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For an initial mass function $dN\propto m^{-\alpha }dm$ on $0.1<m/M_{\odot }<100$, complete loss of material initially above $10M_{\odot }$ removes fractionThe impulsive disruption by stellar mass loss threshold is $F=1/2$, with unique finite root $\alpha \simeq 1.79101236$. Decreasing $\alpha$ shifts mass to larger stars and increases $F$, so the idealized system is globally unbound for $\alpha <{\alpha }_{\mathrm{crit}}$. The derivation requires rapid, well-mixed loss and absence of a confining external potential.

First specify the dynamical estimate: assume an initially isolated virialized system, stars of different masses well mixed with the same velocity distribution, and ejecta escaping rapidly compared with a stellar crossing time, without imparting kicks to surviving stars. If $f$ is the retained stellar mass fraction, the original virial theorem gives $2T+W=0$, where $T$ is kinetic energy and $W<0$ is Newtonian gravitational potential energy. Immediately after loss, positions and velocities are unchanged, soThus impulsive disruption by stellar mass loss makes the energy of the whole remaining system nonnegative when at least half the initial mass is removed.

For the power-law initial mass function, the fraction removed is a mass fraction, not a number fraction:For integer slopes, $F(0)\simeq 0.990$, $F(1)\simeq 0.901$, $F(2)=0.333$, and $F(3)\simeq 0.009$. Interpolating between $\alpha =1$ and $2$ puts the threshold around $1.7$--$1.8$. Solving $F=1/2$ more accurately givesThe rearranged equation has a spurious root at $\alpha =2$ introduced when the vanishing denominator was cleared; the logarithmic limit $F(2)=1/3$ excludes it. Increasing $\alpha$ shifts the mass distribution towards lower masses, so the top-heavy side, $\alpha <{\alpha }_{\mathrm{crit}}$, has nonnegative remnant energy in the impulsive model; equality is marginal. One can see strict monotonicity from the covariance obtained by differentiating $F$: $F^{\prime }=-\mathrm{Cov}(1_{x>10},\ln x)<0$ under the normalized mass-weighted distribution.

The initial mass function alone does not guarantee complete disruption. If losses occur slowly, the adiabatic mass-loss expansion law gives expansion rather than the instantaneous half-mass threshold, and positive total energy can still leave a bound core after selective escape. The stated value is therefore the conventional global impulsive estimate, with no external confining Newtonian gravitational potential.