Libya: What will the UN allies do next?
The coalition may have assessed its first strikes on Saturday night as successful, but this is only the start of what could still be a long process.
Overnight from Sunday into Monday, more bombing raids took place on targets such as air defences - including more strikes from the sea by US and British submarines using Tomahawk missiles.
The stated aim is to degrade Colonel Gaddafi's military capability - specifically his jets and attack helicopters - so it can no longer pose a threat to Libyan civilians.
For the coalition, fire-power that has been drawn together in this alliance of the willing, it should be a relatively quick task.
The Pentagon says the no-fly zone is already in force, although there is clearly still some work to be done.
The hope is that direct strikes on the armed forces still loyal to the Libyan leader will fast affect their morale and encourage them to flee or defect.
The Pentagon claims that Gaddafi's forces are "under significant stress and suffering from both isolation and a good deal of confusion".
But what happens if the rebels now go on the offensive against Gaddafi's troops? That remains unclear.
The Pentagon would not say whether air-strikes would hit Libyan troops fighting back against a rebel assault.
And the coalition is walking a fine line over its intentions towards Col Gaddafi himself.
The UN resolution allows the use of ''all necessary means" to protect civilians - but does not specify regime change as an end, even if that is what many who backed it are hoping for.
So is Col Gaddafi a target of the strikes?
Vice-Admiral Bill Gortney, at the Pentagon, said only that he was "not on its target list" - though others argue that as the commander of forces still seen to be attacking civilians, he is a legitimate target.
And the question of ground troops also appears to remain open. British ministers all insisted on Sunday there were "no plans" for any, though none explicitly ruled them out.
The UN resolution bans an occupying force, but troops to keep a peace, or aid civilians, might be a different matter.
Much will depend on what the rebels do next - something out of the coalition's control.
As it progresses, the coalition's military action is rapidly giving breathing space to the rebels, so that perhaps a more coherent leadership and structure can emerge.
However, with an arms embargo in force, it is not clear whether they would have the wherewithal to challenge Gaddafi's forces if they do not crumble - especially in Tripoli and the rest of the west - nor how widespread support for the rebels would be there.
One thing the allies will be keen to avoid is the breakdown of order and the chaos that emerged in the aftermath of the initial military action in Iraq.
And if Col Gaddafi's regime were to crumble quickly after 41 years in power, that could unleash a wave of score-settling and old tribal enmities - it is far from obvious how much would unite the east and the west of the country in agreeing new leadership.
For all those reasons, Arab backing for, and participation in, the mission remain crucial, as well as in the end-game when it comes.
So Qatar's offer on Sunday of four Mirage aircraft to join with the French in patrolling the no-fly zone will have come as a boost, not least after the head of the Arab League expressed his concerns over the scale of the coalition air-strikes.
His words seemed to suggest that Arab support for the military mission remains in some ways ambivalent.
The US has said it expects to hand over control of the operation to a coalition headed by France, Britain or Nato in a matter of days, though it is not yet certain whether Nato would then take command, as is currently being discussed.
The military operation may relatively quickly turn into something more routine, with ongoing patrols to ensure that the no-fly zone holds.
But the danger is that the result of enforcing the UN resolution could be a stalemate, something Washington has already acknowledged, while President Barack Obama admitted today that "no-one can say for certain how this change will end".
Nor how long it might take.