Kepler's Laws — Orbital Simulator

Two-body Newtonian elliptical-orbit simulator with Kepler-equation solver

Astronomical presets

Binary mode

Uses μ = G(M+m) and enables barycentric visualization. Required when masses are comparable (stellar binaries).

Central mass

Body at the focus. Sun, Earth (for satellites), or primary star.

5.028e-950.28

Semi-major axis

Average of periapsis and apoapsis. Sets the size and period of the orbit.

6.685e-9668.5

Eccentricity

0 = circular orbit; near 1 = highly elongated ellipse (comets).

—

00.99

Argument of periapsis

In-plane orientation of the ellipse. Does not affect the invariants; only rotates the visual geometry.

0360

Newtonian two-point-body model: no third-body perturbations, no general relativity, no radiation pressure or atmospheric drag. Orbits are strictly planar.

Orbital map

0.0%

Speed

Live state

0.9833AU

|v|

30.29km/s

0°

dA/dt

22280010¹⁰ m²/s

g(r)

0.006135m/s²

Orbital invariants

Period T

0.9999

years

365.2 days

Specific energy ε

-443.7

MJ/kg

Specific angular momentum h

4.456e+6

10⁹ m²/s

Orbit geometry

Periapsis r_p

0.9833

Apoapsis r_a

1.017

Semi-minor axis b

0.9999

Semi-latus rectum p

0.9997

Extreme speeds

Maximum speed v_max

30.29

km/s

Minimum speed v_min

29.3

km/s

Fundamentals & Explanation

The Three Empirical Laws and Their Newtonian Origin

Between 1609 and 1619, Johannes Kepler derived three regularities of planetary motion from Tycho Brahe's high-precision Mars observations, marking the first formal break with the Aristotelian-Ptolemaic geocentric model of perfect circular orbits. Sixty years later, Newton proved all three are an exact consequence of the two-body problem under universal gravity F ∝ Mm/r². Today they are read as a manifestation of deep symmetries (Noether, 1918): rotational isotropy → conservation of angular momentum → 2nd law.

1st law (orbits): planets travel on ellipses with the central body at one focus.
2nd law (areas): the radius vector sweeps equal areas in equal times.
3rd law (harmonic): the square of the period is proportional to the cube of the semi-major axis.

Historical Note: Kepler stated the 2nd and 1st laws first (1609, Astronomia Nova) and the 3rd a decade later (1619, Harmonices Mundi). He worked without knowing the Sun's mass nor the constant G — those came with Newton (1687) and Cavendish (1798). The form T² = (4π²/GM)·a³ is therefore Newtonian, not strictly Keplerian.

Geometry of the Ellipse and Kepler's Equation

An ellipse is the locus of points whose summed distance to two foci is constant; the central body sits at onefocus while the other stays empty. The body's position in time is parametrised by three related angles — the mean (M), eccentric (E) and true (ν) anomalies — linked by the transcendental Kepler equation.

Geometry (a, e)

b = a\sqrt{1-e^2}, \; r_p = a(1{-}e), \; r_a = a(1{+}e)

Kepler's Equation

M = E - e\,\sin E

No closed-form algebraic solution exists. The simulator solves it via Newton-Raphson with a tolerance of 10⁻¹², seeded with Danby's starter for quartic convergence (e < 0.95):
E₀ = M + e·sin(M)·(1 + e·cos(M))

Once E is known, the body's coordinates and the true anomaly follow analytically: x = a(cosE − e), y = a√(1−e²)·sinE, tan(ν/2) = √((1+e)/(1−e))·tan(E/2).

Third Law: From Kepler to Newton

Kepler stated T² ∝ a³ empirically. Newton derived the exact coefficient from his law of gravitation, showing it depends on the sum of masses:

T^2 = \frac{4\pi^2}{G(M_1+M_2)}\,a^3

In the reduced case (m ≪ M), the (M+m) factor collapses to M and the law reduces to its Keplerian form. For binary systems with comparable masses (stellar binaries, Pluto-Charon), the M+m factor is essential — the simulator exposes it via toggle. When enabled, both bodies are drawn orbiting their common barycentre.

System	m/M	Importance of (M+m)
Sun – Earth	3.0 × 10⁻⁶	Negligible. Error in T < 10⁻⁶.
Sun – Jupiter	9.5 × 10⁻⁴	Small but measurable: ~0.05% in T.
Pluto – Charon	0.122	Relevant. The barycentre lies outside Pluto.
Sirius A – Sirius B	0.49	Indispensable. Both stars orbit the common barycentre.

Conserved Invariants and Vis-Viva

The orbit is fully determined by two conserved quantities: the specific mechanical energy (ε) and the specific angular momentum (h), both functions of the geometric parameters (a, e) alone, where μ = G(M+m):

Energy (ε)

\varepsilon = -\frac{\mu}{2a}

Ang. Momentum (h)

h = \sqrt{\mu\,a(1-e^2)}

Conservation of h directly implies the 2nd law: dA/dt = h/2 = const. Combining both invariants yields the vis-viva equation, the central formula of astrodynamics:

v(r) = \sqrt{\mu\left(\tfrac{2}{r} - \tfrac{1}{a}\right)}

Yields v_max at periapsis and v_min at apoapsis; the ratio v_max/v_min = (1+e)/(1−e). Halley ranges from 54 km/s at perihelion to 0.9 km/s at aphelion — a 60× factor.

Physical Limits of the Model

Relativistic Limit (v→c, r→r_s)

Newtonian gravity is exact only when v ≪ c and r ≫ r_s = 2GM/c². Mercury precesses 43"/century beyond the Keplerian prediction — Einstein explained it with general relativity (1915). The simulator warns when v_max > 0.1c or when the periapsis falls below 100·r_s.

N-Body Limit

Only two point masses are modelled. Third-body perturbations (Jupiter on the inner planets), quadrupole effects (Earth's J₂ disturbs low satellites), radiation pressure and atmospheric drag require numerical integrators such as those used by JPL Horizons. The discovery of Neptune (Le Verrier, 1846) was made precisely by analysing such residual perturbations on Uranus.

Closed and Planar Orbits Only

Limited to e ∈ [0, 0.99]. Parabolic (e=1) and hyperbolic (e>1) trajectories — typical of probe flybys or non-recurring comets — require the hyperbolic equation M = e·sinh(F) − F. Inclination (i) and ascending-node longitude (Ω) are not modelled either; orbits are projected onto the simulator plane.

Limit Cases (e)

e = 0: Circle. v constant, r = a, dA/dt trivially uniform.

e → 1: Degenerate ellipse. v_max/v_min grows unbounded; Halley with e=0.967 reaches ~60×.

Numerical Validation

The 2nd law requires dA/dt = h/2 = const at every instant. The solver maintains this to ~10⁻¹⁰ relative. Presets calibrated against JPL Horizons ephemerides (DE441/DE440).

Constant	Value	Source
G	6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻²	CODATA 2022
M_☉	1.989 × 10³⁰ kg	IAU 2015 B3
1 AU (exact)	149 597 870 700 m	IAU 2012 B2
M_⊕	5.9722 × 10²⁴ kg	NASA NSSDCA

Academic References

[1] Kepler, J. (1609). Astronomia Nova; (1619). Harmonices Mundi, Liber V. (Original statement of the three laws).
[2] Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica, Book I, Prop. XI–XV. (Derivation from F ∝ 1/r²).
[3] Vallado, D. A. (2013). Fundamentals of Astrodynamics and Applications(4th ed.). Microcosm Press. §2.2 (Kepler's equation) and §2.7 (vis-viva).
[4] Danby, J. M. A. (1992). Fundamentals of Celestial Mechanics (2nd ed.). Willmann-Bell. §6.6 (quartic starter for Newton-Raphson).
[5] Murray, C. D. & Dermott, S. F. (1999). Solar System Dynamics. Cambridge University Press.
[6] JPL Horizons System. ssd.jpl.nasa.gov/horizons. (DE441/DE440 ephemerides used in presets).

Kepler's Laws — Orbital Simulator

Orbital map

Live state

Orbital invariants

Orbit geometry

Extreme speeds

Fundamentals & Explanation

The Three Empirical Laws and Their Newtonian Origin

Geometry of the Ellipse and Kepler's Equation

Third Law: From Kepler to Newton

Conserved Invariants and Vis-Viva

Physical Limits of the Model

Relativistic Limit (v→c, r→r_s)

N-Body Limit

Closed and Planar Orbits Only

Limit Cases (e)

Numerical Validation

Academic References

You may also like:

The Three Empirical Laws and Their Newtonian Origin

Geometry of the Ellipse and Kepler's Equation

Third Law: From Kepler to Newton

Conserved Invariants and Vis-Viva

Physical Limits of the Model

Relativistic Limit (v→c, r→r_s)

N-Body Limit

Closed and Planar Orbits Only

Limit Cases (e)

Numerical Validation

Academic References