ISA 1976 Formulas

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ISA 1976 Formulas

Tropopause pressure [kPa] vs altitude [m] m ISA11 po To λ m * + To / (g - R λ * /^* :

Troposphere pressure kPa vs altitude [m] m ISA25 p1 g 1 * m z1 - (* R T1 * / - exp * :

kPa m ISA11 ≡ m solve maple To 1 - kPa po / ln R * λ * g / - exp + (* λ /

kPa m ISA25 ≡ m solve maple g z1 * - kPa p1 / ln R * T1 * + g / -

kPa kPa :

g 9.806645 : R 287.0528 : po 101.325 : p1 22.6321 : z1 11000 : To 288.15 : T1 216.65 : λ 0.0065 - : 81 sys

g z1 * - kPa p1 / ln R * T1 * + g / 82696827375 - 155474972810000 kPa * 226321 / ln * + (* 1961329 /

To 1 - kPa po / ln R * λ * g / - exp + (* λ / 5763001 - 932921640 kPa * 4053 / ln * 49033225 / exp + (* 13 / -

ISA 1976 atmosphere . . . [0 . . . 25000 . . . kPa, m]

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g 9.806645 : R 287.0528 : po 101.325 : p1 22.6321 : z1 11000 : To 288.15 : T1 216.65 : λ 0.0065 - : 81 sys

m ISA11 po To λ m * + To / (g - R λ * /^* : m ISA25 p1 g 1 m z1 - (* * R T1 * / - exp * : 21 sys

The two ISA 1976 segments x m11 10 x ≤ (x 11000 ≤ (& cases : x m25 111000 x ≤ (x 25000 ≤ (& cases : 21 sys

kPa x ISA11 x m11 * x ISA25 x m25 * 110001281100032 - 22 mat 2000100 Troposphere 12 blue 15 mat 1400050 Tropopause 12 red 15 mat 51 sys :

Atmospheric kPa vs altitude [m]

kPa

kPa switch region 11000 ISA11 11000 ISA25 25000 ISA25 31 mat 22.632 22.632 2.489 31 mat

kPa 1 :

1500 ISA11 84.556

Drummondville [QC]
saturation  98.78 °C 365 ISA11 kPa 97

609.6 ISA11 94.213

ISA altitude[kPa] ... maple_solve kPa alt11 To 1 - kPa po / ln R * λ * g / - exp + (* λ / : kPa alt25 g z1 * - kPa p1 / ln R * T1 * + g / - : 21 sys

sanity check 1500 ISA25 alt25 500 ISA11 alt11 21 mat 150050021 mat

x kPa25 1 2.489 x ≤ (x 22.632 ≤ (& cases : x kPa11 1 22.632 x ≤ (x 101.325 ≤ (& cases : 21 sys

altitude x alt25 x kPa25 * x alt11 x kPa11 * 22.632 16000 22.632 4000 - 22 mat 820000 Troposphere 12 blue 15 mat 488000 Tropopause 12 red 15 mat 101.325 0 + 40 black 15 mat 22.632 11000 + 50 black 15 mat 3213000 commercial 12 black 15 mat 81 sys :

ISA 1976 atmospheric altitude [m] vs kPa

altitude

lowest/highest local barometric pressures recorded

lowest local kPa Typhoon
ISA datum
highest local kPa Mongolia 87 alt11 101.325 alt11 108.57 alt11 31 mat 12670586 - 31 mat

OACI[11 km]... explicit form  ISA 1976 kPa OACI 576300140 kPa * 4053 / ln 490332259329216 / / exp - (* 13 / :

Physics ISA 1976 kPa ISA1976 To 1 - kPa po / ln R * λ * g / - exp + (* λ / :

g 9.806645 : R 287.0528 : po 101.325 : p1 22.6321 : z1 11000 : To 288.15 : T1 216.65 : λ 0.0065 - : 81 sys

85 ISA1976 1457

... Tropopause ... 576300140 kPa * 4053 / ln 490332259329216 / / exp - (* 13 / To 1 - kPa po / ln R * λ * g / - exp + (* λ / ≡

490332259329216 / 5.25587841464921

kPa Δ kPa OACImod kPa ISA1976 - : x λ 1 22.632 x ≤ (x 101.325 ≤ (& cases : 21 sys

OACI[kPa] truncated as published kPa OACIreduced 576300140 kPa * 4053 / ln 5.255 / exp - (* 13 / :

kPa Δoriginal kPa OACIreduced kPa ISA1976 - :

m 1 :

22.632 Δoriginal m 1.59

490332259329216 / 5.255 OACI * ≡

Mutual deviation ISA 1976/OACI

x Δoriginal x λ * 22.632 3 22.632 0.5 - 2 2 mat 101.325 0 + 40 red 1 5 mat 3 1 sys

Comments/conclusion . . .

1. ISA1976 is physics . . . NOT arguable .

2. published OACI truncates ISA 1976.

3. NASA is freak @ datum [0 . . 101.325]

OACI, ISA1976 safe up to 12 km.

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Geopotential atmosphere . . . [°C/ km]

]]> Utilities

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 Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 61 line :

L Δ N xd i 1 N range X i 1 + el L i Δ * + : for X 21 line :

Grad 01120324752618615 56.5 - 56.5 - 44.5 - 2.5 - 2.5 - 20.5 - 86.3 - 28 mat transpose :

laps rate °C /m 56.5 15 + 11000 / 0.0065

X Grad 1 col : Y Grad 2 col : 12 mat h 0 0.5 172 xd : h f X Y h linterp : 31 sys

Data i 1 h rows range °C i el h i el f eval : for h °C augment 21 line :

km Grad . 14 red plot :

Cumul Data Σ :

Geopotential °C @ km [altitude]

Data 11 12 11 72 - 2 2 mat 20 12 20 72 - 2 2 mat km 4 1 sys

Cumulative integration

Cumul 11 228 - + 20 red 1 5 mat 25 1007 - + 20 red 1 5 mat 3 1 sys

°C 1 :

Cumul 23 row Cumul 51 row Cumul Cumul rows row 31 mat °C 11228 - 12 mat 251007 - 12 mat 863146 - 12 mat 31 mat

Global altimetric Formulas

segments [1, 2] . . . ISA 1976 [0 . . . 25 km

segment [3] . . . . . . NASA as is [25 . . . 60 km]

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g 9.806645 : R 287.0528 : po 101.325 : p1 22.6321 : z1 11000 : To 288.15 : T1 216.65 : λ 0.0065 - : 81 sys

m pressure po To λ m * + To / (g - R λ * /^* 0 m & (m 11000 ≤ (& p1 g 1 * m z1 - (* R T1 * / - exp * 11000 m ≤ (m 25000 ≤ (& NOT ISA 1976 [0 ... 25000 m cases :

kPa altitude To 1 - kPa po / ln R * λ * g / - exp + (* λ / 22.632 kPa ≤ (kPa 101.325 ≤ (& g z1 * - kPa p1 / ln R * T1 * + g / - 2.4886 kPa ≤ (kPa 22.632 ≤ (& NOT ISA 1976 [0 ... 25000 m cases :

explicit ISA [25000 ... 11000] g z1 * - x p1 / ln R * T1 * + g / - 82696827375 - 155474972810000 x * 226321 / ln * + (* 1961329 / - ≡

explicit ISA [11000 ... 0] To 1 - x po / ln R * λ * g / - exp + (* λ / 5763001 - 932921640 x * 4053 / ln * 49033225 / exp + (* 13 / - ≡

25000 pressure 2.4886

2 altitude 102 altitude 21 mat NOT ISA 1976 [0 ... 25000 m NOT ISA 1976 [0 ... 25000 m 21 mat

2.4886 altitude 25000

25000 pressure altitude 25000

sanity from other data source x upper 100014189 - 21660125 x * 311 / ln 11.388 / - exp * + (* 299 / :

x up 1 0.02794 x ≤ (x 2.4886 ≤ (& cases :

barometric kPa kPa 22.632 :

m_kPa kPa kPa altitude 0 round 12 mat : 22.632 1100012 mat

x λ 1 2.4886 x ≤ (x 101.325 ≤ (& cases :

pointer 10 - 2500088250008825000 kPa kPa altitude 42 mat :

atmos25km pointer x altitude x λ * x upper x up * 2.4886 25000 + 60 black 15 mat 101.325 0 + 40 black 15 mat 018300 + 150 green 15 mat 1219000 Global 7500 60000 ft 12 green 15 mat 22.632 11000 + 40 black 15 mat 2612000 commercial 12 black 15 mat 91 sys :

atmos25km

green NASA [25000 .. 60000] ,,, red ISA 1976 [0 .. 25000]

measured [kPa] => altitude [m] m_kPa 22.632 1100012 mat

Gloal 7500 crusing ... 7.2351 kPa 60000 ft * 18288 m * ≡

p low 20001440714404100 p * 10129 / ln 5.256 / exp * - (* 649 / : p mid 1000017310020 p * 453 / ln * - (* 157 / : p high 100014189 - 21660125 p * 311 / ln 11.388 / - exp * + (* 299 / : 31 sys

NASA(kPa)... from Maple/solve piecewise/global reserved for upper segment.

x bound 1 0.02794 x ≤ (x 2.4886 ≤ (& cases :

global x low x bound * x mid x bound * x high x bound * 31 sys :

global 0.02794 60000 + 50 black 1 5 mat 2.4886 25000 + 50 black 1 5 mat 3 1 sys

ISA 1976 m(kPa)... SI-Units MSL Mean Sea Level g 9.806645 : (acceleration of gravity m/sec² ≡ R 287.0528 : (realgas constant air m²/°Ksec² ≡ po 101.325 : (datum sea level kPa ≡ p1 22.6321 : (Tropopause kPa 11000 m ≡ z1 11000 : (Tropopause altitude [m] ≡ To 288.15 : (sea level °K ≡ T1 216.65 : (°K @ Tropopause 11000 m ≡ λ 0.0065 - : (temperature laps °C/m ≡ 81 sys

56.5 15 + (11000 / °C m / (* - 0.0065 °C m / * -

Troposphere m/kPa g z1 * - kPa p1 / ln R * T1 * + g / - 82696827375 - 155474972810000 kPa * 226321 / ln * + (* 1961329 / - ≡

Tropopause m/kPa To 1 - kPa po / ln R * λ * g / - exp + (* λ / 5763001 - 932921640 kPa * 4053 / ln * 49033225 / exp + (* 13 / - ≡

kPa ISA11 576300140 kPa * 4053 / ln 490332259329216 / / exp - (* 13 / :

native  SI [m]
[m]converted [ft] 22.6321 ISA11 m * 22.6321 ISA11 m ft / (*'ft * 21 mat 11000 m * 36089'ft * 21 mat

m 1 : m m ft / (*'ft * 21 line 3.28083989501312'ft *

m 2 ln : m m ft / (*'ft * 21 line 2.2741'ft *

m 2 ln : m m ft / (* 21 line 2.2741

As needed export ISA 1976 data set

g 9.806645 : R 287.0528 : po 101.325 : p1 22.6321 : z1 11000 : To 288.15 : T1 216.65 : λ 0.0065 - : 81 sys

L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx - (dx i * + : for U 31 line :

L 2.4886 : H 101.325 : N 100 : 13 mat X L H N xd : 21 line

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 :

kPa ISA g z1 * - kPa p1 / ln R * T1 * + g / - 2.4886 kPa ≤ (kPa 22.6321 ≤ (& To 1 - kPa po / ln R * λ * g / - exp + (* λ / 22.6321 kPa ≤ (kPa 101.325 ≤ (& NOT ISA 1976 cases :

data X X ISA vectorize 1 round vectorize augment :

data data rows row 101.325 012 mat

XY data . 10 black plot :

XY x ISA 101.325 0 + 50 blue 1 5 mat 3 1 sys

4. NASA Altimetric Formula . . .

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Atmospheric kPa <= m ...dressed as given ... 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 :

kPa 1 :

0 NASA kPa 101.4009

101.4 ISA11 m * 6.2 m * -

maple Inverse NASA

y 101.29 288.14 0.00649 h * - 288.08 / (n^* ≡ h solve maple 20001440714404100 y * 10129 / ln n / exp * - (* 649 /

y 22.65 1.73 0.000157 h * - exp * ≡ h solve maple 1000017310020 y * 453 / ln * - (* 157 /

y 2.488 141.89 0.00299 h * + 216.6 / (m -^* ≡ h solve maple 100014189 - 21660125 y * 311 / ln m / - exp * + (* 299 /

pilot chart correction . . .

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t t : (local temperature °C ≡ λ 0.0065 : (ISA temperature laps rate °C/m ≡ H runway datum above sea level ≡ m 1 : (meter unit ≡ h kPa altimeter reading ≡ 51 sys

otherwise formulated

'h' ISA(kPa) altitude

't' local °C from Pt sensor

Local °C flight altitude
add to  altimeter  reading [m] h t H C h 15 t λ H * + (- 273 t λ H * + (+ 0.5 λ * h H + (* - / * H + :

sanity NO correction 300150 C 0

add to altimeter reading 30050 - 0 C m 87.828

landing altitude 30050 - 20 C m 107.627

pressure[kPa] from altimeter

kPa ISA11 5763001 - 932921640 kPa * 4053 / ln * 49033225 / exp + (* 13 / - : kPa t H Flight kPa ISA11 kPa ISA11 t H C + : 31 sys

altimeter reading [m]

landing altitude [m]

't' [°C]from the airport "OR" as measured at aircraft approaching landing runway.

'kPa'Altimeter reading

. . . in short . . .

1. altimeter reads ISA flying altitude from ISA11(kPa)

2. Flight(kPa) adds/subtracts corrected altitude [m]

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85 ISA11 85150 Flight 85300 Flight 85450 Flight 85600 Flight 8530 - 0 Flight 8545 - 0 Flight 8560 - 0 Flight 81 mat m * 1457 m * 1457 m * 1384 m * 1318 m * 1258 m * 1733 m * 1849 m * 1982 m * 81 mat

Table III-1-4-1 a

..... -10,210 is in error .....

M x y Spline A M 112 M cols submatrix : B M 2 M rows 11 submatrix : C M 2 M rows 2 M cols submatrix : j 1 A length 2 range CC j el B C 1 C rows j j submatrix y linterp : for A transpose CC x linterp 51 line :

'x' direction

TableIII °C/Altimeter 60901201501802102402703004506009001200150005510101015151520253550708510 - 10101515202525303045609012015020 - 101520252530354045658513017021530 - 1520253035404555608511517023028540 - 15253040455060657511014522029036550 - 203040505565758090135180270360450715 mat :

'y'

Landing u 1500601470 range : i 1 u rows range C i el TableIII u i el 50 - Spline : for u C 0 round vectorize augment 31 line :

Landing 1 1 sys

runway [m] H 101.325

Local °C flight altitude correction
add [m] to  altimeter reading [h] h t H C h 15 t λ H * + (- 273 t λ H * + (+ 0.5 λ * h H + (* - / * :

x c 1 22.632 x ≤ (x 101.325 ≤ (& cases :

x correct 10 x ≤ (x 1500 ≤ (& cases :

x f 1 87.5 x ≤ (x 101.325 ≤ (& cases :

x 30 - H C x correct * x 50 H C x correct * 2 1 sys

add[m]Alti'r reading

t 15 °C * <

p 92.5 : t 50 - : H 0 : 13 mat

@ runway [0 m]

Corrected landing approach [m alt]

x t H Flight x f * p 0 p p t H Flight 101.325 p t H Flight 101.325 0 p 0 5 2 mat p p t H Flight . 15 black 1 5 mat 3 1 sys

t 15 °C * >

x 15 H Flight x c * x 50 - H Flight x c * x 50 H Flight x c * 3 1 sys

subtract[m]Alt'r reading

ISA ..
flight p ISA11 p t H Flight 21 mat 76298721 mat

kPa 87.5 : t 15 : H 0 : 13 mat

Landing x 15 H Flight x f * x 60 - H Flight x f * x 60 H Flight x f * kPa kPa t H Flight + 50 black 15 mat 41 sys :

Landing approach altitude correction [m]

Landing

kPa t H Flight 1220

kPa 87.5

x Δ x 60 - H Flight x 15 H Flight - (x f * :

compensation [m] kPa Δ 438

Difference landing altitude correction [m]

x Δ kPa Δ + 101.325 kPa Δ + 40 red 1 5 mat kPa 2 kPa Δ * + 40 red 1 5 mat 3 1 sys

uncompensated [m] kPa alt11 1220

Added actual compensation kPa alt11 kPa Δ + 1658

uncompensated [m] kPa alt11 1220

When temperature is LESS than ISA, an aircraft will be LOWER than the altimeter reading. For example, if the OAT is - 40 °C then for a 2000 ft indicated altitude the true altitude is 1520 ft thus resulting in a lower than anticipated terrain separation and a potential obstacle-clearance hazard.

indicated altitude 2000 ft * m 609.6

609.6 463.3 - 146

true altitude 1520 ft * m 463.3

Altimetric pressure kPa 94.213 :

m 1 :

uncompensated [m] kPa alt11 m 610

Excess altitude Δ kPa 40 - 0 Flight kPa 150 Flight - (: m 145

True actual landing True kPa alt11 Δ - : m 464