Republicans and Democrats are already sharply divided over picking a successor to Justice Antonin Scalia. In fact, the divisions are far greater than they were, especially this early in the process, for Obama’s previous two nominees, Sonia Sotomayor and Elena Kagan. In part, this is because the nomination fight is happening in an election year. Yet there’s also another, more fundamental question at stake. The nomination of a successor to Scalia may shift politics on the Supreme Court in ways that the Sotomayor and Kagan nominations did not.

To understand why, you need a bit of background about the politics of the Supreme Court and Supreme Court nominations. In an article published by the American Journal of Political Science, Bryon Moraski and one of us (Shipan) set out a basic framework for understanding these nominations. The central idea is pretty straightforward. Presidents will try to select nominees who, if confirmed, will pull the court closer to their own preferences. But presidents can’t act alone. As Sarah Binder already has noted, presidents are constrained in their choices by the Senate, which needs to confirm any nominee.

However, they also are constrained by what political scientists would call “the distribution of ideological preferences” of the justices themselves. In more everyday language, we might think of the nine justices as being arrayed on a left-right scale, with the most liberal justice at the leftmost point, and the most conservative justice at the rightmost point. When the justices vote according to their ideological positions, then the median justice – the one at the middle – is the one who is most influential. If he or she decides to agree with colleagues to the left, then he or she gives a victory to the left. When he or she votes with colleagues on the right, then they win. The median justice is the swing voter.

The current court illustrates how this works. Right now, Justice Anthony M. Kennedy is widely seen as occupying the middle ground on the court, which is why his vote is often seen as being so pivotal. He is the median member of the Court – the 5th justice on a left-to-right scale in a nine-justice court. Neither the Sotomayor nor Kagan nominations changed this – instead they swapped out one liberal-leaning justice for another liberal-leaning justice. The figure below uses Mike Bailey’s most recent ideology scores to show the ideological distribution of Supreme Court justices, along with President Obama.

What Moraski and Shipan first pointed out was that even if a president was not constrained by the Senate, he or she could move the median only as far as the next closest justice. That is, all he or she can do is shift the swing vote one justice to the left, or one justice to the right. In the current context, if Obama could get a nominee approved who shared his ideology, then the new median would be Kagan. Obama might want to move the court even further to the left, but he just can’t do it. Again, even if he could nominate and somehow get approved a strongly liberal nominee, the median (at least in the short term) could move over only one position to the left – that is, to Kagan, the first justice to the left of the current median.

Meanwhile, Senate Majority Leader Mitch McConnell (R-Ky.) will figure that if he refuses to act on any nominee Obama sends forward, then the next occupant of the White House might be a Republican who would nominate a justice more like Scalia. This would mean that Kennedy would continue to be the median. The relevant calculation for McConnell, therefore, is the distance between the ideological positions of two potential median justices: Kagan, if Obama selects the next justice, or Kennedy if (say) President Cruz selects the next justice. The figure below shows measures of that ideological distance – between what would happen if a president got his way and what would happen if the opposing party in the Senate got its way – for all nominations over the past six decades in which the president has had the potential to pull the court closer to his own position. As the figure shows, the distance for this current nomination is greater than that for any nomination in recent decades.

Now, if the Senate took our simple theory seriously, Republicans don’t have to worry too much about Kagan becoming the new median, because the Republican-controlled Senate simply wouldn’t approve a nominee who would produce such a shift. In fact, the Senate has little incentive to approve any nominee who would shift the court to the left.

However, there’s a problem for the Senate. In a recent paper, we found that presidents are much more powerful than the simple Moraski-Shipan model suggests. In fact, presidents regularly end up being able to use appointments to pull the court toward their preferred ideology – even when the Senate should be able to stop them from doing so. They can do this by drawing upon their political capital – leaning on members of Congress to get their support, trying to rally public opinion, and so on. Given this, there’s the possibility that the president might be able to pull the court toward his preferred ideology – perhaps even as far as Kagan.

There’s always a lot at stake in presidential nominations to the court. These political science tools – which help us to think systematically about the crucial role of the median justice on the Supreme Court – tell us why there’s even more at stake this time around.

David Cottrell is a postdoctoral fellow with the Program in Quantitative Social Science at Dartmouth College.

Charles Shipan is the J. Ira Harris professor of social science at the University of Michigan.