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Let $\mu =M_p/M_{\star }$, $q=\sqrt{1-\beta }$, and restrict first to $0\le \beta <1$. The radiation-pressure coefficient reduces the grain's central gravitational parameter to $G{M}_{\star }{q}^2$, whereas the planet's is approximately $G{M}_{\star }$. Their mean motions obey $n/n_p=q(1+ϵ{)}^{-3/2}$. Setting this ratio to $j/(j+1)$ gives the radiation-pressure shift of a mean-motion resonance:It lies interior to the planet exactly when $q(j+1)<j$, or $j>q/(1-q)$. Equality puts the frequency commensurability at the planet's radius. For $\beta =0$ every such commensurability is exterior.

There is a frame qualification in the printed encounter formula. The circular inertial speed of the grain at $a_p(1+{ϵ}_1)$ is $v_1=q{v}_p(1+{ϵ}_1{)}^{-1/2}$. Subtracting the planet's inertial velocity at conjunction gives the local planet-relative inertial speedThis is the velocity used in the question's two-body gravitational assist calculation. In a frame actually rotating about the star at $n_p$, the grain's tangential velocity is instead $v_1-n_p{a}_p(1+{ϵ}_1)$, with magnitude $v_p[1+{ϵ}_1-q(1-{ϵ}_1/2)]$. In particular its nonradiating shear is $3v_p{ϵ}_1/2$, rather than $v_p{ϵ}_1/2$. Thus the requested expression uses a local translating, inertial encounter convention despite the source's rotating-frame wording. The following derivation consistently uses that convention, and its coefficients are an encounter estimate rather than an exact rotating-frame result.

Take impact parameter $b\simeq a_p{ϵ}_1$ and a small gravitational scattering angle $\theta$. Expanding the supplied hyperbolic scattering relation gives $\theta \simeq 2G{M}_p/(b{v}_{\infty }^2)$. Initially the relative tangential velocity is $-v_{\infty }$; after the deflection it has tangential component $-v_{\infty }\cos \theta$. The relative speed is unchanged in the local hyperbolic Kepler orbit, but the stellar-frame kinetic energy changes. Taking the dot product with the planet's velocity givesThe specific orbital energy of the grain is $-G{M}_{\star }(1-\beta )/(2a)$. Comparing it immediately before and after the encounter at the same position givesConsequentlyThe encounter increases the grain's semi-major axis within this geometry. These steps assume ${ϵ}_1\ll 1$, $\theta \ll 1$, and an orbital kick small enough for the subsequent linearization.

Choose the initial conjunction longitude as zero. The synodic period is $2\pi /(n_p-n)$, and the planet's longitude accumulated by the next conjunction isThese are unwrapped longitudes, counting complete revolutions. The second uses the new mean orbital period while neglecting the encounter's immediate phase offset and the difference between true and mean conjunction. A comparison reduced modulo $2\pi$ could not implement the stated one-turn sensitivity test.

For $\beta =0$, $v_{\infty }\simeq v_p{ϵ}_1/2$, so $\delta ϵ\simeq 32{\mu }^2{ϵ}_1^{-5}$ and ${\lambda }_c\simeq 4\pi /(3ϵ)$. Conjunction-longitude sensitivity to an encounter givesRequiring this to exceed $2\pi$ gives ${ϵ}_1^7<64{\mu }^2/3$. At a high-index first-order mean-motion resonance, ${ϵ}_j=(1+1/j{)}^{2/3}-1\simeq 2/(3j)$, henceThe interpretation of the resonance-overlap encounter estimate is loss of coherent encounter phases: a small kick changes the next synodic encounter time by more than one planetary revolution. Resonant protection by repeatedly meeting at controlled orbital phases can then fail, allowing irregular kicks and diffusion. This sensitivity criterion motivates an unstable region; it is not a proof that every orbit or resonant phase in it must escape.

For a quantitative sketch with radiation, put $h=1-q(1-ϵ/2)$ and $d=1-q(1+ϵ{)}^{-3/2}$. Differentiating the conjunction formula and substituting the kick givesEvaluate this at the positive ${ϵ}_j$ and take $\mathcal{S}=1$ as the critical-index curve. At small $\beta$, increased encounter speed weakens planetary scattering, moving the threshold to larger $j$. As $\beta$ rises further, the radiation-pressure shift of a mean-motion resonance brings each commensurability closer to the planet, strengthening the close encounters. The curve turns over and approaches the exterior-limit boundary $j=q/(1-q)$. Larger planetary mass produces stronger kicks and a lower critical index. These competing effects are shown below; the curves retain the factors in $\mathcal{S}$, so their $\beta =0$ intercepts differ slightly from the leading high-$j$ formula. The boundary for disappearing exterior commensurabilities is geometric; divergence of the encounter approximation near $ϵ=0$ should not be read as a precise physical scattering rate.

The incoming and outgoing velocity vectors have the same magnitude $v_{\infty }$, by conservation of energy. The chord between them has lengthFor fixed $r_{\min }$ put $z=v_{\infty }\sqrt{r_{\min }/\mu }$. Then $|\Delta \mathbf{v}|=\sqrt{\mu /r_{\min }}2z/(1+z^2)$, and $(z-1{)}^2\ge 0$ proves $2z/(1+z^2)\le 1$. Equality occurs at $v_{\infty }=\sqrt{\mu /r_{\min }}$.

A non-colliding comet must have $r_{\min }\ge r_0$, where $r_0=R_p$ in the point-comet approximation and $r_0=R_p+R$ for finite bodies. Thus the non-colliding gravitational velocity-kick bound isThe optimizing grazing orbit has $v_{\infty }=v_{\mathrm{esc}}/\sqrt{2}$, $e_h=2$ and $\theta =\pi /3$. If contact itself is excluded, this is a supremum approached from larger $r_{\min }$. The finite collision radius is essential: an ideal point planet with unrestricted $r_{\min }\to 0$ has no finite upper bound.

A constant Galilean boost by the planet's stellar orbital velocity leaves the vector difference unchanged. It changes the comet's incoming and outgoing heliocentric energies, which is the basis of a gravitational assist.

Let $\chi =v_{\mathrm{esc}}/v_k$, with $v_k=\sqrt{G{M}_{\star }/a_p}$. The Safronov number is $\Theta ={\chi }^2/2$. In the instantaneous planet frame let the asymptotic vectors be ${\mathbf{u}}_{\mathrm{in}}$ and ${\mathbf{u}}_{\mathrm{out}}$; their magnitudes agree. In the stellar frame the corresponding specific orbital energy change during a short encounter isThe quadratic relative-speed terms cancel. A comet with semi-major axis comparable to $a_p$ has stellar binding energy per unit mass of order $v_k^2/2$. Consequently $\chi$ measures whether a single strong encounter can change a substantial fraction of that binding.

Large escape-to-orbital-speed ratios favour ejection; small ratios favour collision or accretion during repeated encounters. For $\chi \gg 1$, a suitably oriented gravitational assist can eject the comet before it strikes the planet. For $\chi \ll 1$, most individual kicks are too weak; many close passages may be needed, providing repeated chances of collision.

The crossover is of order unity, not a sharp theorem about each orbit. The initial binding energy, encounter speed and geometry matter: an already nearly parabolic comet may be ejected by a low-$\chi$ planet, while an unfavourably directed kick can make a comet more tightly bound even when $\chi$ is large.

In a planetary gravitational assist, the incoming and outgoing relative speeds have the same magnitude. If $\mathbf{u}$ is the planet-frame velocity, the heliocentric specific orbital energy change is consequentlynot an arbitrary addition of $|\Delta \mathbf{u}|$ to the heliocentric speed. The initial binding energy per unit mass is $G{M}_{\star }/(Q+q)\simeq G{M}_{\star }/Q$.

For a weak deflection, $b\gg b_{90}=G{M}_p/v_{\infty }^2$, the hyperbolic scattering formula gives $|\Delta \mathbf{u}|\simeq 2G{M}_p/(b{v}_{\infty })$. Its direction is perpendicular to the incoming relative velocity. At the planet, in the near-parabolic prograde limit,On the favorable side of the planet the energy gain is approximately $2G{M}_p{v}_p\kappa /(b{v}_{\infty })$. Equating this to the binding energy gives the ejection impact parameterOnly one sign of impact parameter gains energy for each radial branch. Noncolliding ejections have $R_p<|b|<b_{\mathrm{ej}}$, giving width $(b_{\mathrm{ej}}-R_p{)}_{+}$, where $x_{+}=\max (x,0)$. For $b_{\mathrm{ej}}\gg R_p$, the planetary ejection of a comet rate is approximately ${\Gamma }_{\mathrm{ej}}={\Sigma }_c{v}_{\infty }{b}_{\mathrm{ej}}$. With ${\mu }_p=M_p/M_{\star }$ this givesThe scaling applies when $b_{90}\ll b_{\mathrm{ej}}\ll r_H$ so that the local two-body encounter and weak-deflection approximations are consistent. The favorable-side coefficient assumes a coplanar orbit; averaging over other encounter geometries changes it. Cumulative diffusion by many subthreshold encounters is not included in this one-encounter estimate.

In the planetocentric frame, the incoming relative velocity ${\mathbf{w}}_i$ is prograde and the encounter outside the planet bends it inward through $\theta$, without changing $w=|\mathbf{w}|$. In the inertial frame $\mathbf{v}={\mathbf{v}}_p+\mathbf{w}$, soSubstitution of the previous result yields the negative gravitational assist energy kick

A gravitational assist ejects a weakly bound comet when the increase in its specific orbital energy exceeds its binding energy. For near-parabolic coplanar encounters, the weak-deflection ejection impact parameter scales as $(M_p/M_{\star })Q$. Combining this with the phase-mixed comet surface density gives a one-encounter ejection time proportional to $Q^{1/2}$, within the local two-body encounter approximation.

A local weak gravitational assist estimate for a nonradiating circular grain gives $\delta ϵ\simeq 32{\mu }^2{ϵ}^{-5}$, where $\mu =M_p/M_{\star }$. Conjunction-longitude sensitivity to an encounter then gives $|\delta \Lambda |\simeq 128\pi {\mu }^2/(3{ϵ}^7)$. Requiring the change to exceed one full turn and using $ϵ\simeq 2/(3j)$ gives $j_c\simeq (2/729{)}^{1/7}{\mu }^{-2/7}$. This motivates loss of coherent resonant conjunctions, rather than proving that every initial phase escapes. The coefficient depends on the encounter convention and the approximations; near-planet Hill equations require a consistent rotating-frame shear instead of mixing local inertial and rotating velocities.

For a planet on a circular stellar orbit, a common convention is $\Theta =\frac{1}{2}(v_{\mathrm{esc}}/v_k{)}^2$. The escape-to-orbital-speed ratio measures the size of a strong gravitational assist relative to the stellar binding energy of a nearby comet. Large values tend to favour ejection over collision; small values favour repeated weak encounters with more collision opportunities. This is a statistical comparison, dependent on the initial orbit and encounter geometry, rather than a universal outcome threshold.