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transcript
Astronom
ische Waarneem
technieken(Astronom
ical Observing T
echniques)
10thLecture: 2
4 Novem
ber 2
010
Based
on “Observational A
strophysics” (S
pringer) by P. Lena,
Wikiped
ia, ESO, and
“astronomical spectroscopy” b
y Massey &
Hanson
Conte
nt:
1.Form
ation of Spectral Lines
2.General Principle of a S
pectrometer
3.Dispersers: G
ratings and Filters
4.Advanced
Spectrom
eter Concepts
5.Spectral Line A
nalysis
Form
ation of S
pectra
l Line
s
Macroscopically, th
e received rad
iation can be ch
aracterized by th
e specific intensity I(v,θ
) at frequencyνand
direction
θand
polarization.
Microscopically, th
e transition betw
een two energetic states E
1 , E2
requires the em
ission or absorption of a ph
oton of frequency:
Energy levels could
be due to splitting at several fund
amental levels:
h
EE
12
0
−=
ν
()
()
I
JJ
JE
2
12
+=h
Excita
tion Processe
s•Electronic transitions d
ue to the ch
ange of the principal quantum
num
bers of th
e electronic states (visible).
•Electronic fine structure transitions d
ue to the coupling of electron
spin and nuclear spin.
•Electronic h
yperfine structure transitions due to th
e interaction of the nuclear m
agnetic moment w
ith the m
agnetic field of th
e electron.
•Molecular transitions such
as rotational (change in angular m
omentum
) •Molecular transitions such
as rotational (change in angular m
omentum
) and
vibrational (ch
ange in vibrational energy) transitions*, requiring
dipole m
oment and
moment of inertia I (
near-far-IR
).
•Nuclear lines d
ue to nuclear excititations or electron-positron
annihilation (
MeV range)
•Transitions in solid
s (ices) due to vib
rations phonons (
near-far-
IR).
* rotational transitions are generally weaker and
often coupled to vib
rationaltransitions
vibrational
transitions split further: com
plex structure of vib
rational-rotationaltransitions.
Excita
tion Processe
s –Energy
Range
s
Three General T
ypes of S
pectra
Continuous spectrum
Emission line spectrum
Absorption line spectrum
Physica
l Processe
s causing a
Line
-Shift
•Doppler effect: th
e emitter is in m
otion relative to the ob
server with
a relative line-of-sight velocity com
ponent v|| . T
he resulting frequency
shift is:
Doppler im
agingc v
c v
c v
||
0
||
2/1
2 2||
0
1
1
1ν
νν
≈
−
−
−=
∆
•(norm
al) Zeem
an effect: magnetic field
splits line in three
components (th
e linearly polarized πcom
ponent at ν0and
the tw
o elliptically polarized
σcom
ponents at ±Δνwith :
•Einstein effect: a strong gravitational field
s causes a redshift of th
e ligh
t:
Bm
eB
e
10
10
4.1
4⋅
==
∆π
ν
2
2/1
21
21
Rc
GM
Rc
GM
≈−
−=
∆ν ν
The Basic Priciple
Main ingred
ients of a spectrometer:
1.A slit
(ontwhich the ligh
t of the telescope is focused
)
2.A collim
ator(diverging
parallel/collimated
light)
3.A disperser
(to spectrally disperse th
e light)
4.A cam
era (to focus the spectrum
onto the detector)
Characte
ristics of a Spectrom
eter
•the spectral resolution or spectral resolving pow
eris:
Δλis called
a spectral resolution element.
•the instrum
ental profile P(v) broad
ens a theoretically infinitely
narrow line to th
e observed
line width:
()
()0
0ν
νδ
ν−
=I
()
()
()ν
νν
0I
PI
∗=
λ λ∆=
R
Usually the instrum
ental profile determines the spectral resolution
element, w
hich is typically Nyquist-sam
pled.
•the beam
étenduedeterm
ines the ligh
t gathering pow
er of the
instrument. Larger étend
ues require larger dispersive elem
ents (A)
or highly inclined
beam
s (Ω).
•the transm
issiondeterm
ines the th
roughput
()
()
()ν ν
νη
in
ou
t
I I=
For unresolved
lines, both the S/N and
the line/continuum
increases with increasing resolution:
Spectra
l Resolution a
nd S/N
Mo
de
l sp
ectra
of C
2 H2
at 9
00
K a
nd
HC
N a
t 60
0K
(assu
me
d D
op
ple
r bro
ad
en
ing ~
4 k
m/s
) at
diffe
ren
t sp
ectro
gra
ph
reso
lutio
ns (fig
ure
pro
vid
ed b
y F
. La
hu
is).
R=
20
00
R=
50
00
0
General Principle
of a Grating
Use a d
evice that introd
uces an optical path
difference = fangle to th
e surface
The cond
ition for constructive interference is given b
y the grating equation:
()β
αλ
sinsin
±⋅
=a
m
m= ord
er of diffraction
λ= wavelength
a = distance b
etween equally spaced
grooves
a
Gratings are usually operated
in a collimated
beam
at the pupil.
The m
aximum resolution is given b
ywhere N
is the num
ber of
(illuminated
) periods (grooves), and
the angular d
ispersion is.
mN
R=
a = distance b
etween equally spaced
grooves
α= angle of incom
ing beam
β= angle of reflected
beam
a md
d~
/λ
θ
Blaze Angle
Generally, th
e energy of the beam
diffracted
by a period
ic structure is uniform
ly distrib
uted over th
e different ord
ers m.
If we ob
serve only one arbitrary ord
er this is very inefficient.
For b
lazed gratings th
e directions of constructive interference and specular reflection coincide :
()
2
2
αβ
θθ
αβ
α−
=⇒
+=
+B
B
Advantage:
•High efficiency
Disadvantage:
•Blaze angle θ
B(and
hence b
laze wavelength
λB ) are fix
ed by construction.
Free Spectra
l Range
…
A light bulb seen through a transm
issivegrating, show
ing three diffracted orders. m = 0
corresponds to direct transmission;
colors with increasing w
avelengths (from
blue to red) are diffracted at increasing angles. S
ource: Wikipedia
Different d
iffraction orders overlap w
ith each
other:
The free spectral range is th
e largest wavelength
range for a given order th
at does not overlap th
e same range in an ad
jacent order.
()
()λ
βα
λ′
+=
+=
1sin
sinm
am
mfree
λλ
λλ
′= ′
−=
∆
…and Cross-
Dispe
rsionTo spatially separate th
e orders and
avoid overlap, an ad
ditional
optical element w
ill be need
ed:
A low
-dispersion prism
/grating with a d
ispersion direction
perpendicular to th
at of the high-dispersion grating
hig
h d
ispersio
n
cross
dispersion
hig
h d
ispersio
n
Echelle Gratings
α=
β=
θ
a md
d~
/λ
θTo get h
igh dispersion
one could either
increase the
groove density, or
use large groove periods (a >> λ) and
a large angle of incid
ence, and operate at a very h
igh ord
er of diffraction (m
>~50).
If α= β
=θLittrow
configuration
α=
β=
θ
Θ=
sin2
am
Bλ
In Littrowconfiguration th
e grating equation becom
es:
Grism
s
Grism
= transmission G
Rating + prIS
M
For a given w
avelength and
diffraction ord
er the refraction of grating
and prism
may com
pensate each oth
er and the optical ax
is remains
(almost) unch
anged.
Advantages:
•ideal to b
ring in and out of a collim
ated
beam
(“filter wheel”)
beam
(“filter wheel”)
•red
uces coma (if in non-collim
ated beam
)
Disadvantage:
•difficult to m
anufacture (either b
y replication and
gluing or by direct ruling.
•can b
e quite “bulky” (
filter w
heel)
Inte
rference
(Transm
ission) Filte
rs
Principle: interference layers deposited
on a substrate.
The transm
ission is maximal w
here
ππ
λk
dn
22
21
=+
•spectral resolution typically R
~ few
-1000
•need
s often multiple interference layers
•filters are often tilted
with respect to th
e optical axis to
avoid reflections
shift of λ
0
•wavelength
s farther from
λ0(for w
hich the ab
ove equation is also satisfied
) need a b
locking or absorb
ing filter.
Fabry-Perot E
talon
The transm
ission is:
and has transm
ission peaks where
()
()
1
2
2
2
0co
s2
sin1
41
1
−
−+
−=
id
kr r
r
rI
Iπ
d mk
2=
Two parallel plates (F
abry-Perot etalon) of h
igh
reflectivity rand
transmission t = 1-r.
Here, m
is the ord
er of the interferom
eter, dis th
e separationof th
e plates, and
Δk= 1/2
dthe free spectral range.
d2
The perform
ance of a Fabry-Perot is ch
aracterized by:
1.The finesse
,
2.The resolution
, and
3.The m
aximum through
put (S = illum
inated area of th
e etalon).
r rF
−=
1 π
mF
k kR
=∆
=
R SU
π2=
OH Suppre
ssion Spectrogra
phs
OHS filter out th
e wavelength
s of atmosph
eric OH lines, w
hich
contribute th
e major part of th
e near-IR background
.
http://sub
arutelescope.org/Introduction/instrum
ent/img/O
HS_concept.gif
Multi-
Object S
pectrogra
phs
Use num
erous “slits” in the focal plane
simultaneously
multiple source pick-ups
using fibers
or mirrors.
Need
s different slit m
asks for d
ifferent fields.
Hectospe
c (SAO) w
ith rob
otic fiber positioning
Align all spectra on th
e same
detector:
Inte
gral F
ield Spectrogra
phs
Cut an area on th
e sky in several adjacent slices or sub
-portions, realign them optically into one long slice and
treat it as a long slit spectrograph.
JWST-MIRI im
age slicer:
Echelle Spectrogra
phs
Operation in h
igh ord
er pre-d
isperser essential
Example: E
SO’s V
LT instrum
ent CRIRES:
The ruled echelle
grating of the SOFIA Facility
Spectrom
eter AIRES. Two im
ages of the engineer are seen reflected from
the facets of the grooves that are at angles of 9
0 degrees from
each other.
Fourie
r Transform
Spectrom
eter (1
)
•The FTS or M
ichelson interferom
eter is a two-w
aveinterferom
eter (as opposed
to a grating with Nwaves from
Ngrooves ).
•The signal is an interferogram
. It is the Fourier transform
of the
spectrum of th
e object.
•For each
setting of the spectrom
eter arm length
(x) all
spectral elem
ents contribute to th
e signal (“ spectral multiplex
ing”).
•FTS are ax
isymmetric
and particularly suited
to observe ex
tended
sources.
Fourie
r Transform
Spectrom
eter (2
)
()
()
kxI
xI
π2co
s1
2
0+
=
If x is the difference in path
length the intensity of a
monoch
romatic w
ave of intensity I0and
wave num
berk
is:
Then, a source w
ith a spectral d
istribution I
0 ( k)in th
e
range [k1 ,k
2 ] has:
()
()(
)dk
kxk
Ix
I
kk
π2co
s1
2 121
0+
=∫
Moving th
e mirror in m
any small steps across x
m , the source spectrum
in Moving th
e mirror in m
any small steps across x
m , the source spectrum
in the frequency d
omain, I
0 ( k), can be recovered
via inverse Fourier
transform:
Finite interval
[-xm /2
, +xm /2
] resolution is d
egraded to
()
()
()
()
)(
sinc
00
kx
kI
xI
xI
FT
kI
m∗
=−
=′
00
kx
k
kR
m=
∆=
2 MG
=
Whole integration tim
e used for each
spectral element –
as compared
to
a Fabry-Perot spectrom
eter S/N gain
of
(Mis th
e number of spectral elem
ents).
This is called
the Fellgett (or m
ultiplex) ad
vantage.
Pros and Cons of th
e Diffe
rent T
ypes
Spectrom
eter
Advanta
ges
Disa
dvanta
ges
Long-slit•relatively sim
ple high
through
put•easy to calib
rate
•only one ob
ject at a time
•inefficient use
of detector
space
Echelle
•high spectral resolution
•efficient use of d
etector•challenging grating/optics
•limited
instantaneous∆λ
Integral field•instantaneous 2
D info
•ideal for resolved
objects
•com
plex optics
•single ob
jects only
Multi-ob
ject•up to th
ousands of spectra
•com
plex mech
anisms to select
Multi-ob
ject•up to th
ousands of spectra
•ideal for spectral surveys
•com
plex mech
anisms to select
fields
•fibretransm
ission limits ∆
λ•com
pact objects/regions only
Fabry-Perot
•ideal for large ob
jects•high spectral resolution
•more com
pact than F
TS
•not practical for large ∆
λ•line and
continuumobserved
at different tim
es calib
ration•need
s pre-disperser
Fourier-transform
(FTS)
•Fellgett
advantage: G
=√M/2
•good
for extend
ed sources
•requires low
RN detectors
•high
resolution wide interval
•difficult in cryo
instruments
Qualita
tive Feature
s of a Spectrum
(1)
The line profile is ch
aracterized by:
•the FWHMΔλ
•the center w
avelength or line position λ
0
•the flanks
•the sym
metry
of the flanks
•the wings
λ0
λ
λ
λ
Qualita
tive Feature
s of a Spectrum
(2)
The line intensity d
escribes th
e total power contained
within th
e line and
can be ch
aracterized by eith
er:
•by th
e continuum-sub
tracted total
line intensityI (= ab
solutemeasurem
ent)
oror•by th
e equivalent width, which
expresses th
e integrated line flux
as a rectangular w
indow of th
e continuum
strength at th
at wavelength
(= relativemeasurem
ent).
Measuring S
pectra
l Line
Inte
nsity
The m
ost common m
ethods are:
•by num
erical integration of the line profile:
•by fitting a G
aussianif th
e line profile
is determ
ined by Doppler b
roadening or th
e instrumental profile
(“unresolved line”) [see b
elow].
()
()
−
−=
2
2
0
2ex
p2 1
σ νν
σπ
νφ
G
()
[]
()
Nf
dI
Ilin
e
c=
−∫
νν
•by fitting a Lorentzian
if the line
profile is given by collisions, w
here Δ
νL =1/π
τ, with τthe m
ean time
betw
een collisions.
•by fitting a V
oigt profilewhich is a convolution of
Gaussian and
Lorentzian profile (= most general
case).
()
()
()
22
02
/2 1
L
LL
νν
νν
πν
φ∆
+−
∆=()
()
()ν
φν
φν
φL
GV
∗=
Optim
al E
xraction
Extracting th
e spectral information from
the dispersed
light on a real
detector is non-trivial:
Usually, spectral resolution elem
ents cover more th
an one pixel
the
information sh
ould be weigh
ted accord
ing to the S/N per pix
el:
where S
is the sum
med signal, B
is the background
, and Cis th
e
()
()
()
()
()
()λ
λλ
λλ
∑
∑−
⋅=
i
i i
i
i
W
BC
W
S
where S
is the sum
med signal, B
is the background
, and Cis th
e detected
signal per pixel i.
STIS spectrum
of an O star (M
assey et al. 2004):
top: standard extraction; b
ottom: optim
al extraction.