See "Energy management operations for the Insight solar-powered mission at Mars" https://ieeexplore.ieee.org/document/7943965/

Article is paywalled but the figures suggest that all of this analysis was done before the current dust storm, and the maximum tau value shown in the figures was about 5 IIRC.

It looks like the design team is, reasonably, planning to survive under then worst-case conditions:

"The lander and its unique suite of instruments - a 3-axis
precision seismometer and a self-drilling heat probe both
deployed with a robotic arm onto the Martian soil - have been
designed to operate and/or survive while maintaining a
margined battery-charging energy balance every sol, even
under the combined effect of worst-observed atmospheric dust
opacity (“tau”), worst-case tilt of the solar arrays and a
conservative dust accumulation rate on the solar panels that
assumes no wind cleanings dur
ing the entire mission. '

A major design challenge for dust storms could be the Mars sample return lander and fetch rover. The current working date for landing, 2026, puts the landing in the dust storm period. Both the lander and fetch rover would be solar powered. Presentations on design considerations for both highlight the need to survive the dust storm.

The lander will have a need to keep the fuel in the Mars ascent vehicle above a minimum temperature, likely creating a design challenge in the case of a major dust storm.

The presentations on the fetch rover discuss the need to survive a prolonged dust storm without including any radioisotope heating units in the rover.

As I indicated, though, that appears to have been only the worst-observed prior to the recent dust storm.

Insight will land on November 26, 2018. That is virtually simultaneous with the next southern summer solstice – November 28.

A previous monster dust storm, in 1971, began about three weeks before that southern summer solstice.

This one began about six months before southern summer solstice.

So one thing we can hope for, at minimum, is that Insight will get through nearly an entire martian year before the next dust storm would likely begin.

Is the radio science experiment seriously compromised if it completes a bit less than one martian year?

It seems like the baseline failure rate of missions to the surface of Mars is much higher than the risk Insight faces due to a dust storm, but it's a good issue to bring up. Unlike purely engineering-based risks, this one (given solar panels for power) is beyond our control. It seems like landing right at southern summer solstice is a fortuitous design (or chance circumstance), though.

Note: If Insight had launched at its originally planned date of March 2016, it would have landed in September 2016 and had approximately 18 months of operations before this dust storm. The 2018 launch date reduced the risk. (Which, in hindsight, is more than risk for the voided 2016 dates.)

Opportunity parked from sol 2795 to 2963 to conduct a 5 month mars radio science experiment. I remember at the time that it was said that extending this period would not have provided any more accurate radio science data unless opportunity parked for two full earth years. From this I presumed that insight would need to operate for at least two years to better opportunity's radio science data.

An information packed update on the dust storm in general and Opportunity in particular.
http://www.planetary.org/explore/space-top...ity-sleeps.html

"In the Dust Storm, the mighty Dust Storm, the rover sleeps tonight"

Thanks Serpens for that link. It seems hopeful that the rover may be waking up before too long.
In the early days of the MER mission, the ground ops team used to play "wake-up" songs for the rovers.
http://www.mit.edu/~jfc/MER%20Soundtrack.html
That tradition seems to have faded away after the prime mission, but given the long sleep now, it might be appropriate to revive it just once when the rover does wake, for public outreach purposes.
Possible choices (recycled from the list in the link) might be:

Oh What a Beautiful Mornin'
Soak Up the Sun
I Can See Clearly Now
Wake Up Little Susie [Oppy]
...

Good to see some tau estimates in that report, based on MARCI data:
Of course there's nothing like ground truth, but this does sound promising.

And some values from MSL:

So, at what point does it become concerning that we have not yet re-established contact with Oppy?

There has to be some point at which orbital and telescopic data would indicate tau in Meridiani is at a low enough level that, if Oppy is going to be recoverable, we would begin to expect to hear from it. My guess is that a fair number of people working on Oppy have a good idea of what this timeframe is -- or, failing a timeframe, at least at what level of clearing we ought to start reasonably expecting contact -- but as of yet, I've seen nothing, not even speculation, as to when this might occur.

Again, not so much against a date as against a tau level -- at what level of clearing do y'all think JPL begins to seriously expect to regain contact, if such will be possible?

-the other Doug

I think the article suggests tau needs to be near 2 before expectations of success rise significantly. It could be sooner (or even later), depending on cleaning events and other specifics, but that is the gap between hope and expectation. While lack of communication is always frustrating, I do not think that it becomes a specific source of concern until well into next month; maybe even later. Then, Callas described a long process of bringing the rover back to a normal mode of operations.

After the 2007 peak, communications occurred consistently after tau dropped to around 3.8, and mobility after 2.5. But the rover's mode and state of charge were in entirely different places then.

Yeah, from the report we had:

Obviously huge uncertainty in any such projections...

Despite trying as hard as I could to figure out exactly what the recovery process might look like from public information, I haven't been able to figure this out from the papers or what various project people are quoted as saying.

EDIT: I did run across this -- http://www.planetary.org/explore/space-top...mer-update.html -- which has more detail about the need for "sweep and beep" than I had previously run across.

That infers you know the solar array dust factor - which we do not.

Well, it's somewhere between 0 and 1. One could figure out based on season and a range of dust loading what the tau needed to get into "solar groovy" (1.1A of production) would be -- then it's a matter of hitting the narrow window when the rover would be responsive to commanding in that mode.

What I don't understand about solar groovy is what the battery SoC has to do with it -- is the 1.1A after whatever the battery charging is using? What happens if the battery just refuses to charge?