1
Wavelength
•
Wavelength alone distinguishes types of light

At visible wavelengths – short wavelengths are blue; long are red

Wavelength, color, and energy of a photon are all the same thing
λ∗ν=c
hc
E=hν=
λ
2
Photon Production
●
●
Accelerated charges (typically electrons) produce photons.
Synchrotron radiation (from electrons spiraling in a magnetic
field) may be conceptually the most easily motivated.
●
Consider whirling an electron around at the end of a string.
●
An observer at a distance sees a varying electromagnetic field.
3
Photon Production
●
●
Accelerated charges (typically electrons) produce photons.
Synchrotron radiation (from electrons spiraling in a magnetic
field) may be conceptually the most easily motivated.
●
Consider whirling an electron around at the end of a string.
●
An observer at a distance sees a varying electromagnetic field.
4
Sorting Light – Filters and Spectra
•
Light can be sorted and/or restricted by wavelength.
5
Spectra
•
Light can be sorted and binned by wavelength. The resulting
spectrum can be projected on a screen or plotted on a graph.
6
Two Fundamental Types of Spectra
•
•
Spectra can be from one of two classes

Continuous – a smoothly varying distribution of all colors

Discrete – emission (or absorption) at precise wavelengths
Often a spectrum is a combination of both
7
The Solar Spectrum
8
Continuous Spectra: Thermal Radiation
•
Any hot object glows

The hotter the object the brighter and bluer the glow
9
The Nature of Temperature
•
Temperature is a measure of the energy of motion of particles in a
gas or in a solid.

In a gas the particles (atoms or molecules) are independently flying
about colliding with one another or with the walls of the chamber.



At high temperature the particles move quickly. At low temperatures
they are sluggish.
In a solid the particles are vibrating in place.
The lowest possible temperature is the point at which all thermal
energy has been removed – absolute zero.
10
The Nature of Temperature
•
Temperature is a measure of the energy of motion of particles in a
gas or in a solid.

In a gas the particles (atoms or molecules) are independently flying
about colliding with one another or with the walls of the chamber.



At high temperature the particles move quickly. At low temperatures
they are sluggish.
In a solid the particles are vibrating in place.
The lowest possible temperature is the point at which all thermal
energy has been removed – absolute zero.
11
Continuous Spectra: Thermal Radiation
•
Any hot object glows

The hotter the object the brighter and bluer the glow
12
Continuous Spectra: Thermal Radiation
•
Dense spheres of gas (stars) are good approximations to
blackbodies as well.

The hot stars below are blue. Cooler ones are yellow and red.
13
The Planck Equation
●
The Blackbody/Planck equation defines, for a given temperature,
the spectrum of emergent energy per unit time into a unit solid
angle (i.e. the specific intensity) from a unit area of a blackbody
per unit frequency.
2hν
B ν (T ) =
2
c
3
1
( )
hν
kT
e −1
In general, we care about the amount of energy launched into a given solid
angle from a unit area of a blackbody
Watts = B ν (T ) Δ ν Δ Ω Δ Area
14
Statistical Mechanics 101
●
Energy injected into a coupled/interacting system (imagine a network of springs
or a gas of colliding atoms) tends to distribute itself evenly amongst the degrees
of freedom of the system.
●
●
●
A typical degree of freedom has energy, ½ kT.
Bulk system properties – e.g. the equilibrium temperature, the distribution of
velocity of particles in a gas - are dictated by statistics/probabilities.
States which require higher energy are less probably populated by the factor
ni α e
●
Ei
−
kT
However, each energy may have many identical configurations corresponding
to that energy
ni = (density of states) x e
Ei
−
kT
15
Solid Angle
●
●
A solid angle is the three-dimensional equivalent of a two
dimensional angle – basically a cone defined by its apex angle.
Solid angle is measured in units of steradians, where there are
exactly 4π steradians on a full sphere (41,253 square degrees)
●
For small cone apex angles the solid angle,
2
θ
Ω=π
4
In spherical coordinates a differential unit of solid angle is
d Ω = sin θ d θ d ϕ
Ω,
is given by:
16
Visualizing Solid Angle
17
Filter Profiles – Delta(λ) vs. Delta(ν)
18
Maybe the Most Important Thing You'll Ever
Learn in Astronomy
c
c
ν=
so d ν = 2 d λ
λ
λ
( )
2hν
B ν (T ) =
2
c
3
2 hc
B λ (T ) =
5
λ
1
( )
( )
−2
−1
watts m Hz sr
hν
kT
−1
e −1
2
1
e
hc
λ kT
−1
−2
−1
watts m m sr
−1
19
The Planck Function
20
The Planck Function
http://resources.jorum.ac.uk/xmlui/bitstream/handle/123456789/957/Items/S381_1_009i.jpg
21
Continuous Spectra: Thermal Radiation
•



The equations below quantitatively summarize the light-emitting
properties of solid objects.
The hotter the object the “bluer” the glow.
The Sun (6000K) peaks in the middle of
the visible spectrum
(0.5 micrometers /
500 nanometers)
Room temperature objects (300K) peak
deep in the infrared (10 um).
Wien's Law



The hotter the object the “brighter” the glow.
The power emitted from each square
centimeter of the surface of a hot object
increases as the fourth power of the
temperature.
Double the temperature and the emission
goes up 16 times!
Stefan-Boltzman Law
22
Motivations for Derivation
●
Stefan-Boltzman Law
●
●
Integrate over all wavelengths and 2 solid angle to get emergent
total flux.
Wien's Law
●
Take the derivative vs. λ and set equal to zero.
●
Must be solved iteratively, not analytically.
●
Peaks at hc/λkT=5.
2 hc
B λ (T ) =
5
λ
2
(
1
e
hc
λ kT
−1
)
23
The Planck Function – Two Extremes
http://resources.jorum.ac.uk/xmlui/bitstream/handle/123456789/957/Items/S381_1_009i.jpg
Two Extremes
●
2 hc
λ5
(
1
e
hc
λ kT
−1
)
Rayleigh Jeans
- Ray
●
Long wavelengths for a given temperature
●
Longward of the Wien's Law peak
2 ckT
B λ (T ) =
4
λ
●
Bλ (T ) =
2
hν =
The exponential dominates
2
hc
≪k T
λ
because e x = 1+ x for x≪1
Wien tail
●
hν =
2 hc
B λ (T ) =
e
5
λ
−
hc
λ kT
hc
≫k T
λ
24
25
26
Sunspots and Thermal Radiation
•
Sunspots are relatively cooler regions of the Sun's 6000K surface.

Being only about 1000K cooler than their surroundings, they do glow
brightly, but due to the strong, T4, dependence of a hot solid object's
brightness on its temperature they appear dark.
27
Thermal Radiation and Circumstellar Disks
28
Thermal Radiation and Circumstellar Disks
29
Submillimeter Galaxies
●
●
The study of the first galaxies in
the distant universe benefits from
the fact that much of the stellar
radiation gets reprocessed by dust
via absorption and re-emission at a
temperature of around 30K, thus a
peak wavelength around 100um.
Cosmological redshift moves this
peak into the radio/submillimeter
part of the spectrum.
●
Galaxies actually become
“brighter as they become
more distant in a given radio
band.
30
31
Statistical Mechanics 101
●
Energy injected into a coupled/interacting system (imagine a network of springs
or a gas of colliding atoms) tends to distribute itself evenly amongst the degrees
of freedom of the system.
●
●
●
A typical degree of freedom has energy, ½ kT.
Bulk system properties – e.g. the equilibrium temperature, the distribution of
velocity of particles in a gas - are dictated by statistics/probabilities.
States which require higher energy are less probably populated by the factor
ni α e
●
Ei
−
kT
However, each energy may have many identical configurations corresponding
to that energy
ni = (density of states) x e
Ei
−
kT
32
Statistical Mechanics 101
●
Energy injected into a coupled/interacting system (imagine a network of springs
or a gas of colliding atoms) tends to distribute itself evenly amongst the degrees
of freedom of the system.
●
●
●
A typical degree of freedom has energy, ½ kT.
Bulk system properties – e.g. the equilibrium temperature, the distribution of
velocity of particles in a gas - are dictated by statistics/probabilities.
States which require higher energy are less probably populated by the factor
ni α e
●
Ei
−
kT
However, each energy may have many identical configurations corresponding
to that energy
cats skinned = (ways to skin a cat) * (probability of skinning)ys
33
One Dimensional Velocity Distribution in a Gas
For example
m
P (v) d v = 4 π
2 π kT
(
)
3/ 2
v
2
(e
−m v
2kT
2
)
34
Simple kT scalings
●
●
●
For reference 1um wavelength corresponds
to 2x10-19J = 1.24 eV
hc
E=hν=
λ
Consider a room temperature blackbody
●
T=300K
●
kT = 4x10-21 J
=
0.03 eV → 40um
Right ballpark but the Wien's law suggests something closer to
10um
●
This difference is simply due, in part, to the statistical weight of
the density of states in the Blackbody equation shaping the curve
and distorting it from a simple Boltzmann distribution.
35
Spherical Blackbodies (and Cows)
●
The emergent flux from each square meter (watts/m2) of a
blackbody is T4
●
The surface area of a sphere is 4R2
2
4
L=(4 π R )(σ T )
Stars are good approximations to blackbodies.
Measure Flux of a star and its distance
(not so easy)....
Determine its temperature from the Wien Law
and you can estimate it's size.
36
Equilibrium Temperature of Planets
●
The Sun's Luminosity is 4x1027 Watts. The Inverse Square Law
says that Solar flux drops off with distance as R2.
Luminosity
Flux=
2
4π R
37
Equilibrium Temperature of Planets
●
A spherical planet or asteroid presents a circular cross section to
the Sun's light. The intercepted energy per unit time is:
input = (1− A)π r
Flux=
Luminosity
4 π R2
2
L sun
4π R
r is the radius of the object
R is the distance to the sun
A is the “albedo” of the object (its reflectivity)
2
38
Equilibrium Temperature of Planets
●
The planet/asteroid radiates with emissivity, ε
2
4
output = 4 π r σ T ϵ
Flux=
where ε is the emissivity (think of it as the
radiative efficiency) and would in general be
equal to (1-A), however both are wavelength
dependent and if you are absorbing visible light
but emitting infrared the two terms can be quite
different.
Luminosity
4 π R2
input = (1− A) π r
2
L sun
4πR
2
r is the radius of the object
R is the distance to the sun
A is the “albedo” of the object (its reflectivity)
39
Equilibrium Temperature of Planets
●
The planet/asteroid radiates with emissivity, ε
T=
(
L star (1− A)
4πϵR
2
)
1/4
the temperature of an object falls off as
the square root of its distance from its
star and depends weakly on the
luminosity of the star (one-quarter
power)
Flux=
Luminosity
4 π R2
input = (1− A) π r
2
L sun
4πR
2
r is the radius of the object
R is the distance to the sun
A is the “albedo” of the object (its reflectivity)
ε is the emissivity
40
Equilibrium Temperature of Planets
T=
(
L star (1− A)
4πϵR
2
)
1/4
1
T =279K
√ R AU
Flux=
Luminosity
4 π R2
input = (1− A) π r
2
L sun
4πR
2
r is the radius of the object
R is the distance to the sun
A is the “albedo” of the object (its reflectivity)
ε is the emissivity
41
Asteroid Radiometry
●
Asteroids can have quite different visual reflectivity, but their
emissivities are similar, typically close to ε=1. Infrared flux
measurements are used to pin down asteroid sizes.