When NASA launched twin Pioneer spacecraft toward the outer solar system in the early 1970s, there was no dilly-dallying.

Rocketing away from Earth at more than 32,000 miles (51,000 km) per hour, Pioneer 10 reached Jupiter in just 21 months. Pioneer 11 followed about a year later, then boomeranged out to Saturn. By 1979, their planet-hopping days were over.

But even as they coasted outward toward the stars, the craft continued to radio their whereabouts and a trickle of scientific data back to Earth — and that's when dynamicists noted something very odd: neither spacecraft was as far away as expected. Instead, it was as if some unknown force were adding a little extra tug back toward the Sun.

Over the years, many theorists have weighed in on the possible cause of the "Pioneer anomaly." Logically, a few speculations focused on measurement errors, propellant leaks or some unanticipated property of the spacecraft.

Others explored the Pioneers' interactions with the solar wind, solar radiation pressure, or interplanetary particles. And more than a few were "out there," conjuring a pull imparted by unseen masses, variations in Newtonian physics, and controversial notions of altered space-time.

Five years ago, after many years of painstakingly dredging up old tracking data and reconstructing the probes' trajectory, Slava Turyshev (Jet Propulsion Laboratory) announced that some (but not all) of the retarding force was due to heat radiating away from the spacecraft unevenly.

Now four Portuguese physicists have taken a closer look at how the Pioneers radiated their heat. Key to their analysis is a technique used in computer-graphic programs known as Phong shading. It keeps track of light's diffuse and specular reflections off a surface and uses polygons to model curved surfaces.

Led by Frederico Francisco of Lisbon's Instituto de Plasmas e Fusão Nuclear, the team finds that waste heat from each spacecraft's main equipment compartment was reflecting off the back of the 9-foot-wide (2.7-m) paraboloidal antenna that always pointed inward toward the Sun and Earth.

Just how much heat was bouncing around remains unclear: "The main difficulty in dealing with this problem has always been the lack of sufficient and reliable information for a detailed engineering modeling of the spacecraft," they write. However, by assuming a range of plausible values, they calculate that the tiny force resulting from those reflections closely matches the Pioneers' observed deceleration.

"Unless new data arises," they conclude, "the puzzle of the anomalous acceleration of the Pioneer probes can finally be put to rest."