Electromagnetic Spectra
AST443, Lecture 13
Stanimir Metchev
Administrative
• Homework 2:
– problem 5.4 extension: until Mon, Nov 2
• Reading:
– Bradt, chapter 11
– Howell, chapter 6
• Tenagra data:
– see bottom of Assignments & Exams section on course
website
– M11 (B+V), M52 (B+V+R), HD 209458b (R, all data taken)
– expect M37 (B+V+R) data tonight
– remaining: Hyades (B+V+R)
2
Outline
• Overview
– color-magnitude and color-color diagrams
– spectral classification
• Electromagnetic spectra
– optically thin, synchrotron, and blackbody
emission
– electronic line transitions
• Stellar diagnostics
– atmospheres: temperature, pressure, abundance
– binarity
3
Color-Magnitude Diagram
4
Extinction and Reddening: CCD
•
Legend:
– arrow: AV = 5 mag
extinction
– solid line: main
sequence + giants
– dotted line: substellar
models
– crosses: known
brown dwarfs
– solid points: brown
dwarf candidates
Metchev et al. (2003)
AV = 5 mag
5
OBAFGKM + LT
higher
ionization
potential
species
6
Color-Magnitude Diagram
7
Outline
• Overview
– color-magnitude and color-color diagrams
– spectral classification
• Electromagnetic spectra
– optically thin, synchrotron, and blackbody
emission
– electronic line transitions
• Stellar diagnostics
– atmospheres: temperature, pressure, abundance
– binarity
8
Radiation (Lecture 12)
•
specific intensity Iν
–
–
–
•
spectral flux density Sν
–
–
–
•
F = ∫ Sν d ν
[erg s–1 cm–2] or [W m–2]
power P
–
–
–
•
Sν = ∫ Iν dΩ
[erg s–1 cm–2 Hz–1] or [Jy] or [W m–2 Hz–1]
Sλ = Sν c/λ2
[erg s–1 cm–2 nm–1]
point sources, integrated light from extended sources
flux density F
–
•
dE = Iν dt dA dν dΩ
[erg s–1 cm–2 Hz–1 sterad–1] or [Jy sterad–1]
1 Jy = 10–23 erg s–1 cm–2 Hz–1 = 10–26 W m–2 Hz–1
surface brightness of extended sources (independent of distance)
P = ∫ F dA = dE / dt
[erg s–1] or [W]
received power: integrated over telescope area
luminosity: integrated over area of star
conversion to photon counts
–
energy of N photons: Nhν
9
Extinction and Optical Depth
(Lecture 4)
•
Light passing through a medium can be:
– transmitted, absorbed, scattered
•
extinction at frequency ν over distance s
dLν(s) = –κν ρ Lν ds = –L dτν
Lν = Lν,0e–τ = Lν,0e–κρs =Lν,0e–s/l
Aν = 2.5 lg (Fν,0/Fν) = 2.5 lg(e)τν = 0.43τν mag
– medium opacity κν [cm2 g–1], density ρ [g cm–3]
– optical depth τν = κν ρs [unitless]
– photon mean free path: lν = (κν ρ)–1 = s/τν [cm]
AV = mV – mV,0
10
Neutral Atoms and Molecules Are
Strong Wavelength-Dependent
Absorbers
11
Electronic Transitions
• bound-free
• free-bound
• free-free (bremsstrahlung)
12
Examples of Continuum Spectra
• optically thin thermal radiation
• synchrotron radiation (non-thermal)
• blackbody (optically thick) thermal
radiation
13
Optically Thin Bremsstrahlung
• optical depth << 1
• hot plasma:
– free electrons accelerated in near-collisions with massive
ions
– large accelerarion due to Coulomb force: radiation
j(" ,T) # Z 2 n e n iT $1 2e$h" kT [W m–3 Hz –1 ]
– continuum spectrum:
Z
– atomic number (charge number) of ions
n e,n i
– number densities [m–3 ] of electrons, ions
– spectrum is flat at low (radio) ν (i.e., ~independent of ν)
• occurrence:
! tube
– x-rays in dentist’s
– shocks in supernova remnants
– stellar coronae (~1,000,000 K)
14
Synchrotron Radiation
• charged electrons spiraling in a B field
• spiraling motion means acceleration, hence
radiation
• relativistic electrons can emit x-ray to gammaray photons
– beaming in direction of travel
• spectrum reflects energy distribution of
radiating electrons
– power law: I = Kνα
(α < 0)
[W m–2 Hz–1 sterad–1]
15
Blackbody Radiation (Lecture 4)
2h" 3
1
I(" ,T) = 2 h" kT
c e
#1
• Planck law
– specific intensity
• Wien displacement law
T λmax= 0.29 K cm
!
• Stefan-Boltzmann law
F = σ T4
– energy flux density
– [erg s–1 cm–2]
2# 5 k 4
"=
= 5.67 $10%5 erg cm–2 s–1 K –4
2 3
15c h
• Stellar luminosity
– power
– [erg s–1]
L* = 4 "R*2#Teff4
!
• Inverse-square law
F(r) = L* / r2
16
!
Blackbody Radiation (Lecture 4)
Teff, Sun = 5777 K
T λmax= 0.29 K cm
17
Examples of Continuum Spectra
• optically thin thermal radiation
• synchrotron radiation (non-thermal)
• blackbody (optically thick) thermal
radiation
• see Fig. 11.6 of Bradt, p. 346
18
Radiative Transfer (again)
The optical depth τλ accounts for interaction between photospheric
matter and radiation field.
19
Line Radiation
&1
1)
h" = #E $ R( 2 % 2 +
' n1 n 2 *
!
20
Outline
• Overview
– color-magnitude and color-color diagrams
– spectral classification
• Electromagnetic spectra
– optically thin, synchrotron, and blackbody
emission
– electronic line transitions
• Stellar diagnostics
– atmospheres: temperature, pressure, abundance
– binarity
21
Spectral Lines as
Atmospheric Diagnostics
• chemical content and abundances
– mostly H and He, but heavier “metals” (Z > 2) + molecules
are important sources of opacity
• photospheric temperature
– individual line strength
– line ratios
• photospheric pressure
– non-zero line width
⇒ surface gravity g, mass M*
• stellar rotation
dP
GM r #
= " 2 = "g#
dr
r
equation of hydrostatic equilibrium
– Doppler broadening
!
22
Taking the Stellar Temperature
• individual line strengths
N n " gn e# $ n kT
gn – statistical weight
gn = 2n2 for hydrogen
• line ratios
N n gn #( $ n # $ m ) kT
=
e
N m gm
23
Taking the Stellar Temperature
Teff
•
(Fe II λ5317 / Fe I λ5328) line ratio decreases with decreasing Teff
24
Line Profiles
•
Natural line width (Lorentzian [a.k.a, Cauchy] profile)
–
•
•
Heisenberg uncertainty principle: ∆ν =∆E/h
Collisional broadening (Lorentzian profile)
–
–
–
#E i + #E f
1
1
" natural =
=
+
h /2$
#t i #t f
∆tinteraction > ∆temission
nearby particles shift energy levels of emitting particle
•
•
•
# /2$
2
(" % " 0 ) + # 2 /4
# & Lorentzian FWHM
collisions interrupt photon emission process
∆tcoll < ∆temission ~ 10–9 s
dependent on T, ρ
Pressure broadening (~ Lorentzian profile)
–
–
I" = I0
Stark effect (n = 2, 4)
van der Waals force (n = 6)
dipole coupling between pairs of same species (n = 3)
!
" collisional = 2 #t coll
" pressure % r
&n
; n = 2,3,4,6
!
25
Stark Effect in Hydrogen
•
if external field is chaotic, the energy levels and their differences are smeared →
line broadening
26
Van der Waals Force:
Long-Range Attraction
27
Van der Waals Force:
Long-Range Attraction
28
Line Profiles
•
Natural line width (Lorentzian [a.k.a, Cauchy] profile)
–
•
Heisenberg uncertainty principle: ∆ν =∆E/h
Collisional broadening (Lorentzian profile)
–
–
–
•
∆tinteraction > ∆temission
nearby particles shift energy levels of emitting particle
•
•
•
–
•
#E i + #E f
1
1
" natural =
=
+
h /2$
#t i #t f
!
" collisional = 2 #t coll
" pressure % r
Stark effect (n = 2, 4)
van der Waals force (n = 6)
dipole coupling between pairs of same species (n = 3)
dependent mostly on ρ, less on T
; n = 2,3,4,6
%
1
2
I" =
e 2$
2# $
$ & Gaussian FWHM
emitting particles have a Maxwellian distribution of!velocities
Rotational Doppler broadening (Gaussian profile)
–
&n
(" % " 0 ) 2
Thermal Doppler broadening (Gaussian profile)
–
# /2$
2
(" % " 0 ) + # 2 /4
# & Lorentzian FWHM
collisions interrupt photon emission process
∆tcoll < ∆temission ~ 10–9 s
dependent on T, ρ
Pressure broadening (~ Lorentzian profile)
–
–
•
I" = I0
radiation emitted from a spatially unresolved rotating body
kT
mc 2
"rotational = 2# 0 u /c
"thermal = # 0
!
29
!
Line Profiles: Rotational Broadening
30
Iν
Line Profiles
ν
profiles normalized to the same total area
31
Line Profiles
•
Natural line width (Lorentzian [a.k.a, Cauchy] profile)
–
•
Heisenberg uncertainty principle: ∆ν =∆E/h
Collisional broadening (Lorentzian profile)
–
–
–
•
∆tinteraction > ∆temission
nearby particles shift energy levels of emitting particle
•
•
•
–
•
!
" collisional = 2 #t coll
" pressure % r
Stark effect (n = 2, 4)
van der Waals force (n = 6)
dipole coupling between pairs of same species (n = 3)
dependent mostly on ρ, less on T
&n
; n = 2,3,4,6
(" % " 0 ) 2
%
1
2
I" =
e 2$
2# $
$ & Gaussian FWHM
emitting particles have a Maxwellian distribution of!velocities
Rotational Doppler broadening (Gaussian profile)
–
•
#E i + #E f
1
1
" natural =
=
+
h /2$
#t i #t f
Thermal Doppler broadening (Gaussian profile)
–
# /2$
2
(" % " 0 ) + # 2 /4
# & Lorentzian FWHM
collisions interrupt photon emission process
∆tcoll < ∆temission ~ 10–9 s
dependent on T, ρ
Pressure broadening (~ Lorentzian profile)
–
–
•
I" = I0
radiation emitted from a spatially unresolved rotating body
Composite line profile: Lorentzian + Gaussian = Voigt profile
!
kT
mc 2
"rotational = 2# 0 u /c
"thermal = # 0
32
!
Line Profiles
•
Natural line width (Lorentzian [a.k.a., Cauchy] profile)
–
•
Heisenberg uncertainty principle: ∆ν =∆E/h
Collisional broadening (Lorentzian profile)
–
–
–
•
∆tinteraction > ∆temission
nearby particles shift energy levels of emitting particle
•
•
•
–
•
!
" collisional = 2 #t coll
" pressure % r
Stark effect (n = 2, 4)
van der Waals force (n = 6)
dipole coupling between pairs of same species (n = 3)
dependent mostly on ρ, less on T
&n
; n = 2,3,4,6
(" % " 0 ) 2
%
1
2
I" =
e 2$
2# $
$ & Gaussian FWHM
emitting particles have a Maxwellian distribution of!velocities
Rotational Doppler broadening (Gaussian profile)
–
•
#E i + #E f
1
1
" natural =
=
+
h /2$
#t i #t f
Thermal Doppler broadening (Gaussian profile)
–
# /2$
2
(" % " 0 ) + # 2 /4
# & Lorentzian FWHM
collisions interrupt photon emission process
∆tcoll < ∆temission ~ 10–9 s
dependent on T, ρ
Pressure broadening (~ Lorentzian profile)
–
–
•
I" = I0
radiation emitted from a spatially unresolved rotating body
Composite line profile: Lorentzian + Gaussian = Voigt profile
!
kT
mc 2
"rotational = 2# 0 u /c
"thermal = # 0
33
!
Example: Pressure Broadening
of the Na D Fine Structure Doublet
34
Line Profiles: Equivalent Width (EW)
λ1
"2
EW =
$ (F
$
"1
", cont
"2
"1
λ2
# F", line )d"
F", cont d"
35
Lorentzian Line Profile at
Increasing τ
simulation
for the Hα
line profile
36
Lorentzian Line Profile at
Increasing τ
simulation
for the Hα
line profile
saturation
at τ > 5
37
Lorentzian Line Profile at
Increasing τ
simulation
for the Hα
line profile
38
Lorentzian vs. Gaussian Line
Profiles: Small τ
simulation
for the Hα
line profile
39
Lorentzian vs. Gaussian Line
Profiles: Large τ
simulation
for the Hα
line profile
• core more
sensitive to
Gaussian
parts
• wings
more
influenced
by Lorentzian parts
40
Curve of Growth: Dependence of Line
Equivalent Width W on Column Density N
• N ≡ integral of number density of absorbing atoms or
molecules along line of sight [cm-2]
– for small N, W ∝ N
• linear part of the curve of growth
– for larger N,
W " ln N
• after the Gaussian core bottoms out
• flat part of the curve of growth
– for even larger N,
W" N
•!after the absorption by the Lorentzian wings becomes strong
• square root part of the curve of growth
• There is a different curve of growth, W(N), for each
spectral line
!
41
Universal Curve of Growth
• the ratio of W to Doppler line width Δλ depends upon
the product of N and a line’s oscillator strength f in the
same way for every spectral line (e.g. Unsöld 1955).
1
re
a
qu
s
flat
t
v # 2kT
o
o
W
"
#
r
"# = # =
log $
0
%
c c m
W" N
W " ln N
r
' &! (
a
e
lin
1
W "N
!
!
1
0
m: absorber particle mass
1
!
2
3
!
4
log (Nf )
42
Alkali (Na, K) lines
in visible spectra of
late-L and T dwarfs
become saturated!
(Kirkpatrick 2005)
43
Curve of Growth:
Determining Abundances
• Measure W for a lot of lines (each with distinct, known
f) of a number of atomic or ionic species.
• Plot W/∆λ against xNf where:
– N is the column density of one species
– x is the relative abundance of the atomic species that gives
rise to the line (ratio of number density of that species to the
number density of the first species),
• Adjust x, N, and ∆λ until the points fit the universal
curve of growth.
• Then one knows these three quantities for each
species.
44
Outline
• Overview
– color-magnitude and color-color diagrams
– spectral classification
• Electromagnetic spectra
– optically thin, synchrotron, and blackbody
emission
– electronic line transitions
• Stellar diagnostics
– atmospheres: temperature, pressure, abundance
– binarity
45
Spectroscopic
Binary
(a)
• double-lined (SB2)
– spectra of both stars visible
(d)
(a)
(b)
(b)
(c)
(c)
(d)
(d)
• single-lined (SB1)
– only spectrum of brighter star visible
46
Example: SB1
47
Example: SB2
48
Radial Velocity vs. Time for
Double-lined SB in a Circular Orbit
49
Radial Velocity vs. Time for Doublelined SB in Elliptical Orbit (e = 0.4)
50
51 Peg Ab is an SB1 !
• first planet
detected around a
main-sequence
star
– primary SpT: G2 V
• Mp sin i = 0.47
MJup
(Mayor & Queloz 1995)
51