Deducing Temperatures and Luminosities of Stars(and other objects…)

Review: Electromagnetic Radiation

• EM radiation is the combination of time- and space- varying electric + magnetic fields that convey energy.

• Physicists often speak of the “particle-wave duality” of EM radiation.– Light can be considered as either particles (photons) or as waves, depending

on how it is measured

• Includes all of the above varieties -- the only distinction between (for example) X-rays and radio waves is the wavelength.

Gamm

a Ray

s

Ultrav

iolet

(UV)

X Ray

s

Visib

le Lig

ht

Infra

red (I

R)

Microwav

es

Radio

wav

es

10-15 m 10-6 m 103 m10-2 m10-9 m 10-4 mIncreasing wavelength

Increasing energy

Electromagnetic Fields

Directionof “Travel”

Sinusoidal Fields

• BOTH the electric field E and the magnetic field B have “sinusoidal” shape

Wavelength of Sinusoidal Function

Wavelength is the distance between any two identical points on a sinusoidal wave.

Frequency of Sinusoidal Wave

Frequency: the number of wave cycles per unit of time that are registered at a given point in space. (referred to by Greek letter nu])

is inversely proportional to wavelength

time

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

“Units” of Frequency

meterscyclessecondsecondmeters

cycle

cycle1 1 "Hertz" (Hz)

second

c

Wavelength is proportional to the wave velocity v.Wavelength is inversely proportional to frequency. e.g., AM radio wave has long wavelength (~200 m), therefore it

has “low” frequency (~1000 KHz range). If EM wave is not in vacuum, the equation becomes

Wavelength and Frequency Relation

v

cwhere v and is the "refractive index"n

n

Light as a Particle: Photons Photons are little “packets” of energy. Each photon’s energy is proportional to its

frequency. Specifically, energy of each photon energy is

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

Planck’s Radiation Law• Every opaque object at temperature T > 0-K (a human, a

planet, a star) radiates a characteristic spectrum of EM radiation – spectrum = intensity of radiation as a function of wavelength

– spectrum depends only on temperature of the object

• This type of spectrum is called blackbody radiation

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

Planck’s Radiation Law• Wavelength of MAXIMUM emission max

is characteristic of temperature T

• Wavelength max as T

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

Sidebar: The Actual Equation

• Complicated!!!!– h = Planck’s constant = 6.63 ×10-34 Joule - seconds– k = Boltzmann’s constant = 1.38 ×10-23 Joules -K-1

– c = velocity of light = 3 ×10+8 meter - seconds-1

2

5

2 1

1hc

kT

hcB T

e

Temperature dependence of blackbody radiation

• As temperature T of an object increases:– Peak of blackbody spectrum (Planck function) moves

to shorter wavelengths (higher energies)

– Each unit area of object emits more energy (more photons) at all wavelengths

Sidebar: The Actual Equation

• Complicated!!!!– h = Planck’s constant = 6.63 ×10-34 Joule - seconds– k = Boltzmann’s constant = 1.38 ×10-23 Joules -K-1

– c = velocity of light = 3 ×10+8 meter - seconds-1

– T = temperature [K] = wavelength [meters]

2

5

2 1

1hc

kT

hcB T

e

Shape of Planck Curve

• “Normalized” Planck curve for T = 5700-K– Maximum value set to 1

• Note that maximum intensity occurs in visible region of spectrum

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

Planck Curve for T = 7000-K

• This graph also “normalized” to 1 at maximum

• Maximum intensity occurs at shorter wavelength – boundary of ultraviolet (UV) and visible

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

Planck Functions Displayed on Logarithmic Scale

• Graphs for T = 5700-K and 7000-K displayed on same logarithmic scale without normalizing– Note that curve for T = 7000-K is “higher” and peaks “to the left”

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

Features of Graph of Planck Law T1 < T2 (e.g., T1 = 5700-K, T2 = 7000-K)

• Maximum of curve for higher temperature occurs at SHORTER wavelength : max(T = T1) > max(T = T2) if T1 < T2

• Curve for higher temperature is higher at ALL WAVELENGTHS More light emitted at all if T is larger– Not apparent from normalized curves, must examine

“unnormalized” curves, usually on logarithmic scale

Wavelength of Maximum EmissionWien’s Displacement Law

• Obtained by evaluating derivative of Planck Law over T

(recall that human vision ranges from 400 to 700 nm, or 0.4 to 0.7 microns)

3

max

2.898 10meters

KT

Wien’s Displacement Law

• Can calculate where the peak of the blackbody spectrum will lie for a given temperature from Wien’s Law:

(recall that human vision ranges from 400 to 700 nm, or 0.4 to 0.7 microns)

3

max

2.898 10meters

KT

• Wavelength of Maximum Emission is:

(in the visible region of the spectrum)

3

max

2.898 100.508 508

5700m m nm

max for T = 5700-K

• Wavelength of Maximum Emission is:

(very short blue wavelength, almost ultraviolet)

max for T = 7000-K

3

max

2.898 100.414 414

7000m m nm

Wavelength of Maximum Emission for Low Temperatures

• 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”

3

max

2.898 101.45 1450

2000m m nm

Why are Cool Stars “Red”?

(m)

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

max

Visible Region

Less light in blueStar appears “reddish”

• T >> 5000-K (say, 15,000-K), wavelength of maximum “brightness” is:

“Ultraviolet” region of the spectrum

Star emits more blue light than red appears “bluish”

3

max

2.898 100.193 193

15000m m nm

Wavelength of Maximum Emission for High Temperatures

Why are Hotter Stars “Blue”?

(m)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

max

Visible Region

More light in blueStar appears “bluish”

Betelguese and Rigel in Orion

Betelgeuse: 3,000 K(a red supergiant)

Rigel: 30,000 K(a blue supergiant)

Blackbody curves for stars at temperatures of Betelgeuse and Rigel

Stellar Luminosity• Sum of all light emitted over all wavelengths is the

luminosity– brightness per unit surface area– luminosity is proportional to T4: L = T4

– L can be measured in watts• often expressed in units of Sun’s luminosity LSun

– L measures star’s “intrinsic” brightness, rather than “apparent” brightness seen from Earth

82 4

Joules5.67 10 , Stefan-Boltzmann constant

m -sec-K

Stellar Luminosity – Hotter Stars• Hotter stars emit more light per unit area of its

surface at all wavelengths– T4 -law means that small increase in temperature T

produces BIG increase in luminosity L– Slightly hotter stars are much brighter (per unit

surface area)

Two stars with Same Diameter but Different T

• Hotter Star emits MUCH more light per unit area much brighter

Stars with Same Temperature and Different Diameters

• Area of star increases with radius ( R2, where R is star’s radius)

• Measured brightness increases with surface area

• If two stars have same T but different luminosities (per unit surface area), then the MORE luminous star must be LARGER.

How do we know that Betelgeuse is much, much bigger than Rigel?• Rigel is about 10 times hotter than Betelgeuse

– Measured from its color

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

• But the two stars have equal total luminosities Betelguese must be about 102 (=100) times

larger in radius than Rigel– to ensure that emits same amount of light over entire

surface

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

• Two otherwise identical stars (same radius, same temperature same luminosity) will still appear vastly different in brightness if their distances from Earth are different

• Reason: intensity of light inversely proportional to the square of the distance the light has to travel– Light waves from point sources are surfaces of

expanding spheres

Sidebar: “Absolute Magnitude”

• Recall definition of stellar brightness as “magnitude” m

• F, F0 are the photon numbers received per second from object and reference, respectively.

100

2.5 logF

mF

Sidebar: “Absolute Magnitude”

• “Absolute Magnitude” M is the magnitude measured at a “Standard Distance”– Standard Distance is 10 pc 33 light years

• Allows luminosities to be directly compared– Absolute magnitude of sun +5 (pretty faint)

10

102.5 log

F pcM m

F earth

Sidebar: “Absolute Magnitude” Apply “Inverse Square Law”

• Measured brightness decreases as square of distance

2

2

2

110 10 distance

10pc1distance

F pc pc

F earth

Simpler Equation for Absolute Magnitude

2

10

10

distance2.5 log

10pc

distance5 log

10pc

M m

m

Stellar Brightness Differences are “Tools”, not “Problems”

• If we can determine that 2 stars are identical, then their relative brightness translates to relative distances

• Example: Sun vs. Cen– spectra are very similar temperatures, radii almost

identical (T follows from Planck function, radius R can be deduced by other means)

luminosities about equal– difference in apparent magnitudes translates to relative

distances– Can check using the parallax distance to Cen

Plot Brightness and Temperature on “Hertzsprung-Russell Diagram”

http://zebu.uoregon.edu/~soper/Stars/hrdiagram.html

H-R Diagram

• 1911: E. Hertzsprung (Denmark) compared star luminosity with color for several clusters

• 1913: Henry Norris Russell (U.S.) did same for stars in solar neighborhood

Hertzsprung-Russell Diagram

http://www.anzwers.org/free/universe/hr.html

90% of stars on Main Sequence10% are White Dwarfs<1% are Giants

“Clusters” on H-R Diagram

• n.b., NOT like “open clusters” or “globular clusters”

• Rather are “groupings” of stars with similar properties

• Similar to a “histogram”

H-R Diagram

• Vertical Axis luminosity of star– could be measured as power, e.g., watts

– or in “absolute magnitude”

– or in units of Sun's luminosity:star

Sun

L

L

Hertzsprung-Russell Diagram

H-R Diagram• Horizontal Axis surface temperature

– Sometimes measured in Kelvins. – T traditionally increases to the LEFT– Normally T given as a ``ratio scale'‘– Sometimes use “Spectral Class”

• OBAFGKM– “Oh, Be A Fine Girl, Kiss Me”

– Could also use luminosities measured through color filters

“Standard” Astronomical Filter Set

• 5 “Bessel” Filters with approximately equal “passbands”: 100 nm– U: “ultraviolet”, max 350 nm

– B: “blue”, max 450 nm

– V: “visible” (= “green”), max 550 nm

– R: “red”, max 650 nm

– I: “infrared, max 750 nm

– sometimes “II”, farther infrared, max 850 nm

Filter Transmittances

200 300 400 500 600 700 800 900 1000 1100

0

10

20

30

40

50

60

70

80

90

100U

V

B

R

I

II

U,B,V,R,I,II Filters

Wavelength (nm)

Transmission (%)

Visible Light

UB V

R III

Wavelength (nm)

100

50

0

200 300 400 500 600 700 800 900 1000 1100

Tra

nsm

ittan

ce (

%)

Measure of Color

• If image of a star is:– Bright when viewed through blue filter– “Fainter” through “visible”– “Fainter” yet in red

• Star is BLUISH

and hotter (m)

0.3 0.4 0.5 0.6 0.7 0.8

Visible Region

L(s

tar)

/ L

(Sun

)

Measure of Color

• If image of a star is:– Faintest when viewed through blue filter– Somewhat brighter through “visible”– Brightest in red

• Star is REDDISH

and cooler

(m)

0.3 0.4 0.5 0.6 0.7 0.8

Visible Region

L(s

tar)

/ L

(Sun

)

How to Measure Color of Star• Measure brightness of stellar images taken

through colored filters– used to be measured from photographic plates– now done “photoelectrically” or from CCD images

• Compute “Color Indices”– Blue – Visible (B – V)– Ultraviolet – Blue (U – B)– Plot (U – V) vs. (B – V)