 Why does Gile volunteer for the ship in the end?
But with the Procyon shuttle arriving yesterday, there was no telling what might turn up. For eighty years now, they’d been sending out the little ships that vanished at greater than the speed of light, equipped with every conceivable device to make them return automatically after taking pictures of wherever they arrived. This was the first hope they’d found that the century-long trips between stars in the ponderous shuttles might be ended and he should have been filled with excitement at Jordan’s hasty preliminary report. He had no desire to spend ninety years cooped up with a bunch of callow young emigrants, even in one of the improved Exodus shuttles. And even if Exodus ever got the super-light drive working, there was no reason he should give up his work. The discovery that men could live practically forever had put an end to most family ties; sentiment wore thin in half a century—which wasn’t much time now, though it had once seemed long enough. That missile reached Sirius in less than ten days. It took two accidents to get the thing back to us—and with a half-light-year miss that delayed it about twenty years before the Procyon shuttle heard its signal. Pre-setting a course may take centuries, if we can ever master it. Even with Sirius expecting the missiles and ready to cooperate. Heroism had belonged to those who knew their days were numbered, anyhow. Dubbins was at his side almost at once, with a hand under his arm to guide him toward his suite.
 He accepts his mortality, and is willing to spend his last years on the chance to see his family.
 He hasn’t accepted his  mortality, and is trying to make up for it with an act of heroism.  He knows that no one else will volunteer, and feels responsible to do so.  He fears the end of his life, and wants to try to see his family before he passes.  He accepts his mortality, and is willing to spend his last years on the chance to see his family.