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ASTR 5110, Majewski [FALL 2013]. Lecture Notes

ASTR 5110 (Majewski) Lecture Notes


PHOTOMETRIC DATA REDUCTION:

FROM INSTRUMENTAL TO TRUE MAGNITUDES

References:

  • Bely, The Design and Construction of Large Optical Telescopes, Section 1.4.

  • Lena, Observational Astrophysics, Section 2.3.

  • Buil, CCD Astronomy, Section 5.3.

  • Kitchin, Astrophysical Techniques, Section 3.2.

  • Birney, Chapter 9.

  • Harris et al. 1981, PASP, 93, 507).
  • Mike Bolte's notes: http://www.ucolick.org/~bolte/AY257/ay257_2.pdf.


ATMOSPHERIC BACKGROUND, ABSORPTION AND SCATTERING

Simply counting photons in apertures, or doing profile fitting, is not the end of the story when it comes to doing photometry.

In general, when we deal with photometric data (whether single aperture or two-dimensional) there are several steps we must undertake before we obtain true apparent magnitudes:

Steps:
  1. Subtraction of background sources (including sky flux and dark current).
  2. Correction for atmospheric extinction.
  3. Reduction to a standard system (account for differences in your effective filter system compared to the standard one).
Note that if we are doing relative photometry -- measuring magnitude differences -- we do not usually need to worry about (2) because it will have more subtle effects, and, under some circumstances, you can also ignore (3).


1. Background Subtraction

We have already discussed this on the previous page, but just as a reminder, we need always to remember that our measurements of star flux also include contributions of other "flux" from both the sky as well as the dark current of the camera).

Thus, with either single aperture or two-dimensional photometry we must gauge the per pixel per second contributions from these two sources and subtract the appropriate amounts of these backgrounds from our star flux measurements.

To do this correctly, we must:

  • Measure flux in areas of sky that are free of stars, galaxies, etc.
    • For a single aperture photometer this means periodically nodding the telescope to "blank sky" positions.
    • For a CCD image, sample "clear" parts of the image.
  • Subtract from star flux measures the contribution of dark count and sky background, scaling appropriately for the number of pixels and the integration time.

    Best to do this subtraction in flux (i.e., counts), NOT magnitudes, to make sure you do not get confused by the logarithm.

Sources of Sky Background

"Background sources" are those that contribute to the detected light but do not originate in the source we care about, and include natural sources in the sky, atmospheric emission, thermal emission from your environment and instruments, and side effects from the detector.

A schematic layout of various background sources. From Bely.
Sources of sky background include:

  • moonlight -- very important in the optical

      Note that at professional observatories it is common to divide the available observing nights into categories based on the amount of moonlight.

      Each observatory has its own definitions, but a typical classification is:

    • dark time -- time when moon is less than 1/4 full (within about 3-4 days of New Moon).

      Most valued time for working on very faint objects where signal can easily be swamped by moonlight background. Darkest time generally awarded to most competitive (dark time) observing proposals.
    • bright time -- when the moon is more than half full (i.e., time within about a week of Full Moon).

      Worst time to work on faint object photometry because of high noise from bright sky background.

      At bright time of the month usually schedule either: (1) work on bright objects, particularly spectroscopy, or (2) infrared observations, which are not as affected by the moonlight compared to optical.

      Note, sometimes beneficial to request less desirable, bright time because less demand, easier to get.
    • grey time -- the rest of the month, time near First or Third Quarter Moon.
    At Apache Point Observatory (and some other sites now), the NIGHTS are divided into halves in order that lunar rise/set times can be accounted for in the awarding of telescope time.

    • This effectively makes more dark time available, since first/third quarter nights when moon rises/sets at midnight can have at least 1/2 of the night counted as "dark".

    The following table shows the dramatic changes in the brightness of the sky by moon phase at an otherwise dark, high altitude site:

    Note:

    • "optical regime" -- below 1 micron highly affected by moonlight.

      Table does not account for brightening of sky as angle to moon decreases.

    • "nonthermal infrared regime" -- from 1-2.5 microns, is dominated by narrow OH emission lines (see below).

    • "thermal regime" -- beyond about 2.5 microns background dominated by thermal emission from the Earth's atmosphere (see below).

  • airglow (emission from excited molecules in atmosphere)...

    Earth atmosphere has a circadian rhythm due to pattern of ionization/recombination and excitation/de-excitation of molecules/atoms/ions in atmosphere.

    • Beyond 60 km altitude, we get increasing dissociation of molecules (e.g., O2, N2, H2O) and excitation of both radicals (O2, OH) and atoms (O I, Na I, H I) due to solar UV photons in the daytime.

      ("Ionosphere" is ~100-500 km above the Earth.)

      From Roach & Gordon, "Light of the Night Sky".
    • At >100 km altitude, the density is low enough that radiative de-excitation can dominate over collisional de-excitation.

    • But these excited oxygen/nitrogen ions have small de-excitation probabilities, metastable excited states ("forbidden lines") -- emission can occur hours later.

      • In the daytime this emission is swamped by scattering of sunlight.

      • But in the night time we have airglow from the de-excitation by emission.

        In principle, there is a decay in the brightness of these emission lines from the beginning to the end of the night.

        But there is also potential for emission varying with latitude, sky position and time due to additional and random injection of ions into atmosphere from the solar wind traveling down Earth magnetic field lines (especially near magnetic poles -- aurora borealis).

        Examples of variation of the 6300 A OI emission over the course of nights with different patterns. From Roach & Gordon, "Light of the Night Sky".
      • Airglow can be imaged as ripples of emission across the sky...

        The ripples in this all-sky image are from variations in the airglow across the sky. Airglow is often mistaken for high cirrus clouds. Note that this is a VERY BROADBAND image and it still shows airglow variations emitted by just the few emission lines, showing how much they can dominate the total night sky background from the ground. Concam image from www.concam.net/phenomena.html.

        ... including occasional dramatic contributions of background from aurora borealis/australis
      • Aurora image from http://science-education.pppl.gov/SummerInst/INDEX.HTML.

        If you have ever watched the aurora borealis ("Northern Lights") you have a sense for the minute-level timescale of these variations.

    • In the optical, a dominant contribution comes from [OI] line transitions (along with the NaD doublet at 5893A).

      Optical and NUV line transitions from metastable excited OI. From Roach & Gordon, "Light of the Night Sky".
      The main atmospheric fluoresces in the visible and NUV (this does not include the important OH radical fluorescence in the NIR).

      Optical and NUV line transitions. The emission line intensities are measured in Rayleighs, where 1 Rayleigh = 106 photons cm-2 s-1 ster-1
      = [1.58 x 10-11 / λ (nm) ] W cm-2 ster-1. From Lena, "Observational Astrophysics".
      The following shows the optical sky background surface brightness in spectral form at Kitt Peak National Observatory and usual surface brightness units (but in AB magnitudes!). Note in addition to the atmospheric lines the presence of Hg and Na lines from artificial lighting (see below).

      From O'Connell's ASTR511 Lecture Notes, Lecture 7.
      The same thing from a darker site (La Palma, Canary Islands) and in microJanskys. Note the reduced contribution from artificial lighting and the placement of the optical passbands. From Wikipedia.


    Infrared Airglow: OH Emission

    Starting at about 6000A we get increasing contributions of radiation by excited levels of the OH- radical, known as the Meinel bands:

    From http://www.astrossp.unam.mx/estandar/standards/NightSky/skylines.html. Other zoomed in pieces of this spectrum may be found there.

    The excitation is through the reaction at 75-100 km altitude:

    H + O3 ---> OH- + O2

    • The lines are brightest in the H band (see image below).

    • The brightness of these lines varies with time (on scales of 5-15 minutes) on large angular scales (corresponding to spatial periods of 10s of km) due to the passage of gravity waves in the ionosphere.

    • The typical variation is 10% but as much as 50% is possible.

    • See here for movies showing the time and space variation of the non-thermal NIR background with time.

    This OH emission dominates the non-thermal infrared regime. The ramp up of the thermal infrared regime, is also shown in this plot:

    From O'Connell's ASTR511 Lecture Notes, Lecture 7.

  • Thermal background

    Comes from the fact that the cold atmosphere still emits as a blackbody of several hundred Kelvin.
  • From http://wxpaos09.colorado.edu/radiation/emissionspec.gif.
    This plot shows the combined non-thermal NIR and thermal regime of background emission:

    From http://www.gemini.edu/science/maxat/comparative/comparative.html.
    Note that the thermal background is so high and variable with position and time that it cannot generally be treated as we do in the optical and near-infrared (i.e., by taking long images -- which would saturate anyway -- and using nearby blank sky annuli).

    • Background fluctuates on minute timescales or less due to atmospheric turbulence and temperature drift in the telescope.

    • Most astronomical sources are much fainter than the range of these fluctuations, so any small error in the estimation of the background is a huge problem in accurately measuring the signal.

    • Chopping repeatedly (typically at 3-10 Hz) between the course and nearby blank sky is the only way to accurately monitor the temporal variations, usually with the secondary mirror.

      From Bely, Chapter 1.
    • Note that one generally also nods the telescope pointing about once a minute. This is to correct for differences in the thermal background from internal optics that take place when one chops -- one sees the high emissivity surfaces in the telescope+detector optics slightly differently, and need to account for this.

  • scattered sunlight in Earth atmosphere (during twilight)
  • REFERENCE: Kaler Chapter 7.6.

    Image from http://www.cacr.caltech.edu/Publications/techpubs/brochure/resource.html.
    Note that we define a few "official" types of twilight, depending on the position of the Sun relative to the horizon:

    Image from Wikipedia.
    • 18 degree twilight = "Astronomical Twilight": First/last hints of twilight --> fully dark sky
    • 12 degree twilight = "Nautical Twilight": Sailors could see the horizon but still see many stars for fixing position with sextant
    • 6 degree twilight = "Civil Twilight": Twilight defined for legal reasons (e.g., when you need to have headlights on with your car
    The brightness of the twilight sky comes from the scattering of sunlight to our line of sight.

    • When the sun has just set, it still illuminates the upper atmosphere directly.

    • Because the amount of scattered sunlight is proportional to the number of scatterers (i.e. air molecules) in the line of sight, the intensity of this light decreases rapidly as the sun drops further below the horizon and illuminates less and less of the atmosphere above your horizon.

    Amount of air still illuminated after sunset at the horizon (above the Sun) in terms of the airmass, where the zenithal airmass is defined to be 1 and the horizon has a total airmass of 38 (see below). From Wikipedia.
    Of course the length of twilight depends on your latitude, because of the angle of the setting sun with respect to your horizon:

    • At the equator the Sun generally moves almost perpendicular to the horizon and it can go from day to night in as little as 20 minutes.

    • Twilight gets longer at higher latitudes, taking sometimes hours near the Arctic/Antarctic Circles.

    • At the poles, twilight can be weeks long.

    The number of daylight hours as a function of latitude and date of the year. From Wikipedia.

  • scattered sunlight off dust in solar system:
    • zodiacal light
    • Due to dust grains orbiting Sun and concentrated to the ecliptic plane.

      Two effects:

      • scatters sunlight -- produces a component with near solar spectrum.

      • thermal emission -- with a blackbody spectrum.

      From Lena, Chapter 2.

      Thus there is a dip in the emission between these two regimes at 3.5 microns where we can observe sources outside the solar system with lowest possible background (the "cosmological window").

      From Bely, Chapter 1.

      Images from the DIRBE experiment on the COBE satellite at various infrared wavelengths. The Galactic plane is horizontal across the middle of each panel. As may be seen, for <3.5 microns, starlight dominates the images. At longer wavelengths the images are dominated by the S-shaped ecliptic plane, where interplanetary dust heated by the Sun emits strongly. Note how strongly this interferes with exploring the infrared universe beyond.

      Not uniform in the sky, but maximum toward the Sun and directly away from the Sun due to backscattering (the "gegenschein").

      The minimum in the zodi-light is at 60 deg ecliptic latitude due to interplay of minimum of zodi-dust and temperature of the dust.

      From Bely, Chapter 1.

      In the optical, one can see the zodiacal light at dark sites as a band across the sky (that is NOT the Milky Way!) or as a bright spot in the sky.

      Zodiacal light is sunlight reflecting off of dust particles in the ecliptic plane of the solar system (hence the name "zodiacal"). It is easiest to see in September, October just before sunrise from a very dark location. Concam image from www.concam.net/phenomena.html.

      Another view of zodiacal light, from http://www.oneminuteastronomer.com/1049/false-dawn/. The notion of "false dawn" has to do with the fact that the ` zodiacal light is best seen near sunrise/sunset, when the particles are best backscattering. This was a phenomenon of great importance in the ancient Islamic tradition, where the timing of daily prayers could be thrown off by assuming the zodiacal light was the actual first appearance of dawn. Muhammed the prophet warns about this mistake and explains how to distinguish false and real dawn.
      Two more views of of the zodiacal light. Left image by D. Malin, Anglo-Australian Observatory. Right image is zodiacal light as seen from Paranal, from Wikipedia.

    • gegenschein -- zodiacal light at maximum directly opposite the Sun.
    • Gegenshein is the same as zodiacal light, but it is the point where the reflection of sunlight is maximal in the opposite direction from the Sun. Image by H. Kukushima, D. Kinoshita and J. Watanabe (NAOJ).

    • Note that zodiacal dust is the prime background source for most space observing, whereas airglow is the primary (night time) contribution to the background for ground-based sites.

      From Lena, Observational Astrophysics.

  • unresolved galaxy/starlight (especially in plane of Milky Way)
  • This all-sky image shows unresolved stars from the Milky Way as a band of light. Concam image from www.concam.net/phenomena.html.

    The surface brightness of the sky from cosmical sources (i.e. not solar system or terrestrial sources) was measured by the Pioneer 10 satellite once past the zodiacal disk.

    • This was done in two bands:

      • "blue" -- 3950-4850 Å

      • "red" -- 5900-6900 Å

    • The results were reported in an old method of reporting surface brightness, S10(V), which is the equivalent number of sun-like stars (G2 V) of V=10 per square degree (or 27.78 mag arcsec-2).

      From Allen's Astrophysical Quantities, Fourth Edition by Arthur Cox.
  • clouds -- will reflect back light from the Earth as well as Sun, Moon.

    This all-sky image shows the increased background from clouds. Concam image from www.concam.net/phenomena.html.

  • scattered light from light pollution


Light Pollution

Light pollution is the bane of all who wish to preserve the beauty of the night sky, but is a severe and growing problem for astronomers.

There are increasingly fewer places on Earth now where man-made light can be avoided.

The Earth at night.

The United States at night. Note interstates highly visible.

Obviously, this is a growing problem with time:

View of Los Angeles from the Mt. Wilson Observatory in 1908 (top) and 1988 (bottom). Images from Astronomical Society of the Pacific.

From Bob O'Connell's ASTR511 Notes, Lecture 7.

Central Virginia and Albemarle County, with strict lighting ordinances, is one of the darkest remaining on the East Coast.

The East Coast of the United States at night.

Much of the worst scattered light problems are very nearby.

  • At McCormick Observatory, lights from the University itself are the worst problem (especially on football nights!).

    City lights also bad of course.

  • At Fan Mountain, Charlottesville to North and (in winter) ski runs to west.

Dealing with light pollution often requires political solutions because of reluctance to address the issue due to concerns over:

  • Public safety
  • Cost of changing things for the better.
Both of these are misperceptions that requires educating the public, especially government people, and demonstrating that the needs for astronomy (and dark skies generally) can be met with compromises that are sensible things to do anyway.

  • Any light going up into the sky is not being used for anything -- it is wasted energy.
  • An unshielded light like these is not only bad for dark skies, but is extremely wasteful. For the fixture on the left, about 20% of the light goes upward and does no useful illuminating, while another 20% goes sideways and contributes to glare. The fixture on the right is much worse!

  • A proper lighting fixture directs all light where it is needed -- downward -- and means less energy is needed to get the same illumination.
  • Examples of two types of good lighting fixtures that keep light pointing downward to where it is needed, and not upward, where it is a waste and light pollutant.

  • Astronomers prefer the use of low pressure sodium vapor lamps because the emitted light is in well defined, narrow spectral lines that are easy to avoid.

    These lamps are also more cost-effective for the same amount of luminance as other light bulbs.

    The low pressure sodium lamp (top) emits in a few narrow emission lines in the yellow-orange part of the spectrum. In contrast, the high pressure sodium lamp (middle) and mercury vapor lamp emits all over the visible spectrum. Left images are from about 400 nm on left to 870 nm on right. Right images are from about 400 nm on left to 700 nm on right.
    Recall again how these artificial lighting sources affect the integrated surface brightness of the sky:

    From O'Connell's ASTR511 Lecture Notes, Lecture 7.
  • The International Dark-Sky Association is a non-profit organization dedicated to preserving dark skies.




2. Atmospheric Extinction and Correcting for its Effects

REFERENCE: Chapter 9 of Birney.

Sources of Atmospheric Extinction

The atmosphere not only adds background to your measurements but also removes source flux.

    The net effect is a decrease in S/N.

Light incident on the Earth's atmosphere from an extraterrestrial source is diminished by passage through the Earth's atmosphere.

Thus, sources will always appear less bright when observed below the Earth's atmosphere than above it.

Atmospheric extinction at optical wavelengths is due primarily to two phenomena:

  1. Absorption:

    From http://wxpaos09.colorado.edu/radiation/absorbspec.gif.

    • On IR side primarily water vapor, CO2.
    • On UV side primary absorption is ozone O3
    • Solar spectrum before and after passing through the Earth's atmosphere, with major absorbers identified. Image from Robert A. Rohde, http://scienceblogs.com/startswithabang/2012/08/14/how-many-colors-are-really-in-a-rainbow/ .

    • Note that in a broader context, atmospheric absorption of electromagnetic radiation determines what astronomy can be ground-based (i.e. done from the surface of the Earth.
    • Atmospheric extinction across the electromagnetic spectrum. Figure modified from Observational Astronomy by Lena, Lebrun and Mignard.

      As may be seen, most EM radiation is blocked by the Earth, and only a few atmospheric windows allow certain wavelengths through:

      • Optical
      • Various near and mid-infrared ranges
      • Near infrared transmission of the atmosphere. Note the dependence of the transmission on the amount of foreground water vapor. From Bob O'Connell's ASTR511 lecture notes, Lecture 14.
        Note that the transmission windows in the NIR and MIR have defined the nominal ground-based IR photometric bands.

        Adapted from McClean, Infrared Astronomy.
        Note several K bands defined:

        • Short wavelength side defined by the atmospheric transmission.

        • Long wavelength side defined by the desire to limit thermal background contribution, which starts to rise quickly at about 2.3 microns.

          Adapted from McClean, Infrared Astronomy.
        • Different observatories at different altitudes have different sensible windows.

          • K ' defined for Mauna Kea, which has better transmission on blue side and so is shifted to the blue.

          • Kshort or Ks was invented by Mike Skrutskie for the 2MASS project at Mt. Hopkins.

          • In both cases the red side was chopped earlier than the nominal Johnson K band to minimize thermal background.

      • Mid- and far-infrared windows and bands
      • Adapted from McClean, Infrared Astronomy.
        Note the Infrared Astronomical Satellite (IRAS) bands, which include some windows available on the ground, but others not available.

      • Millimeter and radio wavelengths
      From http://coolcosmos.ipac.caltech.edu/cosmic_classroom/ir_tutorial/images/transmission.jpg.
    • As may be seen, increasing elevations open up new atmospheric windows:

      • On high mountains and Antarctic plateau (where atmosphere is compressed -- equivalent to higher altitude at warmer latitudes): More mid- and far- infrared wavelengths accessible.

        The Atacama Large Millimeter Array (ALMA), to be placed in the high altiplano in Chile at 16,400 feet, consists of 64 antennae, each 12-m across. It will operate between 10 mm and 350 microns, from this high, dry location. ALMA is being constructed on the UVa campus and will be operated from here at the headquarters of the National Radio Astronomy Observatory. As of 2011, early observations with ALMA are underway.

        The Antarctic plateau is the dryest desert on Earth, hence astronomers have begun to take advantage of Antarctic sites, such as this one at the South Pole, to observe some atmospheric windows that are only open with low atmospheric water vapor content.

      • Airplanes to 12 km: Far infrared and sub-mm wavelengths. NASA Airplane Observatories: Lear Jet (1960s-1970s), Kuiper Airborne Observatory (1971-1995), SOFIA (soon).
      • The Kuiper Airborne Observatory operated a 36-inch telescope in the belly of a Lockheed C-141 jet. At 39,000 feet (12 km) the KAO flew above over 99% of the Earth's water vapor, opening a number of infrared windows.

        NASA's Stratospheric Observatory for Infrared Astronomy (SOFIA) is a Boeing 747SP jet with a special fuselage modified to accommodate a 2.5-m telescope. As of 2011, SOFIA is operational and is making a Cycle 1 call for proposals for first scheduled observations in August 2012.

      • High altitude balloons and rockets to 30 km and higher: Near UV, x-ray, gamma ray.

        (LEFT) A gamma-ray balloon instrument being launched over Alice Springs, Australia. The balloon and the telescope float at 40 km above the Australian Outback for one to two days, as data is sent to the ground. The telescope falls safely to Earth via parachute at the end of the flight. Image and caption from http://www.astro.umd.edu/academics/brochure/highe.html. (RIGHT) Much of early x-ray astronomy was done from sounding rockets, like this one. Image from http://www.astro.psu.edu/xray/rockets/.
      • Space-based astronomy of course opens all possible EM windows.
      • Chandra (x-ray), Hubble Space Telescope (Near UV, optical, Near IR), and IRAS (infrared) astronomy satellites.


  2. Scattering: Two mechanisms depending on size of scatterer
    • Molecular scattering: scattering radius a << λ

      • Mainly Rayleigh scattering (elastic -- energies of scattered photons preserved), which has a scattering cross-section as function of wavelength given by:

        where n = index of refraction; N = # molecules / volume

      • As may be seen, the wavelength dependence is such that the blue is affected much more.
      • Clearly, the density of molecules in atmosphere (N) is a function of pressure, temperature, also therefore altitude, etc.
      • So all molecular absorption and scattering is affected/driven by climate / weather.

    • Aerosol scattering: scattering radius a >~ λ/10

      • Scattering from fine dust, water droplets, ice particles, pollution.
      • Mie scattering is light scattered by "large" [relative to wavelength of light] spheres), and the strict theory accounts for Maxwell's equations in the context of all kinds of reflections (external and internal) and surface waves on the scatterer, polarization, etc.

        Full theory requires computers to work out all effects.

        Mie scattering cross-section goes as

        σ π a2

      • This scattering also affects blue more than red light, but the cross-section falls off only as 1/λ so that in fact it is the main source of red extinction.
      • Mie scattering puts a lot of the scattered power in the forward scattering direction.

        From http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/blusky.html.

      • The sky seems more/deeper blue when you look at greater angles from the Sun because that is mostly Rayleigh scattering, while close to the Sun the sky appears "whiter" because this is primarily Mie scattering.

        REFERENCE: Kaler Chapter 13.14.

        More dramatic effect when lots of particulates in air; in clean air near poles of Earth, effect disappears.

        Also reason why mist and fog (Mie scattering) is whitish.

        Image showing progression of sky color to deep blue away from Sun. From http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/blusky.html.

        From http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/blusky.html.

      • Mie scattering involving water droplets and ice crystals responsible for a number of atmospheric effects relating to rainbows, halos, etc.

      • Left: rainbow from http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/rbowp.html#c1. Right: Glory surrounding the shadow of an airplane. From useful website on Mie scattering, http://www.philiplaven.com/p1.html.

      • From the point of view of photometry, scattering results in lost light from our source of interest (recall from point spread function image of a star how scattering puts starlight to large radii).
      • Aerosols are highly variable from night to night.
        • air pollution
        • dust = "haze"
        • volcanic ash (can be horrible)
        • dust storms
      • Example of extinction variation due to changes in nightly aerosol content of the atmosphere above the observatories on the Canary Islands.

      • Plots show the number of visual magnitudes lost for observations at the zenith (= 1 airmass, see below).

        • Seasonal dust storms blown over islands from Sahara desert.

        • Long term (years long) effects of volcanic activity.


Differential Effects in Atmospheric Extinction/Scattering and the Airmass

REFERENCE: Kaler Chapter 13.15.

Above we showed that there are wavelength-differential effects effects in atmospheric extinction and scattering.

You intuitively know that there are also altitude-differential effects to extinction and scattering...

  • E.g. Sun is a lot less bright (and observable with human eye) at sunrise/sunset than at noon -- more atmospheric scattering/absorption as you look through more air.
  • Obviously, all wavelength-differential effects of atmospheric scattering/absorption increase as the amount of atmosphere between you and source increases.

    Sun at sunrise/sunset appears redder because blue light more affected (blue light, 4000 Å, 16X more Rayleigh scattered than red light, 8000 Å), and therefore more removed from direct beam.

  • ASIDE: By the way, if Rayleigh scattering so strong a dependence on wavelength (proportional to 1/[λ4]), then why isn't the sky ultraviolet?

    1. Sun not strong emitter at these wavelengths
    2. O3 absorption of what is emitted by Sun at these wavelengths
    3. Eye not as sensitive to these wavelengths


To calculate proper magnitudes on an absolute scale, need to correct for extinction in photometry

  • Convention - quote mags as seen at "top of atmosphere"
  • Need to understand how many magnitudes of flux lost per given amount of atmosphere looked through
  • Obviously, depends on (altitude above horizon) = (90o-angular distance from zenith)
  • Definition of terms:

    • z, zenith distance is the angle between zenith and source, equal to 90o-altitude.

    • One airmass is the amount of atmosphere seen at zenith, z=0o.

    • X is the total atmosphere looked through, calculated in units of airmasses.
    Then, in a plane parallel approximation:

    cos z = 1/X

    X = 1/cos z = sec z

    plane parallel approx. ok for z < 60o

  • More exactly, with spherical shell atmosphere corrections:
  • Note the following equation for zenith distance, which makes use of the astronomical triangle.

    where:
      φ is your latitude in degrees

      δ is the declination of the source in degrees

      h is the hour angle of the source, converted from time units to degrees



The above equation gives an airmass as a function of z curve that looks like this (apologies that this figure uses M for the more standard variable name of airmass, X ! ):


Airmass (M=X) as a function of zenith angle, z. Because of the large variation in extinction, the graph is divided into two parts. The airmass values given outside the left axis go with the dashed curve, and those inside it with the solid curve. At z=0o (overhead) we look through a unit airmass, and below about z=30o the airmass increases linearly with a trigonometric function (the secant). Above z=30o the curvature of the atmosphere becomes important, and X=M is increasingly less than secant z, rising to a maximum of 38. From Kaler's The Ever Changing Sky.

From a practical standpoint, you can see that, for example:

  • At z=60o you look through 2 airmasses (obtainable from plane parallel approximation).
  • At z=71o you look through 3 airmass.
  • At limit of z=90o you look through 38 airmasses.

Pathlengths in the Earth's atmosphere. (a) In a plane- parallel atmosphere, that is, one whose top is parallel to a flat Earth, star 2, with a zenith distance of 60 o, is seen through twice the airmass as star 1 overhead; star 3, with z = 19o, is seen through triple the airmass. (b) Because of the curvature of the Earth and its atmosphere (not drawn to scale), the maximum airmass is 38 times that at the zenith. From Kaler's The Ever Changing Sky.


Observing Run Preparation -- Airmass Charts

Because of the strong dependence of atmospheric absorption/scattering = light loss as a function of zenith distance, prepared astronomers always attend their observing runs with airmass charts for the targets they intend to study.

The airmass chart shows the number of airmasses to each source as a function of time of the night:

  • The airmass curves are obtained directly from the above equation for airmass as a function of z, which, in turn is given by the equation above in terms of:

  • Hour angle h, which is of course dependent on the source's R.A. and the sidereal time.

  • The declination, δ, of the source.

  • The latitude of the observatory, φ.

An example of an airmass chart for a typical observing run is shown below:

Obviously, it is best to observe each source when it is at the smallest possible number of airmasses, because this is when it is brightest.

Note, that degradation of the image by atmospheric seeing effects also increases with airmass, so it observing at high airmass results in a double hit -- fainter source flux and more degraded image from seeing!

Actually, there can be a triple hit if the atmospheric contribution to the background is large, because larger airmasses can contribute more background (e.g., thermal infrared and airglow).

THOUGHT PROBLEM: Why are the curves for each object symmetric about some particular point of time in the night? What is happening at that time for each object?


Correcting Photometry for Atmospheric Extinction and Bouguer's Law of Attenuation

REFERENCE: See Chapter 9 of Birney for discussion of this material, especially pp. 145-155, or Kitchin Section 3.2.

We have seen that we lose more flux from a source if we look through more atmosphere.

  • The amount of flux lost as a function of number of airmasses is an "optical depth" type of a problem, which always goes as an exponential.

    I.e., the amount of flux that gets through the atmosphere goes down as f = fo e(-X / τ), where τ is an optical depth and, in this case, X is the number of airmasses where the flux goes down by 1 / e.

  • However, since magnitudes are defined in a logarithmic way, this means that in magnitudes, the loss per airmass is actually more or less linear!

  • The linear relationship between loss of brightness in magnitudes and airmass is known as Bouguer's Law.

  • The constant of proportionality, k, in Bouguer's Law is called the extinction coefficient and is something we must always solve for in photometric work.

  • To solve for the extinction coefficient k, which we need to turn our observed or instrumental apparent magnitude, m, into a true apparent magnitude, mo, we must monitor a set of stars as they change their airmass (position with respect to the zenith) and apparent brightness (in magnitudes).
  • From Kitchin, Astrophysical Techniques.

  • Of course, from Earth we cannot observe at less than 1 airmass, and so we must extrapolate Bouguer's law to 0 airmass to get to the above-atmosphere value (where magnitudes are defined).

  • Note extinction is wavelength dependent (as example of setting sun shows and as we have seen in this plot shown one more time!) ...

    ... so obviously k is k(λ).

    This means we have a different extinction coefficient for each wavelength, and therefore each type of filter we observe with. Thus, there are multiple Bouguer's Laws we need to solve for:

  • Obviously, k must be steeper as we go from red to blue, and we find that roughly we have the following values of k (Δm / airmass) for standard broadband filters:
  • U -- 0.50 mag/X -- observe only at lowest possible X
    B -- 0.25 mag/X  
    V -- 0.20 mag/X  
    R -- 0.10 mag/X  
    I -- 0.05 mag/X -- most tolerant of observations at high X

  • Because of the differences in extinction coefficient with wavelength, we see that for star colors there is also an airmass effect and one can write a color extinction coefficient as follows:

    Obviously, the trend is for stars to become redder with airmass (recall the Sun!).

  • Of course, things are never as easy as they should be, and Bouguer's Law for broadband filters is no exception.

    The above color effects also occur within each filter bandpass.

    Because each broadband filter can have a significant difference in wavelength, Δλ, from one side to the other of the passband it will also have a difference in actual extinction/airmass on the blue and red side of the filter passband.

    Thus, a blue star will actually experience slightly more than the average extinction in a particular broadband filter, and a redder star will experience less than the average extinction in the same broadband filter, because of differences in how the star's energy is distributed from the blue to red side of the filter!

    Not only are there differences between your net filter and that used in the set-up of the standard photometric system (e.g. the net filters used by Arlo Landolt), but because a broadband filter has Δλ and blue/red stars have different effective SED, they fill different proportions of λ in Δλ. The above figure is from Mike Bolte's notes: http://www.ucolick.org/~bolte/AY257/ay257_2.pdf.

    This is effectively like the stars having slightly different extinction coefficients in the filter, corresponding to a different effective wavelengths for the same filter (figures below).
    The right figure is from Kitchin, Astrophysical Techniques.

    This effect tends to be more pronounced in the bluer broadband filters (WHY??).
  • Can solve for the color effect by considering k to have two parts, a "mean effect'' part and a "differential star color'' part:
  • k = k' + k"c

    mo = mX - k'X - k"cX

    co = c - kc'X - kc"cX

    To get the values of k' and k":

    • Get observations of same (type of) star at different airmasses -- k'.
    • Use blue/red pairs at different X -- k".
  • In the simplest case, where we ignore color effects, one could solve Bouguer's Coefficient (k λ) to airmass by monitoring the change in "instrumental magnitude" of any star.
    • Recall that "instrumental magnitude" refers to the raw, uncalibrated magnitude given by your instrument.

  • Technically, one could even solve for the color effects in Bouger's constant:
  • with some random stars of assorted (but initially unknown) colors

  • More typically, however, we use stars of known color (and magnitude) - i.e. standard stars - to determine
  • Note, one has to derive kλ anew for every observing run (better, every night) because it varies due to prevailing conditions.

3. Reduction to a Standard System

  • In addition to correcting for airmass, one also needs to account for any differences between your equipment (telescope + detector + filter) and the standard ("universal" established - usually by one person with one set of standard stars) system of equipment (i.e. standard bandpass).

    From Mike Bolte's notes: http://www.ucolick.org/~bolte/AY257/ay257_2.pdf.

    That is, unless you match every detail of the observing system with that of the person who set up the standard filter scheme, you must always correct your instrumental measurements to that system in a way that accounts for differences in your effective bandpass compared to the standard bandpass.


  • One does this by actually observing stars of known color and magnitude and solving an equation like:
  • mtrue = minst + bo + b1C + b2C + ...

    to account for difference in magnitudes seen for different colored stars in your system compared to standard system.

    The above equation is for magnitudes already corrected for the airmass effects of Bouguer's Law!

  • Thus, in the old days of single aperture photometry, one had to go through long, laborious sequence of observations and corrections to get your instrumental magnitudes on an accepted, standard magnitude system:
    1. Airmass correction -- observe set of stars at different airmasses and solve for k'
    2. Airmass*color correction -- observe blue/red pairs of stars at different airmasses and solve for k''
    3. Color, Color2 terms for filter differences with standard -- observe standard set of stars of different colors
    4. Reiterate because all observations you have made for one phenomenon have been affected by the other!
    As you can imagine, with single aperture photometry one would spend much of the night observing calibration/standard stars (1 star at a time)!

    • "1/3 RULE" -- spend 1/3 of night on standard/calibration stars
  • Now - with CCDs, one can simultaneously access many standard stars in one field of view having a range of colors + magnitudes.

    Among the most important standardized lists of these photometric calibration fields are the:
    • Landolt (most famous standards in the North hemisphere and along equator (cf. 1992, AJ, 104, 340), and generally found at least once per hour of right ascension.
    • Graham "E Fields" (southern hemisphere standard fields) 1982, PASP, 94, 244
    (Recall too, in the old photographic plate days, that astronomers often exposed one half of a photographic plate on the "north polar sequence", which was up all year long for northern observatories.)

    Observing these standard star fields kills many birds with one stone:

    • First, observe many standard stars in one field over different airmasses.
    • Second, combine Bouguer constant solution and standard calibration at same time:
    • Recall (atmosphere correction):

      m0,inst = mX - k'X - k"cX

      and (standard passband calibration):

      mtrue = m0,inst + b0 + b1c + ...

      combine to:

      mtrue = mX + a0 + a1X + a2c + a3cX + a4c2 + ...

    • Use many standard stars to find a0, a1, a2, ... at observed X and known mtrue, c.
    • Since each star, i, satisfies an equation like this, system of polynomial equations exists:

    • Since this is a series of polynomial equations, it can be solved with a matrix inversion methodology (see, e.g., Harris et al. 1981, PASP, 93, 507).
    The following figures show the typical progression of a solution to a transformation equation using observations of a series of standard stars with known V magnitudes and B-V colors (figures from Mike Bolte's web lectures -- http://www.ucolick.org/~bolte/AY257/ay257_2.pdf -- show a very commonly found figure in any paper doing photometric calibration).

    In the following, the V and color = B-V are the known values for the individual stars, the X are the airmasses at which the stars were observed, and "time" is the time the stars were observed (shows changes in transparency over a night).

    Each panel shows the residual, given by (V-v), where V = (true V mag) and v = (instrumental V mag).

    • First, we see the residuals when only a zero point, a0 is applied to convert from instrumental to real magnitudes.

      The RMS residual is 0.055 magnitudes.

      Note the clear trend in the residuals with airmass. Clearly, were we to include a term in airmass, we would get a much better transformation from v to V.

    • Now we include a term for airmass. The RMS is clearly better!

      But now a trend in the residuals as a function of color becomes apparent.

    • Now, accounting for the color term, we get an even better transformation from v to V.


Some Important Photometry Observing Rules

  1. Always observe standard stars with colors bracketing the colors of target stars you are calibrating
  2. Always observe standard stars with airmasses spanning the airmasses of your target observations
  3. Use only very clear (photometric) weather! Absolutely no visible clouds.
  4. When possible, use bluer filters at lower X, save higher airmass observations for redder, more airmass-tolerant filters if possible.
  5. Generally work X<1.5, no more than X=2 (z <= 60o)
  6. Large telescopes + CCDs so efficient, easy to saturate standard stars
    • Use short exposures but not too short or face

      (a) scintillation errors (> 5-10 sec. best), and

      (b) shuttering errors (errors in the timing of the shutter typically become proportionally larger with shorter integration times).

      Note difference in shuttering methods, and consider, e.g., how the following shutter will yield variable exposure times in an image:

      Traditional leaf shutter operation, which can yield exposure time variations across the field of view if placed near the focal plane. From http://armycommunications.tpub.com/SS0507/SS05070041.htm .
      Even illumination for shutters near the focal plane can be obtained by ``curtain shutters", which typically have an opening between two moving blades moving in the same direction that sweep across a detector to expose and then cover the detector successively to ensure that every pixel is exposed the same amount of time.

      Example of an astronomical curtain shutter, designed to ensure even illumination (in this case yielding variation in exposure time less than 1msec). Shutters like this typically alternate between three positions, creating sweeps from left-to-right alternating with sweeps from right-to-left from one exposure to the next. Image of the BUSCA shutter from university of Bonn, from http://www.astro.uni-bonn.de/~ccd/busca/ca_newsletter/ .
      Click here to see an image of the world's largest shutter, made for the Dark Energy Camera.


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All material copyright © 2002, 2005, 2007, 2009, 2011, 2013 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 5110 at the University of Virginia.