Measuring the temperature of light

F. Herrmann; Institut für Theoretische Festkörperphysik, Universität, 76128 Karlsruhe

Abstract
Light emitted by a black body has a temperature, the same temperature as the body it is coming from. In this article we will present a method of showing students that sun light on the earth has a very high temperature and how this temperature can be measured by means of a normal thermometer, before they learn anything about statistics and the Planck distribution.

Zusammenfassung
Licht, das von einem schwarzen Strahler emittiert wird, hat eine bestimmte Temperatur: dieselbe Temperatur wie der Körper, von dem es herkommt. In dem Artikel wird ein Weg vorgestellt, wie man Schülern oder Studenten zeigen kann, daß Sonnenlicht auf der Erde eine sehr hohe Temperatur hat und wie diese Temperatur mit einem gewöhnlichen Thermometer gemessen werden kann, noch bevor den Lernenden etwas über Statistik und Planck-Verteilung bekannt ist.

Light emitted by a black body has a temperature, the same temperature as the body it is coming from. Thus, sunlight has a temperature of 6000 K; light coming from an incandescent lamp has about 2800 K; the infrared light emitted by a hot iron may have 500 K and the light emitted by the planet earth has roughly 300 K. After being emitted, the light can change its temperature, e. g. in the process of an isentropic expansion. An example is the cosmic background radiation which at the time of its decoupling from matter 106 years after the big bang had a temperature of 104 K and after about 1010 years of isentropic expansion has now only 2.7 K.
The temperature of a particular light can be deduced from its spectrum. In all the above-mentioned cases the light is Planck distributed, and since the temperature is a parameter of the Planck distribution, the temperature can be determined when the distribution is known.
However, explaining a temperature of a system by means of a distribution is rather complicated and can be discussed only at an advanced level. Moreover, deducing the temperature from a spectrum instead of measuring it directly by means of a thermometer may cause the student to believe that one is dealing with a somewhat generalized temperature concept and not with the same temperature which was introduced at the beginning of the thermodynamics course and measured with a normal mercury thermometer or a thermocouple.
In this article we will present a method of showing students that sun light has a very high temperature and how this temperature can (at least in principle) be measured by means of a thermometer, before they learn anything about statistics and the Planck distribution.
The light we receive from the sun is emitted by what we perceive as the sun's surface. Since this surface has a temperature of 6000 K, the light coming from there should be expected to have the same temperature, i. e. 6000 K. However, this conclusion doesn't appear to be very plausible: Shouldn't every body that is exposed to sunlight immediately burn? And why does a thermometer that is exposed to sunlight not indicate 6000 K? To answer these questions we have to take a closer look at the process of measuring a temperature. To measure the temperature of a system A by means of a thermometer B, A and B have to be set into thermal contact. Accordingly, it should indeed be sufficient to hold the thermometer into the sunlight in order to measure the light's temperature. If this is done, the temperature value displayed by the thermometer is higher than when the thermometer is placed into the shade. However, it is far below the expected 6000 K.
When trying to measure the temperature in this manner, we forgot, that the thermometer is in thermal contact not only with the sunlight but simultaneously with the air. Since the air is not in thermal equilibrium with the sunlight, air and light will have different temperatures. What then will the thermometer display?
This question is similar to the following: Which potential difference will a voltmeter display that is simultaneously connected with two batteries of different open circuit voltages, Fig. 1?


Fig. 1: The voltmeter displays a potential difference that is situated between the open circuit voltages of both batteries.

Here, it is clear that the voltmeter displays a potential difference that is situated between both open circuit voltages but whose exact value depends on the internal resistances of the batteries.
In the same way we can conclude that the thermometer is making a compromise between the temperatures of the various systems to which it is connected, and it is impossible to deduce either of the systems' temperatures it is in contact with.
However, in our case it is easy to remedy: All we have to do is to eliminate the air around the thermometer by placing it, for instance, into a transparent, evacuated container. The result is that the temperature displayed by the thermometer is higher than before but still remains far below 6000 K. What mistake have we been making this time? Let us ask once more the question about which system the thermometer is in thermal contact with. The answer is, of course, with the light of the sun. There remains, however, another competitor, that we have overseen up to now, i. e. the thermal radiation coming from all directions except from the small solid angle occupied by the sun. From 99.9995 % of all directions comes invisible infrared light of ambient temperature, i. e. approximately 300 K. Only from the remaining 0.0005 % of directions does hot sunlight come. Thus, the thermometer has to again make a compromise and this compromise comes out very much in favor of the ambient 300 K radiation.
Again we look for a solution. We have to make sure that sunlight doesn't only come from 0.0005 % of the directions. Rather it has to come from all directions, from the whole solid angle. To realize that is, at least in principle, not difficult. One has to place mirrors and/or prisms around the thermometer in such a way that sunlight falls on the thermometer's sensor from all directions. A continuous mirror that does the job is called a light concentrator (Welford and Winston 1989).
The exact form of a perfect concentrator will not be discussed here. However, using a parabolic mirror or a lens is a good attempt at achieving the same effect as a concentrator. Indeed, in this way one reaches easily a temperature of about 1000 K and, with some more effort, several thousands of Kelvins. A perfect mirror arrangement would indeed lead to 6000 K. (Of course, our normal thermometers can't sustain such a high temperature).
Thus, the sunlight's high temperature can be deduced from the familiar experience made when a burning glass or a parabolic mirror increases the temperature of a body to a high value. Lenses, parabolic mirrors and concentrators, can be used to set a body in thermal equilibrium with light coming from directions within a small solid angle.
Our discussion shows that it doesn't make much sense to say (as it is often done) that the temperature "in the sun" is, for example, 45 degrees centigrade. A thermometer exposed to sunlight displays neither the temperature of the sunlight, nor that of the air or that of the ambient infrared radiation. Rather, the indicated temperature value is no more than the result of a hardly interpretable compromise between the temperatures of these systems.


References
Welford W T andWinston R 1989 High Collection Nonimaging Optics, (Academic Press), p. 55