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.