NavList:

I asked Dominic Ford, an astrophysicist who runs the excellent web site In the Sky https://in-the-sky.org/ how accuratly we know satellite positions. Here is our conversation. If you are not interested in the (done to death here maybe) debate about GNSS vulnerability you dont need to read the second reply! The bottom line is a few hundred metres for position of typical satellites. Maybe that is good enough for a method to use on an overcast day on a long passage. One thing I didnt think of is that some scenarios that knock out GNSS might disturb the orbits of all the other Satellites.

Dominic writes:
The accuracy of satellite positions varies considerably. Navigation
satellites such as GPS / GLONASS are known to metre precision (the
military may well know them to much better precision).

However, spacecraft tracking is an expensive business. Unless you
actively track them, gravitational perturbations and chaos kick in and
you lose accuracy quickly. For many satellites there's no reason to
spend money tracking them, and their positions are probably only good
to a few hundred metres. At low altitudes (e.g. the ISS and almost all
CubeSATs) the situation is particularly bad, because of atmospheric
drag. The ISS loses several hundred metres of altitude per day and
typically thrusts up into a higher orbit every couple of weeks.
CubeSATs, of course, don't have propulsion, so they fall to Earth
after a few months.

Even where spacecraft are tracked, very few have positions measured to
better accuracy than one metre, because that's extremely hard to do.
The Earth's surface is subject to low-frequency seismic oscillations,
and the Earth's rotation speed fluctuates. That means you have
systematic errors arising from not knowing where your tracking
stations are. To add to your woes, general relativistic time dilation
(with altitude) is easily measurable when you're tracking an object
moving at 30 km/s to centimetre precision. Your clock on your
satellite goes at a different speed for your clock down below....

There's a team at BAE systems in Chelmsford who specialise in doing
this stuff for ESA, but their services don't come cheap!

The predictions on my website are not that accurate, I hasten to add.
They're probably good to around the nearest kilometre, as that was all
I needed.. I use a quick-and-dirty Javascript routine to calculate
positions from orbital elements. There are lots of small scale
corrections that I only did as accurately as was needed for a pretty
diagram!

Hope that helps!

Bill replies:

Thanks Dominic, that is really useful. Do you mind if I quote you on
NavList where we were discussing this?

I think it means we (seafarers) still have to keep up our celestial
navigational skills in case  Kessler syndrome some other calamity
knocks the navigation satellites out!

That said navigation to within 1 nautical mile is as good as we can do
with celestial so even a radius of uncertainty of a few miles is OK
for a back up system, ie how far away you can see land visually or by
radar.

I am not teaching problem solving by computer this year but another
year I might make them do orbital calculations. I am teaching Euler's
method to Chemical Engineers and it was fun to find out he invented it
for the Lunar distance method for Longtitude. 

> Thanks Dominic, that is really useful. Do you mind if I quote you on
> NavList where we were discussing this?

No problem at all!

> I think it means we (seafarers) still have to keep up our celestial
> navigational skills in case  Kessler syndrome some other calamity
> knocks the navigation satellites out!

Yes -- our reliance on GPS is interesting and scary. Recently a
computer failure in Moscow knocked GLONASS completely offline for a
few hours (the satellites were sent faulty data about their
positions). A similar error in the US could knock GPS offline, with
global consequences. To give one trivial example, Thameslink trains
use GPS to determine when the train is in a platform. The doors won't
open without a GPS link.

This dependence on an American military system (which could be turned
off at the flick of a switch in Washington) is what's led the EU to
feel it needs to spend billions of Euros on Galileo, even though it's
a totally redundant copy of GPS.

As well as Kessler, another interesting scenario is a repeat of the
Carrington Event (1859). We'd probably get 1-3 days' notice, and then
we'd be looking at many satellite failures all at once. We'd also
almost certainly see serious damage to the National Grid, taking
months to repair. Interesting to ask how the UK would cope with a
month-long nationwide power cut. It almost happened in 2012, but the
storm just missed us.