|MadSci Network: Astronomy|
The melting point of the various elements can be looked up (try webelements.com), and will be given in temperature units. So to answer your question, we need to consider just what is meant by "temperature". Basically, it is a measure of the energy of thermal motion. When we measure air temperature, we are really measuring how fast, on average, the molecules of air are moving around. That's pretty easy to picture in a gas or liquid, but even in a solid, where the atoms are generally fixed in place, they can still vibrate somewhat. The greater that vibration, the hotter the solid.
Next, we need to consider just how we heat something up (or cool it down). Perhaps you remember learning about the three ways that heat energy can be transferred: radiation, convection, and conduction. Radiation is the transport of heat through space without the involvement of matter. Convection is the transport of heat via a fluid, which could be a gas or liquid. Conduction is the transport of heat through solids. If you add energy to an object with one of these methods, it warms up. If the object gives up energy, it cools down.
When you are inside the Sun, heat can be transported by both radiation and convection. Some convection can also occur in the thin atmosphere above the surface of the Sun (recognizing that there are various definitions for "surface" in this case). But once you are any appreciable distance from the Sun, practically all heat is transferred by radiation alone. And this is where things get tricky, because several different properties of the elements need to be considered, as well as the energy output of the Sun.
If you place a solid sample of an element at some distance from the Sun, the temperature it will reach depends on how efficiently it absorbs radiation and how efficiently it radiates it away. The equilibrium temperature is determined by a balance of this absorption and re-radiation. Mathematically, when we set the equilibrium condition of power absorbed equal to power radiated, we get
alpha * P * Across = epsilon * sigma * Teq^4 * Atotal
alpha = coefficient of absorption
P = power density (1350 watts per square meter at the Earth)
Across = cross sectional area of the object
epsilon = coefficient of emission
sigma = Stefan-Boltzmann constant (5.6704e-8 kg s^-3 K^-4)
Teq = equilibrium temperature in kelvins
Atotal = total surface area of the object
(This is a special case of the Stefan-Boltzmann Law, for an object that isn't a perfect blackbody.)
One thing we can see from this is that the shape of your element sample and possibly its orientation will affect the result, since the ratio of the cross sectional area to the total surface area depends on both. If we assume the sample is a sphere, which has a ratio of total area to cross sectional area of 4 regardless of orientation, and solve for Teq we get
Teq = 278 * (alpha / epsilon)^0.25
Or more generally, since the power density varies with distance from the Sun by the inverse square,
Teq = (278 * (alpha / epsilon)^0.25) / R^0.5
where R is the distance from the Sun in astronomical units.
While this equation is easy enough to solve, you will probably have difficulty finding values for alpha and epsilon for many elements. I suspect they aren't even well known in all cases. I use this formula to calculate the temperature of meteoroids in space. Except in the case of iron meteoroids, these are typically complex silicates, not pure elements. Nevertheless, it is interesting to compare some values. The temperature of a sphere of iron at Earth's distance from the Sun (R=1) will be 366 K (93°C). At Mercury (R=0.38) it will be 594 K (321°C). It will reach its melting point of 1811 K at 0.04 AU, or 3.7 million miles from the Sun. If we look instead at a sphere of a light colored silicate near the Earth, its temperature will be about 234 K (-39°C), a lot colder than the iron sample. Even at Mercury, the silicate will only be at 380 K (107°C). The reason for these large differences is that iron is a much better absorber than it is a radiator (alpha / epsilon = 3), while the silicate radiates more efficiently than it absorbs (alpha / epsilon = 0.5).
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