Path to a distant star

Suppose we have a space ship that can go fast enough (0.9c) and we have plenty of time, and we want to travel to a star on the other side of the galaxy. How do you plot a path to navigate to that star? Given the distances and the number of stars in the way, which all affect the shape of the gravitational field, is it even possible to compute a reasonable path to follow? Do we know the trajectories and masses of the stars along the way with enough precision? Should we shoot outside the galactic plane and come back closer to it again as we get closer to the star? Stay in the galactic plane? Something else?

• Is there a particular star you urgently need to get to? – safesphere May 28 '18 at 4:39
• Good question. One with a habitable planet might come in handy quite soon... – Frank May 28 '18 at 23:06

If you are flying about our sun at 1 AU (i.e. the same distance as Earth) you will get away at hyperbolic trajectory if you have a velocity of just 42.11 km/s. This is over 6400 times less than 0.9c. That means that with very high precision you may describe your trajectory as a straight line. Even if you touch the surface of the sun you will be deflected only by about couple arcseconds. However the galaxy is rather empty and unless you are VERY unfortunate so that the target will be almost eclipsed by some other star you will never approach anything massive closer than thousands AU and may basically forget about gravity altogether. Even in the worst case scenario in which your target is eclipsed by Proxima Centauri, starting an arcsecond off course will get you at the distance of 0.2 AU where you will not be influenced by its gravity at any significant measure. You may still need to account for a galactic gravity but we're still talking about small corrections to the initial angle. So don't expect any fancy maneveuring, your path will be almost ideally straight. All you really have to worry is to know the velocity of the star with enough precision to predict its trajectory over the years but if you're really able to do fancy maneveuring at 0.9c all you will have to do is to slightly change your angle as you get closer based on more recent data.

• Well - sure - but given the time it's going to take to travel, you need to predict at least the trajectory of the star you are trying to reach, which I don't believe is trivial: first, we probably don't know the speed of that star very precisely, second, numerical simulations over very long periods of time might diverge to the point of being useless... I think the practical problem is not as simple as you make it sound. – Frank May 28 '18 at 23:09
• @Frank You know, words "practical problem" and starship movong at 0.9c don't fit together much at this time. Even producing a probe moving at 1% of c is a science fiction now while given more resources on a dedicated research we can greatly iresearchhe data and predictions for the movement of nearby stars. Anyway, while the typically stars move faster than anything in our solar system those speeds are still miniscule at the scales we are talking about and the accelerations are even smaller. The ability to correct some arcssmallermakes it really easy. You observe the star at the start,and then – OON May 29 '18 at 6:46
• @Frank ...and mid-course. The difference between the initial observed star position and the one you see in the middle of your journey will allow you to predict the star position as a problem like "train goes from A to B". This will be adequate for journeys taking hundreds of years. It's much harder when you can't make any corrections mid-course. Name me a star you like and I will give you an estimate how precise we can be with our current knowledge – OON May 29 '18 at 6:51
• @Frank If you wait for tomorrow I can add a detailed analysis for the simple shootings and mid-course corrections for the stars I like:) – OON May 29 '18 at 6:59

If you travel close to $c$ you will basically follow the same path as a light ray. This will indeed be bent by gravity, but for most paths through the galaxy this is not close enough to a star that there will be any effect (stars are lightseconds across, distances between them lightyears - and since it is cross-sections we care about, you get to square this ratio).

If stars were just moving linearly, the aiming problem would be simple: the position would be $\mathbf{x}(t)=\mathbf{x_0}+\mathbf{v_x}t$, the position of the spacecraft would be $\mathbf{y}(t)=\mathbf{y_0}+\mathbf{v_y}t$, and you would just have to solve $\mathbf{x}(t)=\mathbf{y}(t)$ to find $\mathbf{v_y}$ and $t$ - a set of 3 linear equations with 4 unknowns, so you have some freedom to select velocity and arrival time (other constraints such as relativistic time dilation and fuel cost obviously would matter).

But stars are orbiting, so $\mathbf{v_x}$ is changing over time. In the solar system where planets and spacecraft follow conic sections finding a trajectory that ends up at the destination is a nonlinear problem with a somewhat messy solution. The galactic version is essentially the same, but now with a different gravitational potential. Presumably there is a similar solution,

One can imagine sending out a bundle of possible trajectories with slightly different velocity vectors; at any given time they form a surface or volume $\{\mathbf{y}_{\mathbf{v}_y}(t)$}. The star follows a curve through spacetime, $\mathbf{x}(t)$. Whenever the curve intersects the surface/volumes there is a possible solution. In reality this is solved using numerical methods.

These methods in the solar system take gravitation from other bodies into account; the galactic version would have to do something similar. This might require estimating the effect of dark matter blobs, which we currently are bad at. But as we map the galaxy more we will be able to plot trajectories better.

• I am wondering if uncertainties in measurements and numerical calculations will not render the calculation of the trajectory useless - the time scales involved are quite long. I heard numerical simulations even for the solar system over a few hundred years become problematic. – Frank May 28 '18 at 23:12
• @Frank - The solar system is fairly strongly interacting, so perturbations build up over time and are strongly increased by possible near passes (but practically modern simulations about most of the solar system has timescales in millions of years or more now). The galaxy case only covers a few hundred thousand years of transit. Still, you can view the uncertainty turning the trajectory from a line into a (bent) cone of possibilities. A good control system would keep it from widening too much. – Anders Sandberg May 29 '18 at 6:25