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ASTR 3130, Majewski [SPRING 2015]. Lecture Notes

ASTR 3130 (Majewski) Lecture Notes


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:

  1. Subtraction of background sources (including sky flux and dark current).
  2. Correction for atmospheric extinction.
  3. Reduction to a standard system.
Note that if we are doing relative photometry -- measuring magnitude differences -- we do not need to worry about (3), and, under some circumstances, you can also ignore (2).

In our experiment on globular cluster RR Lyraes we will be doing differential photometry and will not account for either (2) or (3), to keep things simple.

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.

    Practically it is 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 include:

  • moonlight -- very important

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

    • dark time -- time 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. Awarded to most competitive observing proposals.
    • bright time -- time within about 3-4 days 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 observatories 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 ~2 weeks of the month, time near First or Third Quarter Moon.
  • airglow (photochemical luminescent emission from atoms and molecules in the Earth's atmosphere excited by the Sun's daytime UV radiation, typically Na, OH, O, O2, from about 50-300 km above Earth's surface)...

    The ripples in this all-sky image are from variations in the airglow across the sky. Airglow is often mistaken for high cirrus clouds. Concam image from

    Airglow image taken in Yosemite National Park by Kristal Leonard-Ferrara ( ).
    Airglow spectrum compared to sunlight, from .
    ... including occasional dramatic contributions of background from aurora borealis/australis (also caused by excited atoms, but auroral excitation is by collisions with energetic particles, and seen closer to the Earth's magnetic poles)
  • Aurora image from

  • scattered sunlight in Earth atmosphere (during twilight)
  • REFERENCE: Kaler, The Ever-Changing Sky, Chapter 7.6.

    Image from
    Note that we define a few "official" types of twilight, depending on the position of the Sun relative to the horizon:

    • 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).

  • scattered sunlight off dust in solar system:
    • zodiacal light
    • REFERENCE: Kaler, The Ever-Changing Sky, Section 12.6.

      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

      Another view of zodiacal light. Image by D. Malin, Anglo-Australian Observatory.

    • gegenschein -- zodiacal light is 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. Image by H. Kukushima, D. Kinoshita and J. Watanabe (NAOJ).

  • unresolved 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

  • thermal background (especially in the infrared) -- from the fact that the cold atmosphere still emits as a blackbody of several hundred Kelvin.
  • 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

  • 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.

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

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.

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!).

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.

  • For example, 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. Right images is from about 400 nm on left to 700 nm on right.
  • 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 7 of Birney et al.

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 below the Earth's atmosphere than above it.

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

  1. Absorption:

    • On UV side primary absorption is ozone O3.
    • On IR side water vapor, CO2, methane.
    • Solar spectrum before and after passing through the Earth's atmosphere, with major absorbers identified. Image from Robert A. Rohde, .

    • 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 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
      • Millimeter and radio wavelengths
    • 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 operates between 10 mm and 350 microns, from this high, dry location. Much of ALMA was constructed on the UVa campus and it is being operated from here at the headquarters of the National Radio Astronomy Observatory.

        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 (2010-).
      • 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. It began taking data in 2010.

      • 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 are sent to the ground. The telescope falls safely to Earth via parachute at the end of the flight. Image and caption from (RIGHT) Much of early x-ray astronomy was done from sounding rockets, like this one. Image from
      • 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 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.


      • 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


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

      • Left: rainbow from Right: Glory surrounding the shadow of an airplane. From useful website on Mie scattering,

      • From the point of view of photometry, scattering results in lost light from our source of interest (note point spread function image of star shown above -- shows how scattering puts starlight to large radii).
      • Aerosols 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: Birney et al. Chapter 7, 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 has so strong a dependence on wavelength (∝ 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 magnitudes 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 = 0 o.

    • 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 we will become more clear when we discuss the astronomical triangle later in the semester:

      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 (more accurately obtained from spherical shell atmosphere model).
  • At limit of z=90o you look through 38 airmasses (only correctly derived with spherical shell formula).

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., infrared).

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 7 of Birney for discussion of this material.

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 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).
    • QUESTION: What limitations are there on doing this with a ground-based observatory?

  • 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 (Δ>m / 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.

    Because each broadband filter can have a significant difference, Δλ, 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 broadband filter, and a 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!

    Because a broadband filter has Δλ and blue/red stars have different effective SED, they "use" different proportions of different λ in Δλ. This is effectively like the stars having slightly different extinction coefficients in the filter!

    This effect is most 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 wavelengths -- k'.
    • Use blue/red pairs at different X -- k".
  • In the simplest case, where we ignore color effects, one could solve Bouguer's Coefficient (kλ) 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 -- but later determined iteratively) 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).

    (Draw image of effective bandpass differences between observer and standard system.)

    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, many standard stars in one field of view with many colors + magnitudes, e.g.:
    • Landolt (most famous standards in the northern hemisphere and along equator (cf. 1992, AJ, 104, 340)
    • Graham "E Fields" (southern hemisphere standard fields) 1982, PASP, 94, 244
    Kill 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:

      Can solve with matrix solution.

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 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 scintillation error (> 5-10 sec. best)

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All material copyright © 2002,2006,2008,2012,2015 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 313 and Astronomy 3130 at the University of Virginia.