Spectroscopy for Astronomy

Spectrum: intensity of radiation as a function of wavelength (“dispersed light”)Continuum (e.g., blackbody radiation) and Line Emission

Continuum occurs at all wavelengths: e.g., Bλ(T)

Line emission occurs at specific wavelength: λ = hc/(E2-E1)

Radiation: Continuum & Line Emission

Continuum emission:

Blackbody Radiation (e.g., stellar radiation)

Synchrotron Radiation (e.g., accelerator)

Thermal Free-Free Radiation (= Bremsstrahlung)

Line emission:

Atomic Transition (e.g., H I lines)

Molecular Transition (e.g., OH, CO lines)

Solid-State Feature (e.g., aerosol, …)

Line Emission/Absorption: e.g., Hydrogen

H series (mostly in the visible bands)

e.g. H transition: n = 3 → 2 transition at 656.3 nm.

R: Rydberg constant for hydrogen.

Observed stellar radiation = continuum + line emission (mostly absorption)

Spectrum: intensity of radiation as a function of wavelength (“dispersed light”)Continuum (e.g., blackbody radiation) and Line Emission

Galaxy Spectra: examples

Galaxy: numerous stars and gas clouds

Mixture of continua, absorption, and emission lines

Spectrum: Example for absorption and emission

(in addition to the continuum)

(no continuum)

(no line emission)

Assumptions:

[1] hot source is a pure continuum emission source;

[2] gas is a pure line emission source.

Star: let’s assume it to be a pure continuum source

Gas cloudContinuum source through gas cloud

Colored bars: gas cloud emission lines

Continuum source

Assumptions:

[1] Star is a pure continuum source.

[2] Gas cloud has no continuum emission.

White bars: gas cloud absorption lines

Depending on the relative positions of the sources and observers, spectra appear differently.

Spectrum: Example for absorption and emission

Stellar Spectra

Can you identify any lines?

Temperature (= spectral type) is critical to determine the shape of the stellar spectra.

Stellar Spectra

Stellar Spectra

How do we obtain spectra?

We need to disperse the light. How?

We need a dispersing element.What are the two common dispersing elements?

Spectrograph and Spectroscopy: Dispersing Elements

Prism: based on the wavelength dependence of the index of refraction n()

Grating (= a plate with many grooves): based on the wavelength dependence of the diffraction & interference

Basic Spectrograph Structure

Prism (or Grating)

Slit

Collimator

Dispersing Element

Camera

Detector

The slit spectroscopy is basically imaging the slit along the wavelength. Think about this.

Photons of different wavelengths arrive at the different positions on the detector (e.g., CCD, photographic plate).

So you can tell the wavelengths of the photons.

The slit spectroscopy is basically imaging the slit along the wavelength. Think about this.

Basic Spectrograph Structure

Spectrograph: Prism Spectroscopy

Where are the foci of the lenses?

Spectrograph: Diffraction Grating

Let’s recall the diffraction pattern of a telescope first to refresh …

Single slit

Double slitsDouble slit diffraction pattern: interference of two single slit diffraction complicated, narrower!

Diffraction grating = assembly of numerous slits (=ruled grooves)

Spectrograph: Diffraction Grating

Three slits

Four slits

Slit number more complicated interference pattern

Use the interference to distinguish the wavelength!

Spectrograph: Diffraction Grating

m: order of the principal maxima

Interference of diffractions from multi-slits

Different wavelengths have their maxima at different locations for the same orders!

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Note that the pattern of only two different wavelengths are shown!

This corresponds to the reflection.

Spectrograph: Diffraction Grating

Diffraction Grating (ruled grooves): Multi-slit Interference

IntensityDiffraction Pattern

Interference Pattern

N: number of slits

a: half slit size

d: slit distance

Spectrograph: Diffraction Grating

Wavelength dependence

Transmission grating

Reflection grating

Spectrograph: Diffraction Grating

Diffraction Gratings:Ruled Grooves (= multi slits)

Spectrograph: Diffraction Grating

Grating geometry

transmission

reflection

: incidence angle

b: diffraction angle

: wavelength

q: blaze angle

s: groove spacing

, b: opposite signs at opposite sides

transmission

reflection

Path difference between the incident wavefront and diffracted wavefront? = s (sin + sin b) = mfor constructive interference, m is integer

s

Grating geometry

Grating equation:s (sin + sin b ) = m, m = integer

transmission

reflection

Grating geometry

Grating equation:s (sin + sin b) = m, m = integer

m = 0: = -b (Secular reflection)m > 0: - < bm < 0: - > b

Diffraction gratings

For m = 0, all wavelengths fall at the same place, and thegrating acts as a simple mirror.

Typically higher orders will give weaker spectra: most ofthe power is put into the lower orders.

The separation of wavelengths is greater at higher orders:spectral resolving power increases at higher orders.

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• A grating diffracts light into many orders; one order contains only a fraction of the light

• Rule the grating facets so that the direction of reflection off the facet coincides with the desired order of diffraction.

• Up to 90% of the light can be concentrated into the desired spectral order.

• Most spectrographs use reflection gratings: easier to produce and blaze

“Blaze” a grating

Sawtooth profile (= blazed grating) for high grating efficiency for certain wavelengths and orders

Grating: Angular Dispersion

s

s (sin + sinb) = m

sin b = m/s – sin

A (angular dispersion)

= db/d

= m/(s cosb)

= (sin + sinb) / ( cosb)

Angular dependence of the dispersed light along the wavelength.(db/d)

Grating: Angular Dispersion

s (sin + sinb) = m

A (angular dispersion)

= db/d

= m/(s cosb)

= (sin + sinb) / ( cosb)

: incidence angle

b: diffraction angle

: wavelength

q: blaze angle

s: groove spacing

Angular dispersion is a function of m and s. High groove density ( 1/s) and high spectral order (= m) produce high spectral density (= large angular dispersion).

For given s and , large m is achieved with large and b.

is usually fixed.

How photons of different wavelengths will disperse on the camera focal plane (= detector)?

Linear Dispersion = (Angular Dispersion = A) ×(Camera Focal Length = f2)

(Δff/Δ)?

Focal length of the camera

Grating: Linear Dispersion

Slit at the collimator focal plane

Detector at the camera focal plane

Large linear dispersion → high spectral resolving power; coarse velocity measurement

Small linear dispersion → low spectral resolving power; precise velocity measurement

Grating: Linear Dispersion

So photons of different wavelengths will appear on the different pixels of the detector. “Spectrum”

= reciprocal linear dispersion(or plate factor)

Spectroscopy: Slit Imaging

Spectroscopy: Slit Imaging

White (non-dispersed) light

Colored (dispersed) light

Spectroscopy: Slit Imaging

Spectroscopy: Slit Imaging

Spectroscopy: Slit Imaging

2D Detector

Spectral Direction (= Dispersion Direction)

Spat

ial D

ire

ctio

n

Slit image on the detector along the wavelength

Linear dispersion

Slit spectroscopy is imaging the input slit on the detector along the wavelength

Narrow slits give high spectral resolving power

Reimaging Optics

Input image plane Re-imaged

plane

Collimator lens

Camera lens

Collimator and camera pair forms a reimaging system.

Magnification = (Re-imaged size) / (Input size)

= (Camera focal length) / (Collimator focal length)

= fcam / fcoll

Spectroscopy: Slit Imaging

fcoll

fcam

Projected Slit Width

= [Input Slit Width] × [Grating Magnification] × [fcam / fcoll]

r = (collimated

diffracted beam) /

(collimated incident

beam)

= cosβ/cosα 1

Anamorphic magnification

Spectroscopy: Spectral Resolving Power

R is spectral resolving power;

δ is the spectral resolution element (= projected slit width at the wavelength)

Grating Resolving Power and Rayleigh Criteria

The images formed in two wavelengths that are barely resolved must be separated by q. The light of wavelength + must form its principal maximum of order m at the same angle as that for the first minimum of wavelength in that order.

Nd cosq (= projected grating size) determines the angular separation.

Grating Resolving Power and Rayleigh Criteria

Spectroscopic Resolving Power

Spectrograph: Example

• b from the grating equation (b=21.644°, 23.132°; b=1.488°)

• camera plate scale = 1.75 mm/degree

• linear distance between the 3800 and 4000 Å lines = 1.4881.75 = 2.603mm

• reciprocal dispersion (= plate factor) = 200 Å/2.603 mm (or 76.8 Å/mm)

• focal length ratio = 10.16/39.4 = 0.258; projected slit width = 0.010 mm or 0.77 Å

• CCD of 1024 15-μm pixels: 1 pixel resolution = 0.015 × 76.8 = 1.15 Å/pixel and the wavelength coverage is 1.15 × 1024 = 1180 Å

What will be the plate factor and projected slit width? ( = 5°, = 3800-4000 Å, m = 2; 2 pixels per resolution element)