Rules-of-thumb for splitting double stars with Peterson and Lord algorithm split calculators

fisherka@csolutions.net Rev. 2/3/2008

Go to: Peterson Split Calculator for secondaries between 8.5 and 11.5 magnitudes
Lord Performance Index Split Calculator with Tung's Continuous Performance Indices
Haas Split Table for secondaries brighter than 8.5 mags and components with a 2.0 to 4.0 magnitude difference
Lord Performance Index Split Calculator for secondaries brighter than 13.0 magnitudes

This note discusses an amateur astronomy rule-of-thumb for resolving unequal binaries where the secondary is brighter than magnitude 11. Equal binaries are also discussed. Historical and current major models to predict the splitting of doubles are reviewed.

In the tetralogy of the observing act - eye, telescope, atmosphere and object - modern models tend to focus telescopic resolution performance, seeing and object intensity. By their nature, these models do not include human visual acuity. Beginning double-star observers should be aware that many modern models do not provide practical predictions for about one-half of the double stars that beginners are likely to encounter. One-half of common double star observing lists involve secondaries that are fainter than magnitude 9.0. Most modern double star split calculators do not accurately predict binary resolution where the secondary is below magnitude 9.0.

Double star resolution algorithms could be most improved by focusing future work on adding a visual acuity factor to existing telescopic performance models. If the goal is providing amateurs with practical useable information, further improvement to the accuracy of telescope performance models is of secondary importance.

Nomenclature to describe splitting doubles

Haas (2006) suggests the following nomenclature to describe splitting doubles without numerical physical criteria, but appears to be based on the size of seeing disk. From take-offs from Haas's diagram and resolution descriptions in Rutten (2002), Table 1 suggests nomenclature for describing the degree of a binary split:

How frequently do amateurs encounter photopic doubles brighter than 8.5 magnitudes and scotopic doubles dimmer than 8.5 magnitudes?

About 55% of double stars likely to be encountered by beginning amateur double star observers, as reflected in Table 2, involve secondary components where the magnitude of the secondary is equal to or less than magnitude 9.0 and scotopic vision may be triggered.

In another amateur project, this author prepared a consolidated catalogue of 454 double stars from prominent double star observing lists like the Belmont Society Colorful Doubles List, the AstroLeague Double Star Observing List, Mullaney's Celestial Harvest, selected stars from the USNO 2006 Bright Double Star List, and selected stars from the 6th Orbit Catalogue. The distribution of stars in that consolidated list of doubles lists is shown as follows. 249 out of 454 (55%) secondary components are magnitude 9 or fainter.

Table 2 - Distribution of 454 Doubles by Magnitude of 2nd Component and Separation (Arcsecs) Cohorts in Author's Consolidated List of Common Doubles Lists
Mag. of 2nd component Sep. " < 1 2 4 8 16 32 64 128 256 512 > Total
0 1 1
2 1 1
3 1 1 2
4 1 2 3
5 4 2 1 1 1 1 1 11
6 8 7 5 9 9 4 3 1 2 48
7 11 9 6 13 12 12 3 2 3 71
8 14 7 13 8 7 9 6 3 1 68
9 12 7 6 19 8 6 13 8 1 80
10 > 5 15 15 18 15 34 37 18 11 1 169
Total 51 49 47 70 53 66 64 32 18 4 454

General criteria applicable to equal and unequal binary splits - both stars have to be within a telescope's limiting magnitude

  1. Are both stars visible using a Schaefer TLM Calculator?

Equal binary rules

Equal binary rules apply where the secondary is brighter than magnitude 8.5, the difference in magnitudes is less than 2.0. In this range of conditions, your eyes use photopic-daylight vision. See discussion under "Unequal binary rule-of-thumb", below.

  1. What diffraction criteria do you want to use for the split?
    • Rayleigh's criteria - 140/D_mm: Applies to equal doubles measured at 550nm wavelength. Represents about a 74% dip in maximum intensity.
    • Dawes's criteria - 117/D_mm: Applies to equal doubles measured at 550nm wavelength. Represents about a 3.2% dip in maximum intensity.
    • Sparrow's criteria - 107/D_mm: Applies to equal doubles measured at 550nm wavelength.

    For a 60mm aperture, Rayleigh's criteria is about 2.3"; Dawes is 1.9".

  2. What wavelength of light to you want to measure a split in?
  3. Traditionally this is done at 550nm, but if the star is predominately red or if you are using photographic techniques that reach down to 480nm, Rayleigh's criteria can be refigured at a different wavelength. If you drop the wavelength to 480nm, for a 60mm aperture, Rayleigh's criteria is about 2.0"

  4. Test stars for equal binaries
  5. Argyle, in his 2004 book Observing and Measuring Double Stars, filtered the Fourth Catalog of Interferometric Measurements of Binary Stars for equal doubles, yielding a list of equal binaries whose separation is known with a high degree of accuracy. The following is a subset of stars in Argyle's table ordered by right ascension. Stars selected here have known HD designations. Thus, they should be easy for the amateur to find. If this subset or Argyle's full table does not have suitable targets, the Fourth Catalogue can be accessed directly via the internet, imported into Microsoft Excel, and filtered.

    Table 3 - Selected Argyle's High-Precision Equal Binary Test Stars filtered from the Fourth Catalog of Interferometric Measurements of Binary Stars
    Aperture HD_Id Con J2000 ADS_Id CCDM_Id Common Spec magV1 magV2 Sep PA Epoch
    200mm HD225220 And J000440.10 +341554.4 32 00046 +3416 STF3056; HIP000374 K0 7.02 7.3 0.72 148 2002.0
    200mm HD002471 And J002840.92 +371814.0 382 00287 +3718 A 1504; HIP002252 A5 8.12 8.22 0.54 28 2002.0
    300mm HD003304 Cas J003633.60 +560833.0 504 00366 +5609 A 914; HIP002886 F5 7.97 8.05 0.46 226 2002.0
    200mm HD006886 Psc J010939.12 +234739.8 955 01097 +2348 BU 303; HIP005444 F0 6.65 6.78 0.62 290 2002.0
    400mm HD010196 Cas J014133.89 +624036.1 1318 01416 +6241 KR 12; HIP007895 G8V 7.81 7.88 0.37 300 2002.0
    300mm HD011126 Cas J015119.32 +602112.2 1461 01513 +6021 A 951; HIP008629 B8V 7.98 8.26 0.45 209 2002.0
    300mm HD015328 Cet J022759.88 +015739.2 02280 +0158 KUI 8; HIP011474 K0III 6.45 6.66 0.52 29 2002.0
    300mm HIP014255 Cet J030348.94 -054156.8 03038 -0542 RST4220; HIP014255 F2 8.85 8.9 0.42 338 2002.0
    400mm HD019134J Ari J030526.71 +251518.7 2336 03055 +2515 STF 346; HIP014376 B7Vn 5.45 5.47 0.34 165 2002.0
    300mm HD026882 Cam J041835.88 +602938.0 3105 04186 +6030 STT 75; HIP020105 B9 7.33 7.49 0.38 182 2002.0
    300mm HD029193 Tau J043634.75 +194536.4 3329 04366 +1946 STT 86; HIP021465 A2 7.32 7.34 0.47 32 2002.0
    150mm HD031088 Ori J045245.96 -051711.8 3497 04528 -0517 BU 316; HIP022692 F8 7.71 7.75 0.85 181 2002.0
    400mm HD034807 Aur J052155.37 +393422.4 05219 +3934 COU2037; HIP025060 A2 7.31 7.54 0.37 141 2002.0
    300mm HD044109 Ori J062017.14 +074308.0 4951 06203 +0743 A 2719; HIP030120 B9V 6.76 6.83 0.44 60 2002.0
    090mm HD050700 Mon J065408.52-055108.6 5557 06541 -0551 STF 987; HIP033154 A6Vn 6.39 6.55 1.3 170 2002.0
    300mm HD055726 Gem J071507.94 +255249.1 5918 07151 +2553 BU 1023; HIP035070 F8 8.34 8.52 0.45 298 2002.0
    300mm HD081728 Hya J092714.57 -091325.0 1588 09272 -0913 A 1588; HIP046365 A2V 7.2 7.3 0.4 184 2002.0
    200mm HD088478 Sex J101201.15 -061210.1 7675 10120 -0612 HO 44; HIP049961 F2 7.96 8.27 0.58 212 2002.0
    300mm HD090698 LMi J102858.73 +345148.6 7788 10290 +3452 A 2152; HIP051320 F5 8.52 8.79 0.4 26 2002.0
    200mm HD100808 Leo J113618.00 +274652.3 8231 11363 +2747 STF1555; HIP056601 F0V 5.8 6.01 0.71 315 2002.0
    400mm HD108005 CVn J122424.36 +430515.0 8540 12244 +4305 STT 250; HIP060522 F0 7.88 8.02 0.35 341 2002.0
    150mm HD113415 Vir J130346.08 -203459.2 8757 13038 -2035 BU 341; HIP063738 F7V 5.58 5.62 0.75 312 2002.0
    090mm HD114878 CVn J131247.78 +403014.8 8820 13128 +4030 A 1606; HIP064464 K0 8.81 8.91 1.25 20 2002.0
    300mm HD115002 CVn J131323.76 +525138.9 8825 13134 +5252 A 1607; HIP064517 G5 9.34 9.43 0.47 32 2002.0
    300mm HD134213 Lib J150849.87 -060933.5 9515 15088 -0610 RST4534; HIP074116 F5 8.21 8.2 0.41 9 2002.0
    200mm HD136596 Ser J152059.71 +210406.2 9600 15210 +2104 HU 146; HIP075117 G0 8.82 9.09 0.66 137 2002.0
    200mm HD137588 Dra J152435.30 +541245.7 9628 15246 +5413 HU 149; HIP075425 K0 6.68 6.8 0.6 279 2002.0
    150mm HD192659 Cyg J201421.58 +420614.8 13572 20144 +4206 STT 403; HIP099749 B9IV-V 7.3 7.5 0.94 173 2002.0
    200mm HD202519 Cep J211404.85 +581750.3 14784 21141 +5818 STF2783; HIP104812 A0 7.11 7.34 0.7 15 2002.0
    090mm HD208132J Cep J215137.32 +654510.1 15407 21516 +6545 STF2843; HIP107893 A1m 6.37 6.67 1.4 145 2002.0
    400mm HD213530 Cep J223032.62 +613726.0 16011 22305 +6137 HU 981; HIP111112 A0 6.98 7.23 0.29 223 2002.0
    200mm HD219617 Aqr J231704.97 -135104.0 16644 23171 -1351 BU 182; HIP114962 F8IV 8.16 8.38 0.79 47 2002.0
    300mm HD223672 And J235133.00 +420457.0 17050 23515 +4205 STT 510; HIP117646 A5 7.34 7.41 0.55 312 2002.0
    090mm HD223735 And J235209.65 +433032.0 17063 23522 +4331 BU 728 F8 8.04 8.32 1.2 3 2002.0
    400mm HD224646 Cas J235931.85 +544118.6 17151 23595 +5441 A 1498; HIP118287 F5 7.73 7.77 0.38 75 2002.0

Unequal binary rules-of-thumb - based on human visual acuity

A rule of thumb based on Peterson (1954) is summarized in Figure 1. This unequal binary rule is primarily based on the visual acuity of the human eye. It is often said that the resolution limit for the unaided eye is about 70 arcseconds or one arcminute. This applies to the ability of the eye to resolve extended line pairs - not two points. The ability of normal human vision to separate points is around 120 to 140 arcseconds. "Foveal resolution is approximately 70 [arcseconds] and for clear recognition, twice that, so the minimum magnification needed to resolve diffraction limited separations is approximately x25 per inch of aperture, the so-called Whittaker rule." Lord (1994).

  1. If the secondary is brighter than magnitude 8.5 and the difference between the magnitudes of the primary and secondary are less than 2.0, your eyes are using photopic vision. Apply magnification necessary to raise the true separation of the double above an apparent separation of about 120" or Whittaker's rule of 25x per inch of aperture. The unequal magnitude is not strong factor. Separation generally follows the equal binary rule.
  2. If the secondary is brighter than magnitude 8.5 and the difference between the magnitudes of the primary and secondary are between 2.0 and 4.0 magnitudes, your eyes still are using photopic vision but there is a moderate effect on the eye's resolving power caused by the magnitude difference. Further detail on telescopic resolution of binaries in this magnitude range can be found Hass (2006), as summarized in Table 5 and Table 6, below.
  3. For secondaries dimmer than magnitude 8.5 and brighter than magnitude 11.5, your eyes are using scotopic, or dark-adapted, vision. As the secondaries become fainter than mag 8.5, your eye's visual acuity lessens significantly. Peterson's rule summarizes this effect where scotopic vision is used. Reading the required apparent-field-of-view (AFOV) separation from Figure 1 is the easiest method to decide what magnification is required to split a double, for apertures less than 6", Peterson (1954) provides the following equation for the required true-field-of-view (TFOV) minimum separation:

Figure 1 - Peterson's plot of resolution of binaries in 3" telescope used at 45x

Target_sep = ( 10 ^ [ 5/8 (m2 - TLM + 2.4 ) ] ) * Seeing_sep_limit {Eq. 1}

where telescopic limiting magnitude is computed using the outdated form of:

TLM=9.1+(5*LOG10(D_inches))

and where the seeing separation limit is the greater of 1) the smallest separation limit seen on equal binaries with the telescope-eyepiece combination, or 2) the size of the seeing disk based on current atmospheric turbulence as measured in the eyepiece in the field.

There is rough correlation between the seeing disk size and Antoniadi's scale for 8" to 20" telescopes:

Peterson's rule and Figure 1 comport with general knowledge about human visual acuity limit to separate two points - approximately 120 to 140 arcseconds.

Although reading directly from Figure 1 is the easier method, a Peterson split calculator for the secondaries with magnitudes between 8.5 to 11.5 is provided.

The javascript Peterson split calculator was tested for magnitudes 9.0 through 11.0. Figure 2 plots the calculator results on Peterson's original 1954 data and indicates that this javascript implementation returns results consistent with Peterson's original algorithm.

Since Peterson held magnification constant, his two-part model for bright and faint secondaries measures human visual acuity for faint stars. He also includes an atmospheric turbulence component in terms of a seeing disk.


Figure 2 - Plot of test data from javascript implementation of Peterson's algorithm over Peterson's 1954 chart

Since Peterson held magnification constant and only used a 3" telescope, his model does not adequately incorporate factor for the second player in the observing tetralogy - a telescopic performance factor. As Peterson noted, the accuracy of his scotopic performance curve is limited to 3" apertures. Peterson stated that he prepared two other curves for larger apertures and that the slope of the scotopic curve was steeper for larger apertures. This indicates that Peterson's model did not capture all relevant telescopic performance factors.

Unequal binary rule-of-thumb - based on telescope performance

Another approach to the problem of resolving unequal binaries is to focus on telescope performance instead of the response of human eye. Haas (2006) recently published a table of double split resolution limits after this technique:

Table 5 - Haas (2006) Double resolution limit table (TFOV arcsec)
deltaMag Aperture (mm) 60 100 150 200 250 PrimaryLimit SecondaryLimit
0.0 2.0 1.2 0.8 0.6 0.5 4.5 4.5
0.5 2.0 1.2 0.8 0.6 0.5 4.5 5.0
1.0 2.2 1.3 0.9 0.8 0.6 4.5 5.5
1.5 2.5 1.4 1.0 0.9 0.7 4.5 6.0
2.0 3.2 1.9 1.2 1.1 0.9 4.5 6.5
2.5 3.5 2.0 1.4 1.3 1.1 4.5 7.0
3.0 3.7 2.3 1.6 1.5 1.3 4.5 7.5
3.5 4.4 2.4 1.8 1.6 1.5 4.5 8.0
4.0 4.5 2.6 2.0 1.9 1.6 4.5 8.5

Figure 3 - Overlap between Peterson's plot and Haas Table

Haas's table was based on an extraction of data from a French photometry magnitude publication, Revue des Constellations. Haas's table concerns stars brighter than magnitude 6.5 and secondaries down to 4 magnitudes fainter than the primary. The table is limited to brighter stars within those magnitude brackets - secondaries between 4.5 and 8.5 magnitudes and primaries brighter than 6.5 magnitudes. Her limits are presented in terms of the magnitude difference between primary and secondary components and the aperture of the telescope. The table excludes current seeing conditions - the size of the seeing disk - as a factor.

Assuming the "average" brightness of the primaries are 4.5 magnitudes, Haas's table correlates well with the photopic vision portion of Peterson's chart. See Figure 3, above.

Haas's tabular presentation of resolution data follows Treanor (1947). Treanor looked at resolution data for bright equal (average 4.5 mags) doubles, faint equal doubles (average 9.5 mags) and unequal doubles in those ranges using telescopes larger and smaller than 15 inches of aperture. Treanor standardized the resolution data for telescopes of differing sizes by reducing the observed split to a standard scalar. That scale index was the observed resolution divided by the Dawes limit of the telescope for equal binaries. Haas's data in Table 5 can be recast into that standardized format. In Haas's table (Table 5), the first row approximates the Dawes limit for bright equal binaries.

In the following table, the Dawes limit in the first row of each column is Table 5 used as a divisor for all cells beneath it. This standardizes the data across differing apertures:

Table 6 - Haas (2006) Double resolution limit - Standardized performance ( Obs. resolution arcsec / Dawes limit )
deltaMag Aperture (mm) 60 100 150 200 250 PrimaryLimit SecondaryLimit
0 1.0 1.0 1.0 1.0 1.0 4.5 4.5
0.5 1.0 1.0 1.0 1.0 1.0 4.5 5
1 1.1 1.1 1.1 1.3 1.2 4.5 5.5
1.5 1.3 1.2 1.3 1.5 1.4 4.5 6
2 1.6 1.6 1.5 1.8 1.8 4.5 6.5
2.5 1.8 1.7 1.8 2.2 2.2 4.5 7
3 1.9 1.9 2.0 2.5 2.6 4.5 7.5
3.5 2.2 2.0 2.3 2.7 3.0 4.5 8
4 2.3 2.2 2.5 3.2 3.2 4.5 8.5

Looking at Table 6, it is evident that there are two performance groups: telescopes less than 6 inches of aperture and greater than 6 inches of aperture. Treanor used 15 inches as the cut-off for cohort groups; Petersen noted a performance difference at 6 inches of aperture. Grouping averaging data within each 6 inch cohort gives Table 7. Table 7 is graphically presented in Figure 4, which is similar in format to Treanor's 1947 figure of telescopic performance:

Table 7 - Haas (2006) Double resolution limit - Standardized performance - grouped data
deltaMagAperture<=100(mm)>100(mm) PrimaryLimit SecondaryLimit
0 1.0 1.0 4.5 4.5
0.5 1.0 1.0 4.5 5
1 1.1 1.3 4.5 5.5
1.5 1.2 1.5 4.5 6
2 1.6 1.8 4.5 6.5
2.5 1.7 2.2 4.5 7
3 1.9 2.6 4.5 7.5
3.5 2.2 2.8 4.5 8
4 2.3 3.2 4.5 8.5

Figure 4 - Haas (2006) data recast in Treanor (1947) format - seeing limits the potential diffraction limited performance of large apertures

Treanor (1947) stands for the proposition that telescopic performance measured by aperture alone has an effect on the ability to resolve fainter secondaries that is separate from the Dawes limit.

At first glance, Figure 4 seems counter-intuitive. Why would telescopes of a larger aperture have a different performance index (have a higher resolved split / Dawes limit ratio) than smaller apertures? Why would a standardized performance index seemingly be worse in larger apertures that have a smaller Airy disks and that, in common experience, split smaller double star separations? Treanor noted that:

As the graph shows, large telescopes tend to give high values of the ratio, the implication being that they perform, on the whole, less efficiently than the smaller instruments. This last fact is not altogether surprising, since it is well known that, owing to atmospheric turbulence, large telescopes attain their theoretical limit only in exceptionally favorable weather conditions. (Treanor (1947) at 257.)

Treanor (1947) also means that any accurate prediction telescopic double star resolution performance must include a factor for atmospheric seeing.

Modern complex algorithms for splitting binaries

Modern mathematically complex models tend to focus on unifying seeing and telescopic performance into one algorithm, but by design do not include the effect of human visual acuity. As a consequence amateurs should be aware that these models are not intended to accurately model your experience of splitting doubles where the secondary is faint and dark-adapted scotopic vision is involved, i.e. - secondaries below magnitude 9.0. As noted above, that may occur over 50% of binaries that beginning double star observers are likely to encounter.

Chris Lord's Performance Index Model

Lord's performance index model is implemented in Paul Rodman's (iLanga, Inc.) Astroplanner, versions 1.58 and above.

  • How it works
  • Chris Lord's work from the late 1990s expands on earlier work of Treanor and Peterson. Lord uses indices to represent groups of telescope and seeing characteristics. These characteristic coefficients are then used to particularize a general model of stellar object intensity. Lord's algorithm is based on the following equation:

    Chris Lord's Contrast Algorithm - another telescopic performance model

    In paper originally authored in 1979, Lord explored another model for the resolution of unequal binaries - a contrast index model. Unlike his Performance Index Model that incorporates Stevens's power law, Lord's Contrast Index Model is solely a telescopic performance algorithm built around Rayleigh's criteria of telescopic resolution power. The impact of human visual acuity to resolve faint secondaries in not part of this model by design.

    • Theory behind Lord's contrast model
    • When two Airy disks of unequal brightness overlap, the ability to resolve a split depends on two factors - (1) the relative distance between the centers of the disks and the (2) relative brightness of the disks. Combined, these two factors define the contrast between the two overlapping stars.

      In the case of unequal binaries, it is possible for the first diffraction ring of the brighter primary to overwhelm the full-width half-maximum (FWHM) disk. Figure 7. The lack of any difference in contrast renders the fainter secondary unresolvable.

      Lord sought to model the parameters of this contrast relative separation and relative intensity into two indices. The ratio of the distances between the disk centers (r1) and the size of the FWHM disk (r2) are combined into an index. The contrast between the relative brightness of the two stars (I1, I2) is combined into an index. These two indices are then used to modify a basic telescopic performance equation: Rayleigh's criteria.


      Figure 7 - Schematic of two overlapping Airy disks for equal and unequal binaries
    • Lord's implementation of a contrast model
    • Lord 1979 modifies the familiar Rayleigh criteria for telescopic performance:

      sep_arsec = ( 1.22 * lambda_mm / Dmm ) radians * k_arcsec / D_mm {Eq. 8}

      where k_arcsec converts from radians to arcsecs with:

      k_arcsec = 360 degrees /2PI() radians * 60 arcmins per degree * 60 arcsecs per arcmin = 206265 arcsecs / radian {Eq. 9}

      (Lord's paper uses focal length divided by f-ratio (f/#) to express the reciprocal of aperture in his modified Rayleigh equation. 1/D_mm is used here because it is the form more familiar to amateurs.)

      Depending on wavelength of light applied and units used, gathering the coefficients in Rayleigh's criteria reduces to a few forms well known to amateurs:

      sep_arcsecs = 138 / D_mm Rayleigh's criteria in millimeters {Eq. 10}

      sep_arcsecs = 5.43 / D_mm Rayleigh's criteria in inches {Eq. 11}

      sep_arcsecs = 116 / D_mm Dawes's criteria in millimeters {Eq. 12}

      sep_arcsecs = 4.56 / D_inches Dawes's criteria in inches {Eq. 13}

      These equations are in the general form:

      sep_arcsecs = n / D_mm {Eq. 14}

      Lord's contrast model hypothesizes that an intensity factor (F_i) and a separation factor (F_r) could be used to modify Rayleigh's criteria:

      sep_arcsecs = ( F_i * F_r * n ) / D_mm Dawes's criteria {Eq. 15}

      For F_i, Lord uses the relative intensity of the primary and secondary expressed using a minor modification to a well-known magnitude-brightness equation:

      F_i = 100 ^ (0.1 * (Mag_secondary - Mag_primary) ) {Eq. 16}

      For F_r, Lord uses:

      F_r (rho') = Sqrt( 1 - kappa' ) {Eq. 17}

      where kappa' = 1 / ( 1 - contrast index or gamma' ) {Eq. 18}

      Lord provides other conversion questions to find gamma' from the intensity of the primary and secondary components (I1, I2) that are not repeated here.

      The result is that using two parameters - the magnitude difference between the primary and secondary and aperture of the telescope - it is possible to predict the minimum separation required to resolve unequal binaries. Putting it all together reduces to Lord's Contrast Algorithm:

      sep_arcsecs = (100^(0.1*deltaMag)) * 1.01 * rho' * lambda * 206265 / D_mm {Eq. 19}

      Collecting coefficients in Eq. 19 allows one to generate a table in the generic from of:

      sep_arcsecs = n / D_mm {Eq. 20}

      - which Lord provides on pages 12 and 13 of his 1979 paper.

      Lord's n coefficient table and his coefficient graph were replicated in an Excel spreadsheet, although modifications to Eq. 18 were required to Lord's method for computing kappa' in order to obtain the results shown in Lord's table on page 12:

      Table 11 - Lord's Contrast Algorithm n factors with predicted minimum separations for a 3" telescope - Antoniadi I seeing assumed
      delta_V n Lord's_n D_mm Sep
      0.0 114.6 115 76.2 1.5
      1.0 168.9 169 76.2 2.2
      2.0 230.4 230 76.2 3.0
      3.0 303.5 302 76.2 4.0
      4.0 392.8 391 76.2 5.2
      5.0 503.1 499 76.2 6.6
      6.0 640.6 640 76.2 8.4
      7.0 812.6 804 76.2 10.7
      8.0 1028.5 1016 76.2 13.5
      9.0 1299.9 1282 76.2 17.1
      10.0 1641.6 1646 76.2 21.5

      Figure 8 - Lord's contrast model - n values by delta magnitudes - for use in sep_arcsecs = n / D_mm

      With respect to incorporation of seeing in the model, Lord notes, "[i]t is also assumed the telescope is a good one and the seeing is perfect (Antoniadi I)." At pages 15, 17 and 18 of his 1979 paper, Lord suggests how the model might be supplemented for seeing. His general discussion is not expressed as an additional algrebraic term in the general contrast index model, Eq. 19.

      The Excel spreadsheet reconstructing Lord's contrast index model is available for those wishing to study Lord's contrast model in more depth. (The modification required to match Lord's table is that kappa' was found from 1/gamma' and not 1/(1-gamma'). All of the coefficients computed in Table 10 match Lord's computed coefficients to less than +-1%.)

      Lord's n coefficients have the appeal of simplicity. Look-up the magnitude difference of your double's components, find the n coefficient and divide by your aperture.

    • Comparison of Lord's Contrast Index Model with Peterson (1954)
    • Like his Telescopic Performance Index algorithm, Lord's Contrast Index Model Lord focuses on telescopic performance. By design it does not include a factor for the reduction of visual acuity due to scotopic vision where the secondary is less than magnitude 9.0.


      Figure 9 - Lord's contrast model plotted against Peterson (1954)
    Luis Arugelles's LIDAC model

    Luis Arugelles developed fuzzy logic software, "Luis ArgŁelles Difficulty Index Calculator (LIDAC), which returns a difficulty index for splitting doubles. This algorithm is based on neural net logic and not physics or human visual physiology, so it is not reviewed here. Like Chris Lord's Performance Index, LIDAC is also implemented in Paul Rodman's (iLanga, Inc.) Astroplanner. Arugelles' LIDAC is worth downloading and trying - either in his original application or in Rodman's Astroplanner. Arugelles also has an extensive double star amateur astronomy project called "The Spirit of 33". The "The Spirit of 33" project is a catalogue of 33 doubles in each of the 88 known constellations.

    Is the future of double split algorithms in improving prediction of telescopic resolution or in adding a visual acuity factor?

    An double split algorithm that would have the most practical value to an amateur is one that incorporates all of the key factors in the tetralogy of the observing event, e.g. in the form:

    sep_arcsecs = F_v * F_tp * F_seeing * F_obj {Eq. 21}

    where -

    sep_arcsecs = F_v = human visual acuity factor {Eq. 22}

    sep_arcsecs = F_tp = telescopic performance factor {Eq. 23}

    sep_arcsecs = F_seeing = atmospheric turbulence or seeing factor {Eq. 24}

    sep_arcsecs = F_obj = object intensity factor {Eq. 25}

    Formulation of a visual acuity factor is beyond this amateur author's ability. Schaefer (1990) did considerable work on this factor in formulating his telescopic limiting magnitude algorithm. However for demonstration purposes, a simple visual acuity factor based on a reciprocal Stevens's power law can be used to approximate such a factor and shows how it might improve the utility of modern double splitting algorithms to amateurs. In this demonstration example, Lord's performance index algorithm is used to capture telescopic performance, seeing and object intensity:

    sep_arcsecs = F_v * [ Lord's Performance Index Calculator ] {Eq. 26}

    where, the four factor - visual acuity is captured using a reciprocal Stevens's power law:

    F_v = (1 + (0.0001064 * (1 / 10^( magSecondary / - 2.5 ))) {Eq. 27}

    The effect of the visual acuity or scotopic vision factor on Lord's Performance Index model is shown by table and chart as follows:

    Table 11 - Demonstration of scotopic vision factor applied to 3" telescope to predict the minimum separation in arcsecs needed to resolve a double
    magPri magSec deltaMag Lord_PIM F_v Minimum_sep
    4.5 5.8 1.3 2 1.0 2.0
    4.5 6.5 2 2.4 1.0 2.5
    4.5 7.5 3 3 1.1 3.3
    4.5 8.5 4 3.8 1.3 4.8
    4.5 9.5 5 4.8 1.6 7.8
    4.5 10.5 6 6.1 2.6 15.7
    4.5 11.5 7 7.7 4.9 38.1

    Figure 10 - Lord's performance index model modified with a scotopic visual acuity factor plotted against Peterson (1954)

    Testing your own eye's resolution

    It is also useful to test the resolving ability of your eyes to rule them out as a factor in splitting doubles. Although stars are point objects, you can test the resolving power of your eyes against line charts and the normal human eye linear resolving power of approximately 70 arcseconds. The References section includes resources to obtain, download and print either USAF 1951 line resolution charts or the Koren 2003 Lens Test Chart. Print one of the charts using the highest resolution available on your printer and tape the chart about 8-10 feet away to a wall. Measure the distance from the chart to your chair. Your objective is to determine the angular size of the smallest resolvable black and white line-pair. Use the half angle formula to determine the angular size of one white and black line pair. This will only give a rough measurement of the resolving ability of your eyes, since the charts are printed on a laser printer, instead of plastic film.

    Disclaimers

    This note is amateur astronomer product. Corrections to any errors are welcomed and appreciated.

    References:

    Arguelles, Luis. 2006. The Spirit of 33 Project. (Website). The Spirit of 33

    Arguelles, Luis. 1999. Fuzzy Splitting. (LADIC software). http://www.carbonar.es/s33/Fuzzy-splitting/fuzzy-splitting.html

    Argyle, R. (ed). 2004. Tables 2.1-2.7, Test Stars for Binocular through 600mm Telescopes. In Observing and Measuring Visual Double Stars. Springer. ISBN 1-85233-558-0 http://www.springer.com/

    Haas, S. Jan. 2002. Enjoying Unequal Double Stars. Sky & Telescope (Lord's nomogram)

    Haas, S. 2006. Double Stars for Small Telescopes. Sky Publishing. pp. 5-6

    Hartkopf, William I., Brian D. Mason, & Gary L. Wycoff (USNO). Sept. 2005. Fourth Catalog of Interferometric Measurements of Binary Stars. (Webdatabase) http://ad.usno.navy.mil/wds/int4.html

    Koren, Norman. 2006. Understanding Image Sharpness. (Koren 2003 lens test chart). http://www.normankoren.com/Tutorials/MTF5.html

    Lord, Chris. 1979 (Rev. ____). Contrast & Definition. Brayebrook Observatory. http://www.brayebrookobservatory.org/BrayObsWebSite/BOOKS/contrast%26definition.pdf

    Lord, Chris. 1994. Nomogram for Telescopic Resolution of Unequal Binaries. Brayebrook Observatory. http://www.brayebrookobservatory.org/BrayObsWebSite/HOMEPAGE/BRAYOBS%20PUBLICATIONS.html

    Lord, Chris. 1994. A Report on the Analysis of the Telescopic Resolution of Unequal Binaries. Brayebrook Observatory. http://www.brayebrookobservatory.org/BrayObsWebSite/BOOKS/TELESCOPIC%20RESOLUTION.pdf

    Lord, Chris. Jan. 2008. Personal Communication.

    Monaghan, R. Lens. Undated. Resolution Testing. (USAF 1951 Test Charts) http://medfmt.8k.com/mf/resolution.html

    Peterson, H.H. Sept. 1954. S&T. p. 396

    Rodman, Paul, iLanga, Inc. 2007. Astroplanner (software)

    Schaefer, B.E. Feb. 1990. Telescopic Limiting Magnitude. PASP 102:212-229 http://adsbit.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1990PASP..102..212S

    Tung, Brian. 8/14/2006. Personal Communication

    Treanor, P.J. 1946. On the Telescopic Resolution of Unequal Binaries. The Observatory. 66:265. http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1946Obs....66..255T

    _________. ____. USAF 1951 Test Chart. http://members.cox.net/lenstestr1/AIRFORCECHART.jpg

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