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Author Topic: Calculating staggered arrival times  (Read 1324 times)

Tai Dai Cultist

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Calculating staggered arrival times
« on: 16 February 2013, 01:29:41 »
Forewarning: Math majors may want to avoid this post.  I'm quite sure my math is sound, but my amateurish techniques may easily offend.  Or amuse.  Either way, I DO welcome constructive criticisms :) If anyone wants to check my work vs the formulae in StratOps, do remember to apply the modifications to them for one-way travel NOT including a mid-point turnover.  The entire context of what we're calculating is the last leg of an intransit burn, where the ships have drives pointed towards the destination and are slowing to their eventual stop at the intended site/battlefield.

First of all: What is this process? It is a combination of methods and formulae that allow you to fairly quickly calculate how far apart your forces will arrive to a common transit destination if you split them into staggered waves.
In other words, if you break a force of ASFs and/or combat dropships off from your main fleet, I am providing you a relatively quick and easy way to calculate just HOW much more early the harder-thrusting force will arrive. 

If you're still with me, here's the structure of how it works*:
1. Decide on transit speed of main fleet/transports.
2. Decide how many minutes prior to arrival you want to split off the vanguard.
3. Decide how hard the vanguard will brake.
4. Determine initial velocity and how far it is to destination.
5. Determine how long the braking maneuver takes, and how much distance it covers.
6. Determine how far the vanguard will coast before beginning braking maneuver.
7. Determine for how long the vanguard will coast.
8. Determine how far behind the fleet arrives behind the vanguard.
*- there's actually a much faster way, but this shows how the math works.  the shortcut will follow at the end of the post.

Running the machine in reverse:
A. Decide on how many minutes you want the vanguard to arrive ahead of the main fleet/transports.
B. Decide on how hard your vanguard will brake.
C. Determine how much time the vanguard spent coasting.
D. Determine how much time the vanguard spent braking.
E. Determine the initial velocity.
F. Determine the time of separation from main fleet.

A couple of important things before I explain how it works:
It uses time measured in turns (60 seconds) and hexes (18,000 meters).  This allows me to also use the TW velocity scale.  The entire point of this Sausage Maker is to provide numbers useful to a space battle game, so turns, hexes, and velocity are ideal. I also use acceleration in terms of thrust (1 thrust = 1/2 G)

There's no mathematical problem I can see because the formulae in StratOps confirm that 1 hex being 18,000 meters isn't an arbitrary decision.. They're dead on accurate if each thrust point equals one half of one Gee.

Using the process to 'frontways', to determine how many minutes apart the vanguard and main fleet/transports arrive:

1. Decide on transit speed of main fleet/transports.
2 thrust would be standard and the obvious pick.  However I do want to stress that the math works just fine no matter what thrust you pick, so if you're doing a 1.5G (3 thrust) or 2G (4 thrust) burn you're still covered.  For an example, let's assume standard 1G, or 2 thrust.

2. Decide how many minutes prior to arrival you want to split off the vanguard.
The math will allow you to use fractions of minutes, but there's little point in beginning the process this way as the space battle operates in full minute increments.  In our example, let's say 60 minutes.

3. Decide how hard the vanguard will brake.
The math works whatever you decide, but in order to arrive at the destination FASTER than the main fleet you have to brake at greater thrust.
Example: Let's say you're sending in a force that includes heavy fighters that have max safe thrust of 5, so that's what you pick.

4. Determine initial velocity and how far it is to destination.
Multiplying the turns remaining by thrust gives current velocity.
Performing (turns remaining(turns remaining+1)/2)thrust gives current distance in hexes.
If you track fuel in your game, the velocity will equal fuel points expended.
Example: 60turns X 2thrust = 120 velocity.  (60(61)/2)2 = 3660 hexes to the destination.  120 fuel points will be consumed by each vessel in the vanguard.

5. Determine how long the braking maneuver takes, and how much distance it covers.
Divide the intial velocity by the vanguard's thrust to determine how many turns their braking maneuver will last.
Perform (braking turns(braking turns+1)/2)thrust gives the distance the braking maneuver covers in hexes.
Example: 120 velocity will be turned into 0 velocity at 5 thrust in 24 turns.  (24(25)/2)5 = 1500 hexes covered during braking maneuver.

6. Determine how far the vanguard will coast before beginning braking maneuver.
Subtract the distance in step 5 from the distance in step 4 and that gives you the distance the vanguard will coast before beginning their braking maneuver.
Example: While braking the vanguard covered 1500 of the 3660 hexes of total distance.  That leaves 2160 hexes to be covered while coasting.

7. Determine for how long the vanguard will coast.
Divide the distance determined in step 6 by the velocity determined in step 4.  This gives the number of turns the vanguard will spend in zero g.
Our running example: 120 velocity will cover 2160 hexes in 18 turns.

8. Determine how far behind the fleet arrives behind the vanguard.
Add the time determined in step 5 with step 7, then subtract the time from step 1.  This result will determine how many turns the vanguard will arrive ahead of the main fleet.
Our example finally solved: The vanguard spends 24 turns braking, 18 turns coasting, and has already been on the battlefield for 18 turns by the time the main fleet/transports arrive.


Running the Sausage Maker in reverse:
As you'll see in a successive post where I provide pre-crunched numbers for various values, there are some ratios that remain unchanging.  A LOT of them when the transit speed is 2 thrust.  These ratios can be used to run the machine backwards to use a desired gap in turns into determining the 'initial' break-off point.

A. Decide on how many minutes you want the vanguard to arrive ahead of the main fleet/transports.
Arbitrary choice on your part.
Example: Let's say you'd rather a bit more time to ensure your vanguard can smash all opposition.  You'd prefer a nice round 30 minute window to secure a safe patch of orbital space for your vulnerable transports.

B. Decide on how hard your vanguard will brake.
Another arbitrary choice on your part.
Example: Let's say you want to use that same vanguard composition, so 5 thrust again.

C. Determine how much time the vanguard spent coasting.
So long as your vanguard coasts up to the point of beginning its braking maneuver, this time is always identical to the time you picked in step A.
Example: We know that the vanguard will spend 30 minutes coasting at a so-far unknown velocity for 30 minutes.

D. Determine how much time the vanguard spent braking.
For every point of thrust higher than 2 when transit is 2, the time in A&C is 25% of D.
If you're better at ratios than I am, you should be able to easily extrapolate them from the charts below for non-standard transit burns.
Example: 5 thrust is 3 points higher than 2, so 30 is 75% of the number we seek. 30 / .75 = 40 turns of braking.

E. Determine the initial velocity.
Multiply thrust in B by time in D.
Example: 40 turns of 5 thrust each = initial velocity of 200

F. Determine the time of separation from main fleet.
Divide velocity in E by 2 and you have the number of minutes prior to arrival on the battlefield that the initial separation was.
Example: 200v is turned into 0v at 2 thrust in 100 minutes. 

Using the Sausage Maker in forward gear to prove the numbers that came out in reverse:
100 minute separation at 2 thrust transit = 200 initial velocity. (100(101)/2)2 = 10100h distance to destination.
200v / 5thrust = 40 turns braking. (40(41)/2)5 = 4100h covered during braking maneuver.
10100h - 4100h = 6000h coasting.  Divided by 200v = 30 turns of coasting.
40 turns of braking + 30 turns of coasting = 70 turns of transit, compared to 110 turns for main fleet.  Vanguard arrives 30 minutes ahead of fleet.

*THE SHORTCUT*
Steps 1-3 as normal.
Then divide the initial velocity by vanguard thrust.
Subtract that time from step 1.
Halve THAT result. 
Boom you're done with the lag time on the main fleet.  (but the full blown affair shows how the shortcut is still correct)
« Last Edit: 16 February 2013, 02:13:39 by Tai Dai Cultist »

Tai Dai Cultist

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Re: Calculating staggered arrival times
« Reply #1 on: 16 February 2013, 01:34:56 »


Standard 1G transit charts:
If launched 1 hour prior to start of game, fighters will expend 120 fuel pre-game and arrive:
Ahead of 2nd wave by:     After turns braking:    After turns Coasting @ zero thrust:
0 turns                            60t @ 2 thrust            0 (akin to flying alongside the 2nd wave whole way in)
10 turns                          40t @ 3 thrust            10 turns
15 turns                          30t @ 4 thrust            15 turns
18 turns                          24t @ 5 thrust            18 turns           
20 turns                          20t @ 6 thrust            20 turns               
22.5 turns                       15t @ 8 thrust            22.5 turns

If launched 45 minutes prior to start of game, fighters will expend 90 fuel pregame and arrive:
Ahead of 2nd wave by:    After turns braking:    After turns coasting @ zero thrust:
7.5 t                              30t @ 3 thrust            7.5t
11.25t                           22.5t @ 4 thrust         11.25t
13.5t                             18t @ 5 thrust            13.5t
15t                                15t @ 6 thrust            15t           
16.875t                          11.25t @ 8 thrust       16.875t

If launched 30 minutes prior to start of game, fighters will expend 60 fuel pregame and arrive:
Ahead of 2nd wave by:    After turns braking:    After turns coasting @ zero thrust
5t                                  20t @ 3 thrust           5t
7.5t                               15t @ 4 thrust           7.5t
9t                                  12t @ 5 thrust           9t
10t                                10t @ 6 thrust           10t
11.25t                            7.5t @ 8 thrust          11.25t

If launched 15 minutes prior to start of game, fighters will expend 30 fuel pregame and arrive:
Ahead of 2nd wave by:    After turns braking:    After turns coasting @ zero thrust:
2.5t                               10t @ 3 thrust            2.5t
3.75t                             7.5t @ 4 thrust           3.75t
4.5t                               6t @ 5 thrust              4.5t   
5t                                  5t @ 6 thrust              5t
5.625t                            3.75t @ 8 thrust         5.625t   


1.5G transit charts:

If launched 1 hour prior to start of game, fighters will expend 180 fuel pre-game and arrive:
Ahead of 2nd wave by:     After turns braking:    After turns Coasting @ zero thrust:
7.5 turns                         45t @ 4 thrust           7.5 turns
12 turns                          36t @ 5 thrust           12 turns           
15 turns                          30t @ 6 thrust           15 turns
18.75 turns                     22.5t @ 8 thrust         18.75 turns

If launched 45 minutes prior to start of game, fighters will expend 135 fuel pregame and arrive:
Ahead of 2nd wave by:    After turns braking:    After turns coasting @ zero thrust:
5.625t                           33.75 @ 4 thrust        5.625t
9t                                  27t @ 5 thrust           9t
11.25t                            22.5t @ 6 thrust        11.25           
~14.06t                         16.875 @ 8 thrust       ~14.06

If launched 30 minutes prior to start of game, fighters will expend 90 fuel pregame and arrive:
Ahead of 2nd wave by:    After turns braking:    After turns coasting @ zero thrust
3.75t                              22.5t @ 4 thrust        3.75t
6t                                  18t @ 5 thrust            6t
7.5t                                15t @ 6 thrust           7.5t
9.375                              11.25t @ 8 thrust      9.375

If launched 15 minutes prior to start of game, fighters will expend 45 fuel pregame and arrive:
Ahead of 2nd wave by:    After turns braking:    After turns coasting @ zero thrust:
1.875t                           11.25t @ 4 thrust        1.875t
3t                                  9t @ 5 thrust              3t   
3.75t                             7.5t @ 6 thrust           3.75t
4.6875t                          5.625 @ 8 thrust        4.6875t   


2G transit charts:

If launched 1 hour prior to start of game, fighters will expend 240 fuel pre-game and arrive:
Ahead of 2nd wave by:     After turns braking:    After turns Coasting @ zero thrust:
6 turns                            48t @ 5 thrust           6 turns           
10 turns                          40t @ 6 thrust           10 turns
15 turns                          30t @ 8 thrust           15 turns

If launched 45 minutes prior to start of game, fighters will expend 180 fuel pregame and arrive:
Ahead of 2nd wave by:    After turns braking:    After turns coasting @ zero thrust:
4.5t                               36t @ 5 thrust            4.5t
7.5t                               30t @ 6 thrust            7.5t           
11.25t                            22.5 @ 8 thrust          11.25t

If launched 30 minutes prior to start of game, fighters will expend 120 fuel pregame and arrive:
Ahead of 2nd wave by:    After turns braking:    After turns coasting @ zero thrust
3t                                  24t @ 5 thrust            3t
5t                                  20t @ 6 thrust            5t
7.5t                               15t @ 8 thrust            7.5t

If launched 15 minutes prior to start of game, fighters will expend 60 fuel pregame and arrive:
Ahead of 2nd wave by:    After turns braking:    After turns coasting @ zero thrust:
1.5t                               12t @ 5 thrust            1.5t   
2.5t                               10t @ 6 thrust            2.5t
3.75t                             7.5t @ 8 thrust           3.75t

Tai Dai Cultist

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Re: Calculating staggered arrival times
« Reply #2 on: 16 February 2013, 01:54:41 »
What's the point of providing all this?

Admittedly, it's probably niche.  I went through the bother of working it out for use in ATOW games so that I would have a basic idea of these numbers rather than just making ETAs up off the top of my head without any basis in what the numbers actually would be.

I don't imagine that TW space battles would really be enhanced by having staggered waves of fighters arriving on the battle mat.  Nor would it be a reasonable way to bring surprise force, as their approach should be readily apparent despite being 'off board'.

 In my own opinion, sending a vanguard ahead to fight before the transports arrive isn't just a good idea, it's a Great idea.  If the defenders refuse battle until the transports arrive, they've ceded the high ground and you get free whacks as they come up to orbit.  If they try to bypass your vanguard, you get free whacks at ther tailpipes for a few rounds.  It's a win/win.

I shared all this because I feel it'd be useful info to have in setting up scenarios.  The attackers must aggressively clear the battlespace of hostiles by X amount of turns, because that's when the vulnerable transports start arriving..

 

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