Previous Topic: Reduction of CCD Data Lecture Index Next Topic: Filters and Color Indices

ASTR 5110, Majewski [FALL 2015]. Lecture Notes

ASTR 5110 (Majewski) Lecture Notes


PHOTOMETRY: MAGNITUDES AND FLUXES

This is material that should be pretty familiar to you from undergraduate studies. If not, you should make sure to know this soon!

SOME REFERENCES:

  • Rieke, The Detection of Light, Section 1.1.

  • Lena, Observational Astrophysics, Section 3.3.

  • Kitchin, Astrophysical Techniques, Section 3.1.

  • Birney, Observational Astronomy, Chapter 7.

  • Mihalas & Binney, Galactic Astronomy, Section 2.6.

  • Binney & Merrifield, Galactic Astronomy, Section 2.3.

  • Ridpath's Norton's 2000.0, pp. 132-135.

  • and many other sources, including basic introduction to astronomy texts if you need some brushing up on the concept of magnitudes!


Photometry and Fluxes: Definitions

  • Photometry - measuring the amount of light received from an object
    • Total light (energy) given off by a star
      • l = luminosity, in photons / sec, or in erg / sec

    • Light flux received on Earth from the same star -- i.e., a point source.
      • φ = photon flux, in photons sec-1 cm-2,

        or

        f = flux, in erg sec-1 cm-2

    • For light flux received on Earth from an extended (i.e. resolved) source, like a nebula or galaxy -- we measure flux coming from a solid angle, called a surface brightness, often written as Σ or μ :
      • Σ = SB, in photons sec-1 cm-2 steradian-1,

        or

        Σ = SB, in erg sec-1 cm-2 steradian-1

    • WARNING: The definitions of things like "luminosity", "flux", "flux density", etc. used by astronomers are not the same as those used in other disciplines (e.g., our "luminosity" is called "flux" in other sciences, etc.) -- see Chapter 1 of Rieke about these differences.

  • The astronomical flux quantities are usually quoted for at the top of the Earth's atmosphere.

  • Note that in the above definitions we have given no specificity as to the wavelengths or energies of the individual photons or light.

    • In the above cases we are referring to bolometric quantities, which refers to a sum over all frequencies.

    • Practically this is a very hard thing to measure, since it requires measuring the entire EM spectrum of a source, which is impossible with a single type detector.

    • Thus, bolometric fluxes have to be inferred by knowledge of the physics producing the luminous source for which you have partial information, or pieced together from observations at all parts of the EM spectrum.

  • More commonly we work with fluxes at specific energies.

    • Monochromatic fluxes are defined as the fluxes within infinitesimally small bands --

      (ν, ν + δν) or (λ, λ + δλ) -- and are quoted either in wavelength or frequency units:

      • fν units: erg sec-1 cm-2 Hz-1.

      • fλ units: erg sec-1 cm-2Å-1.

      The above quantities are also referred to as spectral flux densities.

    • Note that because (prove this to yourself), fνδν = fλδλ, and ν = c / λ, one finds (prove to yourself!):
      ν fν = λ fλ

    • The functions fν(ν) and fλ(λ) are referred to as the spectral energy distribution (SED) of the source.

    • When referring to monochromatic fluxes, it is common to use units of Janskys (Jy) (borrowed from radio astronomy):

      • 1 Jy = 10-26 W m-2 Hz-1.

      • 1 Jy = 10-23 erg sec-1 cm-2 Hz-1.

  • The actual energy flux received by an Earthbound instrument really given by:
  • OR:

    where:

    fλo = stellar flux incident on Earth atmosphere

    Tλ = transmission of atmosphere

    Rλ = efficiency of telescope + detector

    Sλ = transmission function of filter

  • Of course, fλo depends not only on how intrinsically bright something is -- its luminosity -- but on its distance d:
  • The received flux from a source falls off inversely proportional to the square of the distance.


Magnitudes

Magnitudes are a brightness scale, a logarithmic representation of the spectral flux density.

  • A device that allows an easy way to intercompare sources with immense ranges in flux density.

  • But a bit arcane, not readily intuitive.

  • Now several definitions used.

  • Should be thought of as "band-specific", where "band" means a set range of λ.

History of the concept of magnitudes:

  • Roots of idea extend to second century B.C. Greek astronomer Hipparchus.

  • Ptolemy's Almagest
    • Catalog of ~1000 naked eye stars.
    • 6 "magnitude" classes:
      • 1 = brightest
      • 6 = faintest
  • But revisions required and made in last few centuries.

    • For example, want to put Sun, Moon, bright planets on same scale -- need to extend scale to brighter than magnitude 1.
    • Later, telescope invented - need to extend scale to > 6 mag.
    • 1850 -- N. R. Pogson (British astronomer) notices that, because eyes work logarithmically, the classical magnitude scale corresponds roughly to set ratios of brightness between successive magnitudes.

      • Also notes that mag 6 is about 100X fainter than mag 1.
      • Since Δm = 5 appears to be 100x ratio in brightness, and
      • Pogson proposes to formalize scale so that ratios between successive magnitudes are exactly 2.5119.

      • Thus, two stars of integrated band fluxes (i.e., the integral over some [λ, λ + δλ] of the spectral flux density) f1 and f2 have a magnitude difference given by
      • Note that the above equation also shows that fractions of magnitudes are possible for stars with brightnesses in between two integer magnitudes.

  • Thus, star brightnesses are quoted on a magnitude scale that is:

    • logarithmic -- Each magnitude is a ratio of ~2.5X brightness.

      Each 5 magnitude difference is 100X brightness ratio.

    • "backwards" -- Brighter objects have smaller magnitude values.

    • open-ended -- As we probe deeper, find fainter stars of larger magnitudes

    Not to insult your intelligence, but just because it has a few interesting things on it:

  • Comments about naked eye stars:

    • Brightest stars are about 0 magnitude (defined by summer star Vega in constellation Lyra -- see below).

    • Sun, Moon, Venus, Mars, Jupiter have negative magnitudes.

    • The faintest visible naked eye stars are about 5-6 magnitude. But actual limiting magnitude depends very much on prevailing conditions (clouds, moonlight, air and especially light pollution).

    • The "bowl" of the Little Dipper is a handy naked eye reference for the magnitude scale, because it contains one star each of approximately magnitude 2, 3, 4 and 5.

    • There are only 11 stars brighter than magnitude 1 visible from Charlottesville but 1630 stars brighter than magnitude 5.

        Faintest objects yet detected (Hubble Space Telescope): >29 magnitude = over 1 billion times fainter than visible to eye.


    Relative Photometry

    Measuring the brightness of one star compared to another is called relative photometry.

    Because observations in astronomy are difficult to do absolutely (they are typically made under "field conditions", where Earth atmospheric transmission varies with time and with equipment having a wide range of wavelength sensitivities), it is more natural to make measurements by comparison of one source to another with the same equipment and close in time.

    • Relative brightnesses are much more accurately determinable than "absolute fluxes".

      • Note that the absolute flux of the Sun at visual wavelengths is still not established to better than a few percent.

        Comparison of magnitudes and fluxes derived for the Sun. From Bessell et al. (1998, A&A, 333, 231).
    • Suggests use of sets of widely agreed upon "standard stars" to serve as references.

    We have seen the Pogson magnitude system which gives the following permutations of the same equations above:
    • Δ m = 5 is a 100x ratio in brightness
    • Thus,

    • In these equations we are always comparing stars and are on a relative scale.
    • When no consideration of 0 point of magnitude system, only brightness ratios (magnitude differences) -- we call this relative photometry

    Absolute Photometry

    When we care about the magnitudes (and fluxes) of stars on a set, universal scale, this is absolute photometry.

    (Don't confuse this with "absolute magnitudes", which is different and addressed below.)

    Setting up a universal magnitude system means defining for the equation

    a set flux value f2 corresponding to m2.

    Then we can evolve to a definition of magnitudes that looks like:

    Several approaches to this problem have now evolved.


    • The Vega-Based System

      One method is to pick one star in the sky as the universal reference.

      The traditional convention has been to choose the star Vega ( Lyrae) because:

      • It is easily observable (in the northern hemisphere -- where most astronomy was being done when the system was set up).

      • As a hot star, Vega has a relatively smooth SED fairly resembling a BB.

      Thus, by convention, Vega was decided to define 0 mag in every bandpass.

      Then:
    • The advantage of this system is that one can use any measure of brightness for f* -- e.g., different detectors with different filters yielding different spectral response -- and one can obtain the magnitude of the test star simply by using the same equipment and atmospheric conditions to measure fVega .

      The disadvantages of this system are:

      • The zero point of the magnitude system (i.e., "const" in the above equation), being defined by Vega's SED

        (fν, Vega ), will be different for each filtered set of wavelengths (i.e., bandpass) observed.

      • One is forced always to compare to Vega, or some other star already carefully calibrated to Vega. But:

        • Vega is too bright to observe for most telescope+detector combinations.

        • Vega is not accessible in the Southern Hemisphere.

        • Even in the Northern Hemisphere, Vega is not up all year.

        • Polar sequences of calibrators were commonly set up in the early days of photometry, because at least the same stars were always accessible in a particular hemisphere (and at nearly the same airmass all of the time from a given site).

      • Not as straightforward to intercompare data in different bandpasses.

        • To do physics, must always go through the SED of Vega to get ergs sec-1 cm-2 Hz-1 for a star (and assign this monochromatic flux to one specific, representative wavelength in the bandpass).

        • For example, in the Vega-based magnitude system, we have for the V passband and all stars with an SED like Vega:

        • BUT, the effective wavelength of the V passband actually shifts based on the SED of the source (e.g., becoming redder than 5450 A for later type stars) -- see next webpage.

        • Requires tedious and sensitive calibrations to produce tables like the following:

          Calibration of various bandpasses by Bessell et al. (1988, A&A, 333, 231).

          NOTE IMPORTANT ERROR: The values in the fourth row are flipped with those in the fifth row of the table.

      • Prone to universal changes, or at least increasing vagaries:

        • Even now, the system has shifted with improved measurements of the Vega SED.

          E.g., Vega's V magnitude is now actually V = 0.03.

          (This may seem odd, but the actual zero-points of the "Vega-based" system are now defined by a large set of secondary standards calibrated to the former measures of Vega. Easier to change magnitude of Vega than all other stars in the system.)

        • There have been suggestions in the literature that Vega's SED may be slightly variable.


    • The AB Magnitude System

      An alternative magnitude scale gaining popularity is the "AB" or "ABν" magnitude system, which is not based on Vega, but instead assumes the CONSTANT in the above equation is always the same for all filtered magnitudes.
      • Popularity grew out of work in the 1970s to do spectrophotometry (precise measurement of fν SEDs) of stars on the Hale 200-inch telescope (e.g., see Oke 1974, ApJS, 27, 21).

      • Defined as:

        mν = -2.5 log10 fν - 48.6

      • Simpler -- don't always have to know what Vega is doing to define the magnitudes.
      • Directly obtain the SED of the source (i.e. fν) when know mν.

      • Usually write "AB" subscript for AB magnitudes, e.g., "V AB".
      • BUT: Now have to make sure you understand the responsivity of your telescope/detector system as a function of wavelength for a given source, or have a well AB-system calibrated source against which you can compare measured, instrumental fluxes of your desired source.


    • The "STMAG" System

      • Analogous to AB system, but for fλ.

      • Defined as:

        mλ = -2.5 log10 fλ - 21.1

      • Used now for Hubble Space Telescope photometry (giving name "STMAG").

    • Intercomparison of Magnitude Systems

      • All three magnitude systems are designed with the zero-points yielding identical magnitudes for Vega at the same wavelength, intended to be the effective wavelength of the Johnson V band (the curve shown at the bottom of the plot below).

      • The three curves at the top correspond to the three standard SEDs that define the zeropoint for mλ in the three magnitude systems.

      • Note how the zeropoints for the AB and STMAG magnitude equations appear in the definitions of the Vega-based magnitude equations in the footnote to the above Bessell et al. table.

        The point of the AB and ST magnitude systems is to make all of the constants in the above table the same for all filters for the row fν (for AB mag system) or fλ (for ST mag system).


    Magnitudes and Distances

    • Now, assume m1 and m2 are for same star, different distance, d. Then:
    • Example :
    • Star with magnitude m2 farther by 10, 100x fainter, 5 mags fainter (larger)


    Absolute Magnitudes and the Distance Modulus

    • Apparent magnitude (m) -- magnitude of star as observed
      • Note that the apparent magnitude of a star depends not only on its luminosity, but its distance, because its observed flux is given by:
      • f = l / (4*Pi*d 2)

      • THOUGHT PROBLEM: Where does the above relation come from?

    • In some cases, we are interested in separating the luminosity and the distance effects. For example, consider comparing the luminosities of stars when placed at the same distance.
    • We can compare these luminosities as ratios of fluxes or as differences on the logarithmic magnitude scale.

      We define an absolute magnitude as the apparent magnitude of star if placed at a distance of 10 parsecs.

      • To distinguish apparent magnitudes from absolute magnitudes, we write the latter as capital "M".
      • A parsec is 3.25 light years.

    • The value of absolute magnitudes comes from the fact that if somehow we can guess the absolute magnitude, M, of a star or other object, and we measure its apparent magnitude, m, we can determine the distance to that star by:
    • The difference between the apparent and absolute magnitude of an object, (m-M ), is called the "distance modulus" of that object, and, as we can see, is directly related to the distance of the object.

    • Observational astronomers, like me, tend to think in distance modulus, because once we have memorized the absolute magnitude of a certain type of star or a certain feature in a color-absolute magnitude diagram, we can instantly gauge the distance to these features in some stellar population by just knowing the apparent magnitudes of the memorized standard candle.

      Alternatively, if we know how far away some object (e.g., a cluster, a galaxy) is, we can immediately assess how faint in apparent magnitudes some type of star or color-magnitude feature is in that object.


    • One of the most important problems in astronomy has to do with determining the absolute magnitudes, M, for objects in order that we can estimate distances.
      • A standard candle is a certain kind of object (star, galaxy, or other object) that:

        1. makes its identity known by some easily observable characteristic, and
        2. has a definable absolute magnitude.
        Standard candles are extremely valuable in astronomy, because if we find one, we can estimate its distance.
      • Different ways of identifying "standard candles" that have supposedly constant M:
        • Variability

          E.g. certain kinds of pulsational variable stars like RR Lyrae or Cepheids vary in brightness in easily identifiable ways:

          RR Lyraes have been key to gauging distances on kiloparsec scales (i.e. Local Group), and Cepheids on Megaparsec scales (the latter critical to establishing the local Hubble relationship).

          Supernovae explode with a reasonably well-defined light profile:

          Recently used in very high redshift systems to establish the presence of dark energy.

        • Note, obviously these variable stars are not constant in brightness, but, in the case of the pulsational variables, the mean brightness is presumed to be constant, and in the case of supernovae, one needs to monitor and fit the light curve, which is assumed to have a standard form.

        • Spectral type

          E.g. We can take a spectrum of the star and see what kind of star it is: Spectral type (OBAFGKM) and luminosity class (I,II=supergiant, III=red giant, IV=subgiant, V=main sequence).

          For example, the Sun is a G2V star (G type star, 2/10's of the way to being a K star, and of luminosity class V=main sequence type).

          All stars of same spectral type + luminosity class should be of same luminosity/absolute magnitude.

          Getting distances in this way is called measuring a spectroscopic parallax.
        • Colors

          Identify "spectral type" of the star by its color, gauged by photometry in different filters, (which is like very coarse spectroscopy).

          Once we have the spectral type, and an assumed luminosity class of the star, in principle you know the absolute magnitude, and then can get the distance through the distance modulus.

          This method of deriving a distance is called measuring a photometric parallax.
        • Unfortunately, not as good as spectroscopic parallaxes, since colors alone can give ambiguities.

          • For example, red stars can be either very luminous red giants or very dim red dwarfs. Making a mistake in confusing the two can lead to distance errors off by factors of 100 or more.
          • There are many kinds of blue stars, from blue supergiants to white dwarfs. Errors in proper identification can lead to distance errors off by factors of 10,000 or more.
          • The hope is that such large errors in distance can be readily identified through other means (or by "sanity checking" that these erroneous results make sense).

        • Morphology

          E.g., Assume all globular clusters of certain concentration are same absolute magnitude.

          Or, assume all galaxies of one type (e.g., spirals with a certain disk to bulge ration) have the same absolute magnitude.
        • Ensembles

          E.g. Assume that the third brightest galaxy in cluster of galaxies are typically all about name M; or the tenth brightest giant star in globular cluster is of fixed M.

      • How do we get the absolute magnitudes of different standard candles??

        • This is the basis of the Cosmic Distance Scale.
        • The "bottom rung" of distance ladder is trigonometric parallaxes.
        • The top part of the diagram above shows the Earth at two different times, and the triangle formed with a nearby star and these two positions of the Earth. The bottom part shows two pictures of the nearby star projected onto more distant stars taken from the two sides of the Earth's orbit. The moving platform of the Earth provides a kind of stereo view of the nearby star over the course of the year, Just like your two eyes allow you to have depth perception. If you stare at the lower images at the right distance and merge mentally the two pictures from the two sides of the Earth's orbit, you will see the nearby star standing in the foreground of the more distant stars in 3-D (or else you will get a headache).
          • This geometric method is perhaps the most reliable method for estimating distances.
          • But only good to 100 parsecs (McCormick Observatory), 1000 parsecs (HIPPARCOS Mission), but hope for 10,000 parsecs in coming years (Gaia).

            A NASA mission was planned that would have enabled parallaxes to be measured accurately to 100,000 parsecs; but this project, the Space Interferometry Mission, was unfortunately just canceled by NASA at the end of 2010.
          • To get distances of more distant objects, have to use less direct means using the distance modulus equation.
        • An example of how the "Ladder" builds is as follows:

          • We obtain parallax of a nearby RR Lyrae star and obtain its absolute magnitude.

          • Find an RR Lyrae in globular cluster. Get distance to globular cluster by apparent magnitude of member RR Lyrae stars. Thereby derive cluster's absolute magnitude.
          • Find a similar globular cluster in the Andromeda galaxy, measure its apparent magnitude and thereby get the distance to Andromeda, and derive the absolute magnitudes for Andromeda-like spiral galaxies.

            Alternatively, find an RR Lyrae in Andromeda Galaxy, etc.
          • Find an Andromeda-like galaxy for which we now can estimate the distance, but which has a supernovae explode in it. Gauge the absolute magnitude of the peak of a supernova lightcurve.
          • Use supernovae of similar type (light curve) to get distances of even more distant things.
          • .
        • A problem with this continuing extrapolation is that it gets shakier and shakier with every step up the ladder.

          • Good parallaxes extremely valuable. McCormick refractor responsible for about 1/3 of ground-based parallaxes before HIPPARCOS.
          • E.g., In 1980s the distance to the Hyades star cluster was found to be 10% off. Since this cluster is used as a low rung on the distance ladder, the entire universe consequently expanded by this 10% systematic change.

    • Robert Trumpler (1930) showed the existence of interstellar absorption by comparing distances of clusters from the brightnesses of their stars to geometric distances from the cluster sizes (i.e., assuming a standard linear size for open clusters). The latter method always gave closer distances.

      • Therefore, stars get dimmer both due to distance and because some light gets absorbed, scattered by dust along line-of-sight.
      • Worse problem near Galactic plane.
      • To account for this dust extinction we can write the distance modulus equation more accurately as:

      • If a distance modulus for a system is being given after correcting for the dust extinction, that distance modulus is written with a "0" subscript: (m - M )0.

      • Determining A is an extremely difficult thing to do.

    • We are often working in a certain filter system and want to understand the distance modulus in this filter system, so it is important to identify the filter; e.g.:
    • Generally, when working in this way, one is not correcting for the dust extinction, but simply stating what the difference in the observed and absolute magnitudes are in a given filter system.

      Unless it is explicitly designated with the addition of a "0" subscript, the reddening is embedded in the distance modulus as written this way.

      Note that the extinction by dust affects each filter differently, and this would be the point of specifying the filters explicitly.

      However, once the reddening is known and can be accounted for, there is actually no need to specify a filter with the distance modulus, since (m - M )0 is the same for all filters.


    THOUGHT PROBLEMS:

    • Explain various ways that one might establish the absolute magnitude of an entire galaxy. The distance to a cluster of galaxies?
    • Supernovae are extremely rare -- they happen on average once a century per galaxy. How are their absolute magnitudes derived?

    Previous Topic: Reduction of CCD Data Lecture Index Next Topic: Filters and Color Indices

    Flux/distance figure modified from http://zebu.uoregon.edu/~js/ast122/lectures/lec03.html. HR diagram from http://www.jb.man.ac.uk/distance/life/sample/stars/. All other material copyright © 2002, 2005, 2007, 2009, 2011, 2015 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.