How Big is the Federation?

[aramis]

I got to thinking, "How big should an F&E hex really be?"

And that lead me to thinking, "How big was the Enterprise's 5YM?"

And that lead me to thinking about the SFTM map and it's error.

If we assume that

• WF 6 is cruise speed (SFTM)
  
• Speed in C= WF^3
  
• The 5YM went 5/8 of a circle in 5 years
  
• that the majority of the 5YM time was spent on the edge
  
• that we need to account only for the inbound and outbound legs (ignoring the two trips to Earth: Tomorrow is Yesterday and Assignment Earth)
  
• That non-cruise time is made up for by travel above cruise speed
  

And knowing that circumference =2*Pi*R

we can work out the radius of the federation.

WF6 = 216C
5Y= 1080LY
distance = 2+(10/8 * Pi) = 2+3.92 = 5.92 radii
radius = 182LY

There are several other things we can do that adjust the figures...
count the inbound/outbound for TiY and A:E as a single or as two trips, figure high speed "distance routes" are at WF+1 or WF+2, and figure that the SFTM got the cruise wrong, and it was actually 6.5 or 7. Or we can figure with Speed(C)=WF^(10/3) as for the lower end of the TNG/DS9/Voyager scale.

WF	Straight	1 Inbound	2 inbound	Fast Straight	Fast 1 Inb	Fast 2 trips
6	182.22	136.24	108.79	331.11	247.57	197.69
6.5	231.67	173.22	138.32	432.36	323.28	258.14
7	289.35	216.35	172.76	553.51	413.86	330.48

Fast means using WF^(10/3)
Inbound Trips are the two episodes on earth.


And, for hex sizes, same categories and calculations

[tr][td]WF	Straight	1 Inbound	2 inbound	Fast Straight	Fast 1 Inb	Fast 2 trips
6	19.18	14.34	11.45	34.85	26.06	20.81
6.5	24.39	18.23	14.56	45.51	34.03	27.17
7	30.46	22.77	18.19	58.26	43.56	34.79

[/td][/tr][/table]

There are some issues, tho. If, perhaps, it was out to Klingon space, then over to Tholian, then to Romulan, back in, and then to gorn, back in, and over to Kzinti space, it may be out, two trips, and back in, but be only 1/4 or so of the rim of the federation, instead of just over half.

Fed Radius, short rim time:

[tr][td]WF	Straight	1 Inbound	2 inbound	Fast Straight	Fast 1 Inb	Fast 2 trips
6	302.45	193.87	142.65	549.59	352.28	259.22
6.5	384.54	246.49	181.37	717.66	460.01	338.49
7	480.29	307.86	226.53	918.75	588.91	433.33

[/td][/tr][/table]
Fed Space Hex, short rim time.

[tr][td]WF	Straight	1 Inbound	2 inbound	Fast Straight	Fast 1 Inb	Fast 2 trips
6	31.84	20.41	15.02	57.85	37.08	27.29
6.5	40.48	25.95	19.09	75.54	48.42	35.63
7	50.56	32.41	23.85	96.71	61.99	45.61

[/td][/tr][/table]

[blackbat242]

OK, a couple things.

1. The shows consistently showed cruise as 2 WF lower than the current "max"... which was WF 8 the first season and WF 9 by season 3.

So sometime in season 2 cruise did indeed become WF7.



2. We know that NCC-1701 was not just traveling around the circumference of the Federation, since many other episodes show them deep inside Federation territory (Journey to Babel, The Ultimate Computer, Amok Time*, etc).

They also ran medical relief trips between Starbases and Federation worlds (The Galileo Seven), and visited Federation colonies in trouble (Operation: Annihilate! and The Devil in the Dark).


So they were actually going out to the border, then back in, over to a couple of Federation worlds, out again, and so on.



* Vulcan is described as being 16 light years from Earth. ST IV: The Voyage Home gives a 1.6 hour travel time between the two in the captured Klingon ship.

[aramis]

Which only shrinks it further, BlackBat.

Let's see how fast that vulcan run is...

[tr][td]8765.76	1 LY in LH	365.24 *24
140252.16	16 LY in LH	*16
876576	Speed in C	/1.6
44.4	Warp Factor,	^(1/3)

[/td][/tr][/table]
Wow, that's fast.

[blackbat242]

Yeah... many times details like that are way off from what the various formulae would indicate.

I got that distance and time from here (near the bottom, in the discussion of the "Alternate reality" (Abramsverse) warp factors):
en.memory-alpha.org/wiki/Warp_factor


Note Abrams' travel time:

Directly after the ship had accelerated to and attained maximum warp, Captain Christopher Pike ordered Pavel Chekov to give an announcement of the mission to the crew. At the end of the broadcast, Chekov stated that the ship would arrive within three minutes.

[aramis]

If it's JJVerse, it's not trek.