**Introduction**:Area and volume do not scale equally. Volume increases more quickly than surface area. This has a profound effect of the evolution of planetary surfaces, and we will use this macro example to study luminosity.

**Theory**: Luminosity is the transfer of energy in or out of a system by electromagnetic radiation. On a human level the band of radiation that accomplishes this is primarily infrared, the heat you feel without contact from something warm:

Figure 1

Of the extremely wide range of wavelengths extant in the EM spectrum - 10^{4}m to 10^{-14}m - IR occupies a relatively narrow three-order-of-magnitude band, and it is primarily caused by temperatures in the range of 3 - 3000K, with human body temperature lying right in the (logarithmic) middle at 310K. At the bottom you have roughly the average temperature of the universe, bumping up against the Cosmic Microwave Background, and at the top you have the temperature of a typical incandescent light bulb (where legally permitted). Any object with a surface temperature in this range will radiate primarily in the infrared.

The mathematical definition of luminosity is:

Equation 1

where *A* is area, *T* is temperature in Kelvins, sigma is the Stefan-Boltzmann constant with value 5.67010E-8^{} Wm^{−2}K^{−4}, and epsilon is the *emissivity. *Emissivity is a dimensionless scale factor between 0 and 1, 0 being no transmission and 1 being 100% emission/absorption, and it corresponds to white (0) and black (1). We will examine *blackbody radiation* is a future lab.

The Zeroth Law of Thermodynamics states, colloquially, that heat flows from hot to cold, and that is true, but only when considering net heat flow. There is no prohibition for a body to absorb heat while it also emits heat. Therefore we should modify equation 1 thusly:

Equation 2

Note the fourth-power relation: if T_{out} > T_{in}, then T^{4}_{out} >>>> T^{4}_{in} and equation 2 reduces to equation 1.The temperatures in our experiment today will be fairly close, so equation 2 should be used.

There are many applications of luminosity, from stellar physics to TV remotes to weapons control, but this usage is interesting because it is unexpected. Inspect the three planets below:

Figure 2

Earth, Mars, and Mercury, their relative sizes intact. You can see that the Earth (water removed for clarity) is covered with lines (drawn in for emphasis), whereas smaller Mars has just the one partial line, roughly equatorial, and smallest Mercury has no lines, save for the ray emanating from an impact crater. These lines mark the borders of tectonic plates. Tectonic plates are like bits of cracked eggshell that can be forced to move around at the surface of a planet. What forces them to move? Escaping heat from deep inside the planet; as it cracks the surface, heat in the form of lava pushes the plates into and sometimes underneath each other. The heat escaping from the interior of the Earth is ultimately responsible for earthquakes, volcanoes, and a lot of mountain building. (The next time the ground shakes, you'll know what to blame!) Where does this escaped heat end up? In space; planets lose heat through their surfaces by radiation into space.

All three planets formed roughly the same time, about 4.5 billion years ago, and then had equally hot interiors. Big Earth still has lots of plate activity, smaller Mars used to have plate activity but it stopped billions of years ago, and smallest Mercury had practically no plate tectonics. Can there be a connection?

- To examine the effect of surface area on cooling and the consequences of that phenonema

- small sphere (6.35 cm)
- large sphere (12.7 cm)
- graduated cylinder
- ring stands
- Bunsen burners
- funnel
- alcohol thermometers
- timer

N.B. You may be wondering why we are not using the Pasco Data Studio with some temperature probes for this experiment. Experience has shown that the probes and/or software tends to 'glitch' during these long runs, and these glitches require the student to start from scratch. For this lab, thermometers are more reliable.

- Set up the spheres (full of water), tripods, and Bunsen burners as shown below:

Figure 3

- Insert the thermometers into the spheres.
- They are alcohol-based, so if they break there are no hazardous materials. But
*don't*test their fragility!

- They are alcohol-based, so if they break there are no hazardous materials. But
- Matches work better than strikers for lighting the burnerss. Make sure there is enough airflow in the burner so that the flame is blue.
- It will take about 15 minutes for the temperature to reach 90 degrees, your starting temperature.
- The two spheres may not reach the starting temperature at the same time. Back off the flame under the small sphere when it reaches 90 degrees so the other can catch up.
- Boiling water will bubble out of the hole around the thermometer, so be prepared!

- Once both spheres are at the
*same*starting temperature, turn off the gas on the bench and the data taking can commence. This is best done "live" in Excel.- Use three columns: one for time and one for each sphere's temperature.
- Every thirty seconds, record each temperature.
- Continue for 1 hour (120 readings for each sphere).
- T
_{in}is the ambient temperature of the room; be sure to note this as well.

- Be sure that your benches' gas is off!
- Using the gloves, return the spheres to the sink area and, using the canned air, empty them.
- Include the two temperature graphs in your report.

- Luminosity is heat flow per unit time which, of course, is power. Using your now formidable expertise at curve fitting find an appropriate function in time for the changing T
_{out}for each sphere. Your polynomial function needs to fit well only over the domain of times recorded. Remember the constants of the fitted function includes area, emissivity, and the Stefan-Boltzman constant.

Equation 3

- Integrate this function over time to find the total heat emitted for each sphere.
- There is an alternate way to check your results via integration but using earlier thermodynamic principles. Find
*E*this second way. - Compare the two total energy results with the amount of heat lost using the non-calculus Q = mc delT. Account for any discrepancies.
- Be sure to include uncertainty!

- Why do you think large mammals, like elephants, hippos, and rhinos, that live in the hot regions of the Earth have little or no fur, but small mammals, like monkeys and shrews, have fur?
- Given that planets can cool only through their surfaces, why does the Earth have active plate tectonics while Mars and Mercury have none? Use your data/results to justify your answer.