Spectroscopy

Spectroscopy is the science of breaking the light up into its component wavelengths (or frequencies), generally through use of a dispersive element. The resolution R = λ/δλ R may range from as low as a few tens in grism spectra to millions. Spectroscopy gives a spectrum, which is a number of adjacent measures of the brightness as a function of wavelength.

Filter photometry with narrowband filters may be used to obtain higher resolutions than some kinds of spectroscopy. It consists of observing a series of single points rather than a dispersed spectrum.

Astrophysics of Spectroscopy

  • Continuua. Spectra of continuum emission can determine whether or not the emission is thermal, and if not, the nature of the spectral energy distribution (SED). Filter photometry is appropriate for many kinds of continuua, but spectra are needed to show the absence (or irrelevance) of lines.
  • Lines. Emission or absorption lines (and edges) are formed by bound-bound or bound-free electronic transitions in ions, atoms, and molecules.

    Line Broadening

  • Natural line broadening gives rise to Lorentzian-shaped lines, with width of order 10-4A.
    Lorentzians are of the general form I(Δλ) = A/(Δλ2+B), where A and B are constants.
  • Pressure Broadening is due to pertubations of energy levels by nearby atoms, by the Stark effect, Zeeman broadening, etc. The line profile is Lorentzian when thin, and develops a saturated core when thick.
  • Thermal Broadening. Due to thermal motions. Gaussian profile.
    I(λ) = I(λ0) e-mc2(Δλ)2/2kTλ2
    The half width is (2kT loge2/mc2)0.5λ0.
  • Turbulent Broadening is due to flows and convection. This includes macroturbulence and microturbulence. It produces a Gaussian profile with a half width = (V2 loge2/c2)0.5λ0.
  • Combined line profile. A sum of Gaussians is another Gaussian. A sum of a Gaussian and a Lorentzian is a Voigt profile.
  • Rotational broadening. Rotation of a uniform disk gives a profile
    I(λ) = I(λ0) (1-(c2Δλ2)/(V2λ02))0.5
    The extrema are at ± V sin i.

    Center of Mass Motions

    Double lines give away a double-lined spectroscopic binary (SB2).

    A single-lined spectroscopic binary (SB1) is given away by varying line velocities.

    Radial Velocities.

    Expansion velocities of nebulae.

    Spectroscopic Measurements

  • Equivalent Widths. The integral of
    (Fc - Fl)/Fc
    where Fc is the flux in the continuum and Fl is the flux in the line. Units are wavelength. Absorption lines are positive; emission lines are negative. The equivalent width is the width of a rectangular black line with the same area as the observed line.
  • Line widths.

    Diffraction Fundamentals

    Light passing through an aperture is diffracted.
    Light passing through 2 or more apertures interferes with itself.

  • Reflection grating geometry. Each step acts as its own aperture.


  • d=slit width
  • s=distance between apertures
  • N=number of apertures (grooves)
  • The first term represents diffraction by a single slit.
  • The second term is the interference pattern of N apertures

    Simplifying, the expression above becomes

    I(θ)/I(0) = sin2(Δ)/Δ2 sin2(Nδ)/sin2(δ).

    Let δ = mπ + P (P is the phase). In the limit that P goes to 0, I(θ)/I(0) = N2

    The fringe pattern has zeros where Nδ = m'π, where m' ≠ mN (m, m' are integers). m is the order number (m=0 is the specular reflection, or the zero-order image). The places where m' = mN are the fringe maxima. They occur at sin(θ)=(m'λ)/(Nd)

    The spectral resolution R is the distance between successive zeros, or R = 2λ/N d cos(θ).

    A grating has the unfortunate property of diffracting light into many overlapping orders. The free spectral range, the wavelength difference between two overlapping points, is sin-1(mλ1/d) = sin-1((m+1)λ2/d), or Σ = λ1 - λ2 = λ2/m.

    For small m, orders can be separated with filters (short-pass, long-pass, or order-sorting). Where m is large, as in an echelle system, care must be taken to separate the orders.

    The slit width does not degrade the resolution so long as the slit width S < λf/(Nd cos(θ)), where f is the focal length of the camera.

    Why use a slit?

    Slitless spectrographs are used for surveying emission line sources, or for emission line spectra of extended objects, such as the Sun

    Spectrographs

    Types of spectrographs:
  • Rowland Circle. Useful in the laboratory, but not compact enough for telescopes.
  • Ebert Spectrograph. Uses flat gratings.
  • Littrow Spectrograph, Uses flat gratings.
  • Wadsworth Spectrograph. Use of curved gratings permits compact design.
  • The optical layout of the HST/GHRS spectrograph.
  • The optical layout of the HST/STIS spectrograph.

  • Basic echelle design
  • Echelle closeup


  • Examples of echelle spectra. The upper spectrum is of the far UV continuum of the white dwarf/K2V binary V471 Tauri. The lower emission line spectrum is of the pre-main sequence star RU Lupi. Both spectra are from the HST/STIS using the E140M grating. Wavelength coverage is about 1100 - 1700 Angstroms.

    Spatially-Resolved Spectroscopy

    Spectroscopy classically involves observing one object at a time, and so is inefficient, as compared to imaging photometry.

    Long Slit Spectroscopy

    Spectroscopy is often done through a long slit. A long slit permits spatially-resolved spectroscopy in one dimension on the sky. Slits have lengths set by the size of the detector and the plate scale, and typically cover a few arcminutes on ground-based spectrographs. The HST/STIS long slit is 52" long in the optical (26" in the UV) in first order. To prevent order overlap, echelle spectrographs generally have very short slits (0.06" on HST/STIS; a few arcsec on the ground).

    Slits can often be rotated to line up objects in the slit (to improve observing efficiency).

    The light entering at each point along the slit gives its own spectrum.



  • Long slits allow one to do sky-subtraction, because unless you are observing a large extended object that completely fills the slit, there will be regions of blank sky. This is very useful on moonlit nights.

    Multi-object Spectroscopy

    One way to take advantage of many objects in the telescope field of view (such as a cluster of galaxies, or a star cluster) is to use optical fibers to direct the light from many targets to the detector. An example is the WIYN-HYDRA spectrograph at Kitt Peak. You can observe literally hundreds of object simultaneously. Light losses in the fibers are more than regained in the observational efficiency.

    Integral Field Spectroscopy

    The IFU is similar in concept to the fiber-fed multi-object spectrograph. In this case, one wants 2-dimensional sampling of the spectra of an extended object. A bundle of fibers covers the object, giving spatial resolutions of the projected size of the fibers. See this site for an example of how an IFU works.

    It is very important to know which spectrum (on the detector) matches back to which fiber position (on the sky).

    Calibrations

    Wavelength Calibrations

    To determine the wavelength scale, you must observe a wavelenth calibrator. This is generally an arc lamp mounted on the side of the telescope (Neon produces strong lines in the red; He-Ar or Th-Ar is used in the blue).

    Each line has a known wavelength. Measure the position of the line in pixels. This gives table of wavelengths as a function of pixel number. Fit this with an analytical function (for example, a fourth order polynomial works well for the RC spectrograph data) to determine the wavelength that corresponds to each pixel.

    He-Ar arc lamp (3800-4400 A)
    Neon arc lamp (6000-7500 A)

    It is common to linearize the spectrum - that is, to interpolate the spectrum to a linear wavelength scale.

    Spectrophotometric Calibrations

    Spectrophotometry involves determining either the absolute flux from an object, or the correct relative spectral flux distribution.

  • Observations through a slit or a fiber are generally lossy - not all the light gets through the slit or fiber (A wider slit or fiber decreases the spectral resolution and lets more background light in).
  • The grating blaze function affects the observed count rate.

    To correct for these, observe a spectrophotometric standard star. These are stars with well-determined spectral flux distributions (ergs cm-2 s-1 A-1) as a function of wavelength.
    At each wavelenth, measure the count rate (counts s-1 A-1). The ratio of the true spectral flux distribution to the observed count rate gives a wavelength-dependent conversion factor (ergs cm-2 per count).

    If the night is photometric, and the guiding is good, you can multiply all your spectra by this conversion factor to recover the true spectral flux. If the night is not photometric, you can still use this conversion factor to recover the relative flux distribution, since clouds tend to be gray absorbers.

    The spectrophotometric standard LTT 4364 before and after calibration.

    What makes a good spectrophotometric calibrator?

  • A reasonably bright star (why waste time doing calibrations?)
  • A star with a featureless continuum.