Air mass
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.
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.
You will need to integrate (h(a,z,x)) where (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 6371 km.
- Task
-
- 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.
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
with Ada.Text_IO; use Ada.Text_IO;
with Ada.Numerics.Generic_Elementary_Functions;
with Ada.Integer_Text_IO; use Ada.Integer_Text_IO;
procedure Main is
subtype double is Long_Float;
package double_io is new Ada.Text_IO.Float_IO (double);
use double_io;
package Elementary_Double is new Ada.Numerics.Generic_Elementary_Functions
(Float_Type => double);
use Elementary_Double;
-- degrees to radians
Deg : constant := 0.017_453_292_519_943_295_769_236_907_684_886_127_134;
-- Earth radius in meters
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. effectively 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
Tested with Agena 5.0.3 Win32
scope # Air mass - translated from the Pluto sample
local constant re := 6371000; # radius of earth in meters
local constant dd := 0.001; # integrate in this fraction of the distance already covered
local constant fin := 1e7; # integrate only to a height of 10000km, effectively infinity
# returns x degrees expressed in radians
local constant toRadians := proc( x :: number ) :: number is return Pi * ( x / 180 ) end;
# returns the maximum of a and b
local constant rmax := proc( a :: number, b :: number ) :: number
is return if a >= b then a else b fi end;
# The density of air as a function of height above sea level.
local constant rho := proc( a :: number ) :: number
is return exp( -a / 8500 ) end;
# a = altitude of observer
# z = zenith angle (in degrees)
# d = distance along line of sight
local constant height := proc( a :: number, z :: number, d :: number ) :: number is
local constant aa := re + a;
local constant hh := sqrt( aa * aa + d * d - 2 * d * aa * cos( toRadians ( 180 - z ) ) );
return hh - re
end; # height
# Integrates density along the line of sight.
local constant column_density := proc( a :: number, z :: number ) :: number is
local sum := 0;
local d := 0;
while d < fin do
local constant delta
:= rmax( dd, dd * d ); # adaptive step size to avoid it taking forever
sum +:= rho( height( a, z, d + 0.5 * delta ) ) * delta;
d +:= delta
od;
return sum
end; # column density
local constant air_mass := proc( a :: number, z :: number ) :: number is
return column_density( a, z ) / column_density( a, 0 )
end; # air_mass
scope
print( "Angle 0 m 13700 m" );
print( "------------------------------------" );
for z from 0 to 90 by 5 do
printf( "%2d %11.8f", z, air_mass( 0, z ) );
printf( " %11.8f\n", air_mass( 13700, z ) )
od
end
end- Output:
Same as the Pluto and Algol 68 samples
BEGIN # Air mass - translated from the Pluto sample #
# Constants. #
INT re = 6371000; # radius of earth in meters #
REAL dd = 0.001; # integrate in this fraction of the distance already covered #
REAL fin = 1e7; # integrate only to a height of 10000km, effectively infinity #
# returns x degrees expressed in radians #
OP TORADIANS = ( REAL x )REAL: pi * ( x / 180 );
# returns the maximum of a and b #
PROC rmax = ( REAL a, b )REAL: IF a > b THEN a ELSE b FI;
# The density of air as a function of height above sea level. #
PROC rho = ( REAL a )REAL: exp( -a / 8500 );
# a = altitude of observer #
# z = zenith angle (in degrees) #
# d = distance along line of sight #
PROC height = ( REAL a, z, d )REAL:
BEGIN
REAL aa = re + a;
REAL hh = sqrt( aa * aa + d * d - 2 * d * aa * cos( TORADIANS ( 180 - z ) ) );
hh - re
END # height # ;
# Integrates density along the line of sight. #
PROC column density = ( REAL a, z )REAL:
BEGIN
REAL sum := 0;
REAL d := 0;
WHILE d < fin DO
REAL delta = rmax( dd, dd * d ); # adaptive step size to avoid it taking forever #
sum +:= rho( height( a, z, d + 0.5 * delta ) ) * delta;
d +:= delta
OD;
sum
END # column density # ;
PROC air mass = ( REAL a, z )REAL: column density( a, z ) / column density( a, 0 );
BEGIN
print( ( "Angle 0 m 13700 m", newline ) );
print( ( "------------------------------------", newline ) );
FOR z FROM 0 BY 5 TO 90 DO
print( ( whole( z, -2 ), " ", fixed( air mass( 0, z ), -11, 8 ) ) );
print( ( " ", fixed( air mass( 13700, z ), -11, 8 ), newline ) )
OD
END
END- Output:
Same as the Pluto sample.
# 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
import ballerina/io;
// constants
const RE = 6371000f; // 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(float a) returns float {
return (-a / 8500.0).exp();
}
function radians(float degrees) returns float {
return degrees * float:PI / 180.0;
}
// a = altitude of observer
// z = zenith angle (in degrees)
// d = distance along line of sight
function height(float a, float z, float d) returns float {
float aa = RE + a;
float hh = (aa * aa + d * d - 2.0 * d * aa * radians(180-z).cos()).sqrt();
return hh - RE;
}
// Integrates density along the line of sight.
function columnDensity(float a, float z) returns float {
float sum = 0;
float d = 0;
while d < FIN {
float delta = DD.max(DD * d); // adaptive step size to avoid it taking forever
sum += rho(height(a, z, d + 0.5 * delta)) * delta;
d += delta;
}
return sum;
}
function airmass(float a, float z) returns float {
return columnDensity(a, z) / columnDensity(a, 0);
}
function F(float f, int size, int prec) returns string {
string s = f.toFixedString(prec);
return size >= 0 ? s.padStart(size) : s.padEnd(size);
}
public function main() {
io:println("Angle 0 m 13700 m");
io:println("------------------------------------");
float z = 0.0;
while z <= 90.0 {
string s1 = F(z, 2, 0);
string s2 = F(airmass(0, z), 11, 8);
string s3 = F(airmass(13700, z), 11, 8);
io:println(`${s1} ${s2} ${s3}`);
z += 5.0;
}
}
- 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
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
#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
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
#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
#include <cstdint>
#include <algorithm>
#include <cmath>
#include <iomanip>
#include <iostream>
#include <numbers>
// Return the density of air at 'height' above sea level
double rho(const double& height) {
return std::exp(-height / 8500.0);
}
// Return the height above sea level at a 'distance' along the line of sight of an observer
// at an 'altitude' viewing at a 'zenith_angle' in degrees from the vertical
double height(const double& altitude, const double& zenith_angle, const double& distance) {
const double earth_radius = 6371000.0; // metres
const double aa = earth_radius + altitude;
const double height = std::sqrt(aa * aa + distance * distance - 2.0 * distance * aa
* std::cos(( 180 - zenith_angle ) * std::numbers::pi / 180.0));
return height - earth_radius;
}
// Return the integral of air density along the line of sight of an observer
// at an 'altitude' viewing at a 'zenith_angle' in degrees from the vertical
double column_density(const double& altitude, const double& zenith_angle) {
const double limit_atmosphere = 10'000'000.0; // metres
const double initial_step_size = 0.001;
double sum = 0.0;
double distance = 0.0;
while ( distance < limit_atmosphere ) {
// step size increases as the atmosphere becmes less dense
const double step_size = std::max(distance * initial_step_size, initial_step_size);
sum += rho(height(altitude, zenith_angle, distance + 0.5 * step_size)) * step_size;
distance += step_size;
}
return sum;
}
// Return the air mass; a measure of the amount of atmosphere between the observer and the object being observed
double air_mass(const double& altitude, const double& zenith_angle) {
return column_density(altitude, zenith_angle) / column_density(altitude, 0.0);
}
int main() {
std::cout << "Angle 0 m 13700 m" << std::endl;
std::cout << "--------------------------------------" << std::endl;
for ( uint32_t zenith_angle = 0; zenith_angle <= 90; zenith_angle += 5 ) {
std::cout << std::setw(2) << zenith_angle
<< std::setw(18) << std::fixed <<std::setprecision(8) << air_mass(0.0, zenith_angle)
<< std::setw(18) << air_mass(13'700.0, zenith_angle) << std::endl;
}
}
- 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
module air_mass_rosetta_code;
import std.math : exp, sqrt, cos, PI;
import std.algorithm : max;
import std.stdio : writeln, writef, writefln;
import std.format : format;
enum double EARTH_RADIUS = 6_371_000.0; // meters
enum double SCALE_HEIGHT = 8_500.0; // meters (atmospheric density scale height)
enum double LIMIT_ATMOSPHERE = 10_000_000.0; // meters (integration upper bound)
enum double INITIAL_STEP_SIZE = 0.001; // meters
// Atmospheric density at `height` meters above sea level.
// @nogc because we don't use any GC features
// @safe because we don't do any unsafe operations
pure nothrow @nogc @safe
double rho(double height)
{
return exp(-height / SCALE_HEIGHT);
}
/**
* Height above sea level (metres) for a point at `distance` along the
* line of sight of an observer at `altitude`, viewing at `zenithAngle`
* degrees from vertical.
*
* Derived from the law of cosines on the triangle formed by the Earth's
* centre, the observer, and the point of interest.
*/
pure nothrow @nogc @safe
double heightAtDistance(double altitude, double zenithAngle, double distance)
{
immutable double aa = EARTH_RADIUS + altitude;
immutable double cosAngle = cos((180.0 - zenithAngle) * PI / 180.0);
immutable double r = sqrt(aa * aa + distance * distance
- 2.0 * distance * aa * cosAngle);
return r - EARTH_RADIUS;
}
/**
* Column density: integral of atmospheric density along the line of sight
* for an observer at `altitude` metres, viewing at `zenithAngle` degrees.
*
* Uses an adaptive step size that grows with distance, keeping the
* integration efficient as density falls off exponentially.
*/
nothrow @nogc @safe
double columnDensity(double altitude, double zenithAngle)
{
double sum = 0.0;
double distance = 0.0;
while (distance < LIMIT_ATMOSPHERE)
{
// Step grows with distance so we don't over-sample the thin upper atmosphere.
immutable double step = max(distance * INITIAL_STEP_SIZE, INITIAL_STEP_SIZE);
// Midpoint rule: sample density at the centre of the current step.
sum += rho(heightAtDistance(altitude, zenithAngle, distance + 0.5 * step)) * step;
distance += step;
}
return sum;
}
/**
* Air mass for an observer at `altitude` metres viewing at `zenithAngle`
* degrees from vertical, normalised to the zenith column density.
*/
nothrow @nogc @safe
double airMass(double altitude, double zenithAngle)
{
return columnDensity(altitude, zenithAngle) / columnDensity(altitude, 0.0);
}
void main()
{
writeln("Angle 0 m 13700 m");
writeln("------------------------------------------");
foreach (zenithAngle; 0 .. 19) // 0..18 inclusive → 0, 5, 10, … 90
{
immutable int angle = zenithAngle * 5;
immutable double seaLevel = airMass(0.0, angle);
immutable double sofia = airMass(13_700.0, angle);
writefln("%2d %18.8f %18.8f", angle, seaLevel, sofia);
}
}
- 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
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;
function Radians(degrees: double): double;
{ Converts degrees to radians}
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
Memo.Lines.Add('Angle 0 m 13700 m');
Memo.Lines.Add('------------------------------------');
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.
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 2 8
print "Angle 0 m 13700 m"
print "------------------------"
for z = 0 step 5 to 90
print z & " " & airmass 0 z & " " & airmass 13700 z
.- Output:
Angle 0 m 13700 m ------------------------ 0 1 1 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
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
!! =============================================================================
! air_mass.f90 -- Astronomical Air Mass Calculation
! =============================================================================
!
! Air mass is a measure of the thickness of atmosphere a ray of light must
! traverse when arriving at an observer from a celestial object. At the
! zenith (z = 0) the air mass is 1.0 by definition. As the zenith angle
! increases the ray travels through progressively more atmosphere.
!
! MODEL
! -----
! We treat the atmosphere as a spherically-symmetric shell around a spherical
! Earth. The density at height h (metres above sea level) follows an
! exponential law:
!
! rho(h) = rho_0 * exp(-h / H)
!
! where H = 8 500 m is the pressure scale height and rho_0 is the sea-level
! density (it cancels out of the ratio).
!
! The line of sight from an observer at altitude a, looking at zenith angle z,
! is parameterised by the path length x measured along the ray from the
! observer. The height h at path length x is given by the law of cosines
! applied to the triangle (centre of Earth, observer, point on ray):
!
! r0 = R_EARTH + a (observer geocentric radius)
! r(x) = sqrt( r0^2 + x^2 + 2*r0*x*cos(z) )
!
! Note: cos(z) is positive for z < 90 (ray pointing upward),
! zero for z = 90 (horizontal), and negative beyond.
!
! h(x) = r(x) - R_EARTH (height above sea level)
!
! AIR MASS INTEGRAL
! -----------------
! The raw column density integral is:
!
! I(a, z) = integral_0^{X_MAX} exp(-h(x) / H) dx
!
! We normalise by the vertical (zenith) integral evaluated analytically:
!
! I_zenith(a) = H * exp(-a / H)
!
! so that air_mass(a, 0) = 1.0 exactly and the result is dimensionless.
!
! NUMERICAL INTEGRATION
! ---------------------
! Composite Simpson's rule with N_STEPS (even) sub-intervals is used.
! X_MAX = 2 000 000 m (2 000 km) is far enough that exp(-h/H) is
! negligibly small beyond the upper boundary.
!
! FLAT-EARTH (PLANE-PARALLEL) APPROXIMATION
! ------------------------------------------
! The classic first-order approximation ignores Earth curvature:
!
! air_mass_flat = 1 / cos(z)
!
! This is accurate for small z but diverges as z -> 90 degrees.
!
! OUTPUT
! ------
! Two tables are printed:
! 1. Sea-level observer (a = 0 m)
! 2. NASA SOFIA aircraft (a = 13 700 m)
!
! Each row shows z, spherical air mass, and the flat-Earth value.
! =============================================================================
program air_mass_program
use iso_fortran_env, only: real64
implicit none
! --------------------------------------------------------------------------
! Physical constants and integration parameters
! --------------------------------------------------------------------------
real(kind=real64), parameter :: R_EARTH = 6371000.0_real64 ! Earth radius (m)
real(kind=real64), parameter :: H_SCALE = 8500.0_real64 ! pressure scale height (m)
real(kind=real64), parameter :: X_MAX = 2000000.0_real64 ! path length limit (m)
integer, parameter :: N_STEPS = 2**17 ! Simpson sub-intervals (131072, must be even)
! Conversion factor
real(kind=real64), parameter :: DEG2RAD = acos(-1.0_real64) / 180.0_real64
! Observer altitudes (m)
real(kind=real64), parameter :: ALT_SEA = 0.0_real64 ! sea level
real(kind=real64), parameter :: ALT_SOFIA = 13700.0_real64 ! SOFIA cruising altitude
! --------------------------------------------------------------------------
! Local variables
! --------------------------------------------------------------------------
integer :: iz
real(kind=real64) :: z_deg, z_rad
real(kind=real64) :: am_sea, am_sofia, am_flat
! --------------------------------------------------------------------------
! Table: sea-level observer
! --------------------------------------------------------------------------
write(*, '(a)') '============================================================='
write(*, '(a)') ' Air Mass Table -- Sea-level observer (a = 0 m)'
write(*, '(a)') '============================================================='
write(*, '(a)') ' z (deg) Spherical Flat (1/cos z)'
write(*, '(a)') ' -------- ---------- ---------------'
do iz = 0, 18
z_deg = real(iz * 5, real64)
z_rad = z_deg * DEG2RAD
am_sea = air_mass(ALT_SEA, z_rad)
if (iz < 18) then
am_flat = 1.0_real64 / cos(z_rad)
write(*, '(f9.1, f13.5, f16.5)') z_deg, am_sea, am_flat
else
! z = 90 deg: flat-Earth value is infinite
write(*, '(f9.1, f13.5, a)') z_deg, am_sea, ' Inf'
end if
end do
write(*, *)
! --------------------------------------------------------------------------
! Table: SOFIA observer
! --------------------------------------------------------------------------
write(*, '(a)') '============================================================='
write(*, '(a)') ' Air Mass Table -- NASA SOFIA (a = 13 700 m)'
write(*, '(a)') '============================================================='
write(*, '(a)') ' z (deg) Spherical Flat (1/cos z)'
write(*, '(a)') ' -------- ---------- ---------------'
do iz = 0, 18
z_deg = real(iz * 5, real64)
z_rad = z_deg * DEG2RAD
am_sofia = air_mass(ALT_SOFIA, z_rad)
if (iz < 18) then
am_flat = 1.0_real64 / cos(z_rad)
write(*, '(f9.1, f13.5, f16.5)') z_deg, am_sofia, am_flat
else
write(*, '(f9.1, f13.5, a)') z_deg, am_sofia, ' Inf'
end if
end do
write(*, *)
! --------------------------------------------------------------------------
! Side-by-side comparison at selected angles
! --------------------------------------------------------------------------
write(*, '(a)') '============================================================='
write(*, '(a)') ' Comparison: Sea level vs SOFIA vs Flat-Earth'
write(*, '(a)') '============================================================='
write(*, '(a)') ' z (deg) Sea level SOFIA Flat (1/cos z)'
write(*, '(a)') ' -------- ---------- ---------- ---------------'
do iz = 0, 18
z_deg = real(iz * 5, real64)
z_rad = z_deg * DEG2RAD
am_sea = air_mass(ALT_SEA, z_rad)
am_sofia = air_mass(ALT_SOFIA, z_rad)
if (iz < 18) then
am_flat = 1.0_real64 / cos(z_rad)
write(*, '(f9.1, f12.5, f12.5, f16.5)') z_deg, am_sea, am_sofia, am_flat
else
write(*, '(f9.1, f12.5, f12.5, a)') z_deg, am_sea, am_sofia, ' Inf'
end if
end do
write(*, *)
write(*, '(a)') 'Notes:'
write(*, '(a)') ' Spherical model: composite Simpsons rule, N = 131 072 intervals'
write(*, '(a)') ' Scale height H = 8 500 m, R_Earth = 6 371 000 m'
write(*, '(a)') ' SOFIA cruising altitude = 13 700 m (~45 000 ft)'
write(*, '(a)') ' Air mass is normalised so that the zenith value = 1.0'
contains
! ==========================================================================
! air_mass(a, z) -- compute air mass for observer at altitude a (m)
! looking at zenith angle z (radians).
!
! Uses composite Simpson's rule to evaluate:
!
! I(a,z) = integral_0^{X_MAX} exp(-h(x)/H_SCALE) dx
!
! then returns I(a,z) / (H_SCALE * exp(-a/H_SCALE))
!
! so that the zenith air mass is 1.0 by definition.
! ==========================================================================
function air_mass(a, z) result(am)
real(kind=real64), intent(in) :: a ! observer altitude above sea level (m)
real(kind=real64), intent(in) :: z ! zenith angle (radians)
real(kind=real64) :: am
real(kind=real64) :: r0 ! geocentric radius of observer (m)
real(kind=real64) :: dx ! step size along ray (m)
real(kind=real64) :: cz ! cos(z), precomputed for efficiency
real(kind=real64) :: integral ! accumulated Simpson sum
real(kind=real64) :: x ! current path length (m)
real(kind=real64) :: h ! height above sea level at x (m)
real(kind=real64) :: w ! Simpson weight (1, 4, or 2)
real(kind=real64) :: zenith_I ! analytic zenith column: H * exp(-a/H)
integer :: i
r0 = R_EARTH + a
dx = X_MAX / real(N_STEPS, real64)
cz = cos(z)
! Simpson's rule: sum weights 1, 4, 2, 4, 2, ..., 4, 1
integral = 0.0_real64
do i = 0, N_STEPS
x = real(i, real64) * dx
! Height at this point via law of cosines
h = sqrt(r0*r0 + x*x + 2.0_real64*r0*x*cz) - R_EARTH
! Clamp h to zero to avoid negative heights at extreme angles
if (h < 0.0_real64) h = 0.0_real64
! Simpson weights
if (i == 0 .or. i == N_STEPS) then
w = 1.0_real64
else if (mod(i, 2) == 1) then
w = 4.0_real64
else
w = 2.0_real64
end if
integral = integral + w * exp(-h / H_SCALE)
end do
integral = integral * dx / 3.0_real64
! Normalise by the analytic vertical column density
zenith_I = H_SCALE * exp(-a / H_SCALE)
am = integral / zenith_I
end function air_mass
end program air_mass_program
=============================================================
Air Mass Table -- Sea-level observer (a = 0 m)
=============================================================
z (deg) Spherical Flat (1/cos z)
-------- ---------- ---------------
0.0 1.00000 1.00000
5.0 1.00381 1.00382
10.0 1.01538 1.01543
15.0 1.03518 1.03528
20.0 1.06399 1.06418
25.0 1.10306 1.10338
30.0 1.15419 1.15470
35.0 1.21998 1.22077
40.0 1.30419 1.30541
45.0 1.41234 1.41421
50.0 1.55280 1.55572
55.0 1.73876 1.74345
60.0 1.99212 2.00000
65.0 2.35200 2.36620
70.0 2.89531 2.92380
75.0 3.79582 3.86370
80.0 5.53886 5.75877
85.0 10.07896 11.47371
90.0 34.32981 Inf
=============================================================
Air Mass Table -- NASA SOFIA (a = 13 700 m)
=============================================================
z (deg) Spherical Flat (1/cos z)
-------- ---------- ---------------
0.0 1.00000 1.00000
5.0 1.00381 1.00382
10.0 1.01538 1.01543
15.0 1.03518 1.03528
20.0 1.06399 1.06418
25.0 1.10306 1.10338
30.0 1.15419 1.15470
35.0 1.21998 1.22077
40.0 1.30419 1.30541
45.0 1.41235 1.41421
50.0 1.55281 1.55572
55.0 1.73877 1.74345
60.0 1.99214 2.00000
65.0 2.35203 2.36620
70.0 2.89537 2.92380
75.0 3.79596 3.86370
80.0 5.53928 5.75877
85.0 10.08116 11.47371
90.0 34.36667 Inf
=============================================================
Comparison: Sea level vs SOFIA vs Flat-Earth
=============================================================
z (deg) Sea level SOFIA Flat (1/cos z)
-------- ---------- ---------- ---------------
0.0 1.00000 1.00000 1.00000
5.0 1.00381 1.00381 1.00382
10.0 1.01538 1.01538 1.01543
15.0 1.03518 1.03518 1.03528
20.0 1.06399 1.06399 1.06418
25.0 1.10306 1.10306 1.10338
30.0 1.15419 1.15419 1.15470
35.0 1.21998 1.21998 1.22077
40.0 1.30419 1.30419 1.30541
45.0 1.41234 1.41235 1.41421
50.0 1.55280 1.55281 1.55572
55.0 1.73876 1.73877 1.74345
60.0 1.99212 1.99214 2.00000
65.0 2.35200 2.35203 2.36620
70.0 2.89531 2.89537 2.92380
75.0 3.79582 3.79596 3.86370
80.0 5.53886 5.53928 5.75877
85.0 10.07896 10.08116 11.47371
90.0 34.32981 34.36667 Inf
Notes:
Spherical model: composite Simpsons rule, N = 131 072 intervals
Scale height H = 8 500 m, R_Earth = 6 371 000 m
SOFIA cruising altitude = 13 700 m (~45 000 ft)
Air mass is normalised so that the zenith value = 1.0
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) }
// Converts degrees to radians
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
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
Adapted from Wren
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
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
Tested with Lua 5.4.6
-- Constants.
local RE <const> = 6371000 -- radius of earth in meters
local DD <const> = 0.001 -- integrate in this fraction of the distance already covered
local FIN <const> = 1e7 -- integrate only to a height of 10000km, effectively infinity
-- The density of air as a function of height above sea level.
local function rho(a) return math.exp(-a / 8500) end
-- a = altitude of observer
-- z = zenith angle (in degrees)
-- d = distance along line of sight
local function height(a, z, d)
local aa = RE + a
local hh = math.sqrt(aa * aa + d * d - 2 * d * aa * math.cos(math.rad(180 - z)))
return hh - RE
end
-- Integrates density along the line of sight.
local function column_density(a, z)
local sum = 0
local d = 0
while d < FIN do
local delta = math.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
return sum
end
local function air_mass(a, z) return column_density(a, z) / column_density(a, 0) end
print("Angle 0 m 13700 m")
print("------------------------------------")
for z = 0, 90, 5 do
local f = "%2d %11.8f %11.8f"
print(string.format(f, z, air_mass(0, z), air_mass(13700, z)))
end
- Output:
Same as the Pluto sample.
R = 6371;
rho[a_] := Exp[-a / 8.5];
height[a_, z_, x_] := Sqrt[(R + a)^2 + x^2 - 2 x (R + a) Cos[(180 - z) °]] - R;
columnDensity[a_, z_] := NIntegrate[rho[height[a, z, x]], {x, 0, Infinity}];
airMass[a_, z_] := columnDensity[a, z] / columnDensity[a, 0];
Table[PaddedForm[airMass[a, z], {10, 8}], {z, 0, 90, 5}, {a, {0, 13.7}}] //
TableForm[#, TableHeadings -> {Range[0, 90, 5], {" 0 m", " 13700 m"}}] &
- Output:
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
import "mathUtil"
import "num"
import "fmt"
mu = mathUtil
// constants
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.
rho = function(a); return num.exp(-a / 8500); end function
// a = altitude of observer
// z = zenith angle (in degrees)
// d = distance along line of sight
height = function(a, z, d)
aa = RE + a
c = cos(mu.degToRad(180 - z))
hh = sqrt(aa * aa + d * d - 2 * d * aa * c)
return hh - RE
end function
// Integrates density along the line of sight.
columnDensity = function(a, z)
sum = 0
d = 0
while d < FIN
delta = mu.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
airmass = function(a, z)
return columnDensity(a, z) / columnDensity(a, 0)
end function
print "Angle 0 m 13700 m"
print "------------------------------------"
z = 0
while z <= 90
zs = fmt.int(z, 2)
am1 = fmt.num(airmass(0, z), 11, 8)
am2 = fmt.num(airmass(13700, z), 11, 8)
fmt.print "{} {} {}", zs, am1, am2
z += 5
end while
- 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
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
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
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
-- Constants.
$define RE = 6371000 -- radius of earth in meters
$define DD = 0.001 -- integrate in this fraction of the distance already covered
$define FIN = 1e7 -- integrate only to a height of 10000km, effectively infinity
-- The density of air as a function of height above sea level.
local function rho(a) return math.exp(-a / 8500) end
-- a = altitude of observer
-- z = zenith angle (in degrees)
-- d = distance along line of sight
local function height(a, z, d)
local aa = RE + a
local hh = math.sqrt(aa * aa + d * d - 2 * d * aa * math.cos(math.rad(180 - z)))
return hh - RE
end
-- Integrates density along the line of sight.
local function column_density(a, z)
local sum = 0
local d = 0
while d < FIN do
local 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
end
return sum
end
local function air_mass(a, z) return column_density(a, z) / column_density(a, 0) end
print("Angle 0 m 13700 m")
print("------------------------------------")
for z = 0, 90, 5 do
local f = "%2d %11.8f %11.8f"
print(string.format(f, z, air_mass(0, z), air_mass(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
""" 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
#Equation for height was worked out by hand using the cosine rule and quadratic formula
h <- function(a, z, x, r) -r + sqrt((a+r)^2 + x^2 + 2*x*(a+r)*cos(z))
rho <- function(a) exp(-a/8.5)
airmass_absolute <- function(a, z) {
integrate(function(x) rho(h(a, z, x, 6371)), 0, 10000)$value
}
airmass_relative <- function(a) function(z) {
airmass_absolute(a, z)/airmass_absolute(a, 0)
}
angles <- seq(from = 0, to = 90, by = 5)
call_angles <- function(f) sapply(angles*pi/180, f)
airmasses <- data.frame(
"Angle" = angles,
"a_0m" = call_angles(airmass_relative(0)),
"a_13700m" = call_angles(airmass_relative(13.7))
) |> print(row.names = FALSE)
- Output:
Angle a_0m a_13700m
0 1.000000 1.000000
5 1.003810 1.003810
10 1.015385 1.015385
15 1.035177 1.035178
20 1.063991 1.063991
25 1.103059 1.103060
30 1.154190 1.154191
35 1.219981 1.219982
40 1.304189 1.304192
45 1.412342 1.412346
50 1.552804 1.552810
55 1.738759 1.738769
60 1.992120 1.992137
65 2.351997 2.352027
70 2.895314 2.895373
75 3.795824 3.795961
80 5.538858 5.539281
85 10.078962 10.081160
90 34.329811 34.366666
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 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 ─────┴─────────────────────────────────────────────────────────────
≪ → 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 }
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
$ 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(z lpad 4);
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
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
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
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