SAN FRANCISCO — NASA scientists have finally cracked a decades-old spaceflight mystery, figuring out why the Pioneer 10 and Pioneer 11 interplanetary probes began to slow as they sped far from the sun.

The cause of the so-called Pioneer Anomaly, they said, is heat coming from the electrical current flowing through the probes’ instrument and power systems. This heat pushed back on the spacecraft, causing them to decelerate slightly, according to a new study published in the Physical Review Letters.

“The effect is something like when you’re driving a car and the photons from your headlights are pushing you backward,” lead author Slava Turyshev of NASA’s Jet Propulsion Laboratory (JPL) in Pasadena, Calif., said in a statement. “It is very subtle.”

Pioneer 10 and Pioneer 11 launched in 1972 and 1973, respectively. They were the first spacecraft to fly through the main asteroid belt, and the first to study Jupiter up-close. The probes kept on cruising after their Jupiter encounters, speeding toward Saturn and beyond. Pioneer 10 and Pioneer 11 will eventually exit the solar system, but they likely will not be the first to do so. The Voyager 1 spacecraft, which is about 17.8 billion kilometers from Earth, is believed to be on the verge of entering interstellar space.

In the early 1980s, mission scientists noticed the Pioneer spacecraft were slowing down unexpectedly. But they dismissed this as a transient phenomenon resulting from dribbles of propellant left in the probes’ lines, researchers said.

The issue did not go away, however. In 1998, when Pioneer 10 and Pioneer 11 were more than 13 billion kilometers from the sun, a team of researchers calculated that the spacecraft were decelerating, albeit very slightly.

The scientists could not explain the slowdown, so they raised the possibility that some new type of physics that contradicted Albert Einstein’s General Theory of Relativity may be responsible.

In 2004, Turyshev started trying to get to the bottom of the Pioneer Anomaly. He could not rely on any new data, because the two spacecraft had stopped communicating with Earth; scientists last received a signal from Pioneer 10 in 2003 and Pioneer 11 in 1995.

So Turyshev and his colleagues searched through old Pioneer telemetry, copying digitized files from JPL navigators who helped steer the Pioneers and receiving more information from other NASA centers.

They also found more than a dozen boxes of magnetic tapes stored under a staircase at JPL, researchers said, and worked with a Canadian programmer to create software that could read the tapes and clean up Pioneer data.

In the end, Turyshev and his team collected more than 43 gigabytes of information, a huge trove for the 1970s.

After poring over all this information, the scientists were able to trace the Pioneer Anomaly to the heat coming from the spacecraft’s power and electrical systems. Further, they determined that the issue was not affecting other spacecraft nearly as much, mostly because of differences in design.

“The story is finding its conclusion because it turns out that standard physics prevail,” Turyshev said. “While of course it would’ve been exciting to discover a new kind of physics, we did solve a mystery.”