## måndag 19 april 2010

### Black Body Temperatures of the Planets

Let us test Stefan-Boltzmann's Black Body Radiation Law  Q = c T^4 = cTTTT on the planets in the Solar system assuming similar absorption and emission. The radius of the Sun is about 2.3 light seconds. The radiation from the Sun is diluted at a distance of R (light seconds) with the factor (r)^2 where r = R/2.3. Taking into account that the area of a disc is 1/4 of a that of a sphere with the same radius, the radiation from the Sun reaching a planet at distance R is diluted  by the factor 4 r^2, which determines the effective temperature T_P of the planet  by the relation

4r^2 = (T_S/T_P)^4

where T_S = 5778 K is the effective temperature of the Sun. We get
1. Earth: distance D = 500 light seconds = 1 AU, r =220, T_P= 5778/21 = 275 K (288 K)
2. Mars: D =1.5, r = 330,  T_P = 275 / 1.22 = 229 K (232 K)
3. Jupiter: D = 5.2, T_P = 275/2.28 = 120 K ( 134 K)
4. Saturn: D = 9.5, T_P = 275/3.08 = 90 K (103 K)
5. Uranus: D = 19.2, T_P = 63 K ( 73 K)
6. Neptune: D = 30.1, T_P = 48 K ( 63 K)
7. Venus: D = 0.72, T_P = 312 K (465 K)
8. Mercury: D = 0.39, T_P = 443 K ( 400 K)
where in parenthesis we give observed mean surface temperatures. We see a close fit for Mars, which has a thin atmosphere of 95% CO2. The atmospheres of the Earth and Venus increase surface temperatures substantially above the black body temperature. The very cold Saturn, Uranus and Neptune give off more heat than receieved, from some internal source of heat. Jupiter is estimated to give off about 2 times what is received from the Sun. The day and night temperatures of Mercury vary between -100 K and 700 K and a mean temperature is difficult to define.

For the Earth, the 275 K fits well with the temperature of the stratopause, while the surface temperature of an Earth without an atmosphere is commonly estimated to the lower value -18 C (instead of 2 C) as a result of certain assumptions on albedo and emissivity modifiying the black body aspects. So the -18 C is speculation and not observation.

The most interesting feature is the close fit for Mars, with seemingly small heating from
its CO2 atmosphere.

#### 4 kommentarer:

1. 喜樂的心是健康良藥，憂傷的靈使骨枯乾。........................................

2. Solar radiation incident at top of atmosphere = 1367W/m^2 (approximately).