[Spaceships] Voyage Time Calculation with Different Drives
 

This may be glaringly obvious, but could someone please explain to me how to calculate interplanetary (or planet-jumppoint-planet) travel time, when the spaceship uses two different STL reaction drives, especially if they accelerate (or decelerate) simultaneously during some (but not necessarily all) of the time?

Maybe one example to clarify what I mean:

The "Donna Jackson-Morley" is a TL 10 and SM +10 deep space container ship on its way from the port station in Earth orbit to Ceres High Port. The GM rules that the distance is 3 AU.

The Donna has two magsail systems, one fusion rocket system, and one reaction mass tank filled with 20 mps worth of water. The average distance from the sun during the trip is roughly 2 AU, so the magsails can provide an acceleration of 0.0005 G.

After breaking earth's orbit the Donna powers up her magsails and ignites her fusion rocket, and for the first 30 hours she accelerates with a combined 0.0155 G.
After that she turns off the fusion rocket and accelerates only with her magsails until the midpoint at which she turns around and starts decelerating with the magsails.
At the last 30 hours of voyage she decelerates with both the fusion rocket and the magsails.

What is the duration of this trip, and, more importantly, how do I calculate that time?

Thank you!

Yes, this mode of travel is inspired by Nathan Lowell's "Golden Age of the Solar Clipper" series ;-)

Re: [Spaceships] Voyage Time Calculation with Different Drives
 

Basically? Newtonian rules of motion. Brush up, or get somebody good at high-school level physics to solve it for you.

In this case, it's relatively simple. In the first and last 30 hours combined the ship only covers a little more than 1 percent of an AU (1/2 A t^2 each), which I'd discount as negligible, and just treat as starting and ending the flight at 10 MPS rather than 0. Then again, that's not much simplification and I'm using a (google) calculator anyway, so I'll go ahead full detail.

So, not counting the first and last 30 hours, we're starting and ending at ~10.33 mps, and covering ~2.988 AU. Thus the equation for either (post burn) half-flight is 1/2 * 0.0005 G * t^2 + 10.33 mps * t = 1.494 AU.

The quadratic formula tells us that Ax^2+Bx+C = 0 has solutions x = (-B +- sqrt(B^2-4AC))/2A. In this case, A = .0005 G/2 = .0005/2 * 9.8 m/s^2, B = 10.33 mps, and C= -1.494 AU. Plugging that right into my search-bar, units and all (with + for the +-, since we're only interested in positive time), I get t ~= 78.04 days for the half-trip. Double to 156.08 days for the full trip, then add back the 60 hours of thruster burn to bring it up to about 158.6 days.

Re: [Spaceships] Voyage Time Calculation with Different Drives
 

we need to break this down to 4 legs of voyage

I am making the (probably faulty) assumtion that the enpoints do not have a relative speed to each other that affects the travel calculation (so we can use 0 for our starting and ending velocities).


4 legs are
leg 1: 30 hours acceleration on both
leg 2: acceleration on just sails
leg 3: decceleration on just sails
leg 4: 30 hours decceleration on just sails.

Since we are assuming velocity of 0 at start and end we can calulate just the acceleration portions as the deceleration portions will be a mirror of that.

d0 = start point = 0
v0 = start speed = 0 m/s
d1 = distance traveled after first leg
v1 = velocity after first leg
t1 = time of first leg
a1 = acceleration of first leg
d2 = distance traveled after second leg (1.5 AU)
v2 = velocity after second leg
t2 = time of second leg
a2 = acceleration of second leg

over the first leg
d1 = d0 + v0*t1 + 1/2*a1*t1^2
v1 = v0 + a1*t1

simiplificed to
d1 = 1/2 * a1 * t1^2
v1 = a1 * t1

over the second leg
v2 = v1 +a2*t2
d2 = d1 + v1*t2 + 1/2*a2*t2^2

so we get the following

v2 = a1 * t1 + a2 * t2
d2 = 1/2 * a1 * t1^2 + a1 * t1 * t2 + 1/2 * a2 * t2^2

1.5 AU = 224397000000 meters
a1 = 0.0155G * 9.8m/s/s = 0.1519 m/s/s
t1 = 30 hours * 60 min/hour * 60 sec/min = 108000 seconds
a2 = 0.0005 G * 9.8m/s/s = 0.0049 m/s/s

plugging in numbers gives:
224397000000m = 8858808m + 16405.2 m/s * t2 + 0.00245 * t2^2
224388141192m = 16405.2 m/s * t2 + 0.00245 * t2^2


solve for t2 = 6790841.176628518 seconds
= 113180.686277141967 minutes
= 1886.3447712856994 hours
= 78.60 days

so t1 + t2 = 79.85 days

that is 1/2 the trip, so the full trip time would be 159.7 days

having the 2 steps of acceleration does make the calculation more complicated, the formula for time if it was a constant acceleration for the whole trip is t=2*square-root(d/a)

Re: [Spaceships] Voyage Time Calculation with Different Drives
 

I built an excel travel-time calculator a while back. You can find it here. It's in Excel 2007 format.

Anyways, the way you find the travel time is in steps. Plug in the Acceleration, Delta-V, etc for the first leg, then play with the initial burn delta-v to figure out how fast she'll be going at the end of the first leg. Then input that delta-v into the 'initial velocity' box, the same delta-v into the 'final velocity' box, (EDIT: and subtract *twice* the burn distance from the distance you want to go, since the other half of the burn distance is the final burn), and input just the magsail acceleration into the acceleration box, and play with the 'initial burn' box until you get one with a cruise time approaching 0. Add up the burn times from the initial phase, the magsail phase, and that initial phase again (this is the same as the final phase), and you'll get the total travel time.

Re: [Spaceships] Voyage Time Calculation with Different Drives
 

I didn't get this far but i did calculate that the trip on rockets alone would have taken 317 days.

Spoo for thsi trip at least you could have taken out the 2 systems of magsails and repalced one with another fuel tank and doubled your accel and decel times on rockets alone and basically achieved the same results.

....and of course had another system's worth of cargo or whatever. If basically all your trips are going to be of this length I'd make thsi a rocket-only ship.

Moral: complications frequently aren't worth the trouble.

Re: [Spaceships] Voyage Time Calculation with Different Drives
 

Unless you need high precision (e.g., a chase) or have high-acceleration propulsion, you can simplify the calculation by averaging the orbital periods of the two planets and halving the result. This assumes instantaneous velocity changes, circular orbits, and no propulsion during transit, but it's something that can be calculated easily in a few seconds with a table of orbital periods and a pocket calculator. (Minimum Energy Transfer Orbits [PDF] via NASA)

Re: [Spaceships] Voyage Time Calculation with Different Drives
 

Not so much high acceleration as Delta-V significantly above the minimum necessary. Your formula wouldn't work very well for many Transhuman Space vessels which would be commonly capable of spend 50 mps of delta-V over just a trip to Mars (though both accel and decel only take a few hours each).

What I do in such situations is take half the Delta-V used as the velocity and simply divide the the distance of the trip by that.

Re: [Spaceships] Voyage Time Calculation with Different Drives
 

Thank you all!

Well, the reason for this thread is that right now I'm fiddling with spreadsheets to find a jump point setup for my FTL jump drive, where magsails make sense...

I'm trying to come up with a commercial/trading campaign where the trip length (and by that the profit) is as much as possible dependent on skill rolls of as many of the crew/party as possible.
And magsails are the one realistic STL drive where I can imagine that navigator and helmsmen/pilot can have meaningful influence (handwave like crazy: because the solar wind isn't constant, homogeneous, or predictable).

Most other drives are pure math: all you need is a computer and an autopilot to find and execute The Perfect Course.
Not much chance for players who play navigators or pilots/helmsmen to make an impact on the goal of the game -- the profit.

Re: [Spaceships] Voyage Time Calculation with Different Drives
 

<shrug> .0005 G is really sucky. You need _very_ long trips for a constant accel of that minimal magnitude to add up to soemthing useful.

I think your combined goals of "hard science" and "can't be calculted to a fare-thee-well" are in conflict.

Re: [Spaceships] Voyage Time Calculation with Different Drives
 

I am an electrical engineer, but I started out in aerospace engineering and had to go through all these sorts of calculations. This is actually a tremendously complex problem to be attempting for recreational purposes. Consider that the travel distance is a function of the relative positions of the two bodies which will be different each time, and that your initial velocity vector in one orbit is tremendously problematic if you want to leave the plane of the ecliptic. You also need to consider the sphere of influence (SOI) of the other nearby planets which could misdirect or outright delay the flight.

More importantly, note that your actual acceleration is also affected by the solar gravitational pull (about 0.0006 at Earth orbit). While that pull is negligible for high-thrust systems, for your craft it is a big deal! Realistically, you would need to perform an accelerated Hohmann transfer, but unfortunately my copy of Hale's Introduction to Space Flight is currently a few hours away so I do not have the relevant equations on hand.

Personally, I would handwave a solution and then set standards by which each piloting or navigation roll would change it. Perhaps start with a 300 day trip, and allow +/- 3 days for each point the roll is below/above success, with perhaps a 240 day minimum. Realistically, note that piloting this type of maneuver is going to be done by computer - there will be lots of set-up time, few variables, and the kind of precision that a human would be hard-pressed to manage.