Download Lecture 4a - University of Rochester

Electromagnetic radiation
Ast 111 Lecture 4a
❑Magnitudes, stellar and
solar system objects
❑Photons and light
❑The electro-magnetic
spectrum
❑Black bodies
❑Atmospheric absorption
Props: CD, hand held spectrographs, laser pointers
Brightness of starlight
• When we speak of brightness really we are talking
of energy flux, f. Flux is energy per unit area per
unit time received from the star. (erg cm-2 s-1)
• Flux is usually measured over a particular
wavelength or frequency range.
• Astronomers often use a logarithmic scale to
measure brightness. Magnitudes.
• Log scale used for Earthquakes (Richter scale) and
audio (decibels)
Astronomical Magnitudes
m
f f+ +constant
m ==−2.5log
2.5 log
constant
10 10
Magnitudesusually
are usually
defined
for flux
the flux
integrated
Magnitude
defined
for the
integrated
across a
particular
wavelength
band bands.
across particular
wavelength
B blue 4200Å; V visual which is green, 5300Å; R red 7000Å
J, H, K : 1.2,1.6, 2.2 µ m near-IR bands.
The difference
in magnitudes
describes
a ratio
of fluxes.
Difference
in magnitudes
describes
a ratio
of fluxes
m1 − m2 = −2.5log f1 − 2.5log f 2
m1 m2 = 2.5 log10 f1 + 2.5 log10 f2
f1
f1
= −2.5log
= 2.5 log
f 2 10 f
2
Magnitudes (continued)
m=
2.5 log10 f + constant
• The brighter the object the lower than magnitude!
• A change of a factor of100 in flux corresponds to
a difference of 5 mag
• 10% error corresponds to about an uncertainty of
0.1 mag
• The faintest star visible by eye is about 6 mag (in
V band)
• An 8m class telescope will let you observe objects
to about 26th mag in V (photometrically)
Distance Modulus
L
f=
4⇡r2
m=
Flux f depends on the luminosity L of the star
and distance to the star r
Flux: Energy per unit area per unit time
2.5 log f + constant
magnitude
2.5 log L + 5 log r + constant0
✓
◆
r
The Distance modulus
DM = 5 log10
10pc
=
Absolute magnitude (objects outside solar system)
The magnitude the star would have if it were 10 pc away
m = M + DM
M absolute magnitude
m observed magnitude
The Constant ….
The constant defined such that the absolute
magnitude is zero for an A star at 10pc — in all
bands.
A star has a particular luminosity and colors.
Absolute magnitudes for solar
system objects
• For objects outside the solar system, the absolute magnitude, M, is
that which you would detect if the object was 10pc away.
• For solar system objects, the absolute magnitude, H, is the
magnitude of the object, at 1AU away, at zero phase angle.
• Phase angle is the angle measured at the center of an illuminated
body between the light source and the observer.
Phase angle
So the absolute magnitude,
H, is computed as if the
object was illuminated by
the Sun, was 1AU from the
Sun and the observer was
sitting on the Sun (toasty).
Example with Magnitudes
You take a spectrum of a star and find that it has lines
typical of a A star and that its colors are mB-mV=0
mB the observed magnitude at B band
mV the observed magnitude at V band
(consistent with an A star and no extinction!)
You measure a magnitude of V=5
Q: How far away is the star?
Example with Magnitudes
mV=5 observed mag
MV=0 Absolute magnitude because it’s an A star
DM = 5 The difference is the distance modulus
DM = 5 log10 (d/10pc) definition of DM
log10(d/10pc) = 1 solve this!
d/10pc = 10
d = 100pc distance to star
Another Example with Magnitudes
A star has an observed magnitude 10
Consider a binary comprised of two of these stars
What is the observed magnitude of both stars together?
Another Example with Magnitudes
A particular star has observed magnitude 10
Consider a binary comprised of two of these stars
What is the magnitude of both stars together?
mA = -2.5 log10 f + constant = 10.0
m2A = -2.5 log10 (2f) + constant
= -2.5 log10 f - 2.5 log10 2 + constant
= 10.0 - 2.5 log10 2
= 10.0 - 0.75
= 9.25
The Electromagnetic Spectrum
• One of the amazing predictions of Maxwell’s equations
is that electric and magnetic fields are related and can
propagate as electromagnetic waves.
• The speed of light, c, relates the frequency 𝝂 of a photon
to its wavelength 𝝀.
= c/⌫
• Quantum mechanics then led to the discovery that the light
is quantized into photons.
Energy E = h𝝂
momentum p = h𝝂/c
h is Planck’s constant h = 6.6𝗑10-27 erg s
Multi-wavelength astronomy
Region
Wavelength
Frequency
Convention for
wavelength
Convention for
flux
Radio
>1mm
<3x11Hz
Hz, mm
Jy
Infrared
700nm-1mm
3x1011-4.3x1014Hz µm
Mag or Jy
1-300 µm
Visible
400-700nm
4.3x10147.5x1014Hz
nm or Ǻ
Mag
UltraViolet
10-400nm
0.1-100eV
7.5x10143x1016Hz
Ǻ or eV
erg/cm2/s
X-ray
0.1-10nm
0.1-20keV
3x10163x1018Hz
keV
erg/cm2/s
Gamma
Ray
>20keV
>3x1018Hz
MeV
erg/cm2/s
Conventions
1Jy (Jansky) = 10-23 erg cm-2 s-1 Hz-1
used in radio/infrared
1Hz = 1 cycle per second
1ev = 1.6 𝗑 10-12 erg
used in X-ray/high energy
Magnitude system defined such that an A star has
zero absolute magnitude (all bands!)
Atmospheric Absorption
Transmission windows
• It’s difficult and expensive to observe from space.
• Most previous astronomy has been done from the ground.
• Many filter bands and radio receivers are designed to be
restricted to atmospheric windows of transmission.
• Atmospheric windows exist (where the atmosphere is
transparent) in the radio, submillimeter, near+mid-infrared
and optical.
• The atmosphere is opaque in the far-infrared, UV, X-ray,
and low frequency radio. Astronomy at these wavelengths
is done from space.
• Note that the atmosphere can also be a source of radiation.
Spectra
• When we look at an astronomical object, we would like to
know how much energy or light it gives of at different
wavelengths or frequencies.
• The flux as a function of wavelength is known as a
spectrum.
• We measure the flux per unit wavelength or frequency
range.
• fv is energy per unit are per unit time per unit frequency
range. (erg cm-2 s-1 Hz-1)
• fλ is energy per unit are per unit time per unit wavelength
range. (erg cm-2 s-1 Å-1 or keV-1 or µm-1)
Origin of Continuum, Emission-line,
and Absorption Spectra
Origin of Continuum, Emission, and Absorption Spectra
Spectroscopic Gratings
Spectroscopic Gratings
Question:
Why do spectrographs
need a slit?
normal
incidence
Spectroscopic Gratings
Why do gratings need a slit?
Answer:
To restrict angle of incoming light
Spectroscopic resolution is reduced with
wider slits
Demo: CD with different laser pointers, different angles
Demo: hand held spectrographs: fluorescent vs sunlight
Hydrogen Emission and Absorption
The spectrum of hydrogen is
important in astronomy because
most of the Universe is made of
hydrogen.
Emission or absorption processes in
hydrogen give rise to series, which
are sequences of lines corresponding
to atomic transitions, each ending or
beginning with the same atomic
state.
The Balmer Series involves
transitions starting (for absorption)
or ending (for emission) with the
first excited state of hydrogen. In
visible spectrum.
The Lyman Series involves transitions that start or end with the ground
state of hydrogen, lines in the UV.
Quantum Mechanics
• We see in spectra that many systems emit
light at particular energies.
• Consequently the energy of these systems
can only be at particular values; we say the
energy is quantized.
• The theory of quantum mechanics explains
this.
Bound Free Radiation
•
•
•
If an electron is ejected
from a Hydrogen atom, its
final energy can take a
range of values, depending
upon its free momentum.
The photon emitted when
this happens can be at a
range of continuous possible
energy values or
wavelengths.
We refer to this as a boundfree transition.
• When an electron jumps from one bound energy level to another (boundbound transitions) then only a few wavelengths are possible.
• Spectra can be smooth (continuum) or full of emission lines or can have
components of both.
Radiation observed at different
wavelengths
• The energy of the light emitted corresponds
to energy changes in the object emitting.
• Small changes in energy levels correspond
to low energy photons.
• Large changes in energy levels correspond
to high energy photons.
Continuous vs discrete spectrum
If there are many possible energy states:
Continuous spectrum
If there are only a few possible energy states:
discrete spectrum
Examples of emission processes
Wavelength
Emission lines
Continuum radiation
Radio
HI spin flip transition
Synchrotron radiation
Infrared
Rotational and Vibrational
molecular transitions
Dust emission
Optical
UV
Atomic transition lines
Atomic transition lines from
ionized atoms
Atomic transition lines from
highly ionized atoms
Bound-free, free-free
Free-free emission
Nuclear exited state
transitions, electron positron
annihilation
Compton scattering
X-ray
Gamma Ray
Free-free
Black Bodies
• A body is in thermal equilibrium when it freely exchanges
energy with its surrounding and a steady state is reached where
there is no net energy flow.
• To maintain a steady state the body must emit radiation at the
same rate that it is absorbed.
• A black body is an object that absorbs equally well radiation at
all wavelengths.
• A black body is described by one parameter, the temperature, T.
• The hotter the body, the more energy it gives off.
• The shape of the spectrum is something that you can derive in a
statistical physics class (Compute the average number of
photons in a square cavity per degree of freedom in thermal
equilibrium).
Black Bodies
Described by 1 parameter, the temperature T
Thermal
Simplest possible description of emission
Stefan-Boltzmann law
Curves only depend on a
single parameter, T
Total luminosity L emitted per unit
surface area A is proportional to T4
T is temperature
The Stefan Boltzmann law:
L
= T4
A
The Stefan Boltzmann constant
𝝈 = 6.7 𝗑 10-5 erg cm-2 K-4 s-1
The luminosity of a sphere with
radius R
L = 4⇡R2 T 4
frequency
Wien Displacement Law
The peak intensity occurs at a
The
peak intensity
occurs
at
wavelength
that
is inversely
a wavelength
thattois the
inversely
proportional
proportional
temperature
to the temperature.
λmaxT = 2.90 × 10−1 cm K
Relation between temperature and
spectrum:
X-rays ---- 106K plasma
Optical---- 104K ionized gas
Radio ---- 100K molecular gas
frequency
The peak intensity occurs at
a wavelength that is inversely proportional
to the temperature.
λmaxT = 2.90 × 10−1 cm K
Wien Displacement Law
On a log log plot!
Planck’s radiation law
2h⌫ 3
1
B⌫ (⌫, T )d⌫ = 2
c e kh⌫
BT
2hc2
1
B ( , T )d = 5
hc
kB T
e
d⌫
1
⌫=
c
d⌫ =
d
1
Units:
energy/time/area/solid angle/Hz
or
energy/time/area/solid angle/
wavelength
Spectral radiance, specific
intensity
frequency
c
2
d
Planck’s radiation law
3
2h⌫
1
B⌫ (⌫, T )d⌫ = 2
c e kh⌫
BT
d⌫
1
Triumph of
statistical physics
h Planck’s constant
kB Boltzmann’s constant
c speed of light
• At large frequencies
Wien limit
B⌫ / ⌫ 3 e
h⌫
kB T
frequency
d⌫
1
Wien
• At small frequencies (low
energy, large
⌫2
B⌫ /
wavelengths)
T
Rayleigh-Jeans limit
(to derive expand
exponential to first order)
Rayleigh-Jeans
Planck’s law
limits
2h⌫ 3
1
B⌫ (⌫, T )d⌫ = 2
c e kh⌫
BT
On a log log plot
Rayleigh Jeans
power law with slope
dependent on temperature
Wein limit
exponential drop
Thermal spectrum
Black body assumes emission/absorption at all possible
wavelengths
Continuum spectrum
Stars are well approximated by black bodies
Line emission/absorption can be characterized by a gas in
thermal equilibrium
Emission can still be thermal but not give a continuum
spectrum
There are also non-thermal emission processes
Review
• Spectrum, electromagnetic radiation, Hydrogen
atom, emission lines, continuous spectrum,
atmospheric windows, black body, Planck’s
constant.
• Magnitudes, Distance Modulus.
• Definitions of Absolute Magnitude for solar system
objects and for Galactic and extra-galactic objects.
• Wien’s displacement law, Stefan-Boltzmann law.