Vol. Science · Astronomy · The sphere of influence
A secondary on a circular orbit carries a sphere inside which it, not the primary, dominates a test particle. The common approximation for a low-mass secondary is rH ≈ a · (m / (3M))1/3. a is the secondary’s semimajor axis, m its mass, M the primary’s mass. When μ = m / (M + m) is much less than 1, the same neighborhood is rH ≈ a · (μ / 3)1/3. That second form is the Hill approximation on the Lagrange desk. Masses, radii, and semimajor axes are the NASA NSSDCA fact sheets, the same figures the escape, Hohmann, Lagrange, and Roche desks already use. For the Sun and Earth the first form is 1,496,565 km (2.151 solar radii, 0.010004 of Earth’s semimajor axis). The μ form is 1,496,563 km. The Moon’s semimajor axis, 384,400 km, is 0.257 of Earth’s Hill radius, so the Moon sits inside. Jupiter’s sphere is 53,147,220 km. Mars’s is 1,084,138 km, and Phobos at 9,378 km is deep inside it, while the Roche desk puts that same orbit between the fluid and rigid limits. Hill is who dominates a test particle. Roche is where a moon of a given density shreds.
Classroom schematic. One plane, top-down from the north, so the orbit runs counterclockwise. The primary sits at the center. The secondary rides a dashed circle of radius a. The Hill sphere is a circle of radius rH centered on the secondary. The full view and the lower panels do not share a scale. Body disks on the full view are marks when a world would otherwise be a speck. On the Hill panel, the circle and a moon’s orbit, when that orbit is large enough to draw, share one scale. A world smaller than a few pixels is a mark, not its radius. Mars–Phobos adds a third panel for Phobos and the Roche band, and that panel has its own scale. The Hill circle is off that panel. Not an ephemeris. Not a mission-design tool.
From the north · orbit counterclockwise · schematic
Pair
The formulas
rH ≈ a · (m / (3M))1/3. Circular orbit. Secondary mass m much smaller than primary mass M. a is the secondary’s semimajor axis about the primary. This is the headline number on every preset. It is a classroom approximation, not a fitted measurement from a spacecraft.
rH ≈ a · (μ / 3)1/3, with μ = m / (M + m). The same neighborhood when μ is tiny, because m / (M + m) and m / M then nearly agree. This is the form the Lagrange desk prints beside its numerical L1 and L2. For Sun–Earth the two forms round to 1,496,565 km and 1,496,563 km. For Sun–Jupiter they round to 53,147,220 km and 53,130,319 km. The gap is the difference between the two approximations. It is not a measurement error.
The inputs. Sun: mass 1,988,400 × 1024 kg, volumetric mean radius 695,700 km. Earth: mass 5.9722 × 1024 kg, volumetric mean radius 6,371.000 km, semimajor axis 149.598 × 106 km. Moon: mass 0.07346 × 1024 kg, semimajor axis 0.3844 × 106 km. Jupiter: mass 1,898.13 × 1024 kg, volumetric mean radius 69,911 km, semimajor axis 778.479 × 106 km. Saturn: mass 568.32 × 1024 kg, volumetric mean radius 58,232 km, semimajor axis 1,432.041 × 106 km. Mars: mass 0.64169 × 1024 kg, volumetric mean radius 3,389.5 km, semimajor axis 227.956 × 106 km. Phobos, in the Mars satellites table: mass 10.6 × 1015 kg, mean orbital distance 9,378 km. This desk keeps the Earth Fact Sheet’s 5.9722 even where another sheet’s Earth column differs.
Hill is not Roche. Hill is where the secondary dominates the primary’s tide for a test particle. Roche is where a satellite of density ρs cannot hold itself together. The Roche desk’s fluid distance is 2.44 R times the cube root of the density ratio, and the rigid distance uses 1.26. For Mars and Phobos those distances are 10,541 km and 5,443 km. Phobos’s 9,378 km sits between them, and also deep inside Mars’s Hill sphere of 1,084,138 km. Both statements use the fact sheets. They do not contradict.
Where this shows up
Sun and Earth. Earth’s Hill radius is 1,496,565 km, 2.151 times the Sun’s volumetric mean radius, and 0.010004 of Earth’s own semimajor axis. In Earth radii that is 234.903. The μ form, 1,496,563 km, is the number on the Lagrange desk. The Moon at 384,400 km is 0.257 of the sphere. The Earth–Moon button draws that orbit. The Sun–Earth button is the sphere itself.
Earth and the Moon. The preset is still Earth’s sphere about the Sun. The Moon is the comparison, not the secondary in the formula. The Moon’s own Hill sphere, with Earth as the primary and the Moon as the secondary, is a different arithmetic: 61,525 km on the m/(3M) form and 61,274 km on the μ form. The Lagrange desk prints that 61,274 km beside Earth–Moon L1. This page does not draw the Moon’s own sphere. It draws the sphere the Moon sits inside.
Sun and Jupiter. Jupiter’s Hill radius is 53,147,220 km, 76.394 solar radii and 0.068271 of Jupiter’s semimajor axis. That is 0.355 of Earth’s semimajor axis, from dividing the two fact-sheet kilometer figures. NASA’s account of the Trojan asteroids puts one swarm nearly 60° ahead of Jupiter, at L4, and another at L5. Those points sit on Jupiter’s orbit. They are not inside the Hill circle drawn around Jupiter. The Lagrange desk draws L4 and L5.
Mars and Phobos. Mars’s Hill radius is 1,084,138 km, 1.558 solar radii and 0.004756 of Mars’s semimajor axis. Phobos’s mean orbital distance, 9,378 km, is 0.00865 of that sphere, about 320 Mars radii in from the edge to a moon at 2.767 Mars radii. The close panel draws the Roche band from the same coefficients as the Roche desk: fluid 10,541 km, rigid 5,443 km, Phobos between them. Phobos’s own Hill sphere, Mars as primary, comes out near 17 km on this formula, on the order of the sheet’s axes 13.0 × 11.4 × 9.1 km. The point-mass picture is a poor fit at that size. This preset does not draw that circle.
Sun and Saturn. Saturn’s Hill radius is 65,405,072 km, 94.013 solar radii and 0.045673 of Saturn’s semimajor axis. The μ form is 65,398,842 km. Rhea’s semimajor axis on the Roche desk is 527,040 km, which is 0.00806 of this sphere, so Rhea sits inside it. The Roche question for Rhea is the other way: 527,040 km is outside both shredding distances. Hill says Saturn dominates the Sun out here. Roche says Rhea is too far from Saturn to be pulled apart.
Literacy · a sphere, not a mission
Sources · public NASA pages
Classroom picture of the Hill sphere. rH ≈ a (m / (3M))1/3, and the μ form beside it. Sun–Earth: 1,496,565 km and 1,496,563 km. The Moon at 384,400 km is inside. Sun–Jupiter: 53,147,220 km. Mars–Phobos: Mars’s sphere is 1,084,138 km and Phobos at 9,378 km is deep inside, between the Roche distances 10,541 km and 5,443 km. Sun–Saturn: 65,405,072 km. Circular, low-mass, one radius. Not an ephemeris, and not a mission-design tool. Earthrise preview: NASA / Bill Anders (public domain).
Keep going
The shredding distance
Fluid and rigid limits. Rhea and the Moon sit outside both. Phobos sits between them.
The five points
The μ form of this radius, beside a numerical L1 and L2. Trojans at L4 and L5.
The speed
Circular orbit and escape at one altitude. √2 apart. Same fact-sheet worlds.
Neighborhood
Where the Sun, Earth, Mars, Jupiter, and Saturn sit. Still a schematic.
One Hill radius for a primary and a secondary, drawn as a circle around the secondary. The numbers come from NASA fact-sheet masses, radii, and mean orbits. The picture is a classroom schematic, not an ephemeris and not a mission-design tool.
| Sun–Earth | The default. Sun 1,988,400 × 1024 kg, Earth 5.9722 × 1024 kg, separation 149.598 × 106 km. rH is 1,496,565 km. The μ form is 1,496,563 km. |
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| Earth–Moon | The same Earth sphere about the Sun. The Moon’s semimajor axis, 384,400 km, is drawn inside it. That orbit is 0.257 rH. The Moon is not the secondary in the formula on this button. |
| Sun–Jupiter | Jupiter’s mass is 1,898.13 × 1024 kg and its semimajor axis is 778.479 × 106 km. rH is 53,147,220 km, 0.068271 of that orbit. The Trojan swarms sit on the orbit, about 60° ahead and behind, not inside this circle. |
| Mars–Phobos | Mars’s sphere about the Sun is 1,084,138 km, from mass 0.64169 × 1024 kg and semimajor axis 227.956 × 106 km. Phobos at 9,378 km is 0.00865 of that. The close panel is the Roche band: fluid 10,541 km, rigid 5,443 km, Phobos between. |
| Sun–Saturn | Saturn’s mass is 568.32 × 1024 kg and its semimajor axis is 1,432.041 × 106 km. rH is 65,405,072 km. Rhea’s 527,040 km orbit sits inside this sphere and outside both Roche limits. |
| m / (3M) | rH ≈ a · (m / (3M))1/3. Headline on every preset. a, m, and M are the fact-sheet values named above. Rounded to the nearest kilometer. |
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| μ / 3 | rH ≈ a · (μ / 3)1/3, μ = m / (M + m). Printed beside the headline. This is the Lagrange desk’s Hill approximation. The two forms agree when μ is tiny. |
| Primary radii | rH divided by the primary’s volumetric mean radius. For these presets the primary is the Sun, so the unit is solar radii. |
| Units of a | rH / a. Earth’s figure is 0.010004. Jupiter’s is 0.068271. That is the fraction of the orbit the sphere occupies. |
| Moon / r_H | On Earth–Moon and Mars–Phobos, the moon’s mean distance divided by the planet’s Hill radius. Less than 1 means the moon’s orbit is inside the sphere. |
| Presets | Tab to a pair button. Enter or Space selects it. Left and Right arrows step through the five pairs when focus is on a pair button, the drawing, or the page background. The pressed button is marked aria-pressed. |
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| Play | Walks the secondary counterclockwise, as seen from the north. A moon mark, when one is drawn, walks its own circle. About 12 seconds per turn. A time-lapse, not the real period. Press again to pause. Enter or Space on the button does the same. |
| Touch | Pair buttons, Play, the question mark, and Close are at least 44 pixels on a side. The drawing does not drag. Tap a pair. |
| This dialog | The question mark opens it. Escape closes it. Tab stays inside until you close it. Click the dim page behind the card to close it. |
| Full view | Primary at the center. Dashed circle: the secondary’s orbit. The Hill circle rides with the secondary when it is large enough to see. Disks are marks if the true radius would be a speck. |
| Hill panel | The secondary’s neighborhood, magnified, own scale. It does not spin with Play, except the moon mark. The dashed circle is rH. Earth–Moon draws the Moon’s orbit inside it. Phobos’s orbit is too small for this panel. |
| Close panel | Mars–Phobos only. Phobos’s orbit, the fluid Roche arc, and the rigid Roche arc, on their own scale. The Hill circle is far outside this panel. The caption says so. |
| N | North, the view. Motion runs counterclockwise, the same sense as the other top-down desks. |