**Watch the Brian Cox video**of a Herschel-type measurement in Death Valley using an umbrella, a thermometer, a watch, and a coffee can filled with water.**The SI data**from the measurement:- The time to raise the temperature of the coffee-can water by \(\Delta T = 1K\) (\(1^oC\)) is found by Cox to be \(t = 700 s\) which is a little less than 12 minutes.
- The coffee can diameter is \(D = 12.5 \thinspace cm\), its height is \(H = 16 \thinspace cm\).
- Then its cross-sectional area is \(A = 1.2 \times 10^{-2} m^2\) and its volume is \(V = 2 Liter\).

**The energy needed to heat some water**can be calculated from water's known "volumetric heat capacity", \(C_V\), which is the energy needed per unit increase in temperature per unit volume.- So the energy which went into the our coffee-can's water is: \(E_{in} = C_V \thinspace \Delta T \thinspace V = \left(\dfrac{4.2 kJ}{K L}\right) \times (1K) \times (2L) = 8.4 kJ\).
**The Solar Constant**\(C_S\) is the energy from the sun arriving at the earth's surface per unit area per unit time.- Our coffee-can value for the Solar Constant is: \(C_S = \dfrac{E_{in}}{A \thinspace t} = \dfrac{1 \thinspace kJ}{m^2 s} \equiv \dfrac{1 \thinspace kW}{m^2}\).

**The energy radiated from the sun**per unit time, \(E_{sun, out}\), is just the solar constant times the area of a complete spherical surface imagined at the earth's radius from the sun.- \(E_{sun, out} = C_S \times (4\pi r^2) = \dfrac{1 \thinspace kJ}{m^2 s} \times 4\pi (1.50 \times 10^{11} m)^2 = 2.8 \times 10^{26} J/s\).
**The number of photons emitted by the sun**per unit time.- We get the number of photons by dividing the amount of sun-emitted total energy by the energy per photon.
- The energy of a photon of wavelength \(\lambda\) is: \(E_\lambda = h \thinspace c / \lambda\) where \(h\) is Planck's Constant and \(c\) is the speed of light.
- Following the class in the video description, we assume \(\lambda = 700 \thinspace nm = 7.0 \times 10^{-7}m\) for the wavelength of some sort of "average" photon from the sun.
- The energy of a \(700 \thinspace nm\) photon is (see (b) above): \(E_{700 \thinspace nm} = \dfrac{(6.3 \times 10^{-34}J \thinspace s) \times (3.0 \times 10^8 m/s)}{7.0 \times 10^{-7} m} = 2.7 \times 10^{-19} J\).
- Then the rate of photon emission by the sun is:

\(N_{sun,out} \approx \dfrac{2.8 \times 10^{26} J/s}{2.7 \times 10^{-19} J} \approx 1 \times 10^{45}\) photons\(/s\).

**Activities**- Suppose you could actually count the emitted photons for 1 second. You might get something like this:
1,383,279,502,884,197,169,399,375,105,820,974,944,592,307,816 photons/s.That's an integer. Would you ever get a fraction of a photon in the number for precisely 1 second?
- How could you correct for the earth's atmosphere above the can and for the spectrum?
- A photon has momentum \(p = \dfrac{h}{\lambda}\). When a photon is absorbed in the eye, conservation of momentum says its momentum goes somewhere. Where? How much fractional change occurs in the velocity of the momentum-receiving object?
- A stream of photons from the sun is called a "solar wind." For what objects does it visibly alter their momentum?
- Is the emission of photons from the sun perfectly constant in time? If it varies, is the variation small or large? Gradual or sudden?
- Will the sun ever run out of energy and stop emitting photons?
- Practice until you are able to present the above calculation of the solar photon emission rate from the Death Valley observation. Be prepared to show that the numerical constants used, along with a few words on their measurement, can be found on the WWW.
- Examine the "page source" for this web page and see how scientists use MathJax-sponsored Latex mark-up to present beautiful, easily-edited, mathematical expressions on a web page. Use the same techniques to produce an html file of text that includes math with fractions. Display and print it from within the browser.

- Suppose you could actually count the emitted photons for 1 second. You might get something like this:

Last change: 1/23/2014

Peter Signell, Project Physnet, www.physnet.org

Peter Signell, Project Physnet, www.physnet.org