Orbits from Newton's Law · Deriving the Ellipse

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1 · The solved equation for every orbit

The derivation above ends at one master formula — the polar equation of a conic with the Sun at a focus. A single number, the eccentricity $e$ fixed by the launch energy, decides which of the three shapes you get. Here are the three closed-form solutions, each exact:

Ellipse — a planet

0 ≤ e < 1 · bound · E < 0

$$r(\theta)=\frac{a(1-e^{2})}{1+e\cos\theta}$$

Closes on itself. Semi-minor $b=a\sqrt{1-e^{2}}$; the Sun sits at a focus, distance $ae$ from centre. Period $T=2\pi\sqrt{a^{3}/GM}$. A circle is the special case $e=0$.

Parabola — a one-way comet

e = 1 · marginal escape · E = 0

$$r(\theta)=\frac{p}{1+\cos\theta}$$

The threshold case: the body reaches infinity with exactly zero speed left. Speed everywhere equals the local escape speed $v=\sqrt{2GM/r}$. Never returns.

Hyperbola — a flyby

e > 1 · unbound · E > 0

$$r(\theta)=\frac{a(e^{2}-1)}{1+e\cos\theta}$$

An interstellar visitor like 1I/ʻOumuamua. Swings once past the Sun and leaves along an asymptote at angle $\theta_\infty=\arccos(-1/e)$, keeping excess speed $v_\infty=\sqrt{GM/|a|}$.

$$e=\sqrt{1+\frac{2E\,h^{2}}{G^{2}M^{2}}},\qquad E=\frac{v^{2}}{2}-\frac{GM}{r},\qquad h=r^{2}\dot\theta=|\mathbf r\times\mathbf v|$$

one launch, two conserved numbers (energy E and angular momentum h) → the eccentricity → the shape

2 · The same law, at every level

One equation of motion, four readings. A middle-schooler sees why a faster throw makes a longer orbit; a graduate student sees a symplectic integrator conserving the Hamiltonian across a chaotic N-body flow. None of them is being told a fib — they are watching the same $Gm_1m_2/r^2$.

Amiddle school

Ellipses from one force. Throw a ball sideways hard enough and it never lands — it falls around the Earth. Slow → a tight circle; faster → a stretched ellipse; too fast → it escapes. The Sun sits at a focus, never the middle.

Bhigh school

Two bodies, both moving. The Sun isn't nailed down — it too is pulled. Two masses swing about a shared balance point, the barycentre, tracing twin ellipses. This wobble is how we discover exoplanets.

Ccollege

The three-body problem. Add one more mass and the tidy formulas die (Poincaré, 1890). Most triples are chaotic, yet exact periodic solutions exist: Lagrange's rotating triangle and the figure-eight, three equal masses on one shared curve.

Dgraduate

The N-body cluster. Scale the same pairwise sum to a star and its planets, or a hundred stars collapsing into a cluster. No orbit is a clean ellipse, yet energy, momentum, and angular momentum stay conserved to the integrator's precision.

3 · Verification computed live

Each row is integrated by this page at load and checked against the closed-form result Newton's law implies. The values come from the same engine that drives the animation, and were reproduced offline in Python to the precision shown.

Test	this page	reference	status
Circular-orbit speed $v_{\rm circ}=\sqrt{GM/r}$	…	exact	…
Kepler III $T^2GM/4\pi^2a^3$ over 3 eccentricities	…	= 1	…
Vis-viva $v^2$ vs $GM(2/r-1/a)$ at peri/apo	…	agree	…
Energy conservation, Verlet, one bound orbit	…	drift → 0	…
Angular momentum (Kepler II, equal areas)	…	drift → 0	…
Two-body barycentre stays fixed (momentum)	…	= 0	…
Figure-eight returns to start after period T	…	closes on itself	…

4 · Boundary of validity mandatory

What the picture is allowed to claim — and where it stops

Honesty tier A (direct): the derivation is exact algebra, and every trajectory is the Newtonian equation of motion integrated with velocity-Verlet, whose energy error stays bounded. Scenes A and B use zero softening, so Kepler's three laws are reproduced to the precision in §3.

Exact for one- and two-body

A single body about a fixed mass, and two bodies about their barycentre, are the complete Kepler problem: conic orbits, Kepler I–III, and vis-viva all hold exactly and are verified in §3.

valid · closed-form Kepler problem

Point masses, no size

Every body is an idealised point — no radii, tides, spin, or physical collisions; bodies pass through one another rather than merging.

idealisation · point particles

Newton, not Einstein

Pure inverse-square gravity. GR's corrections — Mercury's 43″/century precession, light bending — are absent. Valid for weak fields and $v\ll c$.

scope · non-relativistic

Softening in the N-body scene

Scenes C and D add a small $\varepsilon$ so $r^3\to(r^2+\varepsilon^2)^{3/2}$ to keep close encounters stable. Scenes A and B and the derivation use $\varepsilon=0$.

numerical · ε > 0 in C/D

Chaos limits prediction, not conservation

For three or more bodies the flow is chaotic: two runs a hair apart diverge exponentially. The conserved quantities stay trustworthy; a specific far-future path does not.

caveat · sensitive dependence

No dissipation

No gas drag, radiation, or gravitational-wave emission. Orbits here neither decay nor circularise.

omitted · non-gravitational forces

Bottom line. Believe the derivation, the ellipses, the barycentre wobble, the figure-eight, and the conserved quantities completely. Read the chaotic three- and N-body paths as faithful examples, not thousand-orbit predictions, and treat relativity, finite size, and dissipation as the deliberately-omitted next layers.

References

Newton 1687. Philosophiæ Naturalis Principia Mathematica. — universal gravitation; Kepler's laws derived from it.
Kepler 1609 / 1619. Astronomia Nova & Harmonices Mundi. — the three empirical laws.
Binet (early 1800s). The Binet equation $u''+u=GM/h^2$ — the substitution $u=1/r$ used in the derivation.
Poincaré 1890. Sur le problème des trois corps. Acta Math. 13. — no general closed form; the birth of chaos.
Chenciner & Montgomery 2000. A remarkable periodic solution of the three-body problem. Ann. Math. 152, 881. JSTOR — the figure-eight.
Moore 1993. Braids in classical gravity. Phys. Rev. Lett. 70, 3675. — the figure-eight initial data used here.
Hairer, Lubich & Wanner 2006. Geometric Numerical Integration. Springer. — why velocity-Verlet conserves energy on average.
Murray & Dermott 1999. Solar System Dynamics. Cambridge. — vis-viva, orbital elements, the two-body reduction.

Orbits from Newton's Law of Gravitation

Watch the ellipse fall out of the force law animated

Give it a sideways push and it orbits forever

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