# Air mass

Air mass is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

In astronomy air mass is a measure of the amount of atmosphere between the observer and the object being observed. It is a function of the zenith angle (the angle between the line of sight an vertical) and the altitude of the observer. It is defined as the integral of the atmospheric density along the line of sight and is usually expressed relative to the air mass at zenith. Thus, looking straight up gives an air mass of one (regardless of observer's altitude) and viewing at any zenith angle greater than zero gives higher values.

You will need to integrate ${\displaystyle \rho }$(h(a,z,x)) where ${\displaystyle \rho }$(h) is the atmospheric density for a given height above sea level, and h(a,z,x) is the height above sea level for a point at distance x along the line of sight. Determining this last function requires some trigonometry.

For this task you can assume:

•   The density of Earth's atmosphere is proportional to exp(-a/8500 metres)
•   The Earth is a perfect sphere of radius 6731 km.

•   Write a function that calculates the air mass for an observer at a given altitude   a   above sea level and zenith angle   z.
•   Show the air mass for zenith angles 0 to 90 in steps of 5 degrees for an observer at sea level.
•   Do the same for the   NASA SOFIA infrared telescope,   which has a cruising altitude of 13,700 meters   (about 8.3 miles),
it flies in a specially retrofitted Boeing 747 about four flights a week.

## 11l

Translation of: Python
V DEG = 0.017453292519943295769236907684886127134
V RE = 6371000
V dd = 0.001
V FIN = 10000000

F rho(a)
‘ the density of air as a function of height above sea level ’
R exp(-a / 8500.0)

F height(Float a; z, d)
‘
a = altitude of observer
z = zenith angle (in degrees)
d = distance along line of sight
’
R sqrt((:RE + a) ^ 2 + d ^ 2 - 2 * d * (:RE + a) * cos((180 - z) * :DEG)) - :RE

F column_density(a, z)
‘ integrates density along the line of sight ’
V (dsum, d) = (0.0, 0.0)
L d < :FIN
V delta = max(:dd, (:dd) * d)
dsum += rho(height(a, z, d + 0.5 * delta)) * delta
d += delta
R dsum

F airmass(a, z)
R column_density(a, z) / column_density(a, 0)

print("Angle           0 m          13700 m\n "(‘-’ * 36))
L(z) (0.<91).step(5)
print(f:‘{z:3}      {airmass(0, z):12.7}    {airmass(13700, z):12.7}’)
Output:
Angle           0 m          13700 m
------------------------------------
0         1.0000000       1.0000000
5         1.0038096       1.0038096
10         1.0153847       1.0153848
15         1.0351774       1.0351776
20         1.0639905       1.0639909
25         1.1030594       1.1030601
30         1.1541897       1.1541908
35         1.2199808       1.2199825
40         1.3041893       1.3041919
45         1.4123417       1.4123457
50         1.5528040       1.5528102
55         1.7387592       1.7387692
60         1.9921200       1.9921366
65         2.3519974       2.3520272
70         2.8953137       2.8953729
75         3.7958235       3.7959615
80         5.5388581       5.5392811
85        10.0789622      10.0811598
90        34.3298114      34.3666656


Translation of: C
with Ada.Text_IO; use Ada.Text_IO;

procedure Main is
subtype double is Long_Float;
package double_io is new Ada.Text_IO.Float_IO (double);
use double_io;
(Float_Type => double);
use Elementary_Double;

Deg : constant := 0.017_453_292_519_943_295_769_236_907_684_886_127_134;

Re : constant := 6_371_000.0;

-- integrate in this fraction of the distance already covered
Dd : constant := 0.001;

-- integrate only to a height of 10000km. efectively infinity
Fin : constant := 10_000_000.0;

function rho (a : double) return double is (Exp (-a / 8_500.0));

function height (a : double; z : double; d : double) return double is
aa : double := Re + a;
hh : double :=
Sqrt (aa * aa + d * d - 2.0 * d * aa * Cos ((180.0 - z) * Deg));
begin
return hh - Re;
end height;

function column_density (a : double; z : double) return double is
sum     : double := 0.0;
d       : double := 0.0;
d_delta : double;
begin
while d < Fin loop
-- adaptive step size to avoid it taking forever
d_delta := Dd * d;
if d_delta < Dd then
d_delta := Dd;
end if;
sum := sum + rho (height (a, z, d + 0.5 * d_delta)) * d_delta;
d   := d + d_delta;
end loop;
return sum;
end column_density;

function air_mass (a : double; z : double) return double is
(column_density (a, z) / column_density (a, 0.0));

z : double := 0.0;
begin
Put_Line ("Angle     0 m              13700 m");
Put_Line ("------------------------------------");
while z <= 90.0 loop
Put(Item => Integer(z), Width => 2);
Put (Item => air_mass (0.0, z), Fore => 8, Aft => 8, Exp => 0);
Put (Item => air_mass (13_700.0, z), Fore => 8, Aft => 8, Exp => 0);
New_Line;
z := z + 5.0;
end loop;

end Main;

Output:
Angle     0 m              13700 m
------------------------------------
0       1.00000000       1.00000000
5       1.00380963       1.00380965
10       1.01538466       1.01538475
15       1.03517744       1.03517765
20       1.06399053       1.06399093
25       1.10305937       1.10306005
30       1.15418974       1.15419083
35       1.21998076       1.21998246
40       1.30418931       1.30419190
45       1.41234169       1.41234567
50       1.55280404       1.55281025
55       1.73875921       1.73876915
60       1.99212000       1.99213665
65       2.35199740       2.35202722
70       2.89531368       2.89537287
75       3.79582352       3.79596149
80       5.53885809       5.53928113
85      10.07896219      10.08115981
90      34.32981136      34.36666557


## AWK

# syntax: GAWK -f AIR_MASS.AWK
# converted from FreeBASIC
BEGIN {
dd = 0.001  # integrate in this fraction of the distance already covered
DEG = 0.017453292519943295769236907684886127134 # degrees to radians
RE = 6371000 # Earth radius in meters
print("Angle          0 m      13700 m")
for (z=0; z<=90; z+=5) {
printf("%5d %12.8f %12.8f\n",z,am_airmass(0,z),am_airmass(13700,z))
}
exit(0)
}
function am_airmass(a,z) {
return am_column_density(a,z) / am_column_density(a,0)
}
function am_column_density(a,z,  d,delta,sum) { # integrates density along the line of sight
while (d < 10000000) { # integrate only to a height of 10000km, effectively infinity
delta = max(dd,(dd)*d) # adaptive step size to avoid it taking forever
sum += am_rho(am_height(a,z,d+0.5*delta))*delta
d += delta
}
return(sum)
}
function am_height(a,z,d,  aa,hh) {
# a - altitude of observer
# z - zenith angle in degrees
# d - distance along line of sight
aa = RE + a
hh = sqrt(aa^2 + d^2 - 2*d*aa*cos((180-z)*DEG))
return(hh-RE)
}
function am_rho(a) { # density of air as a function of height above sea level
return exp(-a/8500.0)
}
function max(x,y) { return((x > y) ? x : y) }

Output:
Angle          0 m      13700 m
0   1.00000000   1.00000000
5   1.00380963   1.00380965
10   1.01538466   1.01538475
15   1.03517744   1.03517765
20   1.06399053   1.06399093
25   1.10305937   1.10306005
30   1.15418974   1.15419083
35   1.21998076   1.21998246
40   1.30418931   1.30419190
45   1.41234169   1.41234567
50   1.55280404   1.55281025
55   1.73875921   1.73876915
60   1.99212000   1.99213665
65   2.35199740   2.35202722
70   2.89531368   2.89537287
75   3.79582352   3.79596149
80   5.53885809   5.53928113
85  10.07896219  10.08115981
90  34.32981136  34.36666557


## BASIC

### BASIC256

Translation of: FreeBASIC
global RE, dd, LIM
RE  = 6371000  #Earth radius in meters
dd  = 0.001    #integrate in this fraction of the distance already covered
LIM = 10000000 #integrate only to a height of 10000km, effectively inLIMity

print "Angle     0 m              13700 m"
print "------------------------------------"
for z = 0 to 90 step 5
print rjust(z,2); "      "; ljust(airmass(0, z),13,"0"); "      "; ljust(airmass(13700, z),13,"0")
next z
end

function max(a, b)
if a > b then return a else return b
end function

function rho(a)
#the density of air as a function of height above sea level
return exp(-a/8500.0)
end function

function height(a, z, d)
#a = altitude of observer
#z = zenith angle (in degrees)
#d = distance along line of sight
AA = RE + a
HH = sqr(AA^2 + d^2 - 2*d*AA*cos(radians(180-z)))
return HH - RE
end function

function column_density(a, z)
#integrates density along the line of sight
sum = 0.0
d = 0.0
while d < LIM
delta = max(dd, (dd)*d)  #adaptive step size to avoid it taking forever:
sum += rho(height(a, z, d+0.5*delta)) * delta
d += delta
end while
return sum
end function

function airmass(a, z)
return column_density(a, z) / column_density(a, 0)
end function


### FreeBASIC

#define DEG 0.017453292519943295769236907684886127134  'degrees to radians
#define RE  6371000  'Earth radius in meters
#define dd  0.001     'integrate in this fraction of the distance already covered
#define FIN 10000000 'integrate only to a height of 10000km, effectively infinity
#define max(a, b) iif(a>b,a,b)

function rho(a as double) as double
'the density of air as a function of height above sea level
return exp(-a/8500.0)
end function

function height( a as double, z as double, d as double ) as double
'a = altitude of observer
'z = zenith angle (in degrees)
'd = distance along line of sight
dim as double AA = RE + a, HH
HH = sqr( AA^2 + d^2 - 2*d*AA*cos((180-z)*DEG) )
return HH - RE
end function

function column_density( a as double, z as double ) as double
'integrates density along the line of sight
dim as double sum = 0.0, d = 0.0, delta
while d<FIN
delta = max(dd, (dd)*d)  'adaptive step size to avoid it taking forever:
sum += rho(height(a, z, d+0.5*delta))*delta
d += delta
wend
return sum
end function

function airmass( a as double, z as double ) as double
return column_density( a, z ) / column_density( a, 0 )
end function

print "Angle     0 m              13700 m"
print "------------------------------------"
for z as double = 0 to 90 step 5.0
print using "##      ##.########      ##.########";z;airmass(0, z);airmass(13700, z)
next z

Output:
Angle     0 m              13700 m
------------------------------------
0       1.00000000       1.00000000
5       1.00380963       1.00380965
10       1.01538466       1.01538475
15       1.03517744       1.03517765
20       1.06399053       1.06399093
25       1.10305937       1.10306005
30       1.15418974       1.15419083
35       1.21998076       1.21998246
40       1.30418931       1.30419190
45       1.41234169       1.41234567
50       1.55280404       1.55281025
55       1.73875921       1.73876915
60       1.99212000       1.99213665
65       2.35199740       2.35202722
70       2.89531368       2.89537287
75       3.79582352       3.79596149
80       5.53885809       5.53928113
85      10.07896219      10.08115981
90      34.32981136      34.36666557


### True BASIC

Translation of: FreeBASIC
FUNCTION max(a, b)
IF a > b then LET max = a else LET max = b
END FUNCTION

FUNCTION rho(a)
!the density of air as a function of height above sea level
LET rho = exp(-a/8500)
END FUNCTION

FUNCTION height(a, z, d)
!a = altitude of observer
!z = zenith angle (in degrees)
!d = distance along line of sight
LET aa = re+a
LET hh = sqr(aa^2+d^2-2*d*aa*cos((180-z)*deg))
LET height = hh-re
END FUNCTION

FUNCTION columndensity(a, z)
!integrates density along the line of sight
LET sum = 0
LET d = 0
DO while d < lim
LET delta = max(dd, (dd)*d)     !adaptive step size to avoid it taking forever:
LET sum = sum+rho(height(a, z, d+.5*delta))*delta
LET d = d+delta
LOOP
LET columndensity = sum
END FUNCTION

FUNCTION airmass(a, z)
LET airmass = columndensity(a, z)/columndensity(a, 0)
END FUNCTION

LET deg = .0174532925199433       !degrees to radians
LET re = 6371000                  !Earth radius in meters
LET dd = .001                     !integrate in this fraction of the distance already covered
LET lim = 10000000                !integrate only to a height of 10000km, effectively infinity
PRINT "Angle     0 m              13700 m"
PRINT "------------------------------------"
FOR z = 0 to 90 step 5
PRINT  using "##      ##.########      ##.########": z, airmass(0, z), airmass(13700, z)
NEXT z
END


## C

Translation of: FreeBASIC
#include <math.h>
#include <stdio.h>

#define DEG 0.017453292519943295769236907684886127134  // degrees to radians
#define RE 6371000.0 // Earth radius in meters
#define DD 0.001 // integrate in this fraction of the distance already covered
#define FIN 10000000.0 // integrate only to a height of 10000km, effectively infinity

static double rho(double a) {
// the density of air as a function of height above sea level
return exp(-a / 8500.0);
}

static double height(double a, double z, double d) {
// a = altitude of observer
// z = zenith angle (in degrees)
// d = distance along line of sight
double aa = RE + a;
double hh = sqrt(aa * aa + d * d - 2.0 * d * aa * cos((180 - z) * DEG));
return hh - RE;
}

static double column_density(double a, double z) {
// integrates density along the line of sight
double sum = 0.0, d = 0.0;
while (d < FIN) {
// adaptive step size to avoid it taking forever
double delta = DD * d;
if (delta < DD)
delta = DD;
sum += rho(height(a, z, d + 0.5 * delta)) * delta;
d += delta;
}
return sum;
}

static double airmass(double a, double z) {
return column_density(a, z) / column_density(a, 0.0);
}

int main() {
puts("Angle     0 m              13700 m");
puts("------------------------------------");
for (double z = 0; z <= 90; z+= 5) {
printf("%2.0f      %11.8f      %11.8f\n",
z, airmass(0.0, z), airmass(13700.0, z));
}
}

Output:
Angle     0 m              13700 m
------------------------------------
0       1.00000000       1.00000000
5       1.00380963       1.00380965
10       1.01538466       1.01538475
15       1.03517744       1.03517765
20       1.06399053       1.06399093
25       1.10305937       1.10306005
30       1.15418974       1.15419083
35       1.21998076       1.21998246
40       1.30418931       1.30419190
45       1.41234169       1.41234567
50       1.55280404       1.55281025
55       1.73875921       1.73876915
60       1.99212000       1.99213665
65       2.35199740       2.35202722
70       2.89531368       2.89537287
75       3.79582352       3.79596149
80       5.53885809       5.53928113
85      10.07896219      10.08115981
90      34.32981136      34.36666557


## Delphi

Works with: Delphi version 6.0
Translation of: GO
const RE  = 6371000; { radius of earth in meters}
const DD  = 0.001;   { integrate in this fraction of the distance already covered}
const FIN = 1e7;     { integrate only to a height of 10000km, effectively infinity}

function rho(a: double): double;
{ The density of air as a function of height above sea level.}
begin
Result:=Exp(-a / 8500);
end;

begin
Result:= degrees * Pi / 180
end;

function Height(A, Z, D: double): double;
{ a = altitude of observer}
{ z = zenith angle (in degrees)}
{ d = distance along line of sight}
var AA,HH: double;
begin
AA := RE + A;
HH := Sqrt(AA*AA + D*D - 2*D*AA*Cos(Radians(180-z)));
Result:= HH - RE;
end;

function ColumnDensity(A, Z: double): double;
{ Integrates density along the line of sight.}
var Sum,D,Delta: double;
begin
Sum := 0.0;
D := 0.0;
while D < FIN do
begin
delta := Max(DD, DD*D); { adaptive step size to avoid it taking forever}
Sum:=Sum + Rho(Height(A, Z, D+0.5*Delta)) * Delta;
D:=D + delta;
end;
Result:= Sum;
end;

function AirMass(A, Z: double): double;
begin
Result:= ColumnDensity(A, Z) / ColumnDensity(a, 0);
end;

procedure ShowAirMass(Memo: TMemo);
var Z: integer;
begin
Z:=0;
while Z<=90 do
begin
Memo.Lines.Add(Format('%2d      %11.8f      %11.8f', [z, airmass(0, Z), airmass(13700, Z)]));
Z:=Z+5;
end;
end;

Output:
Angle     0 m              13700 m
------------------------------------
0       1.00000000       1.00000000
5       1.00380963       1.00380965
10       1.01538466       1.01538475
15       1.03517744       1.03517765
20       1.06399053       1.06399093
25       1.10305937       1.10306005
30       1.15418974       1.15419083
35       1.21998076       1.21998246
40       1.30418931       1.30419190
45       1.41234169       1.41234567
50       1.55280404       1.55281025
55       1.73875921       1.73876915
60       1.99212000       1.99213665
65       2.35199740       2.35202722
70       2.89531368       2.89537287
75       3.79582352       3.79596149
80       5.53885809       5.53928113
85      10.07896219      10.08115981
90      34.32981136      34.36666557

Elapsed Time: 189.304 ms.



## EasyLang

Translation of: FreeBASIC
func rho a .
return pow 2.718281828459 (-a / 8500)
.
func height a z d .
AA = 6371000 + a
HH = sqrt (AA * AA + d * d - 2 * d * AA * cos (180 - z))
return HH - 6371000
.
func density a z .
while d < 10000000
delta = higher 0.001 (0.001 * d)
sum += rho height a z (d + 0.5 * delta) * delta
d += delta
.
return sum
.
func airmass a z .
return density a z / density a 0
.
numfmt 8 2
print "Angle   0 m      13700 m"
print "------------------------"
for z = 0 step 5 to 90
print z & "   " & airmass 0 z & " " & airmass 13700 z
.

## Factor

Translation of: FreeBASIC
Works with: Factor version 0.99 2021-02-05
USING: formatting io kernel math math.functions math.order
math.ranges math.trig sequences ;

CONSTANT: RE 6,371,000     ! Earth's radius in meters
CONSTANT: dd 0.001         ! integrate in this fraction of the distance already covered
CONSTANT: FIN 10,000,000   ! integrate to a height of 10000km

! the density of air as a function of height above sea level
: rho ( a -- x ) neg 8500 / e^ ;

! z = zenith angle (in degrees)
! d = distance along line of sight
! a = altitude of observer
:: height ( a z d -- x )
RE a + :> AA
AA sq d sq + 180 z - deg>rad cos AA * d * 2 * - sqrt RE - ;

:: column-density ( a z -- x )
! integrates along the line of sight
0 0 :> ( s! d! )
[ d FIN < ] [
dd dd d * max :> delta   ! adaptive step size to avoid taking it forever
s a z d 0.5 delta * + height rho delta * + s!
d delta + d!
] while s ;

: airmass ( a z -- x )
[ column-density ] [ drop 0 column-density ] 2bi / ;

"Angle     0 m              13700 m" print
"------------------------------------" print
0 90 5 <range> [
dup [ 0 swap airmass ] [ 13700 swap airmass ] bi
"%2d %15.8f %17.8f\n" printf
] each

Output:
Angle     0 m              13700 m
------------------------------------
0      1.00000000        1.00000000
5      1.00380963        1.00380965
10      1.01538466        1.01538475
15      1.03517744        1.03517765
20      1.06399053        1.06399093
25      1.10305937        1.10306005
30      1.15418974        1.15419083
35      1.21998076        1.21998246
40      1.30418931        1.30419190
45      1.41234169        1.41234567
50      1.55280404        1.55281025
55      1.73875921        1.73876915
60      1.99212000        1.99213665
65      2.35199740        2.35202722
70      2.89531368        2.89537287
75      3.79582352        3.79596149
80      5.53885809        5.53928113
85     10.07896219       10.08115981
90     34.32981136       34.36666557


## Go

Translation of: FreeBASIC
package main

import (
"fmt"
"math"
)

const (
RE  = 6371000 // radius of earth in meters
DD  = 0.001   // integrate in this fraction of the distance already covered
FIN = 1e7     // integrate only to a height of 10000km, effectively infinity
)

// The density of air as a function of height above sea level.
func rho(a float64) float64 { return math.Exp(-a / 8500) }

func radians(degrees float64) float64 { return degrees * math.Pi / 180 }

// a = altitude of observer
// z = zenith angle (in degrees)
// d = distance along line of sight
func height(a, z, d float64) float64 {
aa := RE + a
hh := math.Sqrt(aa*aa + d*d - 2*d*aa*math.Cos(radians(180-z)))
return hh - RE
}

// Integrates density along the line of sight.
func columnDensity(a, z float64) float64 {
sum := 0.0
d := 0.0
for d < FIN {
delta := math.Max(DD, DD*d) // adaptive step size to avoid it taking forever
sum += rho(height(a, z, d+0.5*delta)) * delta
d += delta
}
return sum
}

func airmass(a, z float64) float64 {
return columnDensity(a, z) / columnDensity(a, 0)
}

func main() {
fmt.Println("Angle     0 m              13700 m")
fmt.Println("------------------------------------")
for z := 0; z <= 90; z += 5 {
fz := float64(z)
fmt.Printf("%2d      %11.8f      %11.8f\n", z, airmass(0, fz), airmass(13700, fz))
}
}

Output:
Angle     0 m              13700 m
------------------------------------
0       1.00000000       1.00000000
5       1.00380963       1.00380965
10       1.01538466       1.01538475
15       1.03517744       1.03517765
20       1.06399053       1.06399093
25       1.10305937       1.10306005
30       1.15418974       1.15419083
35       1.21998076       1.21998246
40       1.30418931       1.30419190
45       1.41234169       1.41234567
50       1.55280404       1.55281025
55       1.73875921       1.73876915
60       1.99212000       1.99213665
65       2.35199740       2.35202722
70       2.89531368       2.89537287
75       3.79582352       3.79596149
80       5.53885809       5.53928113
85      10.07896219      10.08115981
90      34.32981136      34.36666557


## Java

Translation of: FreeBASIC
public class AirMass {
public static void main(String[] args) {
System.out.println("Angle     0 m              13700 m");
System.out.println("------------------------------------");
for (double z = 0; z <= 90; z+= 5) {
System.out.printf("%2.0f      %11.8f      %11.8f\n",
z, airmass(0.0, z), airmass(13700.0, z));
}
}

private static double rho(double a) {
// the density of air as a function of height above sea level
return Math.exp(-a / 8500.0);
}

private static double height(double a, double z, double d) {
// a = altitude of observer
// z = zenith angle (in degrees)
// d = distance along line of sight
double aa = RE + a;
double hh = Math.sqrt(aa * aa + d * d - 2.0 * d * aa * Math.cos(Math.toRadians(180 - z)));
return hh - RE;
}

private static double columnDensity(double a, double z) {
// integrates density along the line of sight
double sum = 0.0, d = 0.0;
while (d < FIN) {
// adaptive step size to avoid it taking forever
double delta = Math.max(DD * d, DD);
sum += rho(height(a, z, d + 0.5 * delta)) * delta;
d += delta;
}
return sum;
}

private static double airmass(double a, double z) {
return columnDensity(a, z) / columnDensity(a, 0.0);
}

private static final double RE = 6371000.0; // Earth radius in meters
private static final double DD = 0.001; // integrate in this fraction of the distance already covered
private static final double FIN = 10000000.0; // integrate only to a height of 10000km, effectively infinity
}

Output:
Angle     0 m              13700 m
------------------------------------
0       1.00000000       1.00000000
5       1.00380963       1.00380965
10       1.01538466       1.01538475
15       1.03517744       1.03517765
20       1.06399053       1.06399093
25       1.10305937       1.10306005
30       1.15418974       1.15419083
35       1.21998076       1.21998246
40       1.30418931       1.30419190
45       1.41234169       1.41234567
50       1.55280404       1.55281025
55       1.73875921       1.73876915
60       1.99212000       1.99213665
65       2.35199740       2.35202722
70       2.89531368       2.89537287
75       3.79582352       3.79596149
80       5.53885809       5.53928113
85      10.07896219      10.08115981
90      34.32981136      34.36666557


## jq

Works with: jq

Works with gojq, the Go implementation of jq

Preliminaries

def pi: 4 * (1|atan);

def radians: . * pi / 180;

def lpad($len): tostring | ($len - length) as $l | (" " *$l)[:$l] + .; # Input: a number # Output: a string with$digits fractional decimal digits, with proper rounding
def fmt($width;$digits):
. as $in | tostring | index(".") as$ix
| if test("[eE]") then .
elif $ix then pow(10;$digits) as $p | ($in * $p | round | tostring) as$s
| if test("[eE]") then $s else ($s | index(".")) as $ix | if$ix then $s[:$ix + 1] + $s[$ix+1: $ix+1+$digits]
else $s[:-$digits] + "." + $s[-$digits:]
end
end
else . + "." + "0" * digits
end
| lpad($width); Physics # constants def RE: 6371000; # radius of earth in meters def DD: 0.001; # integrate in this fraction of the distance already covered def FIN: 1e7; # integrate only to a height of 10000km, effectively infinity # The density of air as a function of height above sea level. def rho: (-./8500) | exp; # a = altitude of observer (in m) # z = zenith angle (in degrees) # d = distance along line of sight (in m) def height($a; $z;$d):
(RE + $a) as$aa
| (($aa *$aa + $d *$d - 2 * $d *$aa * ((180-$z)|radians|cos) )|sqrt ) - RE; # Integrates density along the line of sight. def columnDensity($a; $z): { sum: 0, d: 0 } | until (.d >= FIN; ([DD, DD * .d] | max) as$delta  # adaptive step size to avoid it taking forever
| .sum = .sum + ((height($a;$z; .d + 0.5 * $delta))|rho) *$delta
| .d += $delta ) | .sum ; def airmass(a; z): columnDensity(a; z) / columnDensity(a; 0); "Angle 0 m 13700 m", "------------------------------------", ( range(0; 91; 5) | "\(lpad(2)) \(airmass(0; .)|fmt(11;8)) \(airmass(13700; .)|fmt(11;8))" ) Output: Angle 0 m 13700 m ------------------------------------ 0 1.00000000 1.00000000 5 1.00380963 1.00380965 10 1.01538466 1.01538475 15 1.03517744 1.03517765 20 1.06399053 1.06399093 25 1.10305937 1.10306005 30 1.15418974 1.15419083 35 1.21998076 1.21998246 40 1.30418931 1.30419190 45 1.41234169 1.41234567 50 1.55280404 1.55281025 55 1.73875921 1.73876915 60 1.99212000 1.99213665 65 2.35199740 2.35202722 70 2.89531368 2.89537287 75 3.79582352 3.79596149 80 5.53885809 5.53928113 85 10.07896219 10.08115981 90 34.32981136 34.36666557  ## Julia Translation of: FreeBASIC using Printf const DEG = 0.017453292519943295769236907684886127134 # degrees to radians const RE = 6371000 # Earth radius in meters const dd = 0.001 # integrate in this fraction of the distance already covered const FIN = 10000000 # integrate only to a height of 10000km, effectively infinity """ the density of air as a function of height above sea level """ rho(a::Float64)::Float64 = exp(-a / 8500.0) """ a = altitude of observer z = zenith angle (in degrees) d = distance along line of sight """ height(a, z, d) = sqrt((RE + a)^2 + d^2 - 2 * d * (RE + a) * cosd(180 - z)) - RE """ integrates density along the line of sight """ function column_density(a, z) dsum, d = 0.0, 0.0 while d < FIN delta = max(dd, (dd)*d) # adaptive step size to avoid it taking forever: dsum += rho(height(a, z, d + 0.5 * delta)) * delta d += delta end return dsum end airmass(a, z) = column_density(a, z) / column_density(a, 0) println("Angle 0 m 13700 m\n", "-"^36) for z in 0:5:90 @printf("%2d %11.8f %11.8f\n", z, airmass(0, z), airmass(13700, z)) end  Output: Angle 0 m 13700 m ------------------------------------ 0 1.00000000 1.00000000 5 1.00380963 1.00380965 10 1.01538466 1.01538475 15 1.03517744 1.03517765 20 1.06399053 1.06399093 25 1.10305937 1.10306005 30 1.15418974 1.15419083 35 1.21998076 1.21998246 40 1.30418931 1.30419190 45 1.41234169 1.41234567 50 1.55280404 1.55281025 55 1.73875921 1.73876915 60 1.99212000 1.99213665 65 2.35199740 2.35202722 70 2.89531368 2.89537287 75 3.79582352 3.79596149 80 5.53885809 5.53928113 85 10.07896219 10.08115981 90 34.32981136 34.36666557  ## Nim Translation of: Wren import math, strformat const Re = 6371000 # Radius of earth in meters. Dd= 0.001 # Integrate in this fraction of the distance already covered. Fin = 1e7 # Integrate only to a height of 10000km, effectively infinity. func rho(a: float): float = ## The density of air as a function of height above sea level. exp(-a / 8500) func height(a, z, d: float): float = ## Height as a function of altitude (a), zenith angle (z) ## in degrees and distance along line of sight (d). let aa = Re + a let hh = sqrt(aa * aa + d * d - 2 * d * aa * cos(degToRad(180-z))) result = hh - Re func columnDensity(a, z: float): float = ## Integrates density along the line of sight. var d = 0.0 while d < Fin: let delta = max(Dd, Dd * d) # Adaptive step size to avoid it taking forever. result += rho(height(a, z, d + 0.5 * delta)) * delta d += delta func airmass(a, z: float): float = columnDensity(a, z) / columnDensity(a, 0) echo "Angle 0 m 13700 m" echo "------------------------------------" var z = 0.0 while z <= 90: echo &"{z:2} {airmass(0, z):11.8f} {airmass(13700, z):11.8f}" z += 5  Output: Angle 0 m 13700 m ------------------------------------ 0 1.00000000 1.00000000 5 1.00380963 1.00380965 10 1.01538466 1.01538475 15 1.03517744 1.03517765 20 1.06399053 1.06399093 25 1.10305937 1.10306005 30 1.15418974 1.15419083 35 1.21998076 1.21998246 40 1.30418931 1.30419190 45 1.41234169 1.41234567 50 1.55280404 1.55281025 55 1.73875921 1.73876915 60 1.99212000 1.99213665 65 2.35199740 2.35202722 70 2.89531368 2.89537287 75 3.79582352 3.79596149 80 5.53885809 5.53928113 85 10.07896219 10.08115981 90 34.32981136 34.36666557 ## Perl Translation of: Raku use strict; use warnings; use feature <say signatures>; no warnings 'experimental::signatures'; use List::Util 'max'; use constant PI => 2*atan2(1,0); # π use constant DEG => PI/180; # degrees to radians use constant RE => 6371000; # Earth radius in meters use constant dd => 0.001; # integrate in this fraction of the distance already covered use constant FIN => 10000000; # integrate only to a height of 10000km, effectively infinity # Density of air as a function of height above sea level sub rho ($a ) {
exp( -$a / 8500 ); } sub height ($a, $z,$d ) {
# a = altitude of observer
# z = zenith angle (in degrees)
# d = distance along line of sight
my $AA = RE +$a;
my $HH = sqrt$AA**2 + $d**2 - 2 *$d * $AA * cos( (180-$z)*DEG );
$HH - RE; } # Integrates density along the line of sight sub column_density ($a, $z ) { my$sum = 0;
my $d = 0; while ($d < FIN) {
my $delta = max(dd, dd *$d);  # Adaptive step size to avoid it taking forever
$sum += rho(height($a, $z,$d + $delta/2))*$delta;
$d +=$delta;
}
$sum; } sub airmass ($a, $z ) { column_density($a, $z) / column_density($a, 0);
}

say 'Angle     0 m              13700 m';
say '------------------------------------';
for my $z (map{ 5*$_ } 0..18) {
printf "%2d      %11.8f      %11.8f\n", $z, airmass(0,$z), airmass(13700, $z); }  Output: Angle 0 m 13700 m ------------------------------------ 0 1.00000000 1.00000000 5 1.00380963 1.00380965 10 1.01538466 1.01538475 15 1.03517744 1.03517765 20 1.06399053 1.06399093 25 1.10305937 1.10306005 30 1.15418974 1.15419083 35 1.21998076 1.21998246 40 1.30418931 1.30419190 45 1.41234169 1.41234567 50 1.55280404 1.55281025 55 1.73875921 1.73876915 60 1.99212000 1.99213665 65 2.35199740 2.35202722 70 2.89531368 2.89537287 75 3.79582352 3.79596149 80 5.53885809 5.53928113 85 10.07896219 10.08115981 90 34.32981136 34.36666557 ## Phix constant RE = 6371000, // radius of earth in meters DD = 0.001, // integrate in this fraction of the distance already covered FIN = 1e7 // integrate only to a height of 10000km, effectively infinity // The density of air as a function of height above sea level. function rho(atom a) return exp(-a/8500) end function // a = altitude of observer // z = zenith angle (in degrees) // d = distance along line of sight function height(atom a, z, d) atom aa = RE + a, hh = sqrt(aa*aa + d*d - 2*d*aa*cos((180-z)*PI/180)) return hh - RE end function // Integrates density along the line of sight. function columnDensity(atom a, z) atom res = 0, d = 0 while d<FIN do atom delta = max(DD, DD*d) // adaptive step size to avoid it taking forever res += rho(height(a, z, d + 0.5*delta))*delta d += delta end while return res end function function airmass(atom a, z) return columnDensity(a,z)/columnDensity(a,0) end function printf(1,"Angle 0 m 13700 m\n") printf(1,"------------------------------------\n") for z=0 to 90 by 5 do printf(1,"%2d %11.8f %11.8f\n", {z, airmass(0,z), airmass(13700,z)}) end for  Output: Angle 0 m 13700 m ------------------------------------ 0 1.00000000 1.00000000 5 1.00380963 1.00380965 10 1.01538466 1.01538475 15 1.03517744 1.03517765 20 1.06399053 1.06399093 25 1.10305937 1.10306005 30 1.15418974 1.15419083 35 1.21998076 1.21998246 40 1.30418931 1.30419190 45 1.41234169 1.41234567 50 1.55280404 1.55281025 55 1.73875921 1.73876915 60 1.99212000 1.99213665 65 2.35199740 2.35202722 70 2.89531368 2.89537287 75 3.79582352 3.79596149 80 5.53885809 5.53928113 85 10.07896219 10.08115981 90 34.32981136 34.36666557  ## Python """ Rosetta Code task: Air_mass """ from math import sqrt, cos, exp DEG = 0.017453292519943295769236907684886127134 # degrees to radians RE = 6371000 # Earth radius in meters dd = 0.001 # integrate in this fraction of the distance already covered FIN = 10000000 # integrate only to a height of 10000km, effectively infinity def rho(a): """ the density of air as a function of height above sea level """ return exp(-a / 8500.0) def height(a, z, d): """ a = altitude of observer z = zenith angle (in degrees) d = distance along line of sight """ return sqrt((RE + a)**2 + d**2 - 2 * d * (RE + a) * cos((180 - z) * DEG)) - RE def column_density(a, z): """ integrates density along the line of sight """ dsum, d = 0.0, 0.0 while d < FIN: delta = max(dd, (dd)*d) # adaptive step size to avoid it taking forever: dsum += rho(height(a, z, d + 0.5 * delta)) * delta d += delta return dsum def airmass(a, z): return column_density(a, z) / column_density(a, 0) print('Angle 0 m 13700 m\n', '-' * 36) for z in range(0, 91, 5): print(f"{z: 3d} {airmass(0, z): 12.7f} {airmass(13700, z): 12.7f}")  Output: Angle 0 m 13700 m ------------------------------------ 0 1.0000000 1.0000000 5 1.0038096 1.0038096 10 1.0153847 1.0153848 15 1.0351774 1.0351776 20 1.0639905 1.0639909 25 1.1030594 1.1030601 30 1.1541897 1.1541908 35 1.2199808 1.2199825 40 1.3041893 1.3041919 45 1.4123417 1.4123457 50 1.5528040 1.5528102 55 1.7387592 1.7387692 60 1.9921200 1.9921366 65 2.3519974 2.3520272 70 2.8953137 2.8953729 75 3.7958235 3.7959615 80 5.5388581 5.5392811 85 10.0789622 10.0811598 90 34.3298114 34.3666656  ## Raku Translation of: FreeBASIC constant DEG = pi/180; # degrees to radians constant RE = 6371000; # Earth radius in meters constant dd = 0.001; # integrate in this fraction of the distance already covered constant FIN = 10000000; # integrate only to a height of 10000km, effectively infinity #| Density of air as a function of height above sea level sub rho ( \a ) { exp( -a / 8500 ) } sub height ( \a, \z, \d ) { # a = altitude of observer # z = zenith angle (in degrees) # d = distance along line of sight my \AA = RE + a; sqrt( AA² + d² - 2*d*AA*cos((180-z)*DEG) ) - AA; } #| Integrates density along the line of sight sub column_density ( \a, \z ) { my$sum = 0;
my $d = 0; while$d < FIN {
my \delta = max(dd, (dd)*$d); # Adaptive step size to avoid it taking forever$sum += rho(height(a, z, $d + delta/2))*delta;$d   += delta;
}
sum; } sub airmass ( \a, \z ) { column_density( a, z ) / column_density( a, 0 ) } say 'Angle 0 m 13700 m'; say '------------------------------------'; say join "\n", (0, 5 ... 90).hyper(:3batch).map: -> \z { sprintf "%2d %11.8f %11.8f", z, airmass( 0, z), airmass(13700, z); }  Output: Angle 0 m 13700 m ------------------------------------ 0 1.00000000 1.00000000 5 1.00380963 1.00380965 10 1.01538466 1.01538475 15 1.03517744 1.03517765 20 1.06399053 1.06399093 25 1.10305937 1.10306005 30 1.15418974 1.15419083 35 1.21998076 1.21998246 40 1.30418931 1.30419190 45 1.41234169 1.41234567 50 1.55280404 1.55281025 55 1.73875921 1.73876915 60 1.99212000 1.99213665 65 2.35199740 2.35202722 70 2.89531368 2.89537287 75 3.79582352 3.79596149 80 5.53885809 5.53928113 85 10.07896219 10.08115981 90 34.32981136 34.36666557 ## REXX Translation of: FreeBASIC /*REXX pgm calculates the air mass above an observer and an object for various angles.*/ numeric digits (length(pi()) - length(.)) % 4 /*calculate the number of digits to use*/ parse arg aLO aHI aBY oHT . /*obtain optional arguments from the CL*/ if aLO=='' | aLO=="," then aLO= 0 /*not specified? Then use the default.*/ if aHI=='' | aHI=="," then aHI= 90 /* " " " " " " */ if aBY=='' | aBY=="," then aBY= 5 /* " " " " " " */ if oHT=='' | oHT=="," then oHT= 13700 /* " " " " " " */ w= 30; @ama= 'air mass at' /*column width for the two air_masses. */ say 'angle|'center(@ama "sea level", w) center(@ama commas(oHT) 'meters', w) /*title*/ say "─────┼"copies(center('', w, "─"), 2)'─' /*display the title sep for the output.*/ y= left('', w-20) /*Y: for alignment of the output cols.*/ do j=aLO to aHI by aBY; am0= airM(0, j); amht= airM(oHT, j) say center(j, 5)'│'right( format(am0, , 8), w-10)y right( format(amht, , 8), w-10)y end /*j*/ say "─────┴"copies(center('', w, "─"), 2)'─' /*display the foot separator for output*/ exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ airM: procedure; parse arg a,z; if z==0 then return 1; return colD(a, z) / colD(a, 0) d2r: return r2r( arg(1) * pi() / 180) /*convert degrees ──► radians. */ pi: pi= 3.1415926535897932384626433832795028841971693993751058209749445923078; return pi rho: procedure; parse arg a; return exp(-a / 8500) r2r: return arg(1) // (pi() * 2) /*normalize radians ──► a unit circle. */ e: e= 2.718281828459045235360287471352662497757247093699959574966967627724; return e commas: parse arg ?; do jc=length(?)-3 to 1 by -3; ?=insert(',', ?, jc); end; return ? /*──────────────────────────────────────────────────────────────────────────────────────*/ cos: procedure; parse arg x; x= r2r(x); a= abs(x); numeric fuzz min(6, digits() - 3) hpi= pi*.5; if a=pi then return -1; if a=hpi | a=hpi*3 then return 0; z= 1 if a=pi/3 then return .5; if a=pi*2/3 then return -.5; _= 1 x= x*x; p= z; do k=2 by 2; _= -_ * x / (k*(k-1)); z= z + _ if z=p then leave; p= z; end; return z /*──────────────────────────────────────────────────────────────────────────────────────*/ exp: procedure; parse arg x; ix= x%1; if abs(x-ix)>.5 then ix= ix + sign(x); x= x-ix z=1; _=1; w=z; do j=1; _= _*x/j; z=(z+_)/1; if z==w then leave; w=z; end if z\==0 then z= z * e() ** ix; return z/1 /*──────────────────────────────────────────────────────────────────────────────────────*/ sqrt: procedure; parse arg x; if x=0 then return 0; d= digits(); numeric digits; h= d+6 numeric form; parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g= g * .5'e'_ % 2 m.=9; do j=0 while h>9; m.j= h; h= h%2 + 1; end /*j*/ do k=j+5 to 0 by -1; numeric digits m.k; g= (g+x/g)*.5; end /*k*/ numeric digits d; return g/1 /*──────────────────────────────────────────────────────────────────────────────────────*/ elev: procedure; parse arg a,z,d; earthRad= 6371000 /*earth radius in meters.*/ aa= earthRad + a; return sqrt(aa**2 + d**2 - 2*d*aa*cos( d2r(180-z) ) ) - earthRad /*──────────────────────────────────────────────────────────────────────────────────────*/ colD: procedure; parse arg a,z; sum= 0; d= 0; dd= .001; infinity= 10000000 do while d<infinity; delta= max(dd, dd*d) sum= sum + rho( elev(a, z, d + 0.5*delta) ) * delta; d= d + delta end /*while*/ return sum  output when using the default inputs: angle| air mass at sea level air mass at 13,700 meters ─────┼───────────────────────────────────────────────────────────── 0 │ 1.00000000 1.00000000 5 │ 1.00380963 1.00380965 10 │ 1.01538466 1.01538475 15 │ 1.03517744 1.03517765 20 │ 1.06399053 1.06399093 25 │ 1.10305937 1.10306005 30 │ 1.15418974 1.15419083 35 │ 1.21998076 1.21998246 40 │ 1.30418931 1.30419190 45 │ 1.41234169 1.41234567 50 │ 1.55280404 1.55281025 55 │ 1.73875921 1.73876915 60 │ 1.99212000 1.99213665 65 │ 2.35199740 2.35202722 70 │ 2.89531368 2.89537287 75 │ 3.79582352 3.79596149 80 │ 5.53885809 5.53928113 85 │ 10.07896219 10.08115981 90 │ 34.32981136 34.36666557 ─────┴─────────────────────────────────────────────────────────────  ## RPL Translation of: FreeBASIC ≪ → a ≪ a NEG 8500 / EXP ≫ ≫ ‘RHO’ STO ≪ 'RHO(√(aa^2+D^2-2*aa*D*COS(180-Z))-re)' EVAL ≫ ‘COLD’ STO ≪ 6371000 3 PICK OVER + → a z re aa ≪ DEG z 'Z' STO 'COLD' { D 0 1E7 } 1E-7 ∫ DROP 0 'Z' STO 'COLD' { D 0 1E7 } 1E-7 ∫ DROP / ≫ ≫ ‘AM’ STO  ≪ { } 0 90 FOR z z + z AM + 5 STEP ≫  Output: 1: { 0 1 5 1.00380686363 10 1.01537368745 15 1.03515302646 20 1.06394782383 25 1.10299392042 30 1.15409753978 35 1.21985818096 40 1.30403285254 45 1.41214767977 50 1.55256798138 55 1.73847475415 60 1.99177718552 65 2.35157928673 70 2.89478919419 75 3.79512945489 80 5.53784169364 85 10.0771111633 90 34.3235064081 }  ## Rust Translation of: FreeBASIC const RE: f64 = 6371000.0; // Earth radius in meters const DD: f64 = 0.001; // integrate in this fraction of the distance already covered const FIN: f64 = 10000000.0; // integrate only to a height of 10000km, effectively infinity fn rho(a: f64) -> f64 { // the density of air as a function of height above sea level (-a / 8500.0).exp() } fn height(a: f64, z: f64, d: f64) -> f64 { // a = altitude of observer // z = zenith angle (in degrees) // d = distance along line of sight let aa = RE + a; let hh = (aa * aa + d * d - 2.0 * d * aa * (180.0 - z).to_radians().cos()).sqrt(); hh - RE } fn column_density(a: f64, z: f64) -> f64 { // integrates density along the line of sight let mut sum = 0.0; let mut d = 0.0; while d < FIN { // adaptive step size to avoid it taking forever let mut delta = DD * d; if delta < DD { delta = DD; } sum += rho(height(a, z, d + 0.5 * delta)) * delta; d += delta; } sum } fn airmass(a: f64, z: f64) -> f64 { column_density(a, z) / column_density(a, 0.0) } fn main() { println!("Angle 0 m 13700 m"); println!("------------------------------------"); for a in (0..=90).step_by(5) { let z = a as f64; println!( "{:2} {:11.8} {:11.8}", z, airmass(0.0, z), airmass(13700.0, z) ); } }  Output: Angle 0 m 13700 m ------------------------------------ 0 1.00000000 1.00000000 5 1.00380963 1.00380965 10 1.01538466 1.01538475 15 1.03517744 1.03517765 20 1.06399053 1.06399093 25 1.10305937 1.10306005 30 1.15418974 1.15419083 35 1.21998076 1.21998246 40 1.30418931 1.30419190 45 1.41234169 1.41234567 50 1.55280404 1.55281025 55 1.73875921 1.73876915 60 1.99212000 1.99213665 65 2.35199740 2.35202722 70 2.89531368 2.89537287 75 3.79582352 3.79596149 80 5.53885809 5.53928113 85 10.07896219 10.08115981 90 34.32981136 34.36666557  ## Seed7 Translation of: FreeBASIC  include "seed7_05.s7i";
include "float.s7i";
include "math.s7i";

const float: DEG is 0.017453292519943295769236907684886127134; #degrees to radians
const float: RE is 6371000.0;   #Earth radius in meters
const float: dd is 0.001;       #integrate in this fraction of the distance already covered
const float: FIN is 10000000.0; #integrate only to a height of 10000km, effectively infinity

const func float: rho (in float: a) is
#the density of air as a function of height above sea level
return exp(-a / 8500.0);

const func float: height (in float: a, in float: z, in float: d) is func
#a is altitude of observer
#z is zenith angle (in degrees)
#d is distance along line of sight
result
var float: r is 0.0;
local
var float: AA is 0.0;
var float: HH is 0.0;
begin
AA := RE + a;
HH := sqrt( AA ** 2.0 + d ** 2.0 - 2.0 * d * AA * cos((180.0 - z) * DEG) );
r := HH - RE;
end func;

const func float: columnDensity (in float: a, in float: z) is func
#integrates density along line of sight
result
var float: sum is 0.0;
local
var float: d is 0.0;
var float: delta is 0.0;
begin
while d < FIN do
delta := max(dd, dd * d); #adaptive step size to avoid taking it forever
sum +:= rho(height(a, z, d + 0.5 * delta)) * delta;
d +:= delta;
end while;
end func;

const func float: airmass (in float: a, in float: z) is
return columnDensity(a, z) / columnDensity(a, 0.0);

const proc: main is func
local
var integer: zz is 0;
var float: z is 0.0;
begin
writeln("Angle     0 m              13700 m");
writeln("------------------------------------");
for zz range 0 to 90 step 5 do
z := flt(zz);
write(airmass(0.0, z) digits 8 lpad 15);
writeln(airmass(13700.0, z) digits 8 lpad 17);
end for;
end func;
Output:
Angle     0 m              13700 m
------------------------------------
0.0     1.00000000       1.00000000
5.0     1.00380963       1.00380965
10.0     1.01538466       1.01538475
15.0     1.03517744       1.03517765
20.0     1.06399053       1.06399093
25.0     1.10305937       1.10306005
30.0     1.15418974       1.15419083
35.0     1.21998076       1.21998246
40.0     1.30418931       1.30419190
45.0     1.41234169       1.41234567
50.0     1.55280404       1.55281025
55.0     1.73875921       1.73876915
60.0     1.99212000       1.99213665
65.0     2.35199740       2.35202722
70.0     2.89531368       2.89537287
75.0     3.79582352       3.79596149
80.0     5.53885809       5.53928113
85.0    10.07896219      10.08115981
90.0    34.32981136      34.36666557


## Swift

Translation of: Go
import Foundation

extension Double {
var radians: Double { self * .pi / 180 }
}

func columnDensity(_ a: Double, _ z: Double) -> Double {
func rho(_ a: Double) -> Double {
exp(-a / 8500)
}

func height(_ d: Double) -> Double {
let aa = 6_371_000 + a
let hh = aa * aa + d * d - 2 * d * aa * cos((180 - z).radians)

return hh.squareRoot() - 6_371_000
}

var sum = 0.0
var d = 0.0

while d < 1e7 {
let delta = max(0.001, 0.001 * d)

sum += rho(height(d + 0.5 * delta)) * delta
d += delta
}

return sum
}

func airMass(altitude: Double, zenith: Double) -> Double {
return columnDensity(altitude, zenith) / columnDensity(altitude, 0)
}

print("Angle     0 m              13700 m")
print("------------------------------------")

for z in stride(from: 0.0, through: 90.0, by: 5.0) {
let air = String(
format: "%2.0f      %11.8f      %11.8f",
z,
airMass(altitude: 0, zenith: z),
airMass(altitude: 13700, zenith: z)
)

print(air)
}

Output:
Angle     0 m              13700 m
------------------------------------
0       1.00000000       1.00000000
5       1.00380963       1.00380965
10       1.01538466       1.01538475
15       1.03517744       1.03517765
20       1.06399053       1.06399093
25       1.10305937       1.10306005
30       1.15418974       1.15419083
35       1.21998076       1.21998246
40       1.30418931       1.30419190
45       1.41234169       1.41234567
50       1.55280404       1.55281025
55       1.73875921       1.73876915
60       1.99212000       1.99213665
65       2.35199740       2.35202722
70       2.89531368       2.89537287
75       3.79582352       3.79596149
80       5.53885809       5.53928113
85      10.07896219      10.08115981
90      34.32981136      34.36666557

## Wren

Translation of: FreeBASIC
Library: Wren-math
Library: Wren-fmt
import "./math" for Math
import "./fmt" for Fmt

// constants
var RE  = 6371000  // radius of earth in meters
var DD  = 0.001    // integrate in this fraction of the distance already covered
var FIN = 1e7      // integrate only to a height of 10000km, effectively infinity

// The density of air as a function of height above sea level.
var rho = Fn.new { |a| (-a/8500).exp }

// a = altitude of observer
// z = zenith angle (in degrees)
// d = distance along line of sight
var height = Fn.new { |a, z, d|
var aa = RE + a
var hh = (aa * aa + d * d - 2 * d * aa * (Math.radians(180-z).cos)).sqrt
return hh - RE
}

// Integrates density along the line of sight.
var columnDensity = Fn.new { |a, z|
var sum = 0
var d = 0
while (d < FIN) {
var delta = DD.max(DD * d) // adaptive step size to avoid it taking forever
sum = sum + rho.call(height.call(a, z, d + 0.5 * delta)) * delta
d = d + delta
}
return sum
}

var airmass = Fn.new { |a, z| columnDensity.call(a, z) / columnDensity.call(a, 0) }

System.print("Angle     0 m              13700 m")
System.print("------------------------------------")
var z = 0
while (z <= 90) {
Fmt.print("$2d$11.8f      \$11.8f", z, airmass.call(0, z), airmass.call(13700, z))
z = z + 5
}

Output:
Angle     0 m              13700 m
------------------------------------
0       1.00000000       1.00000000
5       1.00380963       1.00380965
10       1.01538466       1.01538475
15       1.03517744       1.03517765
20       1.06399053       1.06399093
25       1.10305937       1.10306005
30       1.15418974       1.15419083
35       1.21998076       1.21998246
40       1.30418931       1.30419190
45       1.41234169       1.41234567
50       1.55280404       1.55281025
55       1.73875921       1.73876915
60       1.99212000       1.99213665
65       2.35199740       2.35202722
70       2.89531368       2.89537287
75       3.79582352       3.79596149
80       5.53885809       5.53928113
85      10.07896219      10.08115981
90      34.32981136      34.36666557


## XPL0

Translation of: FreeBASIC
define DEG = 0.017453292519943295769236907684886127134;  \degrees to radians
define RE = 6371000.;   \Earth radius in meters
define DD = 0.001;      \integrate in this fraction of the distance already covered
define FIN = 10000000.; \integrate only to a height of 10000km, effectively infinity

function real Max(A, B);
real A, B;
return (if A>B then A else B);

function real Rho(A);
real A;
[   \the density of air as a function of height above sea level
return Exp(-A/8500.0)
end; \function

function real Height( A, Z, D );
real A, \= altitude of observer
Z, \= zenith angle (in degrees)
D; \= distance along line of sight
real AA, HH;
[   AA:= RE + A;
HH:= sqrt( AA*AA + D*D - 2.*D*AA*Cos((180.-Z)*DEG) );
return HH - RE;
end; \function

function real Column_density( A, Z );
real A, Z;   \integrates density along the line of sight
real Sum, D, Delta;
[   Sum:= 0.0; D:= 0.0;
while D<FIN do
[Delta:= Max(DD, (DD)*D); \adaptive step size to avoid it taking forever:
Sum:= Sum + Rho(Height(A, Z, D+0.5*Delta))*Delta;
D:= D + Delta;
];
return Sum;
end; \function

function real Airmass( A, Z );
real A, Z;
[   return Column_density( A, Z ) / Column_density( A, 0. );
end; \function

real Z;
[Text(0, "Angle     0 m              13700 m^M^J");
Text(0, "------------------------------------^M^J");
Z:= 0.;
while Z<=90. do
[Format(2, 0);  RlOut(0, Z);
Format(8, 8);   RlOut(0, Airmass(0., Z));
RlOut(0, Airmass(13700., Z));  CrLf(0);
Z:= Z + 5.;
]
]
Output:
Angle     0 m              13700 m
------------------------------------
0       1.00000000       1.00000000
5       1.00380963       1.00380965
10       1.01538466       1.01538475
15       1.03517744       1.03517765
20       1.06399053       1.06399093
25       1.10305937       1.10306005
30       1.15418974       1.15419083
35       1.21998076       1.21998246
40       1.30418931       1.30419190
45       1.41234169       1.41234567
50       1.55280404       1.55281025
55       1.73875921       1.73876915
60       1.99212000       1.99213665
65       2.35199740       2.35202722
70       2.89531368       2.89537287
75       3.79582352       3.79596149
80       5.53885809       5.53928113
85      10.07896219      10.08115981
90      34.32981136      34.36666557