Learning Objectives u Spectrometers or Spectrographs objective prism

Learning Objectives u Spectrometers or Spectrographs: - objective prism - slit spectroscopy - diffraction grating - line grating - blazed grating - spectral resolution - echelle grating - multi-object spectrographs - integral field spectrographs u Spectral Line Profiles and Shifts: - natural broadening - pressure broadening - Doppler effect - thermal or Doppler broadening

Measuring Spectra u There a number of ways in which the spectra of celestial objects can be measured. u One way is to place a prism just in front of the telescope objective, known as an objective prism.

Measuring Spectra u An example of objective prism spectra of the Hyades open star cluster. u What are the advantages and disadvantages of objective prism surveys? - spectra of all objects can be measured in a single exposure - spectra of different objects can overlap u In this course, we consider spectrographs using diffraction gratings.

Spectrograph u Basic components of a spectrograph are: - entrance aperture, which can be a slit or circular apertures (optical fibers) - collimator, which is an optical element to make light parallel and hence have the same angle of incidence over the surface of the dispersing element - dispersing element (e. g. , prism), which separates light to different wavelengths - spectrum imager, which focuses the spectrum onto a detector - detector, which in modern times is a CCD

Spectrograph

Slit Spectroscopy u If you are measuring the spectrum of one object that is unresolved or in its entirety, it would not matter what shape aperture you use so long as the aperture only lets through light from that object. F ane l P l oca

Slit Spectroscopy What if you want to measure the spectrum as a function of position in an extended object? One way is to use a slit, which of course restricts the spectrum to just locations along the slit orientation position u λ

Diffraction Gratings u A diffraction grating is a collection of equally spaced transmitting or reflecting elements (grooves) separated by a distance comparable to the wavelength of light. u A simple diffraction grating consists of lines (opaque to light) scratched onto a glass surface, so that only the smooth surface transmits (transmission grating) or reflects (reflection grating) light.

Line Gratings u A simple transmission (line-diffraction) grating. Only 2 grating apertures are shown in this example; in practice, there can be many thousands of grating apertures. u Maxima occur when the path difference between adjacent rays, d sin θ, is an integer multiple of the wavelength, λ, of light. m=2 m=1 m=0 m = -1 m = -2 mλ (m – ½ ) λ (m (m

Line Gratings u m is known as the order of the maxima in the diffracted pattern. E. g. , m = 0 is the zeroth order, |m| = 1 is the first order, etc. u Disadvantage of this type of grating is that much of the light falls at the zeroth order where no spectrum is produced, and the light intensity drops quickly towards higher orders. Note also that higher order maxima are more closely separated, so that high order maxima can overlap. m=2 m=1 m=0 m = -1 m = -2 mλ (m – ½ ) λ (m (m

Line Gratings u m is known as the order of the maxima in the diffracted pattern. E. g. , m = 0 is the zeroth order, |m| = 1 is the first order, etc. u Disadvantage of this type of grating is that much of the light falls at the zeroth order where no spectrum is produced, and the light intensity drops quickly towards higher orders. Note also that higher order maxima are more closely separated, so that high order maxima can overlap. m=3 m=2 m=1 m=0 m = -1 m = -2 m = -3

Line Gratings u A simple reflection (line-diffraction) grating. u Sign convention: angles on the side of the incident ray relative to the grating normal are defined to be positive, whereas angles on the side of the reflected ray relative to the grating normal are defined to be negative. u Maxima occur when the path difference between adjacent rays, (d sin θi + d sin θm), is an integer multiple of the wavelength, λ, of light; i. e. , d sin θi + d sin θm = mλ. d

Line Gratings u Is the any advantage of this design over a transmission grating? Hint: law of reflection states that -θm = θi (recall sign convention); at which order is most of the light concentrated? d

Blazed Gratings u In modern diffraction gratings, grooves are cut into the surface to give a regular geometrical shape. Such gratings are called blazed gratings, and have the advantage of causing most of the diffracted light energy to be concentrated in a certain direction away from the zeroth order.

Blazed Gratings u Definition of groove spacing and angles in a blazed diffraction grating. Note that most of light is now reflected not to m = 0 but instead to a higher m order.

Blazed Gratings u Ray 2 has to travel an extra distance d sin θi compared to ray 1 to reach their respective grooves. u On reflection, ray 2 travels a lesser distance d sin (θr) compared to ray 1. u Path difference between rays 2 and 1 is d sin θi + d sin (θr) = mλ for constructive interference. (Recall that θr is negative by the sign convention. ) θi θr

Blazed Gratings u d sin θi + d sin (θr) = mλ = 0 when θr = -θi. This is the zeroth order where no spectrum is produced. In a blazed grating, relatively little light is diffracted into the zeroth order. u Illuminate grating such that the angle of incidence θi and the blaze angle together gives rise to the highest light intensity at the desired order m. θi θr

Blazed Gratings u At progressively higher orders m, angular separation between adjacent orders decreases so that maxima at different wavelengths can overlap. u Use filter to restrict input wavelength range and thus prevent order overlap.

Blazed Gratings u I will leave it as an exercise for you to show that the same mathematical relationship applies to a transmission (blazed) grating.

Blazed Gratings u Path difference between rays 2 and 1 is d sin θi + d sin (θr) = mλ for constructive interference. (Recall that θr is negative by the sign convention. ) u Path difference between rays 2 and 1 is d sin θi + d sin (θr) = m(λ/2) for destructive interference. u In between these extremes, we get partially constructive interference θi θr

Line Gratings u Maxima occur when the path difference between adjacent rays, d sin θ, is an integer multiple of the wavelength, λ, of light. u Minima occur when the path difference between adjacent rays, d sin θ, is an integer multiple of a half wavelength, λ/2, of light. u In between these extremes, we get partially constructive interference. m=2 m=1 m=0 m = -1 m = -2 mλ (m – ½ ) λ (m (m

Spectral Resolution u Even if incident spectral line has no width in wavelength, observed spectral line has a finite width in wavelength owing to properties of spectrograph. u Analogous to Rayleigh’s criterion for angular resolution, Rayleigh’s criterion for spectral resolution is defined as the minimum wavelength separation, Δλ, at the same order, m, such that an interference maximum at wavelength λ + Δλ falls on an interference minimum at wavelength λ. m=2 m=1 m=0 m = -1 m = -2

Dispersion and Resolving Power u The ability of a grating to produce spectra that permit precise measurement of wavelengths is determined by two intrinsic properties of the grating: - the width (or sharpness) of the lines, quantified as the resolving power of a grating R = / - the separation between spectral lines that differ by a small amount , quantified as the dispersion of a grating D = / θo+Δθo

Dispersion u u u To derive the dispersion, D = / , of a spectrograph, recall that maxima occur at: d sin θ = mλ (m = 0, 1, 2, …) Differentiating with and as variables gives d cos d = m d or d cos = m The dispersion D = / = m / (d cos ) With larger D implying higher dispersions, a spectrograph’s dispersion therefore depends on the separation, d, between slits and the order, m, used. It is independent of the number of rulings, N. In practice, there is a limit to how small d can be, and how large an order, m, that can be used due to overlap between adjacent orders.

Resolving Power u u Recall that for principal maxima, require d sin = m between adjacent slits, and therefore N d sin = Nm between first and last slit (altogether N + 1 slits). Thus for the minima just beyond the principal maxima, require N d sin( + ) = Nm + (m = 0, 1, 2, …) Notice that, for larger N and m, is smaller.

Resolving Power u Recall that for principal maxima, require d sin = m between adjacent slits, and therefore N d sin = Nm between first and last slit (altogether N + 1 slits). u Thus for the minima just beyond the principal maxima, require N d sin( + ) = Nm + (m = 0, 1, 2, …) u Expanding the sin term, sin( + ) = sin cos + cos sin = sin + cos for small angles u Solving or for , recalling that d sin = m , gives = / ( N d cos )

Resolving Power u Substituting the expression for the dispersion of a spectrograph (implicit in this substitution is that the principal maxima of one wavelength falls on a minima of an adjacent wavelength, known as Rayleigh’s resolution criterion) D = / = m / d cos gives the resolving power R = / = N m Larger R implies higher resolving power. Thus, the resolving power R = / of a spectrograph depends only on the number of rulings, N, and order used, m. (It is independent of the separation between rulings, d, although for gratings of equal dimensions smaller separations between rulings provide greater total number of rulings. ) θo+Δθo

Spectral Resolution u Using Rayleigh’s criterion for angular resolution, the resolving power of a diffraction grating (irrespective of its exact design) R = λ/Δλ = m. N, where m is the order and N the number of grating apertures. u To get higher spectral resolutions, use more grating apertures (there is practical limit) and higher order m (susceptible to order overlap; alternatively, restricted wavelength range of spectrum). u Recall that d sin θi + d sin (θr) = mλ. To use very high orders m, need very large θi and θr (close to 90 o). θr

Echelle Grating u An echelle grating is a particular type of blazed grating that has a very large blaze angle to allow very large θi and θr to be used, and hence achieve large R. u Recall that large m is suspectible to order overlap. To separate out different orders, a cross disperser (another grating) acts perpendicular to the echelle grating dispersion direction.

Echelle Grating

Echelle Grating u Spectrum of the Sun produced by an echelle spectrum.

Multi-Object Spectrographs u In modern astronomy, we are often interested in taking spectra of hundreds to thousands of objects (e. g. , stars, galaxies) at the same time. u We use a multi-object spectrograph, which directs light from the focal plane of the telescope using optical fibers (having circular apertures) to spectrographs. u E. g. , the LAMOST multi-object spectrograph employs 4000 fibers at the focal plane that can be individually moved, and a bank of sixteen spectrographs.

Integral Field Spectrographs u What if you want to measure the spectrum of an extended object at all positions at the same time? Use an integral field spectrograph. u Several designs have been employed. A simple design is a matrix of optical fibers analogous to a multi-object spectrograph. u E. g. , PPAK on the 3. 5 -m telescope on Calar Alto has 331 target fibers surrounded by 36 sky fibers.

Learning Objectives u Spectrometers or Spectrographs: - prism - diffraction grating - line grating - blazed grating - spectral resolution - echelle grating - multi-object spectrographs - integral field spectrographs u Spectral Line Profiles and Shifts: - natural broadening - pressure broadening - Doppler effect - thermal or Doppler broadening

Natural Broadening u The Heisenberg uncertainty principle dictates that we cannot know the position and velocity of particles (objects) to arbitrarily high precision (foundation of quantum physics). u Instead of orbits being at precise radii from the nucleus, orbits are “fuzzy; ” i. e. , decreasing probability of finding an electron away from the most probable radius. Precise orbits Fuzzy orbits

Natural Broadening u Transitions between the same two energy levels in different atoms can produce photons with slightly different energies.

Natural Broadening u Transitions between the same two energy levels in different atoms can produce photons with slightly different energies. u The resultant is that spectral lines are not infinitely sharp in wavelength or frequency, but has a spread in wavelength/frequency described by a Lorentzian profile.

Pressure Broadening u In describing atoms/ions/molecules so far, we have not considered electrostatic forces between adjacent atoms/ions/molecules. Such forces perturb electron orbits. The resulting broadening of spectral lines is known as pressure broadening. u Pressure broadening also gives rise to a Lorentzian profile. u Pressure broadening can be important in stellar atmospheres (e. g. , mainsequence stars), but typically not important in the interstellar medium.

Doppler Effect u Doppler effect of sound due to an approaching or receding source can be expressed in terms of: 1) the speed of the source relative to the wave-carrying medium (air); and 2) the speed of the observer relative to the wave-carrying medium. u Equation for Doppler effect of sound: where ν is the speed of sound waves in air, νs is the speed of the source relative to air (+ away from observer), and νo is the speed of the observer relative to air (+ towards source).

Doppler Effect u The speed of light is a constant, c, for all observers. Doppler effect of light is cased by a combination of: 1) geometrical effects (just like the Doppler effect of sound); and 2) time dilation (an effect of special relativity). u Equation for Doppler effect of light: where ν is the speed of the source relative to the observer (+ away from observer). u Because of the expansion of the Universe, distant galaxies are receding away from us. The recession velocity of galaxy is expressed in terms of redshift, z:

Thermal or Doppler Broadening u Motion of particles due to their thermal energy result in collisions that produce a particular distribution of speeds. More particles are traveling at lower than higher speeds. Furthermore, roughly equal number of particles are traveling towards as away from us.

Thermal or Doppler Broadening u The distribution of speeds along the line of sight is a Gaussian (or normal) distribution, giving rise to a spectrum with a Gaussian profile. u Full-width half-maximum (FWHM) of thermal or Doppler broadening:

Thermal or Doppler Broadening u The addition of a Lorentzian to a Gaussian profile is a Voigt profile. u In general, spectral lines have Voigt profiles.

Learning Objectives u Slit Spectroscopy: - setup and observations - data reduction - extracting spectrum - wavelength calibration - extinction correction - flux calibration - removing sky lines

Slit Spectroscopy u Steps for doing slit spectroscopy using a CCD: 1. Center the object within slit and take spectrum. 2. Reduce CCD frames with bias, dark, and flat (previously covered) 3. Extract spectrum of object 4. Take a spectrum of the standard lamp for wavelength calibration 5. Correct for Earth’s atmospheric extinction and interstellar extinction (previously covered) 6. Flux density calibration with standard stars 7. Remove sky lines.

Setup and Exposure 1. Center the object within slit and take spectrum. u For a point source (e. g. , a star), a slit of width smaller than the FWHM of the seeing is chosen (i. e. , the narrower the better). u What is the disadvantage of using such a narrow slit?

Setup and Exposure 1. Center the object within slit and take spectrum. u For a point source (e. g. , a star), a slit of width smaller than the FWHM of the seeing is chosen (i. e. , the narrower the better). u What is the advantage of using such a narrow slit?

Setup and Exposure 1. Also take spectrum of standard lamp for wavelength calibration. u In optical astronomy, arc lamps containing gas of one or more elements (e. g. , Hg, C, Zn, Th. AR) are usually used for wavelength calibration. The gas is ionized by an electrical arc, and produces a rich spectrum.

Setup and Exposure 1. Also take spectrum of standard lamp for wavelength calibration. u In optical astronomy, arc lamps containing gas of one or more elements (e. g. , Hg, C, Zn, Th. AR) are usually used for wavelength calibration. The gas is ionized by an electrical arc, and produces a rich spectrum.

Setup and Exposure 1. Also take spectrum of standard lamp for wavelength calibration. u Of course, the wavelength of each line from the standard lamp has previously been measured in the laboratory to high precision.

Setup and Exposure 1. Finally, also observe spectrophotometric standard star for flux density calibration. u Spectrophotometric standards are bright stars with accurate measurements of their spectra: flux density as a function of wavelength HR 9087

Spectral Extraction 3. Extract spectrum of object. u The spectrum of a point source recorded on CCD is a strip extending along the dispersion/spectral direction. Ideally the strip should be a rectangle aligned with the CCD rows/columns, but in practice the strip is curved depending on optics. u Obtain spectrum of the object by integrating few pixels near source along the spatial direction (over gwidth) and subtracting background. Need to correct for “missing light” arising from integrating only a finite width (similar to aperture correction in photometry). Spatial direction u CCD frame Dispersion/Spectral direction

Wavelength Calibration 4. Use standard lamp spectrum for wavelength calibration. u After extracting a spectrum, we need to put the data on a wavelength scale. u From the standard lamp spectrum, create a mapping of the spectral direction pixel number and wavelength. u A good diffraction grating will generate an almost linear mapping. Arc lamp frame Target frame

Flux Density Calibration 6. Flux density calibration. u At this point, spectrum is in units of photon-counts sec-1 Å-1. u Compare with photon-counts sec-1 Å-1 from spectrophotometric standard to convert to ergs-1 sec-1 cm 2 Å-1. Standard star frame Target frame

Sky Lines 7. Remove sky lines (including lines from artificial man-made source). u Apart from spectral lines from target star in the slit, spectral lines produced by Earth’s atmosphere, or scattered by the Earth’s atmosphere from man-made light sources, also are imprinted on the spectrum of the source. u Flux density of sky lines generally vary in position and with time. u Removed by fitting a profile to the sky lines in the background spectrum and subtracting from the target spectrum, or as a final resort blanking out affected parts of the target spectrum. From artificial light source