Polar Equations of Conics & Celestial Orbital Geometry

1. Unified Focus-Directrix Derivation

A remarkable triumph of analytic geometry is that all non-degenerate conic sections (circles, ellipses, parabolas, and hyperbolas) can be described by a single unified equation in polar coordinates when one focus is chosen as the pole $O$.

Let the pole $O$ be the focus of the conic, and let the polar axis be chosen along the axis of symmetry, perpendicular to directrix $D$. Let the directrix $D$ be located at a distance $d$ to the left of the pole, so its Cartesian equation is $x = -d$, or in polar coordinates: $$r \cos(\pi - \theta) = d \iff -r \cos \theta = d \iff r \cos \theta = -d$$ For any point $P(r, \theta)$ on the conic, the distance to the focus is $SP = r$. The perpendicular distance from $P$ to the directrix is: $$PM = d + r \cos \theta$$ By the universal conic definition $SP = e \cdot PM$: $$r = e(d + r \cos \theta) = ed + er \cos \theta$$ $$r(1 - e \cos \theta) = ed \iff \frac{ed}{r} = 1 - e \cos \theta$$ Defining the semi-latus rectum $l \equiv ed$ (the value of $r$ when $\theta = \pi/2$): $$\mathbf{\frac{l}{r} = 1 - e \cos \theta}$$ If the directrix is chosen to the right ($x = +d$), the equation becomes: $$\mathbf{\frac{l}{r} = 1 + e \cos \theta}$$ If the axis of the conic is tilted by an angle $\alpha$ relative to the polar axis: $$\mathbf{\frac{l}{r} = 1 + e \cos(\theta - \alpha)}$$

1. Conic Morphing via Eccentricity $e$

In the universal polar equation $\frac{l}{r} = 1 + e \cos \theta$, the geometric nature of the curve is determined purely by the parameter $e$:

• Circle ($e = 0$): $$\frac{l}{r} = 1 \iff r = l$$ The radial distance is constant for all $\theta$, representing a circle of radius $l$ centered at the pole.
• Ellipse ($0 < e < 1$): Because $e < 1$, the denominator $1 + e \cos \theta > 0$ for all $\theta \in [0, 2\pi)$. The curve is closed and bounded: $$\text{Periapsis (closest approach): } \theta = 0 \implies r_{\min} = \frac{l}{1 + e}$$ $$\text{Apoapsis (furthest distance): } \theta = \pi \implies r_{\max} = \frac{l}{1 - e}$$ The major axis length is $2a = r_{\min} + r_{\max} = \frac{l}{1+e} + \frac{l}{1-e} = \frac{2l}{1 - e^2} \implies l = a(1 - e^2)$.
• Parabola ($e = 1$): $$\frac{l}{r} = 1 + \cos \theta = 2 \cos^2(\theta/2) \implies r = \frac{l}{2}\sec^2(\theta/2)$$ As $\theta \to \pm \pi$, $r \to \infty$. The curve is open, escaping to infinity along a single direction.
• Hyperbola ($e > 1$): The denominator $1 + e \cos \theta$ vanishes when $\cos \theta = -1/e$. The directions $\theta_0 = \pm \arccos(-1/e)$ define the directions of the asymptotes! The curve splits into two branches extending to infinity.

1. Equation of the Chord Joining Two Points

Let $P(\alpha - \beta)$ and $Q(\alpha + \beta)$ be two points on the conic $\frac{l}{r} = 1 + e \cos \theta$. The straight line passing through both points is: $$\mathbf{\frac{l}{r} = e \cos \theta + \sec \beta \cos(\theta - \alpha)}$$ Notice: When $\theta = \alpha - \beta$: $\frac{l}{r} = e\cos(\alpha - \beta) + \sec\beta\cos(-\beta) = e\cos(\alpha - \beta) + 1$, which satisfies the conic equation! When $\theta = \alpha + \beta$: $\frac{l}{r} = e\cos(\alpha + \beta) + \sec\beta\cos(\beta) = e\cos(\alpha + \beta) + 1$, which also satisfies the conic equation!

2. Equation of the Tangent Line

Taking the limit as $\beta \to 0$, the points $P$ and $Q$ coalesce at $\theta = \alpha$. Since $\sec(0) = 1$, the equation of the tangent to the conic at $\theta = \alpha$ is: $$\mathbf{\frac{l}{r} = e \cos \theta + \cos(\theta - \alpha)}$$ This is the universally famous polar tangent formula!

3. Perpendicular Focal Chords Theorem

Let $PSQ$ be a focal chord passing through the pole. The extremities are at $\theta = \alpha$ and $\theta = \alpha + \pi$. $$SP = r_1 = \frac{l}{1 + e \cos \alpha}, \qquad SQ = r_2 = \frac{l}{1 + e \cos(\alpha + \pi)} = \frac{l}{1 - e \cos \alpha}$$ Adding their reciprocals: $$\frac{1}{SP} + \frac{1}{SQ} = \frac{1 + e \cos \alpha}{l} + \frac{1 - e \cos \alpha}{l} = \mathbf{\frac{2}{l} = \text{Constant Everywhere!}}$$ Theorem: The semi-latus rectum $l$ is the harmonic mean of the segments of any focal chord in any conic!

1. Confocal Conics

A family of conics having the same foci is termed confocal. In Cartesian coordinates, the confocal family through foci $(\pm c, 0)$ is: $$\frac{x^2}{a^2 + \lambda} + \frac{y^2}{b^2 + \lambda} = 1 \quad (\lambda \in \mathbb{R})$$ In polar coordinates with a common focus at the pole, the confocal family with a common axis is: $$\frac{l}{r} = 1 + e \cos \theta$$ where $l$ and $e$ vary such that the second focus $S'$ is fixed.

2. Orthogonal Intersection Theorem

Fundamental Theorem of Confocal Conics: Through any point $P(x_0, y_0)$ in the plane (not on the axes), there pass exactly two conics of a confocal family: one ellipse and one hyperbola. Furthermore, these two conics intersect each other at strictly right angles ($90^\circ$) at point $P$! This property is the foundation of elliptic coordinate systems used to solve Laplace's and Helmholtz's equations in mathematical physics.

1. Kepler's First Law and Newton's Gravitational Potential

In celestial mechanics and orbital astrophysics, a satellite or planet orbiting a central gravitational mass $M$ under Newton's inverse-square gravitational force $\mathbf{F} = -\frac{G M m}{r^2}\hat{\mathbf{r}}$ obeys the Binet differential equation: $$\frac{d^2 u}{d\theta^2} + u = \frac{G M}{h^2} = \frac{\mu}{h^2}$$ where $u = 1/r$, $\mu = GM$ is the gravitational parameter, and $h = r^2 \dot{\theta}$ is the specific angular momentum. The exact general solution of this linear differential equation is: $$u = \frac{\mu}{h^2}(1 + e \cos(\theta - \omega)) \iff \mathbf{r(\theta) = \frac{p}{1 + e \cos(\theta - \omega)}}$$ where $p = h^2/\mu$ is the semi-latus rectum and $e$ is the orbital eccentricity!

2. The Energy-Eccentricity Correspondence (Vis-Viva Equation)

The total specific orbital energy $\mathcal{E} = \frac{1}{2}v^2 - \frac{\mu}{r}$ is conserved along the trajectory: $$\mathcal{E} = -\frac{\mu^2(1 - e^2)}{2h^2} = -\frac{\mu}{2a}$$ The sign of the orbital energy determines the conic geometry:

• Bound Elliptical Orbit ($\mathcal{E} < 0, \; 0 \le e < 1$): Periodic planetary orbits (Keplerian orbits).
• Parabolic Escape Trajectory ($\mathcal{E} = 0, \; e = 1$): Critical escape velocity $v_{\text{esc}} = \sqrt{2\mu/r}$. The spacecraft possesses just enough energy to escape to infinity with zero residual velocity.
• Hyperbolic Flyby / Interstellar Trajectory ($\mathcal{E} > 0, \; e > 1$): Gravity-assist planetary flybys and interstellar objects (e.g., 1I/'Oumuamua and 2I/Borisov). The spacecraft escapes to infinity with hyperbolic excess speed $v_\infty = \sqrt{2\mathcal{E}} = \sqrt{\mu/a}$.