plotG(vx,vy,char,size,color)

609320ee-d495-47f6-b55c-62484c32e956 35 9 12 decimal

&[DATE] &[TIME] - &[FILENAME]

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Altimetric pressure => km => kPa

XY 0.5 - 107.477 0 101.325 0.5 95.461 1 89.876 1.5 84.559 2 79.501 2.5 74.691 3 70.121 3.5 65.78 4 61.66 4.5 57.752 5 54.048 5.5 50.539 6 47.217 6.5 44.075 7 41.105 7.5 38.299 8 35.651 8.5 33.154 9 30.8 9.5 28.584 10 26.499 10.5 24.54 11 22.699 11.5 20.984 12 19.399 12.5 17.933 13 16.579 13.5 15.327 14 14.17 14.5 13.1 15 12.111 15.5 11.197 16 10.352 16.5 9.571 17 8.849 17.5 8.182 18 7.565 18.5 6.994 19 6.467 19.5 5.979 20 5.529 422 mat :

X XY 1 col : Y XY 2 col : 12 mat header km kPa ΔkPa 13 mat : Pts XY . 12 black plot : 31 line

x kPa11 66.316994 3988.071258 x 2.403022 + 1379.737008 x 28.965069 + 539.638243 x 13.217861 + / - / + / - :

β 0.02264 127.8954 6.35874 31 mat :

x β kPa20 β 1 el β 2 el x β 3 el / - exp * + :

Pressure [kPa] @ geopotential x = [km]

x kPa11 x β kPa20 11 50 11 0 2 2 mat 20 50 20 0 2 2 mat Pts 5 1 sys

Altimetric Pressure kPa @ Geopotential km

x kPa11 x β kPa20 21 mat 1 - x ≤ (x 11 ≤ (& 11 x ≤ (x 20 ≤ (& 21 mat ≡

Above your head is not homogeneous. The first layer up to 11 km is rational. The next layer is expDecay. It was so recognized by experimenters and so reproduced for 'ρ & kPa' from agreed samples. The two segments are declared exact.

Δ11 i 124 range δ11 i el Y i el X i el kPa11 - : xy i el XY i el : 21 line for δ11 21 line :

Δ20 i 2442 range δ20 i 24 - el Y i el X i el β kPa20 - : xy i 24 - el XY i el : 21 line for δ20 21 line :

Residuals header XY Δ11 Δ20 stack augment stack :

Residuals 11 mat km kPa ΔkPa 0.5 - 107.477 0.002 - 0 101.325 0 0.5 95.461 01 89.876 0 1.5 84.559 02 79.501 0 2.5 74.691 03 70.121 0 3.5 65.78 04 61.66 0 4.5 57.752 05 54.048 0 5.5 50.539 06 47.217 0 6.5 44.075 07 41.105 0 7.5 38.299 08 35.651 0 8.5 33.154 09 30.8 0 9.5 28.584 010 26.499 0 10.5 24.54 0.001 11 22.699 0 11.5 20.984 012 19.399 0 12.5 17.933 0.001 - 13 16.579 0 13.5 15.327 014 14.17 0 14.5 13.1 015 12.111 0 15.5 11.197 016 10.352 0 16.5 9.571 017 8.849 0 17.5 8.182 0.001 18 7.565 0.001 18.5 6.994 019 6.467 0 19.5 5.979 0.001 - 20 5.529 0433 mat 11 mat

kPa [20 km . . . 500 km]

]]> plotG(vx,vy,char,size,color)

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

hPa 20 55.3 301140350 0.9 60 0.25 100 0.0004 200 0.0000013 300 0.0000002 400 0.000000044 500 0.000000011 102 mat :

hPa hPa 16 range 12 range el :

Thiele Table

1. Thiele Table ... Thiele(X,Y,Δ) => 'K'

X Y Δ Thiele ε 1018 -^: n X rows : i 1 n range M i 1 el Y i el : for j 2 n range i j n range dM M i j 1 - el M j 1 - j 1 - el - : dM dM abs ε < ε M i j el X i el X j 1 - el - dM / : if : 21 line for for i 1 n range c i 1 el X i el : c i 2 el M i i el : 21 line for Δ 0 ≡ c M if 61 line :

Cfr(j,K,X,x)

j K X x Cfr Optimiz Symbolic j rows(K)-2 N j 1 + el K j 1 + el x X j 1 + el - K j 2 + el / + : c j el N j 1 + el : a j el K j el x X j el - c j el / + : jj j 1 - (1 range c jj el a jj 1 + el : a jj el K jj el x X jj el - c jj el / + : 21 line for a 1 el 71 line :

Plot data & support points

X hPa 1 col : Y hPa 2 col : Kvec X Y 0 Thiele : 31 line

Pts i 1 data rows range p i el . : for data 1 col data 2 col p augment 21 line :

X Kvec 1 col : K Kvec 2 col : 21 line

support hPa . 14 black plot :

Thiele 'K' convergents Kvec 20 55.3 30 0.225734 - 40 63.826171 - 50 4.120423 60 10.343297 100 69.141182 - 62 mat

K 55.3 0.225733634 - 63.826170799 - 4.120423276 10.343296891 69.14118169 - 61 mat

Activate Thiele

K 55.3 0.225733634 - 63.826170799 - 4.120423276 10.343296891 69.14118169 - 61 mat :

j Kvec rows 2 - :

j 4

x g j K X x Cfr :

x g confrac x convert maple 62500000 x * 4077905753 / - 1104547646594252873604444808981266069045285607952144000000000 / + 183502509070258435072797995774200728473947478826466735068617542502245232699098075283142216000000000000000000000000000 x 242706415173820906672104500462991759342042101314029327836829371400016007104395466809316511496611874855083107160279841548965679241402669052588775198880228107202000000000000000000 / + 3432030698505130768095690679606618178112406470672951249091384678181858373900171339730770825847780881438675297495445596230718363720421478953117706479926605143101658668902757418854793533891883267785815428328404953110697533954187609893742908936904784293929434664397972093814641920000000000000000000000000000000 x 283388880063750795772949689034441710915229454064274145336368241117253752095627773401003628103374029119198309680291389578991530586694027446800000000000000000 / - (* / + (* / -

x J_Frac 0.015326494 x * - 4.151357199 + 338.252073024 x 16.217057959 + 796.264182051 x 38.608311339 - / + / - :

View/export Thiele continued fraction from symbolic expansion ... optimiz NONE x g 55310 / x 20 - 112866817500000000 / - x 30 - 638261707991000000000 / - x 40 - 1030105819250000000 / x 50 - 103432968911000000000 / x 60 - 6914118169100000000 / - / + / + / + / + / +

x c 116 x ≤ (x 100 ≤ (& cases :

x J_Frac x g ≡

60 10 60 10 - 2 2 mat x J_Frac x c * support 3 1 sys

Accepted function

hPa 20 55.3 301140350 0.9 60 0.25 100 0.0004 200 0.0000013 300 0.0000002 400 0.000000044 500 0.000000011 102 mat :

0.00053968551356535

60 J_Frac 0.250000005

support hPa . 14 black plot :

Global up 500 km ... above 85 km use linterp

x hPa500 55 x 20 - 2272727271000000000 / - x 30 - 15888888889250000000 / - x 40 - 40961673291000000000 / x 50 - 103595910191000000000 / x 60 - 701239649811000000000 / - x 100 - 692512099250000000 / - x 200 - 1179600992162500000 / x 300 - 3866966740000000 / x 400 - 584461 - / + / + / + / + / + / + / + / + / + :

x c 120 x ≤ (x 500 ≤ (& cases :

85 5 85 5 - 2 2 mat x hPa500 ( x c * support 3 1 sys

https://www.grc.nasa.gov/WWW/K-12/airplane/atmosmet.html

Atmospheric kPa <= m

h NASA 101.29 288.14 0.00649 h * - 288.08 / (5.256^* 1000 - h ≤ (h 11000 ≤ (& 22.65 1.73 0.000157 h * - exp * 11000 h ≤ (h 25000 ≤ (& 2.488 141.89 0.00299 h * + 216.6 / (11.388 -^* cases :

h NASA 5.53 - 0 ≡ h 1500025000 solve 20000

x NASA 11000 40 11000 10 - 2 2 mat 25000 10 25000 10 - 2 2 mat 3 1 sys

(N/sq.m)/100=kPa

alt . . . . . . altitude in kilometers.

sigma . . . density divided by sea-level density.

delta . . . . pressure divided by sea-level pressure.

theta . . . . temperature divided by sea-level temperature.

temp . . . . temperature in kelvins.

press . . . . pressure in newtons per square meter.

dens . . . . density in kilograms per cubic meter.

a . . . . . . . the speed of sound in meters per second.

visc . . . . viscosity in 10**(-6) kilograms per meter-second.

k.visc . . . kinematic viscosity in square meters per second.

N/m²=Pa

http://www.pdas.com/atmosTable1SI

]]>

kPa [20 km . . . 86 km]

]]> plot(data,char,size,color)

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

data from Ralph Carmichael July 2017 hPa 20 55.3 301240 2.87 50 0.8 60 0.22 70 0.052 86 0.0037 72 mat :

Thiele Table

1. Thiele Table ... Thiele(X,Y,Δ) => 'K'

X Y Δ Thiele ε 1018 -^: n X rows : i 1 n range M i 1 el Y i el : for j 2 n range i j n range dM M i j 1 - el M j 1 - j 1 - el - : dM dM abs ε < ε M i j el X i el X j 1 - el - dM / : if : 21 line for for i 1 n range c i 1 el X i el : c i 2 el M i i el : 21 line for Δ 0 ≡ c M if 61 line :

Cfr(j,K,X,x)

j K X x Cfr Optimiz Symbolic j rows(K)-2 N j 1 + el K j 1 + el x X j 1 + el - K j 2 + el / + : c j el N j 1 + el : a j el K j el x X j el - c j el / + : jj j 1 - (1 range c jj el a jj 1 + el : a jj el K jj el x X jj el - c jj el / + : 21 line for a 1 el 71 line :

X hPa 1 col : Y hPa 2 col : 12 mat support hPa . 14 black plot : Kvec X Y 0 Thiele : 31 line

X Kvec 1 col : K Kvec 2 col : 21 line

x c 120 x ≤ (x 100 ≤ (& cases :

Thiele 'K' convergents Kvec 20 55.3 30 0.230947 - 40 66.438952 - 50 2.601821 60 12.38037 70 135.975056 - 86 0.630825 72 mat

K 55.3 0.230946882 - 66.438952297 - 2.601821246 12.380370058 135.97505603 - 0.630824516 71 mat

Activate Thiele

K 55.3 0.230946882 - 66.438952297 - 2.601821246 12.380370058 135.97505603 - 0.630824516 71 mat :

j Kvec rows 2 - :

j 5

x g j K X x Cfr :

Explode Thiele ... export

View/export Thiele continued fraction from symbolic expansion ... optimiz NONE x g 55310 / x 20 - 115473441500000000 / - x 30 - 664389522971000000000 / - x 40 - 1300910623500000000 / x 50 - 6190185029500000000 / x 60 - 13597505603100000000 / - x 70 - 157706129250000000 / / + / + / + / + / + / +

60 10 60 10 - 2 2 mat x g x c * support 3 1 sys

Accepted function