Download Deducing Temperatures and Luminosities of Stars

Review: Electromagnetic Radiation
Increasing energy
)
t
)
UV
gh (IR
t(
R
Li
le
d
s
o
a
y vi
le are
a
m
b
R
ra
si Infr
am
X Ult
Vi
s
ay
Deducing Temperatures and
Luminosities of Stars
(and other objects…)
G
10-15 m
10-9 m
10-6 m
10-4 m
M
w
ro
ic
es
av
10-2 m
o
di
Ra
w
es
av
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
Sinusoidal Fields
• BOTH the electric field E and the magnetic
field B have “sinusoidal” shape
Direction
of “Travel”
Wavelength λ of Sinusoidal Function
Frequency ν of Sinusoidal Wave
λ
time
z Wavelength λ is the distance between any two
identical points on a sinusoidal wave.
1 unit of time
(e.g., 1 second)
z Frequency: the number of wave cycles per unit of
time that are registered at a given point in space.
(referred to by Greek letter ν [nu])
z ν is inversely proportional to wavelength
1
“Units” of Frequency
⎡ meters ⎤
c⎢
⎣ second ⎥⎦ = ν ⎡ cycles ⎤
⎢⎣ second ⎥⎦
⎡ meters ⎤
λ⎢
⎥
⎣ cycle ⎦
⎡ cycle ⎤
= 1 "Hertz" (Hz)
1⎢
⎣ second ⎥⎦
Wavelength and Frequency Relation
z Wavelength is proportional to the wave velocity v.
z Wavelength is inversely proportional to frequency.
z e.g., AM radio wave has long wavelength (~200
m), therefore it has “low” frequency (~1000 KHz
range).
z If EM wave is not in vacuum, the equation
becomes
ν=
where v =
v
λ
c
and n is the "refractive index"
n
Planck’s Radiation Law
Light as a Particle: Photons
z Photons are little “packets” of energy.
z Each photon’s energy is proportional to its
frequency.
z Specifically, energy of each photon energy is
• 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
E = hν
Energy = (Planck’s constant) × (frequency of photon)
h ≈ 6.625 × 10-34 Joule-seconds = 6.625 × 10-27 Erg-seconds
http://scienceworld.wolfram.com/physics/PlanckLaw.html
Planck’s Radiation Law
• Wavelength of MAXIMUM emission λmax
is characteristic of temperature T
• Wavelength λmax ↓ as T ↑
Sidebar: The Actual Equation
B (T ) =
2hc 2
λ5
1
e
hc
λ 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
λmax
http://scienceworld.wolfram.com/physics/PlanckLaw.html
2
Temperature dependence
of blackbody radiation
Wien’s Displacement Law
• 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
• Can calculate where the peak of the blackbody
spectrum will lie for a given temperature from
Wien’s Law:
λmax [ meters ] =
2.898 × 10−3
T [K]
(recall that human vision ranges from 400 to 700 nm, or
0.4 to 0.7 microns)
Colors of Stars
Colors of Stars
• Star “Color” is related to temperature
• If T << 5000 K (say, 2000 K), the wavelength of
the maximum of the spectrum is:
– If star’s temperature is 5000 K, the wavelength of
the maximum of the spectrum is:
λmax
λmax =
2.898 × 10−3
=
m 0.579µ m = 579nm
5000
2.898 × 10−3
m 1.45µ m = 1450nm
2000
(in the “near infrared” region of the spectrum)
(in the visible region of the spectrum)
• The visible light from this star appears “reddish”
Why are Cool Stars “Red”?
Colors of Stars
Less light in blue
Star appears “reddish”
• If temperature >> 5000-K (say, 15,000-K),
wavelength of maximum “brightness” is:
λmax =
0.4
0.5
0.6
0.7
0.8
λ (µm)
0.9
1.0
1.1
1.2
2.898 × 10−3
m 0.193µ m = 193nm
15000
1.3
λmax
“Ultraviolet” region of the spectrum
Star emits more blue light than red ⇒appears “bluish”
Visible Region
3
Betelguese and Rigel in Orion
Why are Hotter Stars “Blue”?
Betelgeuse: 3,000 K
(a red supergiant)
More light in blue
Star appears “bluish”
0.1
0.2
0.3
λmax
0.4
0.5
0.6
0.7
0.8
0.9
1.0
λ (µm)
Rigel: 30,000 K
(a blue supergiant)
Visible Region
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 = σ T4
Joules
⎛
⎞
−8
, Stefan-Boltzmann constant ⎟
⎜ σ ≈ 5.67 × 10
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
4
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.
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”
• “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 ) ⎥⎦
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
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”
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 ⎠
5
Simpler Equation for Absolute
Magnitude
⎡⎛ distance ⎞ 2 ⎤
M = −2.5 × log10 ⎢⎜
⎟ ⎥+m
⎢⎣⎝ 10 pc ⎠ ⎥⎦
⎡ distance ⎤
= −5 × log10 ⎢
⎥+m
⎣ 10 pc ⎦
Plot Brightness and Temperature
on “Hertzsprung-Russell Diagram”
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. α 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 α Cen
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
http://zebu.uoregon.edu/~soper/Stars/hrdiagram.html
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
6
H-R Diagram
Hertzsprung-Russell 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
H-R Diagram
“Standard” Astronomical Filter Set
• Horizontal Axis ⇒ surface temperature
–
–
–
–
• 5 “Bessel” Filters with approximately equal
“passbands”: ∆λ≈ 100 nm
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
–
–
–
–
–
–
U: “ultraviolet”, λmax ≈ 350 nm
B: “blue”, λmax ≈ 450 nm
V: “visible” (= “green”), λmax ≈ 550 nm
R: “red”, λmax ≈ 650 nm
I: “infrared, λmax ≈ 750 nm
sometimes “II”, farther infrared, λmax ≈ 850 nm
Filter Transmittances
Measure of Color
100
R
B
II
I
• If image of a star is:
– Bright when viewed through blue filter
– “Fainter” through “visible”
– “Fainter” yet in red
V
U
50
• Star is BLUISH
and hotter
0
L(star) / L(Sun)
Transmittance (%)
Visible Light
0.3
0.4
0.5
0.6
0.7
0.8
λ (µm)
200
300
400
500
600
700
800
Wavelength (nm)
900
1000
1100
Visible Region
7
Measure of Color
How to Measure Color of Star
• Measure brightness of stellar images taken
through colored filters
• If image of a star is:
– Faintest when viewed through blue filter
– Somewhat brighter through “visible”
– Brightest in red
L(star) / L(Sun)
• Star is REDDISH
and cooler
– used to be measured from photographic plates
– now done “photoelectrically” or from CCD
images
• Compute “Color Indices”
0.3
0.4
0.5
0.6
λ (µm)
0.7
0.8
– Blue – Visible (B – V)
– Ultraviolet – Blue (U – B)
– Plot (U – V) vs. (B – V)
Visible Region
8