Vol. Science · Astronomy · The five points
Two bodies on circular orbits have five points where a third, tiny mass can sit still in the frame that rotates with them. L1, L2, and L3 lie on the line through the primary and the secondary. L4 and L5 complete equilateral triangles, 60° ahead of the secondary and 60° behind. The mass parameter is μ = m₂ / (m₁ + m₂). A Hill-sphere approximation puts L1 and L2 a distance r ≈ a · (μ / 3)1/3 from the secondary. That is an approximation. This desk also solves the collinear force-balance equation and prints that distance beside it, in kilometers and in units of the separation a. Masses and semimajor axes are the NASA NSSDCA fact sheets. With those inputs, Sun–Earth L1 is 1,491,557 km from Earth and L2 is 1,501,538 km. Earth–Moon L1 is 326,380 km from Earth and 58,020 km from the Moon. The year of one solar circle is the Kepler desk. The speed at one radius is the Escape desk. The half-ellipse between two circles is the Hohmann desk. The sphere around the secondary, rH ≈ a · (m / (3M))1/3 beside this μ form, is the Hill desk.
Circular restricted three-body classroom schematic. Two bodies on circles about their barycenter, and a test mass with no pull of its own. One plane. Top-down, from the north, so the pair’s motion runs counterclockwise. Body disks are marks, not radii, so the picture is not to scale where a world is drawn. L1 through L5 are drawn at the computed positions. For Sun–Earth, L1 and L2 sit almost on Earth in the full view; the lower panel magnifies that neighborhood, and that panel has its own scale. Not mission-planning data. Not an ephemeris. Not a halo orbit.
Rotating frame · from the north · L4 leads
Pair
Highlight a point
The formulas
μ = m₂ / (m₁ + m₂). m₁ is the primary, m₂ the secondary. Sun–Earth uses the Sun Fact Sheet mass 1,988,400 × 1024 kg and the Earth Fact Sheet mass 5.9722 × 1024 kg. Earth–Moon uses that same Earth mass and the Moon Fact Sheet mass 0.07346 × 1024 kg. Sun–Jupiter uses the Sun’s mass and the Jupiter Fact Sheet mass 1,898.13 × 1024 kg. The separation a is the secondary’s fact-sheet semimajor axis: Earth 149.598 × 106 km, Moon 0.3844 × 106 km, Jupiter 778.479 × 106 km.
r ≈ a · (μ / 3)1/3. This Hill-sphere approximation is one distance from the secondary, used here for both L1 and L2. It is labeled as an approximation on every readout. It is closer when μ is tiny. For Sun–Earth it and the numerical L1 differ by about 5,000 km. For Earth–Moon the Hill figure is 61,274 km from the Moon and the numerical L1 is 58,020 km. The Hill desk prints that μ form beside rH ≈ a · (m / (3M))1/3. For Sun–Earth those two round to 1,496,563 km and 1,496,565 km. Jupiter’s sphere on that desk is 53,147,220 km, a neighborhood around Jupiter, not the Trojan clouds at L4 and L5.
The collinear balance. In units where a = 1 and the mean motion is 1, the primary sits at x = −μ and the secondary at x = 1 − μ. A point on that line is in equilibrium when x − (1−μ)(x − x₁)/|x − x₁|³ − μ(x − x₂)/|x − x₂|³ = 0. That condition is a quintic in the distance from the secondary. This page solves it by bisection: L1 between the bodies, L2 beyond the secondary, L3 beyond the primary. L4 and L5 are the equilateral points, each one separation from both bodies, so the angle at either body is 60°.
Routh. In this model L4 and L5 are linearly stable when μ(1−μ) < 1/27. That is a mass ratio m₁/m₂ greater than 24.96, the threshold this page computes from that inequality. L1, L2, and L3 are unstable. A spacecraft there needs station-keeping. NASA’s Webb orbit page calls L1, L2, and L3 metastable: an object drifts away unless it maintains its position, for example with small periodic rocket thrust. The same page says objects at L4 and L5 orbit those points with no assistance. The mass-ratio test is the classroom criterion. The three pairs on this desk all pass it.
Where this shows up
Sun–Earth L2. NASA’s Webb mission page says the James Webb Space Telescope orbits the Sun 1.5 million kilometers from Earth at L2. The Webb orbit page says the same distance, and says Webb orbits around L2 rather than sitting on the point. This desk’s arithmetic, from the fact-sheet masses and 149.598 × 106 km, puts numerical L2 at 1,501,538 km from Earth.
Sun–Earth L1. The same Webb orbit page says L1 is 1.5 million km from Earth toward the Sun, and names SOHO among the solar observatories there. NASA’s SOHO mission page says SOHO was placed 1.5 million kilometers from Earth in an elliptical Lissajous orbit around the L1 libration point. This desk’s numerical L1 is 1,491,557 km from Earth. The same Webb page says some small asteroids are known at Sun–Earth L4 and L5.
Sun–Jupiter L4 and L5. NASA’s account of the Trojan asteroids says Max Wolf’s 1906 asteroid stayed nearly 60° in front of Jupiter, at Jupiter’s L4, and that another swarm was found at L5. Those asteroids lead or follow Jupiter in its orbit around the Sun. On this desk L4 and L5 are the equilateral points of the Sun–Jupiter pair.
Literacy · a point, not a mission
Sources · public NASA pages
Classroom picture of the five Lagrange points. Sun–Earth L1 is 1,491,557 km from Earth and L2 is 1,501,538 km, from the fact-sheet masses and Earth’s 149.598 × 106 km semimajor axis. Earth–Moon L1 is 326,380 km from Earth and 58,020 km from the Moon. The Hill radius is the approximation, printed beside those roots. Circular, coplanar, a massless third body. Not a halo, and not mission-planning data. Earthrise preview: NASA / Bill Anders (public domain).
Keep going
The year
P² = a³ in years and AU. How long one solar circle takes.
The speed
Circular orbit and escape at one altitude. √2 apart.
The coast
Two burns and half an ellipse. Earth to Mars is 258.88 days.
The sphere
The same μ approximation, as a sphere around the secondary. Earth’s is 1,496,565 km. Jupiter’s Trojans sit outside it.
Neighborhood
Where the eight planets sit. Day and local time, still a schematic.
Five equilibrium points of two bodies on circular orbits, drawn in the rotating frame. Distances come from NASA fact-sheet masses and semimajor axes. The picture is a classroom schematic, not a trajectory to fly.
| Sun–Earth | The default. Sun 1,988,400 × 1024 kg, Earth 5.9722 × 1024 kg, separation 149.598 × 106 km. Numerical L1 is 1,491,557 km from Earth. Numerical L2 is 1,501,538 km. The Hill approximation is 1,496,563 km. |
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| Earth–Moon | Earth 5.9722 × 1024 kg, Moon 0.07346 × 1024 kg, separation 0.3844 × 106 km. Numerical L1 is 326,380 km from Earth and 58,020 km from the Moon. The Hill approximation is 61,274 km from the Moon, so the lower panel shows the approximation and the root apart. |
| Sun–Jupiter | Jupiter’s mass is 1,898.13 × 1024 kg and its semimajor axis is 778.479 × 106 km. L4 and L5 are where NASA places the Trojan asteroids, 60° ahead of Jupiter and 60° behind. |
| All five, L1–L5 | Highlights one point, or all of them. You can also tap a point on the drawing. L1, L2, and L3 are the collinear points. L4 leads. L5 trails. |
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| Play | Turns the whole pattern counterclockwise, as seen from the north. The five points stay fixed relative to the two bodies. That is the rotating frame, shown against a fixed page. About 12 seconds per turn. A time-lapse, not a year. Press again to pause. |
| Full view | Barycenter at the cross. Primary and secondary on the line of centers. The secondary’s orbit is the circle. L4 and L5 sit on the equilateral triangles. Body disks are marks, not radii. |
| Lower panel | The secondary’s neighborhood, magnified, always drawn along the line of centers. It does not spin with Play. The dashed circle is the Hill approximation. The dots are numerical L1, toward the primary, and numerical L2, outward. The scale is not the scale of the view above. |
| N | North, the view. Motion runs counterclockwise, the same sense as the other top-down desks. |
| L1 and L2 faces | The numerical distance from the secondary, in km. The line under each face is that distance in units of a, and the Hill approximation in km. The Hill figure is the same for both points. The roots are not. |
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| From the primary | Center to center, along the line. Earth–Moon L1 from Earth is the primary distance: 326,380 km. The 58,020 km figure is from the Moon. |
| Routh | Passes when m₁/m₂ is greater than 24.96, from μ(1−μ) < 1/27. All three pairs on this desk pass, so L4 and L5 are linearly stable in the model. L1–L3 stay unstable either way. |
| Barycenter | μ·a from the primary’s center. Compared with the fact-sheet volumetric mean radius: inside the Sun for Sun–Earth, inside Earth for Earth–Moon, outside the Sun’s volumetric radius for Sun–Jupiter. |