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If not, when can a white dwarf be more massive than a neutron star?

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The smallest, precisely measured mass for a neutron star is now $1.174 \pm 0.004 M_{\odot}$ - Martinez et al. (2015). The theoretical lower limit is more like $0.1M_{\odot}$, but there are no obvious formation channels to produce such an object. See https://physics.stackexchange.com/questions/143166/what-is-the-theoretical-lower-mass-limit-for-a-gravitationally-stable-neutron-st

The highest mass for a stable white dwarf (commonly called the Chandrasekhar mass) is theoretically about $1.39 M_{\odot}$ for a helium or carbon white dwarf (and a little bit lower for oxygen or neon white dwarfs), but can be increased somewhat by rotation. The observation of type Ia supernovae is strong circumstantial evidence that this limit is reached and then exceeded, probably by mass transfer onto a smaller white dwarf. The most massive, probably single, white dwarf known/measured is "WD 33" in the cluster NGC 2099 and has a mass of $1.28^{+0.05}_{-0.08}\ M_{\odot}$ (Cummings et al. 2016).

So, both observationally and theoretically, the maximum mass of a white dwarf is higher than the minimum mass for a neutron star.

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No, the two limits are not the same - there is some range of masses that both white dwarfs (WDs) and neutron stars (NSs) can have.

The Chandrasekhar mass limit suggests that WDs cannot be more massive than about $1.4\,M_\odot$. However, this is true for non-rotating WDs. Rapidly rotating WDs might be as massive as $2\,M_\odot$. Accretion in a binary system could both spin up a WD and increase its mass.

NSs that have not undergone accretion are regularly observed with masses around $1.3\,M_\odot$, while NSs who have 'recycled pulsars' have an observed mean mass of $\sim 1.5\,M_\odot$ according to the same paper.

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