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-------------------- 2488 - SUN - how hot is it?
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- How hot is the Sun? How do we know that? How can we possibly measure the temperature of the Sun? Actually, there are a couple of interesting ways you could do it. You could take a regular thermometer to the Sun but you would have to land there at night otherwise you would be burnt to a crisp.
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- Another way is to measure the energy of the light that is striking the Earth. A bolometer, or a wattmeter, would measure the intensity of radiation striking the surface of the Earth to be 1,300 watts per square meter.
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- But, the atmosphere has absorbed a lot of the Sun’s energy. So, let’s take a high altitude balloon up and measure at the top of the atmosphere. We again measure the energy reaching Earth.
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- We know that radiation energy decreases as the square of the distance from the source. We calculate the total energy from the Sun. Then, we calculate the energy from the surface of the Sun.
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- We can use Stefan-Boltzman’s law in physics that found the energy density to be proportional to the 4th power of the temperature to find the temperature of the surface.
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----------------- Energy Density = 6.5 * 10^7 watts / m^2
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- Here is how we measure it. A bolometer is the instrument used to measure the energy of electromagnetic radiation. The reading we get in the balloon is 1,400 watts per square meter.
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- The Sun’s energy reaching the Earth is 1,400 watts per square meter.
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- A bolometer measures energy in electromagnetic radiation. The first bolometer was invented in America in 1878 by Sam Langley. Langley’s bolometer was two strips of platinum covered with lampblack to make them good absorbers of light. One strip was covered and the other exposed to the radiation. The lampblack strip would heat up changing its resistance to electric current in the platinum strip.
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- A Wheatstone bridge and a galvanometer were used to measure the change in resistance and therefore the change in temperature. In astronomy, in order to achieve the highest sensitivity the bolometer is cooled to near absolute zero, -273 degrees C. The galvanometer is calibrated in watts / square meter. The reading is 1,400 watts / m^2.
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- The Earth is 93 million miles away from the Sun. The energy falls off as the inverse square of the distance. You can visualize this by thinking of a sphere surrounding the Sun that has a radius of 93 million miles, or 1.5*10^11 meters. The total surface area of this sphere is 4*pi* radius^2. The total surface area is 13 * (1.5 * 10^11)^2 = 13 * 2.3*10^22 = 28 * 10^22 meters^2.
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- And, we only measured one square meter to be 1,400 watts / m^2. Therefore, the total energy from the Sun is 40 * 10^25 watts.
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--------------- Energy of the Sun = 40^10^25 watts
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- To get the Energy Density on the surface of the Sun we need to take this total energy and divide by the surface area of the Sun. Area = 4*pi*r^2. But, what is the radius of the Sun?
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- We can measure the diameter of the Sun as seen from Earth in arc seconds and that is 0.57 degrees. This is the same as the Moon, thus allowing a total solar eclipse when they pass each other in the sky.
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- Again the distance to the Sun is 1.5*10^11 meters. The diameter of the Sun is to 0.57 degrees as the circumference of the circle, 2*pi*1.5*10^11 meters is to 360 degrees. The diameter is therefore = 14 * 10^8 meters.
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- Or, you could use trigonometry where the sine of the angle 0.57 degrees is 0.01. The sin (.057) * hypotenuse = 0.01 * 1.5*10^11 = 15 * 10^8 meters. So, the radius of the Sun is 7*10^8 meters, or the diameter is 14 * 10^8 meters
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- The surface area of the Sun is 4*pi*(7*10^8)^2 = 13 * 49 * 10^16 = 616 * 10^16 m^2.
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- The energy density on the surface of the Sun = 40*10^25 watts / 616 * 10^16 m^2
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---------------- Energy Density = 6.5 * 10^7 watts / m^2
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- Ok, back to Stefan-Boltzman’s law that states the energy density of a source to be directly proportional to the 4th power of its temperature. The constant of proportionality between energy and temperature is called the Stefan-Boltzman constant.
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- The Stefan-Boltzman constant measured in the laboratories here on Earth = 5.7 *10^-8 watts/m^2/Kelvin^4.
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---------- Energy Density = 5.7 *10^-8 watts/m^2/Kelvin^4 * ( Temperature)^4
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---------- 6.5 * 10^7 watts / m^2 = 5.7 *10^-8 watts/m^2/Kelvin^4 * ( Temperature)^4
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----------- T^4 = 1140 * 10^12 Kelvin^4
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----------- T = 5800 Kelvin
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- So, the temperature of the surface of the Sun is 5,800 Kelvin, or, 5,527 Centigrade, or, 9,980 Fahrenheit. Most often we just say the surface of the Sun is 10,000 F.
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- There is another way to get this temperature and that is by using Wien’s law in physics that states the peak wavelength of a radiating source is inversely proportional to temperature.
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- While we are up in the balloon let’s measure the energy of radiation at several frequencies. As we increase the frequencies in the measurement the energy intensity increases up to a peak at some frequency maximum, then decays at the higher frequencies. The wavelength at the freq. max. that we get is 500 nanometers.
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- That particular wavelength is the color yellow-green. Again, experiments done here on Earth can determine the constant of proportionality between maximum wavelength and temperature. The constant is called Wien’s Constant = 0.0029 meter * Kelvin.
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----------------- Wavelength Maximum = 0.0029 meter*Kelvin / Temperature
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----------------- 500 * 10^-9 meters = 0.0029 meter*Kelvin / Temperature
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----------------- T = 5800 Kelvin
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- The surface temperature again is 5,800 degrees Kelvin
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- If you want to get a constant named after you all you have to do is find one of these relationships where one variable is proportional to another. You plot these on a graph and then calculate the slope of the curve.
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- The slope is the rate of change of one variable versus the other and that is the constant of proportionality. Josef Stefan, (1853-1893) in Vienna, Austria, in 1879, at the age of 26, Stefan determined that radiation energy was directly proportional to the 4th power of temperature.
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----------------- Energy = constant * T^4
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- He plotted measurement data on a graph. Drew a straight line between the points. Calculated the slope of the line to be 5.7*10^-8 for units of watts, meters, and Kelvin.
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------------------ Energy = 5.7 *10^-8 * T^4
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- Ludwig Edward Boltzman ( 1844-1906) deduced the same law using thermodynamic principles and the constant became known as the Stephan-Boltzman Constant = 5.7*10^-8 watts/m^2 / Kelvin^4.
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- Wilhelm Wien (1864-1928) Munich, Germany determined that the intensity of radiation at its maximum was at a wavelength inversely proportional to temperature. In 1893 at the age of 29 Wien found that wavelength maximum = constant / Temperature.
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- Plotting measured data of intensity maximums versus temperature, the curve gave a negative slope of 0.0029 meter*Kelvin for the constant.
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---------------- Wavelength maximum = 0.0029 m*K / Temperature
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--------------- Frequency maximum = (5.8*10^10 / sec*K) * Temperature. This is the constant you get if you plot energy intensity versus frequency instead of wavelength. The energy peak is directly proportional to frequency, but Wien, and wavelength are easier to remember.
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- And ,0.0029 is easier than 58,000,000,000 cycles per second to remember. So, most often you see Wien’s constant expressed as 0.0029 wavelength * Kelvin.
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- In the above calculations we often used a “constant of proportionality“. A constant of proportionality is a measure of a rate of change of one variable versus another. A rate of change is a derivative of the function versus the independent variable. A constant of proportionality is different for each set of units you use in the measurement.
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- A fun example is that Johannes Kepler (1571 - 1630) discovered that the planets time of orbit squared is directly proportional to the radius of orbit cubed, period^2 = radius^3.
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- If you choose the units to be years and astronomical units, AU’s , which is the distance from Earth to the Sun, 93,000,000 miles, then the constant of proportionality is 1. This choice is not a good idea if you want to get a constant named after you. 1 has already been taken. For example:
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------------ The original 5th planet was Ceres. It has a period of 4.6 years and a radius of 2.77 AU. If you measure one you can always calculate the other.
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------------- p^2 = r^3
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------------- (4.6 years)^2 = (2.77 AU)^3
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------------- 21 years^2 = 21AU^3 * 1 / years^2/AU^3
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- If you choose the units to be meters and seconds and the mass of the Sun is much, much bigger than the mass of Ceres, then the different constant of proportionality becomes:
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---------------- 3*10^-19 sec^2/m^3, instead of 1 years^2/AU^3
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--------------- Ceres period of 4.6 years = 14.5*10^7 seconds
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-------------- Ceres orbit radius = 4.14 * 10^11 meters.
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--------------- p^2 = 3*10^-19 * r^3
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--------------- (14.5*10^7 seconds)^2 = 3*10^-19 * (4.14*10^7)^3
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-------------- 210*10^14 = 210*10^14
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- Sue enough Kepler’s formula works.
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- This second constant of proportionality comes from Newton’s formula:
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-------------- p^2 = 4*pi^2 * r^3 / G(M1+M2)
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--------------- The constant of proportionality for the force of gravity, G = 6.7*10^-11 m^3/kg*sec^2
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-------------- The mass of the Sun is 2*10^30 kg
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-------------- p^2 = 3*10^-19 sec^2 / m^3 * r^3
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- We did all that to learn:
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- The mass of the Sun to be 2,000,000,000,000,000,000,000,000,000,000 kilograms
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- And, The surface temperature is 5,800 degrees Kelvin.
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- I know it is more math than you can shake stick at.
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- November 18, 2019 2488 858
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--------------------- Monday, November 18, 2019 --------------------
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