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<../ptolemy/ptolemy_sculpture.html>


  Magnitudes or the Magnitude System

De-htmlized text version of this page <./index.txt>

apparent magnitude, absolute magnitude, short star catalogue
<./star_catalogue.html>, magnitude difference to intensity ratio a
conversion table <./mag_conv.html>

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      Starlight
      Star bright
      I wish ...

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      <../constellation/ptolemy_venice_1515.html>           The
      invention of the *MAGNITUDE SYSTEM* is often attributed to
      *HIPPARCHOS* (2nd century BC), but this is only a conjecture. The
      system probably did precede *PTOLEMY* (circa 100--175)
      <../ptolemy/ptolemy.html> (To-16 <../../../writer/toomer.html>).
      In any case, Ptolemy effectively established the traditional
      semi-qualitative system in the star catalogue of his * ALMAGEST*
      <../../../reference/toomer.html>. He presented the longitudes,
      latitudes, and magnitudes of 1022 fixed stars divided into 48
      constellations and a handful of nebulae (No-113
      <../../../reference/north.html>; To-14
      <../../../writer/toomer.html>).

      <./ptolemy_magnitude.png>           The magnitude system was a
      description of *STAR BRIGHTNESS IN 6 CLASSES*. 1st magnitude
      described the brightest stars; 2nd magnitude the 2nd brightest,
      and so on down to 6th magnitude which described the faintest that
      could normally be seen by the naked eye
      <../glossary/naked_eye_star.html>. Ptolemy's catalog was certainly
      based on earlier work, but he probably made some additions
      himself. The magnitude system is only semi-quantitative since
      measurements were by the naked eye and assignment of magnitude is
      based on comparison of stars. When the *TELESCOPE* was invented in
      1608 (No-328 <../../../reference/north.html>, Lecture 4.8
      <../../../course/c_ast101/lecture/lec004.html>), vastly many stars
      fainter than 6th magnitude were observed and the magnitude was
      eventually extended higher values (lower brightnesses).

      <./eye.html>           In the *19TH CENTURY* it became possible to
      measure the intensity of light (i.e., energy per unit time per
      unit area) in different wavelength bands. (In fact in the 19th
      century the modern concept of *ENERGY* was first invented and so
      this ability couldn't have come any earlier.) It was then
      discovered that the *TRADITIONAL MAGNITUDE SCALE* corresponded to
      an approximately *LOGARITHMIC* measure of intensity. The
      psycho-physical response of the eye to intensity is, in fact,
      approximately logarithmic at least under some conditions: see the
      not too informative logarithms <./eye.html> note on eye
      response</a>. See also <a href=>.

                The logical thing to do would be to abandon traditional
      magnitudes and define a new magnitude scale based on intensity
      measurements using *BASE 10 LOGARITHMS* with brighter stars having
      higher magnitudes. But no that's not what was done.

      <./magnitude_plot.png>           A 19th century chap named *NORM
      R. POGSON*---and he's lucky I think that his name is merely almost
      forgotten rather than execrated---in 1856 invented a new system
      that retained near consistency with the past. He (or someone else
      about then) noted that 5 traditional magnitudes were about a
      factor 100 increase in intensity in what astronomy defines to be
      the optical wavelength band of light, and so he (i.e., Pogson)
      fixed * 5 MAGNITUDES* as corresponding exactly to a *FACTOR OF 100
      INCREASE IN INTENSITY* on his new logarithmic scale and retained
      that the smaller magnitude corresponds to the brighter object with
      an odious minus sign. His formula for the magnitude M of an object
      of intensity I is


     M=-2.5 * log(I)  + CONSTANT ,

      where the log function is to *BASE 10* and the *CONSTANT* which
      sets the zero-point depends on the particular wavelength band you
      are looking at: the zero-point constants are part of the arcana of
      astronomy and not just anyone gets told what they are.

                The magnitude difference between object of intensity I_1
      and I_2 is


     M_2-M_1=-2.5 * [log(I_2) - log(I_1)] 

            =-2.5 *log(I_2/I_1)   ,

      using the standard logarithm result that the addition/subtraction
      of logarithms is the multiplication/division of their arguments.
      As constructed


     if I_2/I_1=100,

     then log(I_2/I_1)=2  , and

     M_2 - M_1 =  -5   .

      So a factor of a 100 increase in intensity causes a decrease in
      magnitude by a factor 5. The brighter object has the lower
      magnitude. This *WRONG-WAYNESS* of the magnitude scale is a
      constant insult to *RATIONAL CONVENTION* in the opinion of yours
      truly---but no one ever listens to me.

                The inverse formula for obtaining intensity ratio from
      magnitude is


     I_2/I_1  =   10**[-.4(M_2-M_1)]  

      A difference of 1 magnitude corresponds to an intensity change by
      a factor of


     2.511886431509580... (which is irrational I suppose)

   =2.512 approximately

   =2.5 even more approximately .

      The 2.511886431509580... number is in fact the base of the
      astronomical magnitude logarithmic scale, but we never use that
      base explicitly: we always use standard base 10 logarithms with
      the multiplicative factor 2.5.

                As an aid to *CONVERSION* from a magnitude difference to
      an intensity ratio a *CONVERSION TABLE* <./mag_conv.html> is
      provided.

      <../earth/nasa_earth_001.html>           Now the magnitude we
      measure directly for a star or other astro-body is called
      *APPARENT MAGNITUDE*. Apparent magnitude is based on the actually
      intensity we measure on Earth <../earth/earth.html>. Apparent
      magnitude is what Ptolemy measured and we see. Apparent magnitude
      depends on both the luminosity of the star (energy per unit time
      in the wavelength band of interest) and distance through the
      inverse-square law of light <../glossary/inverse_square.html>. To
      get a direct measure of luminosity, we define *ABSOLUTE
      MAGNITUDE*. Absolute magnitude is the apparent magnitude of the
      astro-body as measured at 10 parsecs (or about 32.62 lyr) (Cl-8
      <../../../clayton.html>). see parsec <../glossary/parsec.html>.

                For interest, we present a short star catalogue
      <./star_catalogue.html> of the brightest nearby stars with
      apparent and absolute magnitudes.

      * Sources *

         1. Cl-7ff <../../../writer/clayton.html>
         2. No-113 <../../../writer/north.html>
         3. Se-12ff <../../../writer/seeds.html>
         4. To-14ff <../../../writer/toomer.html>