There's no agreed upon definition of "Prediction" "the planets" or "the orbits", so its any random number between 1M and 10B years under wildly divergent criteria.
Orbits have a large number of degrees of freedom and the chaos noise amplification effect is different for each, so given 15M of noise in starting conditions the point along the orbit of the earth is completely unconstrained in about 100M years, but the diameter of the orbit is very tightly constrained in comparison. In 100M years it'll be pretty easy to draw an ellipse and completely impossible to predict where along that ellipse the earth will be, but it'll be somewhere along that ellipse.
There's no analogy in physics, but there is an analogy in numerical simulation, that its like predicting electron orbits. I can do organic chemistry and build lasers all day without knowing where any specific electron is located, does that mean prediction of electron orbits is impossible or that its irrelevant to most practical issues? The specific mechanism for electron orbits is a quantum mechanical limitation whereas the orbit problem is statistical undersampling of noisy data.
Given the services of a measurement calibration lab and sufficient budget I can predict very accurately what comes out of a screw making factory tomorrow given today's sample data without continuously supervising the factory overnight, but I can't calibrate the screw making factory today, let it run 100M years, and usefully statistically predict what the factory will try to ship on one given day 100M years from now.
Orbits have a large number of degrees of freedom and the chaos noise amplification effect is different for each, so given 15M of noise in starting conditions the point along the orbit of the earth is completely unconstrained in about 100M years, but the diameter of the orbit is very tightly constrained in comparison. In 100M years it'll be pretty easy to draw an ellipse and completely impossible to predict where along that ellipse the earth will be, but it'll be somewhere along that ellipse.
There's no analogy in physics, but there is an analogy in numerical simulation, that its like predicting electron orbits. I can do organic chemistry and build lasers all day without knowing where any specific electron is located, does that mean prediction of electron orbits is impossible or that its irrelevant to most practical issues? The specific mechanism for electron orbits is a quantum mechanical limitation whereas the orbit problem is statistical undersampling of noisy data.
Given the services of a measurement calibration lab and sufficient budget I can predict very accurately what comes out of a screw making factory tomorrow given today's sample data without continuously supervising the factory overnight, but I can't calibrate the screw making factory today, let it run 100M years, and usefully statistically predict what the factory will try to ship on one given day 100M years from now.