= Solution
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, so
$$
T'=fT,\qquad W'=f^2W,\qquad
E'=fT+f^2W=|W|f\left(\frac12-f\right).
$$
Thus <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:
$$
F(\alpha)=\frac{\int_{10}^{100}x^{1-\alpha}dx}{\int_{0.1}^{100}x^{1-\alpha}dx}
=\frac{100^{2-\alpha}-10^{2-\alpha}}{100^{2-\alpha}-0.1^{2-\alpha}}
\quad(\alpha\ne2),\qquad F(2)=\frac{\ln10}{\ln1000}=\frac13.
$$
For integer slopes, $F(0)\simeq0.990$, $F(1)\simeq0.901$, $F(2)=0.333$, and $F(3)\simeq0.009$. Interpolating between $\alpha=1$ and $2$ puts the threshold around $1.7$--$1.8$. Solving $F=1/2$ more accurately gives
$$
2\,10^{2-\alpha}=100^{2-\alpha}+0.1^{2-\alpha},\qquad
\boxed{\alpha_{\rm crit}\simeq1.7910.}
$$
The 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 \b[the top-heavy side, $\alpha<\alpha_{\rm crit}$, has nonnegative remnant energy in the impulsive model]; equality is marginal. One can see strict monotonicity from the <covariance> obtained by differentiating $F$: $F'=-\operatorname{Cov}(\mathbf1_{x>10},\ln x)<0$ 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>.
Back to article page
