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Now I am solving the problem about the transition from $V$ (= -13 in 400-700nm) to $m_{Bol}$.

V is the full moon's apparent magnitude in 400-700mm(i.e., visible light), and $m_{bol}$ is the bolometric apparent magnitude of the moon.

So, I should know about the Bolometric Correction of the moon(=$m_{bol}-V$), but there is only the Bolometric Correction about 'stars' not 'moon' in google.

I wonder whether I can obtain the moon's Bolometric Correction from the star's Bolometric Correction.

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Yes

To compute a bolometric correction you need to know the spectrum of the object, so that you can calculate the fraction of the total flux of the object in the wavelength range of interest. Given that the Moon in non-luminous and is visible only due to the light reflected from the Sun, I would suggest a good approximation for the Moon would be to take the Sun's spectrum (or just use the Sun's bolometric corrections).

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    $\begingroup$ Ah, I didn't think about it. Thank you for your good comment. $\endgroup$
    – BAO
    Sep 5, 2022 at 8:40
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    $\begingroup$ This is a good first-order approximation, assuming that the Moon is equally gray in all wavelengths of interest (where the Sun has significant emission) - AND - completely neglecting Moon's own thermal radiation (up to ~400K are to be expected at the surface). $\endgroup$
    – fraxinus
    Sep 5, 2022 at 15:22

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