The only way to project the path of an object in the solar system is to computationally model the solar system and then run the simulation forward. Obviously space programs like NASA and ESA have such models; that's how they get space probes like New Horizons or Rosetta to their destinations.
Interestingly, I've read that these simulations use Newtonian physics, not general relativity, to model the solar system. At interplanetary speeds, the error between the two is just too small to matter.
So if you want to project the orbit for the "lifetime" of the Roadster, you'd have to just run the simulation forward until it finds a collision or the sun burns out.
This would take a lot of energy and time... it would not be worth it.
The main reason is that interplanetary space is very empty in general, so without running any simulation we can guesstimate that an accidental collision is very unlikely.
Also, probably no one really cares if the Roadster hits an asteroid 513 million years from now.
Finally, a physical model simulating that many objects is chaotic--even eensy weeny teeny tiny little errors will compound over time and produce a divergence from reality. I doubt anyone has characterized this drift beyond the horizon needed to plan missions--a few decades at most. So even if you ran the model forward for 513 million model years, there might be no way to constrain the accuracy of the final predicted location of the Roadster. Heck, maybe the difference between Newton and GR would matter at these time scales.
There's also just forces that will be hard to characterize in a model, eg thermal recoil where the heat radiating off an object produces a tiny acceleration in the opposite direction. The roadster has black panels that will heat up more than the rest of the body work. Then there's other tiny forces like the solar wind. All tiny but over huge time spans their influence keeps widening the error bounds on the simulation.
Interestingly, I've read that these simulations use Newtonian physics, not general relativity, to model the solar system. At interplanetary speeds, the error between the two is just too small to matter.
So if you want to project the orbit for the "lifetime" of the Roadster, you'd have to just run the simulation forward until it finds a collision or the sun burns out.
This would take a lot of energy and time... it would not be worth it.
The main reason is that interplanetary space is very empty in general, so without running any simulation we can guesstimate that an accidental collision is very unlikely.
Also, probably no one really cares if the Roadster hits an asteroid 513 million years from now.
Finally, a physical model simulating that many objects is chaotic--even eensy weeny teeny tiny little errors will compound over time and produce a divergence from reality. I doubt anyone has characterized this drift beyond the horizon needed to plan missions--a few decades at most. So even if you ran the model forward for 513 million model years, there might be no way to constrain the accuracy of the final predicted location of the Roadster. Heck, maybe the difference between Newton and GR would matter at these time scales.