Previous table, 7G Acceleration for 100 Days, describes daily progress of vessel accelerating from zero velocity at 7g.
After 100 days of 7g acceleration, a vessel travels total distance of 9,842 AUs (.155 Light Year, LY). It gained velocity of about 150 AU/day (.866 c, 86.6% light speed).
Assume 7G vessel cruises at this velocity until the optimal time/distance to start decelerating at 7g. 
Decel.
Time(t)  Spot
Velocity (V_{t})  Spot
Distance (d_{t})  Daily
Dist. (d_{Δ})  7G
Dist.(d_{7G})  Total
Fuel (F) 
0 days  149.93 AU/dy  86.59% c  9,811.1 AU  0.1551 LY  0.0 AU  9,811.1 AU  27.974% GW_{0} 
1 days  149.46 AU/dy  86.32% c  9,661.7 AU  0.1528 LY  149.4 AU  9,960.5 AU  28.210% GW_{0} 
2 days  148.98 AU/dy  86.04% c  9,512.8 AU  0.1504 LY  148.9 AU  10,109.4 AU  28.445% GW_{0} 
3 days  148.48 AU/dy  85.76% c  9,364.4 AU  0.1481 LY  148.4 AU  10,257.9 AU  28.679% GW_{0} 
4 days  147.98 AU/dy  85.46% c  9,216.5 AU  0.1457 LY  147.9 AU  10,405.8 AU  28.913% GW_{0} 
5 days  147.46 AU/dy  85.17% c  9,069.1 AU  0.1434 LY  147.4 AU  10,553.2 AU  29.146% GW_{0} 
6 days  146.94 AU/dy  84.86% c  8,922.2 AU  0.1411 LY  146.9 AU  10,700.1 AU  29.378% GW_{0} 
Given 

 d_{t1}  d_{t}  d_{Δ} + Σd_{Δ}  1(1ε∇)^{100+t} 
To accomplish the resupply mission, the 7G vessel must slow down to match the 1G vessel cruise velocity (.644c = 111.5 AU/day). This table describes daily progress throughout the 48¼ days required to do this.

Interstellar 7G Vessel
In this example, 7G vessel starts its 100 day acceleration at day 260 of 1G vessel’s acceleration.
On day 360, 7G vessel stops propulsion, and cruises at 149.9 AUs per day (.865c).
100 days of 7G acceleration takes vessel to 9,842 AU, .155 LY. 
xDecel.
Time(t)  Spot
Velocity (V_{t})  Spot
Distance (d_{t})  Daily
Dist. (d_{Δ})  7G
Dist.(d_{7G})  Total
Fuel (F) 
7 days  146.40 AU/dy  84.56% c  8,775.8 AU  0.1388 LY  146.4 AU  10,846.5 AU  29.609% GW_{0} 
8 days  145.86 AU/dy  84.24% c  8,630.0 AU  0.1365 LY  145.8 AU  10,992.3 AU  29.840% GW_{0} 
9 days  145.30 AU/dy  83.92% c  8,484.7 AU  0.1342 LY  145.3 AU  11,137.6 AU  30.070% GW_{0} 
10 days  144.73 AU/dy  83.59% c  8,340.0 AU  0.1319 LY  144.7 AU  11,282.3 AU  30.299% GW_{0} 
11 days  144.15 AU/dy  83.26% c  8,195.8 AU  0.1296 LY  144.1 AU  11,426.4 AU  30.527% GW_{0} 
12 days  143.56 AU/dy  82.91% c  8,052.3 AU  0.1273 LY  143.6 AU  11,570.0 AU  30.755% GW_{0} 
Given 

 d_{t1}  d_{t}  d_{Δ} + Σd_{Δ}  1(1ε∇)^{100+t} 

Points a thru f map out following events:
ⓐ After 360 days, resupply vessel ends 100 days of 7G propulsion and starts cruise at 149.9 AU/day.
ⓑ After 365.25 days of 1G acceleration, primary vessel cruises at 111.5 AU/day.
ⓒ After 400 days, resupply vessel is about 12,000 AU behind primary vessel.
ⓓ After 500 days, resupply vessel closes the gap to slightly over 8,000 AU.
ⓔ 600 days, gap closes even more to just over 4,000 AU.
ⓕ 684 days, gap shrinks to about 1,000 AU in prep for 7G vessel’s deceleration to match speed of primary vessel.

Decel.
Time(t)  Spot
Velocity (V_{t})  Spot
Distance (d_{t})  Daily
Dist. (d_{Δ})  7G
Dist.(d_{7G})  Total
Fuel (F) 
13 days  142.96 AU/dy  82.57% c  7,909.3 AU  0.1251 LY  143.0 AU  11,712.9 AU  30.982% GW_{0} 
14 days  142.34 AU/dy  82.21% c  7,767.0 AU  0.1228 LY  142.3 AU  11,855.3 AU  31.208% GW_{0} 
15 days  141.72 AU/dy  81.85% c  7,625.3 AU  0.1206 LY  141.7 AU  11,997.0 AU  31.433% GW_{0} 
16 days  141.08 AU/dy  81.48% c  7,484.2 AU  0.1183 LY  141.1 AU  12,138.1 AU  31.658% GW_{0} 
17 days  140.42 AU/dy  81.10% c  7,343.7 AU  0.1161 LY  140.4 AU  12,278.6 AU  31.881% GW_{0} 
18 days  139.76 AU/dy  80.72% c  7,203.9 AU  0.1139 LY  139.8 AU  12,418.3 AU  32.105% GW_{0} 
Given 

 d_{t1}  d_{t}  d_{Δ} + Σd_{Δ}  1(1ε∇)^{100+t} 

Decel.
Time(t)  Spot
Velocity (V_{t})  Spot
Distance (d_{t})  Daily
Dist. (d_{Δ})  7G
Dist.(d_{7G})  Total
Fuel (F) 
19 days  139.08 AU/dy  80.32% c  7,064.8 AU  0.1117 LY  139.1 AU  12,557.5 AU  32.327% GW_{0} 
20 days  138.38 AU/dy  79.92% c  6,926.4 AU  0.1095 LY  138.4 AU  12,695.9 AU  32.549% GW_{0} 
21 days  137.67 AU/dy  79.51% c  6,788.7 AU  0.1073 LY  137.7 AU  12,833.6 AU  32.770% GW_{0} 
22 days  136.95 AU/dy  79.10% c  6,651.7 AU  0.1052 LY  137.0 AU  12,970.6 AU  32.990% GW_{0} 
23 days  136.21 AU/dy  78.67% c  6,515.4 AU  0.1030 LY  136.3 AU  13,106.9 AU  33.210% GW_{0} 
24 days  135.46 AU/dy  78.24% c  6,379.9 AU  0.1009 LY  135.5 AU  13,242.4 AU  33.428% GW_{0} 
Given 

 d_{t1}  d_{t}  d_{Δ} + Σd_{Δ}  1(1ε∇)^{100+t} 


Remaining Slides: Plan for Rendezvous
Profiles will be adjusted later; however, initially, assume acceleration profiles as follows:
Main mission vessel accelerates at 1G for one year; then cruises at constant velocity per following distance equation: d1G = .267 LY + .644c×t
Resupply vessel starts 7G accel at 265.25 days after mission vessels begins. 100 days later (365.25 days into mission), 7G vessels stops accel and starts cruise: d7G = .711 LY + .866c×t
FOR CONVENIENCE: Set t=0 at beginning of mission; thus, the 2nd mission year is the first cruise year. During 1st year of cruise, 7G vessel trails 1G vessel. At cruise start (365.25 days) the 1g pax vessel is .377 LY (23,841.5 AU) from Sol with velocity of .644c (= 111.5 AU/day = 193,066 km/sec). 7G vessel trails 1G vessel (.155 LY =9,811 AU from Sol); but it has much greater velocity, (.866 c = 149.9 AU/day = 259,620 km/sec) and will catch up.
d1G = .267 LY + .644c×t = .711 LY + .866c×t = d7G
16,885AU + 111.5AU/day×t = 44,964 AU + 149.9 AU/day; thus, tInt = 731.2 days and dInt= 64,644 AUs. Unfortunately, the vessels have a huge velocity differential of about .22c. Next slide considers the required deceleration of the 7G, resupply vessel.

Decel.
Time(t)  Spot
Velocity (V_{t})  Spot
Distance (d_{t})  Daily
Dist. (d_{Δ})  7G
Dist.(d_{7G})  Total
Fuel (F) 
25 days  134.70 AU/dy  77.79% c  6,245.1 AU  0.0988 LY  134.8 AU  13,377.2 AU  33.646% GW_{0} 
26 days  133.91 AU/dy  77.34% c  6,111.1 AU  0.0966 LY  134.0 AU  13,511.2 AU  33.864% GW_{0} 
27 days  133.12 AU/dy  76.88% c  5,977.9 AU  0.0945 LY  133.2 AU  13,644.4 AU  34.080% GW_{0} 
28 days  132.30 AU/dy  76.41% c  5,845.5 AU  0.0924 LY  132.4 AU  13,776.8 AU  34.296% GW_{0} 
29 days  131.47 AU/dy  75.93% c  5,713.9 AU  0.0904 LY  131.6 AU  13,908.4 AU  34.512% GW_{0} 
30 days  130.62 AU/dy  75.44% c  5,583.1 AU  0.0883 LY  130.7 AU  14,039.1 AU  34.726% GW_{0} 
Given 

 d_{t1}  d_{t}  d_{Δ} + Σd_{Δ}  1(1ε∇)^{100+t} 


Decel.
Time(t)  Spot
Velocity (V_{t})  Spot
Distance (d_{t})  Daily
Dist. (d_{Δ})  7G
Dist.(d_{7G})  Total
Fuel (F) 
31 days  129.76 AU/dy  74.94% c  5,453.2 AU  0.0862 LY  129.9 AU  14,169.0 AU  34.940% GW_{0} 
32 days  128.88 AU/dy  74.43% c  5,324.2 AU  0.0842 LY  129.0 AU  14,298.1 AU  35.153% GW_{0} 
33 days  127.98 AU/dy  73.91% c  5,196.1 AU  0.0822 LY  128.1 AU  14,426.2 AU  35.366% GW_{0} 
34 days  127.06 AU/dy  73.38% c  5,068.9 AU  0.0802 LY  127.2 AU  14,553.4 AU  35.577% GW_{0} 
35 days  126.13 AU/dy  72.84% c  4,942.6 AU  0.0782 LY  126.3 AU  14,679.7 AU  35.788% GW_{0} 
36 days  125.17 AU/dy  72.29% c  4,817.3 AU  0.0762 LY  125.3 AU  14,805.0 AU  35.999% GW_{0} 
Given 

 d_{t1}  d_{t}  d_{Δ} + Σd_{Δ}  1(1ε∇)^{100+t} 

Decel.
Time(t)  Spot
Velocity (V_{t})  Spot
Distance (d_{t})  Daily
Dist. (d_{Δ})  7G
Dist.(d_{7G})  Total
Fuel (F) 
37 days  124.20 AU/dy  71.73% c  4,692.9 AU  0.0742 LY  124.4 AU  14,929.4 AU  36.208% GW_{0} 
38 days  123.20 AU/dy  71.15% c  4,569.5 AU  0.0723 LY  123.4 AU  15,052.8 AU  36.417% GW_{0} 
39 days  122.19 AU/dy  70.57% c  4,447.1 AU  0.0703 LY  122.4 AU  15,175.2 AU  36.626% GW_{0} 
40 days  121.15 AU/dy  69.97% c  4,325.7 AU  0.0684 LY  121.4 AU  15,296.6 AU  36.833% GW_{0} 
41 days  120.10 AU/dy  69.36% c  4,205.4 AU  0.0665 LY  120.3 AU  15,416.9 AU  37.040% GW_{0} 
42 days  119.02 AU/dy  68.74% c  4,086.1 AU  0.0646 LY  119.3 AU  15,536.1 AU  37.246% GW_{0} 
Given 

 d_{t1}  d_{t}  d_{Δ} + Σd_{Δ}  1(1ε∇)^{100+t} 


Decel.
Time(t)  Spot
Velocity (V_{t})  Spot
Distance (d_{t})  Daily
Dist. (d_{Δ})  7G
Dist.(d_{7G})  Total
Fuel (F) 
43 days  117.92 AU/dy  68.11% c  3,968.0 AU  0.0627 LY  118.2 AU  15,654.3 AU  37.452%GW_{0} 
44 days  116.80 AU/dy  67.46% c  3,850.9 AU  0.0609LY  117.1 AU  15,771.4 AU  37.657%GW_{0} 
45 days  115.66 AU/dy  66.80%c  3,735.0 AU  0.0591 LY  115.9 AU  15,887.3 AU  37.861%GW_{0} 
46 days  114.49 AU/dy  66.12% c  3,620.2 AU  0.0572 LY  114.8 AU  16,002.1 AU  38.065%GW_{0} 
47 days  113.30 AU/dy  65.44% c  3,506.6 AU  0.0554 LY  113.6 AU  16,115.7 AU  38.268%GW_{0} 
48 days  112.09 AU/dy  64.74% c  3,394.2 AU  0.0537 LY  112.4 AU  16,228.0AU  38.470%GW_{0} 
Resupply Vessel Matches Cruise Velocity of Baseline "Pax" Vessel (1.0 yr of 1G Accel.)
at exactly the time/distance of the intercept. 
48¼days  111.78 AU/dy  64.56% c  3,366.3 AU  0.0532 LY  27.9 AU  16,256.0 AU  38.520%GW_{0} 
Given 

 d_{t1}  d_{t}  d_{Δ} + Σd_{Δ}  1(1ε∇)^{100+t} 


Decel.
Time(t)  Spot
Velocity (V_{t})  Spot
Distance (d_{t})  Daily
Dist. (d_{Δ})  7G
Dist.(d_{7G})  Total
Fuel (F) 
52 days  106.98AU/dy  61.79%c  2,966.6 AU  0.2636LY  107.3AU  16,667.7AU  39.272%GW_{0} 
53 days  105.64AU/dy  61.01%c  2,860.5AU  0.2652LY  106.0AU  16,773.7AU  39.471%GW_{0} 
Resupply Vessel Matches Cruise Velocity of Slower "Pax" Vessel (0.9 yr of 1G Accel.)
at exactly the time/distance of the intercept. 
53.4 days  105.10AU/dy  60.70%c  2,818.4AU  0.2659LY  42.0 AU  16,815.7AU  39.551%GW_{0} 
54 days  104.27AU/dy  60.22%c  2,755.7AU  0.2662LY  62.6 AU  16,836.3AU  39.669%GW_{0} 
Given 

 d_{t1}  d_{t}  d_{Δ} + Σd_{Δ}  1(1ε∇)^{100+t} 

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