It challenges four decades of constitutional doctrine and is based on disputed scientific theories.

Yet a push to ban abortion at 20 weeks after conception, on the theory that the fetus can feel pain at that point, has emerged as a potent new tactic of the anti-abortion movement. Advocates saw the potential of such a measure because it taps into public concern about late-stage abortions, appears to alter the rules only incrementally, and claims to be rooted in science.

"Any time we talk about developmental landmarks of the unborn child, anything showing that the unborn child is a member of the human family – that gets the public to take a closer look at abortion," said Mary Spaulding Balch, the state policy director of the National Right to Life Committee, who is widely seen as the architect of 20-week legislation.

The 20-week ban was first adopted in 2010 in Nebraska, where conservatives aimed to rein in one well-known abortion doctor. A pain-based abortion limit has now been enacted in a dozen states, most recently in Texas, and a bill to impose one nationally passed the Republican-controlled House in June. Some recent polls, while affirming public support for legal abortion overall, suggest that a majority of people would draw the line at 20 weeks of pregnancy.

Abortion rights advocates call the pain argument duplicitous and say the laws will surely be declared unconstitutional. But they have also been forced to mobilize against 20-week bills in state after state, and they credit their opponents with skillful marketing.

"These laws are cloaked in the language of two-week increments, rather than banning abortion at conception or other more radical measures," said Suzanne B. Goldberg, the director of the Center for Gender and Sexuality Law at Columbia University. "They are cutting back on women’s constitutional rights, but less dramatically, so they trigger less alarm across society."

In the three states where the bans have been legally challenged, the courts blocked them. In the standard laid out by the Supreme Court in Roe v. Wade in 1973 and elaborated on in later decisions, women have a right to an abortion until the fetus is viable outside the womb, around 24 weeks into pregnancy.

But proponents of 20-week bans hope that one of the cases will be accepted by the Supreme Court. Reading into opinions by Justice Anthony M. Kennedy, the likely swing voter, they are hoping for a legal upheaval.

With these bills, the anti-abortion movement is tapping into a powerful strand in the complex tangle of public opinion on abortion. Support for legal abortion drops when people are asked about the later stages of pregnancy.

In a Gallup poll last December, 61 percent of Americans said abortion should be legal in the first three months of pregnancy, but 27 percent said it should be legal in the second three months, and 14 percent in the final three.

That pollsters are now asking about a 20-week limit is evidence of the success in injecting the new threshold into public discourse, said Michael Dimock, the director of the Pew Research Center for the People and the Press.

By any measure, the practical impact of a 20-week ban is small compared with the potential legal and symbolic effects. In all cases but one, in Arizona, the laws ban abortions at the 20th week after fertilization, which is the 22nd week after the last menstrual period, the most common way of describing pregnancy. The estimate of fetal viability at around 24 weeks is also timed from the last menstrual period, so the actual gap between the two approaches is about two weeks, involving several thousand abortions, at most, out of an estimated 1.2 million performed every year.

"It may be a small number of women in those weeks, but for each of them, it matters a great deal," said Nancy Northup, the president of the Center for Reproductive Rights in New York. She noted that the discovery of birth defects, which often occurs in the second trimester, was a common reason for later abortions.

For groups like Northup’s, which is involved in lawsuits to fend off abortion restrictions across the country, keeping fetal viability as the dividing line is essential.

The Supreme Court, including Kennedy, has repeatedly affirmed viability as the point at which the state’s interest in protecting life outweighs a woman’s right to control her body, Northup noted.

"There is no other line that is workable," she said. "It is an appropriate line to draw."

Balch, the Right to Life official, who is also a lawyer, said she had been considering fetal pain as a way to draw attention to "the humanity of the unborn child" since she heard President Ronald Reagan speak about it stirringly in 1984. In a speech to religious broadcasters that year, he said, "Medical science doctors confirm that when the lives of the unborn are snuffed out, they often feel pain – pain that is long and agonizing."

Balch started collecting journal articles by a small minority of scientists who asserted that pain may be sensed earlier in fetal and brain development than commonly assumed.

Most scientists and medical associations say that perception of pain is impossible without brain developments that occur well after 20 weeks.

Even as new theories about pain were emerging, anti-abortion groups were dissecting abortion-related opinions by Kennedy, especially his 2007 opinion upholding a federal ban on a particular procedure that critics call partial-birth abortion.

In that decision, in the interpretation of Balch and others, the court showed that it was open to restricting abortion before fetal viability if there was a compelling moral reason.

But other legal experts like Goldberg and Northup call this a misinterpretation. In upholding the partial-birth ban, they note, the court emphasized that it would not significantly impinge on abortion rights, since other methods were still available. By contrast, a 20-week law would prevent all abortions in the weeks preceding fetal viability.

The theories about fetal pain and possible chinks in Roe v. Wade came together in Nebraska in late 2009.

Earlier that year, Dr. George R. Tiller, who was known for performing late-term abortions at his clinic in Kansas, was killed by an anti-abortion activist. Dr. Leroy H. Carhart, in suburban Omaha, said he would try to fill the gap.

Nebraska conservatives wanted to find a way to prevent Carhart from making Nebraska the "abortion capital of the Midwest," recalled Julie Schmit-Albin, the executive director of Nebraska Right to Life.

It was Balch, Schmit-Albin said, who "saw an opening there" and molded the new legal tactic, and Republican legislators made it a priority. (Carhart still has a clinic in Nebraska, but he had to open a new one in Maryland to perform abortions at later than 20 weeks.)

Schmit-Albin and other abortion opponents readily admit that a 20-week ban is a step toward their ultimate goal.

"Our mission is to restore legal protection to unborn life from the moment of conception," she said. "This is a marathon."