Hyperbolic Orbits

Hyperbolic orbit analysis and results

Placeholder — this section is being built more slowly, per the storyboard note. It will summarize the hyperbolic derivation (cos(Φ_c) = 1/e) and the geometric contrast with the elliptical case below.

The Hyperbolic Bridge Relation

Place the cone apex C at the origin with its axis along z, and measure the half-angle \(\Phi_c\) from the axis to the generator. A point on the cone at height z lies at horizontal distance \(z\tan\Phi_c\) from the axis:

\[ x^{2} + y^{2} = z^{2}\tan^{2}\Phi_c \]

(H-1)

Cut the cone with the plane \(x = \rho\), parallel to the axis. Substituting into Eq. H-1 and dividing through gives the hyperbola in standard form:

\[ \frac{z^{2}}{(\rho\cot\Phi_c)^{2}} - \frac{y^{2}}{\rho^{2}} = 1, \qquad a = \rho\cot\Phi_c, \quad b = \rho \]

(H-2)

The center O of the hyperbola is the point \((\rho, 0, 0)\), directly across from the apex, so the orbit's transverse coordinate measured from O is the height z itself. The vertex lies on the generator at height a and horizontal distance ρ, so \(\cot\Phi_c = a/\rho\). With \(c^{2} = a^{2} + b^{2} = \rho^{2}\csc^{2}\Phi_c\), the eccentricity is \(e = c/a = \sec\Phi_c\):

\[ \cos\Phi_c = \frac{1}{e} \]

(H-3)

matching Table F-1. The plane offset then follows from \(\rho = a\tan\Phi_c\) and \(p = a(e^{2}-1)\):

\[ \rho = a\sqrt{e^{2}-1} = \sqrt{ap} = b \]

(H-4)

The offset of the cutting plane from the cone axis is exactly the semi-conjugate axis b.

Hyperbolic orbits in edge view with equal total energy: generators fanning from the apex at different half-angles, a dashed line at height a, and vertical cutting planes rising from where each generator meets that line, with each orbit drawn rotated into the page — Fig. H-1 — Hyperbolic orbits in edge view with equal total energy, all sharing the same a. Each cutting plane lies a distance \(\rho = a\tan\Phi_c = a\sqrt{e^{2}-1}\) from the axis, and each orbit is drawn rotated into the page from its vertex at height a. (Preliminary sketch, to be redrawn in site style.)

Fig. H-1 is the hyperbolic counterpart of the elliptical equal-energy family (Fig. E-4). Holding a fixed holds the energy \(+\mu/2a\) fixed, and the eccentricity then sets the half-angle through Eq. H-3 and the plane offset through Eq. H-4. Low-eccentricity orbits lie on steep generators with planes close to the axis; as \(e \to 1\), \(\Phi_c \to 0\) and the plane approaches the axis itself. As \(e\) grows, the generator flattens and the plane moves outward.

Hyperbola in its plane: center O, foci, point P, focal distances r and r-prime, and r-double-prime from O to P — Fig. H-2 — Hyperbolic orbit in its plane, with center O, focal distances r and r′, and r″ = OP. (Placeholder, to be drawn in site style.)

Slant distance. For a point P on the orbit, the slant distance from the apex is \(R^{2} = x^{2} + y^{2} + z^{2}\). On the cone, Eq. H-1 turns this into \(R^{2} = z^{2}\sec^{2}\Phi_c\), and by Eq. H-3:

\[ R = e\,z \]

(H-5)

The eccentricity is the secant of the half-angle, converting axial height into slant distance. On the near branch the focal distance is \(r = ez - a\), so

\[ R = r + a \]

(H-6)

In-plane view. The perpendicular from C meets the orbital plane at O, so with \(r'' = OP\),

\[ R^{2} = \rho^{2} + r''^{2}, \qquad r''^{2} = (r + a)^{2} - b^{2} \]

(H-7)

and with \(\rho = b\) from Eq. H-4, Eq. H-7 collapses to Eq. H-6.

General form. The focus F lies a distance \(c = ae\) from O. With the true anomaly \(\theta\) measured from periapsis, the periapsis direction from F points toward O, and the law of cosines gives \(r''^{2} = c^{2} + r^{2} - 2cr\cos\theta\). The orbit equation gives \(e\,r\cos\theta = p - r\), so \(c\,r\cos\theta = a(p - r)\), and

\[ R^{2} = r^{2} + 2a(r - p) + CF^{2}, \qquad CF^{2} = \rho^{2} + c^{2} = a^{2} + 2ap \]

(H-8)

where CF is the distance from the apex to the focus. Expanding confirms that Eq. H-8 is identical to \((r + a)^{2}\). This is the general form carried in earlier framework notes, now derived for the hyperbolic case.

Comparison with the ellipse. In the elliptical construction the perpendicular from the apex lands on the focus itself, so there is no in-plane offset and the cross term vanishes, giving Eq. E-9, \(R^{2} = r^{2} + ap\). In the hyperbolic construction it lands on the center, offset from the focus by c. Both are instances of

\[ R^{2} = r^{2} + CF^{2} - 2\,d\,r\cos\theta \]

(H-9)

where d is the in-plane distance from the foot of the apex perpendicular to F: \(d = 0\) for the ellipse and \(d = c\) for the hyperbola.

Elliptical-to-hyperbolic transition

Conic sections — elliptical, parabolic, and hyperbolic — take on a renewed structure in the conical reference frame. The transition from one type to another becomes smooth, perhaps continuous.

Elliptical orbits occupy a fixed 45° half-angle cone; hyperbolic orbits occupy cones with a varying half-angle greater than 45°.

Diagram showing eccentricity families e1 and e2 traced across the elliptical and hyperbolic regions

Figure captions here are carried over directly from the storyboard slide and still need to be reconciled with which image supports which claim before this becomes final.