Advanced "OVER LONG PATH " technical notice
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REMINDER / RAPPEL
(E)
The real meaning of the word "Tz" is for hours "Time Zone", then there
is 24 Tz (for 24 hours) on the horizontal axis. The Earth is a 360 degree
sphere, then one Tz is also 15 degree, and as the Earth circumference is
40.000 Km then one Tz is also about 1666 Km at the equator level, as weel
as for each meridian, but also for each great circle arc round the world.

(F)
La signification reele du mot "Tz" est "Time Zone" (Fuseaux Horaire) donc
il y a 24 Tz (pour 24 heures) dans le sens horizontal. La terre fait 360
degres spherique et donc chaque Tz est aussi egal a 15 degres d'angle, et
comme la terre a une circonference de 40.000 Km chaque Tz est aussi egal
a environ 1666 Km au niveau de l'equateur, mais aussi sur chaque meridien
de meme que pour tout arc de grand cercle faisant le tour de la terre.


Formulas are :
   Arc AB  = ArcCos ((Cos TzA x Cos TzB) + (Sin TzA x Sin TzB x Cos TZ))
 
   Angle A = ArcCos ((Cos TzB - (Cos AB x Cos TzA)) / (Sin AB x Sin TzA))

   Angle B = ArcCos ((Cos TzA - (Cos AB x Cos TzB)) / (Sin AB x Sin TzB))
The Tz World Map


Example between A and B (Mexico with Mauritius)

Mexico    is about LATITUDE 4.5 Tz  then  TzA =  4.5 x 15 =  67.5
Mauritius is about LATITUDE 7.5 Tz  then  TzB =  7.5 x 15 = 112.5 
Time Zone is about         10.5 Tz  then  TZ  = 10.5 x 15 = 157.5  

Then :

Arc AB  = ArcCos ((Cos TzA x Cos TzB) + (Sin TzA x Sin TzB x Cos  TZ))
------             C  67.5   C 112.5     S  67.5   S 112.5   C 157.5
        = 159 degree length

Angle A = ArcCos ((Cos TzB - (Cos AB x Cos TzA)) / (Sin AB x Sin TzA))
-------            C 112.5    C  159   C  67.5      S  159   S  67.5
        = 95 degree angle

Angle B = ArcCos ((Cos TzA - (Cos AB x Cos TzB)) / (Sin AB x Sin TzB))
-------            C  67.5    C  159   C 112.5      S  159   S 112.5
        = 85 degree angle

(E)
This A to B calculated arc is the shortest distance between A and B with
159 degree length. Translated in Tz that is about 159 / 15 = 10.5 Tz.

For that, A is beaming 95 degree toward East (TwE), and B is beaming 85
degree toward West (TwW) at 360 - B = 360 - 85 = 275 degree for Azimuth.

(F)
Ce calcul de l'arc nous a donne le plus court chemin entre A et B avec
159 degres de longueur. Traduit en Tz c'est 159 / 15 = 10.5 Tz environ.

Pour cela, A beam a 95 degres vers l'Est (TwE), quand B beam lui a 85
degres mais vers l'Ouest (TwW) a 360 - B = 360 - 85 = 275 pour l'Azimut.


The longest distance / Le plus long chemin (for example)

(E)
But there is an other path, a longer distance between A and B, that is
24 - 10.5 = 13.5 Tz ... A is beaming at its opposite 180 + 95 = 275
degree, and B is beaming at its opposite 180 - 85 = 95 degree ...

For that, any other way, you must have "LP conditions" with more than
8 Tz for propagation, in fact "LP conditions" are really the SP axis.

Second time, you must have more than 8 daylight hours between A and B,
you must use the complementary angle 180+A and 180-B for Azimuths, and
simply add 12 hours on SP hours to obtain all best LP reciprocal hours.

Remind that LP conditions are a solar event at a local sun time, which
is recurring if propagation conditions are stable (sun and geomagnetic).

(F)
Mais il y a aussi un autre chemin, une plus longue distance entre A et
B, c'est 24 - 10.5 = 13.5 Tz ... A beam alors a l'oppose 180 + 95 = 275
degres, et B beam aussi a son oppose 180 - 85 = 95 degres ...

Pour cela, quoi que vous fassiez, il faut avoir des "conditions de LP"
de plus de 8 Tz, c'est a dire en fait d'avoir au moins le Short Path.

Dans un second temps vous devez avoir 8 heures de jour entre A et B, il 
faut utiliser les angles complementaire 180+A et 180-B pour les Azimuts,
puis ajouter 12 heures aux heures de SP pour avoir les heures de LP.

Gardez en memoire que le LP est lie a des conditions d'ensoleillement,
et que les phenomenes LP sont reproductibles (au decalage horaire pres)
ceci bien sur si les conditions de propagation sont stables.


A visual model between two stations named (A) and (B)

           Minimum 8 H dayligth             Normal daylight   
        <----------------------->          <-------------->

  (B) Sun-up                  Sunset (A) Sun-up      Sunset (B) 

  <-------  more than 12 Tz  --------> <-- less than 12 Tz -->

                                            <- Iz/Oz 4Tz ->
                                         <- Ez/Lz 4 to 8Tz ->
  <----- LP is more than 12 Tz -------> <--- SP 8 to 12Tz --->

  <-------  longest distance --------> <- shortest distance ->


A concrete example with SP = 10 Tz (eg. LP conditions) between (A) and (B)
                                         < LP CONDITIONS 8Tz >

                                    10.00L -> TwE    TwW <- 20.00L
                                       <----  SP=10 Tz  ---->

  (B) Sun-up                  Sunset (A) Sun-up        Sunset (B) 

       (Best LP hours are SP+12H)

  8.00L -> TwE               TwW <- 22.00L  
    <-----------  LP=14 Tz ----------->

Note : 'a' and 'b' are the short path azimuthal direction for A and B stations
(E)
Suppose there are SP conditions +10 Tz TwE around 10.00L for A to B by the
shortest distance, reciprocal hours are -10 Tz TwW around 20.00L for B to A.

The evening, around 22.00L (10.00L + 12H = 22.00L), X can try to beam to the
opposite direction TwW (180+a) to make -14 Tz ... when B will try the same
action but the tomorrow around 8.00L (20.00L + 12H = 8.00L) TwE (180-b). 

But attention, contacts with more than 12 Tz are very difficult, you need
a very good beam antenna, often great power, and do not mistake with signals
coming on the back of your antenna by the shortest distance because, when
there is a very good propagation, all paths are possible (short and long). 
 
(F)
Supposez qu'il y a des conditions de SP de +10 Tz TwE vers 10.00L de A vers B
en courte distance, donnant donc -10 Tz TwW environ vers 20.00L de B vers A.

Le soir vers 22.00L (10.00L + 12H = 22.00L) A peut essayer de beamer dans
l'autre sens TwW (180+a) pour faire -14 Tz, alors que B fera l'inverse vers
les 8.00L mais le lendemain matin (20.00L + 12H = 8.00L) en beamant egalement
a l'inverse, c'est a dire TwE (180-b), pour tenter les +14 Tz avec A.

Mais attention, les contacts a plus de 12 Tz de distance sont tres difficiles
a faire, il faut une tres bonne antenne directive, et parfois de la puissance,
de meme qu'il ne faut pas confondre avec le signal venant par l'arriere de
l'antenne car, dans les cas de grande propagation, tout est possible.

Summary / Resume
TwE = Toward East = Beaming between   0 to 180 degrees (angle for azimuth)
TwW = Toward West = Beaming between 180 to 360 degrees (angle for azimuth)

Tz = Time zone  = One SUN time HOUR (around 1666 Km for 15 degrees angle)

Iz = Intra zone = +4 to -4 Tz in a same hemisphere (local distance)
Oz = Outer zone = +4 to -4 Tz over equator with the other hemisphere
Ez = Extra zone =  4 to  8 Tz in a same hemisphere (that TwE or TwW)
Lz = Long  zone =  4 to  8 Tz over equator with the other hemisphere
SP = Short Path =  8 to 12 Tz shortest long distance less than 20.000 km
LP = Long  Path = 12 to 14 Tz longest distance over than 20.000 km

(E)
The objective is for a best understanding by stations listen you. If you
beam "Short Path" you beam for calling between 8 and 12 Tz (TwE or TwW),
when you beam "Over Long Path" you say that you are reaching especially
for more than 12 Tz in order to avoid unexpected other stations between.

(F)
L'objectif est de vous faire comprendre par vos correspondants. Si vous
beamez "Short Path" vous beamer entre 8 et 12 Tz (vers l'est ou l'ouest),
en beamant "Over Long Path" vous dites que vous recherchez specifiquement
plus de 12 Tz afin que les stations entre deux ne s'intercalent pas ...

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