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Spectrographs
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Literature: Astronomical Optics, Daniel Schneider Astronomical Observations, Gordon Walker Stellar Photospheres, David Gray
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Spectral Resolution d 1 2 Consider two monochromatic beams They will just be resolved when they have a wavelength separation of d Resolving power: d = full width of half maximum of calibration lamp emission lines R = d
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R = 15.000 R = 100.000 R = 500.000
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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 d / = v/c = 3000/3x10 5 = 0.01 R > 100
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R = 3.000 R = 30.000
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35.0000.160100000 60.0000.09130000 100.0000.05310000 140.0000.046000 200.0000.0283000 R th (Ang) T (K) 3. Thermal Broadening of Spectral lines:
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3000001K 1000003G0 1200025F5 375080F0 2000150A0 R1R1 Vsini (km/s)Sp. T. 4. Rotational Broadening: 1 2 pixel resolution, no other broadening
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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.
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6 Isotopic shifts: Example: Li 7 : 6707.76 Li 6 : 6707.92 R> 200.000
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7 Line shapes (pulsations, spots, convection): R=100.000 –200.000 Driven by the need to detect subtle distortions in the spectral line profiles.
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Line shapes due to Convection Hot rising cell Cool sinking lane The integrated line profile is distorted. Amplitude of distortions ≈ 10s m/s
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R = 200.000 R > 500.000
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8 Stellar Radial Velocities: RV (m/s) ~ R –3/2 ( ) –1/2 wavelength coverage R (m/s) 100 000 1 60 000 3 30 000 7 10 000 40 1 000 1200
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collimator Spectrographs slit camera detector corrector From telescope Anamorphic magnification: d 1 = collimator diameter d 2 = mirror diameter r = d 1 /d 2
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slit camera detector corrector From telescope collimator Without the grating a spectograph is just an imaging camera
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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 without disperser with disperser slit fiber
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Spectrographs are characterized by their angular dispersion dd d Dispersing element dd d A =
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f dldl dd d dldl d = f In collimated light
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S dd d dldl d = S In a convergent beam
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Plate Factor P = ( f A ) –1 = ( f ) dd d P = ( f A ) –1 = ( S ) dd d P is in Angstroms/mm P x CCD pixel size = Ang/pixel
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w h f1f1 d1d1 A D f d2d2 w´w´ h´h´ D = Diameter of telescope d 1 = Diameter of collimator d 2 = Diameter of camera f = Focal length of telescope f 1 = Focal length of collimator f 2 = Focal length of collimator A = Dispersing element f2f2
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w h f1f1 d1d1 A D d2d2 f w´w´ h´h´ f2f2 w = slit width h = slit height Entrance slit subtends an angle and ´ on the sky: = w/f ´ = h/f Entrance slit subtends an angle and ´ on the collimator: = w/f 1 ´ = h/f 1
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w ´ = rw(f 2 /f 1 ) = r DF 2 h ´ = h(f 2 /f 1 ) = ´ DF 2 F 2 = f 2 /d 1 r = anamorphic magnification due to dispersing element = d 1 /d 2
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w ´ = rw(f 2 /f 1 ) = r DF 2 This expression is important for matching slit to detector: 2 = r DF 2 for Nyquist sampling (2 pixel projection of slit). 1 CCD pixel ( ) typically 15 – 20 m Example 1: = 1 arcsec, D = 2m, = 15 m => rF 2 = 3.1 Example 2: = 1 arcsec, D = 4m, = 15 m => rF 2 = 1.5 Example 3: = 0.5 arcsec, D = 10m, = 15 m => rF 2 = 1.2 Example 4: = 0.1 arcsec, D = 100m, = 15 m => rF 2 = 0.6
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5000 A 4000 A n = –1 5000 A 4000 A n = –2 4000 A 5000 A n = 2 4000 A 5000 A n = 1 Most of light is in n=0
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bb The Grating Equation m = sin + sin b 1/ = grooves/mm
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dd d = m cos = sin + sin cos Angular Dispersion: Linear Dispersion: d dx d dd = dd = 1 f cam 1 d /d dx = f cam d Angstroms/mm
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Resolving Power: w ´ = rw(f 2 /f 1 ) = r DF 2 dx = f 2 dd d f 2 dd d r DF 2 R = / d = A r 1 d1d1 D = rr A D d1d1 For a given telescope depends only on collimator diameter Recall: F 2 = f 2 /d 1
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D(m) (arcsec) d 1 (cm) 2110 4120 10152 100.526 300.577 300.2538 R = 100.000A = 1.7 x 10 –3
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What if adaptive optics can get us to the diffraction limit? Slit width is set by the diffraction limit: = D R = r A D d1d1 D = A r d1d1 R d1d1 100000 0.6 cm 1000000 5.8 cm
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For Peak efficiency the F-ratio (Focal Length / Diameter) of the telescope/collimator should be the same collimator 1/F 1/F 1 F 1 = F F 1 > F 1/f is often called the numerical aperture NA
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F 1 < F d/1d/1 But R ~ d 1 / d 1 smaller => must be smaller
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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 = 5000 A
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Grating normal Relation between blaze angle , grating normal, and angles of incidence and diffraction Littrow configuration: = 0, = = m = 2 sin A = 2 sin R = 2d 1 tan D A increases for increasing blaze angle R2 echelle, tan = 2, = 63.4 ○ R4 echelle tan = 4, = 76 ○ At blaze peak + = 2 m b = 2 sin cos b = blaze wavelength
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3000 m=3 5000 m=2 40009000 m=1 6000 14000 Schematic: orders separated in the vertical direction for clarity 1200 gr/mm grating 2 1 You want to observe 1 in order m=1, but light 2 at order m=2, where 1 ≠ 2 contaminates your spectra Order blocking filters must be used
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4000 m=99 m=100 m=101 5000 9000 14000 Schematic: orders separated in the vertical direction for clarity 79 gr/mm grating 30002000 Need interference filters but why throw away light? In reality:
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collimator Spectrographs slit camera detector corrector From telescope Cross disperser
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yy ∞ 2 y m-2 m-1 m m+2 m+3 Free Spectral Range m Grating cross-dispersed echelle spectrographs
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Prism cross-dispersed echelle spectrographs yy ∞ –1 y
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Cross dispersion yy ∞ · –1 = Increasing wavelength grating prism grism
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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
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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
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Cross dispersing elements: Pros and Cons Grisms: Pros: Good spacing of orders from red to blue Cons: Low efficiency (40%)
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So you want to build a spectrograph: things to consider Chose R product – R is determined by the science you want to do – 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
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Chose and , choice depends on – Efficiency – Space constraints – „Picket Fence“ for Littrow configuration
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normal
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White Pupil design? – Efficiency – Costs, you require an extra mirror
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Tricks to improve efficiency: White Pupil Spectrograph echelle Mirror 1 Mirror 2 Cross disperser slit
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Reflective or Refractive Camera? Is it fed with a fiber optic? Camera pupil is image of telescope mirror. For reflective camera: Image of Cassegrain hole of Telescope camera detector slit Camera hole Iumination pattern
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Reflective or Refractive Camera? Is it fed with a fiber optic? Camera pupil is image of telescope mirror. For reflective camera: Image of Cassegrain hole camera detector A fiber scrambles the telescope pupil Camera hole ilIumination pattern
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Cross-cut of illumination pattern For fiber fed spectrograph a refractive camera is the only intelligent option fiber 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. Lost light
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Stability: Mechanical and Thermal? HARPS HARPS: 2.000.000 Euros Conventional: 500.000 Euros
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Tricks to improve efficiency: Overfill the Echelle d1d1 d1d1 R ~ d 1 / w´ ~ /d 1 For the same resolution you can increase the slit width and increase efficiency by 10-20%
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Tricks to improve efficiency: Immersed gratings Increases resolution by factor of n n Allows the length of the illuminated grating to increase yet keeping d 1, d 2, small
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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
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Fourier Transform Spectrometer
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Interferogram of a monchromatic source: I( ) = B( )cos(2 n )
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Interferogram of a two frequency source: I( ) = B 1 ( )cos(2 1 ) + B 2 ( 2 )cos(2 2 )
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Interferogram of a two frequency source: I( ) = B i ( i )cos(2 i ) = B( )cos(2 )d –∞ +∞ Inteferogram is just the Fourier transform of the brightness versus frequency, i.e spectrum
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