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An interstellar civilization has superluminal travel which allows for travel between two points, effectively without traversing the space in between so they don't crash into anything between them and their target.

Navigation presents a problem: the visible stars only give a snapshot of their positions years previous and they have moved since then, so the current position of the target must be calculated.

The dangers of incorrect calculations include getting lost or, less commonly, arriving inside a solid object. Getting lost isn't immediately deadly but costs precious time and fuel to detour, assuming you have the instruments necessary to navigate on your own.

What logistical concerns are involved in the civilization calculating a constantly updating map of the exact position of every star, planet, moon, and other navigational hazard in the galaxy?

(This question is inspired by propulsion in the Battlestar Galactica remake, which uses concepts like "jump drives" and "red lines." I did find an analysis, but wasn't sure whether this applies to FTL in general.)

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  • $\begingroup$ after beating the cosmic red light (breaking speed of causality) be thankful that you won't get any ticket so don't push it! alright let's get to business... every celestial objects in known universe including dark matter is govt by the laws of gravity and a subtle variation like a tiny speck of dust gain insignificant amount of kinetic energy via kissing could creates a spiral galaxy somewhere else(galaxies colliding) you sure you still wanna compute? $\endgroup$ – user6760 Oct 15 '16 at 5:04
  • $\begingroup$ Just to clarify: Your question title says "superluminal travel" which implies that the velocity has some finite value in all reference frames (you may be going fast, but you are still moving at some specific real velocity in any given reference frame), but your question body says "instantly" which implies that there exists some reference frame in which $v=\infty$ (because otherwise the time required to travel the distance would, in every reference frame, be non-zero and thus not "instant"). Which is it? $\endgroup$ – a CVn Oct 15 '16 at 11:34
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Summary

What you're trying to do doesn't make sense unless you send a few thousand (or more, depending on sensitivity) mapping drones into the galaxy to keep up-to-date mapping data. The drones would be connected in an FTL sneakernet so you can just ping the nearest drones for data from time to time and have pretty accurate data.

Note that if you're in the thousands of probes range, the data will necessarily be predictions from light thousands of years old. The simulations should be very accurate with advanced math, but things like "the Death Star exploded your planet" won't show up.

Additionally, without extremely good resolution on the drones (not likely due to diffraction limits), there will always be uncertainty about the exact positions of small bodies in the unexplored / infrequently visited solar systems. As such, you'll need to send a scout drone ahead to be absolutely certain there's nothing where you're going in you're in unexplored areas.

However, mapping within explored solar systems would be done routinely so there will be no issues there.

You're pretty safe blindly jumping anywhere you can see, so it's not that huge of a deal anyways1.

If you want one-year "real-time" accuracy, you'll need hundreds of millions to hundreds of billions of probes. Same thing if you want to see all the planets in the system without probes jumping ahead.

I didn't do any math, but the entire network could potentially be setup within days if your mapping drones can jump hundreds of light years at a time. However, it will take some time to calculate precise velocity information to make your models accurate. A few years should be fine, but an advanced civilization could probably do it faster. From there, updates every few decades should be sufficient.

Can't take a snapshot from a single point.

There's no way to take a snapshot of the entire galaxy from a single point in the galaxy. The galaxy itself blocks light from reaching us, plus there's a resolution limit due to diffraction. There's (probably) also the pesky super-massive black hole in the center of the galaxy that's difficult to see past.

That said, you could get pretty good pictures of the galaxy by using a number of probes spread through the galaxy. There's no way you'd get exact positions of all the planets, asteroids, and smaller junk from really long distances, but you could at least know where all the major stuff is.

Highly populated areas would have lots of data available for real-time updates on pretty much anything hazardous. Unexplored areas would require mapping before a jump.

Need lots of probes. How many?

Future technology will improve our resolution to an extent, but let's say 1000 light years is a good range to see most stuff (this page says we can use parallax out to a couple thousand, but let's be conservative). Then we just need to cover the galaxy in a 1000 light year grid.

The galaxy is roughly cylindrical. From Wikipedia, it's about 100k light years in diameter, and 1k light years thick. That's a volume of about $2\pi rh$ $=2\pi(50kly)^2\cdot1kly$ $\approx6\cdot10^{13}ly^3$.

In 3D space, the furthest points in a rectangular grid are $\sqrt{d^2+d^2+d^2}$ $=\sqrt{3d^2}$ $=d\sqrt{3}$. The furthest point should be in the middle of a grid cube. This means our grid needs to be $2\cdot\frac{1000ly}{\sqrt{3}}$ $\approx 1155 ly$ from node to node in order for every star to be within 1000 light years of a station.

Originally, I thought of a bunch of cubic groups. Each group will have eight stations (the corners of the cube), but almost all of those stations will be shared by four or eight cubes. Each cube will be $(1155ly)^3$ $=1.54\cdot10^9ly^3$ in volume. That means we need about $\frac{6\cdot10^{13}ly^3}{1.54\cdot10^9ly^3}$ $\approx 39000$ cubes. Since each cube has eight stations, and most nodes are shared by eight cubes, that's around 39000 stations. Note, however, that this only applies if you're putting a lot of stations much closer together.

It gets a little tricky here. In a 3D grid, each station will be shared by eight cubes. So there's approximately 1 station per cube, as above (technically, the outer edges will have un-shared stations, but they're in the minority). But once your view distance is over 1000-2000 ly (the thickness of the galaxy), we just have a single layer covering everything, so it's a 2D grid. In this configuration, each station is shared by about 4 cubes, so there are 2 stations per cube.

Diagram showing vector distance to galaxy edge from single cube.

The blue dot is a star at the edge of the galaxy. It's maximal distance to a station given a single layer of cubes is given by the green vectors, which are equivalent to summing the maroon, red, and orange vectors. The distance is $\sqrt{m^2+r^2+o^2}$. The maroon and red vectors are both half the distance between stations, or $m=r=\frac{a}{\sqrt{3}}$, and we need the total distance to be less than $a$ (where $a$ is the maximum distance we need). Plug that all in to get:

$a=\sqrt{2\cdot\frac{a^2}{3}+o^2}$
$a^2=\frac{2a^2}{3}+o^2$
$o^2=a^2-\frac{2}{3}a^2$
$o=\frac{a}{\sqrt{3}}$

Total thickness is $2o$ (same distance above and below the cube) plus the height of the cube, $\frac{2a}{\sqrt{3}}$. If $h$ is the galaxy thickness, then:

$h=2\frac{a}{\sqrt{3}}+\frac{2a}{\sqrt{3}}$ $=\frac{4a}{\sqrt{3}}$
$a=\frac{h\sqrt{3}}{4}$ $=433ly$

So one layer of cubes will blanket most of the galaxy (the central regions are a bit thicker) if you're putting enough stations to keep one within 433 light years of every star. In this case, we're no longer comparing volumes, but areas. The galaxy is $\pi r^2$ $=\pi(100000ly)^2$ $=3.14\cdot10^{10}ly^2$ in area. Each cube on the grid is $(\frac{2a}{\sqrt{3}})^2$ $=\frac{4a^2}{3}$. For $a=1000ly$, we need $23550$ cubes. Each station only shares about four nodes, so you need twice as many stations as cubes, $47100$ stations total.

Additionally, if our stations can see far enough, we don't even need cubes. At that point, we just need a 2D plane of stations. We can use the same math as above, except use a zero-height cube.

$h=2\frac{a}{\sqrt{3}}+0$ $=\frac{2a}{\sqrt{3}}$
$a=\frac{h\sqrt{3}}{4}$ $=866ly$

Again, we're comparing areas. Each square has four stations, and most stations are touching four squares, so it's one-to-one. $23550$ stations total.

That's a lot of stations, but considering you're talking about exploring the galaxy, it's not really that bad. Plus, you only need to blanket the parts of the galaxy you intend to explore.

Modifying the number based on different view distances.

Ok, so we can already see more than 1000 light years using parallax methods. What if you want to calculate a different value?

There are three cases. In the case where $a<433ly$ (or you're looking at a spherical galaxy or something), stations is proportional to $a^3$. So take the ratio of new $a$ to calculated $a=1000ly$ above, cube the ratio, then divide that into 39000. For example, using $a=100ly$:

$\frac{39000\text{ stations}}{(\frac{100ly}{1000ly})^3}$ $=\frac{39000\text{ stations}}{\frac{1}{1000}}$ $=39\text{ million stations}$

In the other two cases, where $433ly<a<866ly$ and $a>866ly$, station count is proportional to $a^2$. Same thing, but square the difference.

$\frac{23550\text{ stations}}{(\frac{5000ly}{1000ly})^2}$ $=\frac{23550\text{ stations}}{25}$ $=942\text{ stations}$

That's the entire galaxy with only 942 mapping stations.

How do we update that in "real-time"?

Use the "sneaker net", combined with FTL, like in this question.

Basically, each mapping station has a few small FTL drones that warp back and forth between nearby nodes. A drone's host node gives the drone all its current information via some kind of close-range, wireless (or wired, doesn't really matter) transmission. The drone pops to each "connected" node (the six nearby nodes: up/down, left/right, front/back) and transmits that information to the connected node with the same kind of short-range transmission. At the same time, it would receive information about distance nodes from the connected node. Then the drone pops back to the host node and uploads all the newest data.

You'd need a pretty big dataset to hold all this information, but a society this advanced shouldn't have much trouble with that. Each node has a complete copy of the data at all times. Far away nodes will be slightly out of date, but the FTL nature of the network means they'll be accurate within about $JumpTime\cdot NumberOfNodes$. This is a linear node count; worst case scenario is going to be about $JumpTime\cdot\frac{2\cdot GalaxyDiameter}{NodeDistance}$ (data is coming diagonally across the grid, so it has to jump south, east, south, east, etc., for example, meaning the delay is about twice as long as going straight along the nodes).

For a 2 minute jump time (time to jump, transfer data, jump back, transfer new data) and 5774 ly node distance (corresponding to a 5000 ly view distance), that's $2min\cdot\frac{200000ly}{5774ly}$ $\approx69min$, or about 1 hour. The number is linear, so if you double the jump time, you'll double the lag time.

Also, it depends on how many drones there are per station. The 2 hour number assumes six active drones per station; if you only have one drone popping to 6 stations, that's six times the delay, or about 12 hours. For the 2D grid of cubes, you only need 5 drones per station, and for the 2D square grid, you only need 4 drones per station.

In a highly 3D grid (in a spherical galaxy or similar), worst case is $\frac{3}{2}$ worse, because it's doing south, east, down, south, east, down, etc., for example. Note that a disc-shaped galaxy (like ours) will still use the above numbers, even if you have a lot of nodes. In this case, the number of down jumps will be small compared to the number of south and east jumps, so you can ignore them.

What is "real-time"?

As kingledion notes in a comment, "real-time" isn't really real-time. Each node still has a pretty big lag between the events happening and the light hitting the node. My assumption was that we just needed to see the stars closely enough that we could keep the calculations accurate.

If my assumption is correct (the question seems to indicate that's all we need), then the "real-time" aspect isn't really important. You just need to update once every few decades or centuries to keep your simulations reasonably accurate. This is a good thing, because it means you can get by with far fewer drones and energy requirements.

However, if you want more accurate data, you'd need to either:

A) Have a lot of probes. To keep everything within 1 year accuracy, you'd need to bring the probe distance to 1 light year. That's around $4\cdot10^{13}$, or $40\text{ trillion}$ stations. A lot more than you likely want.

To be fair, there are only 200-400 billion stars in the galaxy, and there's not a really good reason to use more than one station per star except highly traveled FTL routes. So 200-400 billion is a more "reasonable" cap.

B) Have probes that do a lot of hopping around. Depending on your FTL energy requirements, this might be quite difficult. But you can just have (relatively) a few probes that pop from star to star. From this site about cloaking (and how you can't do it in space), they calculate about 4 hours to scan the entire sky for things the size of spaceships.

Our futuristic space probes could likely do it in 30 minutes or less, though they'll need to do three or four scans from different positions, so let's call it 2 hours (probably a lot less). You say these guys can jump hundreds of light years at a time, so a probe should easily be able to hop the five to ten light years between nearby stars in a single jump.

Add the jump time to the scan time (the probes can absorb most of this by running calculations and charging the jump drives while scanning). Let's say jump time is pretty minimal, so the total is 2.5 hours per system.

Now, let's say you want data less than 1 year old. Each probe can jump through $\frac{8760 \frac{h}{yr}}{2.5 \frac{h}{\text{system}}}=3504\text{ systems}$ per year. This means you need around a hundred million probes to cover the galaxy.

Both of these options also have the advantage that you can see all the planets and so forth inside every system. At high cost, of course.

What about those pesky, unexplored systems?

If you're jumping into uncharted territory, you'll want to send a lead drone. The drone pops in, maps the nearby area, then pops back to the main ship with the results. You just want to make sure you don't hit anything, so the drone can be tiny. If the drone doesn't come back, don't teleport there. Send a second drone a few million miles away and try again.

Space is huge. Even stuff inside the solar system is really far apart. The reality is that you could randomly jump around our solar system for the rest of your life and probably die of old age (or mutiny)1. A couple lead drones should be more than sufficient for most crews.

1Derivation of safety statistic.

The Sun is about 99.8% of the solar system's mass. The Sun's density is $1410 \frac{kg}{m^3}$. Ice comets have a density of $0.6 \frac{g}{cm^3}$ $=600\frac{kg}{m^3}$. The Sun's volume is about $1.4\cdot10^{27}m^3$. Double that and you've got way more than the volume of "stuff" in the solar system. The solar system (just counting out to Pluto) is about 7.5 billion km in radius. That's a volume of about $1.8\cdot10^{30}km^3$.

That means about $\frac{1}{1.3\cdot10^{12}}$ of the solar system is stuff. If you make one jump an hour for 60 years, that's 525600 hours. Your probability per jump of hitting something is $P=\frac{1}{1.3\cdot10^{12}}$. The probability of hitting something after N jumps is $p(n)=1-(1-P)^N$. $p(525600)$ $=1-(1-\frac{1}{1.3\cdot10^{12}})^{525600}$ $\approx 4\cdot10^{-7}$. That means about 1 out of 2.5 million people will hit something if they all jump once an hour for 60 years.

Some guys from Reddit come up with something similar.

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  • $\begingroup$ You're multiplying 321,000 cubes by 8 stations, but wouldn't that mean every corner except 8 would have 2-4 stations overlapping right in the same spot? $\endgroup$ – Dan Oct 15 '16 at 5:23
  • $\begingroup$ @Dan: Good catch. Then I went back over the math and noticed other errors. I made my cubes half the height they needed to be, so I was off by a factor of 8. Also, I realized I should take the thickness of the galaxy into account, since any larger view distances mean it's really a 2D graph, instead of 3D. $\endgroup$ – MichaelS Oct 15 '16 at 10:00
  • $\begingroup$ The OP wants an 'up-to-date' map of the galaxy, so I'd assume that would be something like data that is between 1 year and 10 years old. Even the 100 ly stations scenario (39 million stations) means that most of the information in the galaxy is decades old. Based on the update frequency of nautical charts on earth, that is very old. I think you should post a calculation for 1ly or 10ly distance and see how many stations you need (answer: many). $\endgroup$ – kingledion Oct 17 '16 at 0:23
  • $\begingroup$ One more, the FLT data update back to the main datacenter is pretty trivial compared to the update time by light travel from whatever you are mapping. Even at 1ly distance between stations, if data from probe to data center update time is in hours, that is nothing compared to a star to prove update time of up to a year. I only comment because its a great answer, though. $\endgroup$ – kingledion Oct 17 '16 at 0:26
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Predicting the motion of a set of stars into the future is exactly like predicting the weather. Fast computers can run the equations, but the slight errors and omissions in your original measurements will gradually amplify and eventually your predictions will show errors.

Weather predictions begin to fail in a matter of days, but predictions of star motions should last quite a bit longer. Still, a prediction about what is happening "now" 1,000 or a 1,000,000 lightyears away may show some errors.

Also, to be accurate, there is no universal "now" in a universe governed by relativity. Two observers traveling in different directions can both look at a distant star and calculate what they think is "now" over there, but they will reach different conclusions.

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  • $\begingroup$ As to your last paragraph. I think as soon as you hit superluminal speeds, the "now is relative" part of relativity goes away. Regardless, "now" would be calculated relative to the normal motions of stuff in the galaxy, which are similar enough that the differences between relative "now"s are pretty trivial, plus there would be a normalized reference frame to get even more accuracy. $\endgroup$ – MichaelS Oct 15 '16 at 2:37
  • $\begingroup$ @MichaelS: I am not assuming any privileged frames of reference. I am only assuming that you can't travel back to your point of origin before you left. $\endgroup$ – Anonymous Oct 15 '16 at 3:27
  • $\begingroup$ @CharlesGillingham: so does this mean that a concept like the "red line" makes sense? $\endgroup$ – Anonymous Oct 15 '16 at 3:40
  • $\begingroup$ @MichaelS your first sentence is wrong. See this post. How do you think superluminal travel between events in spacetime changes the fact that you can choose an x axis to assign equal time values across space? $\endgroup$ – JDługosz Oct 15 '16 at 6:30
  • $\begingroup$ @JDługosz: There's only one choice of axis that allows light to travel the same speed in all directions for both the preferred frame and the FTL frame. Any other axis and you'll realize that FTL moves you back and forth in time. I suppose you could decide the Aether is moving relative to spacetime. But you still only get two possible timelines, instead of infinity. So if you're calculating from a single reference point, you just need one transform to apply to all FTL travel. $\endgroup$ – MichaelS Oct 15 '16 at 10:05
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Nothing. I expect reconnosince craft to jump ahead into boring empty space and/or make small jumps, for the express purpose of making accurate maps.

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  • $\begingroup$ Manual exploration is far too slow. I mean take a snapshot of the night sky and calculate the current position of every single object in the galaxy that could present a navigational hazard using computers without ever visiting the space in question. $\endgroup$ – Anonymous Oct 15 '16 at 2:15
  • $\begingroup$ @Anonymous: How is manual exploration too slow? As long as the maps are made within a short range of the jump point (say, hundreds to thousands of light years), it should be easy to mathematically calculate the current and future positions of everything in the night sky. And if there are people in that part of the galaxy to need the maps, then at least some of them can keep the maps up-to-date. You wouldn't want to jump into a system that wasn't normally mapped without checking it out first, but each ship could do that on its own. $\endgroup$ – MichaelS Oct 15 '16 at 2:33
  • $\begingroup$ @MichaelS: in this scenario the ships don't have the luxury to survey every system. They need to travel hundreds of light years into a system they never visited before and make sure they don't arrive inside a planet or something. They need to do this many times in quick succession. $\endgroup$ – Anonymous Oct 15 '16 at 3:35
  • $\begingroup$ I think the question is not stating what Anonymous is really wanting to ask, as he seems to be answering his own question in these comments. $\endgroup$ – JDługosz Oct 15 '16 at 6:26
  • $\begingroup$ Thank you. I have adjusted my question to be less... whatever it was? $\endgroup$ – Anonymous Oct 17 '16 at 13:06
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If you've got superluminal drives, then you've got time travel. In a relativistic universe, those are just two different ways of talking about the same capability. So if your galaxy is 30,000 light years across then every decade or so you map it as it was 30,000 years ago and then send the map back in time by 30,000 years to the people who need it.

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You could kinda-sorta do this. However, it would be tricky. Remember, everything moves, and everything interacts. This is known as the n-body problem. As far as we know, there is no closed form solution to these problems. The best we can do is a numeric solution (i.e. simulate the motion of objects). This means errors will crop up.

Of course, the other thing to consider is that 99% of objects out there are too dim to see from a distance. By mass, you can say most of the mass is found in the stars. However, there's plenty of dark objects out there like asteroids. If you don't want to hit one of those, you're going to need more than a visual snapshot of the area.

A better approach would be to define "safe" areas which are well charted. You jump to one of those and then observe the local space before jumping to a less safe spot. That keeps the manual exploration cost down.

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The method of finding the parallax, and thus the distance, to stars involves observing them from Earth or Earth orbit and plotting their apparent positions half a year apart. The distance Earth travels between two observations will make nearby stars seem to change position very slightly against the back ground of farther stars.

The accuracy of observations and distance measurements can be improved by improving the resolution of the instruments and/or by moving the instruments a greater distance between observations than a mere two astronomical units. Or you can use simultaneous observations from widely separated instruments. So by first using widely separated observatories in our own solar system with accurately measured distances between them we can find the distances to nearby stars very accurately.

Then instantly transport observatories to some nearby stars at distances about a million times the diameter of the Earth's orbit. Simultaneous observations from those widely spaced observatories will thus be a million times as accurate as observations from Earth six months apart. That will be enough to make a very accurate three dimensional map of the globular star clusters.

Then instantly transport observatories to distances thousands of times farther from Earth than those nearby stars. The map of the globular star clusters will enable precise positioning of those observatories billions of times farther apart than the spread of the earth's orbit, and thus capable of measuring parallaxes billions of times more precisely.

Observatories thousands of light years "above" and "below" the galactic plane will have views of hundreds of billions of stars in the galaxy, unobstructed by the dust clouds.

Observing each star over a period of years will detect angular motion while spectroscopy will reveal how fast it is traveling toward or away from each observatory. Knowledge of how far away each star is from an observatory will indicate how old the positions are and thus where the star has moved to since its light was emitted.

And that is the procedure for a preliminary mapping of the galaxy to be followed by more detailed surveys of stellar positions. but you would need really super sized telescopes to detect small objects like moons, asteroids, comets, etc. over hundreds or thousands of light years. Detecting them should be left to surveys of individual solar systems.

If space travelers are afraid of running into unseen objects in interstellar space despite the fact that "Space is empty. Very, very, very empty." they might as well stay home and not dare to travel in interstellar space.

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