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A brown dwarf is a deuterium-fusing substellar object, and, in the cases of the larger ones, a lithium-fusing one. They often emit electromagnetic radiation in frequencies that humans cannot see. However, the larger ones tend to emit visible light, if not much of it.

In this scenario, there's an Earth-sized body whose center is 2,000,000 kilometers from the center of a brown dwarf. Assume that the Earth-sized body is functionally identical to Earth in every way except for a lack of cloud cover.

How luminous does the brown dwarf need to be in order to illuminate the Earth-sized body with 0.5 lux/0.5 lumens per square meter of visible light at high noon? Assume that the Earth-sized body has an atmosphere identical to Earth's, except without cloud cover capable of reflecting said visible light.

Note that I'm not referring to radiant flux per unit area here here; radiant flux per unit area is the total of all electromagnetic striking a certain area, whereas I'm referring to luminous flux - that is, the amount of visible light striking a certain area. Certainly, there'll more energy striking the Earth-sized body then what's visible, but, in this question, I'm only referring to visible light.

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    $\begingroup$ Why does it have to be an Earth-sized planet? The rest of the question is ill-defined since the luminosity and temperature of a brown dwarf depend hugely on their age and mass. $\endgroup$
    – ProfRob
    Commented Jan 30, 2022 at 17:32
  • $\begingroup$ @ProfRob Not sure, actually, but I figured that it would just make it easier for the person answering the question. As for the age and mass, I plan on asking another question about that later, once I know how luminous it is. $\endgroup$
    – KEY_ABRADE
    Commented Jan 30, 2022 at 20:18
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    $\begingroup$ The luminosity and luminous flux will vary by many, many orders of magnitude depending on the mass and age. For most brown dwarfs (except those at very young ages) the answer is essentially zero. How close to zero it is will depend on the very poorly modelled and almost entirely immeasurable V-band flux from a cold brown dwarf. $\endgroup$
    – ProfRob
    Commented Jan 30, 2022 at 20:54

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