Solar Brightness

The Bright Light

Introduction

A photometer made by making a grease spot on white paper can be used to compare the brightness of the sun to the brightness of a lamp. By finding a position at which the sun is as bright as the lamp the power output of the sun can be estimated.

Material

• A piece of white typing paper 10 cm square or larger
• a drop of cooking oil
• a meterstick
• a 200 Watt lightbulb and lamp to hold it. (It must be a spherical bulb not a flood light. a 150 Watt bulb can be used in a pinch.)

optional photometer

• two pieces of translucent white plastic at least 2.5 cm (1”) square and 5 mm thick
• a piece of aluminum foil the same size as the plastic
• transparent tape or a rubber band

Assembly

To make a grease spot photometer, place a small drop of food oil in the center of the paper and spread it around into a circle about 2 cm in diameter with your finger.

Optional assembly

To make a plastic photometer, place the aluminum foil between the two pieces of plastic and hold them all together with the rubber band or tape.

To Do and Notice

Place the lamp outside in direct sunlight.

Hold the photometer paper between the lamp and the sun. Move the photometer toward and away from the lamp until the brightness of the lamp is the same as the brightness of the sun.(The grease spot will be as bright as the surrounding paper or when viewing the photometer from its edge, one side will be as bright as the other.)

Measure the distance from the center of the lamp to the photometer.

Notice the difference in color between sunlight and incandescent lamp light.

What’s Going On?

The lamp filament is cooler than the sun. Cooler objects emit proportionately more red light than blue light and so appear redder in color. (The filament is about 2900 K while the sun is about 5800 K.)

The light intensity decreases inversely proportional to the square of the distance. This inverse square proportionality can be used to quantitatively compare the brightness of the bulb and the brightness of the sun.

Math Root

The brightness of a light is how bright it appears to the human eye. The intensity of a light is the power per meter squared carried by the light. If the lamp were to put out 200 Watts of power in visible light, then its intensity at a distance of 1 meter would be 200 Watts spread over the area of a sphere 1 meter in diameter, 200/4pir2. Since the area of a sphere increases as the square of its radius, the intensity of a light source which spreads uniformly decreases as the inverse square of the distance from the center of the source (the radius). When the photometer indicates that the brightness of the lamp is equal to the brightness of the sun then the intensity of the lamp is approximately equal to the intensity of light from the sun.

Intensity of lamp = intensity of sun

power of lamp/distance to lamp2 )= power of sun/distance to sun2

Plamp/D2lamp= Psun/D2sun

or

Psun = PlampD2sun/D2lamp

The distance to the sun is 1.5 x 1011 m.

Example calculation.

If the brightness of a 200 W lightbulb equals the brightness of the sun at a distance of 10 cm then:

Psun = Plamp (1.5 x 1011m)2 /(0.1m)2

= Plamp x 2.25 x 1024

So the sun is over 2 X 1024 times as bright as the 200 Watt bulb or 4 x 1026 Watts.

The power of the sun is actually about 4 x 1026 Watts (The Physics Factbook). Not all of the sun's power is in the visible, however more of the sun's power is in the visible than the lightbulb's power, since hotter objects have a larger fraction of their light in the visible than in the infra-red.

The Solar Constant

The solar constant is the amount of solar power per square meter arriving at the earth.

Taking the power of the sun as 4 x 1026 watts we can calculate the solar constant.

Distribute the solar power over the surface of a sphere with the radius of the earth's orbit. r = 1.5 x 1011m

The area is A = 4 pi r2 so A = 4 pi (1.5 x 1011m)2 = 3 x 1023 m2 .

The solar constant is then I = 4 x 1026 watts/3 x 1023 m2 = 1300 W/m2

This is the solar constant above the atmosphere.

The atmosphere removes some of this energy making the solar constant at the surface of the earth 1,000 W/m2

Etc

Here are three complications to this experiment:

The atmosphere of the earth scatters and absorbs sunlight. This will make our estimate of the solar power too small.

This experiment only compares the power of the sun and lightbulb in the visible portion of the spectrum. An incandescent lamp emits less than 3% of the power it consumes as visible light. The lamp thus emits 3% of 200 Watts as light or about 6 Watts. The sun is about twice the temperature as the lightbulb filament. As a result, the sun emits a larger fraction of its total power as visible light. See the activity titled Solar Thermal for a continuation of this activity. In which the total energy output of the sun at all wavelengths is compared to the energy output of the lamp.

The human eye is more sensitive to the spectrum of light emitted by the sun than to the reddish light emitted by the filament. This means that when the brightnesses of the two sources look equal the intensities are not exactly equal.(The intensity of the sun is slightly less than that of the lamp when they look to be the same brightness.)