Vol. Science · Astronomy · The speed
Two speeds belong to one distance from a body’s center. A circular orbit needs v = √(GM/r). Leaving, and never coming back, needs vesc = √(2GM/r). The second is the first times √2. The NASA NSSDCA fact sheets print GM and a volumetric mean radius for the Moon and each planet. This desk uses those two numbers. At altitude 0 on Earth, GM = 0.39860 × 106 km3/s2 and R = 6,371.000 km give 11.186 km/s, the escape velocity the Earth Fact Sheet lists. The Gravity desk names Newton’s cannon and does not compute these speeds. The year of a solar orbit is the Kepler desk. The half-ellipse between two of those circles is the Hohmann desk. Speed anywhere on one ellipse is the vis-viva desk. The neighborhood those orbits sit in is the solar map.
Classroom schematic. The body is a smooth sphere with the volumetric mean radius. The launch is sideways, from rest relative to a body that is not spinning. No air, so no drag. No thrust after the speed is set. The drawing is Newton’s cannon as a diagram: paths are conic sections, clipped when they leave the frame. It is not a launch window, not a trajectory a spacecraft could fly, and not mission-planning data. Miles per hour are a conversion of the computed km/s. They are not a second NASA table.
Sideways launch · body to scale with altitude · no air, no spin
What this means
Escape speed does not care which way you point, or how heavy the projectile is, in this model. The speed is the one that makes kinetic energy equal the depth of the well: ½mv² = GMm/r, so v = √(2GM/r). The projectile’s mass cancels. Direction never enters the energy. The same speed, aimed any way that stays clear of the sphere, still coasts to infinity. Aim into the sphere and you hit it even above escape speed. The drawing fires sideways, the cannon way. Straight up at the same speed is a different path and the same threshold.
A rocket does not have to be at escape speed in one instant. Escape speed is a coasting speed at one place, with the engine already off. A rocket can thrust for minutes and add energy the whole way. It may never pass through the surface escape speed as a single reading. Most launches are not trying to escape. A circular orbit needs less speed, by exactly √2. The vehicle climbs, turns, and settles into that slower circle. This page has no throttle. The slider is the speed at the moment the coast begins.
Air and rotation are left out on purpose. A spinning body gives an eastward launch a head start. An atmosphere takes speed back as drag, and it makes a surface-grazing circle impossible. The fact-sheet notes define escape velocity as the speed at the surface needed to escape, ignoring atmospheric drag, and for a gas giant that surface is the 1-bar level. This sphere uses the volumetric mean radius, because that is the radius that reproduces the sheet’s mean gravity. Jupiter, Saturn, Uranus, and Neptune list an escape speed that matches the equatorial 1-bar radius instead. Both radii are on the sheet. The readout keeps them apart.
Literacy · two speeds, one radius
Sources · public NASA pages
Classroom picture of two speeds at one radius. GM and the volumetric mean radius are the fact sheets’ printed figures. Earth’s surface escape on this page is 11.186 km/s because that is what those two figures produce, and it is what the Earth Fact Sheet lists. Not a trajectory, not a launch, and not mission-planning data. Earthrise preview: NASA / Bill Anders (public domain).
Keep going
The pull
Newton’s cannon, named. Weight, orbits, and the gap this speed formula does not close.
The year
P² = a³ in years and AU. How long a solar orbit takes, not how fast you leave.
The coast
Two burns and half an ellipse between two circular orbits. Earth to Mars is 258.88 days. Not a mission.
The ellipse
Speed anywhere on one Keplerian ellipse. Earth’s perihelion rounds to 30.29 km/s. Not an ephemeris.
Neighborhood
Where the eight planets sit. Day and local time, still a schematic.
The difference
The same pull, weaker on the far side. Two bulges, not a harbor table.
A classroom sphere and a sideways launch. Circular speed is √(GM/r). Escape speed is √(2GM/r). The picture ignores air and spin. It is not a trajectory to fly.
| Moon through Neptune | Each button loads that fact sheet’s GM and volumetric mean radius. Altitude returns to the surface so the speeds are the surface speeds. The sheet’s listed escape velocity stays in the readout for comparison. |
|---|---|
| Earth | GM 0.39860 × 106 km3/s2 and volumetric mean radius 6,371.000 km. Surface escape speed is 11.186 km/s, the figure the Earth Fact Sheet lists. |
| Custom | Type a mass in 1024 kg and a radius in km. GM is NIST’s 2022 CODATA G, 6.67430 × 10−11, times that mass. There is no fact-sheet escape speed to compare. Typing Earth’s mass does not reproduce Earth’s button: the button uses the sheet’s GM, not G times mass. |
| Altitude | Height above the volumetric mean radius, from 0 to twice that radius. The distance in the formulas is radius plus altitude. Both speeds fall as you climb. Their ratio stays √2. |
|---|---|
| Launch speed | The coasting speed at that altitude, from 0 to twice escape speed. The bright path is this slider. The faint paths stay put so the four cases remain visible. |
| Falls back | Sets a speed below circular that intersects the sphere. |
| Circular | Sets √(GM/r). |
| Ellipse | Sets 1.2 times circular speed. Closed, and it misses the sphere. The far end may leave the frame. |
| Escape | Sets √(2GM/r). The path opens and does not return. |
| Sphere | The body, to scale with the launch radius. A higher altitude makes the sphere smaller in the frame. The size is the volumetric radius, not a photograph. |
|---|---|
| Arrow | Sideways, at the top of the sphere or of the altitude. Longer means faster, up to a cap so escape does not grow an arrow off the page. |
| Four paths | A fall that hits, a dashed circle, a longer ellipse, and an open escape curve. Your launch is the thick gold path. Paths that would run far are cut at the frame. |
| Circular and escape | km/s from the formulas, and mph from those km/s times 3,600/1.609344, rounded. The mph is a conversion, not the fact sheet’s miles-per-second column. |
|---|---|
| √2 ratio | Escape speed divided by circular speed. It stays √2 when you move altitude or change the body. |
| Gravity here | GM/r² in m/s². At the surface it is the sheet’s mean surface gravity. |
| Sheet escape | The escape velocity printed on the fact sheet, for a preset. On Jupiter, Saturn, Uranus, and Neptune that print matches the equatorial 1-bar radius. This sphere’s surface escape is the volumetric one. The line under the numbers says which is which. |