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Calculating Jump Vectors

This article originally appeared in the March/April 2020 issue.

Space is huge, and planets and even stars are small in relation. When Andromeda collides with the Milky Way in 4.5 billion years it is estimated that with over 1.3 trillion stars between them there will be less than two direct stellar collisions.

Jumping to a star system even one parsec distant is not simply pointing the ship at the star and engaging the jump drive. Not only are the stars moving with respect to each other (about 100km/s on average), but the light from that star is several years old. If one doesn’t know which direction the star is moving the chance of hitting even the 100D limit of the star itself is near zero (at 1pc it will have moved up to 69 AU in 3.26 years at 100km/s). Navigation in Traveller is not usually detailed beyond some notes about difficulty in calculating the math to open the jump space bubble. The computer needs to do things in just the right way based on the ship’s jump grid and requires a lot of time and calculations is about all that is covered. The start of this process is obviously the coordinates, and thus direction to and distance of the destination. This might seem like the easy part, but is just the opposite. So navigation is just putting in the destination to the computer and it can, given an accurate current location of the starship (more on this another time), compute a heading and distance to where the star should be. Taking into account to avoid any known masses along the route in either the source or target systems, it can then perform the even more complicated jumpspace calculations. Despite how small the target is we can assume the technology is more than up for the task if we have navigational data.

Navigation data normally obtained by the scout surveys provides an ephemeris of all star motions in the Imperium relative to each other; this will account for almost all the trips that a normal ship will take. What happens if there is no data? That is where things get interesting. Some examples of where navigational data might be missing are:

The next section outlines some of the difficulties in determining where a star is located without proper stellar data. Thus charts for otherwise un-surveyed systems would have some monetary value. This brings up several new adventure ideas:

Angles used in determining the distance to or the location of a star are often expressed as “mas”, milli-arc-seconds (an arc-second is 1/3600 of a degree, so 1/1000 of that). A 1 mas angle measured from the earth to the moon subtends 1.8m; that is, if you are off course by 1mas, by the time you reach the moon you are 1.8 meters from where you were targeting. Of course, like stars, the moon is moving, too, making things difficult. Extending that to 1 parsec, 1 mas subtends 150,000 km (1/1000 of an AU by definition).

Resolution of telescopes is an important part of determining where a star is. While large 8-10 m ground telescopes can have a resolution of 30mas, a space based telescope can fare much better and approach the theoretical maximum (Dawes’ limit) expressed as 11.6/Diameter in cm for arc-second resolution. Thus the 8m ground telescope in space would have a resolution of 14.5mas. However there are ways to get even better resolutions. Interferometry can be used on very stable platforms (e.g., accuracy of 1 micrometer in a 100m array) to get resolution to 1mas which would be difficult to achieve in a spacecraft due to vibrations. Thus planets and deep space platforms can calculate good star positions.

Distance to a star is measured by parallax, which is the angle measured to the star from two widely-separated points (e.g., opposite sides of a planet’s orbit around the system primary). We compare the relative position of the star from each of those locations. Given that a large planetary telescope array can resolve to about 1 mas that gives the position of the star to about ± 0.0005 parsec. But space is huge, so this is a variance of about 100 AU. For example, decades of observations or Proxima Centauri with huge expensive telescopes lets us calculate its distance from Earth to be 4.2441 ± 0.0011 light years, or a margin of error of ±70AU, a lot more than the 100D limit of most stars. While not possible today, in Traveller the base line can be almost any distance and one would not need to wait 6 months to take the second reading. Two telescope platforms at the distance of Jupiter (around the planet and at the L3 point) would have a baseline of 10.5 AU giving 5× better accuracy or ±20AU. However, travelling 10 AU would still take over a week at 2G.

The alternative to being accurate with the distance to the star is to be accurate for position, as long as your course is accurate one can plan to jump “past” the star, hit the 100D limit and precipitate out. Unfortunately the trade-off is that it takes time to determine the movement of the star. If the telescope has a 10 mas resolution, the star needs to move 1.5 million km before it appears on a different pixel in the telescope, you would want at least a few such movements before you could have some confidence in saying where the star will be 3, 6 or more years in the future (as the light from the star is several years old). At 50km/s the 1.5 million km movement will take 8 hours, so a day will give you a rough estimate of where to travel to reach the star.

The problem is that ships (except perhaps lab ships) don’t usually carry 8-10 meter telescopes, in fact optical sensors are usually not described in any detail. A 120mm telescope (current cost of about $500 – which could be assumed to be in normal sensors) would have an accuracy of about 1000 mas (1 arc-second), a $100K 700 mm telescope (definitely a line item in a starship design) about 175 mas. At 17 to 100 times less resolution the 8 hour time period per measureable position change becomes 6 to 30 days. So 18 to 90 days of observation to get a good idea where the star will be after the jump. Ships don’t have that sort of time. Luckily the slower the star, and thus harder to measure means that it will have travelled less distance too.

For almost all adventures that take place inside of the Imperium jumps can just be something that automatically succeeds with the navigator being able to reduce the distance from the destination at jump breakout with skill checks. (That exact value will be covered next article on jump masking.) However, for jumping without charts either at the edge of the Imperium or to a purposely “hidden” system, the following basic rules can be used to determine how extremely far away from the target the player’s ship will be when it drops out of jumpspace (more detailed rules will be in a future article on exploratory navigation).

Basic Rules for Jumping Without Charts

If the system location is purposely altered then the people who did so will have altered the stellar data in a systematic fashion, which will be known to those who are authorized to visit the system. The PCs may know that the alteration results in coming out of jump a known distance away, but not the direction of the error; alternatively, they may only know that the data is altered. In this case the players need to determine how long they would like to observe the star before jumping.

If the ship has drop tanks or is capable of jumping twice the distance to the star it will take less time to jump there, measure an accurate position from several hundred AU away (wherever they came out of jump), jump back, and jump again with accurate data than do the measuring from a distance. The downside of this is the extra expense and time of using three jumps to accomplish what would be done in one jump with accurate data.

Secretly roll 1D to determine the proper motion of the star: (1 = low, 2-5 = medium, 6 = high). Allow the PCs to determine how long they wish to spend observing the target star. They will have accurate information after the periods shown in Table 1:

Table 1: Observation Period for Accurate Stellar Position Determination
Low Proper Motion Medium Proper Motion High Proper Motion
32 days 16 days 7 days

After the indicated periods of observation, the proper motion of the star will be known (in reality it is a spectrum of values not just three), and the players can adjust their estimate of the position.

Roll 1D and multiply by the square of the distance in parsecs (p) to the target star. Then, multiply by the proper motion factor from Table 2:

Table 2: Proper Motion Factor for Calculating Deviation
Low Proper Motion Medium Proper Motion High Proper Motion
0.05 0.19 0.67

This is the deviation (d), in AU. 100/d2 is the base percent chance to hit the 100D limit. (Alternatively, use 20/d for main sequence stars and 4/d for dwarf stars). For each multiple of the minimum time given above with military sensors or 3 times the minimum time for civilian sensors the chance should be doubled; halve the chance if they haven’t met the minimum time.

The navigator then rolls to determine if they hit the 100D limit of the star, in which case normal travel times apply. If they miss, they will be 10AU×p×(2D6+3) AU from the star. If the ship was able to make observations with a large baseline in the origin system, divide this distance by the number of weeks spent observing the target star.

Example: The target star is 2 parsecs away. The referee determines secretly the star has a medium proper motion. The players observe for 1 week and see that it is not measurably moving, so decide to wait another 9 days after which they can discern movement in their optics. They decide not to wait 7 total weeks to get the extra bonus with their civilian sensors so they jump. The referee rolls a 4 for the deviation calculation, so 0.19×4×22 = 3.04 AU. (10/3.04)2 is 10.8%. If they had waited the extra 32 days to get good data it would have doubled to 21.6%—probably not enough better to make the extra delay worthwhile, so likely a good call. If they had not waited the 16 days to start, the chances of hitting the 100D limit would have been halved to 5.4%. If they are lucky, they exit at 100D and have a quick journey to their destination. Most likely they missed the star and with a roll of 6 would come out 10×2×(6+3), divided by 2 for the 2 weeks of observation, for a final “miss” distance of 90 AU. Travel time is now 27 days for a total of 43 days. If they had military sensors and observed the target star for an extra 16 days, the chance would have doubled twice, to 43% and with 4 weeks of observation they would have been only 45 AU off target with the same roll, and 13 days of travel to come out with the same worst case (45 days) of total time and a much lower chance of failure. A Navigation skill check could be added to this as well to make the trip a little shorter (or longer, on a failure).

Travel with star charts is easy and can be ignored for any journey that isn’t to a system with bad data. However without star charts a slow deliberate observation can still get you close – but remember to pack extra supplies.