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Mercury's orbital speed is 48 km/s or about 10 AU/year. Wouldn't probes accelerate towards the Sun if we give them a little push from LEO, and only 0.6-1.6 AU from here take a big gravity assist from Mercury? Its orbital period is only 3 months, so we could frequently choose any direction of its graviational assist we need to get to any other planet (compared to Jupiters 11 year orbit).

How large fraction of Mercury's orbital speed could a probe get through gravity assist, if it passed really nearby? And could a gas giant capture into its orbit a probe with such high speed? Would the heat at Mercury be a challenge for a flyby probe?

I don't think Mercury has been used for gravity assists inspite of its vicinity and high speed, so what's the catch?

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It takes more than a little push to drop a probe to Mercury. A 1 AU x .387 AU heliocentric orbit would be moving 22.25 km/s at aphelion. Earth moves about 30 km/s. So Vinfinity for Trans Mercury Insertion would be 7.78 km/s. From LEO the burn would be about be about 5.5 km/s.

At perihelion the ship would be traveling 57.5 km/s wrt sun. It'd be traveling a hyperbolic path wrt to mercury. Vinfinity wrt to Mercury is about 9.2 km/s. If Mercury periapsis were 0 km from Mercury's surface, turning angle of hyperbola would be about 10.2 degrees.

I think the Mercury gravity assist would be good for 1.7 km/s at best. Which wouldn't be sufficient to boost aphelion to Mars, much less Jupiter.

From LEO, Trans Jupiter Insertion is about 6.3 km/s

Regarding a gas giant's gravity capturing the probe: An earth to Jupiter Hohmann would be a hyperbola wrt to Jupiter. Vinfinity wrt to Jupiter would be about 6 km/s. From Jupiter's view point, it'd see the distant probe approaching at 6 km/s. It'd accelerate as it fell into Jupiter's well, then after Jupiter periapsis as it climbs up the well. Eventually it'd slow to 6 km/s wrt to Jupiter. From Jupiter's point of view, the probe would have the same velocity outgoing as it does incoming. It is with regard to the sun that Jupiter's gravity changes the velocity.

Jupiter capture might be possible through one of Jupiter's moons changing the probes path, or maybe if the probe's periapsis passes through Jupiter's atmosphere. But Jupiter's gravity alone would not change a hyperbola to an ellipse.

Here's a pic showing how a hyperbola's velocity approaches Vinf as it grows more distance from the central body:

enter image description here

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    $\begingroup$ The picture is from my orbital mechanics coloring book: amazon.com/Conic-Sections-Celestial-Mechanics-Coloring/dp/… $\endgroup$
    – HopDavid
    Mar 28, 2014 at 4:24
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    $\begingroup$ Gravity assists by other planets can give much more delta-V (~10 km/s at Jupiter). $\endgroup$
    – Hobbes
    Mar 28, 2014 at 17:46
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    $\begingroup$ There are two important factors for the gravity assist: Vinfinity and turning angle. delta V is 2 * sin(turning angle/2) * Vinfinity. Here's a pic: 3.bp.blogspot.com/-MrY__qORzRQ/UWyyc1o2HjI/AAAAAAAAAZw/… $\endgroup$
    – HopDavid
    Mar 28, 2014 at 19:31
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    $\begingroup$ I suspect that the amount of delta-V a planet can impart depends on the size of its gravitational field. Jupiter will start pulling on the probe from a very long distance. You'd have to get much closer to Mercury to begin to feel its effects. I'll have to dig deeper to get a real answer, though. $\endgroup$
    – Hobbes
    Mar 28, 2014 at 20:50
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    $\begingroup$ @MagicOctopusUrn That's because it's out of print. But I'm working on a second edition. Thanks for your interest! $\endgroup$
    – HopDavid
    Jul 24, 2019 at 0:27

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