A gedanken on being able to navigate back to Earth

Question

Imagine a sphere centered at Earth with a radius of 4 light years. Suppose we pick a random point on the surface of this sphere and drop someone, let us say, "Joe" there. Will Joe be able to come back to earth? (Ignore all things ignorable).

Would the answer be different if the radius of the sphere was 40 light years or 400 lightyears?

EDIT: We can ignore if we can travel 4 light years in a lifetime. I was interested in if we could look at the stars and constellations in the sky to get back to Earth. So assume navigation only by looking at the space.

Answer 1

Answering the corrected question, "I was interested in if we could look at the stars and constellations in the sky to get back to earth. So assume navigation only by looking at the space." The answer is yes, with the right equipment and data Joe could.

The stars that are visible are scattered in three dimensions, so a shift of 4 ly will cause a huge parallax shift in the positions of the nearer stars and progressively smaller changes in the positions of more distant ones and extragalactic objects will move only imperceptibly.

So: Use your telescope (it doesn't need to be a terrifically big one, though it should be stable) to photograph the sky. Use the software you just happen to have along to compare those photos with data from Earth.

First, you identify the extragalactic objects -- very easy as they will have shifted very little with respect to one another -- and compare their observed positions with what's in your catalogs: you now have a coordinate system which is aligned with Earth's.

Next you work your way inwards and look for easy to identify, but moderately distant objects: things like the Orion nebula, the Pleiades, specific globular and open clusters, etc. They will have shifted positions relative to the distant stars due to parallax, and those parallaxes will give you a pretty good estimate of your position.

Finally, you start looking at stars. Because you now know roughly where you are (to better than a light year's accuracy) you can use the known 3D positions of the stars and estimate where they would be shifted to. You'll very quickly discover that there's only one position in space that's consistent with observation. This should get you accuracy of .01 to .1 light year -- basically the limitation will be the systematic errors in your 3D star positions in the catalog. (Pretty small in the era of Gaia.)

Head towards the Sun. Stop when closer and measure again if needed.

(Note that this gets harder the further away from the Sun you start, since our 3D start positions get rougher the further away you go. from 400 ly or 4000 ly, you d almost certainly have to make several stops, first getting close enough to the Sun that you can ID specific stars and then getting close enough that you can pick out the Sun from all the similar stars.)

Answer 2

You can challenge yourself to do such a thing by using a program such as Celestia or Spaceengine.

During a period of depression many years ago, I would repeatedly (using Celestia) travel to a random star about ~5000 ly from earth and try to find my way back. It's not always easy, but this is how I generally did it:

First, look for Orion. The stars that make up Orion are very bright and (mostly) in the same rough area, although obviously, it will look different from different angles. I imagine that any hypothetical civilisation in a 10,000 ly radius from us will feature some form of orion as a major constellation. You can confirm Orion by identifying Betelgeuse relatively easily. The stars in orion are about 1000-2000 ly away from earth, so that provides a good starting point.

Next, look for the Pleiades. This compact star cluster is about 400 ly from earth, and should look distinct from any other star cluster in the vicinity of Orion.

Next, look for the Hyades. This is a less bright, more diffuse star cluster that you need some form of parallax to spot. The Hyades are about 140 ly from Earth, but the red star Aldebaran will be nearby. Since Aldebaran and the Hyades both form part of the constellation Taurus, drawing a line through them will point roughly in the direction of the Sun. Aldebaran is about 65 ly away from earth, which is getting us close.

The next bit is tricky. You can continue to identify nearby bright stars (Arcturus at 36 ly, Fomalhaut and Vega at 25 ly), but I found that doesn't help terribly much. What you're looking for is a set of four relatively dim stars (the Sun, Alpha Centauri, Sirius and Procyon) that form a sort of elongated tetrahedron. This can be tricky as there is at least one other set of relatively nearby stars that also sort of look like this (whose names escape me at the moment). Once you find this elongated tetrahedron, you're basically done, as the Sun is the dimmest of the four stars.

This is achieved through being able to click on the stars to identify them. Practically speaking, if you're in a ship that can travel multi-thousand light year distances in a human lifetime, you probably also have a star database on hand that lets you identify stars based on their spectral properties.

I imagine finding your way back without conclusively identifying stars would be much trickier, though still possible.

EDIT: Here is a Spaceengine forum discussion detailing others' methods of finding their way back to earth.