Substellar temperatures comparable to that of the Earth

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To find a substellar temperature comparable to Earth's near Rigel, one must consider its high surface temperature of approximately 12,000 K and its large radius. The term "substellar temperature" is often confused with "subsolar temperature," but they generally refer to similar concepts regarding temperature thresholds for habitability. The radiation intensity from Rigel decreases with the square of the distance, suggesting that a planet would need to be about 140 million miles away to match Earth's radiation levels. This distance calculation is crucial for understanding potential habitability in exoplanetary systems around massive stars like Rigel. The discussion emphasizes the need for precise mathematical models to accurately describe these temperature and distance relationships.
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How far from Rigel would we have to be to find a substellar temperature comparable to that of the Earth?

I know that the surface temperature of Rigel is about 12,000 K and the radius is R = 35*Radius of Sun.

Is the substellar temperature the same idea as subsolar temperature? This is the thing that confuses me the most. Also, what sort of mathematical description can we derive for this "substellar temperature"?
 
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The radiation level near a star falls off as the inverse square of the distance from the star. Therefore, since the sun has a surface temp. around 6,000 K, I presume that a planet around Rigel would need to be about 140 million miles away to receive the same radiation intensity as the earth.
 

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