Deducing Temperatures and

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16 Νοε 2013 (πριν από 4 χρόνια και 8 μήνες)

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Deducing Temperatures and
Luminosities of Stars

(and other objects…)

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

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.

10
-
15
m

10
-
6
m

10
3
m

10
-
2
m

10
-
9
m

10
-
4
m

Increasing wavelength

Increasing energy

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

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

time

1 unit of time

(e.g., 1 second)

“Units” of Frequency

meters
cycles
second
second
meters
cycle
cycle
1 1 "Hertz" (Hz)
second
c
n

 
 
 
 

 
 
 
 
 
 

 
 

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
n

c
where 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 = h
n

Energy = (Planck’s constant)
×

(frequency of photon)

h

6.625
×

10
-
34

Joule
-
seconds =
6.625
×

10
-
27

Erg
-
seconds

Every opaque object at temperature T > 0
-
K (a human, a
planet, a star) radiates a characteristic
spectrum
of EM

spectrum = intensity of radiation as a function of wavelength

spectrum depends
only

on temperature of the object

This type of spectrum is called

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

Wavelength of MAXIMUM emission

max

is characteristic of temperature T

Wavelength

max

as T

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

max

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
1
hc
kT
hc
B T
e

Temperature dependence

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
1
hc
kT
hc
B 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

T
1

< T
2

(e.g., T
1

= 5700
-
K, T
2

= 7000
-
K)

Maximum of curve for higher temperature
occurs at SHORTER wavelength

:

max
(T = T
1
) >

max
(T = T
2
) if
T
1

< T
2

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

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

[

[

3
max
2.898 10
meters
K
T

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 10
meters
K
T

Wavelength of Maximum Emission is:

(in the visible region of the spectrum)

3
max
2.898 10
0.508 508
5700
m 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 10
0.414 414
7000
m 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 10
1.45 1450
2000
m 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 blue

Star 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 10
0.193 193
15000
m 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 blue

Star 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
T
4
:
L
=

T
4

L
can be measured in
watts

often expressed in units of Sun’s luminosity
L
Sun

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

8
2 4
Joules
5.67 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

T
4
-
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 (

R
2
,

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 10
4

(=10,000) times more energy
per unit surface area

than Betelgeuse

But the two stars have equal
total luminosities

2

(=100) times

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,

F
0

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

10
0
2.5 log
F
m
F
 
  
 
 
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
10
2.5 log
F pc
M m
F earth
 
   
 
 
 
Sidebar: “Absolute Magnitude”
Apply “Inverse Square Law”

Measured brightness decreases as square of
distance

2
2
2
1
10
10
distance
10pc
1
distance
F pc
pc
F earth
 
 
 
 
 
 
 
 
 
 
Simpler Equation for Absolute
Magnitude

2
10
10
distance
2.5 log
10pc
distance
5 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

identical (
T

R

can
be deduced by other means)

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 Sequence

10% 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
100
U
V
B
R
I
II
U,B,V,R,I,II Filters
Wavelength (nm)
Transmission (%)
Visible Light

U

B

V

R

I

II

Wavelength (nm)

100

50

0

200 300 400 500 600 700 800 900 1000 1100

Transmittance (%)

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(star) / 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(star) / 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)