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In this answer to Why does JWST have such a big Blind Spot? I argue that this space telescope primary relationship with time is that it strives to look way back in it and so as long as a given direction is visible sometime during the year and it doesn't matter that at any moment only about 40% of $4 \pi$ SR is viewable (polar angles 85° to 135° wrt the Sun, see below).

But now I'm wondering

  1. If there are some types of transient events where an infrared telescope like this (0.6 to 28 microns I think?) could offer something that Hubble or ground observatories couldn't. For that I've already asked For what types of transient events would the JWST be ideal or even uniquely-suited for observing? (If it could look in time)

But here I'd like to ask:

Question: What is the fraction of the time that the JWST could view a short transient event on-demand as a function of position on the celestial sphere?"

The author of that question gives a few examples just to illustrate the idea of transient events

Over a 6 month half-orbit of the sun, the entire sky is observable. But what happens if an interesting event occurs in the “blind spot”? Like Shoemaker-Levy or the arrival of Rama? A gravitational wave event?

and (at least) one of those really did "shine" in the thermal infrared!

JWST's field of regard (FOR) from the linked question and originally from JWST field of regard (FOR) Also see How much of the sky can the JWST see?.

JWST's field of regard (FOR)

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    $\begingroup$ Note the timescale for Target of Opportunity (ToO) observations for both HST and JWST are long for a substantial fraction of transient science. Non disruptive ToOs, which already require significant additional justification in the proposal, take at least 2 weeks before execution. Disruptive ToOs, of which there can only be 8 triggers per cycle, take 48 hours (Source: JWST ToO restrictions) $\endgroup$
    – astrosnapper
    Commented Dec 7, 2021 at 17:18
  • $\begingroup$ Space Telescope Science Institute video Webb's Field of Regard $\endgroup$
    – uhoh
    Commented Aug 23, 2022 at 14:38
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    $\begingroup$ @astrosnapper That link is now dead. But there's general ToO info on jwst-docs.stsci.edu/methods-and-roadmaps/… $\endgroup$
    – PM 2Ring
    Commented Dec 28, 2022 at 19:45
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    $\begingroup$ There's now a new link for Cycle 2 but your link looks to be the better and more general/stable one @PM2Ring $\endgroup$
    – astrosnapper
    Commented Dec 28, 2022 at 23:05

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Depends largely on the targets elevation/altitude/latitude above the solar ecliptic.

Best case is for targets that are within 5° of the poles of the solar ecliptic. These targets never leave the Field of Regard for JWST.

JWST operates in an ecliptic coordinate framework, there are 2 small continuous viewing zones (CVZs) centered at each of the ecliptic poles (see Figure 6). The 85° solar exclusion zone then determines the radius of the allowed CVZs to be essentially 5°, although any observation approaching the 85° limit will have additional limitations due to safety considerations. 1

Blind time increases for targets farther from the poles of the solar ecliptic.

Worst case is for targets that are directly on the solar ecliptic.

Field of Regard "Width": 50° (135° - 85°)
Field of Regard period: 1 sidereal year

The target will pass through the Field of Regard twice in a given period.

50° / 360° * 1 sidereal year = 50.7300504 days

By Angle:

 In:  50°
Out:  90° anti-solar blind spot
 In:  50°
Out: 170° sun shield blind spot
Repeat...

By Time:

 In:  50.7300504 days
Out:  91.3140908 days anti-solar blind spot
 In:  50.7300504 days
Out: 172.482171  days sun shield blind spot
Repeat...

I guess this isn't the precise function you are asking for, but maybe it provides someone else insights to discover the function. If I manage to do it I will update this answer.

Update: Here is a plot of said function against ecliptic latitude. Graph of the function against ecliptic latitude

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    $\begingroup$ I just realized that JWST is in the ecliptic coordinate system and not in the equatorial coordinate system like all other telescopes. I suppose astronomers are already used to translating between all the astronomical coordinate systems. $\endgroup$
    – Gabriel
    Commented Mar 4, 2022 at 21:06

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