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Suppose we look (above the Earth's atmosphere) at the wavelength ($=\frac c f)$ spectrum emitted by the Sun:

enter image description here

This shows that the Sun is approximately a black body with a temperature of about $5525(K).$ Now If we place a black body at a distance $l$ from the Sun will the radiation coming from the Sun (after a while, depending on $l$) causes this black body to have a temperature of $5525(K) $ too (by means of the blackbody radiation corresponding to the Sun), or will this only happen when the black body is completely surrounded completely by a material at $5525(K) $ (implying that the black body we place somewhere around the Sun is in a state of dynamical equilibrium)? Somewhat like a thermometer put in interstellar space will show a temperature of $2.7(K)$ because it's surrounded at all sides by the CMBR.

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    $\begingroup$ Have you thought that the earth is in a fist approximation a black bod an $l$ distance from the sun? $\endgroup$ – anna v Jan 15 at 16:57
  • $\begingroup$ Yes, that's why I thought that a dynamic equilibrium would develop: at daytime, the Earth absorbs energy, while at nighttime, the Earth radiates energy away. I can remember that I once asked in this question physics.stackexchange.com/questions/245691/… if the rotation of the Earth was a necessary condition for life to develop. Thanks for the comment. $\endgroup$ – descheleschilder Jan 15 at 17:38
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Assuming the black body doesn't produce heat and we don't focus radiation towards it, it needs to be completely surrounded by a material at 5525K and reach thermal equilibrium to have a temperature of 5525K. If it is instead somewhere around the sun it will be in a dynamic equilibrium, having temperature lower than 5525K. If it has a non-infinite thermal coductivity its "dark" side will be even colder. The reason for this dynamic equilibrium is that idealized black bodies, in addition to being perfect absorbers are also perfect emmitters, this might raise some eyebrows at first, but it actually makes sense when you think about it keeping in mind the second law of thermodynamics. If it was absorbing without emitting, it would just gather energy in one place, reducing the entropy.

Of course I'm talking about reasonable time scales. You can go overboard and claim it will eventually reach the Sun's temperature some time during the heat death of the universe, but that is more about the sun dying and cooling down than it is about the Sun heating up our black body.

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