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Gravitation & Orbits
From falling apples to planets, satellites, black holes and dark matter. 50 topics, from beginner to advanced. Open the interactive version with labs and quizzes
Gravitation explains why apples fall and planets orbit. Newton's law of universal gravitation gives the attractive force between any two masses, and it explains the acceleration due to gravity on Earth, how g changes with altitude and depth, and the motion of satellites and planets. The area ends with escape velocity, orbital energy and a brief look at black holes and dark matter.
How the topics connect
Gravitation starts with the inverse-square law and explains everything from falling objects to planets, satellites and tides. Kepler's laws describe orbits, energy and angular momentum predict how they behave, and general relativity extends the picture to black holes, gravitational waves and cosmology.
Real-world uses
Calculate the orbital speed and period needed to place a satellite in low Earth orbit or a geostationary orbit.
Detect small variations in g caused by underground rock density, a technique used in mineral and oil exploration.
Estimate the mass of a planet, star or black hole from the orbit of a moon or companion star.
Work out escape velocity and transfer orbits to plan launches and interplanetary trips.
Satellite operators calculate orbital periods and speeds to place weather, communication and GPS satellites in the right orbits.
Mission planners use escape speeds, transfer orbits and gravity assists to reach other planets with limited fuel.
Universal law of gravitation
Every mass attracts every other with a force that falls off as inverse square of distance.
F = Gm₁m₂/r²Every pair of masses attracts with a force F = Gm₁m₂/r², along the line joining them. It was Newton's insight that the same law governs falling apples and orbiting planets. The force is always attractive and falls off as the square of the distance.
Earth–Moon: F ≈ 2×10²⁰ N with M = 6×10²⁴ kg, m = 7.3×10²² kg, r = 3.84×10⁸ m.
r is the distance between centres, not surfaces.
Gravitational constant G
A tiny constant measured first by Cavendish.
G = 6.67×10⁻¹¹ N·m²/kg²G = 6.67×10⁻¹¹ N·m²/kg² sets the strength of gravity. It is tiny, which is why gravity is only noticeable for huge masses. Cavendish measured it in 1798 with a torsion balance.
Two 1 kg masses 1 m apart attract with 6.7×10⁻¹¹ N.
G is a constant everywhere, g is not.
Acceleration due to gravity
Field strength at Earth's surface.
g = GM/R²At a planet's surface g = GM/R². It does not depend on the falling mass. For Earth g ≈ 9.8 m/s²; on Mars about 3.7.
Mass and radius of the Moon give g ≈ 1.6 m/s².
g varies with location; G does not.
Variation of g with altitude
g decreases as one moves away from Earth.
g' = g(R/(R+h))²At height h above the surface g' = g(R/(R + h))² ≈ g(1 − 2h/R) for small h. It decreases slowly.
At 400 km (ISS) g' ≈ 8.7 m/s², still 89% of the surface value.
Astronauts are weightless due to free fall, not because gravity vanishes.
Variation of g with depth
Inside the Earth, g falls linearly toward the centre.
g' = g(1−d/R)Inside a uniform Earth only the mass inside radius r attracts, so g' = g(1 − d/R), falling linearly to zero at the centre.
At half the radius g' = 4.9 m/s².
Real Earth is denser at the core, so g first slightly increases.
Effect of Earth's rotation on g
g is smaller at the equator due to centrifugal effect.
g' = g − ω²R cos²λRotation reduces effective gravity by the centrifugal term ω²R cos²λ. It's zero at the poles and about 0.034 m/s² at the equator. Also, Earth bulges at the equator.
Apparent weight is 0.3% less at the equator.
The effect is small but measurable.
Gravitational field
Force per unit mass at a point.
E = F/m = −GM/r²The gravitational field g(r) = F/m is the force per unit mass at a point, directed toward the source. For a point mass |g| = GM/r².
The field at the Moon's surface due to the Moon is 1.6 N/kg.
Field is a vector; add vectorially.
Gravitational potential energy
Negative energy of attracting masses.
U = −GMm/rFor two masses U = −GMm/r, with zero at infinity. It is negative because work must be done to separate them. Near the surface ΔU ≈ mgh.
U for a 1 kg object on Earth's surface = −62.6 MJ.
mgh works only for small height changes.
Gravitational potential
Potential energy per unit mass.
V = −GM/rV = −GM/r is the potential energy per unit mass. The field is the negative gradient of V. Potential differences give the work per unit mass.
On Earth's surface V = −62.6 MJ/kg.
Potentials are scalars, so simply add.
Field of a spherical shell
Zero inside, like a point mass outside.
Outside a uniform shell the field is as if all mass were at the centre; inside it is exactly zero. Newton proved this by integration.
There's no net gravity inside a hollow planet of uniform shell.
Applies to spherically symmetric shells.
Field inside a uniform sphere
Grows linearly with distance from the centre.
g ∝ rInside, only the mass within radius r counts: g(r) = GMr/R³, proportional to r. At the surface it matches GM/R².
Halfway to the centre: g is half.
This describes a uniform density; real planets differ.
Escape velocity
Minimum speed to leave a planet without returning.
v_e = √(2GM/R)Escape speed v_e = √(2GM/R) is the minimum speed to reach infinity with zero speed left, ignoring air. It follows from ½mv² = GMm/R.
Earth 11.2 km/s, Moon 2.4 km/s, Sun 618 km/s.
It's independent of the direction and the object's mass.
Escape velocity of Earth
Approximately 11.2 km/s.
v_e ≈ 11.2 km/sAbout 11.2 km/s (40,300 km/h). Rockets don't need this at launch; they climb gradually and need more energy over time.
Moon missions need less than full escape energy.
Applies to a projectile with no further propulsion.
Orbital velocity
Speed of a circular orbit.
v_o = √(GM/r)For a circular orbit, gravity supplies the centripetal force: GMm/r² = mv²/r, so v = √(GM/r). Low Earth orbit speed is about 7.8 km/s.
At 400 km altitude: 7.67 km/s.
Higher orbits are slower.
Satellite energy
Total energy of a circular orbit is negative.
E = −GMm/2rA circular orbit has K = GMm/2r, U = −GMm/r, so E = −GMm/2r = −K. Losing energy via drag lowers the orbit but speeds the satellite up.
Dragging a satellite makes it faster.
Total energy is negative for bound orbits.
Orbital period
Time for one revolution.
T = 2π√(r³/GM)T = 2π√(r³/GM). The period depends only on the orbital radius and the central mass, not the satellite's mass.
At r = 6770 km: T ≈ 92 minutes.
r is measured from Earth's centre.
Geostationary satellites
Orbit in 24 hours above the equator, appearing fixed in the sky.
Orbit with a 24 h period at about 35,786 km above the equator (r = 42,164 km). It appears fixed in the sky, perfect for TV and weather.
Needs to be above the equator, moving eastward.
Only one altitude fits.
Polar and low Earth orbits
Used for imaging and the International Space Station.
Low orbits (200–2000 km) give high resolution and short periods; polar orbits pass over both poles and eventually scan the entire planet.
Earth-observation satellites at about 700 km.
Low orbits suffer drag.
Weightlessness in orbit
Astronauts are in free fall, so apparent weight is zero.
Astronauts and spacecraft fall around the Earth together, so the contact force is zero: "apparent weightlessness". Gravity at the ISS is still 89% of the surface value.
Dropped objects appear to float.
Weightlessness ≠ no gravity.
Kepler's first law
Planets orbit the Sun in ellipses with the Sun at a focus.
orbitPlanets move on ellipses with the Sun at one focus. Circles are special cases. Newton showed this comes from an inverse-square force.
Earth's orbit eccentricity 0.017 is almost circular.
The Sun is at a focus, not the centre.
Kepler's second law
The line from Sun to planet sweeps equal areas in equal times.
dA/dt = L/2mA line from the Sun to a planet sweeps equal areas in equal times, a consequence of angular momentum conservation, so planets move fastest at perihelion.
Earth moves at 30.3 km/s in January and 29.3 km/s in July.
Holds for any central force.
Kepler's third law
Square of the period is proportional to cube of semi-major axis.
T² ∝ a³T² ∝ a³: with T in years and a in AU, T² = a³. More precisely T² = 4π²a³/GM.
Mars: a = 1.52 AU gives T = 1.88 years.
Applies to bodies around the same central mass.
Elliptical orbit energy
Total energy depends only on the semi-major axis.
E = −GMm/2aFor an ellipse E = −GMm/2a, with a the semi-major axis. All orbits with the same a have the same energy, whatever their eccentricity.
A comet's long ellipse has the same energy as a circular orbit of radius a.
Not −GMm/2r with the current r.
Eccentricity
A measure of how elongated an orbit is.
e = c/ae = c/a measures how stretched an ellipse is: 0 for a circle, near 1 for long thin ellipses; e = 1 is a parabola (escape), >1 hyperbola.
Halley's comet e = 0.97.
e ≥ 1 means unbound.
Perihelion and aphelion
Closest and farthest points to the Sun; speed is highest at perihelion.
Closest and farthest points: r_p = a(1 − e), r_a = a(1 + e). Speed is greatest at perihelion; angular momentum is conserved: r_p v_p = r_a v_a.
Halley's comet: 0.59 AU to 35 AU.
Seasons on Earth are not caused by perihelion.
Orbital transfers
Hohmann transfer is the most fuel-efficient two-burn orbit change.
The Hohmann transfer uses two burns along an ellipse tangent to both circular orbits. It is the most fuel-efficient two-impulse transfer.
Low Earth orbit to geostationary needs about 3.9 km/s of Δv.
Burning prograde raises the opposite side of the orbit.
Gravity assist
A spacecraft steals momentum from a planet to speed up.
A spacecraft passing a moving planet can gain or lose speed relative to the Sun, taking momentum from the planet (negligibly changing it).
Voyager used Jupiter and Saturn.
It's an elastic collision at a distance.
Tides
The Moon's gravity gradient stretches Earth's oceans.
The Moon's gravity is stronger on the near side of Earth than the far side; this difference (gradient) stretches the oceans into two bulges, giving two high tides per day. The Sun adds spring and neap effects.
Tidal range ≈ 1 m in the open ocean.
Tides come from differences in gravity, not gravity alone.
Tidal locking
Tidal friction makes rotation match orbit, as with our Moon.
Tidal friction slows a body's rotation until its rotation period equals its orbital period, so it shows the same face. The Moon is locked to Earth.
Pluto and Charon are mutually locked.
Takes millions to billions of years.
Roche limit
Distance within which tides tear a satellite apart.
Inside about 2.44 R(ρ_planet/ρ_moon)^{1/3}, tides tear apart a fluid satellite; this likely created Saturn's rings.
For Earth–Moon-like densities, about 18,000 km.
Solid bodies with cohesive strength can survive closer.
Lagrange points
Five positions where a small body co-moves with two large bodies.
Five points where a small body can stay fixed relative to two large masses. L1, L2, L3 are unstable on a line; L4 and L5 are stable if M₁/M₂ > 24.96. The James Webb telescope orbits Sun–Earth L2.
Trojan asteroids at Jupiter's L4/L5.
Unstable points need station-keeping.
Two-body problem
Reduces to one body with reduced mass.
Two masses orbit their common centre of mass. The relative motion is a one-body problem with reduced mass μ and total mass M in the force. Kepler's laws hold with M = m₁ + m₂.
Binary stars of equal mass orbit a point midway.
The central body also moves.
Effective potential
Combines gravity with the centrifugal barrier.
U_eff = −GMm/r + L²/2mr²For radial motion, add a centrifugal barrier: U_eff = −GMm/r + L²/2mr². Minimum gives the circular orbit; energies between give ellipses.
A comet with low L plunges close to the Sun.
Only the radial motion appears in U_eff.
Binary stars
Two stars orbiting their common centre of mass.
Two stars orbiting a common centre. The orbital data give masses via Kepler's third law: M₁ + M₂ = 4π²a³/GT².
Sirius A and B: T = 50 years.
Mass ratio from the distances to the centre of mass.
Weighing the Sun and planets
Orbital data give masses of central bodies.
M = 4π²a³/GT²From a planet's orbital radius and period, M = 4π²a³/GT². With Earth's a = 1.496×10¹¹ m and T = 1 yr, M☉ ≈ 2×10³⁰ kg.
Jupiter's moons give Jupiter's mass.
You can find a body's mass from the motion of satellites.
Gravitational lensing
Mass bends light, forming arcs and multiple images.
A mass bends the path of light, deflecting it by 4GM/(c²b) (twice Newton's value). Lensing produces multiple images or rings, and shows dark matter.
Eddington's 1919 eclipse deflection of 1.75″.
Lenses can magnify distant galaxies.
Gravitational waves
Ripples in spacetime from accelerating masses, first detected in 2015.
Accelerating masses produce ripples in spacetime travelling at c. LIGO detected two black holes merging in 2015 via a length change of 10⁻¹⁸ m.
Neutron star merger GW170817 had an optical counterpart.
Detectable only for extreme events.
Black holes
Objects so dense that not even light escapes the horizon.
r_s = 2GM/c²A black hole has an event horizon where the escape speed equals c. Anything inside cannot send signals out. Formed by the collapse of massive stars.
Sagittarius A* is 4 million solar masses.
They are not cosmic vacuum cleaners; far away they gravitate like any mass.
Schwarzschild radius
The event horizon radius of a non-rotating black hole.
r_s = 2GM/c²r_s = 2GM/c² is the horizon radius of a non-rotating black hole. For the Sun, 3 km; for Earth, 9 mm.
A 10 solar mass black hole has r_s = 30 km.
It is a radius for the horizon, not the singularity.
Neutron stars and white dwarfs
Compact stellar remnants supported by degeneracy pressure.
White dwarfs (up to 1.4 M☉) are supported by electron degeneracy pressure; neutron stars by neutron degeneracy and nuclear forces, about 20 km across with density 10¹⁷ kg/m³.
A teaspoon of neutron star is about 10⁹ tonnes.
Beyond ~2–3 M☉ collapse to a black hole.
Dark matter evidence
Galaxy rotation curves show more gravity than visible matter explains.
Orbital speeds in galaxies stay flat far out instead of falling like √(1/r), and cluster lensing exceeds visible mass, implying extra non-luminous matter.
The Bullet Cluster separates the gas from the mass.
The nature of dark matter remains unknown.
Newton's shell theorem
A spherically symmetric body attracts as if its mass were at its centre.
A spherically symmetric shell attracts external bodies as if its mass were at the centre and exerts no force inside. This allows planets to be treated as point masses.
Inside a hollow sphere you float freely.
The theorem needs spherical symmetry.
Gravitational self-energy
Energy needed to assemble a sphere of mass.
U = −3GM²/5RAssembling a uniform sphere from dust requires binding energy U = −3GM²/5R. It powers the early heating of stars and planets.
The Sun's self-energy ≈ −2.3×10⁴¹ J.
Compare to the Sun's total luminosity over time.
Inertial vs gravitational mass
Their equality is the equivalence principle.
Inertial mass measures resistance to acceleration; gravitational mass measures the response to gravity. Their equality (tested to 10⁻¹⁵) is why all bodies fall equally.
Eötvös and torsion balance experiments.
This equality is the starting point for general relativity.
Equivalence principle
Gravity and acceleration are locally indistinguishable.
In a small freely falling lab you cannot detect gravity, and in an accelerating rocket you feel a force identical to gravity. Einstein built general relativity on this.
A light beam bends in an accelerating lift, so also in gravity.
Applies locally.
Orbital precession
Mercury's orbit precesses faster than Newton predicts, explained by general relativity.
Mercury's perihelion advances by 574″/century; Newtonian effects of other planets explain 531″, and general relativity adds the missing 43″.
This was an early test of general relativity.
Extra effects come from curved space.
Galaxy and cosmic structure
Gravity builds stars, galaxies and clusters from small density differences.
Tiny initial density differences in the early universe grew by gravity into stars, galaxies and clusters, forming a cosmic web of filaments and voids.
Simulations like Millennium reproduce it.
Dark matter provides the scaffold.
Space elevators and tethers
Concept using orbital mechanics to lift loads without rockets.
A cable from the equator to above geostationary altitude (about 36,000 km) could lift loads without rockets. Needs materials with huge strength-to-weight such as carbon nanotubes.
A counterweight beyond geostationary orbit keeps it taut.
Not yet feasible.
Gravity on other planets
Surface gravity depends on mass and radius.
Surface gravity g = GM/R²: Mercury 3.7, Venus 8.9, Mars 3.7, Jupiter 24.8, Moon 1.6 m/s². Weight changes but mass doesn't.
A 60 kg person weighs 222 N on Mars.
Gravity doesn't depend on mass alone but also radius.
Spacecraft launch energy
Energy needed to reach orbit vs escape.
To reach low orbit one needs about 32 MJ/kg of kinetic energy plus 9 MJ/kg potential, but rocket inefficiencies and gravity losses make the needed Δv about 9.4 km/s.
Escape requires 62.6 MJ/kg.
The rocket equation makes small increases expensive.
Frequently asked questions
Why do astronauts feel weightless on the International Space Station?
They are not outside gravity. The station and the astronauts are both falling freely around Earth, so there is no contact force pressing them against the floor.
Does g change with altitude?
Yes. g decreases with distance from the centre of the Earth, roughly as 1/r squared, so it is slightly smaller on a mountain top than at sea level.
What is escape velocity?
It is the minimum speed needed for an object to leave a planet's gravitational pull without further propulsion. On Earth it is about 11.2 km/s and it does not depend on the mass of the escaping object.
Why do astronauts float if gravity still acts on them?
They are in continuous free fall around Earth. Gravity is still about 90 percent as strong at station height, but they and the station fall together, so nothing presses on them.
Is Newton's gravity still used?
Yes, for most everyday and spaceflight calculations. General relativity is needed for very strong gravity, such as near black holes, and for precise GPS timing.
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