Vol. Science · Astronomy · The flyby
A planet does not catch a probe and throw it. In the planet’s frame the flyby is an unpowered hyperbola. Far away, on the way in and on the way out, the speed relative to the planet is the same excess, v∞. What changes is the direction. The eccentricity is e = 1 + rp v∞² / μ. The turn of that excess vector is δ = 2 arcsin(1/e). μ is the planet’s GM. rp is periapsis, measured from the planet’s center, and this desk will not put it inside the volumetric mean radius. In the Sun’s frame the planet is already moving at Vp. Add Vp to the incoming excess and that is the inbound heliocentric velocity. Rotate the excess by δ and add Vp again and that is the outbound one. The magnitude of the heliocentric change is |Δv| = 2 v∞ sin(δ/2) = 2 v∞ / e. The energy is exchanged with the planet’s orbit. For the planet the change is tiny. This page does not invent a spacecraft mass to divide it by. μ and the volumetric mean radius for Earth, Venus, Mars, and Jupiter are the NASA NSSDCA fact sheets, the same figures the Escape desk uses. Vp is the circular speed at each fact-sheet semimajor axis, from the Sun Fact Sheet GM, 132,712 × 106 km3/s2. It is not the sheet’s mean orbital velocity. The Earth cartoon uses a classroom excess of 5 km/s and a periapsis 400 km above the volumetric radius, so rp = 6,771 km. That 400 km is the classroom altitude on the Oberth desk. The fact sheet does not print it. With Earth’s GM, 0.39860 × 106 km3/s2, the eccentricity is 1.4247, the turn is 89.16°, and the speed at periapsis is 11.947 km/s. Vp is 29.785 km/s. Aimed so the turn lies along the planet’s motion, heliocentric speed goes from 26.515 km/s to 33.484 km/s. The assist magnitude is 7.019 km/s. Pass ahead of the planet and those two speeds swap. Venus, Mars, and Jupiter start at the same 5 km/s and the same periapsis ratio, rp/R = 6,771/6,371. Jupiter’s turn is 160.50°. Mars’s is 37.60°. The difference is μ. Pricing that 7.019 km/s with the rocket equation, at the rocket desk’s classroom 450 s and g0 = 9.80665 m/s², leaves mf/m0 = 0.2038. The flyby itself burns nothing. The half-ellipse between two circles is Hohmann. Speed on one ellipse is vis-viva. The neighborhood where the planet, not the Sun, dominates a test particle is the Hill desk.
Classroom schematic. Point-mass planet, unpowered hyperbola, planet velocity held constant across the flyby. The aim puts the heliocentric Δv along Vp, which raises or lowers speed as much as this δ allows. It is not a B-plane target. No drag, no oblateness, and no third body. Not a mission planner, not JPL Horizons, and not patched-conics software. This page does not do patched conics. The 5 km/s excess is a classroom example. The matched periapsis ratio is Earth’s volumetric radius plus that 400 km classroom altitude. Earth, Venus, and Mars have atmospheres this picture does not model. Jupiter’s equatorial 1-bar radius is 71,492 km, which is 1,581 km outside the volumetric mean radius. A periapsis between those two is inside the 1-bar surface the fact sheet uses for escape speed.
Planet frame · motion to the right · the hyperbola bends v∞
Classroom preset
Which side of the planet
Altitude above the volumetric mean radius, in km
Enter an excess above 0.5 km/s, and an altitude of zero or greater.
Earth · classroom cartoon
v∞ 5.000 km/s · 400 km above the volumetric radius
Turn · δ
89.16°
e = 1.4247
Assist · |Δv|
7.019 km/s
2 v∞ / e
Heliocentric in
26.515 km/s
V_p + v∞ in
Heliocentric out
33.484 km/s
V_p + v∞ out
Pass behind Earth. v∞ is 5 km/s on the way in and on the way out. The hyperbola turns it 89.16°. Added to V_p, heliocentric speed goes from 26.515 km/s to 33.484 km/s. The magnitude of that change is 7.019 km/s.
V_p is the circular speed at Earth’s semimajor axis. The sheet’s mean orbital velocity is 29.78 km/s. The 400 km altitude is a classroom example. Earth’s air is not in this model.
The picture people reach for
The planet does not catch and throw the probe like a baseball. The hyperbola rotates v∞ while the planet keeps moving. In the planet’s frame the speed at infinity is the same before and after. Only the direction changes. Heliocentric speed changes because that rotated excess is added to a velocity the planet already had. Slow the probe in the planet frame and you have invented a burn. This page has no burn.
Pass behind, along the orbit, and the outbound excess points more nearly with Vp. The craft leaves faster than it arrived. Pass ahead and the outbound excess points more nearly against Vp. The craft leaves slower. Both are the same hyperbola. The sign is which side you aimed for. The planet’s orbit gains whatever the craft loses, and loses whatever the craft gains. Dividing that by a spacecraft mass is a mission. This desk does not do it.
What the formula is
e = 1 + rp v∞² / μ. μ is GM of the planet. rp is the closest approach, from the center. v∞ is the speed relative to the planet, far away. On a hyperbola the specific energy is v∞²/2, so the same speed comes back out. A larger μ, or a closer periapsis, makes e closer to 1 and the bend sharper. Jupiter’s GM is 126.687 × 106 km3/s2. Earth’s is 0.39860 × 106. At the same v∞ and the same rp/R, Jupiter turns harder.
δ = 2 arcsin(1/e). That is the angle between the incoming excess and the outgoing excess. The asymptotes sit at true anomaly θ∞ = arccos(−1/e), and δ = 2θ∞ − 180°. When e = 2, δ is 60°. When e is barely above 1, the turn is nearly 180° and the excess almost reverses. The speed at periapsis is vp = √(v∞² + 2μ/rp). That is the fast point. It is not a burn.
v = Vp + v∞. Vp on this desk is √(μSun / a), a circle at the fact-sheet semimajor axis. The sheet’s mean orbital velocity is a different print, and the readout keeps both. This aim sets the incoming and outgoing excess symmetric about the line perpendicular to Vp, so the heliocentric change lies along the planet’s motion. Then |vout|² = Vp² + v∞² + 2 Vp v∞ / e and |vin|² = Vp² + v∞² − 2 Vp v∞ / e. The specific-energy change at that same a is Vp times the assist magnitude. A different aim would spend some of δ on direction instead of speed. This page shows the speed.
The rocket line is a comparison, not a burn. The rocket desk writes Δv = Isp · g0 · ln(m0/mf) with g0 = 9.80665 m/s². The 300 s and 450 s figures are that desk’s classroom examples. They are not engines. Putting this page’s |Δv| into that equation asks what an ideal stage would spend to buy the same change of speed in one burn, with no gravity losses and no drag. The flyby does not spend it. The 7.019 km/s Earth cartoon, at 450 s, leaves mf/m0 = 0.2038. The fraction uses the unrounded assist magnitude. It is not a propellant load.
Words
Classroom picture of one unpowered hyperbola. Earth’s 5 km/s cartoon at 400 km turns 89.16° and the heliocentric change is 7.019 km/s, from 26.515 km/s to 33.484 km/s. Jupiter, at the same excess and the same periapsis ratio, turns 160.50°. Mars turns 37.60°. Fact-sheet GM and volumetric radii. Vp is a circular speed, not the sheet’s mean. The 5 km/s excess and the 400 km altitude are classroom examples. Not a mission planner, not Horizons, and not patched conics. Earthrise preview: NASA / Bill Anders (public domain).
Keep going
The speed
Circular orbit and escape at one altitude. √2 apart. The same fact-sheet GM and volumetric radius. Not this flyby.
The coast
Two burns and half an ellipse. Earth to Mars is 5.594 km/s. This page does not splice a flyby into that coast.
The burn
The same impulsive Δv buys more energy where the craft is already fast. A flyby is not that burn.
The propellant
What an ideal stage would spend for this |Δv|. At 7.019 km/s and a classroom 450 s, mf/m0 = 0.2038. The flyby burns none of it.
The ellipse
Speed anywhere on the heliocentric coast. v = √[μ (2/r − 1/a)]. Not the planet-frame hyperbola.
The sphere
Where the planet, not the Sun, dominates a test particle. Jupiter’s sphere is 53,147,220 km. This page stays in the planet frame.
One unpowered hyperbola and the velocity addition that follows. The planet frame turns v∞. The Sun frame adds V_p. Not a mission planner, not Horizons, and not patched conics.
| Earth | GM 0.39860 × 106 km3/s2. Volumetric mean radius 6,371.000 km. Starts at a classroom 5 km/s and 400 km altitude. That altitude is an example. The fact sheet does not print it. |
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| Venus | GM 0.32486 × 106 km3/s2. Volumetric mean radius 6,051.8 km. Same classroom excess. Periapsis ratio matched to Earth’s 6,771/6,371. |
| Mars | GM 0.042828 × 106 km3/s2. Volumetric mean radius 3,389.5 km. Same excess, same ratio. The turn is the small one. |
| Jupiter | GM 126.687 × 106 km3/s2. Volumetric mean radius 69,911 km. Same excess, same ratio. The turn is the large one. The equatorial 1-bar radius is 71,492 km. |
| Arrow keys | With a planet button focused, Left and Right move through Earth, Venus, Mars, and Jupiter. With a side button focused, they switch behind and ahead. They do nothing while a field is focused, and nothing while this dialog is open. |
| Raise · pass behind | Periapsis on the trailing side. V_p points right, so the closest point is to the left of the planet. Heliocentric speed rises. |
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| Lower · pass ahead | Periapsis on the leading side, to the right of the planet. Heliocentric speed falls. |Δv| is the same number. |
| v∞ | Hyperbolic excess, in km/s. The slider starts at 0.5 to 15. Typing a larger number extends it, up to 80. 5 km/s is the classroom example. It is not a mission approach speed. |
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| Periapsis | Altitude above the volumetric mean radius. Zero is the soft floor. The slider is cubic out to 20 radii, so the low altitudes are reachable. The field accepts up to 80 radii. |
| 5 km/s | Puts the excess back on the classroom example. It does not move the altitude. |
| Matched ratio | Puts r_p/R back on 6,771/6,371, the Earth 400 km cartoon scaled to this planet. It does not move v∞. |
| Planet frame | The sphere, the hyperbola, and the dashed asymptotes. The sphere is the volumetric radius. If it would be a speck, it is drawn larger and the hint says so. |
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| V_p | The planet’s circular velocity, pointing right, along the orbit. Its length in the left panel is a marker. The right panel scales it with the other velocities. |
| Sun frame | V_p, then the two excess vectors from its tip, then the heliocentric inbound and outbound from the origin. The arc is δ. |
| Turn | δ in degrees. The line under it is e. |
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| Assist | The magnitude of the heliocentric velocity change, 2 v∞ / e. Not a burn. |
| Heliocentric in / out | Speed before and after, in the Sun frame, for this aim. |
| V_p and the sheet mean | V_p is √(μ_Sun / a). The sheet mean is the fact sheet’s mean orbital velocity. They are printed apart on purpose. |
| 450 s and 300 s | The ideal rocket remaining fraction for this |Δv|, using the rocket desk’s classroom specific impulses and g0 = 9.80665 m/s². Not a propellant load. |
| Air | Earth, Venus, and Mars have atmospheres. A low periapsis is not a corridor. Jupiter’s 1-bar surface is the equatorial radius, 1,581 km outside the volumetric floor. The cadence says when the periapsis is inside it. |
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| Patched conics | Not done. The hyperbola is not stitched to an arrival ellipse or a departure ellipse. |
| A mission | No spacecraft, no ephemeris, no Horizons state, no named flyby. The presets are cartoons computed from the fact sheets. |
| ? | Opens this dialog. Escape closes it. Tab stays inside until it closes. |
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| Escape | Closes the dialog and returns focus to the ? button. |