Taking the temperature of the Sun
Student Brief
The surface of the Sun is hot. This experiment aims to find out how hot.
The heat from the sun radiates out into the solar system. By the time it gets to us it has spread out into a sphere the size of the Earth’s orbit.
The idea of this experiment is that you can capture this spread out heat in a cup of water.
If you know what part of the Sun’s spread out heat you have caught in your cup you can calculate the total heat output of the Sun.
Once you have that piece of information it is possible to work out how hot the surface of the Sun must be.
There’s quite a lot of maths needed to do the calculations but you are going to use a computer spreadsheet to crunch the numbers for you.
You need to concentrate on getting a good set of measurements to plug into the spreadsheet.
What will mess up your experiment?
Clouds reflect the Sun’s energy and will make it appear cooler – if it’s overcast forget it!
The atmosphere will absorb the energy. People closer to the equator will see the Sun through a thinner layer of heat absorbing atmosphere at mid day and the Sun will appear hotter. (A measure of a stick’s shadow compensates for this in the spreadsheet.)
The water could evaporate
Light could reflect off the water
Light could reflect off the cup
Heat could radiate/convect/conduct out of the cup
Heat could radiate/convect/conduct into the cup (from sources other than the Sun – eg you hand)
The top of the cup will be at an angle to the Sun unless you are in the tropics. (A measure of a stick’s shadow compensates for this in the spreadsheet.)
So, there are lots of factors to mess up the experiment but lots of opportunity to try to design them out and probably lots to evaluate at the end too.
Taking the Temperature of the Sun  Method
Equipment
Plastic cup, insulating material, sticky tape, cling film, thermometer, measuring cylinder, stopwatch, 2 x metre rules, black ink, graph paper, results sheet, access to a computer to use the spreadsheet

Insulate the plastic cup on the outside (eg bubble wrap).

Measure the outdoor temperature.

Use water which is a couple of degrees below the outdoor temperature.

Accurately measure the amount of water needed to fill the cup ¾ full.

Add three drops of black ink to the water (so that it absorbs the sunlight).

Put on a cling film top (to stop evaporation).

Push a thermometer through the cling film.

Place the cup outside in a sunny position.

Start the timer and take the temperature every minute.

Gently stir with the thermometer before taking each temperature.

Continue until it is several degrees above the outside temperature.
You also need to find out the angle of the Sun. Do this by holding the end of a metre stick between your thumb and index finger and dangle it just above the ground. Get someone else to use another metre stick to measure the shadow length.
Taking the Temperature of the Sun  Results sheet
Volume of water

cm^{3}

Length of shadow

mm

Weather conditions

Clear/Average/Hazy


Time (s)

Temperature (^{o}C)

0


60


120


180


240


300


360


420


480


540


600


660


720


780


840


900


960


1020


1080


1140


1200


Now plot a graph of temperature against time (in seconds)
Look at the part of the graph that is a straight line both side of the outside temperature and draw a line of best fit through it. Take the temperature rise and time between two points on the graph and plug them into the spreadsheet.
Unless you are on a mountain top in the tropics with the Sun directly overhead, the sunlight will be partly absorbed by the atmosphere and striking your cup top at an angle. Straight overhead is called the Zenith. You need to find out the angle the Sun is off the Zenith. Plug your shadow stick measurements into the Zenith Angle mini sheet. Mark the Zenith angle it calculates on the transmission graph below. Draw a line up to the local weather condition line. Now draw a horizontal line across to find out the proportion of sunlight that should be getting into your cup. If you are using this as a Word doc you can resize the box on the right to help take readings from the graph.
Plug the figure into the main spread sheet or start crunching numbers yourselves.
Some quantities in the calculation are shown in this diagram
How the numbers are crunched
If you find maths scary don’t read this! It’s really only for those that enjoy difficult maths.
First calculate the energy transfer rate by using the specific heat capacity (how much energy 1g of something needs to heat up by 1oC) of water. Luckily at our level of accuracy, 1g of water = 1cm^{3}.
(^{o}C/s) x 4.2(J/cm^{3}/^{o}C) x vol. water (cm^{3}) = Watts
You now need to find the Effective Area of the cup.
First find the area of the top of the cup (A cup) using πr^{2} (in metres).
To compensate for the Zenith angle (θ) first find the angle using the length of the stick (L) and the shadow (S).
tan(θ) =S
L
Now compensate by multiplying the area of the cup by the cosine of the Zenith angle A = A cup . cos(θ)
Now you need to find out the amount of energy striking each square metre of the Earth’s surface .
Divide the Watts absorbed by the cup by its effective area. You now have a measurement in Watts/m^{2}
You now need to compensate for the atmospheric absorption at your zenith angle.
Read this off the supplied graph as described in the main method. Multiply your Watts/m^{2} figure by this compensation factor
You have now calculated what is called the Solar Constant (E). This is a measure of the number of watts striking one square metre of surface at a distance of one astronomical unit (1.5 x10^{11}m) from the Sun.
You now need too now find out the area of that one Astronomical Unit sphere 
4πr^{2} where r =1.5 x10^{11}m
= 2.83e+23
Now multiply the area of the Earth orbit sized sphere by your Solar constant figure to find out the energy output of the Sun – its Luminosity.
2.83e+23 E = Luminosity
It is at this point you will find out how far out you are as the figure is really 3.825 x 10^{23}W
Now that you know how much total energy the Sun puts out, you can also work out how much energy is given out by each square meter of the solar photosphere (ESun). Since the radius of the Sun is about 7.0 x10^{8}m, we divide the total energy output by its surface area:
ESun= Luminosity/4π(7.0 x10^{8}m)^{2}
Next you need to convert the output to the temperature of the surface.
The Sun radiates energy according to the StefanBoltzmann law: the power radiated per unit area (ESun) is proportional to the fourth power of its absolute temperature T measured in degrees Kelvin:
ESun= σ T4
where σ (called the StefanBoltzmann constant) is known from experiments to be 5.69E10^{8 }watts/(m^{2}K^{4})
So
T = (ESun/ σ)^{1/4}
You finally have the temperature of the Sun in ^{o}K
0^{o} Kelvin = 273^{o} Celsius (i.e. Absolute Zero temperature)
273^{o} Kelvin = 0^{o} Celsius
So to convert Kelvin to Celsius subtract 273
Check your answer with the spreadsheet
Diagrams and graphs courtesy of Sommers Bausch Observatory, Department of Astrophysical & Planetary Sciences, University of Colorado
Austrian Scientist first to
calculate the temperature of the Sun
Joseph Stephan 18351893 Ludwig Boltzman 18441906
The first person to take the temperature of the Sun was Joseph Stephan in 1884. He had done lots of experiments in his laboratory that showed that hot objects gave off heat in a particular relationship to their temperature. His student Ludwig Boltzman worked out the mathematics needed to calculate the temperature. Stephan was then able to measure the approximate temperature of the photosphere of the Sun as 5430°C, the first sensible estimate.
