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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.

A typical globular cluster has undergone appreciable two-body relaxation, since its stellar relaxation time is $t_{\mathrm{rel}}\sim 0.1(N/\ln N)t_{\mathrm{cross}}$: for $N\sim 10^5$ and stellar crossing time $t_{\mathrm{cross}}\sim 10^6\mathrm{yr}$ this is about $10^9\mathrm{yr}$, shorter than its age, allowing a relatively relaxed, tidally limited structure. A galaxy with $N\sim 10^{11}$ and $t_{\mathrm{cross}}\sim 10^8\mathrm{yr}$ instead has $t_{\mathrm{rel}}\sim 4\times 10^{16}\mathrm{yr}$ and behaves as a collisionless stellar system, retaining differences from formation, galaxy mergers, and star formation. Relaxation erases more memory in clusters than in galaxies, although real globular clusters also differ in concentration and evolutionary state, so structural similarity is approximate.

The stellar relaxation time is the time on which accumulated discrete gravitational encounters change a typical star's velocity by an amount comparable to its original velocity. It concerns two-body relaxation, rather than the much faster orbital evolution in a smooth gravitational potential. A collisionless stellar system requires this time to greatly exceed the duration being studied.

In the straight-line impulse approximation, an encounter at relative speed $u$ has transverse acceleration $Gmb/(b^2+u^2{t}^2{)}^{3/2}$. ThusA small-angle gravitational encounter needs $\Delta v_{\perp }\ll u$. The transition to order-one deflections is therefore $b\sim Gm/u^2$, ignoring factors such as the equal-mass relative deflection. For the spherical estimate $u\sim v$,This is the lower cutoff of the weak-scattering estimate; closer encounters actually occur and require strong-scattering treatment.

The number of encounters in time $T$ with impact parameters in $[b,b+db]$ is $2\pi bdbnuT$. Independent random transverse directions make mean kicks cancel while their variances add. ConsequentlyEach logarithmic interval contributes equally, giving the Coulomb logarithm in stellar dynamics. Defining a deflection time by variance $v^2$ and setting $u=v$ gives $v^3/(8\pi G^2{m}^{2}n\log \Lambda )$. The paper instead uses an order-one normalization three quarters of this estimate:Both have the same physical scaling. The specified characteristic speed, the word “comparable”, the velocity-distribution average and strong-encounter cutoff do not fix that numerical coefficient uniquely. The crude equal-speed impulse calculation must not be claimed to determine $3/(32\pi )$ exactly.

For a self-gravitating, approximately virialized system, the virial theorem gives $v^2\sim GNm/R$. With $n=3N/(4\pi R^3)$, $\Lambda \sim N$ and the stellar crossing time $t_{\mathrm{dyn}}=R/v$, substitution into the paper's convention givesAn externally dominated gravitational potential or a different structural constant changes this substitution. For an N-body simulation lasting $T=K{t}_{\mathrm{dyn}}$, a tolerable fractional velocity-squared diffusion $\epsilon \ll 1$ requires $T/t_{\mathrm{rel}}\lesssim \epsilon$, henceThere is no unique smallest $N$ without $K$, the error tolerance and the force prescription. For scale, equality at $t_{\mathrm{rel}}/t_{\mathrm{dyn}}=100$ occurs near $N=7.1\times 10^3$; $N=10^5$ gives about $1.1\times 10^3$ crossing times. A calculation lasting 100 crossing times therefore needs substantially more than the first threshold to have negligible relaxation. Gravitational softening can raise the effective cutoff and reduce artificial scattering, but it also sets the spatial force resolution.

Small-angle gravitational encounters occur at rate $2\pi bdbnu$ and supply kicks $2Gm/(bu)$. Independent kick variances add to $d⟨\Delta v^2⟩/dt=8\pi G^2{m}^{2}n\ln \Lambda /u$. Setting the accumulated variance to $u^2$ and using virial scaling gives a stellar relaxation time of order $0.1N/\ln N$ stellar crossing times, with the coefficient dependent on profile and crossing convention.

The stellar crossing time is a characteristic time to traverse a system, $t_{\mathrm{dyn}}\sim R/v$. In a virialized self-gravitating system, $v^2\sim GM/R$, so $t_{\mathrm{dyn}}\sim \sqrt{R^3/(GM)}$. It is distinct from the much longer spherical stellar relaxation time. Radius versus diameter and characteristic-speed conventions change order-one factors.