Japanease version 
                                        April 2002
                                        by K.Funakoshi  
  On the Resolution Curve for unequal brightness Double Stars

  Although famous Dawes' Limit(116"/D) apply to equal brightness double stars, there are
many uneqal brightness double stars in the celestial. Then, I think about the extension of
Dawes' Limit to the unequal brightness double stars.
1.Relation between difference magnitude and separation angle of Airy disk
  and diffraction ring

  Figure 1(A) shows separation angle and difference magnitude of Airy disk and diffraction
 ring in the case of 4-inch refractor. The brightness data of diffraction rings when the bright-
 ness of Airy disk is zero magnitude are written in monthly astronomical magazine "Gekkan
 Tenmon 1994 july : Diffraction image and Double Star by Mr.Youich Kimura".
  Figure 1(B) shows the plotted graph where X-axis is difference magunitude,Y-axis is sepa-
 ration angle of Airy disk and diffraction rings. For example, difference of brightness between
 Airy disk and first diffraction ring is 4.47 magnitude, and separation angle is 1.7 arc second
 in case of 4-inch refractor.  Then, the point of (x,Y)=(4.47,1.7) is plotted .(@ of Figure1(B))
 Similarly, considering to second-, third-,forth-diffraction ring, we plotted A B C ,and so on.
     
      
       Figure 1(A)(B)

2.Resolution limit of unequal brightness double stars
  The resolution limit of unequal brightness double stars is based on the concept that it depend
  on the relation of primary star's diffraction ring brightness and secondary star's brightness.
 Figure 2 shows the relation of primary star's k-th diffraction ring and secondary star's Airy disk.
 I think that if separation angle and brightness of primary star's k-th diffraction ring and second-
 ary star's Ariy disk are nearly equal, it is resolution limit of unequal brightness double star.
           
      ΔMk:difference magnitude of primary star's Ariy disk and k-th diffraction ring
          (k=1,2・・)
      Δm:difference magnitude of primary star and secondary star
      Tk:separation angle of primary star's Ariy disk and k-th diffraction ring
        (arc second)(k=1,2・・)
      t:separation angle of primary star and secondary star
        (arc second)
      C:constant number

   Figure 2 relation of primary star's k-th diffraction ring and secondary star's Airy disk

  In Figure 2, resolution limit is
    ΔMk=C×Δm and Tk=t  (k=1,2・・) ・・・・・・(1)       
  wehe C is defined by observation.

3.Resolution curve for 4-inch refractor(provisional version)
  We think cartesian coordinate, X-axis is diffrence magnitude,Y-axis is separation angle.
  Then, we plot the points calculated by above equation (1). The resolution curve is drawing
  by these points and Dawes' Limit point(X,Y)=(0,1.16).  Figure 3 shows resolution curve for
  4-inch refractor (in case of C=0.9).  C=0.9 is defined by Orion's trapezium C,F star's sepa-
  ration as resolution limit of 4-inch refractor.
   
      Figure 3  Resolution curve for 4-inch refractor

   In case of applying other apeature D(mm), using above resolution curve L as basic curve,
   the resolution curve of apeature D refractor is L×100/D.

4.Functional expression of resolution curve
  We consider the resolution curve by Least Square method for 4-inch refractor.
  As shown in Figure 4, 4-th order Polynomial approximation is
    y=(2E-05)x^4+0.0102x^3+0.0042x^2-0.034x+1.1

  [Complement] Above Polynomial is also expression as follows;
   the resolution curve of apeature D refractor is
    y=((2E-05)x^4+0.0102x^3+0.0042x^2-0.034x+1.16")*100/D
     =(0.002/D)*X^4 + (1.02/D)*X^3 +(0.42/D)*X^2 -(3.4/D)*X +116"/D
     =116"/D -(3.4/D)*X +(0.42/D)*X^2 + (1.02/D)*X^3 +(0.002/D)*X^4
     ↓    ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄↓ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄
 
Dawes' Limit term  Extension term for unequal brightness double star resolution

     
     Fig.4 4-th order Polynomial approximation

 Therefor, resolutional condition of apeature Dmm refractor is as follows;
  When x is difference of magnitude and t is separation angle,
   t≧((2E-05)x^4+0.0102x^3+0.0042x^2-0.034x+1.1)*100/D →resolution
   t<((2E-05)x^4+0.0102x^3+0.0042x^2-0.034x+1.1)*100/D →non resolution

5.Resolution curve for central obstraction optics
  In the case of central obstraction optics, brightness of diffraction rings differ to
  the non obstraction optics. Therefor, we draw the resolution curve for central
  obstraction optics in consideration of it. the blue line curve of Figure 5 shows
  resolution curve for 4-inch centrarl obstraction optics. It looks like very queer
  curve.  
  
     Fig.5 Resolution curve for central obstraction optics

6.Comparison to observation data
  We compare the resolution curve to Takahashi FS-128 and Meade 25.4cm Schmidt
  Cassegrain.
 (1)Observation data by FS-128 (Figure 6)
  This data is sent from Mr.Miyazaki resident in Nara city ,Japan. According to Mr.Miyazaki,
  this data depend on FS-128 and observed in the suburbs of Nara city, fairly middle level
  light-polluted skies. Red line curve of figure 6 shows resolution curve.
   
   Figure 6 Observation data by FS-128

 (2)Observation data by 25.4cm Schmidt Cassegrain (Figure 7)
  This data is used from "Double Star for Maniac observer" which is HP of Mr.Nakai resident
 in Hiroshima ,Japan. Red line curve of figure 7 shows resolution curve of central obstraction.
    
   Figure 7 Observation data by 25.4cm Schmidt Cassegrain

7.Double Star resolution Decision Tool (Trial version)
  Figure 8 shows "double star resolution decision tool for refractor" which is
  based on 4-th order polynomial approximation.
  <Input data>
   ・Apeature
   ・Target double star's separation angle and difference of magnitude
   ・Environmental factor: seeing and obseving ability
  <Output data>
   ・Decision of the target double star's resolution
  
   Figure 8 Double Star resolution Decision Tool

  (Above Tool's download is here )

8.Conclusion
  Above resolution curve and resolution decision tool are still provisional.
  It is necessary many observational feed back in oder to make accurate
  resolution tool.


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