Deducing Temperatures and
Luminosities of Stars
(and other objects…)
Review: Electromagnetic Radiation
Increasing energy
10-15 m
10-9 m
10-6 m
10-4 m
10-2 m
103 m
Increasing wavelength
• 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.
Electromagnetic Fields
Direction
of “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 n of Sinusoidal Wave
time
1 unit of time
(e.g., 1 second)
 Frequency: the number of wave cycles per unit of
time that are registered at a given point in space.
(referred to by Greek letter n [nu])
 n is inversely proportional to wavelength
“Units” of Frequency
 meters 
c

second

 n
 meters 


 cycle 
 cycles 
 second 
 cycle 
1

1
"Hertz"
(Hz)

 second 
Wavelength and Frequency Relation
 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
n
v

c
where v  and n is the "refractive index"
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 = hn
Energy = (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 
max
http://scienceworld.wolfram.com/physics/PlanckLaw.html
Sidebar: The Actual Equation
B T  
2hc

2
1
5
e
hc
 kT
1
• 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
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
B T  
2hc

2
1
5
e
hc
 kT
1
• 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]
Shape of Planck Curve
http://csep10.phys.utk.edu/guidry/java/planck/planck.html
• “Normalized” Planck curve for T = 5700-K
– Maximum value set to 1
• Note that maximum intensity occurs in visible
region of spectrum
Planck Curve for T = 7000-K
http://csep10.phys.utk.edu/guidry/java/planck/planck.html
• This graph also “normalized” to 1 at maximum
• Maximum intensity occurs at shorter
wavelength 
– boundary of ultraviolet (UV) and visible
Planck Functions Displayed on
Logarithmic Scale
http://csep10.phys.utk.edu/guidry/java/planck/planck.html
• 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”
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 Emission
Wien’s Displacement Law
• Obtained by evaluating derivative of Planck Law
over T
2.898 10
max [ meters  
T [K
3
(recall that human vision ranges from 400 to 700 nm, or
0.4 to 0.7 microns)
Wien’s Displacement Law
• Can calculate where the peak of the blackbody
spectrum will lie for a given temperature from
Wien’s Law:
2.898 10
max [ meters  
T [K
3
(recall that human vision ranges from 400 to 700 nm, or
0.4 to 0.7 microns)
max for T = 5700-K
• Wavelength of Maximum Emission is:
max
3
2.898 10

m
5700
0.508 m  508nm
(in the visible region of the spectrum)
max for T = 7000-K
• Wavelength of Maximum Emission is:
max
3
2.898 10

m
7000
0.414 m  414nm
(very short blue wavelength, almost ultraviolet)
Wavelength of Maximum
Emission for Low Temperatures
• If T << 5000-K (say, 2000-K), the wavelength of
the maximum of the spectrum is:
max
3
2.898 10

m 1.45 m  1450nm
2000
(in the “near infrared” region of the spectrum)
• The visible light from this star appears “reddish”
Why are Cool Stars “Red”?
Less light in blue
Star appears “reddish”
0.4
0.5
0.6
0.7
0.8
 (m)
Visible Region
0.9
1.0
1.1
1.2
1.3
max
Wavelength of Maximum
Emission for High Temperatures
• T >> 5000-K (say, 15,000-K), wavelength of
maximum “brightness” is:
max
3
2.898 10

m 0.193 m  193nm
15000
“Ultraviolet” region of the spectrum
Star emits more blue light than red appears “bluish”
Why are Hotter Stars “Blue”?
More light in blue
Star appears “bluish”
0.1
0.2
0.3
max
0.4
0.5
0.6
 (m)
Visible Region
0.7
0.8
0.9
1.0
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
Joules


8


5.67

10
,
Stefan-Boltzmann
constant


m 2 -sec-K 4


– 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
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
m  2.5  log10  
 F0 
• F, F0 are the photon numbers received per second
from object and reference, respectively.
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)
 F 10 pc  
M  2.5  log10 
m
 F  earth  
Sidebar: “Absolute Magnitude”
Apply “Inverse Square Law”
• Measured brightness decreases as square of
distance
2
 1 
2


F 10 pc 
10 pc 
 distance 




2
F  earth  
1
  10 pc 


 distance 
Simpler Equation for Absolute
Magnitude
 distance  2 
M  2.5  log10 
 m
 10 pc  
 distance 
 5  log10 
m
 10 pc 
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
“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”
90% of stars on Main Sequence
10% are White Dwarfs
<1% are Giants
http://www.anzwers.org/free/universe/hr.html
H-R Diagram
• Vertical Axis  luminosity of star
– could be measured as power, e.g., watts
– or in “absolute magnitude”
Lstar
– or in units of Sun's luminosity:
LSun
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
U,B,V,R,I,II Filters
Transmission (%)
100
100
U
Visible Light
Transmittance (%)
90
V
R
80
B
II
I
R
I
II
70
50
V
B
60
U
50
40
30
20
10
0
0
200
200
300
300
400
400
500
500
600
600
700
700
800
800
Wavelength (nm)
Wavelength
(nm)
900
900
1000
1000
1100
1100
Measure of Color
• If image of a star is:
• Star is BLUISH
and hotter
L(star) / L(Sun)
– Bright when viewed through blue filter
– “Fainter” through “visible”
– “Fainter” yet in red
0.3
0.4
0.5
0.6
 (m)
Visible Region
0.7
0.8
Measure of Color
• If image of a star is:
• Star is REDDISH
and cooler
L(star) / L(Sun)
– Faintest when viewed through blue filter
– Somewhat brighter through “visible”
– Brightest in red
0.3
0.4
0.5
0.6
 (m)
Visible Region
0.7
0.8
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)