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"Most difficult", as in the minimum $\Delta v$ required, the objective being capture into an orbit around the object.

Looking through some Delta-v maps, I would suspect the answer to be some inner moon of Jupiter or Saturn, but those maps are not including every solar system object, they don't always take full advantage of all the routes available in the patched conics approximation, mostly ignore flyby routes, and often simplify the aerobrake options available.

Thus, I'm looking for the worst case when also considering the more complex trajectories. It should be practical to execute within a couple of decades though.

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    $\begingroup$ Getting close to the sun is pretty hard in terms of delta-v. It looks like it takes 29km/s to hit the sun, but only 12km/sec to leave the solar system. $\endgroup$ – zeta-band Sep 17 '19 at 18:30
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    $\begingroup$ Once you consider flyby routes, it takes quite a bit less than that. Also, an escape is equivalent to hitting the Sun, as you can do a very small manoeuvre when far away, and fall back into the sun. But anyway, this question is about achieving orbit, and achieving a Sun orbit just requires escaping the Earth. $\endgroup$ – SE - stop firing the good guys Sep 17 '19 at 18:50
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A couple of decades is long enough to reach more or less anywhere in the solar system by launching onto a Venus transfer and then using two or three Venus and Earth gravity assists to get to Jupiter and a Jupiter assist to get to your destination. For some destinations you can miss out the trip to Jupiter, so to a rough approximation, you can arrive anywhere on a Hohmann transfer orbit from Venus, Earth or Jupiter for the same cost (the cost of the original boost to Venus transfer).

So now you are left with trying to stop and get into orbit. For anything with an atmosphere you can (in principle) use aerocapture to get into orbit. That covers any planet except Mercury and Pluto, although this is hard and untried technology. You can also use aerocapture at the primary to get you onto a local Hohmann tranfer orbit to any moon of any of those, reducing your arrival velocity. That suggests that the hardest target is either going to be Mercury, or a small asteroid or comet. If there is one, I'd guess at an asteroid in a highly inclined orbit as close to the Sun as possible. You're going to have to go out to Jupiter for the plane change and then get rid of all that velocity on the way back in. You won't be able to aerobrake at Earth or Venus because you won't be in their orbital plane.

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    $\begingroup$ This illustrates the intuition behind it very well. Do you have any candidates for asteroids that have this kind of orbit? $\endgroup$ – SE - stop firing the good guys Sep 17 '19 at 23:54
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    $\begingroup$ A search of the JPL small bodies database ssd.jpl.nasa.gov reveals a fair few asteroids with these kinds of orbits (perihelion < 0.5 AU, semi-major axis < 1 AU, inclination > 40 degrees for example). I haven't the time or skills to work out which of those is going to be hardest to match orbits with. $\endgroup$ – Steve Linton Sep 18 '19 at 7:10
  • $\begingroup$ I think I will just go ahead and accept this. Unless someone can show that there's some place else that's more difficult to reach, this is the most promising candidate. $\endgroup$ – SE - stop firing the good guys Sep 18 '19 at 19:05
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Among objects hanging around Earth's orbit, a surprising candidate is 2010TK$_7$ (Wikipedia), famous for being the first (and currently, the only) known Earth trojan. One might expect that a trojan asteroid would be easy to reach from the associated secondary body (Earth in this case), but this is true only if the trojan stays in the orbital plane. In reality, 2010TK$_7$ librates far from the Earth orbital plane and requires $\Delta v=9.4\text{ km/s}$ to reach it, more than double the $\Delta v$ for some other near-Earth objects that stay close to our orbital plane.

Here's the orbital path relative to the sun in a corotating frame with Earth (blue dot, lower left): Animated gif of 2010 TK7.

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    $\begingroup$ Here's a visualization I found helpful: en.wikipedia.org/wiki/File:Animation_of_2010_TK7.gif $\endgroup$ – Magic Octopus Urn Sep 23 '19 at 21:02
  • $\begingroup$ Yes I've seen that. Can this be incorporated into the answer, and if so how? Thanks! $\endgroup$ – Oscar Lanzi Sep 23 '19 at 21:04
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    $\begingroup$ Ah, I take it back, the reason you can't add that specific one to the answer is because it's freakin' huge! A link may work best here. Feel free to roll back that edit. $\endgroup$ – Magic Octopus Urn Sep 23 '19 at 21:10
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    $\begingroup$ Edit looks good. Lanzi's Law: Everything that's really good on SE is either freakin' huge or behind a freakin' paywall. $\endgroup$ – Oscar Lanzi Sep 23 '19 at 21:17
  • $\begingroup$ @JCRM thoughts on What would be proper site hygiene for large image hot linking? $\endgroup$ – uhoh Sep 23 '19 at 23:36
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I can't identify the object, only describe it: A comet that is orbiting retrograde, as short a period as you can find.

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