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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