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My textbook on Astrophysics says the following about accretion (translation):

Assume we have a particle with mass $m$ that falls on a neutron star; $R\approx 10$ km, $M\approx 1.4M_{\text{sun}}$, so $v_{ff}=2GM/R\approx 0.64c$, and $E_{kin}=1/2mv^2\approx0.2mc^2$. The particle falls with a considerable fraction of the speed of light on the surface of the neutron star, and has a kinetic energy that is a considerable fraction of its rest-mass energy ($mc^2$). The source of this kinetic energy is the gravitational potential energy of the particle. When the particle gets slowed down, this energy is converted to heat and radiation. If the gravitational potential energy is emitted like this, then de mass os the neutron star only increased by $0.8m$.

I don't understand why the mass isn't just $1 m$. Because at the beginning (at $\infty$) the particle has no gravitational potential energy, so its only energy is its rest-mass energy $mc^2$. But then it gains energy, so it's total energy would be $mc^2 + 0.2mc^2$. When it is slowed down at the surface, it loses this acquired energy, so its total energy should again be $mc^2$, and therefore the neutron star did gain $mc^2$ instead of $0.8mc^2$.

So what mistake am I making?

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As you wrote yourself "this energy is converted to heat and radiation".

Especially the radiation will not be captured by the neutron star, I will simply radiate away as the particle falls towards the neutron star.

As for the heat, it will also radiate way, but over longer time-scales.

So if you subtract the radiation and heat, I suspect you are left with an energy of $0.8mc^2$.

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