Visualize all three Kepler laws on a single orbit animation. Vary eccentricity and semi-major axis to see the ellipse shape, constant areal velocity, and T² ∝ a³ relationship in one view. For Newton-Raphson numerical solving see the Kepler equation simulator; for satellite design see orbital mechanics.
Yellow: Sun (one focus). Blue: planet. The orange swept area illustrates Kepler's 2nd law (constant areal velocity).
1st law (ellipse law): every planetary orbit is an ellipse with the Sun at one focus. In polar form:
$$r = \frac{a(1-e^2)}{1 + e\cos\theta}$$$r$: distance from the Sun, $a$: semi-major axis, $e$: eccentricity, $\theta$: true anomaly. Perihelion: $r_p = a(1-e)$. Aphelion: $r_a = a(1+e)$.
2nd law (equal areas): the line from the Sun to the planet sweeps equal areas in equal times. From angular-momentum conservation:
$$\frac{dA}{dt} = \frac{1}{2}r^2\dot{\theta} = \frac{L}{2m} = \text{const}$$where $L = mr^2\dot{\theta}$. The areal velocity is $h = L/m = \sqrt{GMa(1-e^2)}$ ($G$: Newton constant, $M$: solar mass).
3rd law (harmonic law): the square of the period is proportional to the cube of the semi-major axis:
$$T^2 = \frac{4\pi^2}{GM}\, a^3$$In solar-system units ($T$ in years, $a$ in AU), this collapses to $T^2 = a^3$.
Satellite orbit design: GPS satellites at ≈20,200 km altitude (≈3.36 R_E) have a ≈12 h period; the ISS at ≈420 km has a ≈92 min period (≈16 orbits per day). The 3rd law gives both numbers immediately.
Hohmann transfers: a minimum-energy trajectory between two coplanar circular orbits. Earth-to-Mars takes ~259 days; this is a direct application of Kepler's 3rd law.
"The Sun sits at the centre of the ellipse" is wrong. The Sun is at one of the two foci. The centre-to-focus distance is $a e$. For Earth ($e \approx 0.017$) the Sun is nearly central, but for Mars ($e \approx 0.093$) the offset is clearly visible.
This simulator solves the two-body problem only. Real orbits suffer perturbations from other planets, and Mercury's perihelion precession requires general relativity for a complete description.
1st law (ellipse): $r = \dfrac{a(1-e^2)}{1+e\cos\theta}$
2nd law (equal areas): $\dfrac{dA}{dt} = \dfrac{L}{2m} = \text{const}$
3rd law: $T^2 \propto a^3$ (in AU and years, $T^2 = a^3$).
Industry: communications and weather satellites are placed in geostationary orbit by transferring from a low Earth orbit through an intermediate ellipse and circularising at apogee — Kepler's laws size every leg of that manoeuvre.
Education and research: introductory aerospace and astronomy courses use this kind of simulator to make speed variation along the ellipse tangible. Students compare the simulator's $T$ and apsidal speeds against the analytic results.
CAE workflow: dedicated tools such as STK or GMAT propagate orbits with high precision; this simulator is a sketch-level tool for understanding the geometry before committing to a full numerical propagation.
Mars orbit: semi-major axis a=1.524 AU, eccentricity e=0.093. Simulator yields orbital period T≈1.88 years (matches observed value). Perihelion distance r_p=1.381 AU gives velocity v_p≈24.1 km/s; aphelion r_a=1.666 AU yields v_a≈21.9 km/s. Areal velocity remains constant at 0.0253 AU²/year throughout orbit, confirming Kepler's second law (equal areas swept in equal times).