The air mass coefficient defines the optical path through the Earth’s atmosphere , expressed as a relative ratio to the path length vertically upwards, ie at the zenith . The air mass coefficient can be used to help characterize the solar spectrum after solar radiation has traveled through the atmosphere. The air mass coefficient is commonly used to characterize the performance of solar cells under standardized conditions, and is often referred to as “AM” followed by a number. “AM1.5” is almost universal when characterizing terrestrial powergenerating panels.
Description
Solar radiation closely matches a body radiator at about 5,800 K. ^{[1]} As it passes through the atmosphere, sunlight is attenuated by scattering and absorption ; the more atmosphere through which it passes, the greater the attenuation .
As the sunlight travels through the atmosphere, chemicals interact with the sunlight and absorb certain wavelengths changing the amount of shortwavelength light reaching the Earth’s surface. A further active component of this process is water vapor, which results in a wide variety of absorption bands and many wavelengths, while molecular nitrogen, oxygen and carbon dioxide add to this process. By the time it reaches the Earth’s surface, the spectrum is strongly confined to the far infrared and near ultraviolet.
Atmospheric scattering plays a role in removing higher frequencies from direct sunlight and scattering it about the sky. ^{[2]} This is why the sky appears blue and the sun – more of the higherfrequency blue light arrives at the observer via indirect scattered paths; and less blue light follows the direct path, giving the sun a yellow tinge. ^{[3]}The greater the distance in the atmosphere through which the sunlight travels, the greater this effect, which is why the sun looks orange or red at dawn and sundown when the sun is traveling very obliquely through the atmosphere – progressively more of the blues and greens are removed from the direct rays, giving an orange or red appearance to the sun; and the sky appears pink – because the blues and greens are scattered over such long paths that they are highly attenuated before arriving at the observing, resulting in characteristic pink skies at dawn and sunset.
Definition
For a path length {\ displaystyle L} through the atmosphere, for the solar radiation incident at angle {\ displaystyle z}relative to the normal to the earth’s surface, the air mass coefficient is: ^{[4]}

{\ displaystyle AM = {\ frac {L} {L _ {\ mathrm {o}}} \ approx {\ frac {1} {\ cos \, z}} \,} ( A.1 )
Where {\ displaystyle L _ {\ mathrm {o}}}is the zenith path length (ie normal to the Earth’s surface area) at sea level and{\ displaystyle z} is the zenith angle in degrees.
The air mass number is thus dependent on the Sun’s elevation path through the sky and then varies with the time of day and with the latitude of the observer.
Accuracy near the horizon
The above approximation overlooks the curvature of the earth, and is reasonably accurate {\ displaystyle z}up to around 75 °. A number of refinements have been proposed to more accurately model the path towards the horizon, such as proposed by Kasten and Young (1989): ^{[5]}

{\ displaystyle AM = {\ frac {1} {\ cos \, z + 0.50572 \, (96.07995z) ^ { 1.6364}}} \,} ( A.2 )
A more detailed list of these models is provided in the article by Airmass , for various atmospheric models and experimental data sets. At sea level the air mass towards the horizon{\ displaystyle z}= 90 °) is approximately 38. ^{[6]}
Modeling the atmosphere as a simple spherical shell provides a reasonable approximation: ^{[7]}

{\ displaystyle AM = {\ sqrt {(r \ cos z) ^ {2} + 2r + 1}} \; – \; r \ cos z \,} ( A.3 )
where the radius of the earth {\ displaystyle R _ {\ mathrm {E}}} = 6371 km, the effective height of the atmosphere {\ displaystyle y _ {\ mathrm {atm}}} ≈ 9 km, and their ratio {\ displaystyle r = R _ {\ mathrm {E}} / y _ {\ mathrm {atm}}} ≈ 708.
These models are compared in the table below:
{\ displaystyle z}  Flat Earth  Kasten & Young  Spherical shell 

degree  ( A.1 )  ( A.2 )  ( A.3 ) 
0 °  1.0  1.0  1.0 
60 °  2.0  2.0  2.0 
70 °  2.9  2.9  2.9 
75 °  3.9  3.8  3.8 
80 °  5.8  5.6  5.6 
85 °  11.5  10.3  10.6 
88 °  28.7  19.4  20.3 
90  {\ displaystyle \ infty}  37.9  37.6 
This implies that the atmosphere can be considered to be 9 km, ^{[8]} ie, essentially all the atmospheric effects are due to the atmospheric mass in the lower half of the Troposphere . This is a useful and simple model when considering the atmospheric effects on solar intensity.
Cases
 AM0
The spectrum outside the atmosphere, approximated by the 5,800 K black body, is referred to as “AM0”, meaning “zero atmospheres”. Solar cells used for space power applications, like those on satellite communications are characterized by AM0.
 AM1
The spectrum after traveling through the atmosphere is directly referred to as, by definition, as “AM1”. This means “one atmosphere”. AM1 ({\ displaystyle z}= 0 °) to AM1.1 ({\ displaystyle z}= 25 °) is a useful range for estimating performance of solar cells in equatorial and tropical regions.
 AM1.5
Solar panels of the Earth’s surface are the most effective. Many of the world’s major population centers, and hence, solar installations and industry, across Europe, China, Japan, the United States of America and elsewhere (including Northern India, Southern Africa and Australia) link in temperate latitudes. An AM number representing the spectrum at midlatitudes is therefore much more common.
“AM1.5”, 1.5 atmosphere thickness, corresponding to a solar zenith angle of {\ displaystyle z}= 48.2 °. While the summertime is numbered for midlatitudes during the middle of the day is 1.5, higher figures apply in the morning and other times of the year. Therefore, AM1.5 is useful for representing the overall average for midlatitudes. The specific value of 1.5 has been selected in the 1970s for standardization purposes, based on an analysis of solar irradiance data in the conterminous United States. ^{[9]} Since then, the use of AM1.5 for all standardized testing or terrestrial solar cells or modules, including those used in concentrating systems. The latest AM1.5 standards pertaining to photovoltaic applications are the ASTM G173 ^{[10] }^{[11]} and IEC 60904, all derived from simulations obtained withSMARTS code
 AM2 ~ 3
AM2 ({\ displaystyle z}= 60 °) to AM3 ({\ displaystyle z}= 70 °) is a useful range for estimating the overall average performance of solar cells at high latitudes such as in northern Europe. Similar AM2 to AM3 is useful for estimating wintertime performance in latitudes, eg airmass coefficient is greater than 2 at all times of the day in winter at latitudes as low as 37 °.
 AM38
AM38 is well regarded as being the airmass in the horizontal direction ({\ displaystyle z}= 90 °) at sea level. ^{[6]} However, there is a high degree of variability in the field of solar intensity .
 At higher altitudes
The relative air mass is only a function of the sun’s zenith angle, and therefore does not change with local elevation. Conversely, the absolute air mass, equal to the relative air mass multiplied by the local atmospheric pressure and divided by the standard (sealevel) pressure, decreases with elevation above sea level. For highaltitude solar panels, for example, for an altiplano region, it is possible to use a higher absolute value. other latitudes. However, this approach is approximate and not recommended. It is best to simulate the actual spectrum based on the relative air mass (eg, 1.5) and theactual atmospheric conditions for the specific elevation of the site under scrutiny.
Solar intensity
Solar intensity at the collector reduces with increasing airmass coefficient, but due to the complex and variable atmospheric factors involved, not in a simple or linear fashion. For example, almost all high energy is removed from the upper atmosphere (between AM0 and AM1) and so AM2 is not twice as bad AM1. Furthermore, there is great variability in Many of the factors Contributing to atmospheric attenuation, ^{[12]} Such As water vapor, aerosols, photochemical smog and the effects of temperature inversions . Depending on the level of pollution in the air, overall attenuation can vary by up to ± 70% towards the horizon, especially affecting performance particularly towards the horizon where effects of the lower layers of the atmosphere are amplified manyfold.
One approximate model for solar intensity versus airmass is given by: ^{[13] }^{[14]}

{\ displaystyle I = 1.1 \ times I _ {\ mathrm {o}} \ times 0.7 ^ {(AM ^ {0.678})} \,} ( I.1 )
where solar intensity external to Earth’s atmosphere {\ displaystyle I _ {\ mathrm {o}}} = 1.353 kW / m ^{2} , and the factor of 1.1 is derived assuming that the component is 10% of the direct component. ^{[13]}
This formula fits comfortably within the midrange of the expected pollutionbased variability:
{\ displaystyle z}  AM  range due to pollution ^{[12]}  formula ( I.1 )  ASTM G173 ^{[11]} 

degree  W / m ^{2}  W / m ^{2}  W / m ^{2}  
–  0  1367 ^{[15]}  1353  1347.9 ^{[16]} 
0 °  1  840 .. 1130 = 990 ± 15%  1040  
23 °  1.09  800 .. 1110 = 960 ± 16% ^{[17]}  1020  
30  1.15  780 .. 1100 = 940 ± 17%  1010  
45 °  1.41  710 .. 1060 = 880 ± 20% ^{[17]}  950  
48.2 °  1.5  680 .. 1050 = 870 ± 21% ^{[17]}  930  1000.4 ^{[18]} 
60 °  2  560 .. 970 = 770 ± 27%  840  
70 °  2.9  430 .. 880 = 650 ± 34% ^{[17]}  710  
75 °  3.8  330 .. 800 = 560 ± 41% ^{[17]}  620  
80 °  5.6  200 .. 660 = 430 ± 53%  470  
85 °  10  85 .. 480 = 280 ± 70%  270  
90  38  20 
This is only a few degrees above the horizon. For example, when the sun is over 60 ° above the horizon ({\ displaystyle z}<30 °) The solar intensity is about 1000 W / m ^{2} (from equation I.1 as shown in the above table), where the sun is only 15 ° above the horizon ({\ displaystyle z}= 75 °) the solar intensity is still about 600 W / m ^{2} or 60% of its maximum level; and only 5 ° above the horizon still 27% of the maximum.
At higher altitudes
An approximate model for an increase in altitude and accuracy to a few kilometers above sea level is given by: ^{[13] }^{[19]}

{\ displaystyle I = 1.1 \ times I _ {\ mathrm {o}} \ times [(1h / 7.1) 0.7 ^ {(AM) ^ {0.678})} + h / 7.1] \,} ( I.2 )
Where {\ displaystyle h} is the solar collector’s height {\ displaystyle AM}is the airmass (from A.2 ) as if the collector was installed at sea level.
Alternatively, given the significant practical variabilities, the homogeneous spherical model could be applied to estimate AM, using:

{\ displaystyle AM = {\ sqrt {(r + c) ^ {2} \ cos ^ {2} z + (2r + 1 + c) (1c)}} \; – \; (r + c) \ cos z \,} ( A.4 )
where the normalized heights of the atmosphere and the collector are respectively {\ displaystyle r = R _ {\ mathrm {E}} / y _ {\ mathrm {atm}}} ≈ 708 (as above) and {\ displaystyle c = h / y _ {\ mathrm {atm}}} .
And then the table of the appropriate equation ( I.1 or I.3 or I.4 for average, polluted or clean air respectively) can be used in the normal way.
These approximations at I.2 and A.4 are suitable for use at altitudes of a few kilometers above sea level, and they are associated with AMO performance levels at only around 6 and 9 km respectively. By contrast much of the attenuation of the high energy components occurs in the ozone layer – at higher altitudes around 30 km. ^{[20]} Hence these approximations are suitable for estimating the performance of ground based collectors.
Solar cell efficiency
Silicon solar cells are not very sensitive to the portions of the spectrum lost in the atmosphere. The resulting and spectrum at the Earth’s surface area more étroitement matches the bandgap of silicon n silicon solar cells are more efficient than at AM1 AM0. This apparently counterintuitive result arises simply because As illustrated below, even though the efficiency is lower at AM0 the total output power ( P _{out}) for a typical solar cell is still highest at AM0. Conversely, the shape of the spectrum does not have a significant impact on the environment.
AM  Solar intensity  Output power  Efficiency 

P _{in} W / m ^{2}  P _{out} W / m ^{2}  P _{out} / P _{in}  
0  1350  160  12% 
1  1000  150  15% 
2  800  120  15% 
This illustrates the more general item That Given That solar energy is “free”, and where available space is not a limitation, other factors Such As total P _{out} and P _{out} / $ are Often more significant considerations than efficiency ( P _{out}/ P _{in} ).
See also
 Air mass (astronomy)
 Diffuse sky radiation
 Earth’s atmosphere
 insolation
 Mie scattering
 Photovoltaics
 Rayleigh scattering
 Solar cell
 Solar cell efficiency
 Solar energy
 Solar power
 Solar radiation
 Solar tracker
 Sun
 Sun chart
 Sun path
Notes and references
 Jump up^ or more precisely 5,777 K as reported inNASA Solar System Exploration – Sun: Facts & Figuresretrieved 27 April 2011 “Effective Temperature … 5777 K”
 Jump up^ See also the articleDiffuse sky radiation.
 Jump up^ Yellow is thecolor negativeof blue – yellow is the color of what remains after scattering removes some blue from the “white” light from the sun.
 Jump up^ Peter Würfel (2005). The Physics of Solar Cells . Weinheim: WileyVCH. ISBN3527408576.
 Jump up^ Kasten, F. and Young, AT (1989). Revised optical air mass tables and formula approximation. Applied Optics28: 47354738.
 ^ Jump up to:^{a }^{b} The main article Airmass reports in the range 36 to 40 for different atmospheric models
 Jump up^ Schoenberg, E. (1929). Theoretische Photometrie, g) Über die Extinktion des Lichtes in der Erdatmosphäre. InHandbuch der Astrophysik. Band II, erste Hälfte. Berlin: Springer.
 Jump up^ The main articleAirmassreports in the range 8 to 10 km for different atmospheric models
 Jump up^ Gueymard, C .; Myers, D .; Emery, K. (2002). “Proposed reference irradiance spectra for solar energy systems testing”. Solar Energy . 73 (6): 443467. doi : 10.1016 / S0038092X (03) 000057 .
 Jump up^ Reference Solar Spectral Irradiance: Air Mass 1.5NREL retrieved 1 May 2011
 ^ Jump up to:^{a }^{b} Reference Solar Spectral Irradiance: ASTM G173 ASTM retrieved 1 May 2011
 ^ Jump up to:^{a }^{b} Planning and installing photovoltaic systems: a guide for installers, architects and engineers , 2nd Ed. (2008), Table 1.1, Earthscan with the International Institute for Environment and Development, Deutsche Gesellschaft für Sonnenenergie. ISBN 1844074420 .
 ^ Jump up to:^{a }^{b }^{c} PVCDROM retrieved May 1, 2011, Stuart Bowden and Christiana Honsberg, Solar Power Labs, Arizona State University
 Jump up^ Meinel, AB and Meinel, MP (1976). Applied Solar EnergyAddison Wesley Publishing Co.
 Jump up^ TheEarthscanreference uses 1367 W / m^{2}as the solar intensity to the external atmosphere.
 Jump up^ The ASTM G173 Standard Measures solar intensity over the band 280 to 4000 nm.
 ^ Jump up to:^{a }^{b }^{c }^{d }^{e} Interpolated from data in the Earthscan reference using suitable Least squares estimate variants of equation I.1 :
 for polluted air:


{\ displaystyle I = 1.1 \ times I _ {\ matthm {o}} \ times 0.56 ^ {(AM ^ {0.715})} \,} ( I.3 )

 for clean air:


{\ displaystyle I = 1.1 \ times I _ {\ mathrm {o}} \ times 0.76 ^ {(AM ^ {0.618})} \,} ( I.4 )

 Jump up^ The ASTM G173 Standard Measures solar intensity under “rural aerosol loading” ie clean air requirements – Thus the standard value fits étroitement to the maximum of the expected range.
 Jump up^ Laue, EG (1970)The measurement of solar spectral irradiance at different elevations terrestrial,Solar Energy, vol. 13, no. 1, pp. 4350, IN1IN4, 5157, 1970.
 Jump up^ RLF Boyd (Ed.) (1992). Astronomical photometry: a guide, section 6.4. Kluwer Academic Publishers. ISBN 0792316533.