Vol. Science · Astronomy · The ellipse
On a Keplerian ellipse the speed is not one number. It is v = √[μ (2/r − 1/a)]. μ is GM of the body at the focus. r is the current distance from that focus. a is the semimajor axis. At periapsis, rp = a(1−e) and the craft is fastest. At apoapsis, ra = a(1+e) and it is slowest. A circular orbit at the same r would move at vc = √(μ/r). Leaving from that r, and not coming back, takes vesc = √(2μ/r). The year of the ellipse is still Kepler’s P = 2π √(a³/μ). It does not depend on e. For Earth around the Sun, a is 149.598 × 106 km and e is 0.0167 on the Earth Fact Sheet. μ is the Sun Fact Sheet GM, 132,712 × 106 km3/s2. Vis-viva at the printed perihelion, 147.095 × 106 km, is 30.287 km/s. The sheet lists a maximum orbital velocity of 30.29 km/s. At aphelion, 152.100 × 106 km, it is 29.291 km/s. The sheet lists 29.29 km/s. When e = 0 the formula collapses to √(μ/a). When r = a, the speed is √(μ/a) even if the orbit is not a circle. The circle and the escape speed at one altitude are the Escape desk. The half-ellipse between two circles is the Hohmann desk. The year in years and AU is the Kepler desk.
Classroom schematic. Point-mass, two-body, one focus. No drag, no oblateness, and no third body. Not an ephemeris and not a mission planner. Top-down, from the north, so motion runs counterclockwise. The central body sits at the focus, not at the center of the ellipse. The empty focus is marked. The body is a dot unless the orbit sits just above the volumetric radius, as in the 400 km example. The arrow length is proportional to speed. On Earth’s orbit that change is about 3 percent, so the arrow barely shortens. The 400 km altitude is a classroom example. The Earth Fact Sheet does not print it. The ellipse with e = 0.5 is not a real satellite.
From the north · Sun at the focus · arrow length tracks speed
Preset
Distances in × 10⁶ km
Enter a semimajor axis greater than zero, an eccentricity from 0 up to but not including 1, and a distance between periapsis and apoapsis.
Or set the ends
Sun · Earth
147.095 × 10⁶ km
Speed now · √[μ(2/r−1/a)]
30.287 km/s
At this r
Periapsis · vp
30.286 km/s
rp = a(1−e)
Apoapsis · va
29.291 km/s
ra = a(1+e)
Earth’s fact sheet prints the orbit. The Sun’s GM supplies μ.
Drag r from perihelion to aphelion. The arrow length is the speed.
What the formula is
v = √[μ (2/r − 1/a)]. μ is GM of the central body. r is the distance from the focus right now. a is the semimajor axis, half the long axis of the ellipse, not the current distance. The same a fixes the period. Moving r along the ellipse changes the speed and does not change the year.
vp = √[μ (2/rp − 1/a)] with rp = a(1−e). va = √[μ (2/ra − 1/a)] with ra = a(1+e). Periapsis is the near end. Apoapsis is the far end. For a planet around the Sun those ends are perihelion and aphelion. For the Moon they are perigee and apogee.
vc = √(μ/r) is a different orbit: a circle that passes through the same point. vesc = √(2μ/r) is the coasting speed that leaves and does not return. Escape is circular times √2, at that same r. The Escape desk is those two speeds at one altitude. This page is the speed the ellipse already has.
When e = 0, vis-viva collapses to √(μ/a). Then r is a everywhere, 2/r − 1/a becomes 1/a, and the square root is the circular speed. When r = a, v = √(μ/a) even if e is not 0. The craft is not at the center. It is one semimajor axis from the focus, which happens twice on every ellipse that is not a circle. The speed there matches a circular orbit whose radius is a. The period P = 2π √(a³/μ) is that same a, and it ignores e.
The round numbers
The Earth Fact Sheet prints maximum orbital velocity 30.29 km/s and minimum 29.29 km/s. Vis-viva at the printed perihelion and aphelion, with the printed semimajor axis and the Sun’s GM, is 30.287 km/s and 29.291 km/s. Rounded to the hundredth, those are the sheet’s figures. The printed eccentricity 0.0167 does not land exactly on the printed perihelion. a(1−e) is 147.100 × 106 km. The sheet’s perihelion is 147.095 × 106 km. The slider can sit on either distance. Both speeds still round to 30.29 km/s.
The Moon sheet lists 1.082 km/s at mean perigee and 0.970 km/s at mean apogee. Vis-viva with the Earth Fact Sheet GM is 1.076 km/s and 0.964 km/s. The sheet’s note says those distances were used for the maximum and minimum velocities. This desk does not adjust μ to force the match. Mars does match: 26.500 km/s and 21.970 km/s round to the sheet’s 26.50 and 21.97.
Words
Classroom picture of one formula. Earth’s printed perihelion, with the Sun’s GM and the printed semimajor axis, is 30.287 km/s. The sheet lists 30.29 km/s. Aphelion is 29.291 km/s. The sheet lists 29.29 km/s. Point-mass, two-body. Not an ephemeris and not a mission planner. The 400 km altitude is an example, not a fact-sheet line. The e = 0.5 ellipse is not a real body. Earthrise preview: NASA / Bill Anders (public domain).
Keep going
The speed
Circular orbit and escape at one altitude. √2 apart. A cannon diagram, not this ellipse.
The coast
The same vis-viva formula, on the half-ellipse between two circles. Earth to Mars is 258.88 days.
The year
P² = a³ in years and AU. The period comes from a. This page is the speed at one r.
Neighborhood
Where the eight planets sit. Day and local time, still a schematic.
One ellipse, the focus, and the speed at the current r. v = √[μ (2/r − 1/a)]. The picture is a classroom schematic. It is not a trajectory to fly.
| Earth around Sun | a = 149.598 × 106 km, e = 0.0167, from the Earth Fact Sheet. μ is the Sun’s GM. The distance slider runs from the printed perihelion, 147.095 × 106 km, to the printed aphelion, 152.100 × 106 km. Speeds there round to the sheet’s 30.29 and 29.29 km/s. |
|---|---|
| Moon around Earth | Moon sheet semimajor axis, perigee, apogee, and eccentricity. μ is Earth’s GM, 0.39860 × 106 km3/s2. The computed speeds sit next to the sheet’s 1.082 and 0.970 km/s and are not forced to match. |
| Mars around Sun | Mars sheet semimajor axis, perihelion, aphelion, and eccentricity. μ is the Sun’s GM. Perihelion rounds to 26.50 km/s. Aphelion rounds to 21.97 km/s. |
| 400 km LEO | r = 6,371.000 km + 400 km, e = 0, μ = Earth’s GM. A circle, so vis-viva is √(μ/a). The altitude is a classroom example, the same one the Hohmann desk uses. The fact sheet does not print it. |
| Schematic e = 0.5 | Not a real body. a = 20,000 km around Earth, so periapsis is 10,000 km and apoapsis is 30,000 km. The arrow shortens a lot. It is a picture of the formula. |
| Arrow keys | With a preset button focused, Left and Right move through the presets. They do nothing while a field is focused, and nothing while this dialog is open. |
| Semimajor a | Half the long axis. Changing it leaves the preset. The period and both end speeds change. r is kept if it still lies on the new ellipse. |
|---|---|
| Eccentricity e | From 0 to just under 1. Zero is a circle. rp = a(1−e) and ra = a(1+e) follow. |
| Distance r | Clamped between the near end and the far end. On a fact-sheet preset the ends are the printed periapsis and apoapsis when those differ from a(1±e). |
| rp and ra | Type the two ends. a becomes their average. e becomes (ra − rp) / (ra + rp). Apoapsis has to be farther than periapsis. |
| Jump buttons | rp = a(1−e), r = a, and ra = a(1+e). Printed perihelion and aphelion appear when the fact sheet’s distances are not the same rounded figures as a(1±e). |
| Speed now | √[μ (2/r − 1/a)] at the slider. |
|---|---|
| Periapsis and apoapsis | The same formula at rp = a(1−e) and ra = a(1+e). |
| Circular and escape | √(μ/r) and √(2μ/r) at the current r. A different orbit, and the leaving speed, at the same place. |
| Period | 2π √(a³/μ). Days, hours, or minutes. It does not change when you slide r. |
| e = 0 | The three speeds on the ellipse become one speed, √(μ/a). |
| r = a | The speed is √(μ/a) even when e is not zero. |
| ? | Opens this dialog. Escape closes it. Tab stays inside until it closes. |
|---|---|
| Escape | Closes the dialog and returns focus to the ? button. |