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Adopt circular orbits, negligible planetary masses, spherical bodies, uniform stellar brightness and no transit dilution. Interpret the measured duration as the total duration of a circular planetary transit, from first to fourth contact, $T_{14}$. These conventions matter especially for the off-centre crossing: finite planet size cannot automatically be ignored there. Unknown orbital eccentricity or limb darkening would require further information to infer unique stellar radii.

First convert the apparent magnitude difference into a fractional exoplanet transit depth, rather than treating a milli-magnitude as a part in a thousand:Both crossings are non-grazing because their transit impact parameters satisfy $b<1-k$, so the full projected planetary disc can block the star. Equal depths determine equal radius ratios, not equal absolute planet radii.

For orbital inclination $i$ measured from the sky plane, $b=(a/R_{\star })\cos i$. Choosing mid-transit as orbital phase zero, the projected separation obeysAt first and fourth contact, $z/R_{\star }=1+k$. The full crossing duration therefore isFor the numerical scale use solar radius $R_{\odot }=6.957\times 10^8\mathrm{m}$ and nominal solar mass parameter $(GM{)}_{\odot }=1.3271244\times 10^{20}{\mathrm{m}}^3{\mathrm{s}}^{-2}$, with one day equal to $86400\mathrm{s}$. These are stated conversion choices, not additional measurements of the two systems.^[ref]

Kepler's third law gives $a=[G{M}_{\star }{P}^2/(4{\pi }^2){]}^{1/3}$. For a solar-mass, solar-radius host, this gives $a/R_{\odot }=12.3051$ at five days and $6.68023$ at two days. Substitution in the contact formula yieldsThe transit duration of a small planet instead takes both $k\to 0$ and $R_{\star }/a\ll 1$, giving $T\simeq P{R}_{\star }\sqrt{1-b^2}/(\pi a)$. It predicts $3.10\mathrm{h}$ and $1.37\mathrm{h}$, respectively. The difference is appreciable for B because the fractional change in its crossing chord is larger. Durations defined by mid-ingress to mid-egress or by second to third contact should not be silently equated with $T_{14}$.

The original diagram uses a straight projected track to display the contact geometry; the duration formula above retains orbital curvature.

To infer the actual stellar radii, invert the contact formula before applying the supplied mass-radius scaling. Let $A=a/R_{\star }$ and $s=\sin (\pi T_{14}/P)$; thenThe measured durations imply $A_A=8.34661$ and $A_B=7.55743$. Normalize the stipulated constant mass-radius ratio to the Sun, so $M_{\star }/M_{\odot }=R_{\star }/R_{\odot }\equiv r_{\star }$. Kepler's third law then givesThus the stellar radius from transit duration and a linear mass-radius relation isThe same assumed scaling gives $M_A=1.79M_{\odot }$ and $M_B=0.831M_{\odot }$. A's longer crossing requires a lower mean stellar mass density, while B's shorter full-contact crossing requires a higher one. They are not both solar replicas under this model. Their planets have $R_{p,A}=0.1213R_{\odot }$ and $R_{p,B}=0.05633R_{\odot }$.

For a consistently applied point-planet approximation, the duration scales as $T\propto P^{1/3}{r}_{\star }^{2/3}\sqrt{1-b^2}$ and the inferred radii would instead be $R_A\simeq 1.98R_{\odot }$, $R_B\simeq 1.07R_{\odot }$. The latter is close to solar. Consequently the assertion of a substantial non-solar radius for B depends on which duration convention and finite-size approximation are intended. The contact-based values above use the measured depth rather than discarding it. If eccentric orbits are allowed, the transit speed introduces an additional factor depending on eccentricity and orientation, so the two durations alone cannot establish those radii uniquely.

For an outer coplanar planet around B, the common orbital inclination gives $b_2/b_B=a_2/a_B=(P_2/P_B{)}^{2/3}$. A point-planet centre must cross the stellar disc, requiring $b_2<1$. The usual coplanar transit limit is thereforeThis is a limiting value approached from below; a tangent path has no duration. For an outer planet of finite, unspecified radius ratio $k_2$, any overlap instead requires $b_2<1+k_2$, giving $P_2<P_B[(1+k_2)/b_B{]}^{3/2}$. If it had the same ratio $k_2=k$, that geometric limit would be $3.08$ days; requiring full overlap would give $2.52$ days. Because neither its radius nor a photometric detection threshold is supplied, no unique finite-size practical maximum period can be inferred beyond this conditional expression. The stellar radius cancels from the point-planet period-ratio bound.

For B's inner planet, a reflected-light planetary phase curve haswhere $A_g$ is geometric albedo and $\alpha$ is phase angle. Near exoplanet secondary eclipse the visible hemisphere is almost fully illuminated. Using the standard full-phase approximation and the contact-based $A_B$,The exact lost fraction of combined stellar and planetary light is $F_p/(F_{\star }+F_p)$, which changes the result negligibly at this contrast. The corresponding magnitude depth is about $0.0437$ milli-magnitudes. The circular orbit has the same impact distance at eclipse as at transit, and $b_B<1-k$, so the star can hide the entire planetary disc.

The inclination inferred for B is $83.92^{\circ }$, putting closest approach to full phase at $\alpha \simeq 6.08^{\circ }$. A fully precise prediction would therefore also require the reflected-light phase function, which the question does not specify; the boxed value uses $\Phi \simeq 1$. The consistent point-planet duration approximation yields about $56.7\mathrm{ppm}$ instead, through its different inferred $a/R_B$. It is important to keep the geometry used for the radius inference and the reflected-light calculation consistent.

Finally, the stellar radial-velocity semi-amplitude for a low-mass companion on a circular orbit isThe two stars' equal apparent magnitudes make a comparison at similar photon statistics plausible, though spectral-line widths, stellar variability and observing cadence can still differ. With the contact-based stellar estimates,If the planets have comparable mean densities, A is easier: their equal depth then gives $M_{p,A}/M_{p,B}\simeq (R_A/R_B{)}^3\simeq 9.99$, so $K_A/K_B\simeq 4.44$. The larger planetary mass outweighs A's longer orbital period and larger host mass. For equal planetary masses, B is easier by a factor $2.25$ in amplitude. The supplied transit data give planet radii, not masses, so the ranking is conditional on a planetary mass-radius assumption; a unique unconditional ranking would invent information.

The observed population is the parent population filtered by an astronomical selection function and measurement errors. Malmquist bias arises because a flux-limited galaxy selection admits luminous objects from larger volumes: in Euclidean space $d_{\max }\propto L^{1/2}$ and $V_{\max }\propto L^{3/2}$. Thus observed luminosity counts are weighted by $V_{\max }$. Even a nominal standard candle with intrinsic scatter is preferentially observed on its bright side near the limit; ignoring this can underestimate distances. A volume-limited sample or a justified $1/V_{\max }$ weighting helps, subject to completeness, geometry and evolution assumptions. Separately, Eddington bias occurs when noise scatters a steeply rising population across a detection threshold: more faint objects enter than bright objects leave.

Lutz-Kelker bias concerns noisy stellar parallaxes and population/selection weighting. Locally uniform spatial density gives $p(r)\propto r^2$ and, by changing variables, $p(\varpi )\propto {\varpi }^{-4}$ over the model's valid finite range. More distant stars are available to scatter into a parallax-limited sample than nearby stars to scatter out. Inverting an uncertain parallax also introduces a distinct nonlinear-estimation problem. There is no universal magnitude correction for every star: spatial density, luminosity, extinction and how the sample was selected all matter. The classical treatment motivates this issue; it should not replace an explicit generative model.^[ref]

Bayes' theorem combines the measurement likelihood function with a justified prior distribution. For example, with a Gaussian parallax error and consistent distance/parallax units,A proper distance prior for parallax inference might have $p(r)\propto r^2{e}^{-r/L}$ with a justified scale $L$; extending $r^2$ to infinity gives an improper posterior here. Negative measured parallaxes remain valid noisy data. For selection conditioning in Bayesian inference, distinguish individual inference from population inference: if selection is determined entirely by the exact data already conditioned on, it adds no further information about that individual's parameters. For population parameters $\Lambda$, the selected-data likelihood must instead include the detection normalization $\alpha (\Lambda )=\int S(d)p(d\mid \Lambda )dd$. Applying a cut twice is also an error.

Selection can depend on surface brightness, angular resolution, dust, sky coverage, colour and observing cadence as well as total flux. Crowding and unresolved binaries bias distance calibrators. Transit surveys favour large planets and short periods, with an additional geometric transit probability approximately $R_{\star }/a$ for small planets on circular orbits; injection/recovery tests quantify the remaining transit survey completeness. Cluster catalogues select noisy mass proxies from a steep abundance distribution. Planck's 2013 SZ catalogue explicitly characterized detection completeness and reliability rather than identifying every detection with an unbiased mass census.^[ref] Calibrators, artificial-source tests and a forward model of detection, measurement and follow-up are the practical checks on any correction.

Stellar reflex motion. A planet and its host orbit their common barycentre. The star's line-of-sight velocity shifts its absorption lines by $\Delta \lambda /\lambda \simeq v_r/c$ at low speeds. A time series can be fitted withHere $\nu (t)$ is true anomaly, $\varpi$ the argument of periastron and $\gamma$ an arbitrary systemic velocity offset. At small companion mass and known stellar mass, the orbit supplies $M_p\sin i$, not the true planet mass without an inclination constraint. Long monitoring helps distinguish a reproducible Keplerian signal from stellar oscillations, magnetic activity, spots and instrumental trends.

The Lick reference-cell implementation. Starlight passes through a temperature-controlled cell of molecular iodine before entering the Hamilton high-resolution echelle spectrograph. The cell imprints a dense pattern of reference absorption lines on the stellar spectrum, predominantly in the approximately $500$–$630\mathrm{nm}$ interval. These lines share the same exposure and downstream optical path as the stellar light. A calibrated high-resolution Fourier-transform iodine spectrum and an iodine-free stellar template allow the wavelength solution and instrumental line-spread function to be fitted along with the stellar Doppler shift. The method corrects shifts and line-shape changes that a separate calibration exposure would not necessarily follow.^[ref]

A useful forward model for one short spectral segment isfollowed by pixel integration, continuum scaling and a fitted wavelength calibration. Here $S$ is the intrinsic stellar spectrum, $T_{{\mathrm{I}}_2}$ the cell transmission and $\mathcal{P}$ the instrumental line-spread function. The spectra multiply before convolution; simply dividing a recorded blended spectrum by the laboratory iodine spectrum is not equivalent to this model. The iodine pattern is stationary in the observatory frame while stellar lines move relative to it. Separate spectral segments accommodate wavelength dependence of the instrumental profile.

Earth's rotational and orbital velocities must then be removed with a barycentric correction using accurate timing, ideally weighted by photon arrival rate. Many stellar lines constrain a shift smaller than one instrumental resolution element; spectral resolution alone is not the velocity error. Photon statistics, template quality, instrumental-profile modelling and intrinsic stellar variability limit the precision. The method measures changes relative to a template and does not by itself provide a perfectly absolute centre-of-mass velocity. No particular discovery, modern precision floor or historical observing performance is assumed here.

There is no convincing single birthplace of science unless the definition is narrowed in advance. Systematic astronomy and mathematics preceded classical Greece; Greek natural philosophy developed distinctive explanatory and demonstrative ambitions; early modern Europe transformed the relationship between mathematics, experiment and institutions. These are different historical achievements. Treating any one of them as the creation of everything now called science confuses a particular form of inquiry with inquiry's entire history.

A good test begins with surviving practices rather than a list of supposed founding geniuses. The Babylonian astronomical diary VAT 4956 records positions of the Moon and planets relative to named stars, together with other observations, for 568–567 BCE. Such a record requires trained observers, a conventional vocabulary and a means of accumulating observations over time. Its intellectual value is not annulled because astronomy was entangled with divination and royal administration. However, this particular diary should not be credited with everything achieved by later Babylonian predictive astronomy: an observational record is evidence of systematic recording, not by itself evidence of a particular predictive algorithm.^[ref]

Egyptian mathematics provides a different kind of evidence. Problem 50 of the Rhind mathematical papyrus gives a procedure for the area of a circular field: remove one ninth of its diameter and square the remainder. Written in modern algebraic notation, the procedure isComparing this with the modern circle formula $A=\pi d^2/4$ gives the effective approximation $\pi \approx 256/81$. This is our reconstruction of the rule's accuracy, not a claim that its author used our symbol $\pi$ or formulated an abstract theory of that constant. A worked numerical procedure is mathematical knowledge even when it is not presented as a Greek-style proof.^[ref]

The case for Greece therefore needs a more specific claim. In Physics by Aristotle, nature is investigated through principles of motion and change, and explanation requires identifying different kinds of cause. His distinctions between material, formal, moving and final causes constitute an explicit programme of explanation, not merely a collection of useful recipes. Greek mathematics, exemplified by Euclid's Elements, made the organization of results into chains of proof unusually prominent. These traditions help explain why later European scholars gave Greek texts such authority. They do not establish that earlier or non-Greek practitioners lacked reasoning, nor that all Greek explanations were correct.^[ref][ref]

The European sixteenth and seventeenth centuries nevertheless deserve attention as more than the recovery of ancient truths. Galileo Galilei's Two New Sciences by Galileo Galilei joins precise definitions of motion to geometrical demonstrations and reported tests. An inclined plane lengthens a descent sufficiently to compare its distance and time; a mathematical dependence becomes something an apparatus can discriminate. Francis Bacon's Novum Organum by Francis Bacon instead urges a disciplined reconstruction of inquiry through ordered experience and the correction of intellectual prejudices. Their programmes differ, but both show participants deliberately contesting how knowledge should be made. Neither text authorizes the claim that observation or reasoning had only just been invented.^[ref][ref]

Institutional criteria also change the answer. If science means a modern profession with specialized laboratories, credentials and journals, even sixteenth-century Europe is too early for much of it. If it means any successful craft technique, the category becomes so broad that the first surviving written text cannot identify its beginning. If it means public, revisable claims about nature supported by disciplined observation and reasoning, several ancient traditions qualify, with different strengths and social settings. Each definition selects different evidence; none should be smuggled into a geographical verdict.

The strongest conclusion is consequently plural but discriminating. Greece was a major centre of explanatory natural philosophy and demonstrative mathematics; early modern Europe was a major site of their reorganization; neither was the unique beginning of systematic knowledge of nature. This conclusion recognizes real differences in standards of explanation without using proximity to present-day science as the sole measure of past achievement. It also leaves open the historical work of tracing transmission and transformation rather than imagining isolated civilizations independently possessing, or wholly lacking, a timeless scientific essence.

Mid-twentieth-century environmental and population concerns converged around limits to growth, but disagreed over what caused environmental damage and what intervention followed. More people could increase pressure on land, food and water; that did not establish that numbers alone explained depletion or pollution. A strong account follows the transition from resource conservation to chemical contamination and then to explicit global mathematical models, distinguishing observations from conditional projections.

The resource argument was already prominent in 1948. William Vogt's Road to Survival by William Vogt linked human numbers to the productive capacity and degradation of land. Soil loss could reduce future food production, while further population increase enlarged the demands placed on the same resource base. The relevant constraint was not simply the size of the planet: it was the maintenance of the ecological processes on which agricultural production depended. This linked population dynamics with a changing carrying capacity. The argument's policy implications depended on whether one emphasized birth limitation, restoration of land or changes in resource use.^[ref]

The mechanism can be clarified with modern notation. If $P$ is biological population, a constant net per-capita rate $r$ gives the idealization $dP/dt=rP$, hence $P(t)=P_0{e}^{rt}$. A finite food supply cannot support that positive exponential growth indefinitely. But the premise is conditional: observed birth rates, mortality and food production need not remain fixed. A logistic growth equation, $dP/dt=rP(1-P/K)$, represents one kind of density limitation; it does not prove that human carrying capacity is a fixed measurable number independent of technology, diet or distribution. Resource arguments needed empirical work on both population change and the systems supplying people.

The postwar problem was also about the consequences of attempts to sustain human welfare. Increasing agricultural output and suppressing insect-borne disease could reduce mortality or enlarge food supply, while chemical interventions created further ecological effects. Rachel Carson's Silent Spring by Rachel Carson made that feedback concrete. Its Clear Lake example followed DDD through plankton, fish and fish-eating birds: small environmental concentrations could become large tissue residues through exposure and feeding relationships. The account connected intended pest control with unintended damage to other biological populations. It did not depend on proving that the number of human inhabitants alone had caused the damage.^[ref]

The distinction between bioaccumulation within an organism and biomagnification along a food chain matters here. Neither follows simply from dividing a pesticide dose by the number of people. Persistence, uptake, elimination and feeding relations determine exposure. Rachel Carson's 1963 response to critics listed continuing wildlife losses under officially organized spraying and protests by farmers themselves. The historical conflict was therefore over the evidential adequacy and distribution of an intervention's costs, not merely a choice between caring about people and caring about nature.^[ref]

Paul R. Ehrlich and John P. Holdren's 1971 paper explicitly defended the importance of population growth while addressing consumption and technology. Its decomposition of total impact into numbers of people and impact per person also made clear why density alone is inadequate. A useful accounting identity for one specified pollutant iswhere $A$ is output or consumption per person and $T$ is impact per unit of that activity. For a waste stream measured in mass per year, the units multiply consistently to mass per year. This three-factor notation is an analytical extension of the debate, not a quotation of every equation in the 1971 article. It identifies separate questions about abundance, consumption and production techniques; it is not by itself an independently tested causal law.^[ref]

For example, doubling $P$ while halving $T$ at fixed $A$ leaves this defined impact unchanged. Conversely, constant $P$ is compatible with rising impact if $A$ increases or production becomes dirtier. These counterexamples do not show that population is irrelevant. They show why an explanation must state what else is changing. They also explain a political disagreement within environmentalism: concentrating on fertility can obscure high consumption and harmful production, while concentrating exclusively on cleaner technology can overlook the scale of demand.

The Limits to Growth by Donella H. Meadows and others in 1972 joined population, industrial production, food, resources and pollution in a computer computer simulation. Its importance lay in their interaction and in delays: a system could overshoot a sustainable level before feedback sufficiently changed investment or demographic behavior. The resulting curves were conditional scenarios generated by assumptions about those relations, not direct observations of the future. The authors examined changed assumptions and possible stabilization; the report should not be summarized as one unconditional prediction of a uniquely dated collapse.^[ref]

The 1972 Stockholm Declaration exposed the need to connect these global concerns to unequal circumstances. Principle 8 treated economic and social development as necessary to environmental improvement, while principle 16 discussed demographic policies subject to human rights and governmental judgment. Their coexistence matters: a global population total did not erase differences in poverty, consumption or responsibility.^[ref]

Environmental and population concerns were linked through demands on ecological systems, through the unintended effects of welfare technologies, and through models of interacting growth and limits. They never established one agreed diagnosis. The enduring historical issue was which variables, mechanisms and distributions of responsibility an apparently simple claim about too many people concealed.

Put $k=p+1$, $\mu =G{M}_{\star }$ and $n_{\mathrm{pl}}=\sqrt{\mu /a_{\mathrm{pl}}^3}$. To leading order the mean-motion resonance fixes $a=a_{\mathrm{pl}}{k}^{-2/3}$. The specific angular momentum isThe asteroid's instantaneous angular speed is $h/r^2$. Thus the high-eccentricity angular-speed crossover satisfies $h/r_x^2=n_{\mathrm{pl}}$, givingHere $q\ll r_x\ll a$, so a parabolic Kepler orbit gives the local motion accurately. Its polar equation is $r=2q/(1+\cos f)$. Write $f_x=\pi -{ϵ}_x$. Since $1+\cos f_x\simeq {ϵ}_x^2/2$,This is the outward crossing; the inward crossing has true anomaly $-f_x$ modulo $2\pi$.

For the pericentre-to-crossover flight time use $D=\tan (f/2)$. The parabolic Kepler orbit has $r=q(1+D^2)$ and $dt=r^{2}df/h$, whence direct integration gives the Barker equationAt the crossover $D_x\simeq \sqrt{r_x/q}\gg 1$. Consequently the planet's angular displacement isThe error tends to zero as $q/a_{\mathrm{pl}}\to 0$ at fixed $k$; it includes the finite binding energy of the elliptic Kepler orbit as well as the subleading term in the Barker equation.

At the $j$th pericentre passage the asteroid's unwrapped mean longitude is $\lambda =\varpi +2\pi j$. The resonant argument therefore givesThus $\varphi /k$ specifies the planet's direction relative to the asteroid's longitude of pericentre when the asteroid passes pericentre, with $k$ branches separated by $2\pi /k$. It is not the instantaneous true-longitude separation throughout the orbit. At high orbital eccentricity the asteroid quickly sweeps to a direction nearly opposite pericentre, while the planet scarcely moves. During the long outer excursion the asteroid's direction changes slowly and the planet advances substantially. This makes an astronomical conjunction during the outer excursion much more dangerous than an astronomical conjunction close to pericentre.

At the following apocentre, the planet has advanced $\pi /k$. The asteroid's direction in the rotating reference frame is consequentlyThese $k$ apocentre directions are equally spaced. Phase protection of interior integer resonances places the planet in the largest gap between them, rather than at an apocentre direction. For $k=2$, $\varphi =\pi$ puts an apocentre toward the planet, whereas $\varphi =0$ puts the apocentres at $\pm \pi /2$. For $k=3$, $\varphi =0$ puts an apocentre toward the planet, whereas $\varphi =\pi$ puts the apocentres at $\pi /3,\pi ,5\pi /3$. Hence the geometrically favored resonant-argument libration centres in this regime of high orbital eccentricity areThis is an encounter-avoidance argument for likely stability, not a calculation of a resonant Hamiltonian or a claim that every orbit near either centre is stable at arbitrary orbital eccentricity.

The following rotating-frame drawing uses $a_{\mathrm{pl}}=1$, $a=3^{-2/3}$, $q=0.002$ and $\varphi =\pi$. It follows three successive Kepler orbits, which fill one planet orbital period and close the pattern. The blue segment is just the first outward half-orbit. The marked crossover is a maximum of its rotating-frame polar angle, since $d\psi /dt=h/r^2-n_{\mathrm{pl}}$ changes from positive to negative there. The planet stays at $(1,0)$; the three apocentres avoid that direction.

To estimate angular-momentum kicks to nearly radial orbits, note that $q\simeq h^2/(2\mu )$, so an order-one fractional change in $q$ requires $|\Delta h|$ of order $h$. A characteristic distant encounter at radii of order $a_{\mathrm{pl}}$ has tangential acceleration of order $G{M}_{\mathrm{pl}}/a_{\mathrm{pl}}^2$ and duration of order $n_{\mathrm{pl}}^{-1}$. Its torque per unit asteroid mass therefore producesWith the most favorable coherent signs, the crude encounter count isOrder-one factors depend on $p$ and encounter geometry. Uncorrelated signs instead give a random walk count of order $(M_{\star }/M_{\mathrm{pl}}{)}^{2}q/a_{\mathrm{pl}}$ when this is large. A librating phase-protected orbit can suppress the kicks even further.

There is an important limit to calling this a minimum. The question does not specify an encounter impact parameter. For a weak close encounter with impact parameter $b$ and relative speed $u\sim \sqrt{G{M}_{\star }/a_{\mathrm{pl}}}$, the impulse estimate gives $|\delta h|\sim G{M}_{\mathrm{pl}}{a}_{\mathrm{pl}}/(bu)$ and hence $N\sim (M_{\star }/M_{\mathrm{pl}})(b/a_{\mathrm{pl}})\sqrt{q/a_{\mathrm{pl}}}$, capped below by one. In the 2:1 case the Kepler orbit can cross the planet's orbit, so one suitably close encounter can suffice: there is no impact-parameter-independent large lower bound. For $k\ge 3$, $Q\simeq 2a_{\mathrm{pl}}{k}^{-2/3}<a_{\mathrm{pl}}$ bounds the separation below by $a_{\mathrm{pl}}-Q$, and supplies a geometric factor at fixed $k$. The boxed count is the usual orbital-scale, maximally coherent estimate; the assumptions are essential.

Use rotating barycentric coordinates $\mathbf{R}=(x,y,z)$, rotating velocity $\mathbf{w}$, and $\mathbf{\Omega }=\hat{\mathbf{z}}$. Multiplying the three equations by the corresponding velocity components and adding cancels the Coriolis force terms:Thus the Jacobi constant is $C=2U-|\mathbf{w}{|}^2$. The inertial barycentric velocity is $\mathbf{V}=\mathbf{w}+\mathbf{\Omega }\times \mathbf{R}$. If $H_z=(\mathbf{R}\times \mathbf{V}{)}_z$ and $E_b=|\mathbf{V}{|}^2/2-{\mu }_1/r_1-{\mu }_2/r_2$, thenThe inertial energy $E_b$ and inertial angular momentum $H_z$ need not separately be constant. Their combination is constant because the two gravitational sources rotate steadily at unit angular speed.

For the Tisserand parameter take ${\mu }_2\ll {\mu }_1\simeq 1$, and compare the osculating orbital elements well outside a close encounter, where ${\mu }_2/r_2$ and barycentre-to-primary corrections are negligible. ThenThis requires a circular secondary orbit, the restricted three-body problem approximation, and no dissipative force or other perturber. $T$ is conserved to leading order between encounter episodes; the osculating $T$ during a close encounter need not be constant. The exact invariant is $C$.

For clarity the exact two-body energy relation is $a={\mu }_1/(2{\mu }_1/r_1-v_1^2)$, rather than the printed formula without the numerator ${\mu }_1$. At the leading order ${\mu }_1=1$ used for $T$, the two agree; retaining finite ${\mu }_1$ in just part of that formula is inconsistent.

At a close encounter put the secondary radius and speed equal to one at leading order, and denote the particle's incoming relative speed outside the secondary's strong-deflection region by $u$. Conservation of relative speed in the short gravitational encounter givesAn incoming orbit with $a=1$ has $|{\mathbf{v}}_1|=1$ at $r_1=1$ by the vis-viva equation; escape there requires an outgoing speed exceeding $\sqrt{2}$. The triangle inequality therefore gives the necessary single-encounter escape velocity boundEquality only gives a marginal parabolic limit. This argument deliberately gives a weak necessary bound, not a sufficient scattering criterion.

The planet-encounter relative velocity is related to the Tisserand parameter byHere $v_{1,\mathrm{tangential}}=h_z$ at the encounter radius, and the formula also applies to inclined passages at the planet's orbital plane. For the intended prograde coplanar incoming orbit, $I=0$ and $a=1$, so $u^2=2-2\sqrt{1-e^2}$. Applying the preceding necessary bound givesCoplanar alone also allows $I=\pi$. For that retrograde case $u^2=2+2\sqrt{1-e^2}$, and even a circular incoming orbit can be scattered onto an escaping orbit; the orbital eccentricity restriction therefore presumes prograde motion. Also, retaining the outgoing relative-velocity geometry gives the sharper necessary condition $|{\mathbf{v}}_{1,\mathrm{out}}|\le 1+u$, hence $u>\sqrt{2}-1$ and $e>\sqrt{4\sqrt{2}-5}/2$ for the same prograde incoming orbit. This stronger bound is consistent with, and implies, the weaker bound above.

For the Tisserand periapsis bound, first writeA bound orbit capable of another close encounter must intersect the secondary's circular radius, so $q\le 1\le Q$. For fixed $q<1$, the largest possible $T$ occurs at $I=0$. The remaining expression decreases with $Q\ge 1$, sinceThus every such orbit obeysOn $0<q<1$, $g(q)$ increases monotonically from $2$ to $3$. For $2<T<3$ the least accessible pericentre is the unique root of $T=g(q)$: the limiting orbit is prograde and coplanar, with apocentre just at the planet. Introduce $s=\sqrt{3-T}$. Its tangential speed at this apocentre is $1-s$, and its vis-viva equation gives the particularly well-behaved answerEquivalently, setting $w=\sqrt{2q/(1+q)}$ gives $T=2-w^2+2w$ and $w=1-s$, with $q=w^2/(2-w^2)$. Rationalizing the previous result yieldsThe apparent singularity at $T=\sqrt{8}$ is removable; the first form gives $q_{\min }=(\sqrt{2}-1)/2$ there. The limits are $q_{\min }\to 0$ as $T\downarrow 2$ and $q_{\min }\to 1$ as $T\uparrow 3$. The same lower envelope also excludes a smaller pericentre on an escaping trajectory when $2<T<3$. To see this without introducing an apocentre, use the planet-frame velocity sphere: $u=\sqrt{3-T}<1$ forces the tangential component at an encounter to be at least $w=1-u>0$, so total specific angular momentum obeys $h\ge w$. At pericentre the stellar energy obeys $h^2=2q+2E{q}^2$, while $E=h_z-T/2\le h-T/2$. ThusIf $q<q_{\min }=w^2/(2-w^2)$, then $q^2<w\le h$, and the left side is increasing with $h$. Hence $w^2\le 2q+(2w-T)q^2$. But $2w-T=w^2-2$, and the right side is smaller than $w^2$ for $0\le q<q_{\min }$, a contradiction. This establishes the barrier for all Kepler trajectories that reach the encounter radius, including unbound ones.

For $0<T\le 2$, bound encounter-crossing orbits can approach $q=0$ (take $Q\to 2/T$), so the positive root extrapolated from the displayed expression is not a universal barrier. For $T>3$ there is no trajectory reaching a close encounter in this approximation, since $u^2=3-T$ would be negative. Multiple encounters can approach the derived lower envelope by changing energy, angular momentum and inclination while preserving the Tisserand parameter; the invariant alone does not guarantee a particular encounter history reaches it.

At each fixed semimajor axis, let $B=b_{3/2}^2(\alpha )/b_{3/2}^1(\alpha )$ and $Z_f=B{e}_{\mathrm{pl}}$. The planet's longitude of pericentre is the zero of longitude. The secular perturbation equation becomes $\dot{z}=iA(z-Z_f)$. Integrating it with the initial condition $z(0)=0$ givesThe forced eccentricity vector is the constant real $Z_f$; the free eccentricity vector is $-Z_f{e}^{iAt}$. Consequently the complex eccentricity follows a circle in the Argand plane centered at $Z_f$, with radius $Z_f$, starting at the origin and moving counterclockwise. Its initial velocity is downward, $\dot{z}(0)=-iA{Z}_f$. The secular period is $2\pi /A$, andThe longitude of pericentre is undefined at the origin, but the complex eccentricity passes smoothly through it. At $At=\pi$, $z=2Z_f$ and the orbit is maximally eccentric and aligned with the planet.

Using the small-$\alpha$ Laplace coefficient expansion givesAlso $n=n_{\mathrm{pl}}{\alpha }^{-3/2}$ to leading order in the planet-to-star mass ratio. The secular forcing by a distant outer planet is thereforeA relative $O({\alpha }^2)$ correction to $A$ can accumulate a phase error at very long times; the leading-frequency formula is not a uniform-in-time expansion.

If $a_2=a_1(1+\delta )$ with $\delta =\delta a/a_1\ll 1$, then $Z_{f,2}/Z_{f,1}=1+\delta +O({\delta }^2)$ and $A_2/A_1=1+3\delta /2+O({\delta }^2)$. The outer planetesimal therefore has a slightly larger forced eccentricity circle and a shorter secular period. Its accumulated secular phase lead is $\Delta (At)\simeq 3A_{1}t\delta /2$. Initially both circles start at the origin, but their eccentricity vectors progressively lose alignment. Linearizing the exponential in $\delta$ additionally requires $|A_{1}t\delta |\ll 1$.

The orbit crossing is caused by this accumulating differential secular phase, even though individual orbital eccentricities remain small. To first order in orbital eccentricity, the radius of an orbit at fixed inertial longitude $\theta$ isFor an infinitesimally neighboring orbit,This derivative is evaluated within the first-order orbital-shape model; derivatives of the neglected quadratic terms give relative $O(Z_f)$ corrections near crossing. The least separation over longitude is consequently $\delta a[1-|z+a{\partial }_{a}z|]$. The differential secular orbit-crossing criterion is $|z+a{\partial }_{a}z|=1$. Since $Z_f\propto a$ and $A\propto a^{3/2}$,The first term stays bounded by $4Z_f\ll 1$, whereas the second grows in proportion to time. To leading order in $Z_f$, crossing begins when $(3/2)Z_{f}At=1$. ThusMore precisely, in this first-order orbital-shape approximation the exact first crossing lies within a fractional $O(Z_f)$ window of this value, since the magnitude differs from $(3/2)Z_{f}At$ by at most $4Z_f$. Higher-order Laplace coefficients give additional corrections. The local derivation takes the neighboring-orbit limit before the long-time limit; for a finite separation it requires $\delta a/a_1\ll Z_f$ near crossing. A broad continuous disk contains such neighboring orbits. The crossings first appear in directions selected by the phase of $z+a{\partial }_{a}z$, not simultaneously at every longitude.

For the collision energy estimate adopt the phase-averaged mean orbital eccentricity $\overline{e}=4Z_f/\pi \simeq 5\alpha e_{\mathrm{pl}}/\pi$. The specified velocity estimate givesUsing the projectile kinetic energy per target mass convention for specific impact energy, equal-size and equal-density bodies have equal masses and $Q=\frac{1}{2}{v}_{\mathrm{col}}^2$. A catastrophic planetesimal collision requires $Q\gtrsim Q_D^{\ast }$, soThe numerical coefficient is only illustrative because the collision speed was specified only to order of magnitude. If the disruption threshold is instead defined using the centre of mass kinetic energy per combined mass, equal masses give $Q_R=v_{\mathrm{col}}^2/8$ and the coefficient becomes $8{\pi }^2/25$ for a threshold expressed in that convention. The robust result is a lower bound of order $Q_D^{\ast }{a}_{\mathrm{pl}}/(G{M}_{\star }{e}_{\mathrm{pl}}^2)$.

Within the inner-disk approximation, catastrophic collisions require sufficiently large $a_1/a_{\mathrm{pl}}$, not arbitrarily small radii. Although the Keplerian speed rises inward, the secularly induced orbital eccentricity falls faster: $v_{\mathrm{col}}^2\propto \alpha$. The lower bound must be much smaller than one to leave a domain compatible with $\alpha \ll 1$. A small planet orbital eccentricity, a large disruption threshold, or a distant planet can eliminate that domain. The planet mass controls the time to crossing but cancels from the eventual forced eccentricity amplitude and this collision-energy criterion. The threshold alone does not ensure that crossing has occurred within the disk's available lifetime.

Validity and the complete angular equation. The Lagrange planetary equations here describe a planar, circular-planet, small-orbital eccentricity, weakly perturbed exterior mean-motion resonance. Require $M_{\mathrm{pl}}/M_{\star }\ll 1$, $e\ll 1$, one isolated slowly varying resonant argument, perturbation times long compared with an orbital period, and no close planetary encounters. Other resonant harmonics and short-period terms must be negligible; coefficients may be frozen only across a narrow range of semi-major axis. Take $p>0$, $q=1$ or $2$ in a reduced integer ratio. Exactly $e=0$ makes the longitude of periapsis undefined, so the displayed angular variables must then be replaced by nonsingular eccentricity vector components.

Neglecting the perturbation to the accumulated mean longitude gives $\dot{\lambda }\simeq n(a)$; this is an accumulated-phase approximation, not differentiation of a fictitious expression $n(t)t$ while discarding $t\dot{n}$. With $n^2{a}^3=G{M}_{\star }$,Differentiating, using $\dot{n}=-3n\dot{a}/(2a)$ and the supplied orbital element equations, gives the full result within this constant-coefficient model:An equivalent formula replaces the last two terms in brackets by $q^2{C}_r{e}^{q-2}[(p+q)n-p{n}_{\mathrm{pl}}-2q{C}_s]-2q^3{C}_r^2{e}^{2q-4}\cos \varphi$. No derivatives of $C_r,C_s$ are included because they were specified as constants.

A full fixed point requires $\sin \varphi =0$ and $\dot{\varphi }=0$. Hence ${\varphi }_{\ast }=0$ or $\pi$, with its corresponding detuned semi-major axis fixed byAt either point $\dot{a}=\dot{e}=0$. The longitude of periapsis still precesses, so the fixed point refers to the reduced resonant dynamics.

Linear stability analysis and encounter geometry. Linearizing the full equation about a fixed point givesFor $q=2$, $\pi$ is a center equilibrium and zero is a saddle equilibrium. For $q=1$, $\pi$ is always a center equilibrium, while zero is a saddle equilibrium ifAt sufficiently small orbital eccentricity the full truncated equations instead admit a center at zero as well; equality is a degenerate case requiring higher-order analysis. Thus an unconditional instability claim at zero does not follow from the full equation. In the usual fixed-eccentricity weak-resonance regime the inequality holds and the stable libration center is $\pi$ for either order.

The physical explanation is resonance protection. At equal mean longitudes, the conjunction direction relative to periapsis obeys $q(\Lambda -\varpi )=\varphi$ modulo $2\pi$. For $q=1$, $\varphi =0$ places conjunction near periapsis, where the exterior particle comes closest to the planet; $\varphi =\pi$ places it near apoapsis. For $q=2$, zero includes both apsidal conjunctions, one of them near periapsis, whereas $\pi$ places the two conjunction branches near quadrature. The protected arrangements give restoring kicks in the positive-$C_r$ leading-harmonic model. This geometric explanation is conditional on its non-crossing, small-orbital eccentricity approximation; it does not override the low-$e$ term retained in the full stability calculation.

Small resonant-argument librations. Define the nominal resonant semi-major axis and the displaced center byThis follows by expanding $n=n_r[1-3(a-a_r)/(2a_r)]$ and taking $\cos (\pi +\delta )=-1+O({\delta }^2)$. It gives the requested initial semi-major axis to first order in the precession-induced detuning; $e_0=e(0)$ and $\delta =\Delta \varphi$.

For the pendulum approximation of a mean-motion resonance, also require fractional changes in $e$ small across a libration, and precession terms small compared with $n$. In particular $C_r/(n_r{e}_0^3)\ll 1$ for $q=1$ makes the extra full-equation curvature negligible; for $q=2$, small fractional eccentricity changes require $C_r/(n_r{e}_0^2)\ll 1$. Freeze $e=e_0$ in the leading restoring coefficient and setThen $u=\varphi -\pi$ obeys simple harmonic motion. Using $u(0)=\delta$, $\dot{u}(0)=0$ and integrating the remaining Lagrange planetary equations givesThe omitted apsidal modulation comes from higher perturbative orders and $u^2$. All expressions are leading resonant approximations, valid while the omitted fractional changes remain small. The small-libration phase portrait iswith period $2\pi /{\omega }_0$. The semi-major axis half-width is $2a_r{\omega }_0|\delta |/(3p{n}_{\mathrm{pl}})$ and the orbital eccentricity half-width is $q{C}_r{e}_0^{q-1}|\delta |/{\omega }_0$.

Dividing the original $\dot{e}$ and $\dot{a}$ equations gives the exact invariant of their retained terms,Thus the $e$–$a$ plot is a segment of this increasing curve traversed back and forth, not a closed ellipse. Locally its slope is $q/[2(p+q)e_0{a}_c]$.

Finite amplitude and the separatrix. Keep the same weak-resonance approximation but allow $\delta$ to be finite. The leading angular equation is $\ddot{\varphi }={\omega }_0^2\sin \varphi$, or $\ddot{u}=-{\omega }_0^2\sin u$. Multiplication by $\dot{u}$ givesThis resonant pendulum energy is constant in the pendulum approximation of a mean-motion resonance. It is not an exact integral of the earlier full equation when $e$ and its restoring coefficient vary: the $\dot{\varphi }\sin \varphi$ term alone prevents that conclusion in general.