Newtonian Parabolic and Hyperbolic Encounters

An unbound Kepler encounter is most naturally specified far from the attracting mass by a relative speed $v_\infty$ and impact parameter $b$. Conservation laws turn those asymptotic data into the hyperbola’s eccentricity, closest approach, and deflection angle. Gravity focuses trajectories, but conservative point-mass gravity alone cannot permanently capture them.

Required background. Classical and Variational Mechanics provides action principles, canonical variables, and mechanical conservation laws; Newtonian Gravity provides the gravitational potential, Poisson equation, and inverse-square force law. Helpful background. Calculus and Linear Algebra provides multivariable differentiation, linear maps, inner products, and basis changes; Ordinary Differential Equations provides initial-value problems, linearization, and asymptotic solution methods.

Hyperbolic geometry from conserved data

Write $\mathcal G=G(m_1+m_2)$ and work per unit reduced mass. For a nonradial encounter, take $v_\infty\gt0$ and $b\gt0$, so

$$\varepsilon=\frac{v_\infty^2}{2}, \qquad h=bv_\infty.$$

Within this nonradial sector, $\varepsilon\gt0$ proves that the Kepler orbit is hyperbolic.

The eccentricity relation gives

$$e=\sqrt{1+\frac{b^2v_\infty^4}{\mathcal G^2}}\gt1.$$

At infinity, the polar orbit $r=p/(1+e\cos\nu)$ approaches its asymptotes where $1+e\cos\nu_\infty=0$. Comparing the incoming and outgoing directions yields the deflection magnitude

$$\chi =2\arctan\!\left(\frac{\mathcal G}{bv_\infty^2}\right), \qquad 0\lt\chi\lt\pi.$$

Weak encounters have $bv_\infty^2\gg\mathcal G$ and $\chi\simeq2\mathcal G/(bv_\infty^2)$. Slow or close encounters bend more strongly. The sign of the transverse deflection depends on the chosen orientation; $\chi$ above is its positive magnitude.

The head-on value $b=0$ is not an ordinary hyperbola. It has $h=0$ and the degenerate conic value $e=1$ despite $\varepsilon\gt0$; in the ideal point-mass problem the incoming trajectory reaches the singular center rather than scattering smoothly onto an outgoing asymptote. A collision scale or other short-distance model is needed before assigning that limit a physical outcome.

The periapsis follows either from $r_{\rm p}=p/(1+e)$ or energy conservation:

$$r_{\rm p} =\frac{\mathcal G}{v_\infty^2} \left[ \sqrt{1+\left(\frac{bv_\infty^2}{\mathcal G}\right)^2}-1 \right].$$

Solving instead for $b$ gives the gravitational-focusing relation

$$b^2=r_{\rm p}^2 \left(1+\frac{2\mathcal G}{r_{\rm p}v_\infty^2}\right).$$

The first term is geometric; the second is the enhancement caused by gravitational acceleration before closest approach.

Encounter cross sections and “capture”

Suppose a physical process occurs whenever $r_{\rm p}\le R$: collision, tidal disruption, entry into a dissipative envelope, or passage inside a model-dependent strong-field radius. The maximum impact parameter is

$$b_{\max}^2=R^2 \left(1+\frac{2\mathcal G}{Rv_\infty^2}\right),$$

so a uniform incident beam has encounter cross section

$$\sigma=\pi b_{\max}^2.$$

This is an encounter threshold, not by itself a capture threshold. In the conservative Kepler problem, $\varepsilon$ is unchanged and a trajectory arriving with $\varepsilon\gt0$ must leave with $\varepsilon\gt0$. Permanent binding requires energy loss, mass exchange, a third body, radiation, or an absorbing boundary. Calling a close hyperbola “captured” without naming such a mechanism violates energy conservation.

The parabolic boundary

The limit $v_\infty\to0$ at fixed periapsis has $\varepsilon=0$ and $e=1$. It separates bound ellipses from unbound hyperbolas. From $p=r_{\rm p}(1+e)$ and $h^2=\mathcal G p$,

$$h^2=2\mathcal G r_{\rm p}, \qquad v_{\rm p}=\sqrt{\frac{2\mathcal G}{r_{\rm p}}}.$$

The periapsis speed is the local escape speed. There is no finite nonzero asymptotic speed and therefore no finite impact parameter defined by $h=bv_\infty$; the parabolic case must be taken with care rather than obtained by substituting $v_\infty=0$ into every hyperbolic formula.

Worked example and checks

Let $R$ be the sum of two physical radii. If $2\mathcal G/(Rv_\infty^2)=9$, then

$$b_{\max}=\sqrt{10}\,R, \qquad \sigma=10\pi R^2.$$

Gravity has increased the geometric collision cross section by a factor of ten. Dimensional checks are immediate: $\mathcal G/v_\infty^2$ is a length, the argument of the arctangent is dimensionless, and $\sigma$ has units of area.

Two limits also diagnose sign errors. As $G\to0$, $r_{\rm p}\to b$ and $\chi\to0$. As $b\to\infty$ at fixed $v_\infty$, the same straight-line limit is recovered. Newtonian point-particle formulas become unreliable when $r_{\rm p}$ approaches physical radii, relativistic length scales, or a domain where tides and dissipation are important.

The hyperbolic geometry and deflection-angle calculation are developed in Tong 2013, §§4.3.1 and 4.4–4.4.1, pp. 56–66, PDF and, for attractive gravitational encounters, in Peraire and Widnall 2008, Lecture L16, Eqs. (11)–(15), pp. 4–6, PDF.

Transfer task

Show that the hyperbolic periapsis speed satisfies

$$v_{\rm p}^2=v_\infty^2+\frac{2\mathcal G}{r_{\rm p}}.$$

Solution

Specific energy is conserved. At infinity $\varepsilon=v_\infty^2/2$; at periapsis it is $v_{\rm p}^2/2-\mathcal G/r_{\rm p}$. Equating the two expressions and multiplying by two gives the result. It is the Newtonian vis-viva equation specialized to an unbound encounter.

Continue to Perturbed Orbits and Apsidal Precession to learn how weak non-Keplerian effects accumulate from one passage or revolution to the next.

References

• Peraire, Jaime, and Sheila Widnall. “Central Force Motion: Orbits.” MIT OpenCourseWare 16.07 Dynamics, Lecture L16, Fall 2008, version 1.1. Eqs. (11)–(15), pp. 4–6, PDF.
• Tong, David. Dynamics and Relativity. University of Cambridge lecture notes, Lent 2013. §§4.3.1 and 4.4–4.4.1, pp. 56–66, PDF.

Further reading

• NASA Science, “Chapter 3: Gravity and Mechanics”, sections on Keplerian and hyperbolic motion.