field B have “sinusoidal” shape

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Wavelength Wavelength λ λ

Distance between two identical points on wave

λ

One way of describing light is by its wavelength. Wavelength is the distance between the two identical points on the wave. The wave must be steady (no change in the oscillation and no change in its velocity) for it be possible to measure the wavelength. Wavelength is also shorthanded to a Greek letter Lambda.

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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)

The same exact wave can be described using its frequency. Frequency is defined as the number of cycles of the waves per unit time. In the case shown, the frequency would be 1.5, since there are exactly one and a half complete cycles of the wave in the given time. The frequency is inversely proportional to its wavelength. Frequency is denoted by Greek letter “nu”.

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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)

Wavelength and frequency are related to one another by the wave’s velocity. Wavelength is proportional (wavelength increases if velocity increases), and wavelength is inversely proportional to frequency (wavelength decreases if frequency increases). An AM radio wave has a large wavelength, so it has a low frequency (compared to other EM radiation.) In the case of EM radiation, the velocity is the speed of light, denoted by c. the speed of light is as mentioned before, approximately 3x108 meters per second. Using algebra, one can solve for any one of the variables.

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“ “Units Units” ” of Frequency

• f Frequency

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

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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

Now the particle nature of EM radiation. These little packets of light is known as photons. These photons carry a certain energy which is related to its

• frequency. This energy is equal to Planck’s constant (h) multiplied by the

frequency of the photon. By substituting “nu” with the equation in the previous slide, we can get the equivalent equation in terms of wavelength. Planck’s constant is 6.6261 x 10-34 joule second

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Generating Light Generating Light

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

– Heat – Explosions

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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

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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 = −

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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

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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

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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

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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

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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

−

× =

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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

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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

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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”

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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

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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)

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

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

RIgel Betelgeuse

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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

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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 =

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Consider 2 stars with same Consider 2 stars with same diameter and different T diameter and different T’ ’s s

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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

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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

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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

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The The Hertzsprung Hertzsprung-

• Russell Diagram

Russell Diagram

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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

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Star Types Star Types

O B A F G K M