Justice Antonin Scalia’s death, and the resulting vacancy on the U.S. Supreme Court, raise an obvious question: What Obama nominee – if any – would this Senate confirm?

Of course, initial signals suggest that the answer is none. President Obama’s administration faces an intensely polarized Senate held by the opposition party. Moreover, a moderate or liberal replacement for Scalia would shift the ideological center of the court dramatically to the left, potentially the largest such shift in modern times. And Obama also confronts a Republican opposition that smells potential electoral victory next November.

Under these circumstances, the Senate leadership clearly wants to block a vote on an Obama nominee. But suppose Senate Majority Leader Mitch McConnell (R-Ky.) actually allows a nominee to come to the floor. What will happen?

In two recent papers, we examined the voting behavior of senators in Supreme Court nominations since 1937.  The more recent paper examines one theory — discussed by David Cottrell and Charles Shipan — which predicts that all senators more conservative than Justice Anthony Kennedy should vote against a nominee to Kennedy’s left. If so, then Obama should nominate the best confirmable nominee — a Kennedy clone, as suggested by Matthew Yglesias — whom the Senate would approve.

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But we find something different. In fact, senators have frequently been far more accommodating of presidential nominees than this theory predicts. Unsurprisingly, presidents have therefore often been more aggressive than the theory would predict — nominating potential justices who seem ideologically out-of-step with key senators.

Given this historical pattern, does a nominee less conservative than Kennedy have a real chance at confirmation?

In another paper (co-authored with Jee-Kwang Park), we analyzed senators’ actual voting behavior on Supreme Court nominees. We considered how ideologically similar nominees were to each senator, how qualified each nominee was, how much interest groups mobilized in support of each nominee, and how much each nominee would affect the ideological balance on the court.

Using the statistical model in that paper and the ideologies of current senators, we can estimate the support that a given nominee of any ideological persuasion would receive. We’ll assume a high quality nominee and intense interest group mobilization (similar to that in the nomination of Justice Clarence Thomas).

Here are the results:

Here is how to read the figure. The horizontal axis depicts the ideology of the nominee, moving from extremely liberal to moderate  to conservative. For every potential nominee, we estimate how each senator would vote, and then sum the total number of yes votes, which is depicted on the vertical axis.

The five names in black type depict the current median on the court (Kennedy) as well as the court’s four liberals. The short bars at the top of the figure show the ideological locations of every senator, with Democrats in blue and Republicans in red. As the figure makes clear, the Senate is severely polarized, leaving only a narrow band in which to fit a confirmable nominee.

Accordingly, the more conservative the nominee, the less inclined Democrats become to support that person, while Republicans become more inclined. The model predicts that even an ideological twin of Justice Stephen Breyer — the most moderate of the court’s current liberals — would fail to get even a majority of votes in the current Senate. To secure 50 votes, a nominee would have to be substantially more moderate. In the event of a filibuster, the nominee would have to be even more conservative to secure 60 votes.

Still, the model does identify a range of nominees who could thread the confirmation needle. In the graph, the justices in purple serve as reference points for nominees who (roughly speaking) lie between the 50-vote threshold and Kennedy.

The confirmable justices were all appointed before 1975, and all were either conservative Democrats or moderate-to-liberal Republicans. (Recall these measures are based on perceptions of the nominees; Earl Warren was perceived as a moderate Republican at the time of his nomination, although he famously became a reliably liberal justice.)

Nominees like this may not be particularly attractive to liberals today – and, indeed in this era of hyperpolarization, may be even hard to find. Still, if confirmed, one would shift the center of the court considerably to the left.

To be sure, this exercise does not take fully into account the extraordinary present circumstances, which are worse for Obama because of his lame-duck status. Nor do we account for how public opinion affects senators’ votes to confirm or reject a nominee.

So this exercise may be overly optimistic about how many Republicans would support an Obama nominee. On the other hand, the shrewd pick of a nominee with attractive attributes beyond legal qualifications – such as being a woman or an underrepresented minority – could garner more votes than our model predicts.

The statistical evidence won’t allow a definitive prediction, needless to say. But a highly qualified moderate with other attractive attributes might still thread the confirmation needle, if only barely.

Charles Cameron and Jonathan Kastellec are professors in the Department of Politics at Princeton University.