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Unequal Double StarResolution Calculator

by

Chris Lord



A Java Script Calculator version of my famous Double Star Nomogram published in Sky & Telescope, Jan. 2002
by Chris Lord


Unequal Doubles Calculator

Unequal Double Star Split Calculator - Chris Lord (1994) telescopic performance index algorithm for secondaries brighter than 13.0 magnitudes with Fried length seeing modification and critical acuity vision zone warning

K. Fisher fisherka@csolutions.net Rev. 2/19/2008

This is an alpha test version of Lord's proposed Fried seeing modification and TLM critical vision zone warning - to the 1994 nomogram - version 6. This verison is a K. Fisher proposal, modified by Chris to rollback the delta 7 magnitude limit.


Chris Lord Algorithm Unequal Double Split Calculator

Aperture (mm): Magnitude of primary:
Magnitude of secondary: brighter than 2.4 of telescopic limiting magnitude.
Aperture class (mm): Obstruction percent: Seeing (spurious disc radius):

Minimum TFOV separation to resolve: (arcseconds) during periods of stability in turbulent air
Dawes criteria: arcsecs | Rayleigh criteria: arcsecs
Rayleigh blur adjusted disk: arcsecs | Fried criteria: arcsecs

Calculator assumptions: Primary and secondary not within 2.4 magnitudes of the telescopic limiting magnitude.

Table 1 - Antoniadi's 5 point seeing scale
ScaleDescriptionDisk diameter computed from D_mmDisk diameter for 8" (203mm) to 20" (508mm) scopes per the Canadian Weather Service Corresponding Lord's step
I Perfect steadiness; without a quiver. < 0.4"I<0.25 rho
II Slight undulating, with moments of calm lasting for several seconds. ~ 0.4-0.9" II>0.25 rho
III Moderate seeing, with larger air tremors. ~ 1.0-2.0" III<0.50 rho
IV Poor seeing, with constant troublesome undulations. ~ 3.0-4.0"IV<1.00 rho
V Very bad seeing, unsuitable for anything except possibly a very rough sketch.< 4"V> 1.50 rho

Source: Antoniadi scale wikipedia
Canadian Weather Service. http://weatheroffice.ec.gc.ca/astro/seeing_e.html
Lord, C. Jan. 2008. Personal Communication.
Notes: Lord's seeing performance characteristic ( n_e * rho) is expressed as a radius. The third column above is twice that radius - a diameter - in order to be comparable to the Canadian Weather Service table.

Math Appendix

Chris Lord's algorithm is:
  • S = 1.033 * 10 ^ [ 1/n * ( Abs(delta mag) - 0.1 ) ] * rho {Eq 1.}
    • where performance index n is the sum of three indices from the following table with a range between 4.0 and 12.0:
    • Table 2 - Performance indices for Lord's algorithm
      Aperture DmmAperture indexObstruction ratioObstruction indexSeeing - suprious diskSeeing index
      <75 4 0 4 I<0.25p 4
      75-150 3 0.1 3 II>0.25p 3
      151-300 2 0.2 2 III<0.50p 2
      301-450 1 0.33 1.5 IV<1.00p 1
      451-600 0.5 0.4 1 V>1.50p 0.5
      600 0.5-0.25 0.5 0.5

    • where rho is Dawes criteria or rho = 116 / D_mm; and,
    • where the secondary is brighter than the phototopic limit for the telescope's aperture where the photopic limit (m_pht) for any aperture D_mm is:

      m_pt =5logD(mm)-0.68 {Eq. 2}

Versions 2 and 5 simple seeing and magnitude constraint

Critical visual acuity zone warning

Visual acuity degrades exponentially as the secondary comes within 2 magnitudes of a telescope's limiting magnitude. Using Stevenson's basic telescope limiting magnitude relationship of:

Telescopic limiting magnitude = 16 + 5 * log10(D_mm/1000) {Eq. 3}

the user is warned when the secondary comes within 2.4 magnitudes of that limit, that the model may not return useable values. The user is allowed to complete the computation.

The 2.4 range is based on Peterson's 1954 unequal double's algorithm:

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.

That algorithm is based on the rapid decline in visual acuity when the secondary is within 2.4 magnitudes of a telescope's limiting magnitude.

Simple seeing sensitivity scalar

rho in Eq. 1 above is replaced with rho'. rho' is a trigger and sensitivity parameter computed by the following algorithm:

  1. Compute the seeing disk radius using Dawes criteria and Lord's seeing performance characteristic.

    Seeing disk radius arcsecs = n_s * (116 / D_mm) {Eq. 4}

  2. Lookup the seeing characteristic n_s (0.25-1.5) from Table 2.
  3. Compute the blur disk radius using Rayleigh's criteria and Lord's seeing performance characteristic.

    Blur disk radius arcsecs = n_s * (138 / D_mm) {Eq. 5}

  4. If the blur disk is larger than the seeing disk, rho' equals the blur disk, otherwise it is equal to the seeing disk.

    If (Blur disk > Seeing disk ) { rho' = Blur disk; } else { rho' = Seeing disk; } {Eq. 6}

Version 3 Fried seeing disk

About

Fried seeing disk is an expression of scintillation expressed in terms of a telescope's aperture. In perfect seeing, the objective lens acts like a aperture 1.6 times the diameter of the current aperture in average seeing. In poor seeing conditions, a telescope sees at an aperture equivalent to one-quarter the size of the current aperture.

Computation

rho in Eq. 1 above is replaced with rho'. rho' is a trigger and sensitivity parameter. In Version 3, three alternative seeing disk sizes (rho') are computed: Dawes, Rayleigh adjusted for blur, and Fried blur disk. The worst-case or largest of those alternatives are selected as rho'. Computation of the blur-adjusted Rayleigh disk is discussed under Version 2 above. The Fried disk size is computed by the following algorithm:

  1. Lookup the Fried seeing characteristic n_sFried (1.6-0.25) from Table 3.
  2. Compute the Fried seeing disk radius using Rayleigh's criteria.

    Blur disk radius arcsecs = (138 / (D_mm * n_sFried ) {Eq. 7}

  3. Choose the worst case seeing disk between Dawes criteria, the Rayleigh blur adjusted disk (version 2) and a Fried disk (version 3).

    rho' = Max of (Dawes seeing disk, Blur disk, Fried disk ) {Eq. 8}

Contribution of each seeing scale to the computed result

The following table for a 75mm refractor, a 250mm 20% obstructed Newtonian and a 1 magnitude primary-secondary difference, illustrates how, in Lord's model, the Dawes or Rayleigh criteria governs seeing and visual acuity at good and average seeing. In poor seeing, the blurred Rayleigh disk or the Fried disk overwhelms the Dawes and Rayleigh resolution criteria.

Table 5 - Sensivity of the model results to various seeing criteria
D_mm deltaMag Seeing Dawes Rayleigh Rayleigh blur Fried disk Minimum separation per model
75 1 I 1.5 1.8 0.5 1 1.9
75 1 II 1.5 1.8 0.5 1.2 2
75 1 III 1.5 1.8 0.9 3.1 4
75 1 IV 1.5 1.8 1.8 4.1 5.5
75 1 V 1.5 1.8 2.8 6.2 8.4
250 1 I 0.5 0.6 0.1 0.3 0.6
250 1 II 0.6 0.7 0.1 0.4 0.6
250 1 III 0.6 0.7 0.3 0.9 1.4
250 1 IV 0.6 0.7 0.6 1.2 1.9
250 1 V 0.6 0.7 0.8 1.9 3

Source: Calculator, above

References

Fried, D.L. 1965. Statistics of a Geometric Representation of Wavefront Distortion. J. Opt. Soc. Am. 55(11):1427-1435.

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, C. 2008. Limit of Telescopic Photopic Vision. (Web article). http://www.brayebrookobservatory.org/BrayObsWebSite/HOMEPAGE/forum/PhotopicTelLim/PhotopicTelescopicLimit.html#TOP accessed Feb. 2008

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