Does the engine have to fire 3 times, once to set them on their way, once to slow down enough to be captured by the moon and a third time to escape the moon and return to Earth?
In the diagram that foghorn posted (also quoted below), you can see that, after the initial launch (steps 1-3), Artemis was put into a large elliptical orbit, but still an orbit around the Earth (steps 4-8). There will be some "proximity operations" during that large orbit (which lasts for about a day), in which the astronauts will be practicing some parts of a docking manoeuvre (mentioned in item 6 on the diagram).
There is another burn at 8 to increase the radius of the elliptical orbit at perigee (the closest point to the Earth on the orbit, which is at point 9).
At step 9, there is a long "trans-lunar injection" burn, which takes the spacecraft out of the elliptical orbit and puts it on track to the moon. That will happen tomorrow, at around the same time as today's launch. After that burn, the astronauts are going around the Moon, no matter what. There's no turning back.
From there, there are no more engine burns, except possibly for slight course alterations. Gravity does all the work between points 9 and 14.
I will have to see if they have to do any slowing down out of the natural swing and any positioning towards earth.
They won't be slowing down. Basically, the moon's gravity will slingshot them around the moon and put them onto the return path to Earth. Point 12 allows for slight course corrections, if necessary.
Ps. I'm not Patrick Moore here, so I could be wrong.
I'm back... looks like they do a correction burn to put them on course to moon and a burn after swinging round moon to head home.
That is points 10 and 12 in picture below:
View attachment 7419
Does that suggest 9 gives them enough velocity to escape from Earth...
Yes, but the Moon's gravity will put the spacecraft back onto a return trajectory.
... yet by the time they reach the moon Earth’s gravity has slowed them enough that the moon can capture them?
No. The moon won't capture the spacecraft. It is not going into lunar orbit on this mission. In future missions, where NASA
does want to orbit and/or land on the moon, some slowing down will be needed (and then speeding up again to start back towards the Earth).
So no powered deceleration when they get there?
No. This is a "free return trajectory". No power from the spacecraft needed.
But if so they will need to give it a kick to get out of lunar gravity for the return.
Not on this trip. At the moon, the spacecraft will have sufficient velocity to escape the Moon's gravity, once it goes around.
However I see they seem to plan on various correction burns en route, both out and back.
Those are mostly to make sure that the spacecraft is headed in the right direction. Small corrections far away can make enormous differences later on. It's a very long way from the Earth to the Moon (380,000 km).
Evidently they are content that the engine can be restarted at will, then.
As well as the main engine, the spacecraft also has smaller manoeuvring thrusters. For example, at times it is put into a slow rotation to allow for more even heating by the Sun.
That’s different from my recollection of the CSM engine in Apollo.
Apollo also had manoeuvring thrusters.