They are aiming for the distinctive "golf ball" radar domes, and thus creating a diversion.
The "rabbits" are the protesters sent in over the security razor-wire protecting the complex to distract and divert police officers inside.
Dressed in their Greenpeace overalls, adorned with anti-Star Wars slogans, the mission for these men and women is to hare across the grassy complex.
They are aiming for the distinctive "golf ball" radar domes, and thus creating a diversion.
This allows the specialist climbers amongst them to reach their targets, scramble up and chain themselves on.
Anne-Marie Rasmussen, from Denmark, was one of the "rabbits".
She told me: "We raced in all sorts of direction to confuse the police. Some were rugby-tackled pretty quickly, but the boys who mattered got through, and the protest had succeeded."
On two consecutive days it was a push-over.
Carrying their own cameras to record the invasion, the protesters were highly organised.
Their movements were synchronised by two-way radios, cell phones and forward planning.
The eye-catching images were those showing activists dressed up in black costumes designed as mock missiles.
They could be seen skipping and sauntering passed security guards at the main gates of the electronic monitoring station - in the protesters' words, a huge "spy base" - meant to boast a tight security regime.
On Day One more than 100 demonstrators broke through.
Half of them managed to "capture" the high ground on the prominent base water tower and the slippery, sloping roof of a building snuggling under the huge white radar domes.
It was an embarrassment for the security officials caught short by the protest.
There were also red faces in the UK and US Governments, whose personnel, military and civilian, operate the massive complex of underground listening posts at Menwith Hill.
There was no official comment either from the Ministry of Defence (MoD), or the American authorities.
But it was pointed out that the trespass was illegal.
The position of the protesters aloft was said to be dangerous and unsafe, so removing them would be a slow, careful process.
One climber, identified as "Ricko" by the ever-busy publicity machine employed by Greenpeace, stayed aloft through the night, finally succumbing to dehydration and exhaustion.
But, surprise, surprise, early on Wednesday morning a squad of "rabbits" and their better-equipped hares were back over the fence, using a gate they had climbed before at a spot where the razor wire had not been repaired.
The security was swift - but not swift enough.
Three protesters scrambled up security camera pylons.
Among them was Anne-Marie from Denmark, who this time was a climber, not a "rabbit".
She was eventually persuaded to come down by MoD police and rejoined protester-colleagues at the main gates.
She said: "I went over the wire three times yesterday - and was ejected each time.
"We are here to highlight the dangers we believe President Bush's missile ideas will mean for the world - and a banner from a tower inside this base grabs the attention."
It was not without significance that the repetition of the invasion came on 4 July, US Independence Day.
Greenpeace was determined that its demonstration against the Star Wars missile defence proposals - proposals they insist will include Menwith Hill as the" eyes and ears" of the control system - should have an impact in the US as well as in the UK.