Air mass coefficient
Encyclopedia
The air mass coefficient defines the direct optical path length through the Earth's atmosphere
, expressed as a ratio relative to the path length vertically upwards, i.e. 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 cell
s under standardized conditions, and is often referred to using the syntax "AM" followed by a number.
"AM1.5" is almost universal when characterizing terrestrial power-generating panels.
Solar radiation closely matches a black body
radiator at about 5,800 K.
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 though the atmosphere, chemicals interact with the sunlight and absorb certain wavelengths.
Perhaps the best known example is the stripping of ultraviolet
light by ozone
in the upper atmosphere, which dramatically reduces the amount of short wavelength light reaching the Earth's surface.
A more active component of this process is water vapor, which results in a wide variety of absorption bands at 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 between the far infrared and near ultraviolet.
Atmospheric scattering plays a role, removing higher frequencies from direct sunlight and scattering it about the sky.
This is why the sky appears blue and the sun yellow - more of the higher frequency blue light arrives at the observer via indirect scattered paths; and less blue light follows the direct path, giving the sun a yellow tinge.
The greater the distance of atmosphere the sunlight travels through, the greater this effect, which is why the sun looks orange/red at dawn and sundown when the sunlight is traveling very obliquely through the atmosphere - progressively more of the blues and greens are removed from the direct rays, leading to orange or red appearance of the sun; and the sky appears pink - because the blues and greens are scattered over such long paths that they are very greatly attenuated before arriving at the observer, resulting in the characteristic pink skies at dawn and sunset.
through the atmosphere, for solar radiation incident at angle 
relative to the normal to the Earth's surface, the air mass coefficient is:
where
is the zenith path length (i.e. normal to the Earth's surface) at sea level
and
is the zenith angle in degrees.
The air mass number is thus dependent on the Sun's elevation path through the sky and therefore varies with time of day and with the passing seasons of the year, and with the latitude of the observer.
up to around 75°. A number of refinements have been proposed to more accurately model the path thickness towards the horizon, such as that proposed by Kasten and Young (1989):
A more comprehensive list of such models is provided in the main article Airmass
, for various atmospheric models and experimental data sets.
At sea level the air mass towards the horizon (
= 90°) is approximately 38.
Modelling the atmosphere as a simple spherical shell provides a reasonable approximation:
where the radius of the Earth
= 6371 km, the effective height of the atmosphere 
≈ 9 km, and their ratio 
≈ 708.
These models are compared in the table below:
This implies that for these purposes the atmosphere can be considered to be effectively concentrated into around the bottom 9 km, i.e. 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.
The spectrum outside the atmosphere, the 5,800 K black body, is referred to as "AM0", meaning "zero atmospheres". Solar cells used for space power applications, like those on communications satellite
s are generally characterized using AM0.
The spectrum after travelling through the atmosphere to sea level with the sun directly overhead is referred to, by definition, as "AM1". This means "one atmosphere".
AM1 (
=0°) to AM1.1 (
=25°) is a useful range for estimating performance of solar cells in equatorial and tropical regions.
Solar panels do not generally operate under exactly one atmosphere's thickness: if the sun is at an angle to the Earth's surface the effective thickness will be greater. Many of the world's major population centres, and hence solar installations and industry, across Europe, China, Japan, the United States of America and elsewhere (including northern India, southern Africa and Australia) lie in temperate
latitudes. An AM number representing the spectrum at mid-latitudes is therefore much more common.
"AM1.5", 1.5 atmosphere thickness, corresponds to a solar zenith angle of
=48.2°. While the summertime AM number for mid-latitudes during the middle parts of the day is less than 1.5, higher figures apply in the morning and evening and at other times of the year. Therefore AM1.5 is useful to represent the overall yearly average for mid-latitudes. Consequently, the solar industry uses AM1.5 for all standardized testing of terrestrial solar panels.
AM2 (
=60°) to AM3 (
=70°) is a useful range for estimating the overall average performance of solar cells installed at high latitudes such as in northern Europe.
Similarly AM2 to AM3 is useful to estimate wintertime performance in temperate latitudes, e.g. airmass coefficient is greater than 2 at all hours of the day in winter at latitudes as low as 37°.
AM38 is generally regarded as being the airmass in the horizontal direction (
=90°) at sea level.
However, in pactice there is a high degree of variability in the solar intensity received at angles close to the horizon as described in the next section Solar intensity.
Airmass coefficient reduces with increasing altitude above sea level. Therefore solar panels installed at high altitudes, e.g. in an Altiplano
region, would use lower AM numbers than for the corresponding latitude at sea level: AM numbers less than 1 towards the equator, and correspondingly lower numbers than listed above for other latitudes.
For example, almost all high energy radiation is removed in the upper atmosphere (between AM0 and AM1) and so AM2 is not twice as bad as AM1.
Furthermore there is great variability in many of the factors contributing to atmospheric attenuation,
such as water vapor, aerosols, photochemical smog and the effects of temperature inversions.
Depending on level of pollution in the air, overall attenuation can change by up to ±70% towards the horizon, greatly affecting performance particularly towards the horizon where effects of the lower layers of atmosphere are amplified manyfold.
One approximate model for solar intensity versus airmass is given by:
where solar intensity external to the Earth's atmosphere
= 1.353 kW/m2.
This formula fits comfortably within the mid-range of the expected pollution-based variability:
This illustrates that significant power is available at only a few degrees above the horizon, and hence the value of tracking solar collectors
.
One approximate model for intensity increase with altitude and accurate to a few kilometres above sea level is given by:
where
is the solar collector's height above sea level in km and 
is the airmass (from ) as if the collector was installed at sea level.
Alternatively, given the significant practical variabilities involved, the homogeneous spherical model could be applied to estimate AM, using:
where the normalized heights of the atmosphere and of the collector are respectively
≈ 708 (as above) and 
.
And then the above table or the appropriate equation ( or or for average, polluted or clean air respectively) can be used to estimate intensity from AM in the normal way.
These approximations at and are suitable for use only to altitudes of a few kilometres above sea level, implying as they do reduction to AM0 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.
Hence these approximations are suitable only for estimating the performance of ground based collectors.
so silicon solar cells are more efficient at AM1 than AM0. This apparently counter-intuitive result arises simply because silicon cells can't make much use of the high energy radiation which the atmosphere filters out.
As illustrated below, even though the efficiency is lower at AM0 the total output power (Pout) for a typical solar cell is still highest at AM0.
Conversely, the shape of the spectrum does not significantly change with further increases in atmospheric thickness, and hence cell efficiency does not greatly change for AM numbers above 1.
This illustrates the more general point that given that solar energy is "free", and where available space is not a limitation, other factors such as total Pout and Pout/$ are often more important considerations than efficiency (Pout/Pin).
Earth's atmosphere
The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...
, expressed as a ratio relative to the path length vertically upwards, i.e. at the zenith
Zenith
The zenith is an imaginary point directly "above" a particular location, on the imaginary celestial sphere. "Above" means in the vertical direction opposite to the apparent gravitational force at that location. The opposite direction, i.e...
.
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 cell
Solar cell
A solar cell is a solid state electrical device that converts the energy of light directly into electricity by the photovoltaic effect....
s under standardized conditions, and is often referred to using the syntax "AM" followed by a number.
"AM1.5" is almost universal when characterizing terrestrial power-generating panels.
Description
Black body
A black body is an idealized physical body that absorbs all incident electromagnetic radiation. Because of this perfect absorptivity at all wavelengths, a black body is also the best possible emitter of thermal radiation, which it radiates incandescently in a characteristic, continuous spectrum...
radiator at about 5,800 K.
As it passes through the atmosphere, sunlight is attenuated by scattering
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
and absorption
Absorption (electromagnetic radiation)
In physics, absorption of electromagnetic radiation is the way by which the energy of a photon is taken up by matter, typically the electrons of an atom. Thus, the electromagnetic energy is transformed to other forms of energy for example, to heat. The absorption of light during wave propagation is...
; the more atmosphere through which it passes, the greater the attenuation
Attenuation
In physics, attenuation is the gradual loss in intensity of any kind of flux through a medium. For instance, sunlight is attenuated by dark glasses, X-rays are attenuated by lead, and light and sound are attenuated by water.In electrical engineering and telecommunications, attenuation affects the...
.
As the sunlight travels though the atmosphere, chemicals interact with the sunlight and absorb certain wavelengths.
Perhaps the best known example is the stripping of ultraviolet
Ultraviolet
Ultraviolet light is electromagnetic radiation with a wavelength shorter than that of visible light, but longer than X-rays, in the range 10 nm to 400 nm, and energies from 3 eV to 124 eV...
light by ozone
Ozone
Ozone , or trioxygen, is a triatomic molecule, consisting of three oxygen atoms. It is an allotrope of oxygen that is much less stable than the diatomic allotrope...
in the upper atmosphere, which dramatically reduces the amount of short wavelength light reaching the Earth's surface.
A more active component of this process is water vapor, which results in a wide variety of absorption bands at 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 between the far infrared and near ultraviolet.
Atmospheric scattering plays a role, removing higher frequencies from direct sunlight and scattering it about the sky.
This is why the sky appears blue and the sun yellow - more of the higher frequency blue light arrives at the observer via indirect scattered paths; and less blue light follows the direct path, giving the sun a yellow tinge.
The greater the distance of atmosphere the sunlight travels through, the greater this effect, which is why the sun looks orange/red at dawn and sundown when the sunlight is traveling very obliquely through the atmosphere - progressively more of the blues and greens are removed from the direct rays, leading to orange or red appearance of the sun; and the sky appears pink - because the blues and greens are scattered over such long paths that they are very greatly attenuated before arriving at the observer, resulting in the characteristic pink skies at dawn and sunset.
Definition
For a path length through the atmosphere, for solar radiation incident at angle relative to the normal to the Earth's surface, the air mass coefficient is:where
is the zenith path length (i.e. normal to the Earth's surface) at sea levelSea level
Mean sea level is a measure of the average height of the ocean's surface ; used as a standard in reckoning land elevation...
and
is the zenith angle in degrees.The air mass number is thus dependent on the Sun's elevation path through the sky and therefore varies with time of day and with the passing seasons of the year, and with the latitude of the observer.
Accuracy near the horizon
The above approximation overlooks the curvature of the Earth, and is reasonably accurate for values of up to around 75°. A number of refinements have been proposed to more accurately model the path thickness towards the horizon, such as that proposed by Kasten and Young (1989):A more comprehensive list of such models is provided in the main article Airmass
Airmass
In astronomy, air mass is the optical path length through Earth’s atmosphere for light from a celestial source. As it passes through the atmosphere, light is attenuated by scattering and absorption; the more atmosphere through which it passes, the greater the attenuation. Consequently, celestial...
, for various atmospheric models and experimental data sets.
At sea level the air mass towards the horizon (
= 90°) is approximately 38.Modelling the atmosphere as a simple spherical shell provides a reasonable approximation:
where the radius of the Earth
= 6371 km, the effective height of the atmosphere ≈ 9 km, and their ratio ≈ 708.These models are compared in the table below:
 |
Flat Earth | Kasten & Young | Spherical shell |
|---|---|---|---|
| degree | |||
| 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° |  |
37.9 | 37.6 |
This implies that for these purposes the atmosphere can be considered to be effectively concentrated into around the bottom 9 km, i.e. essentially all the atmospheric effects are due to the atmospheric mass in the lower half of the Troposphere
Troposphere
The troposphere is the lowest portion of Earth's atmosphere. It contains approximately 80% of the atmosphere's mass and 99% of its water vapor and aerosols....
. This is a useful and simple model when considering the atmospheric effects on solar intensity.
Cases
- AM0
The spectrum outside the atmosphere, the 5,800 K black body, is referred to as "AM0", meaning "zero atmospheres". Solar cells used for space power applications, like those on communications satellite
Communications satellite
A communications satellite is an artificial satellite stationed in space for the purpose of telecommunications...
s are generally characterized using AM0.
- AM1
The spectrum after travelling through the atmosphere to sea level with the sun directly overhead is referred to, by definition, as "AM1". This means "one atmosphere".
AM1 (
=0°) to AM1.1 (=25°) is a useful range for estimating performance of solar cells in equatorial and tropical regions.- AM1.5
Solar panels do not generally operate under exactly one atmosphere's thickness: if the sun is at an angle to the Earth's surface the effective thickness will be greater. Many of the world's major population centres, and hence solar installations and industry, across Europe, China, Japan, the United States of America and elsewhere (including northern India, southern Africa and Australia) lie in temperate
Temperate
In geography, temperate or tepid latitudes of the globe lie between the tropics and the polar circles. The changes in these regions between summer and winter are generally relatively moderate, rather than extreme hot or cold...
latitudes. An AM number representing the spectrum at mid-latitudes is therefore much more common.
"AM1.5", 1.5 atmosphere thickness, corresponds to a solar zenith angle of
=48.2°. While the summertime AM number for mid-latitudes during the middle parts of the day is less than 1.5, higher figures apply in the morning and evening and at other times of the year. Therefore AM1.5 is useful to represent the overall yearly average for mid-latitudes. Consequently, the solar industry uses AM1.5 for all standardized testing of terrestrial solar panels.- AM2~3
AM2 (
=60°) to AM3 (=70°) is a useful range for estimating the overall average performance of solar cells installed at high latitudes such as in northern Europe.Similarly AM2 to AM3 is useful to estimate wintertime performance in temperate latitudes, e.g. airmass coefficient is greater than 2 at all hours of the day in winter at latitudes as low as 37°.
- AM38
AM38 is generally regarded as being the airmass in the horizontal direction (
=90°) at sea level.However, in pactice there is a high degree of variability in the solar intensity received at angles close to the horizon as described in the next section Solar intensity.
- At higher altitudes
Airmass coefficient reduces with increasing altitude above sea level. Therefore solar panels installed at high altitudes, e.g. in an Altiplano
Altiplano
The Altiplano , in west-central South America, where the Andes are at their widest, is the most extensive area of high plateau on Earth outside of Tibet...
region, would use lower AM numbers than for the corresponding latitude at sea level: AM numbers less than 1 towards the equator, and correspondingly lower numbers than listed above for other latitudes.
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 radiation is removed in the upper atmosphere (between AM0 and AM1) and so AM2 is not twice as bad as AM1.
Furthermore there is great variability in many of the factors contributing to atmospheric attenuation,
such as water vapor, aerosols, photochemical smog and the effects of temperature inversions.
Depending on level of pollution in the air, overall attenuation can change by up to ±70% towards the horizon, greatly affecting performance particularly towards the horizon where effects of the lower layers of atmosphere are amplified manyfold.
One approximate model for solar intensity versus airmass is given by:
where solar intensity external to the Earth's atmosphere
= 1.353 kW/m2.This formula fits comfortably within the mid-range of the expected pollution-based variability:
 |
AM | range due to pollution | ASTM G-173 | |
|---|---|---|---|---|
| degree | W/m2 | W/m2 | W/m2 | |
| - | 0 | 1367 | 1353 | 1347.9 |
| 0° | 1 | 840 .. 1130 = 990 ± 15% | 1040 | |
| 23° | 1.09 | 800 .. 1110 = 960 ± 16% | 1020 | |
| 30° | 1.15 | 780 .. 1100 = 940 ± 17% | 1010 | |
| 45° | 1.41 | 710 .. 1060 = 880 ± 20% | 950 | |
| 48.2° | 1.5 | 680 .. 1050 = 870 ± 21% | 930 | 1000.4 |
| 60° | 2 | 560 .. 970 = 770 ± 27% | 840 | |
| 70° | 2.9 | 430 .. 880 = 650 ± 34% | 710 | |
| 75° | 3.8 | 330 .. 800 = 560 ± 41% | 620 | |
| 80° | 5.6 | 200 .. 660 = 430 ± 53% | 470 | |
| 85° | 10 | 85 .. 480 = 280 ± 70% | 270 | |
| 90° | 38 | 20 |
and airmass coefficient AMThis illustrates that significant power is available at only a few degrees above the horizon, and hence the value of tracking solar collectors
Solar tracker
A solar tracker is a generic term used to describe devices that orient various payloads toward the sun. Payloads can be photovoltaic panels, reflectors, lenses or other optical devices....
.
- At higher altitudes
One approximate model for intensity increase with altitude and accurate to a few kilometres above sea level is given by:
where
is the solar collector's height above sea level in km and is the airmass (from ) as if the collector was installed at sea level.Alternatively, given the significant practical variabilities involved, the homogeneous spherical model could be applied to estimate AM, using:
where the normalized heights of the atmosphere and of the collector are respectively
≈ 708 (as above) and .And then the above table or the appropriate equation ( or or for average, polluted or clean air respectively) can be used to estimate intensity from AM in the normal way.
These approximations at and are suitable for use only to altitudes of a few kilometres above sea level, implying as they do reduction to AM0 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.
Hence these approximations are suitable only 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 spectrum at the Earth's surface more closely matches the bandgap of siliconSilicon
Silicon is a chemical element with the symbol Si and atomic number 14. A tetravalent metalloid, it is less reactive than its chemical analog carbon, the nonmetal directly above it in the periodic table, but more reactive than germanium, the metalloid directly below it in the table...
so silicon solar cells are more efficient at AM1 than AM0. This apparently counter-intuitive result arises simply because silicon cells can't make much use of the high energy radiation which the atmosphere filters out.
As illustrated below, even though the efficiency is lower at AM0 the total output power (Pout) for a typical solar cell is still highest at AM0.
Conversely, the shape of the spectrum does not significantly change with further increases in atmospheric thickness, and hence cell efficiency does not greatly change for AM numbers above 1.
| AM | Solar intensity | Output power | Efficiency |
|---|---|---|---|
| Pin W/m2 | Pout W/m2 | Pout / Pin | |
| 0 | 1350 | 160 | 12% |
| 1 | 1000 | 150 | 15% |
| 2 | 800 | 120 | 15% |
This illustrates the more general point that given that solar energy is "free", and where available space is not a limitation, other factors such as total Pout and Pout/$ are often more important considerations than efficiency (Pout/Pin).
See also
- Air mass (astronomy)
- Diffuse sky radiationDiffuse sky radiationDiffuse sky radiation is solar radiation reaching the Earth's surface after having been scattered from the direct solar beam by molecules or suspensoids in the atmosphere. It is also called skylight, diffuse skylight, or sky radiation and is the reason for changes in the colour of the sky...
- Earth's atmosphereEarth's atmosphereThe atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...
- InsolationInsolationInsolation is a measure of solar radiation energy received on a given surface area in a given time. It is commonly expressed as average irradiance in watts per square meter or kilowatt-hours per square meter per day...
- Mie scattering
- PhotovoltaicsPhotovoltaicsPhotovoltaics is a method of generating electrical power by converting solar radiation into direct current electricity using semiconductors that exhibit the photovoltaic effect. Photovoltaic power generation employs solar panels composed of a number of solar cells containing a photovoltaic material...
- Rayleigh scatteringRayleigh scatteringRayleigh scattering, named after the British physicist Lord Rayleigh, is the elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the light. The particles may be individual atoms or molecules. It can occur when light travels through...
- Solar cellSolar cellA solar cell is a solid state electrical device that converts the energy of light directly into electricity by the photovoltaic effect....
- Solar cell efficiencySolar cell efficiencyThe efficiency of a solar cell may be broken down into reflectance efficiency, thermodynamic efficiency, charge carrier separation efficiency and conductive efficiency...
- Solar energy
- Solar powerSolar powerSolar energy, radiant light and heat from the sun, has been harnessed by humans since ancient times using a range of ever-evolving technologies. Solar radiation, along with secondary solar-powered resources such as wind and wave power, hydroelectricity and biomass, account for most of the available...
- Solar radiation
- Solar trackerSolar trackerA solar tracker is a generic term used to describe devices that orient various payloads toward the sun. Payloads can be photovoltaic panels, reflectors, lenses or other optical devices....
- SunSunThe Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
- Sun chartSun chartA Sun chart is a graph of the ecliptic of the Sun through the sky throughout the year at a particular latitude.Most sun charts plot azimuth versus altitude throughout the days of the winter solstice and summer solstice, as well as a number of intervening days...
- Sun pathSun pathSun path refers to the apparent significant seasonal-and-hourly positional changes of the sun as the Earth rotates, and orbits around the sun. The relative position of the sun is a major factor in the heat gain of buildings and in the performance of solar energy systems...