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Parameterize the trajectory by the decreasing orbital eccentricity. The Poynting–Robertson drag invariant givesFor $Q_0\gg q_0$, $C\simeq 2q_0$. Initially $e\simeq 1$, and the orbit moves almost vertically down a plot of $Q/q_0$ against $q/q_0$: the apocentre distance shrinks rapidly while the pericentre distance changes little. Integrating the high-eccentricity slope givesOnce the orbit has moderate orbital eccentricity, both distances change appreciably. For example $e=1/2$ gives $q\simeq 0.766q_0$ and $Q\simeq 2.30q_0$ in the $Q_0/q_0\to \infty$ limit. Eventually $e\ll 1$, $q\simeq Q\simeq C{e}^{4/5}$, and the trajectory approaches the diagonal $Q=q$ before reaching the origin. Thus the two approximate phases are apocentre contraction at nearly fixed pericentre, followed by nearly circular inward migration. They are a smooth crossover, not two separate exact solutions.

The planetary expansion is a decomposition into secular eigenmodes of conservative linear Laplace-Lagrange secular theory. It assumes small orbital eccentricities, coplanarity, approximately constant semi-major axis values, and averaging over fast orbital phases away from important mean-motion resonances and close encounters. Dissipation, migration, and nonlinear secular effects are absent from this constant-coefficient representation.

The $g_i$ are real eigenfrequencies of the Laplace-Lagrange secular matrix, describing apsidal precession. For mode $i$, the signed real coefficients $e_{ji}$ give its amplitude in planet $j$; their relative magnitudes are fixed by an eigenvector, and their signs distinguish aligned and anti-aligned apsides. The common phase ${\beta }_i$ specifies that mode's orientation at the chosen time origin. A planet's actual orbital eccentricity is $|z_j(t)|$, which generally varies through interference of modes; it is not a sum of the positive mode amplitudes.

To first order in orbital eccentricity, the semi-minor axis is $a+O(e^2)$, while the ellipse's center is displaced from the star by $-a\mathbf{e}$. Thus each orbit is a circle of radius $a$ about $-az$, in complex spatial coordinates.

Write $z=z_f+e_p{e}^{i{\varpi }_p}$ at one secular epoch. The centers of the individual circles lie on a circle of radius $a{e}_p$ about $C_f=-a{z}_f$. Their union is the proper-eccentricity annulusIts full radial width is $2a{e}_p$ and its center is displaced by $a{e}_f$ toward the forced apoapsis direction. Relative to the star, if the forced longitude of periapsis is zero,The middle panel of the preceding figure shows this construction. For fully sampled orbital phases and proper phases, the annulus is occupied throughout, but its density need not be uniform; turning points in the proper radial excursion give integrable edge enhancements. The construction requires $e_f,e_p\ll 1$ and is an instantaneous secular geometry.

In resonant trapping of dust, the resonant torque must compensate the inward Poynting–Robertson drag. At the nearly fixed resonant semi-major axis, ${\dot{a}}_{\mathrm{res}}+{\dot{a}}_{\mathrm{pr}}=0$ requiresSince $|\sin \varphi |\le 1$, a necessary torque-balance condition is $|C|e^q\ge A(1+3e^2)/[a^2(p+q)]$. When the limiting orbital eccentricity is small, this givesEquality allows only the extreme phase $|\sin \varphi |=1$, with no margin for restoring torque; a robust trapped state needs $e>e_{\min }$. This is a necessary condition for resonance capture, not a sufficient probability-of-capture criterion for a migrating particle.

Expand the secular Poynting–Robertson drag rates using $(1-e^2{)}^{-3/2}=1+\frac{3}{2}{e}^2+O(e^4)$ and $(1-e^2{)}^{-1/2}=1+O(e^2)$. Through second order in orbital eccentricity,The second formula follows from $d(e^2)/dt=2e\dot{e}$. The cubic correction in $\dot{e}$ contributes only at fourth order in $d(e^2)/dt$.

The specific angular momentum is $h=\sqrt{G{M}_{\star }a(1-e^2)}$, so the periapsis angular speed isAt the exterior mean-motion resonance, neglecting radiation pressure, $n/n_p=p/(p+q)$. Thus at the limiting orbital eccentricity $e=\sqrt{2q/(5p-q)}$,The condition for overtaking the planet in angle isFor the formal second-order equilibrium, it holds for every physical bound value $0<q<5p/3$. To see this, rewrite $q/p=5e^2/(2+e^2)$, so $p/(p+q)=(2+e^2)/(2+6e^2)$. For $0<e<1$, $\sqrt{1+e}/(1-e{)}^{3/2}>1+2e$, whileThe instantaneous angular motion is therefore prograde relative to the planet near periapsis, although its mean relative motion is retrograde. The range $q\ll p$ is the regime where the eccentricity expansion is most reliable.

The expansion separates naturally into three timescales:

Short-period terms matter for instantaneous osculating elements and encounters, resonant terms near period commensurabilities, and secular terms for long-term evolution away from strong resonance.

In centre-of-mass and relative coordinates, the kinetic energy separates into centre-of-mass motion plus $\mu v^2/2$, while the mutual potential is $-G{M}_1{M}_2/r=-GM\mu /r$. In the centre-of-mass frame the orbital energy is therefore

The force is central, so the specific relative angular momentum $h=r^2\dot{\theta }$ is conserved. With $u=1/r$, the radial equation becomes the Binet equationwhose solution isHere $\theta$ is the true anomaly measured from periapsis and $e$ is the orbital eccentricity. Substitution into the energy, or evaluation at an apsis, givesand hence

Immediately before the supernova the circular relative speed obeys $v^2=GM/a$. The impulsive supernova kick in a binary star changes it toApplying the energy formula just after the explosion at the unchanged separation $r=a$ givesUsing the original circular-speed relation,The post-explosion system is bound exactly when $1+2\alpha \cos \psi +{\alpha }^2<2M^{\prime }/M$. The left side ranges from $(1-\alpha {)}^2$ to $(1+\alpha {)}^2$, so every kick direction remains bound ifwhereas every direction unbinds the stars if

For an unbound orbit, conservation of relative specific energy at infinity gives $V^2/2=v^{\prime 2}/2-G{M}^{\prime }/a$. Therefore

Direct evidence for stellar feedback includes expanding ionized shells and superbubbles around young associations, hot X-ray-emitting gas in supernova remnants, broad or split emission lines, and blueshifted absorption showing cool and warm outflows. P-Cygni profiles reveal massive-star winds, while extraplanar filaments and metal-enriched gas demonstrate transport away from star-forming disks. Indirect evidence includes the galaxy mass--metallicity relation, low baryon fractions and suppressed star formation in dwarf galaxies, chemically enriched circumgalactic gas, and correlations of outflow speed and mass loading with the star formation rate.

Rapid gas removal changes the gravitational potential before stellar and dark-matter orbits can respond adiabatically. Positions and velocities are initially unchanged, but orbital binding energies rise; orbits expand and become more eccentric, and some particles escape. Repeated burst--outflow--reaccretion cycles can irreversibly transfer energy to collisionless matter and turn a central dark-matter cusp into a core. This matters because dwarf-galaxy rotation curves are used to test dark-matter microphysics: a feedback-made core can mimic a non-cold or self-interacting dark-matter signature.

Write the initial potential energy as $W=-aG{M}^2/R$. The virial theorem gives $T=-W/2$. If a well-mixed fraction $ϵ$ remains after instantaneous mass loss, the immediate kinetic and potential energies are $T_a=ϵT$ and $W_a={ϵ}^{2}W$, soAfter revirialization at $R^{\prime }$, $E_f=W_f/2=-aG{ϵ}^2{M}^2/(2R^{\prime })$. Equating energies gives the impulsive mass-loss expansion lawThe remnant is bound only for $ϵ>1/2$; loss of half or more of the gravitating mass disrupts this idealized system.

For many infinitesimal, individually revirialized losses, put $ϵ=1+dM/M$ in the impulsive result. To first order, $dR/R=-dM/M$. Integration yields the adiabatic mass-loss expansion lawSlow loss causes finite expansion for every positive remaining mass fraction and has no sharp disruption threshold.

Finally consider an initially circular orbit of radius $R$ around a point mass $M$. Its speed and specific angular momentum obey $v^2=GM/R$ and $h^2=GMR$. Immediately after $M\to ϵM$, these remain unchanged, while the new specific energy isUsing the orbital eccentricity relation $e^2=1+2E^{\prime }{h}^2/(G^2{ϵ}^2{M}^2)$ givesIt is an ellipse for $ϵ>1/2$, parabolic at $ϵ=1/2$, and unbound for smaller $ϵ$, in agreement with the virial argument.

Here $a_p$ is the planet's semi-major axis, while $a,e,I$ are the particle's semi-major axis, orbital eccentricity, and orbital inclination relative to the planet's plane. The formula assumes the circular restricted three-body problem: the planet-to-star mass ratio is small, the particle has negligible mass, and its motion is approximately heliocentric and Keplerian away from brief encounters. The planet's own orbit is circular.

The Jacobi constant is exactly conserved in that ideal rotating problem. Expressing it in heliocentric orbital elements away from the planet gives the approximately conserved Tisserand parameterThe first term measures the particle's normalized binding energy. The second is twice the component of its specific angular momentum normal to the planet's plane, normalized by $\sqrt{G{M}_{\ast }{a}_p}$.

The radiation-pressure coefficient reduces the dust grain's effective stellar gravitational parameter to $(1-\beta )G{M}_{\ast }$. At the exterior 5:4 mean-motion resonance, $n_d/n_p=4/5$, so mean motion givesThis is a first-order mean-motion resonance. Its leading disturbing-function term is therefore linear in the small orbital eccentricity:Thus its dimensionless strength is of order $(M_p/M_{\ast })e_d$, up to the Laplace-coefficient combination $f_{5:4}$, and its resonant argument varies slowly near commensurability.

During resonant trapping of dust, the mean semi-major axis is stationary. Setting the supplied ${\dot{a}}_d$ to zero givesA real resonant phase exists only when $|\sin \varphi |\le 1$. Hencewhere $e_{d0}$ is the grain's orbital eccentricity when trapping begins. The inequality also has the expected sign $\sin \varphi <0$, allowing the planetary torque to oppose Poynting–Robertson drag.

Keeping one resonant harmonic and treating orbital eccentricity as fixed reduces mean-motion resonance dynamics to a physical pendulum equation. For the inner first-order term $\mathcal{R}=(G{M}_2/a_2)f(\alpha )e\cos \varphi$, leading Lagrange planetary equations give $\ddot{\varphi }\simeq 3j^2{\mu }_2{n}^2\alpha fe\sin \varphi$. Around its stable phase, the small-amplitude libration frequency is $jn\sqrt{3{\mu }_2\alpha |f|e}$. Evolution of eccentricity, other harmonics, and approach to a separatrix change this approximation.

In a high-orbital eccentricity interior $k:1$ mean-motion resonance, the resonant argument $\varphi =k{\lambda }_p-\lambda -(k-1)\varpi$ places successive rotating-frame apocentres at $\pi -[\varphi +(2j+1)\pi ]/k$. Avoiding astronomical conjunction at these slow outer excursions favors $\varphi =0$ for $k=2$ and $\varphi =\pi$ for $k=3$. This geometrical phase protection suggests stable resonant-argument libration centres; actual dynamical stability requires the resonant perturbation problem.

A prograde, equatorial circular orbit with orbital period equal to the planet's spin period. In the ideal model its satellite remains above one equatorial ground location. Nonzero orbital inclination or orbital eccentricity destroys exact stationarity even when the periods agree.

For an attractive inverse-square force, $r=p/(1+e\cos f)$, where $f$ is true anomaly, $e$ is orbital eccentricity, and $p=h^2/\mu$ is the semilatus rectum. The specific angular momentum relation $\dot{f}=h/r^2$ converts this geometric equation into an angular evolution law.

The orbit-averaged Poynting–Robertson drag equations preserve $C=a(1-e^2)e^{-4/5}$ for $0<e<1$. This relates shrinking semi-major axis to decreasing orbital eccentricity; a circular orbit is treated by its separate limiting solution.

For a common semi-major axis $a$ and proper eccentricity $e_p$, randomly distributed proper longitudes of periapsis give an annulus of radii $a(1-e_p)$ and $a(1+e_p)$ about the point displaced from the star by minus $a$ times the forced eccentricity vector, to first order in orbital eccentricity.

The leading uniform stellar radiation pressure is a differential acceleration relative to a planet, of scale $\beta G{M}_{\star }/a_p^2$. Relative to planetocentric gravity at dust radius $a$, its magnitude isThus lying inside a Hill sphere alone does not guarantee small radiative perturbations. Approximately circular tightly bound dust requires this ratio sufficiently small, as well as weak tides; the uniform radiative force can otherwise excite orbital eccentricity or unbind grains.