atpollard said:
Does anybody have handy formulas for the 'golfball' approach (brief acceleration, long coast, brief deceleration)?
I know A WAY to calculate it, but was wondering if there might be other (simpler) ways. With increasing reaction fuel requirements, it seems likely that lower energy concepts would become more acceptable ("OK, so it takes 3 days instead of 1 day, I can live with that.")
The minimum energy (without getting into transfer orbits or gravity slings) would probably be accelerate to escape velocity, cruise, decelerate to a stop. Some quick idea of the tradeoff between reaction mass and travel time for longer accelerations would be useful.
Ishmael gave you the math, but I think I have a better way to represent it in game terms.
CONCEPTS:
The idea of a G-hr as a velocity. This is the velocity your ship will build up by accelerating at 1g for 1 hour.
1 G-Hr = 35.316 km/sec
I have converted this into more useful units:
1 G-Hr = 127138 km/hr = 0.610 AU/month
If you accelerate for 2 hours, you get twice that speed, if you accelerate at 2gs, for 1 hour, you get twice that speed, so it is completely linear in calculating this velocity for a variety of thrust and time intervals.
However, very rarely will you be able to actually travel in a straight line. Almost every transfer will be in an ellipse, even if it is a shallow one, after all your ship is orbiting a planet or the sun and that must be accounted for.
Therefore, I suggest using the following velocities to account for this non-linear travel:
1 G-Hr = 120,000 kph = 0.5 AU/Mo
This seems backwards, but rather than calculating a longer distance, we just assume a slightly lower velocity and the end result (trip time) will be about the same.
Now, we have to consider two different types of maneuvering.
1. Orbital Maneuvers
This is travel from one orbit to another around a planet. For example, travelling from low planetary orbit to a moon or from one moon to another moon about a planet.
2. Interplanetary Maneuvers
This is travel from one planet to another
Each situation uses the same basic math, but because the time scales are so different, certain simplifying assumptions can be made about condition #2.
In each case, I have made an assumption that the acceleration time is the same as the deceleration time. This is not exact, real orbital mechanics must account for the differences in orbital velocity, but that makes things WAY to complicated. As a base assumption, I suggest you add 1 hour and 1 G-Hr of thrust to account for any variations in orbital velocities; this will be more than enough expect for very special circumstances.
So:
CONDITION 1: Orbital Travel
Step 1: Determine Straight Line Distance in Kilometers
You must know the straightline distance between your starting point and your endpoint. For my example, I will use a trip from Low Earth Orbit (LEO) to Earth's moon Luna. Distance is from 360,000 to 401,000 km.
I will select a distance of 380,000 km.
Step 2: Determine G-Hr expenditure for the trip.
This is selected by the captain of the ship. You must specify both the Thrust (Gs) and the Time (Hours)
I will select 1 G-hr of thrust (1 G for 1 Hr)
Step 3: Calculate Cruise Speed
Given my 1G-Hr, I know that my cruise speed is 120,000 kph
Had I used 2 G-Hr, the Cruise Speed would have been 240,000 kph; had I used 0.5 G-Hr Cruise Speed would be 60,000 kph.
Step 4: Determine Base Travel Time
Simply divide your distance by the cruise speed.
Base Time = 380,000/120,000 = 3.17 Hours
Step 5: Determine Acceleration Time and Distance.
It takes time to go from zero velocity to Cruise Speed, you also travel during that acceleration time. Conveniently, by assuming symmetrical thrust at beginning and end, it is possible to figure this out very quickly and easily. The distance travelled during acceleration and deceleration is the same distance that you travel during 1 hour at Cruise Speed. So, to account for this extra time to go the first unit of distance, just add the acceleration time (from Step 2) to the Base Travel Time.
So, in my example, I accelerated for 1 hour
STEP 6: Calculate Total Trip Time
To calculate total trip time, I add 1 hour to the Base Travel Time for the Acceleration Time.
In my example:
Trip Time = 3.17 hours + 1 = 4.17 hours
So, a trip from Earth orbit to Luna using a 1 G-Hr thrust pattern would take 4 hours and 10 minutes. Two of those hours (one at the beginning and one at the end) would be under 1G thrust and the remaining 2 hours 10 minutes would be under Zero-G.
CONDITION 2: Orbital Travel
This is actually a lot simpler, since we are dealing with travel times of WEEKS or MONTHS, we can ignore the acceleration/deceleration time and just assume instantaneous change in velocity to Cruise Speed and only worry about the Base Time.
So, you follow Steps 1 through 4 above and you are done.
For example, assume a trip from Earth to Mars. Earth and Mars can very in distance from 0.52 AU to 2.52 AU.
Step 1: Determine Distance
I will assume 1.5 AU distance
Step 2: Determine Thrust Pattern
I will use the same 1 G-Hr as above, this time though, I don't worry about how I got that, it could be 2G for 0.5 hours or whatever, it doesn't matter.
Step 3: Calculate Cruise Speed
Cruise speed is 0.50 AU/Mo
Step 4: Detemine Travel Time
Travel time is distance divided by Cruise Speed.
Travel Time = 1.5/0.50 = 3.0 Months
If I wanted to reduce that time by accelerating longer, I could.
Lets assume I have 24 G-Hrs of fuel to burn for this trip. I can use 22 G-Hrs for the trip, leaving me 1 G-Hr to match orbital speeds and 1 G-Hr of fuel for emergencies.
22 G-Hrs total means that I can use 11 G-Hrs to accelerate and 11 G-Hrs to decelerate
Step 1: Same Distance: 1.5 AU
Step 2: Thrust Pattern: 11 G-Hrs (11 hrs at 1G)
Step 3: Cruise Speed:
Cruise Speed = 0.50 * 11 = 5.5 AU/Mo
Step 4: Travel Time:
Travel Time = 1.5/5.5 = 0.273 Months = 8.2 Days