Air mass: Difference between revisions
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</pre> |
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=={{header|Python}}== |
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<lang python>""" Rosetta Code task: Air_mass """ |
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from math import sqrt, cos, exp |
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DEG = 0.017453292519943295769236907684886127134 # degrees to radians |
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RE = 6371000 # Earth radius in meters |
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dd = 0.001 # integrate in this fraction of the distance already covered |
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FIN = 10000000 # integrate only to a height of 10000km, effectively infinity |
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def rho(a): |
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""" the density of air as a function of height above sea level """ |
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return exp(-a / 8500.0) |
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def height(a, z, d): |
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""" |
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a = altitude of observer |
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z = zenith angle (in degrees) |
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d = distance along line of sight |
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""" |
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return sqrt((RE + a)**2 + d**2 - 2 * d * (RE + a) * cos((180 - z) * DEG)) - RE |
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def column_density(a, z): |
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""" integrates density along the line of sight """ |
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dsum, d = 0.0, 0.0 |
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while d < FIN: |
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delta = max(dd, (dd)*d) # adaptive step size to avoid it taking forever: |
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dsum += rho(height(a, z, d + 0.5 * delta)) * delta |
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d += delta |
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return dsum |
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</lang>{{out}} |
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<pre> |
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Angle 0 m 13700 m |
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------------------------------------ |
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0 1.000 1.000 |
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5 1.004 1.004 |
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10 1.015 1.015 |
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15 1.035 1.035 |
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20 1.064 1.064 |
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25 1.103 1.103 |
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30 1.154 1.154 |
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35 1.220 1.220 |
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40 1.304 1.304 |
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45 1.412 1.412 |
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50 1.553 1.553 |
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55 1.739 1.739 |
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60 1.992 1.992 |
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65 2.352 2.352 |
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70 2.895 2.895 |
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75 3.796 3.796 |
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80 5.539 5.539 |
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85 10.079 10.081 |
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90 34.330 34.367 |
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</pre> |
</pre> |
||
Revision as of 11:41, 1 November 2021
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 (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 6731 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.
AWK
<lang 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) } </lang>
- 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
C
<lang c>#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));
}
}</lang>
- 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
Factor
<lang factor>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</lang>
- 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
FreeBASIC
<lang 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 </lang>
- 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
<lang go>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))
}
}</lang>
- 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
<lang java>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
}</lang>
- 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
Adapted from Wren
Works with gojq, the Go implementation of jq
Preliminaries <lang jq>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($width);</lang>
Physics <lang jq># constants def RE: 6371000; # radius of earth in meters def DD: 0.001; # integrate in this fraction of the distance already covered def FIN: 1e7; # integrate only to a height of 10000km, effectively infinity
- The density of air as a function of height above sea level.
def rho: (-./8500) | exp;
- a = altitude of observer (in m)
- z = zenith angle (in degrees)
- d = distance along line of sight (in m)
def height($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))" )</lang>
- 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
<lang julia>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
</lang>
- 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
<lang Nim>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</lang>
- 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
<lang perl>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);
}</lang>
- 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
<lang 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
</lang>
- Output:
Angle 0 m 13700 m ------------------------------------ 0 1.000 1.000 5 1.004 1.004 10 1.015 1.015 15 1.035 1.035 20 1.064 1.064 25 1.103 1.103 30 1.154 1.154 35 1.220 1.220 40 1.304 1.304 45 1.412 1.412 50 1.553 1.553 55 1.739 1.739 60 1.992 1.992 65 2.352 2.352 70 2.895 2.895 75 3.796 3.796 80 5.539 5.539 85 10.079 10.081 90 34.330 34.367
Raku
<lang perl6>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);
}</lang>
- 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
<lang rexx>/*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</lang>
- 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 ─────┴─────────────────────────────────────────────────────────────
Rust
<lang rust>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)
);
}
}</lang>
- 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
<lang seed7>$ 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;</lang>
- 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
<lang swift>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)
}</lang>
- 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
<lang ecmascript>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| Math.exp(-a/8500) }
// 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 = Math.max(DD, 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
}</lang>
- 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
<lang XPL0>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.; ]
]</lang>
- 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