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Old 05-19-2020, 04:53 PM   #1
Varyon
 
Join Date: Jun 2013
Default Help with some Space Math (sorta)

I'm working more on my Harpyias setting, thanks largely to the current wave of spaceship-related threads in the GURPS subforum (I'm posting here as my questions aren't GURPS specific), and am running into a few issues roughly mapping out the systems involved.

Briefly, Harpyias is a setting taking place in the systems around some special stars, called aetheric stars in the setting, which have a special interaction with hyperspace that allows reactionless drives to work within roughly 100 AU of them, and said drives allow a ship to transition into hyperspace at the 61.15 AU mark. Travel in hyperspace is seemingly slow - a typical interstellar vessel has a maximum speed around 10 yards per second in hyperspace - but distance in hyperspace maps to a huge distance in real space, such that each yard in hyperspace corresponds to around 0.2 AU in real space. You can only transition between hyperspace and real space when the center of your ship (technically, center of mass, but I'm working under the assumption these are the same) is at the 61.15 AU mark, and overlapping part of your ship with the location of an aetheric star is Very Bad Idea (tm), as doing so will basically tear your ship apart. This essentially means any vessel longer than around 500 yards must enter and exit hyperspace at an angle, and any vessel who's shortest dimension is greater than 500 yards is unable to safely make the transition.

Ideally, I'd like to have all of the core systems be 50 light years apart (requiring around 14 days travel from the typical vessel, above), but this is where I run into the first problem. Unless I'm mistaken, the largest number of points that can all be the same distance from each other is 4, arranged as the tips of a regular tetrahedron (a d4). Is this correct, or is there an arrangement that would allow for more? I feel 4 may be too few, although I might be willing to have something like "lesser core" systems by essentially using each face of the initial tetrahedron as one face of a new regular tetrahedon, allowing for 4 "lesser core" systems.

Beyond the core, there are roughly 3 tiers of outer systems, which I'd like to have better category names for. Tier 1 are systems that are located 50 light years from one core system; depending on geometry, they may be as close as ~75 light years or as far as ~100 light years from another core system (or further, if comparing the system is connected to a lesser core system and you're comparing to another lesser core system). Tier 2 are systems that are located 50 light years from a Tier 1 system, and the furthest, Tier 3, are systems that are located 50 light years from a Tier 2 system. Is anyone aware of any (ideally free) software/web application that could be used to map this? Granted, traveling more than 50 light years at a time in hyperspace is generally a bad idea (hyperspace is basically corrosive; a typical interstellar freighter will just need some time for its armor to stabilize and be polished up after a 50 light year journey, but going much further (or, rather, staying in hyperspace for more than 14 days) will mean the armor starts to slough off, "eaten" by hyperspace, and once the armor is gone the ship and its contents will be torn apart in fairly short order), but vessels can be designed to go further in one trip (by having thicker armor or being designed to be faster) or even just push their luck to do so (arriving missing a few layers of armor), so knowing the system the PC's are going to would only take 3 weeks going straight, rather than the 4 weeks (plus time to stabilize, polish, and perhaps recharge capacitors) it would take by going to an intermediate system on the way would be useful.

Next up are the reactionless drives' performance while in real space. I've got most of the math worked out for them, like how much energy they use, how having more/larger (or fewer/smaller) drives affects velocity, and so forth. I've decided to have them go faster the further one is from the system's sun (technobabble explanation has to do with too-dense of an aetheric wind - think solar wind, but interacting with hyperspace - interfering with the function of the drive, although some aetheric wind is necessary for them to function). Specifically, each aetheric star has 12 consistent orbits of note (one each at 0.25 AU, 0.41 AU, 0.67 AU, 1.11 AU, 1.84 AU, 3 AU, 5 AU, 8.25 AU, 13.6 AU, 22.45 AU, 37 AU, and finally 61.15 AU; all but the last have a randomly-generated planet, asteroid belt, etc, while the last is empty and is where the aetheric wind largely terminates, and where transition to/from hyperspace can occur), and it takes 1 day for a typical freighter to travel from one orbit to the next. For simplicity, I'm treating all of these orbits as circular; maybe they actually are (aetheric systems are already weird, although unless I'm mistaken this would mean there aren't really seasons), or maybe they're all consistent ellipses and travel on the side of the sun where they are closer to it is actually slower. The drives are what GURPS Spaceships calls "pseudovelocity boost drives," which means they generate no real velocity (in fact, using them zeroes your velocity relative to the sun fairly quickly, meaning STL interstellar ships can basically use them to "brake") and reach full speed instantly, with no need to accelerate. As it turns out, this gives an easy equation to determine pseudovelocity at any given distance from the central sun - (distance)/2 gives this in units (miles, AU, whatever was plugged in for distance) per day. A more useful equation is to determine how long it takes to go from one point to another, which is the absolute value of 2*ln(d1) - 2*ln(d0), where d1 is the distance from the sun for your destination, d0 is the distance from the sun for your point of origin. However, this assumes going directly away from or directly toward the sun, when most likely you'd be going at an angle to intersect where your destination is actually located (or will be by the time you reach it, if it is in a stable orbit). I'm not certain how to calculate the time needed in this case.

Considering both velocity and circumference scale linearly with radius, it looks like, if your destination were 90 degrees away from going straight out, traveling 90 degrees in your current orbit and then flying straight out would take just as long as flying straight out and then traveling 90 degrees in your new orbit (traveling along your orbit always takes 4*pi days to make a full circuit with a typical freighter). Considering this, would it be appropriate to simply use the Pythagorean Theorem here? Say your destination is in an orbit 1 AU away from your current location, and is located 45 degrees away from where you'd intersect its orbit flying straight out; for simplicity, assume it isn't moving. The long way to travel to it would be to follow your own orbit for 0.5*pi days, reaching the 45-degree mark, then boosting out for 1 day to reach your destination. Boosting out first, then making the 45-degree trip, would similarly take pi/2 + 1 = ~2.57 days. Going at a diagonal, we use the Pythagorean Theorem to determine it would take ((pi/2)^2 + 1^2)^0.5 = ~1.86 days instead. Does this seem correct?

Another question that pops up, is there a decent way to determine how far a vessel can travel in a given amount of time, such as if after a battle a vessel only has enough power left to boost for an hour, and it's necessary to know how far it can go (and then determine if there are any power stations within that range). The problem, of course, is that the range of possible movement is basically an ellipse, not a circle, due to the drive going at different speeds depend on where you're at. Is the best option just to look at various potential destinations and work out how long it would take to reach each to see which (if any) the vessel can reach before running out of power, or is there some sort of equation that could be utilized?

Finally, the drives can be used to generate real acceleration, but they have an incredibly low top speed when doing so - a typical freighter can generate up to 0.2G, and has a top speed of a mere 20 yards per second. If the drives were to alternate between a pseudovelocity boost and 0.2G acceleration, would this feel like being at 0.1G or 0.2G, or would it just feel like constantly switching from 0G and 0.2G (probably causing some serious nausea)? If the last, would there be a time frame where the difference between this alternating acceleration would feel like constant acceleration (say, if you alternate every picosecond)? I'd like spacers to be able to get some semblance of gravity without having to rely on spin gravity, but I'm not willing to actually have artificial gravity.

Apologies there's a lot going on here, but these are some of the hangups I'm getting into. To summarize:

1) What is a good (preferably free) piece of software or webapp I could used to design a 3-dimensional star chart of sorts for my setting? Bonus points if it has built-in support for determining the distance between any two given points.
2) Given the way my drives function, is using the Pythagorean Theorem to work out how long it takes to reach somewhere that isn't a straight shot away from/toward the sun appropriate? (Bonus question - would the flight path look more like a straight line or an arc? I'm thinking arc, but am uncertain).
3) Is there an easy way to work out how far a vessel could go given a certain amount of time available for travel?
4) Is there a way to alternate between 0G and some acceleration such that it functions like a constant acceleration?
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Old 05-19-2020, 06:20 PM   #2
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Default Re: Help with some Space Math (sorta)

All the same distance apart requires, at a minimum, that the stars should occupy the vertices of a regular polyhedron; if they don't, then by definition, not all of them are equally spaced.

Now, in a tetrahedron, all the sides are equilateral triangles, where every vertex is equidistant from every other vertex. But in a cube, the sides are squares, and the opposite corner of a square is 1.414x as far away as the adjacent corner. Likewise, in a dodecahedron, the sides are pentagons, which have similar issues.

An octahedron and an icosahedron have sides that are equilateral triangles. But in a tetrahedron, every pair of vertices are in the same triangle (in fact, they are two such triangles). In an octahedron, opposite vertices are NOT in the same triangle, so the fact that the triangles are equilateral doesn't help; and likewise in an icosahedron.

So you can't have more than four stars in this arrangement.
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Old 05-19-2020, 06:32 PM   #3
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Default Re: Help with some Space Math (sorta)

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Originally Posted by whswhs View Post
So you can't have more than four stars in this arrangement.
At least, in 3-dimensional flat space.
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Old 05-19-2020, 07:46 PM   #4
Varyon
 
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Default Re: Help with some Space Math (sorta)

Quote:
Originally Posted by whswhs View Post
All the same distance apart requires, at a minimum, that the stars should occupy the vertices of a regular polyhedron; if they don't, then by definition, not all of them are equally spaced.

Now, in a tetrahedron, all the sides are equilateral triangles, where every vertex is equidistant from every other vertex. But in a cube, the sides are squares, and the opposite corner of a square is 1.414x as far away as the adjacent corner. Likewise, in a dodecahedron, the sides are pentagons, which have similar issues.

An octahedron and an icosahedron have sides that are equilateral triangles. But in a tetrahedron, every pair of vertices are in the same triangle (in fact, they are two such triangles). In an octahedron, opposite vertices are NOT in the same triangle, so the fact that the triangles are equilateral doesn't help; and likewise in an icosahedron.

So you can't have more than four stars in this arrangement.
Thank you for this; it's good to have confirmation. Given that there is no arrangement that allows for more than 4 stars, I think I'll go with having 4 Inner Core systems that make up the central tetrahedron, and 4 Outer Core systems that make up points of the additional tetrahedrons, for a total of 8 Core systems. I'll probably end up having more branch-off points for the outer core systems than the inner core ones, but I'll need to wait until I have some way to model that.

EDIT: Another option could be an octahedron or icosahedron, where there's simply no Core systems that have more directly-adjacent core systems than any other. The former would give me 6 Core systems to work with; the latter, 12. Interestingly, I had initially considered 12 Core systems, but that was just a number that sounded OK. Once I get more involved in designing polities, this gives me the options of 6, 8 (4 Inner, 4 Outer), or 12 Core systems.
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Old 05-19-2020, 10:51 PM   #5
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Default Re: Help with some Space Math (sorta)

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Originally Posted by Varyon View Post
1) What is a good (preferably free) piece of software or webapp I could used to design a 3-dimensional star chart of sorts for my setting? Bonus points if it has built-in support for determining the distance between any two given points.
ChView is an old, but still functional (last time I checked, anyway), freeware program for 3D star mapping. It includes range calculations.
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Old 05-20-2020, 06:23 AM   #6
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ChView is an old, but still functional (last time I checked, anyway), freeware program for 3D star mapping. It includes range calculations.
Unfortunately, it appears to not have native Mac support, and it crashes upon attempting to add a star when opening it with wine. A shame, that.
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Old 05-20-2020, 09:09 AM   #7
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Default Re: Help with some Space Math (sorta)

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ChView is an old, but still functional (last time I checked, anyway), freeware program for 3D star mapping. It includes range calculations.

Doesn't work on windows 10 for me. Its a really old program.
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Old 05-20-2020, 10:51 AM   #8
Fred Brackin
 
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Default Re: Help with some Space Math (sorta)

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Unfortunately, it appears to not have native Mac support, and it crashes upon attempting to add a star when opening it with wine. A shame, that.
Um, did you mean to capitalize that "W" to indicate the well-known Windows emulator or are you trying to open the program with a container of vino instead of a computer? :)

Now if it crashes when you try and open it with a computer while you are at the same time you are opening a bottle of wine that would be a serious if inexplicable flaw.
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Old 05-20-2020, 11:01 AM   #9
Varyon
 
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Default Re: Help with some Space Math (sorta)

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Um, did you mean to capitalize that "W" to indicate the well-known Windows emulator or are you trying to open the program with a container of vino instead of a computer? :)

Now if it crashes when you try and open it with a computer while you are at the same time you are opening a bottle of wine that would be a serious if inexplicable flaw.
... are you saying your properly-fermented grape juice is unable to run computer programs? You must be drinking the cheap stuff. Yours at least has augmented reality support, right?

More seriously, back when I used Linux, the command to run Wine was lowercase (at least on my system), and it all the subprocesses (but not the main process, now that I look at it) show up in my Activity Monitor lowercase, so I legitimately thought it was officially lowercase.
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Old 05-20-2020, 05:09 PM   #10
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Default Re: Help with some Space Math (sorta)

ChView is 16-bit windows; win10 only supports 32 and 64 bit.
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