Hi All, Playing Devil's Advocate, I decided to try coming up with a scenario that attempts to maximize the thermal equilibrium temperature of a chondritic meteoroid just prior to encountering the earth's atmosphere. The typical formula for computing the thermal equilibrium temperature for an object without an atmosphere is:
Te = [S0 * (1-A) / (4*epsilon*sigma)] ^ (1/4) where the body is assumed to be spherical (the source of the 4 in the denominator), S0 is the solar constant (mean value 1361 W/m^2), A is the bolometric Bond albedo, epsilon is the meteoroid's emissivity, and sigma is the Stefan-Boltzmann constant (5.670 x 10^-8 W/m^2-K^-4). A, in turn, can be estimated from the following equation: A ~= q * pv where q is the phase integral and pv is the visible albedo. Using Bowell's H, G magnitude system, we can compute q from: q = 0.290 + .684*G The commonly used value for the slope parameter, G, is 0.15, in which case: q = 0.393 A = 0.393 * pv For very dark asteroids (e.g. Trojan asteroids, Hildas, Cybeles), the albedo can be 5% or lower. However, most NEOs have semi-major axes less than 3 a.u. and albedos averaging closer to 20%. The final missing value is the emissivity. For regolith, a range of 0.9-0.95 is often mentioned. However, emissivity and albedo work hand-in-hand (epsilon + pv ~= 1). So if we're going to choose an emissivity of 0.9, we should set the albedo, pv, to 10%. So what is a typical equilibrium temperature for a spherical NEO with 10% albedo, 0.9 emissivity, 1 a.u. from the sun? A = .393*10% = .0393 Te = [1361 * (1-.0393) / (4*0.9*5.67 x 10^-8)]^0.25 = 282.9 K or about 49.6 F So, cool, but certainly not freezing. How can we get a warmer answer? One way is to pick the time of year when the earth is closest to the sun (early January) and the solar constant is higher: about 1414 W/m^2. This raises the temperature in the above example to 285.6 K, or 54.4 F. Still not warm, but warmer. Lowering the emissivity will help, too. Let the albedo increase to 20%, and set the emissivity to 0.8. With the perihelion solar constant, the equilibrium temperature is now up to 291.1 K (64.3 F). Lowering the emissivity further is probably not realistic for most earth-crossing asteroids, so we're at the limit of what we can achieve via S0 and emissivity. However, there *is* a way to get a big increase in the equilibrium temperature which I'll cover in the next installment. --Rob
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