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ASTR 3130, Majewski [SPRING 2015]. Lecture Notes

ASTR 3130 (Majewski) Lecture Notes


PHOTOMETRY: THE MAGNITUDE SYSTEM

REFERENCE: Birney et al. Chapter 5, 10, 14.

Photometry: Definition

  • 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.
      • f = flux, in photons / sec / cm2, or in erg / sec / cm2

    • (Note: For light flux received on Earth from an extended source, like a nebula or galaxy -- measure flux coming from a solid angle:
      • f = flux, in photons / sec / cm2 / steradian, or in f = flux, in erg / sec / cm2 / steradian )

  • Energy flux received at Earth really given by:
  • OR:

    where the quantities in the first equation are functions of wavelength (and similar quantities can be used as functions of frequency, as in the second equation):

    fλo = stellar flux incident on Earth atmosphere

    Tλ = transmission of atmosphere

    Rλ = efficiency of telescope + detector

    Sλ = transmission function of filter

  • fλo depends not only on how bright something is -- its luminosity -- but on its distance d:

Definition of Magnitudes

Magnitudes are a Brightness Scale, a measurement of received fluxes.

  • Use system of brightness measurement with roots extending to second century B.C. by Greek astronomer Hipparchus.

  • Ptolemy - writes "The Almagest":
    • catalog of 1000 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 < 1 mags.
  • 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 fluxes 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 which 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

    NOTE:

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

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

    • The faintest visible with naked eye are about 5-6 magnitude. But (as you have seen in Lab 1 and 2) the actual limiting magnitude depends very much on prevailing conditions (clouds, moonlight, air and especially light pollution).

    • A finding chart useful for determining the limiting magnitude is given in Norton's Star Atlas.

    • The "bowl" of the Little Dipper is a handy 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): 31 magnitude = over 10 billion times fainter than anything visible to unaided eye.


    Relative Photometry

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

  • We have seen the Pogson magnitude system which gives the following permutations of the same equations above:
    • Δm = 5 be 100x ratio in brightness
    • In these equations we are always comparing pairs of stars and are on a relative scale.
    • When no consideration of 0 point of magnitude system, only brightness ratios (magnitude differences) -- call this relative photometry.

    Absolute Photometry

    When we care about the magnitude of a star on a set (e.g., the Hipparchus) magnitude scale, this is absolute photometry.

    • To set up the absolute scale, astronomers need to decide on a certain star to have a specific magnitude.
    • By convention, we choose the star Vega ( Lyrae) to be 0 mag in all filters. Then:
    • where we can use any measure of brightness for f* --- e.g., different filters giving different wavelength ranges observed --- and the constant only depends on measuring the same set of wavelengths in the Vega spectrum.

      NOTE: THE CONSTANT WILL BE DIFFERENT FOR EACH FILTERED SET OF WAVELENGTHS USED, AND IS ALWAYS TIED TO THE SPECTRUM OF VEGA.

    • In effect, the absolute photometry scale means we are always doing relative photometry to the same star -- Vega.
    • ASIDE: An alternative magnitude scale gaining popularity is the "AB magnitude system", which is not based on Vega, but instead assumes the CONSTANT in the above equation is the same for all filtered magnitudes.
      • Simpler -- don't always have to know what Vega is doing to define the magnitudes.
      • If flux is measured in units of ergs sec-1 cm-2 Hz-1, then we have mAB = -2.5 logf - 48.60.
      • Usually write "AB" subscript for AB magnitudes, e.g., "VAB".

    Magnitudes and Distances

    • Now, assume m1 and m2 are for same star, different d. Then:
    • Example :
    • If the star with magnitude m2 is farther by 10x, then it is 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 π d 2)

        where l is the luminosity of the star and d is the distance.

      • 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 utility of the concept 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.

    • We are generally working in a certain filter system, so important to identify the filter; e.g.:

    • 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

          REFERENCE: Chapter 14 of Birney et al. discusses variable stars and how to find their periods (useful to the RR Lyrae lab in this class).

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

          Supernovae explode with a reasonably well-defined light profile:

        • 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, 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" by photometry in different filters, which is like very coarse spectroscopy.

          Unfortunately, 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.
        • 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 100,000 parsecs in the future from space astrometry missions.

            (This was supposed to be done by NASA's Space Interferometry Mission, but this project was canceled in 2010; ESA's Gaia mission will go after parallaxes on ~10,000 parsec scales however.)
          • 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 the parallax of a nearby RR Lyrae star and obtain its absolute magnitude.

          • Find an RR Lyrae in a globular cluster. Get the distance to the 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 that 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 are extremely valuable. McCormick refractor responsible for about 1/3 of ground-based parallaxes up to a decade or so ago.
          • 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.

      An example of the use of a standard candle that resulted in a significant discovery was the work by Trumpler to prove the existence of interstellar dust.

      • Robert Trumpler (1930) showed existence of interstellar absorption from dust 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 the Galactic plane.
        • To account for this dust extinction we can write the distance modulus equation more accurately as:


      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?
      • Does it bother you that variable stars are called standard candles? Why? How can this be?

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    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 © 2002,2006,2008,2012,2015 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.