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Let's assume we have a working FTL drive and thus interstellar travel is possible, either by traveling at very high speeds or by instant teleportation (the two main subtypes of FTL drives in sci-fi). A question arises: how to know where we are going?

All information we have about stars come from their light (or other EM emissions); but since the speed of light is limited, it takes some time to reach us. If we travel to the apparent position of a star which is 50 light years far from us, we'll go where the star was 50 years ago; now multiply that by hundreds or thousands, and you get the picture. A given star might not even exist anymore because it has gone nova, but we simply can't know that before the light from the explosion reaches us.

How can interstellar navigation be achieved, assuming interstellar travel is possible?

(Also note that if we don't have a working FTL drive, the question is exactly the same... but worsened by needing years to get to our destination, at the very high risk of wasting lots of resources to get to a place that isn't there anymore).

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    $\begingroup$ We can generally know from the light from a star that it won’t be going nova for the next few centuries. Novas are not random, and only happen at certain stages of a star’s life-cycle. $\endgroup$ – Mike Scott Aug 3 '18 at 17:11
  • $\begingroup$ Welcome to Worldbuilding, Massimo! If you have a moment, please take the tour and visit the help center to learn more about the site. You may also find Worldbuilding Meta and The Sandbox (both of which require 5 rep to post on) useful. Here is a meta post on the culture and style of Worldbuilding.SE, just to help you understand our scope and methods, and how we do things here. Have fun! $\endgroup$ – Gryphon Aug 3 '18 at 17:19
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    $\begingroup$ Right now it takes years to hit the outer planets, yet we still manage to hit them with probes. How? With math. $\endgroup$ – GrandmasterB Aug 3 '18 at 18:23
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    $\begingroup$ You have a very common misconception here. When you ask where the stars are "now", you are assuming a common definition of "now" between here and the stars. But it is the lack of that very concept of simultaneousness between separated events that is the basis of relativity. Without knowing how your FTL drive works, there is no way to determine when in the worldline of its destination that it will deliver its passengers. $\endgroup$ – Paul Sinclair Aug 4 '18 at 1:44
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    $\begingroup$ @GrandmasterB - The same thing we do every night: take over world building space questions with math. $\endgroup$ – Mazura Aug 4 '18 at 18:03

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Stars don't simply change their orbit.

Most stars will be in an orbit around the galaxy's center of gravity. This orbit might be disturbed by other masses, but the disturbance will only happen very, very slowly.

That's much like the problem of flying to a planet in the solar system, or sending a tight-beam radio transmission there. You need good data about the position and the course of the destination planet (or destination star) and the other objects that perturb the flightpath. For the planet, those are the sun and the other planets. Small asteroids can't significantly affect the planet's orbit. Similar for a star in the galaxy. Rogue planets or brown dwarves won't affect the star's path significantly unless the star comes very, very close to the dark objects.

Just do the math, calculate where the star will be when you get there.

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    $\begingroup$ I like this answer but it uses the "Just" word. You need astronomy and a good understanding of the vector of your destination. $\endgroup$ – jorfus Aug 3 '18 at 22:50
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    $\begingroup$ @jorfus, compared to the stardrive this should be easy. There would be maps, and computer software, and the officers of the ship would be expected to understand the math and be able to do it with pen and paper, in theory, so they can troubleshoot things. $\endgroup$ – o.m. Aug 4 '18 at 4:54
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    $\begingroup$ @jorfus I get what you're saying, but I would expect the people in charge of navigating the ship to have appropriate training and/or degrees to do the job they were hired to do. The OP didn't ask "how would an untrained space-rookie know where to go" after all. $\endgroup$ – Steve-O Aug 4 '18 at 19:50
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There is a complex answer to this that involves general relativity, time travel and maths. I’m gunna ignore that and go for the simple answer where FTL is done by jumping and not ‘going fast’:

Do a series of short jumps.

Jump halfway there. Remeasure. Jump half the distance again. Remeasure. Repeat until you’re happy. There may be times you get halfway there and realise it was a wasted trip. Early seafarers spent years and lives trying to find viable trading routes. Loss is inevitable.

All you can realistically do is try limit it by limiting how much resource you risk in a single go.

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    $\begingroup$ While I agree with your basic approach I think your 50% is way out of line. Look at the quality of the data you have on where the star is (and note that in a FTL world that will be far better than current as you will have observations from other star systems) and how accurate your jump engine is and jump so that you're sure to come up short. Observe and repeat. Shouldn't take more than a few jumps. $\endgroup$ – Loren Pechtel Aug 4 '18 at 19:42
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    $\begingroup$ @LorenPetchel: the 50% was really just an oblique Xeno’s paradox reference. Really I’d pick a distance based on current known data and error margins, possible events en-route and the exact operating regime and resource requirements of the jump tech. The main point was the ‘ correct as you go’ approach. $\endgroup$ – Joe Bloggs Aug 4 '18 at 21:37
  • $\begingroup$ I would be more concerned by arriving too close to the star than too far. $\endgroup$ – Florian F Aug 5 '18 at 17:43
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To summarize a long answer, space is very, very, very transparent. So some of the brightest appearing objects at only one point will be very, very, very far away and measuring the angles between them will give your position on a large scale. Measuring the angles between closer objects will narrow your position down, and so on.

When planning an interstellar voyage, people will aim their star ship at where the destination star will be when they get there, not at where it is now.

There are only few types of events that can destroy a star, and they can usually be predicted thousands, millions, or billions of years in the future, so it will be rare for space travelers to arrive and find their destination no longer exists.

And here are the details of my long answer.

When people shoot at a moving target at a shooting gallery, when hunting, or in war, they "lead the target". They estimate how fast the target is moving sideways and aim a little ahead of the target when they pull the trigger.

Many modern weapons systems have computers to calculate the trajectory of the target and adjust the trajectory of the missile to make sure that it will hit the target. And they do that more or less instantly.

The trajectory of the target star in an interstellar mission might be calculated over and over again over a period of decades to prefect the mission calculations before the mission ever starts.

Astronomers have been measuring the distances to some stars with ever increasing accuracy for over 180 years starting in the late 1830s. The method involves measuring the position of the star with great accuracy a number of times spaced 6 months apart, when the Earth is on opposite ends of its orbit. That makes the apparent position of the star wobble back and forth very slightly, which is called its parallax, and the amount of the parallax gives the distance to the star.

All stars are so distant that even the closest stars have parallaxes less than one arc second - and an arc second is only 0.0000007 of a full circle. Astronomers have been measuring angles of less than an arc second for more than 180 years.

From 1989 to 1993 the Hipparcos satellite of the European Space Agency measured the positions of over 120,000 stars with an average precision of about 0.001 arc second.

The Gaia satellite, also by the European Space Agency, has been measuring the positions of millions of stars and other objects since 2013. The goal is to provide a 3D map of about 1,000,000,000 stars, about 1 percent of all the stars in our galaxy. The accuracy of angle measurement is to be about 20 micro arc seconds.

When travel within our solar system is more advanced, satellites at least as advanced as Gaia will be placed in the leading and trailing Trojan points of the orbits of the four giant planets in our solar system. At any given moment the two satellites in Jupiter's orbit will be separated by about 5.2 times, the two satellites in Saturn's orbit will be separated by about 9.54 times, the two satellites in Uranus's orbit will be separated by about 19.22 times, and the two satellites in Neptune's orbit will be separated by about 30.06 times the total separation between points in Earth's orbit 6 months apart.

The precision of measurements should increase in the same ratio as the length of the baseline increases.

And if a faster than light drive is invented a bunch of manned or automated astrometric observatories will be placed in positions 1 parsec from the Sun in all directions. Since a parsec is 206,265 astronomical units (an astronomical unit is the distance from Earth to the Sun), two observatories on opposite sides from the solar system will have a baseline 206,265 times as long as observatories in Earth orbit or on Earth, and their measurements will thus be 206,265 times as precise.

How will the movements of the stars be discovered? The same ways they have been discovered for a century already.

The movement of a star, relative the the solar system, has two components.

One is the radial velocity toward or away from the solar system. The spectrum of the star will show a doppler shift that shows how fast it is moving toward or away from the solar system. If one knows how far away the star was when the light that reaches us was emitted, and thus how many years the light has traveled, and if the doppler shift in the spectrum tells how fast the star is moving toward or away from the solar system, it is easy to calculate how far away the star is now, or how far away it will be when your starship reaches it.

The other component of a star's motion is the sideways motion of the star relative to the solar system, the proper motion. That is detected by measuring the direction to the star several times over years and noticing any tiny change in the direction. Because the closest stars are likely to have the largest apparent proper motion, astronomers often selected stars with high proper motion for the first measurements of stellar parallax back in the 1830s and 1840s, so proper motion has been measured with increasing accuracy for at least 180 years.

And one of the main missions of the Hipparchos and Gaia astrometric satellites has been to measure the proper motion of many stars much more accurately than before.

So if a star is exactly 100 light years from Earth, it would be exactly 100 times 9,460,730,472,580.8 kilometers, or 946,073,047,258,080 kilometers from Earth.

If a starship can travel 100 times as fast as light for the entire journey, it will take it exactly one year of 365 days (light years are the distance light travels in 365 days) to reach the target star, at a speed of 0.2739726 light years per day, or 99.9999 light days per day - make it an even 100 light days per day, or 2,400 light hours per day, or 144,000 light minutes per day.

If the target star has a fairly reasonable sideways speed or proper motion of about 100 to 500 kilometers per second, it should travel 6,000 to 30,000 kilometers in a minute of 30 seconds, 360,000 to 1,800,000 kilometers in an hour of 3,600 seconds, 8,640,000 to 43,200,000 kilometers in a day of 24 hours or 86,400 seconds, and 3,153,600,000 to 15,768,000,000 kilometers in a year of 365 days or 8,760 hours, or 31,536,000 seconds.

Since a light minute equals 17,987,547 kilometers and a light hour equals 1,079,252,820 kilometers, a distance of 3,153,600,000 to 15,768,000,000 kilometers would equal 2.9220 to 14.4759 light hours distance, or 0.0012175 to 0.0060316 days travel time for the starship, or 0.02922 to 0.1447 hours travel time for the starship, or 1.7532 to 8.685 minutes travel time for the starship.

And that is only if the starship aims at the direction where the star is when it leaves, instead of aiming at the direction where the star will be a year later.

And what if a starship travels at only 1 percent of the speed of light to reach the star 100 light years distant? It will take the starship 10,000 years to reach the destination star, and in that time the proper motion of the star will move it 10,000 times as far to the side as in the earlier example.

Thus the target star will move about 31,536,000,000,000 to 157,680,000,000,000 kilometers, or 3.3333578 to 16.666789 light years, in 10,000 years, which will take the starship about 333.33578 to 1,666.6789 years to travel at one percent of the speed of light.

Thus the importance of calculating the future position of the star and aiming for that future position is proportional to the length of time that the trip will take.

So star ships will tend to aim for a future position of the star instead of its exact present position. The navigators will also be able to observe the apparent position of the star during the voyage, and if they notice any minor errors in the course calculations the ship can adjusts its course during the voyage.

If faster than light space ships "jump" from one point to another in space, without travelling the distance between them, then it seems simple to construct a formula to calculate the time a voyage from one point to another will take.

IMHO the formula should be: (X)Y + (X-1)Z, when X is the number of jumps made in the voyage, Y is the average time that the jumps may take, and Z is the average length of time it takes the ship to recharge its batteries, or recalculate, or for the crew to recover from the stress, or whatever, between each jump. Of course Y and Z can be zero, and X could be anything from one to infinity.

Within a glaxy, the stars that have the largest relative velocity to each other are likely to be on the opposite sides of the galactic center, since they will be travelling in opposite directions as they orbit the center of the galaxy. The Sun has an orbital speed of about 225 kilometers per second, so a star on the opposite side of the galaxy at the same distance from the center should have a velocity difference of about 450 kilometers per second relative to the Sun. Stars closer in could have orbital speeds of 1,000 kilometers per second, so two such stars on opposite sides of the galaxy should have a speed difference of about 2,000 kilometers per second, and so on.

The gravity between the Milky Way Galaxy and the Andromeda Galaxy is pulling them together at a speed of about 110 kilometers per second, and they are expected to collide in about 4,000,000,000 years.

Only about 100 nearby galaxies are approaching our galaxy. The vast majority of galaxies are moving farther apart due to the expansion of the universe. The farther away they are, the greater the velocity difference. Hundreds of kilometers per second, thousands of kilometers per second, tens of thousands of kilometers per second, and so on.

The farther away a distant galaxy is, the longer the light from it took to reach Earth, and the farther away it is now than when the light was emitted.

The oldest electromagnetic radiation detected is about 13,799,000,000 years old and thus was emitted about 13,799,000,000 light years from Earth. And during the 13,799,000,000 years it took that light to reach Earth, the places where it was emitted have moved much farther away from Earth. It is believed the source of that radiation is now about 46,500,000,000 light years from Earth.

So the distance between Earth and the source of that radiation has increased by about 32,701,000,000 light years in the last 13,799,000,000 years. So you can calculate that the distant sources of the oldest known radiation have been moving away from Earth at an average speed of 2.3698 times the speed of light, which is impossible. Actually the distant galaxies are not moving away from each other and Earth, the space between them is increasing in size, so the speed of light limit doesn't apply.

Ahhhh!

Anyway, that shows that very careful calculations would have to be made for a very long space voyage of billions of light years.

A given star might not even exist anymore because it has gone nova, but we simply can't know that before the light from the explosion reaches us.

Actually, stars do not just go nova at any random moment. Novae have causes, and astrophysicists can study a star and determine if it is ever going to go nova, and if so, approximately when. Rigel, or Beta Orionis, for example, is predicted to become a type II supernova in about ten million years.

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  • $\begingroup$ TL;DR Well, not quite true. I did read about a third of your answer before I realised it wasn't going to end any time soon. Perhaps you can say all that a bit more concisely? $\endgroup$ – CJ Dennis Aug 5 '18 at 7:20
  • $\begingroup$ I agree with CJ Dennis, if you can shorten your answer to around 1/3 of its current length you'll have my upvote! As it stands,your answer is over 3 pages and will take the average reader around 9 minutes to read though! $\endgroup$ – TheAverageCanadian Aug 5 '18 at 17:33
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We have a pretty accurate positions and velocities for stars anywhere we could reach even with an FTL system. I don't think there's any real issue. We get a great deal of information from light from stars, including Doppler shifts and we get more from things like parallax, because Earth moves in an orbit and we can compute the position of stars out to a significant distance.

But in any case the first thing anyone would do before sending valuable ships out into the void is check by survey.

You'd built relatively cheap scout ships (probably automated) and send them out to do high accuracy surveys (and you'd be combing data from all of them to get even more accuracy).

Note that a bigger issue is that FTL existing would imply General Relativity was not enough to work with and you'd be depending on the accuracy and maturity of your new theories that explain FTL and (presumably) reduce to general relativity in some limit. Thus you'd be making your maps based on a theory that was not fully tested (in the FTL scenario).

In the non-FTL (General Relativity "still rules") scenario then we've plenty of physics and computing power now to work out accurate maps, or to be more precise, accurate "enough" maps, particularly with the addition of scout ship survey data.

Given we don't know the details of how these other systems are configured, we would not simply power in at max speed. You'd most likely do what sailing ship explorers did in regions which were poorly mapped (or unmapped) - you'd get close enough to survey and them hold your distance until you felt you'd enough information not to be taking bad risks (e.g. finding your ship suddenly navigating through a hard to see region of asteroids that a careful survey would have spotted).

Note also Joe Blogg's answer. This method (jump as far as safe and then jump ever smaller distances to hone in safely on the target) was one that appears in Asimov's great and seminal Foundation series of novels.

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Send probes, and wait for them to return.

Setting aside what has already been said about predicting cosmic events with accuracy.

If you have FTL technology and it is brand new, you may not want to travel very far while your work out the issues with the technology. But once FTL is established and understood, if there has not been an FTL sensory breakthrough you can simply do the following.

Equip several probes with FTL capability. Send these to various locations in and around your destination. Wait for them to return with the data that they observed from jumping to that location and collecting star light for a few hours. You should now be able to form a pretty complete picture of what is going on at your intended destination.

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    $\begingroup$ Apparently SE requires at least 6 characters in an edit, so I'll just leave this here: I'm assuming you meant "now" instead of "not" ;) $\endgroup$ – Steve-O Aug 4 '18 at 19:54
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For high-speed drives, this shouldn't be a problem, as long as you are traveling towards whatever you're aiming for (based on my miniscule understanding of modern physics).

This is because photons can still hit your eyes even if you are going above the speed of light as light as you and the photons are going opposite to one another. Photons going straight towards you can be detected, but photons coming from behind (your origin) can't, since they are going slower than you and therefore cannot hit you.

What this means is that you still have a visual of where you are headed in a high-speed FTL drive. You can then change course based on that.

For jump drives, I assume it will be harder. I would suggest going the route taken by Joe bloggs and the foundation novels, doing micro-jumps and recharting your route based on the observed error.

This is assuming physics and reality don't change dramatically past light speed, of course.

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    $\begingroup$ Constantly adjusting your course will result in a longer path than just going directly to where the star will be. $\endgroup$ – Acccumulation Aug 3 '18 at 21:43
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    $\begingroup$ As o m said, the star is not going to go anywhere by the type you get to it, unless you're traveling to a different Galaxy. However, a point raised in the foundation novels can be referenced here. Close encounters with massive bodies on your journey and gravitational disturbances from our universe can cause you to go off course, which requires constant readjustment. $\endgroup$ – Totillity Aug 3 '18 at 22:02
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You look for Pulsars... which... as their name implies... have a pulse of radiation. This pulse is both constant in the rate of pulses and idiosyncratic. No two pulsars pulse at the same interval of time. It's effectively Space GPS... SPS if you will. You can find Pulsar A, B, and C and triangulate from their position on a 3D plane where in the galaxy you are.

Not only that, but assuming that you worked out the time dilation issues of FTL (i.e. 3 minutes at FTL speed in real life does not mean three minutes of time for your Mission Control Desk Jockey... It's gonna be the better part of a life time depending on just how fast you go) you can also use these Pulsars to create a universal clock between Earth Time and Space Time.

After a quick search, I found that current science has identified that the quickest time between pulses of a Pulsar discovered is 1.4 milliseconds and the longest period of time is 118.2 seconds (1 minute, 58 seconds). The latter (Called AR Scorpii) is unique in that it is the only known White Dwarf Pulsar. A Pulsar known as PSR J0437-4715 is considered to be the most accurate clock ever, as it's Pulse is more accurately timed than even Atomic Clocks (5.75 milliseconds per pulse). This also made it the most accurately located object outside of our solar system. Just an example of three really good choices for real objects that can be used to triangulate based on uniqueness. You can even use the 1:58 gap of AR Scorpii to justify a cool down time on your next jump... so you have a reason for your hot shot pilot and the idiot kid form that stupid back water desert world to dick around while the evil triangle empire ships close in on them... for dramatic tension... and it would be nice if the hot shot pilot didn't flaunt his ignorance of units of measure.

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So, the way I see it, you have a few options:

  1. You have a very in-depth and up-to-date starmap with information about the probability of each star changing state or exploding in a given time-frame, and every time you jump you're taking a calculated (statistical) risk. IT's true that stars don't change their orbit unless acted on by a sufficiently large source of gravity, but it's also true that we don't know the full scope of things comprising space, so it's always possible that your maps are suddenly wrong in some place.
  2. You fire FTL tracers in a range of directions. Each tracer has an entangled atomic computer, with counterparts on your spaceship. You'll get instant information on the status of each tracer, and you'll be able to find out if it hits something and where.
  3. You utilize a a 4th, 5th, or 6th dimensional string to get perfect, instant, up-to-date information about all of the space it comes into contact with (which may be a lot or a little, depending on how you interpret string theory). It's like being a spider and plucking the web to see what's there.
  4. If you've got the technology to peek into other spacial dimensions (like with some kind of sensing array designed to look at the 4D component of the universe), you could potentially check on a broad section of space as easily as being a 3D person looking at a 2D paper world. The issue is that the information you get from this will be extremely difficult for a human to quickly comprehend, because it's coming from a spacial dimension that we aren't equipped to sense. So your computer would have to do all the interpreting. Getting a 4D snapshot and letting computers find empty-straight-lines through space is a good intermediate step on the road to being able to remotely view the current status of all of 3D space without going there.

3 and 4 are pretty powerful, and the technology might impact your universe in a big way that you don't want. I think a good way to handle it would be to combine 1 and 2. You generate a statistical model of where planets and stars might be, based on past data, and then you fire a series of FTL tracers into space to update your model and find the best rout.

ETA: Just thought... if you're jumping into an area of space that you haven't explored, you can still generate a statistical model using whatever information your sensors pick up, and then fire your FTL tracers into it.

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I like OM's answer (you plot the direction of travel and velocity of your destination, calculate how much to lead it based on distance and jump). I think the answer oversimplifies and misses some of the difficulty of astro-navagation. I'll provide an example of how I'd do the astronomical calculation for an instantaneous jump.

You start at "Home Star" you're going to "Destination Star" which is 4 light years away. Your astronomy shows you that Destination Star is moving at in an slightly wider orbit than Home Star. You plot the angle from historical astronomical data. Lets say it's moving at 200mk/s relative to galactic center so you lead it along the known trajectory by that velocity times 4 years for the star's "current location" relative to your time frame. Point there and go. Things are more complicated if you need to account for travel time.

Alternate answer: Your FTL mechanism jumps you into a parallel dimensional space (lets say one that was identical but a billion times smaller and hopefully empty). There is gravitational bleed across dimensions so you can navigate via gravitational distortions from the other side. The stellar map from your universe still works, you just set the scale to a billion to one. You get where you're going, tear a hole between universes and pop out near your destination star. Don't forget to match velocities though.

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True "FTL" actually results in aiming the "wrong way." You actually need to aim before the spot that you observe something is at, because you're going to arrive "early." To see why this happens, you have to consider special relativity. Here's a diagram that demonstrates the problem:

FTL Travel in MS Paint

In this graph the horizontal black line is space; the vertical lines are time (up towards the future, down to the past). Some distance separates you from the target. The blue area represents events that neither party can observe without going FTL, because the light from that event would not arrive in time.

The yellow and green areas represent everything you can observe via light. For example, if something is one light year from you, you'll observe an event one year after it happened. In this chart, imagine that the point where the yellow line crosses the target's time axis represents observing something that happened one year ago. In this regard, the target is one year in the future from your observations.

Similarly, the orange and green areas are the areas is everything the target can observe. When the orange line crosses the your time axis, this represents what you were doing a year ago. Going slower than the speed of light brings you closer to the center of the cones, while going faster takes you towards the edges, with light speed being 45 degrees.

This is where things get awkward; if light takes one year to get from you to the target, but you can arrive in half a year (the red FTL travel line), then you've actually traveled to the target's past, although in the future from your perspective.

This is true for both instantaneous FTL and just really-really-fast FTL. You will end up meeting up with the target in a place relative to the target's past from your observation. Interestingly, anyone watching from the target would end up seeing your trip "in reverse", since the light you reflected near the target would actually arrive before the light near the beginning of your trip.

This is all theoretical, of course, but the main takeaway here is that if you FTL at all, you are going to end up in a time where you can actually partake in the events you've already witnessed (depending on how fast your FTL is). The practical uptake in all this is you don't need to calculate anything if you've been keeping track of where stuff used to be.

Let's say you want to get somewhere that's one light year away, and you want to travel there in six months (2c). At this point, you simply mark a day on the calendar, observe the object's position, wait six months, then take off at where it used to be. Since the light took a year to reach you, you'll end up catching up with the object right as the original observation was taken.

I realize that this answer is similar to the other answers, but I feel that a diagram helps clarify things a bit. You'll need to think about "how FTL" you're actually going, because that will determine the calculations you need to make. This answer also presumes that you want to end up as close as practical in a single shot, if only for the reason that FTL is really expensive and jumping about all day is impractical.

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Even today, we already have an accurate idea of the three-dimensional motion of many of the stars in the Milky Way, especially the ones that are close by (the 50 l.y. example you give). It's easy enough to extrapolate those motions and know where the stars are actually right now.

Here's a movie of stellar motions of two million of the brightest (apparent brightness) stars in the night sky, compiled using data from the Gaia spacecraft. https://www.youtube.com/watch?v=Ag0qsSFJBAk&ab_channel=ESAScience%26Technology

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Eh, it doesn't matter. Even assuming that you are moving at JUST the speed of light, and take 50 years on a return trip the remote star system wouldn't have moved "much" on a galatic scale - about 600 million km, assuming it moves through the galaxy at the same speed our sun does.

Which, again if it were aimed at Sol, would put it about 5 earth orbits out from the sun.

Given the distance, to put it in shooting terms, this is so much smaller than MOA that it would be multiple bullets through one small ragged hole at 100 yards.

All of this of course goes to heck when you start changing how fast the target star is moving, etc. due to black hole proximity, etc.

Sol speed via http://solar-center.stanford.edu/FAQ/Qsolsysspeed.html

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