1.980492e+20
1.980492e+17
0.002697
0.000000000067
m^3/(kg*s^2)
1.980492e+20
1.980492e+17
0.002697
0.000000000067
m^3/(kg*s^2)
The gravitational constant G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² is Newton's constant — the fundamental coupling strength of gravity. It appears in Newton's law of universal gravitation F = Gm₁m₂/r², Einstein's field equations of general relativity, and every calculation involving gravitational forces, orbital mechanics, and black holes.
G is the least precisely known of the fundamental constants. While the speed of light, Planck's constant, and Boltzmann constant are now exactly defined, G is measured experimentally. The current CODATA 2018 value is G = 6.67430 × 10⁻¹¹ m³/(kg·s²) with a relative uncertainty of 2.2 × 10⁻⁵ — five orders of magnitude less precise than most other fundamental constants. Decades of precision experiments using torsion balances (the Cavendish-type experiment) continue to refine this value, but systematic errors from vibrations, residual gas, and electrostatic effects make sub-10 ppm measurements extremely challenging.
Despite its enormous cosmological importance, gravity is by far the weakest of the four fundamental forces. The ratio of gravitational to electromagnetic force between two protons is G × m_p²/(k_e × e²) ≈ 10⁻³⁶ — gravity is 36 orders of magnitude weaker. This hierarchy problem — why gravity is so much weaker than other forces — is one of the deepest unsolved puzzles in physics.
From G, Newton's mass of the Earth and Sun were first determined: using Cavendish's 1798 measurement of G, combined with the known acceleration of gravity g = 9.8 m/s² and Earth's radius, one can calculate M_Earth = gR²/G ≈ 6 × 10²⁴ kg. Similarly, knowing Earth's orbital period and radius gives M_Sun = 4π²a³/(GT²) ≈ 2 × 10³⁰ kg — the classic application of Kepler's third law.
Select the calculation: gravitational force between two masses, orbital period via Kepler's third law T = 2π√(a³/GM), escape velocity v_esc = √(2GM/R), or Schwarzschild radius r_s = 2GM/c². Enter the relevant masses and distances. G = 6.67430 × 10⁻¹¹ m³/(kg·s²) is used (CODATA 2018).
Gravitational force is extremely weak for everyday masses but dominates at astronomical scales due to its infinite range and always-attractive nature. The Schwarzschild radius is the radius at which an object would become a black hole — for Earth this is 8.87 mm; for the Sun it is 2.95 km.
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The Earth-Moon gravitational force is approximately 1.98 × 10^20 N. This force drives tidal effects and keeps the Moon in its ~27.3-day orbit.
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The Sun's Schwarzschild radius is 2.95 km. If all the Sun's mass were compressed into a sphere of radius less than ~3 km, it would become a black hole. The Sun's actual radius is 696,000 km.
Unlike atomic constants measured by quantum effects, G must be measured by observing mechanical forces between macroscopic bodies. The gravitational force between lab-sized masses is extremely small (piconewtons) and is easily masked by seismic vibrations, electrostatic forces, and residual gas pressure. Different precision experiments have obtained values differing by more than their quoted uncertainties, suggesting unidentified systematic effects.
Henry Cavendish's 1798 torsion balance experiment first measured G (though Cavendish framed it as 'weighing the Earth'). Two small lead spheres on a torsion balance were deflected by the gravitational attraction to two larger lead spheres. The angle of deflection revealed G. Modern versions use laser interferometry to measure deflection and achieve uncertainties of ~10 ppm.
Kepler's third law states T² = 4π²a³/(GM), where T is orbital period, a is semi-major axis, and M is the central mass. Newton derived this from his law of gravitation and second law. It allows mass measurements of astronomical objects: if you know a satellite's orbital period and radius, you can calculate the mass of the central body.
The Schwarzschild radius r_s = 2GM/c² is the radius at which a spherically symmetric non-rotating mass becomes a black hole (the event horizon radius). For Earth: r_s = 2 × 6.674 × 10⁻¹¹ × 5.972 × 10²⁴ / (3 × 10⁸)² ≈ 8.87 mm. For a human (70 kg): r_s ≈ 10⁻²⁵ m, far below the Planck length.
General relativity uses the same Newtonian G, but reinterprets it as the coupling strength in Einstein's field equations: Gμν = (8πG/c⁴) Tμν. The combination 8πG/c⁴ ≈ 2.08 × 10⁻⁴³ m/(J) is called the Einstein gravitational coupling constant. GR reduces to Newtonian gravity in the weak-field, slow-motion limit.
Some extensions of physics (Brans-Dicke theory, string theory) predict that G might vary over cosmological time. Lunar laser ranging experiments and planetary radar tracking have constrained dG/dt to less than 10⁻¹³ per year — consistent with no variation. Pulsar timing provides even tighter constraints. No confirmed variation in G has ever been observed.
The Planck length lP = √(ℏG/c³) ≈ 1.616 × 10⁻³⁵ m is the length scale at which quantum gravitational effects become important. It combines G (gravity), ℏ (quantum mechanics), and c (special relativity) — the three pillars of modern physics. Below the Planck length, our current theories break down and a theory of quantum gravity is needed.
Stellar structure is governed by hydrostatic equilibrium: dP/dr = -Gm(r)ρ(r)/r², where the pressure gradient supports the star against gravitational collapse. This differential equation, combined with equations of state and nuclear energy generation, describes everything from stellar evolution timescales to the relationship between stellar mass and luminosity (mass-luminosity relation: L ∝ M³·⁵).
In Planck units (where ℏ = c = kB = 1), Newton's constant G = 1, defining the Planck unit system. The gravitational coupling of two protons is α_G = Gm_p²/ℏc ≈ 5.9 × 10⁻³⁹, compared to the electromagnetic coupling α = e²/4πε₀ℏc ≈ 1/137. The enormous ratio α/α_G ≈ 10³⁶ encapsulates the fundamental weakness of gravity.
NASA uses G with 5-significant-figure precision for most mission planning, which is entirely adequate. The Pioneer anomaly (a small unexplained acceleration of Pioneer 10 and 11 spacecraft) was resolved not as modified gravity but as thermal radiation pressure from the spacecraft's radioisotope thermoelectric generators — a mundane but precisely calculated effect of about 8 × 10⁻¹⁰ m/s².
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