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    Re: Accurate Parallax with numerical examples
    From: Antoine Couëtte
    Date: 2011 Feb 18, 08:16 -0800

    RE :
    http://www.fer3.com/arc/m2.aspx?i=115697
    [NavList] Re: Accurate Parallax Corrected
    From: antoine.m.couette---fr
    Date: 17 Feb 2011 01:21

    Hello to all,

    In the hereabove referenced post, I wrote :

    QUOTE
    2 - MOON CASE ONLY FOR OUR USUAL CELESTIAL NAVIGATION.
    If you observe Body Lower or upper Limb (instead of Body Center) - which is our general case as Navigators - and if you require the Best Accurate Augmented SD, then you cannot (unfortunately) escape at least one iteration. Once Dip + Refraction are taken care of, then from Upper/Lower Limb Height remove/add Geocentric SD (which is immediately available) and from this 1st approximate Topocentric Height enter Step(1) in the Annexed Document. Step(7) yields an updated Augmented SD value. From Upper/Lower Limb Height remove/add such updated Augmented SD value and return to Step(1). At the end of the second pass, Step(7) will yield an updated Augmented SD accurate to much better than 1 arcsecond.
    UNQUOTE

    ******* ******* ******* ******* ******* ******* *******

    In this current post, I wish to further illustrate the points here-above and accordingly give one dual example of an almost "extreme" Moon case as well as one "normal" example, both solved through the algorithm indicated hereabove as earlier detailed in the Annex of http://www.fer3.com/arc/m2.aspx?i=115697 . I will first start from a Lower Limb Height to derive the exact Topocentric CENTER Height and (Topocentric) Augmented SD. I will then proceed from the corresponding Upper Limb Height (i.e. initial LL Height + twice the exact resulting Augmented SD) to retrieve the very same end results, i.e. Topocentric CENTER Height and (Topocentric) Augmented SD.

    Although the iterations are not quite lengthy, once given all first two passes intermediate results (as a courtesy to our NavList Members wishing to crosscheck them), I will afterwards give the relevant end-results only.
    In this entire exercise, all observed angles are freed from both DIP and Refraction.

    *******

    Let us assume that :
    • Dg = 56 (which is close to the minimum Moon-Earth distance in units of Earth Radius), and that
    • we are observing the Moon Lower Height (again already freed from DIP and Refraction effects) at a height HLL = 87°, a very important height for which the difference between SD Geocentric and SD Topocentric is about as big as it could ever be. And that
    • ρ = 0.985

    We can immediately compute the following quantities which depend exclusively on Dg :

    HP equatorial = 1°.023193304 , and
    SDg = 0°.278799665 .

    ******* ******* *******
    1a - EXAMPLE #1 - DUAL EXAMPLE 1 OF 2 STARTING WITH LOWER LIMB

    1a0 - FIRST COMPUTATION

    We compute an initial estimate to Ht as follows :

    Ht0 = HLL + SDg = 87° + 0°.278799665 = 87°.278799665 , with DT=56 and ρ=0.985
    Step(1.0) With Ht0 = 87°.278799665, get p0 = 0°.047845983
    Step(2.0) Hg0 = Ht0 + p0 = 87°.32664565
    Step(3.0) Dg x Cos Hg0 = 2.611946824 and Dg x Sin Hg0 = 55.93905374
    Step(4.0) Dt0 x Cos Ht0 = Dg x Cos Hg0 = 2.611946824 and Dt0 x Sin Ht0 = Dg x Sin Hg0 - ρ = 54.95405374
    Step(5.0) Dt0 = SQRT[(Dt0 x Cos Ht0)+(Dt0 x Sin Ht0)] = 55.01609118 and
    Ht0 = Atan (Dt0 x Cos Ht0)/(Dt0 x Sin Ht0) = 87°.278799667, with a 2 x 10-9 difference with "initial" Ht0 value due to Calculator
    Step(6.0) Sin SDt0 = Sin Dg x (Dg / Dt0) = 0.004952976
    Step(7.0) SDt0a = 0°.283785764 with (SDt0a - SDg) = 17".9500

    *******
    1a1 - ITERATION(1)
    We now start with Ht1 = HLL + SDt0 = 87° + 0°.283785764 = 87°.28378576, and with unchanged Dg=56 and ρ=0.985

    Step(1.1) With Ht1 = 87°.28378576, get p1 = 0°.0047758380
    Step(2.1) Hg1 = Ht1 + p1 = 87°.33115414
    Step(3.1) Dg x Cos Hg1 = 2.607164318 and Dg x Sin Hg1 = 55.93927685
    Step(4.1) Dt1 x Cos Ht1 = Dg x Cos Hg1 = 2.607164318 and Dt1 x Sin Ht1 = Dg x Sin Hg1 - ρ = 54.95427685
    Step(5.1) Dt1 = SQRT (Dt1 x Cos Ht1) + (Dt1 x Sin Ht1) = 55.01608719 and
    Ht1 = Atan (Dt1 x Cos Ht1) / (Dt1 x Sin Ht1) = 87.28378576
    Step(6.1) Sin SDt1 = Sin Dg x (Dg / Dt1) = 0.00492976
    Step(7.0) SDt1a = 0°.283785785 with (SDt1a - SDt0) = 7.56 10-5"

    Note : This is where I recommend to stop, i.e. after ITERATION(1). Just by curiosity, let's iterate one more time to see what happens.

    *******
    1(a)2 - ITERATION(2)
    Starting with Ht2 = HLL + SDt1 = 87° + 0°.283785785 = 87°.28378578, and with unchanged Dg=56 and ρ=0.985
    proceed from Step(1.2) to Step(7.2) and get
    Step(7.2) SDt2a = 0°.283785785. With SDt2a=SDt1a, we have reached the calculator computing accuracy limit.

    Final Result (1a) : SDta = SDt2a = 0°.283785785

    ******* ******* *******

    1b - EXAMPLE #1 - DUAL EXAMPLE 2 OF 2 STARTING WITH LOWER LIMB

    Since HLL = 87°, and since SDt = 0°.283785785, we can accurately compute HUL (Upper Limb) as :
    HUL = HLL + 2 x SDt = 87°.56757157, and
    we will start this second part with Ht0= HUL - SDg = 87°.56757157 - 0°.278799665 = 87°.28877190

    *******
    1(b)0 - ITERATION(0)

    Starting with Ht0 = 87°.28877190, and with unchanged Dg=56 and ρ=0.985
    proceed from Step(1.0) to Step(7.0) and get
    Step(7.0) SDt0b = 0°.283785805 , with (SDt0b - SDg) = 17".9501

    *******
    1(b)1 - ITERATION(1)

    Starting with Ht1 = HUL - SDt0 = = 87°.56757157 - 0°.283785805 = 87°28378576, and with unchanged Dg=56 and ρ=0.985
    proceed from Step(1.1) to Step(7.1) and get
    Step(7.1) SDt1b = 0°.283785785 , with (SDt1 - SDt0) = 7.3080 10-5" and with SDt1b = SDta

    Final Result (1b) : SDtb = 0°.283785785 , and SDtb = SDta , which ensures the consistency and reliability of the Algorithm.

    ******* ******* *******

    2 - EXAMPLE #2
    Let us assume that:
    • Dg = 56 (which is close to the minimum Moon-Earth distance in units of Earth Radius), and that
    • we are observing the Moon Lower Height (once again already freed from DIP and Refraction effects) at a height HLL = 40°, a height for which the parallax in Altitude is still quite significant, and that
    • ρ = 1

    With unchanged D = 56, we have unchanged values for :
    HP equatorial = 1°.023193304 , and
    SDg = 0°.278799665

    *******
    2(b)0 - ITERATION(0)

    We compute an initial estimate to Ht as follows :
    Ht0 = HLL + SDg = 40° + 0°.278799665 = 40°.278799665 , with DT=56 and ρ=1
    proceed from Step(1.0) to Step(7.0) and get
    Step(7.0) SDt0 = 0°.282082448 , with (SDt0 - SDg) = 11"".8180

    *******
    2(b)1 - ITERATION(1)

    Starting with Ht1 = HLL - SDt0 = = 40° + 0°.282082448 = 40°.282082448, and with unchanged Dg=56 and ρ=1
    proceed from Step(1.1) to Step(7.1) and get
    Step(7.1) SDt1 = 0°.282082668 , with (SDt1 - SDt0) = 2.2040 10-7"

    *******
    2(b)2 - ITERATION(2)

    Starting with Ht2 = HLL - SDt1 = = 40° + 0°.282082668 = 40°.282082668, and with unchanged Dg=56 and ρ=1
    proceed from Step(1.2) to Step(7.2) and get
    Step(7.2) SDt2 = 0°.282082668 , with SDt2 = SDt1. We have again reached the calculator computing accuracy limit.

    Final Result (2) : SDt = 0°.282082668

    ******* ******* ******* ******* ******* ******* *******

    CONCLUSION ON THE ALGORITM USED HERE

    If we assume that the "Observer - Earth Center" direction is strictly perpendicular to the Observer's local Horizon, then the iteration algorithm used here will derive "perfect" results in the mathematical sense. In other words and for an Observer on a Sphere this iteration algorithm is mathematically correct (after an infinite number of iterations though …).

    Earth is much better approximated by a revolution Ellipsoid and the line joining Observer and Earth Center is no longer a local vertical in the general case. Since it computes in a vertical local plan containing the Body Center this 2 dimension algorithm accordingly "deviates from the mathematical truth". Nevertheless it DOES take in account the corresponding most (and only) significant parameter, i.e. the actual "ρ" value which has a sensible influence on the distance between Observer and Moon Center. And even under extreme cases, this algorithm converges very fast since after its first iteration (i.e. after 2 computation passes) it reaches an accuracy better than 0.001 arc second for the Augmented SD determination.

    The most important remaining ignored effect when performing computations on a Sphere to "approximate" actual Ellipsoid results is the Ellipsoid Parallax in Azimuth which usually reaches up to 15 or 20 arc seconds for our usual Moon applications, i.e. nothing to really worry about that much ! Under extreme cases Parallax in Azimuth can reach +/ 180° for a finite distance body, a fact (most) always ignored in all Navigation courses, probably because it has no practical consequence (if any) onto our traditional Celestial Navigation. In addition to being Altitude dependent, from an Ellipsoid, the Body-Observer Distance from an Ellipsoid is also azimuth dependent. Again, the resulting change in the Augmented SD is totally insignificant for our CelNav standard applications. However, and as earlier indicated, for accurate artificial satellites work for which the Parallax is MUCH MORE important, taking in account all this geometry requires systematic computations in 3 dimension space.

    *******

    That's all Folks … I think that I have covered everything necessary for our interested Lunarians, and more generally for our curious NavList Members.

    Thanks again to both of you, my Good NavList Friends George Huxtable and Douglas Denny for stirring up my renewed interest towards accurate Parallax and Augmented SD Computations in (local) Horizontal Coordinates, and

    Best Regards to y'all

    Kermit

    Antoine M. "Kermit" Couëtte

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