If the first term is all about making an impact, the second term is about the legacy. For Obama - still weathering an economic storm - gun control has forced itself to the top of his agenda.

Obama's challenge graphic

It was all very different four years ago. Almost two million people flocked to Washington DC, braving the sub-zero temperatures to witness a piece of history. The nation's first black president, taking the oath of office on the steps of the Capitol: promising hope, promising change, promising to change Washington for ever.

The question, Obama declared was "not whether government is big or small, but whether it works." This was to be a new era in post-partisan politics: "an end to the petty grievances and false promises, the recriminations and worn-out dogmas that for far too long have strangled our politics."

January 2013, and the crowds are hardly filling the streets in the same spirit. Washington has never been more divided, politics more polarised, Congress held in such public contempt. But Obama is only the eighth president to win more than 50 per cent of the popular vote for a second term.

According to Ken Duberstein, who served as chief of staff under Ronald Reagan: "In the first term, you're running for election. In the second term, you're running for legacy." It is posterity, not the people, who will be the judge.

The window of opportunity for any president is small: for one entering his second term, it is smaller still - history suggests a year to eighteen months at best, and that is without the looming threats of Congressional gridlock and deadlock and shutting the government down.

The full faith and credit of the United States of America is not a bargaining chip. President Obama

When the president stands on those steps again on Tuesday, to address the nation, there is unlikely to be such a sense of grand ambition, of a leader who is aiming to transform the times in which we live.

But the priorities which Obama will set out are already clear: and top of the agenda is gun control: an issue which he never really talked about in his first four years, but which has burst onto the national consciousness since the terrible tragedy in Connecticut just over a month ago.

The president has already outlined his key proposals: criminal background checks on all gun sales, a ban on assault weapons, strict limits on ammunition magazines, and stronger penalties for gun trafficking.

He signed 23 executive orders to push through as many reforms as possible without needing to go through Congress, and in an op-ed in the Connecticut Post, he appealed to ordinary Americans to support him. "The truth is, there's only one voice powerful enough to make this happen: yours."

If there is one issue which could provide Obama's lasting legacy, it could be this. And he has pledged to use the powers of his executive office: "And then members of Congress I think are going to have to have a debate and examine their own conscience."

But still - taking on the powerful gun lobby will require a supreme political effort, from a president who liberals claim has consistently shied from a fight. He at least has public opinion on his side, but an opposition which is determined, and entrenched.

And looming over it all - round two of the debt ceiling fight. Another economic crisis to scare the markets and force a very real debate over spending and taxation into a partisan corner.

Obama (reuters)

Obama has already warned his opponents that there will be no repeat of the bickering on the Hill that brought the country to the very brink of the fiscal cliff deadline.

He called a surprise press conference to ram home his point: warning the Republicans that refusing to raise the debt ceiling would be "irresponsible and absurd" - before warning: "The full faith and credit of the United States of America is not a bargaining chip.".

But there will still be serious spending cuts to face up to, not just in the defence budget, but in entitlement programmes too, given his commitment to make a $1tr dent in the deficit.

Cutting it fine

In his new treasury secretary, Jack Lew, Obama has an experienced operator - but also a fellow progressive, who will be determined to protect social security and medicare as far as possible, while making the rich pay their fair share.

If the worst happens, and there is no debt ceiling deal by 1 March, the massive spending cuts which threatened to trigger a full blown recession at the turn of the year will come into force. Federal agencies have already been asked to "intensify" their preparations for steep cuts in jobs and services.

But the priorities don't stop there. Comprehensive immigration reform, promised in the first term but never quite delivered, has emerged as the other main hope of legislative achievement.

What the White House wants, is a broad, overarching piece of legislation, which will provide a path for citizenship for America's eleven million undocumented immigrants: this too, Obama has warned, is non-negotiable.

His main hope of success is the urgent need on the part of the GOP to win over those Latino voters who shunned the party so overwhelmingly in the 2012 elections. Florida senator Marco Rubio, a potential Republican presidential candidate in 2016, has already put forward ideas which do not look dissimilar to the administration's.

And then there is diversity: more specifically, Obama's top team, where as the pundits put it, the optics look all wrong. The three most senior appointments - defence, State, and head of the CIA, have all been middle aged white men, Denis McDonough as the next chief of staff would make another.

In his defence, Obama has insisted that in his first four years, high ranking women were some of the most important influences on policy, from Hilary Clinton to his two supreme court appointees. "Fifty per cent of my White House staff were women", he said. "So I think people should expect that record will be built on during the next four years."

Building a government that looks like America, and reflects the new generation that brought him back to power: that too, would be a legacy worth fighting for.

Felicity Spector writes about US politics for Channel 4 News

Tomorrow: Obama's foreign policy challenges.