WASHINGTON — They gave him their hearts when he visited last summer. Now, the question hanging over Europe is how much more they'll give Barack Obama as he returns for the first time as president of the United States.

Obama leaves Tuesday on an eight-day tour. He remains enormously popular in Europe, and the throngs that greeted him last summer as a candidate are likely to grow. With first lady Michelle Obama along, Obama's debut on the world stage as president already is inspiring anticipation of the kind of rock-star reception that greeted John and Jackie Kennedy on their first trip as first couple to Europe in 1961.

Yet Obama also heads into his first overseas trip with grand goals — looking to forge a coordinated global response to the Great Recession, hoping Europe will send more of its sons and daughters to help in an escalating war in Afghanistan, and seeking to restore international cooperation that he thinks suffered in the Bush years.

That will be a tough sell. Publicly, European and world leaders will embrace Obama. But privately, they likely will say no to some of his requests, most notably sending combat troops to Afghanistan, or simply avoid the subject.

"He remains a superstar in European public opinion," said Reginald Dale, a scholar at the Center for Strategic and International Studies, a centrist research organization. Dale noted that Europeans have even more trust in Obama than Americans do, according to a recent poll by the Financial Times newspaper.

"European leaders want to be seen next to Obama, preferably with ... his arms around their shoulders and a big smile, because he's so popular in Europe. And nobody's going to try and raise awkward subjects with him."

Perhaps, but those subjects will be unavoidable as Obama heads first to the United Kingdom, then on to France, Germany, the Czech Republic and Turkey.

LISTEN AS WELL AS LEAD

White House aides said yesterday that Obama is eager, as he heads overseas, to rescue the U.S. and world economy and press an international approach to Afghanistan, but also to "re-energize" the international alliances that have guided world affairs for more than half a century.

He'll do that, he said, by taking a more collaborative style, and less of the "my-way-or-the-highway" approach critics say President George W. Bush embodied.

"The president and America are going to listen in London as well as to lead," said White House Press Secretary Robert Gibbs.

EUROPEAN TOUR

President Obama's trip to Europe will include a whirlwind of diplomacy and good will.

UNITED KINGDOM

His first stop will be at the G-20, a group of 19 major economic powers, plus the European Union, meeting in London.

Obama already has been pushing them for more government spending to stimulate the global economy, as he's doing at home. Many European countries, however, instead are emphasizing tougher regulation of the financial system.

"Even the European Union itself is balking," said Nile Gardiner, a scholar at the Heritage Foundation, a conservative research organization. "I think we are going to see a significant transatlantic divide emerging at the G-20 between the U.S. position of massive stimulus spending and European opposition to that."

Prime Minister Mirek Topolanek of the Czech Republic, the current EU president, last week ripped Obama's free-spending approach as a "road to hell." But his own opposition party pushes for stimulus spending, and last week passed a vote of no confidence in his government.

While in London, Obama also will have a chance to remedy his perceived missteps when he hosted British Prime Minister Gordon Brown in Washington. In one, Obama gave Brown a set of DVDs, ridiculed as a cheap slight by the British press.

White House aides declined yesterday to say what Obama would have with him when he meets Queen Elizabeth II.

FRANCE AND GERMANY

Obama next will attend a meeting of the NATO alliance in Strasbourg, France, and Baden-Baden, Germany.

Officially, the meeting will mark the 60th anniversary of the alliance. Unofficially, the war in Afghanistan will dominate the meeting as the U.S. asks for help.

"President Obama will probably not have much luck in obtaining additional combat forces," said Stephen Flanagan, a scholar at the Center for Strategic and International Studies. "There just isn't the willingness on the part of most of the European allies to do that right now."

Americans already make up more than half of the international forces in Afghanistan. Obama announced on Friday that the U.S. would send another 4,000 troops to help train Afghan forces, atop the 17,500 additional combat troops he already committed. When all are present, U.S. forces will total about 60,000 of the approximately 92,000 NATO troops there.

"American expectations are being lowered, or Europeans are trying to lower American expectations," Flanagan said. "There may be some modest additions, but it's more likely that European governments will be offering trainers for both the Afghan national army and the Afghan police."

CZECH REPUBLIC

In Prague, Obama will attend a meeting of the EU.

There, Obama will give what aides called a major speech on proliferation, including not just the threats from nuclear weapons but also cyber threats and energy security.

Another likely topic will be the U.S. commitment to deploy a missile defense system in the Czech Republic, as the Bush administration proposed.

Ostensibly aimed at protecting against missile launches from Iran, the system also is seen as a defense against Russia — but Moscow sees it as a threat.

"If Obama withdraws on missile defense, which he seems to be doing, then he is going to leave the Czech and Polish governments out there hanging in the wind," said Dale of the center. "They went to great lengths to reach agreement on ... basing those facilities there, even though their public opinions were largely against it."

One possible result: remain vaguely committed to the missile defense while continuing a review of the policy. The challenge: doing it in a way that assures the Czechs and Poles they're not being strung along and convincing the world that the Obama administration isn't caving to pressure from Russia.

TURKEY

Obama will visit Ankara and Istanbul before turning homeward.

Though this isn't the site of Obama's promised speech reaching out to the Muslim world — that will come later in a still-unidentified Muslim capital — Turkey is a Muslim country, and Obama's talk likely will strive to reach that audience.

"Obama will start with a great advantage when he gets to Turkey, because his name is not George Bush. He was extremely unpopular in Turkey, as well as in the Islamic world," said Bulent Aliriza, the director of the Turkey Project at the center.