Here’s the nub of the problem that the U.S. and its allies now face in Libya: They’ve got a very expansive authorization to use force, but on behalf of a very limited objective, i.e., protecting Libyan civilians. However, President Obama and his British and French counterparts have made clear that Qaddafi’s ouster is the strategic purpose of their national policies. At least in theory, if Qaddafi were smart enough to actually suspend his military operations, stop attacking civilians, and seek urgent mediation of a ceasefire through the good offices of the U.N. secretary general, he might indeed be able to stay the hand of allied forces and buy himself some much-needed time to undermine the disparate international coalition that now confronts him. Supplied with the slightest hint of reasonableness from Qaddafi, I bet there’s a decent chance that the likes of Russia, China, and Turkey, to name but a few, would jump at the opportunity to give peace a chance, throw Qaddafi a lifeline, and short-circuit any chance of a successful Western military intervention.

Resolution 1973 actually identifies a ceasefire as its goal. But a ceasefire that leaves Qaddafi in control of at least half of Libya is a formula for long-term stalemate that the U.S. should want no part of. While it averts, temporarily at least, the worst-case outcome of a humanitarian nightmare wherein the rebel stronghold of Benghazi is wiped out, it does little to advance the strategic goal of quickly seeing the bloody-minded Qaddafi replaced by a new order that treats its people with some minimal degree of decency, restores Libya’s place in the world economy, and works with friendly countries to combat extremism. On the contrary, it leaves the U.S. military policing an open-ended face-off between Tripoli and Benghazi; U.S. diplomats fighting a losing battle to maintain the cohesion of what is almost certain to be a fair-weather coalition (the Arab League is already moaning that they asked for a no-fly zone, not a bombardment); and U.S. credibility and prestige eroding each time Qaddafi trumpets to the world that he’s defied Obama’s demand that he go. Quagmire, anyone?

In his statement on Friday, Obama wisely sought to avoid this predicament by establishing “non-negotiable” conditions for a ceasefire that went well beyond merely an end to hostilities. In addition to silencing their guns, he demanded that Qaddafi’s forces withdraw from territory in eastern and western Libya that they’d previously re-taken from the rebels, including areas within striking distance of Tripoli itself. Being forced to undertake such a retreat would subject Qaddafi and his forces to a much-heightened degree of humiliation and demoralization. It would certainly increase the likelihood of mass defections from his ranks, or that someone from his inner circle acts decisively against him. Of course, should Qaddafi choose not to comply, the coalition would be free to continue the systematic dismantling of his military capabilities, which could well trigger the same instinct for self-survival and flight amongst his loyalists.

But assuming that Qaddafi could comply with Obama’s ceasefire demands and survive, then what? Presumably, the rebels would be subject to the ceasefire as well. What would happen if they regrouped and began again to march their forces west toward Qaddafi’s stronghold in Sitre and, eventually, Tripoli? Resolution 1973 authorizes foreign militaries to act to defend civilians from the onslaught of Qaddafi’s forces. But what are the rules of engagement when it’s the rebels who are initiating hostilities and on the offensive? Do Qaddafi’s forces have the right to fight back and defend his regime if the rebels violate the ceasefire? Would the U.S. and its allies fly air cover for an advancing rebel army? That would certainly be consistent with their goal of seeing Qaddafi removed from power. But it would also almost certainly put them at odds with the U.N. resolution that Obama has put so much stock in and made the touchstone of his policy.

The bottom line is that the U.S., its allies, and the Libyan rebels should be doing everything in their power to avoid a ceasefire and drive Qaddafi from power quickly. Obama’s preemptive move to set a very high ceasefire bar, which will be difficult for Qaddafi to accept, was exactly the right thing to do. All the evidence so far also suggests that Qaddafi will be an unwitting ally in this effort, as his forces continue their brutal assaults around the country even as his diplomats blatantly lie that their guns have gone silent. Allied air power must exploit the free hand they currently have to degrade massively Qaddafi’s entire military machine in all its facets, ground, air and naval. Command and control nodes, including the headquarters and bunkers of Qaddafi and his key sons must also be targeted. To borrow a much maligned phrase, “shock and awe” should be the order of the day if we hope to shorten the conflict, limit overall suffering, and achieve our strategic objectives.

As quickly as they are able to regroup, rebel forces should be enabled and encouraged to take advantage of ongoing allied military efforts to resume their march west, re-seizing lost territory and threatening Qaddafi strongholds. At the same time, as daunting as it may be, the rebels should encourage their supporters in Tripoli to resurrect their uprising in the capital. If Qaddafi’s forces move again to repress civilians there, the liberation of Tripoli from the regime’s thugs would then become a legitimate addition to Obama’s ceasefire demands and further grounds for continuing to dismantle his military. And if Qaddafi were to waver in Tripoli, the likelihood that his regime would soon be overwhelmed by the street and rapidly unravel would increase exponentially.

If President Obama acts now with boldness and resolve, a window does exist for avoiding a long-term U.S. military commitment in Libya and achieving a rapid end of Qaddafi’s regime. But the opening is narrow and could close fast. It will likely require the president to demonstrate a degree of ruthlessness on the battlefield, and a willingness to incur a degree of criticism internationally, that have not exactly been his forte to date. One hopes that the dark shadow of a costly, drawn-out military engagement in the Mediterranean without end in sight will provide the president sufficient incentive to overcome whatever hesitation may exist in this regard. If he does, Obama will have succeeded in limiting the suffering of the Libyan people and significantly advancing America’s national interests and global standing, a pretty good day’s work that all Americans would rightly applaud.

— John Hannah is a former national-security adviser to Vice President Dick Cheney.