The Solar Constant: Compute the Energy produced by the Sun.

We here compute the "power" of the Sun using measurements, observations, and geometry.

M = E / c^2

We also calculate via E=Mc^2, how much matter is "transformed" to energy to produce the Sun's luminous output!

 How much solar energy is available in the space above the atmosphere that we can exploit for our use?

For years, we have built and launched man-made satellites to orbit the Earth above the Earth's Atmosphere. Most of these satellites are equipped with solar panels that face the Sun and that gather sunlight and convert its energy to electricity of the appropriate voltage and amperage to operate the satellite's electronics. The energy from sunlight is used to operate multiple devices in the satellite; it's gyros, its sensors, its camera's -- all the paraphernalia the satellite was launched to use.

After years of orbital experience with geosynchronous and near-earth orbiting satellites, we know how much "power" the Sun radiates. With calibrated solar energy collecting devices, we can observe and measure this energy directly.

This graphic below, shows how much of the Sun's energy "lands on" solar collectors above the Earth's atmosphere. By measuring the amount of energy delivered to devices on the satellite, and knowing the solar cells efficiency (around 16%), a very precise quantification of the amount of energy from the Sun can be measured. There are fluctuations in the amount of energy received from the Sun over time.

Image NASA, SOHO

Caption from NASA: Two and a half solar cycles of Total Solar Irradiance (TSI), also called 'solar constant'. This composite, compiled by the VIRGO team at the Physikalisch-Meteorologisches Observatorium / World Radiation Center Davos, Switzerland, shows TSI as daily values plotted in different colors for the different originating experiments. The difference between the minima values is also indicated, together with amplitudes of the three cycles.

 That is what is shown in the "solar constant" images above, from a variety of calibrated satellite instruments aboard SOHO.

Image NASA

NASA TEXT:Spacecraft illustration -- SOHO was launched in December 1995 by an Atlas Centaur rocket and became operational in March 1996. SOHO weighs about two tons and with its solar panels extended stands about 25 feet across. It was launched in December, 1995. SOHO will continue operating well past the next solar maximum in 2001. (Image credit: Alex Lutkus)

----------------------------Now, Some Details

COMPUTING THE POWER OF A STAR

"Our Star"

Okay, we are going to work from the observations at the top of Earth's atmosphere "back" to the radiating surface of the Sun -- and thereby conclude what the current "rough" energy output of the Sun is. Astronomers also can, by brightness measurements, determine the energy output of distant stars. The biggest uncertainty for the other (distant) stars is the distance. For the Sun, this is not the case and astronomers have settled on a value backed by many observations. It is as close as we can get to 1 A.U. at present.

Here are a few of the conceptual relationships we will be using.

The Sun is "not exactly" a point. Mathematics allows you to work with points, and often when you see a distant star at night, it sure looks like a point. But it isn't. The Sun's surface is the radiant source from which most life on Earth is powered and sustained. The Sun is by far the largest body in the Solar System. To determine the energy out put of the Sun, we here treat the Sun as a spherical body with a certain radius, radiating uniformly in all directions. We won't treat it like a point.

For the Inverse Square Law (geometry) a linear doubling of the radius quadruples the area over which radiation is spread.

Radiance = 1 / Radius^2

(Re) Radius of the Earth's orbit = 149,597,870,691 meters
(Rs) Radius of Sun = 695,980,000 meters
(As) Area of the Sun's surface = pi x 4 x (Rs)^2 = 6,087,001,144,794,057,611.981 square meters.

That is 6.087 x 10^18 square meters

(Es) = Energy output of Sun = (Sc) x ((Re) / (Rs))^2

(Sc)Solar Constant = whatever value observed, measured, imagined! (you choose and plug it in)

We calculate the dimunition of solar radiation received compared to the source of the energy in two steps since we are dealing with energy per unit area.

(Re/Rs) = 149,597,870,691 meters / 695,980,000 meters = 214.9456

Where we are located in the solar system defines the Ratio of Earth's orbital radius to the radius of the Sun = 214.9456. To change this linear amount into an area, we "square it" to see how much energy is emanating from the surface of the Sun.

Ratio of area of "Sphere at Earth's Orbit" to "surface area" of Sun = (214.9456)^2 =46,201.6307

Energy output of each square meter of the Sun's surface = (Sc) x 46,201.637 That means a square meter on the Sun's surface is 46,201.637 times as energetic as a square meter at the distance of the Earth. This is an average "geometric" distance.

If we observe and measure the Solar Constant to be 1365.4 watts/second/m^2 like in one of the minimum averages in 1986, for example: 46201.637 x 1365.4 watt/m^ 2=63,083,715.1598 Watt per meter^ 2, so the Sun's area in square meters is multiplied by the energy per square meter.

For a Solar constant of 1365.4 W/m^2 we find that the Sun radiates 63.08 Million Watts/m^2

The Sun has a large surface, so we multiply that wattage/meter^2 by the number of square meters on the Sun's surface. This will yield the total energy produced by the Sun.

6,087,001,144,794,057,611.981meters^2 x 63,083,715 Watts/m^2 = 383,990,646,395,564,847,025,887,550 Watts

6.087 x 10^18 m^2 x 63.08 x 10^6W/m^2 = 3.84 x 10^26 watts

To turn this to ergs, which is what is used by the traditional (not newbie) astronomers, we convert watts to ergs by multiplying by 10,000,000. Therefore: Sun's ouput in ergs = 10^7 ergs/watt x 3.84 x 10^26 watts = 3.84 x 10^33 ergs every second of time. 

As it grows older, the Sun gets brighter; more "luminous"

This still image is a current, "latest image," taken by SOHO today.

SOHO is a jointly operated satellite (Both NASA and ESA) stationed at the (L1) Lagrangian point. SOHO moves around the Sun in step with the Earth, by slowly orbiting around the First Lagrangian Point, where the balanced gravity of the Earth and Sun keep SOHO in an orbit locked to this Earth-Sun line. The L1 point is approximately 1.5 million kilometers away from Earth (about four times the distance of the Moon), in the direction of the Sun. SOHO enjoys an uninterrupted view of our daylight star.

Maybe next version?

It is too bad a good optical package with a 12-inch or so objective was not piggy-backed for a selective magnification full visible light Earth-image to be broadcast back to Earth "live." Al Gore mentioned this at a later time as the full value of SOHO was being recognized.

 
IMAGES: NASA, SOHO

SOHO scientists and technicians are movie makers and pure scientific documentarians--since there are 'movies' that are documents recording actual events on a continuous basis. Many of these movies are spectacular and quite interesting -- to many and even scientists they are just plain fun to watch and spectacular activity is common. Below there are links to movies, and hints of how the Sun and its top layers perform through these images.

An mpeg is here: http://sohowww.nascom.nasa.gov/pickoftheweek/2breaks304.mpg

A Quicktime movie is here: http://sohowww.nascom.nasa.gov/pickoftheweek/2breaks304.mov

NASA: "SOHO was designed for a nominal mission lifetime of two years. In 1997 the mission was extended until 2003 because of its spectacular success. In 2002, a second extension of another four years was granted, that is, through March 2007. This will allow SOHO to cover a complete 11-year solar cycle."

Still going strong, it should be operated until its replacement is built and stationed -- and this time a good Earth Observation package in the visible optical to see the planet Earth, as it would look to a person sitting in space, at L1, watching the Earth in perpetual "daytime."

THE SCIENCE:
Observations and Measurements

Sunspots numbers. Compare the rise in sunspot numbers and the solar energy incident to earth.

Image: NASA

Galileo saw many sunspots, noted the Sun's rotation, and made written records of these numbers -- and all of those observations that have survived to this day are a part of Solar Science. The following graphic is a rendition of data compiled by Judith Lean -- and contains values for the solar constant that are both illustrative and puzzling. They show the Maunder Minimum and best evidenced estimates of the solar constant for the period from year 1500 AD to year 1998 AD.  As a youngster in the 50's and a youth in the early 60's, I remember the very active sunspot period -- from direct observation.

 Image:NASA, Data compiled by Judith Lean, of Naval Research Laboratory.

NASA has a link to the data: Here: http://aom.giss.nasa.gov/SOLAR/SUN.LP

This is a way to look at the solar constant and its variation through time.

Visit this site -- and realize that a few short years ago this information was available in an easier to access form. Thanks to G.W. Bush it has been made intentionally more difficult for you to access.

http://rredc.nrel.gov/solar/old_data/nsrdb/redbook/atlas/

http://rredc.nrel.gov/solar/old_data/nsrdb/redbook/atlas/colorgifs/325.GIF

For insight into what is going on now under the influence of Big Carbon and the Bush Administration this kind of information and data is priceless. You can address the denialists hired by Big Carbon via use of these kinds of data.

For solar constant by proxy means, you can link to this site, noted also above and get a feel for the rise of the value with time.

If you link to it, you can scroll from 1500 AD to 1998 AD, and get a feel for the data Judith Lean assembled. I do not know what she used for her sources. Sometime in the past, the Sun's output was a few watts lower than now. For example, in year 1523 AD the Solar Constant is given as 1363.4656 W/m^2 and in year 1998 AD as 1366.1332 W/m^2. You can see the 11 year solar cycle in recent years, and in the older data, you can see where the 11-year cycle simply did not exist. No real understanding of the sunspot or irradiance cycle yet exists. The Sun is a Variable Star -- but thankfully for Life on Earth -- the variations are small and changes are slow.

By using these energy observations -- and geometry -- it is possible to calculate "roughly" how much matter is transformed to energy by the Sun. From the observed solar constant we can determine the energy output of the Sun.

"Roughly?" Yes.

Although the Sun is about 8 min 20 seconds away at the speed of light -- the energy that springs forth from the Sun into space took thousands of years to climb out of the Sun's core to the surface. The time of energy travel from its transformation from matter to energy is a random radiation walk through the dense core of the Sun -- and thence outward through layer after layer of the Sun's body -- and will be addressed as another topic.  [Note also, that nuetrinos aren't hampered in the least [hardly if at all] by hundreds of thousands of miles of solar gas on their way into space. They pass right out of the Sun once they pop into existence, and even "change" into different varieties as they reach and pass through the Earth.]

Earth's climate and life forms are determined by how much of the energy the Sun emits is retained by the Earth. One of the reason's the Earth is not a frozen-ice-covered-world -- is that the atmosphere is composed of gases that absorb and trap energy that would otherwise be radiated into space. The "constancy" of the solar constant is what allows living things to adapt as the energy environment of the Earth-Sun changes through time.

Far in the future, when the evolving sun is 10% more energetic than now, the Earth's oceans will evaporate into the air. This will begin to occur in a serious way ~ 1100 million years in the future. I added the 10% increase to the Solar Constant in the little list below, at the bottom of the listing.

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The selling of lots and properties upon the Sun's surface.

To get a feel for the immensity of the energy compared to our human scale, we can compare the largest power plant we humans have built with how many square meters of Sun would produce the same amount of energy!

Earth's largest power plants are non-polluting hydroelectric plants which use the Sun's energy indirectly. Water is evaporated (by the Sun,) rises (water vapor is lighter than average air), is transported to altitude and rains or snows to Earth If it returns directly by rain or snow to the ocean, it can't be dammed and exploited, but when it rains or snows on land --

For this exercise, we will use 1367 Watts/m^2 as the Solar Constant where the Sun's output is 63,157,637.7790 Watts per square meter of surface.

THREE GORGES DAM China supposedly with capacity of 20 GW would be equivalent of 316.7 square meters of property on the Sun equal to a square 17.7 meters on a side, an "upper class" American home of 3408.6 square feet of floor space.

ITAIPU Brazil/Paraguay 14.75 GW = 233.5 square meters = a square 15.3 meters on a side or an area of about 2,513 square feet -- about the same as an ample American family suburban dwelling.

GRAND COULEE DAM produces 6.809 GW of power. The surface of the Sun needed to produce that amount of power with 1367 Watts/m2 as the solar constant is 107.81 square meters. A square 10.38 meters on a side -- or a floor space of 1160 square feet. Like a condo or a medium sized apartment.

Image: U S Bureau of Reclamation

So how much "property" on the surface of the Sun would supply all of the Earths energy use at present?

In 2004, the species used 15 TW. But let us assume a little more energy use so we can get momentarily ahead of the curve. . .let's use 18 TW.

How much property on the surface of the Sun would it require to produce 18 TW?

18,000,000,000,000 Watts / 63,157,637.7790 W/m^2 = 285,001.16 square meters, A square 534 meters on a side. This is an area of of about 70.43 acres, approximately 28.5 hectares.

And how much of the Earth surface would be needed to capture the 18TW? With current solar cells?

A simple way to relate this to solar cells on the Earth's surface is to use that 46,201.637 factor, and multiply the 70.43 acres into an area on the Earth's surface. This is 3,253,981.3 acres or 5084.3458 square miles, that area could be contained in a square 71.3 miles on a side.

Okay, but the Earth's surface does not receive the watts of sunlight that reach the top of the Earth's atmosphere. In the tropics the average insolation is around 400 W/m^2, further north or south we are looking at an average in the temperate zone of from 130 W/m^2 to 370W/m^2.

Since we are presently confined to land to collect sunlight in any stable way to get the power on the electrical grid for this gedanken, let's just work with an average of 200 W/m^2 instead of the 1367W/m^2 at the top of the atmosphere, or 200/1367 or overall about 15%. 1/0.15 = 6.7, so 6.7 x 5084.3458 = 33,897.33 square miles. This could be contained in a square 184.11 miles on a side. An area about 1/3 the size of the State of Wyoming.

But if solar cells convert sunlight at only 15% efficiency, we would need 6.7 times as much area. That is 6.7 x 33,897.93 square miles = 227,112.111 square miles.

Okay. An area a little smaller than the State of Texas would supply the 18TW energy needs of the entire planet.  And that is with no real efficiency.

Of course, we could do this. Spread uniformly over the Earth's land surface, this would not be that big a contraption. Right now, there are 6.7 billion people. This works out to 945 square feet of the Earth's surface per person. (Nearly 88 square meters of solar collector for each person -- )

(This is a very approximate figure, derived by determining how many square feet is covered by 227,112.111 square miles and dividing by 6.7 billion. In the developed world we could easily afford this over a year or five and do away with the fossil fuel power generation. Once done Big Carbon just kinda dies. . .in a "peaceful" world.)

We need to work on building the global power network so that sunlight captured on one side of the world is available on the other. Buckminster Fuller had the right idea.

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Image:NASA, SOHO

How much matter is transformed to energy in the Sun every second?

We are going to use values from above for the total energy of the Sun for 1367 W/m^2
as the amount of energy at the top of the atmosphere. And 3.844 x 10^33 ergs as the energy produced by the sun.

E = M c^2 ; re-arranging . . . M = E / c^2

c = speed of light = 299,792,458 meters / second (by Oct 21,1983 definition)

c = 29,979,245,800 centimeters / second

c^2= 898,755,178,736,817,640,000 cm^2/s^2

M = 3.844 x 10^33 ergs / 898,755,178,736,817,640,000 cm^2/s^2

M = 4,277,471,875,492.53 grams and 1 million ( 1,000,000 grams = 1 metric ton)

M = 4,277,471.87549 metric tons

or 4.3 million metric tons of matter are transformed to energy each second.

Keep these relationships in mind as you help fight against Big Carbon's War on America.