Expert: Philip Stahl - 2/22/2010

Question
I am the guy that asked the question about measuring the temperatures on planets in our solar system and those out of our solar system. I am 65 and I am not enrolled in any class. I was at a party this weekend and we got into a discussion about how the sun effects our planets. None of us knew how to measure the temperature on other planets.

Answer
Hello,

Okay, that's fine. The general practical method is using what's called a thermocouple- which now, of course, is done from the convenient environs of close passing spacecraft as opposed to using apparatus from Earth. (You can google 'thermocouple' to learn all about its working). Let's see how to obtain it from first principles.

In terms of obtaining the "effective temperature" at the planet's putative "surface" (for the gas giants this will be ill-defined) one makes use of some basic physics. First, we know for each planet there will be a certain fraction of solar radiation intercepted at that distance. At Earth, it translates to what it called the solar constant k_s or 1360 W/m^2. Obviously, it will differ for other planets more or less distant, being much lower for outer planets, much nigher for inner.

A given solar constant can be found using:

k_s  =  L / (4 pi R^2)

where R is the distance from the sun, and L = 4 x 10^26 W is the solar luminosity

Now, the total radiant energy absorbed by the planet would be:

k_s/ R^2  times the *cross-sectional area* of the planet, which is just the
area of a circle of radius r equal to the planet's:  Or: pi (r^2)

A given planet absorbs some of this energy and reflects the rest. Reflecting
a fraction A (albedo) it must *absorb* a fraction (1 -A).

The total incident energy absorbed each second is then:

E (per sec) =  [(1- A) pi r^2 x k_s]/ R^2


4 pi r^2 (sigma) T^4

where T is the temperature sought (in Kelvin degrees) and 'sigma' is the Stefan-Boltzmann
constant. The above expression is just that for the energy absorbed by the
planet (with given surface area) treated as a black body.

The effective temperature (from 1st principles) would then be found from:


T_eff = {[(1 - A) x k_s]/ 4 (sigma) R^2]}^ 1/4


Note: Tables with 'A' (albedo) are available for the planets (except Pluto, I believe), and R can be found using assorted software programs or tables, so k_s can be computed. Thus, T_eff can be computed for most of the planets.


Hope this helps!