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The region beyond Neptune is full of dozens, if not hundreds, of dwarf planets and possible dwarf planets. Why can't the interstellar probes visit some of them anyway, along their path? The only TNOs ever visited were Pluto (with its satellites) and Arrokoth (former Ultima Thule), the latter of which is a (twin) asteroid rather than a possible dwarf planet.

Other than many bodies' highly inclined orbits, what makes it so unlikely for the Voyager probes and New Horizons to approach another TNO? It's not like you need to come to a thousand miles from a body's surface, a million miles would suffice for many significant observations and discoveries. Are there any trans-Neptunian objects New Horizons will likely approach within 10 million mi distance?

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    $\begingroup$ "Space [...] is big. Really big. You just won't believe how vastly hugely mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist, but that's just peanuts to space." [Ford Prefect, by way of Douglas Adams]. $\endgroup$ – Steve Linton Dec 21 '20 at 14:13
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    $\begingroup$ @SteveLinton I awaited that quote, wondering how soon it would appear! :-) $\endgroup$ – Greenhorn Dec 21 '20 at 15:22
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    $\begingroup$ @Greenhorn Pluto's light, and New Horizons is fast. The combination makes Pluto a terrible target for gravity assists, if New Horizons was also going to arrive at Pluto in a timely manner. $\endgroup$ – notovny Dec 21 '20 at 18:47
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    $\begingroup$ This question combined with your other question about the hole in the Zvezda module make me think that you need to fundamentally reassess your understanding of scale. $\endgroup$ – RonJohn Dec 22 '20 at 5:43
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    $\begingroup$ @RonJohn To be fair, humans are really, really bad with that particular concept. $\endgroup$ – chrylis -cautiouslyoptimistic- Dec 22 '20 at 20:23
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It's a small matter of propulsion, or rather a matter of small propulsion.

Pluto orbits just over 5 billion kilometers from the sun, if you look at the volume of the space between 5 billion km and 8 billion km you get 1.6210618e+30 cubic km, that's 1621061800000000000000000000000 cubic kilometers of space. There may be thousands of objects in that area, even if there's a million each one still has 1621061800000000000000000 cubic kilometers of space.

If we assume that all the objects are in the same ecliptic plane we can look at the 2d figures instead. Looking a the area of a 2d space between 5 and 8 billion km we get 1.2252211e+20 square kilometers, that's 122522110000000000000 square kilometers of space. If you have a million objects of interest in that space each one still has 122522110000000 square kilometers to be in. Of course they are not evenly spaced out, but these figures are intended to give you an idea of how vast these spaces are.

The New Horizons probe has 16 small hydrazine thrusters and carried 76kg of hydrazine at launch, enough for 400 m/s delta-V. It's current speed is over 14,500 m/s, the spacecraft cannot slow down appreciably, all it can do is influence its course a bit to study any objects of interest. Note that despite all the vastness NASA did send New Horizons to study Arrokoth, which is a Kuiper belt object, as it was very close to the spacecraft's course and already known about. Now there's a problem of where to go, there isn't another known target close to the spacecraft's path that it can get to with such a small amount delta-v available. The spacecraft's own instruments may find something as it zooms out of the solar system, if so maybe we will get a chance to see something else.

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    $\begingroup$ @Greenhorn Using units which are well specified over the whole world should be a logical thing to do, the biggest random event in your life should have no influence on that. $\endgroup$ – Arsenal Dec 22 '20 at 14:02
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    $\begingroup$ The random event is where you are born. Astronomy uses SI units, as does any serious science; a number of disasters have been caused by metric/Imperial conversion in the past. $\endgroup$ – pjc50 Dec 22 '20 at 15:09
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    $\begingroup$ @Greenhorn The region proposed in the answer has a volume of $1.6\times 10^30 \mathrm{km}^3$. The volume within 16 million km of a straight path through that volume is $(1.6\times 10^7)^2\pi \times3\times 10^9 < 3\times 10^24\mathrm{km}^3$. That is: the ship passes within 16 million km of roughly 3 parts in a million of the total volume. Thus, for a first approximation, the odds of passing that close to any given object in the volume are around 3 in a million. In other words: you'd need about $230,000$ objects out there to have a $50\%$ chance of passing that close to at least one of them. $\endgroup$ – user3482749 Dec 22 '20 at 20:07
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    $\begingroup$ @Greenhorn looking forward to all those conversion factors between "slugs", "acre-feet", "BTU" etc when you could just have multiples of 10 $\endgroup$ – pjc50 Dec 23 '20 at 9:15
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    $\begingroup$ @Greenhorn, I'm an American with a science degree, and I can say in my nearly 18 years of science and mathematics education in the USA I was taught to use metric the entire time, the only work on imperial measurements was to convert them to metric. About the only time I use imperial measurements is to report my altitude in feet and speed in knots to air traffic control. $\endgroup$ – GdD Dec 23 '20 at 21:29
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Try to imagine a planetary walk. All distances and diameters are scaled down by a factor of 1E-9 or 1 to 1 billion. On such a walk the Sun is a sphere of 1.4 m diameter.

From Sun to Earth you have to walk only 150 meter, Earth is a blue marble of 13 mm diameter.

To Pluto (49.305 AU or 7.37593 billion km) it is 7.37 km to walk. But be careful not to overlook Pluto, it's diameter is only 2.37 mm. The moons of Pluto: Charon is 1.2 mm, for Nix and Hydra you would need a microscope, they are about 50 µm, fifty millionth of a meter. The distance from Pluto to Hydra is only 0.05 m.

225088 Gonggong, Aphelion 101.238 AU (15.145 Tm), Perihelion 33.703 AU (5.042 Tm), diameter 1230 km. On our walk the distance to Gonggong is 5.042 to 15.145 km, it is 1.23 mm small. I love distances in terameter Tm, they are so easy to scale down by 1 billion.

Sedna is very far away, 76.257 AU (11.4079 Tm) to 937 AU (140.2 Tm), it's dimension is about 1000 km. On the planetary walk 11 to 140 km distance and the dimension 1 mm. Distances less than 15 km may be walked in less than 4 hours, but 140 km may take 7 seven days (20 km each day, then rest).

Read Wikipedia about the Sagan Planet Walk, I recommend reading carefully the Table of scaled sizes and distances. Both planet sizes and distances between them are scaled down to one five billionth of its actual size.

The smaller scale proposed by Greenhorn is not useful. If 1 AU (1.495E11 m) is scaled to 3 yard ( 3* 0.914 m), the Earth is 233 µm or 0.233 mm small. Pluto is too tiny to be seen. For a planetary walk both the distance and the dimension of the planets should be visible to the user. If you scale down to some meters only, even larger planets will be invisible.

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    $\begingroup$ What about the TNOs beyond the Pluto-Charon system? That's what the question asks. $\endgroup$ – Camille Goudeseune Dec 22 '20 at 5:38
  • $\begingroup$ The Sagan Planet Walk is quite big. A smaller scale would be more overseeable such as 1 au = 3 yards. $\endgroup$ – Greenhorn Dec 22 '20 at 7:07
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    $\begingroup$ @Greenhorn It wouldn't be overseeable if you used yards. $\endgroup$ – Nobody Dec 22 '20 at 20:10
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    $\begingroup$ @Greenhorn If you want a smaller and more overseeable planet walk, you did not understand the concept of a planet walk. If the model is too small, the diameter of some planets models are too small to see them. $\endgroup$ – Uwe Dec 23 '20 at 0:51
  • $\begingroup$ @Uwe Why do you think I "didn't understand" it? I'm conscious that one couldn't see Ceres or Charon on a too small scale. But definitely the eight recognized planets, Titan and Ganymede, the Sun and ,if included, planet Nine. $\endgroup$ – Greenhorn Dec 23 '20 at 7:35
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For reasons mentioned in the other answers, New Horizons is not going to fly close to many TNOs aside from Pluto. But it can image dozens of objects from a distance using its long-range camera, the Long-Range Reconnaissance Imager (LORRI). From NASA:

During its extended mission in the Kuiper Belt, which began in 2017, New Horizons is aiming to observe at least two-dozen other KBOs, dwarf planets and “Centaurs,” former KBOs in unstable orbits that cross the orbits of the giant planets. Mission scientists study the images to determine the objects’ shapes and surface properties, and to check for moons and rings. The spacecraft also is making nearly continuous measurements of the plasma, dust and neutral-gas environment along its path.

LORRI was also a key instrument in closer-in portions of the New Horizons missions, enabling:

  • the observation of Jupiter's moons from millions of kilometers away during the gravity assist (had the spacecraft flown closer to the giant planet, the deflection in its path would have been too much to reach Pluto). The highlight, at least in popular headlines, was capturing a real-time volcanic eruption on Io.

  • improved images of Pluto while still several Earth-days and millions of kilometers away from the dwarf planet. Without this long-range capability, the slow rotation of Pluto would have prevented large portions of the surface from being imaged.

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