Entrepreneur Elon Musk’s announcement last week accelerating plans for manned flights to Mars ratchets up political and public-relations pressure on NASA’s efforts to reach the same goal.

With Mr. Musk laying out a much faster schedule than the National Aeronautics and Space Administration—while contending his vision is less expensive and could draw primarily on private financing—a debate unlike any before is shaping up over the direction of U.S. space policy.

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