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I've been reading about brown dwarfs, and checking their spectral types, and I noticed that they do not have a luminosity class (V, VI, etc). So I am wondering, do brown dwarfs have a luminosity class, or is it intentionally absent, or is it defaultly V or VI?

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The SIMBAD page for the binary brown dwarf Epsilon Indi B lists the components as T1V and T6V, indicating they use the normal dwarf luminosity class V.

There are also T subdwarfs, e.g. Burningham et al. (2014) give the spectral type of ULAS J131610.28+075553.0 as sdT6.5. This follows the usual convention I've seen where subdwarfs are written with a prefixed "sd" rather than using the Roman numeral luminosity class VI.

Schneider et al. (2020) also indicate that WISEA J041451.67-585456.7 and WISEA J181006.18-101000.5 are extreme subdwarfs, of spectral classes esdT0±1. Again this is indicated by a prefix ("esd") rather than a Roman numeral luminosity class. They also mention 2MASS 05325346+8246465 as being of spectral type esdL7.

Zhang et al. (2018) list a number of L subdwarfs (sdL), extreme subdwarfs (esdL) and ultra-subdwarfs (usdL).

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