Date: Tue, 27 Oct 1998 23:13:28 -0700
From: Christopher Thrash 
Subject: Re: Jump point masking (long)

The probability that a mainworld's jump limit (as seen from all angles) is
masked is proportional to the solid angle (in steradians) subtended by a
disk equal in radius to the primary's jump limit minus the mainworld's jump
limit, at the mainworld's orbital radius, as a fraction of 4*pi steradians.
This neglects parallax, but for the distances involved all lines of
sight/flight are essentially parallel anyway. (Cases where parallax is
important -- microjumps -- are left as an exercise for the interested
reader.) To find the solid angle, divide the area subtended by the disk by
the area of a sphere of orbital radius. Hence:

Area (disk) = 2*pi*a*h, where a is orbital radius (semi-major axis), and h
= a - sqrt(a^2 - (R - r)^2), and R and r are the jump limits of primary and
mainworld, respectively.

Area (sphere) = 4*pi*a^2.

Masked = 2*pi*a*h / 4*pi*a^2 = h/2a = (a - sqrt(a^2 - (R - r)^2))/2a
				  = (1 - sqrt(1 - (R - r)^2 / a^2))/2.

Free = 1 - Masked.

Revised table follows:

Spec.	Dia.	100D	100D	Habit.	Free	Free	Free
Class	(Sols) (au) 	(orb)	Zone	(Sz 0)	(SGG)	(3d)
O5	18.00	16.84	8	-	-	-	
B0	7.40	6.92	7	12	100.0%	100.0%	auto
B5	3.80	3.56	6	9	99.8%	99.8%	auto
A0	2.50	2.34	5	7	98.6%	98.7%	17-
A5	1.70	1.59	4	6	97.6%	97.8%	16-
F0	1.30	1.22	4	5	95.0%	95.5%	15-
F5	1.20	1.12	4	4	85.6%	87.4%	14-
G0	1.05	0.98	3	3	59.4%	69.4%	11-
G5	0.93	0.87	3	2	0.0%	0.0%	no
K0	0.85	0.80	3	2	0.0%	0.0%	no
K5	0.74	0.69	2	0	0.0%	0.0%	no
M0	0.63	0.59	2	0	0.0%	0.0%	no
M5	0.32	0.30	1	-	-	-	-	
M8	0.13	0.12	1	-	-	-	-	

Once again, for GT: Roll twice, once for origin and once for destination.
[Optionally, add +1 for the mainworld satellite of a gas giant.]


Effects on Trade and Commerce:

Using CT Book 6 (Scouts), p. 28, for spectral class frequencies (whether
accurate or not), and looking only at main sequence (size V) stars, only
21.4% of all jumps on average will be to or from unmasked mainworlds, most
of those around class F primaries:

Spectral 	
Class		Free(%)		Stars(%)	Total Free
B0		100.0%		0.0%		0.0%	
B5		99.2%		0.0%		0.0%	
A0		98.6%		1.4%		1.4%	
A5		97.6%		1.4%		1.4%	
F0		95.1%		8.1%		7.7%	
F5		85.8%		8.1%		7.0%	
G0		60.6%		6.6%		4.0%	
G5		0.0%		6.6%		0.0%	
K0		0.0%		8.2%		0.0%	
K5		0.0%		8.2%		0.0%	
M0+		0.0%		51.3%		0.0%	
						21.4%	

Assuming that jumps may freely begin and end *beyond* the 100D limit, the
maximum distance required to clear the primary's jump masking is equal to
the primary's jump limit radius; such a jump skims the edge of the
primary's jump sphere on a tangent line. Distances and travel times (at 1G)
are given below:

Spect. 	Dia. 	100D 	Run 	Stand 
Class	(Sols)	(au)	(hrs)	(hrs)	
O5	18.00	16.84	199.1	281.5	
B0	7.40	6.92	127.6	180.5	
B5	3.80	3.56	91.5	129.4	
A0	2.50	2.34	74.2	104.9	
A5	1.70	1.59	61.2	86.5	
F0	1.30	1.22	53.5	75.7	
F5	1.20	1.12	51.4	72.7	
G0	1.05	0.98	48.1	68.0	
G5	0.93	0.87	45.2	64.0	
K0	0.85	0.80	43.3	61.2	
K5	0.74	0.69	40.4	57.1	
M0	0.63	0.59	37.2	52.7	
M5	0.32	0.30	26.5	37.5	
M8	0.13	0.12	16.9	23.9	


The weighted average maxima, including free jumps, are +25.8 hours (running
jump, at 1G) and +36.4 hours (standing jump, at 1G). These apply equally at
origin and destination. Average times are +21.7 hours and +30.6 hours,
respectively.

Conclusions:  Even worst case, jump masking adds (on average) at most a day
to a day and a half to each end of an interstellar jump.  This reduces
canonical port time from 6-1/2 days to 4-5 days: well within the capability
of most freight liners to discharge and load. Minimum possible turnaround
time (arrival to arrival), using conventional (non-LASH/tender) designs,
goes up from 8-1/2 days (41 trips per year) to 10 days (35 trips per year).
This is closer to the canonical value of 14 days (25 trips per year). Free
traders may have a harder time making 25 trips a year, but who said life
was fair?