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Assume the object is small, spherical, and illuminated only by the Sun, and has some apparent motion. The object is either:

a) white - reflecting 100% perfectly diffusely (assume low temperature)

b) black - absorbing 100% and in uniform thermal equilibrium re-radiating infrared

c) Arrokoth-like - having a similar albedo spectrum (reddish) and temperature (?).
(Is it possible/likely that JWST could find a new target for New Horizons to visit?)

d) Voyager-like - having ~5 kW of RTGs at ~570 K ($\lambda_{max} = 5 \mu m$).
(At what distance could JSWT still see Voyager?)

[Edit responding to @astrosnapper]
It may be a rather artificial question, but assume we know closely where to look, e.g. the object is known to be in the field of view.

I'm adding a quick back-of-the-envelope calculation that answers the last question (JWST & Voyager):

Sun's apparent bolometric magnitude is −26.832 corresponding to 3.0128×10^28 Watts of power. Voyager with 5 kW would have an apparent magnitude of
−26.832 - 2.5log10(5000/(3.0128×10^28)) = 35.1 at 1 AU.
JWST has a limiting magnitude of 34, So Voyager RTGs are not quite visible to JWST at 1 AU and obviously totally invisible at 155 AU :-(

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    $\begingroup$ Presumably you are referring to detecting known things (or at least very common things) ? Both JWST and HST have tiny fields of view, making them terrible for performing surveys for rare and faint objects because it's so expensive in terms of time. The HST search for a post-Pluto flyby target for New Horizons (Stern et al 2018, Section 5) was only granted time after several years' search with 8m telescopes + wide field cameras and coming up empty (due to stellar crowding) $\endgroup$ Jan 15 at 0:59
  • $\begingroup$ @astrosnapper Excellent point. I'll try to clarify/restrict the question $\endgroup$
    – Roger Wood
    Jan 15 at 1:37
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    $\begingroup$ What's a) white and b) black and c) red all over? (A newspaper) $\endgroup$
    – uhoh
    Jan 16 at 4:36
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    $\begingroup$ @uhoh hah! yes. That would be interesting. I'm guessing a newspaper has an albedo of ~0.8 and is spectrally flat. but there's probably not very many in heliocentric orbit. $\endgroup$
    – Roger Wood
    Jan 16 at 19:32
  • $\begingroup$ @uhoh or a bleeding zebra. But back to the question: what's the shortest focal range design for each of these 'scopes? $\endgroup$ Jan 17 at 15:08

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