## 31 July 2008

### Earth temperature 1

The earth's temperature is something I'll probably come back to a few times as it is a lot more involved than you might think. So this is the first part -- the temperature as observed from space.

'Observed from space' shows our first complexity. Satellites can't drop a thermometer into the earth's atmosphere, oceans, etc., to find out what the temperature is. So what can they measure from up there? Fundamentally, they can measure voltages, resistances, currents, counts of an oscillator -- electrical/electronic things like that. Not very helpful at the start. But we can arrange it so that the things we can measure have something to do with what we want to know about.

This is actually how some thermometers work. Think of a traditional old mercury thermometer. It doesn't measure 'temperature', whatever that may be; it measures the length of a thread of mercury. Alcohol thermometers use alcohol instead, but in the same idea. 'Temperature' is the property, then, which makes materials expand (when hotter) or contract (when colder). It was discovered first as a practical matter that materials do expand with temperature in a sense in agreement with our own physical ideas of hot and cold (ex. Mr. Fahrenheit and M. Celsius). So temperature could be equated to expansion of materials. In the 1800s, a firm theoretical basis for which it should be like that (and sometimes not like that at all) was laid down.

For the satellites, a similar process of trying to match up what could be measured to what was desired was involved. The little 'aim it in your ear' thermometers are a little like the satellite method. What they do (satellites more thoroughly, home ear thermometers only in a small color zone) is arrange a detector so that it gets hotter as more radiation falls on it, then measure the resistance/voltage/... of this hotter detector wall.

The satellite detector I've described relies on the Stefan-Boltzmann law to decide temperature -- it measures the energy (which causes the detector to heat up) coming from the earth, and then with the law (Energy = s * T^4, s = the Stefan-Boltzmann constant, wonder why), converts that measured energy to a temperature. That temperature is the 'Black Body' temperature of the earth. If the earth were an ideal black body, the observed amount of energy is what would be seen if the earth were radiating at the given temperature.

We can also make detectors which measure the amount of energy that's within a certain small wavelength interval (blue, for instance). This is how the ear thermometer works, except it uses infrared. Given that observation, and Planck's law (more involved than Stefan-Boltzmann, look it up), we can compute the temperature your inner ear would have to be to be radiating that much energy -- if your inner ear were a black body. It's a fair approximation to one.

In either case, we have a 'Brightness Temperature' -- the temperature the thing you're looking at would have to be to give the observed brightness (energy). In the case of the earth, it is about 255 K, -18 C, 0 F. For Venus it's about 232 K, a good deal colder than the earth. Seriously, check http://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html
to verify, or find some others.

What happened? Venus is supposed to be hot! The earth is seldom as cold as -18 C or 0 F anywhere, much less for a planetary average. Well, Venus is hot (over 400 C at the surface) and the earth's surface is rather warmer (about 33 C or 60 F) than the brightness temperature (black-body equivalent temperature -- same thing).

The thing is, we have to pay attention to what the satellites observe -- radiation. If that radiation comes from the surface, we see a surface temperature. But in general, the radiation comes from somewhere up in the atmosphere. For Venus, it is a very long way from the surface, so very much colder. For the earth it is typically several km (or miles) up from the surface.

If you think this is indirect and complicated, wait 'til we talk about trying to observe the temperature of layers within the atmosphere from space!