Transcript Spectroscopy – Lecture 1

Spectrographs
Literature:
Astronomical Optics, Daniel Schroeder
Astronomical Observations, Gordon Walker
Stellar Photospheres, David Gray
Spectral Resolution
dl
Consider two monochromatic
beams
They will just be resolved when
they have a wavelength
separation of dl
Resolving power:
l1
l2
l
R=
dl
dl = full width of half
maximum of calibration
lamp emission lines
R = 15.000
dl = 0.73 Å
R = 100.000
dl = 0.11 Å
R = 500.000
dl = 0.022 Å
Spectral Resolution
The resolution depends on the science:
1. Active Galaxies, Quasars, high redshift (faint) objects:
R = 500 – 1000
2. Supernova explosions:
Expansion velocities of ~ 3000 km/s
dl/l = v/c = 3000/3x105 = 0.01
R > 100
R = 30.000
R = 3.000
3. Thermal Broadening of Spectral lines:
T (K)
Dlth (Ang)
R
3000
0.028
200.000
6000
0.04
140.000
10000
0.053
100.000
30000
0.091
60.000
100000
0.160
35.000
4. Rotational Broadening:
Sp. T.
1
Vsini (km/s)
R1
A0
150
2000
F0
80
3750
F5
25
12000
G0
3
100000
K
1
300000
2 pixel resolution, no other broadening
5. Chemical Abundances:
Hot Stars: R = 30.000
Cool Stars: R = 60.000 – 100.000
Driven by the need to resolve spectral lines and
blends, and to accurately set the continuum.
6 Isotopic shifts:
Example:
Li7 : 6707.76
Li6 : 6707.92
R> 200.000
7 Line shapes (pulsations, spots, convection):
R=100.000 –200.000
Driven by the need to detect subtle distortions in the
spectral line profiles.
Line shapes due to Convection
Hot rising cell
Cool sinking lane
•The integrated line profile is distorted.
• Amplitude of distortions ≈ 10s m/s
R > 500.000
R = 200.000
8 Stellar Radial Velocities:
sRV(m/s) ~ R–3/2 (Dl)–1/2
R
100 000
60 000
30 000
10 000
1 000
s(m/s)
1
3
7
40
1200
Dl = wavelength
coverage
Spectrographs
Anamorphic magnification:
d1 = collimator diameter
d2 = mirror diameter
r = d1/d2
camera
detector
corrector
From telescope
slit
collimator
From telescope
collimator
corrector
slit
Without the grating a spectograph
is just an imaging camera
camera
detector
A spectrograph is just a camera which produces an
image of the slit at the detector. The dispersing element
produces images as a function of wavelength
slit
without
disperser
with disperser
fiber
with disperser
without
disperser
Spectrographs are characterized by their angular dispersion
Dispersing element
b
l
l + dl
db
A
=
db
dl
In collimated
light
f
dl
dl
dl
=
f
db
dl
In a convergent beam
S
dl
dl
=
S
db
dl
Plate Factor
–1
P = (f A) = (f
db
dl
–1
P = (f A) = (S
db
dl
–1
)
–1
)
P is in Angstroms/mm
P x CCD pixel size = Ang/pixel
f
da
d1
w
d2
db
A
D
h´
h
f
f1
f2
D = Diameter of telescope
f = Focal length of telescope
d1 = Diameter of collimator
f1 = Focal length of collimator
d2 = Diameter of camera
f2 = Focal length of camera
A = Dispersing element
w´
f
da
d1
w
A
D
w = slit width
h = slit height
db
w´
h´
h
f
d2
f1
f2
Entrance slit subtends an
angle f and f´on the sky:
Entrance slit subtends an angle
da and da´on the collimator:
f = w/f
da = w/f1
f´= h/f
da´= h/f1
w´ = rw(f2/f1) = rfDF2
h´ = h(f2/f1) = f´DF2
F2 = f2/d1
r = anamorphic magnification due to
dispersing element = d1/d2
w´ = rw(f2/f1) = rfDF2
This expression is important for matching slit to detector:
2D = rfDF2 for Nyquist sampling (2 pixel projection of slit).
1 CCD pixel (D) typically 15 – 20 mm
Example 1:
f = 1 arcsec, D = 2m, D= 15mm => rF2 = 3.1
Example 2:
f = 1 arcsec, D = 4m, D= 15mm => rF2 = 1.5
Example 3:
f = 0.5 arcsec, D = 10m, D= 15mm => rF2 = 1.2
Example 4:
f = 0.1 arcsec, D = 100m, D= 15mm => rF2 = 0.6
5000 A
n = –2
4000 A
5000 A
4000 A
4000 A
5000 A
n = –1
Most of
light is in
n=0
n=1
4000 A
n=2
5000 A
The Grating Equation
s
bb
f
a
ml
s =
sin a + sin bb
1/s = grooves/mm
Angular Dispersion:
m
db
= s cos b =
dl
Linear Dispersion:
sin a + sin b
l cos b
dx = fcam db
dl
dl db
=
=
dx
db dx
1
1
fcam db/dl
Angstroms/mm
Resolving Power:
dx = f2 db Dl
dl
w´ = rw(f2/f1) = rfDF2
f2
Recall: F2 = f2/d1
db
Dl = rfDF2
dl
rf
dl =
A
R = l/dl =
D
d1
For a given telescope
depends only on collimator
diameter
lA 1
r
f
d1
D
A = 1.7 x 10–3
R = 100.000
D(m)
2
4
10
10
30
30
f (arcsec)
1
1
1
0.5
0.5
0.25
d1 (cm)
10
20
52
26
77
38
Adaptive Optics corrects for the atmospheric
motion and allows one to achieve near
diffraction limit
What if adaptive optics can get us to the diffraction
limit?
Slit width is set by the diffraction limit:
f= l
D
l A D
R=
r
l
R
100000
1000000
d1
A
d1
=
D
r
d1
0.6 cm
5.8 cm
For Peak efficiency the F-ratio (Focal Length / Diameter) of the
telescope/collimator should be the same
1/F
1/F1
F1 = F
F1 > F
1/f is often called
the numerical
aperture NA
d /1
F1 < F
But R ~ d1/f
d1 smaller => f must be smaller
Normal gratings:
• ruling 600-1200 grooves/mm
• Used at low blaze angle (~10-20 degrees)
• orders m=1-3
Echelle gratings:
• ruling 32-80 grooves/mm
• Used at high blaze angle (~65 degrees)
• orders m=50-120
Both satisfy grating equation for l = 5000 A
a
b
q
d
d
q
Grating normal
Relation between blaze angle d,
grating normal, and angles of
incidence and diffraction
Littrow configuration:
m l = 2 s sin d
q = 0, a = b = d
A = 2 sin d/l
A increases
for increasing
blaze angle
R = 2d1 tan d/f D
R2 echelle, tan d = 2, d = 63.4○
R4 echelle tan d = 4, d = 76○
At blaze peak a + b = 2d
mlb = 2 s sin d cos q
lb = blaze wavelength
1200 gr/mm grating
Schematic: orders separated in the
vertical direction for clarity
m=1
l1
6000
14000
m=2
4000
m=3
9000
l2
3000
5000
You want to observe l1 in order m=1, but light l2 at order m=2,
where l1 ≠ l2 contaminates your spectra
Order blocking filters must be used
79 gr/mm grating
In reality:
Schematic: orders separated in the
vertical direction for clarity
9000
14000
m=99
m=100
m=101
5000
9000
4000
5000
2000
3000
Need interference filters but why throw away light?
Spectrographs
camera
detector
corrector
Cross disperser
From telescope
slit
collimator
dl
Free Spectral Range Dl = l/m
m-2
m-1
m
m+2
m+3
y
Dy ∞ l2
Grating cross-dispersed echelle spectrographs
Prism cross-dispersed echelle spectrographs
y
Dy ∞ l–1
Cross dispersion
grism
prism
grating
Increasing
wavelength
Dy ∞ l2 · l–1 = l
Cross dispersing elements: Pros and Cons
Prisms:
Pros:
• Good order spacing in blue
• Well packed orders (good use
of CCD area)
• Efficient
• Good for 2-4 m telescopes
Cons:
• Poor order spacing in red
• Order crowding
• Need lots of prisms for large
telescopes
Cross dispersing elements: Pros and Cons
Grating:
Pros:
• Good order spacing in red
• Only choice for high
resolution spectrographs on
large (8m) telescopes
Cons:
• Lower efficiency than prisms
(60-80%)
• Inefficient packing of orders
Cross dispersing elements: Pros and Cons
Grisms:
Pros:
• Good spacing of orders from
red to blue
Cons:
• Low efficiency (40%)
Important Data reduction issues:
1. Blaze function
2. Scattered Light
3. Reflections
•
„Picket Fence“ or reflected light for Littrow configuration
Spectrum of a White Light Source (Flat Lamp)
Picket fence:
Scattered light
Bias level
of CCD
A cross section across rows of the spectrum of the white light source
Scattered light is light that is scattered into the interorder
spacing of echelle spectrographs. All instruments have
scattered light at some level or another. This must be
removed in the reduction process. Why?
To determine the abundance of an element in the stellar spectrum
you need to measure the equivalent width
w
wl = ∫
Ic
Il
Ic – Il
Ic
dl
Width of a perfectly black
line of rectangular profile that
would remove the same
amount of flux
Id
w
I c + Is
Id + Is
Is
wl = ∫
wl =
Ic + Is – (Il +Is)
Ic + Is
∫
Ic – Il
Ic + Is
dl
dl
Scattered light reduces
equivalent width
So you want to build a spectrograph:
things to consider
•
Chose Rf product
– R is determined by the science you want to do
– f is determined by your site (i.e. seeing)
If you want high resolution you will need a narrow
slit, at a bad site this results in light losses
Major consideration: Costs, the higher R,
the more expensive
Do I need to tilt the grating to make it fit in my room?
normal
a
g
• Reflective or Refractive Camera? Is it fed with a fiber
optic?
Camera pupil is image of telescope mirror. For reflective camera:
slit
camera
Image of
Cassegrain
hole of
Telescope
detector
Camera
hole
Iumination
pattern
• Reflective or Refractive Camera? Is it fed with a fiber
optic?
Camera pupil is image of telescope mirror. For reflective camera:
A fiber scrambles the telescope pupil
camera
Image of
Cassegrain
hole
detector
Camera
hole
ilIumination
pattern
Cross-cut of illumination pattern
fiber
Lost light due
to hole in
mirror
For fiber fed spectrograph a refractive camera is the only
intelligent option
e.g. HRS Spectrograph on HET:
Mirror camera: 60.000 USD
Lens camera (choice): 1.000.000 USD
Reason: many elements
(due to color terms), anti
reflection coatings, etc.
• Stability: Mechanical and Thermal?
HARPS
HARPS: 2.000.000 Euros
Conventional: 500.000 Euros
Tricks to improve efficiency:
Overfill the Echelle
d1
R ~ d1/f
w´ ~ f/d1
d1
For the same resolution you
can increase the slit width
and increase efficiency by
10-20%
Atmospheric Seeing Blurs the Image on Slit
slit
Lost light
But…
R = l/dl =
lA 1
r
f
d1
D
You catch more photons, but a
wider slit means lower
resolution
Into this
Need to turn this
Tricks to improve efficiency:
Image slicing
The slit or fiber is often smaller than the
seeing disk:
Image slicers reformat a circular image into a line
A modern Image slicer
Fourier Transform Spectrometer
Interferogram of a monchromatic source:
I(d) = B(n)cos(2pnd)
Interferogram of a two frequency source:
I(d) = B1(n1)cos(2pn1d) + B2(n2)cos(2pn2d)
Interferogram of a two frequency source:
+∞
I(d) = S Bi(ni)cos(2pnid) =
 B(n)cos(2pnd)dn
–∞
Inteferogram is just the Fourier transform of the brightness
versus frequency, i.e spectrum
Words of Advice
If it is too good to be true it probably isn‘t
Lessons learned:
1. „The Phosphorus Stars“
2. „The Lithium Stars“
3. „The non-pulsating, pulsating A stars“
„You have to be careful that you do not fool yourself and
unfortunately, you are the easiest person to fool“
- Richard Feynman