The Solar System at a Glance
The solar system formed ~4.6 billion years ago from a collapsing molecular cloud. Conservation of angular momentum caused the cloud to flatten into a rotating disk — the solar nebula — from which the planets accreted. The result is a remarkably flat system: all planetary orbital planes lie within a few degrees of the ecliptic plane (Earth's orbital plane).
Mercury
a = 0.387 AU · T = 87.97 d · e = 0.206 · Most eccentric inner planet; extreme temperature swings (−180°C to +430°C)
Venus
a = 0.723 AU · T = 224.7 d · e = 0.007 · Nearly circular orbit; retrograde rotation; hottest planet (464°C surface)
Earth
a = 1.000 AU · T = 365.25 d · e = 0.017 · Orbital unit of measure; one Moon; liquid water
Mars
a = 1.524 AU · T = 686.97 d · e = 0.093 · Two moons (Phobos, Deimos); 6-month journey from Earth
Jupiter
a = 5.203 AU · T = 11.86 yr · e = 0.049 · 95 known moons; Great Red Spot storm; 318× Earth mass
Saturn
a = 9.537 AU · T = 29.46 yr · e = 0.057 · Iconic ring system; least dense planet (floats on water)
Uranus
a = 19.19 AU · T = 84.01 yr · e = 0.046 · 98° axial tilt — orbits on its side; ice giant
Neptune
a = 30.07 AU · T = 164.8 yr · e = 0.010 · Predicted mathematically before discovery; winds to 2100 km/h
Kepler's Three Laws
Johannes Kepler (1609–1619) derived three empirical laws from Tycho Brahe's observations. Newton later proved they follow from the inverse-square law F = GMm/r².
Kepler's First Law (Law of Ellipses):
Each planet orbits the Sun in an ellipse with the Sun at one focus.
r = a(1−e²)/(1 + e·cos ν) (polar equation, ν = true anomaly)
Kepler's Second Law (Law of Equal Areas):
The radius vector sweeps equal areas in equal times — a planet moves faster at perihelion (closest approach) and slower at aphelion (farthest point).
dA/dt = L/(2m) = constant (conserved angular momentum)
Kepler's Third Law (Law of Periods):
T² = (4π²/GM) · a³ (Newton's generalisation)
In solar units: T²/a³ = 1 when T is in years and a in AU (for all planets!)
Orbital Mechanics Data
| Planet | a (AU) | e | i (°) | T (years) | v_orb (km/s) | T²/a³ |
|---|
| Mercury | 0.387 | 0.206 | 7.0° | 0.241 | 47.4 | 1.000 |
| Venus | 0.723 | 0.007 | 3.4° | 0.615 | 35.0 | 1.000 |
| Earth | 1.000 | 0.017 | 0.0° | 1.000 | 29.8 | 1.000 |
| Mars | 1.524 | 0.093 | 1.9° | 1.881 | 24.1 | 1.000 |
| Jupiter | 5.203 | 0.049 | 1.3° | 11.86 | 13.1 | 1.000 |
| Saturn | 9.537 | 0.057 | 2.5° | 29.46 | 9.7 | 1.000 |
| Uranus | 19.19 | 0.046 | 0.8° | 84.01 | 6.8 | 1.000 |
| Neptune | 30.07 | 0.010 | 1.8° | 164.8 | 5.4 | 1.000 |
Orbital Mechanics: From Kepler Equation to Position
To find a planet’s position at time t, this simulation solves Kepler’s Equation:
Step 1 — Mean anomaly: M(t) = 2π · t / T (uniform angular speed)
Step 2 — Eccentric anomaly: Solve M = E − e·sin(E) via Newton-Raphson iteration
Step 3 — True anomaly: tan(ν/2) = √((1+e)/(1−e)) · tan(E/2)
Step 4 — Orbital radius: r = a(1−e·cos E)
Step 5 — 3D position: Rotate by argument of perihelion ω, inclination i, and longitude of ascending node Ω
Hohmann Transfer Orbits
The most fuel-efficient way to move between two circular orbits is a Hohmann transfer ellipse — an ellipse tangent to both orbits. For Earth→Mars:
a_transfer = (r_Earth + r_Mars)/2 = (1.0 + 1.524)/2 = 1.262 AU
T_transfer = (a_transfer)^(3/2) years = 1.262^1.5 = 1.416 years = 259 days (one way)
Launch window: Every 2+ years when Earth-Mars alignment is correct (synodic period 779.9 days)
Frequently Asked Questions
Why do inner planets orbit faster than outer planets?
Kepler’s Third Law — T² ∝ a³ — means period grows faster than distance. This follows from Newton’s gravity: closer planets feel stronger attraction and must move faster to avoid falling into the Sun. Mercury (0.387 AU) travels at 47 km/s; Neptune (30.07 AU) at just 5.4 km/s — a 9× speed difference for 78× the distance.
Why are orbits elliptical?
Newton proved that any F ∝ 1/r² attractive force produces conic-section orbits (Bertrand’s theorem). An ellipse results when total energy E = ½mv² − GMm/r < 0 (bound orbit). Eccentricity e = √(1 + 2EL²/(G²M²m³)) where L is angular momentum. Nearly circular orbits (e ≈ 0) arise when the angular momentum is nearly "optimal" for the energy level — all planets inherited near-optimal conditions from the protoplanetary disk.
What are Lagrange points?
Five positions in the Sun-Earth system where gravity + centrifugal force exactly cancel. L1 (~1.5 Mkm sunward) hosts SOHO and DSCOVR. L2 (~1.5 Mkm anti-sunward) hosts JWST and Gaia. L4 and L5 (60° ahead/behind Earth) are stable for objects — Earth has no Trojans, but Jupiter has ~10,000+ Trojan asteroids at its L4/L5.
How is this simulation computed?
Each planet’s position is computed every frame from its mean anomaly M = 2π·t/T. Kepler’s equation M = E − e·sin(E) is solved numerically (Newton-Raphson, 5 iterations, <10⁻¹⁰ rad accuracy). The eccentric anomaly E gives heliocentric 3D coordinates via the orbital elements (a, e, i, Ω, ω). Projection to canvas uses a simple perspective with adjustable tilt and rotation angles, controlled by mouse drag. The camera system uses 3D rotation matrices rather than full-scene 3D rendering.