INTRODUCTION
Energy from the Sun reaching the Earth drives almost every known physical and biological cycle in the Earth system. By making solar radiation calculations and examining radiation measurements, students can gain a better understanding of many physical cycles and concepts associated with the Earth system.
A detailed study of solar irradiance will give Earth & Space Science and Physics students a better understanding of:
 Solar radiation
 Electromagnetic spectrum
 Mathematical concepts that apply to solar radiation
 Climate variation due to latitude
 Seasonal weather changes
 Global energy balance
 Daily changes in solar radiation
 Changes in solar irradiance due to solar cycles
 Effects of solar irradiance variations on the earth system
This educational brief is designed to serve as a source of background information on solar radiation studies and as a reference for student investigations on this subject. Links to student investigations can be found at the end of this brief. Before beginning a detailed investigation of solar radiation, there are three terms that must be understood.
 Irradiance  The amount of electromagnetic energy incident on a surface per unit time per unit area. In the past this quantity has often been referred to as "flux".
* When measuring solar irradiance (via satellite), scientists are measuring the amount of electromagnetic energy incident on a surface perpendicular to the incoming radiation at the top of the Earth's atmosphere, not the output at the solar surface.
 Solar Constant  The solar constant is the amount of energy received at the top of the Earth's atmosphere on a surface oriented perpendicular to the Sun’s rays (at the mean distance of the Earth from the Sun). The generally accepted solar constant of 1368 W/m^{2} is a satellite measured yearly average.
 Insolation  In general, solar radiation is received at the Earth's surface. The rate at which direct solar radiation is incident upon a unit horizontal surface at any point on or above the surface of Earth. *I will refer to insolation as direct solar radiation at the Earth's surface.
The solar constant is an important value for current studies of global radiation balance & climate models. The problem that faces scientists studying Earth’s radiation budget and climate is that while satellites can “accurately” measure solar irradiance and calculate a solar constant, the surface insolation is much more difficult to assess. When the solar constant is calculated there are four major problems in trying to relate this radiation intensity to its effect on the Earth's surface or surface insolation.
 First, the calculation is made for the top of the atmosphere and not for the surface of the Earth.
 Second, the calculation assumes that the surface receiving the radiation is perpendicular to the radiation.
 Third, the calculation assumes that the surface receiving the radiation is at a mean SunEarth distance.
 Fourth, the calculation assumes that radiation emission from the Sun remains constant.
Trying to relate calculations made for the top of the atmosphere to the surface is a problem because up to 70% of incoming radiation can be blocked by the atmosphere and cloud cover. In attempts to create global energy budget models, scientists must insert estimations for the amount of energy actually reaching the surface.
Assuming that the surface receiving the radiation is perpendicular to the incoming radiation is a problem because this is a rare occasion even at tropical latitudes due to the rotation of the Earth (time of day), tilt of the Earth's axis in relation to the incoming solar radiation (season), and the latitude and orientation of the surface. All of these factors change the angle of the surface receiving the radiation, which changes the intensity of the energy received.
Assuming that the radiation emission of the Sun is constant is a problem because this value fluctuates with cycles in solar activity. NASA satellites have measured incoming radiation since 1978 and have recorded changes in solar irradiance. This data can be accessed on the internet from Goddard Space Flight Center.
SOLAR RADIATION AND THE ELECTROMAGNETIC SPECTRUM
The electromagnetic spectrum consists of the entire range of frequencies and wavelengths at which electromagnetic waves can travel. The electromagnetic spectrum organizes energy types by wavelength and frequency. The peak wavelength of radiation emitted from an object is dependent upon the temperature of the object and can be calculated using the Wien Displacement Law when the temperature of the object is known. (In astronomy these are solid objects such as stars and planets.)
Wien Displacement Law:
 maximum = 2897 / T
maximum = The peak wavelength of energy in
micrometers
T = The temperature of the object radiating energy

 Using this law, the peak wavelength of radiation emitted from an object is inversely proportional to the temperature of the object. The irradiance or radiation output of an object can be calculated using the StefanBoltzman Law when the temperature is known.
StefanBoltzman Law: E = T^{4}
 E = Surface Irradiance of the object
* = Emissivity of the object
= StefanBoltzman Constant (5.67x10^{8 }W/m^{2}K^{4 })
 T = Temperature of the object

 *Emissivity is the factor of how well a surface can absorb and emit energy. Emissivity numbers range from 0 to 1. Very black objects such as charcoal have an emissivity near 1 while shiny objects have an emissivity near 0.

 The Wien Displacement & StefanBoltzman laws strictly apply only to black bodies. Black bodies are capable of absorbing and emitting radiation at all wavelengths. Because the Sun & Earth are not perfect black bodies, applying these laws to them only allows approximate values to be obtained. The fact that the Sun is not a perfect black body is especially important when studying solar cycles. The most significant variations in solar radiation during these cycles occur in the UV & XRay portions of the solar spectrum. In order to compare solar emissions to black body emissions at the same temperature go to the Solar Spectrum/Black Body Graph.
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SOLAR RADIATION ENTERING THE EARTH SYSTEM
In order to study the effects of solar radiation on the Earth system, it is necessary to determine the amount of energy reaching the Earth's atmosphere & surface. Once the surface irradiance of the Sun is determined the amount of energy reaching the top of the Earth's atmosphere can be calculated using the Inverse Square Law. The average amount of energy received on a surface perpendicular to incoming radiation at the top of the atmosphere is the solar constant. (*While this calculation can lead to a better student understanding of the Inverse Square Law, the accepted value is a yearly average from NASA satellite measurements.)
Solar Radiation Striking the top of the Earth's Atmosphere
 The Inverse Square Law is used to calculate the decrease in radiation intensity due to an increase in distance from the radiation source.
 Inverse Square Law: I = E(4x R^{2})/(4x r^{2})
 I = Irradiance at the surface of the outer sphere
E = Irradiance at the surface of the object (Sun)
4 x R^{2} = surface area of the object
4 x r^{2} = surface area of the outer sphere
 In order to calculate the solar constant the following equation is used:
 So = E(Sun) x (R(Sun) / r)^{2
}So = Solar Constant
E= Surface Irradiance of the Sun
R= 6.96 x 10^{5 }km^{ }= Radius of the Sun
r = 1.5 x 10^{8 }km^{ }=Average SunEarth Distance
Insolation: Solar Radiation Striking the Surface
I = S cos Z
 I= Insolation
 S~ 1000 W/m^{2} (Clear day solar insolation on a surface perpendicular to incoming solar radiation. This value actually varies greatly due to atmospheric variables.)
 Z = Zenith Angle (Zenith Angle is the angle from the zenith (point directly overhead) to the Sun's position in the sky. The zenith angle is dependent upon latitude, solar declination angle, and time of day.)
Z = cos^{1 }(sin sin + cos cos cos H)
 = Latitude
H = = Hour Angle = 15^{o} x (Time  12) (Angle of radiation due to time of day. Time is given in solar time as the hour of the day from midnight.)
= Solar Declination Angle
 Solar Declination Angles for the Northern Hemisphere
 Vernal Equinox Mar. 21/22 = 0^{o}
Summer Solstice Jun. 21/22 = +23.5^{o
}Autumnal Equinox Sept. 21/22 = 0^{o
}Winter Solstice Dec. 21/22 = 23.5^{o}
IRRADIANCE DATA SOURCES
 In addition to making calculations for solar irradiation based upon physics concepts, students can access & analyze solar irradiance data that is collected by orbiting satellites and ground based pyranometers. Satellite irradiance data is available from 1978 to the present on the internet. The irradiance data has been collected by the following NASA satellites.
 Nimbus 7 (Earth Radiation Budget) 1978 1993
Solar Maximum Mission: Active Cavity Radiometer Irradiance Monitor I (ACRIM I) 19801989
Earth Radiation Budget Satellite (ERBS) Solar Monitor Measurements 1984 1996
Upper Atmosphere Research Satellite (UARS) ACRIM II Measurements 19911997
 Data and further information related to these satellites is available through the NASA Goddard Space Flight Center Data Archive Center.
SCIENCE VOCABULARY
Investigation #1: Irradiance and the Electromagnetic Spectrum
Electromagnetic Spectrum
Wavelength
Irradiation
Solar Constant
StefanBoltzman Law
Wien Displacement Law
Watt
Investigation #2: Calculating the Solar Constant
Global Energy Balance
Inverse Square Law
Solar Constant
Aphelion
Perihelion
Investigation #3: Variations in Solar Insolation Due to Time of Day, Season, & Latitude
Declination Angle
Zenith
Zenith Angle
Solar Constant
Solar Insolation
Hour Angle
Investigation #4: Solar Activity Cycles and Solar Irradiance
Sunspot
Solar Cycle
Solar Irradiance
Black body
Solar Flare
