Calculating relativistic effects of motion in solar system

[KenJackson]

This question and answer are posed in Kim Stanley Robinson's novel "2312".

"Pauline, if someone had calculated the trajectory of an impactor to hit [an exact spot on the planet Mercury], but they forgot to include the relativistic precession of Mercury in their calculation and only used the classical calculus of orbital mechanics, how far would they miss by? Assume the impactor was launched from the asteroid belt a year earlier."

Pauline said, "The precession of Mercury is 5603.24 arc seconds per Julian century, but the portion of that caused by the curvature of space-time as described by general relativity is 42.98 arc seconds per century. Any trajectory a year in duration, plotted without that factored in, would therefore miss by 13.39 kilometers."

My question is, is this all made up? Or might it be accurate?

 

[PeterDonis]

The effect described is one of the classic solar system tests of General Relativity. See, for example, here:

https://en.wikipedia.org/wiki/Tests_of_general_relativity#Perihelion_precession_of_Mercury

I'm not sure about the calculation of the miss distance, because the number given for the total precession is off by an order of magnitude (unless you misplaced a decimal point when copying).

 

[KenJackson]

Interesting. The numbers are in the novel the way I typed them.

Even though Pauline answered only the precession part of the question, I was fascinated by the possibility of the whole calculation. In recent years, NASA and other countries' space agencies have sent probes to Mars, asteroids and even a comet. But I think they all had thrusters to do course corrections along the way.

But if you throw a rock at Mercury from the asteroid belt, how would you calculate (even in theory) the direction and speed to make it hit a specific spot after a year's travel--with no mid-course corrections!?

 

[PeterDonis]

In recent years, NASA and other countries' space agencies have sent probes to Mars, asteroids and even a comet. But I think they all had thrusters to do course corrections along the way.

They did, but the course corrections are very small; a probe can't carry enough fuel to make large course corrections, so it has to be launched very accurately to begin with. The course corrections are not always needed, but NASA allows for the possibility to be safe.

if you throw a rock at Mercury from the asteroid belt, how would you calculate (even in theory) the direction and speed to make it hit a specific spot after a year's travel--with no mid-course corrections!?

You would have to know the precise positions of the asteroid, the Sun, and the planets at launch, so you could compute the rock's trajectory to the required accuracy. It's tedious, but straightforward; you start out with the asteroid and Mercury moving in the field of the Sun as your first approximation, then just add in effects of other planets until you've taken into account every effect that's large enough to matter. Nowadays computers would do all the grunt work anyway.