Sunday, July 20, 2014

7G TABLE: Decelerate for 48 1/4 Days









VOLUME I:
ASTEROIDAL
VOLUME II:
INTERPLANETARY
VOLUME III:
INTERSTELLAR








To effectively accomplish an interstellar resupply mission, an Artificial Intelligent (AI) vessel could use 7G propulsion to quickly intercept the primary 1G vessel.  However,  the 7G vessel must slow down prior to intercept in order to match the 1G vessel cruise velocity. This table describes daily progress throughout the deceleration duration.
7G Deceleration for 48 ¼ Days 

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 (Vt)
Spot
Distance (dt)
Daily
Dist. (dΔ)
7G
Dist.(d7G)
Total
Fuel (F)
0 days149.93 AU/dy86.59% c9,811.1 AU0.1551 LY0.0 AU9,811.1 AU27.974% GW0
1 days149.46 AU/dy86.32% c9,661.7 AU0.1528 LY149.4 AU9,960.5 AU28.210% GW0
2 days148.98 AU/dy86.04% c9,512.8 AU0.1504 LY148.9 AU10,109.4 AU28.445% GW0
3 days148.48 AU/dy85.76% c9,364.4 AU0.1481 LY148.4 AU10,257.9 AU28.679% GW0
4 days147.98 AU/dy85.46% c9,216.5 AU0.1457 LY147.9 AU10,405.8 AU28.913% GW0
5 days147.46 AU/dy85.17% c9,069.1 AU0.1434 LY147.4 AU10,553.2 AU29.146% GW0
6 days146.94 AU/dy84.86% c8,922.2 AU0.1411 LY146.9 AU10,700.1 AU29.378% GW0
Given
(1 - (1-Δ)100-t) × c
c×(100-t)+ Vt

ln(1-Δ)
dt-1 - dt dΔ + ΣdΔ 1-(1-ε∇)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 (Vt)
Spot
Distance (dt)
Daily
Dist. (dΔ)
7G
Dist.(d7G)
Total
Fuel (F)
7 days146.40 AU/dy84.56% c8,775.8 AU0.1388 LY146.4 AU10,846.5 AU29.609% GW0
8 days145.86 AU/dy84.24% c8,630.0 AU0.1365 LY145.8 AU10,992.3 AU29.840% GW0
9 days145.30 AU/dy83.92% c8,484.7 AU0.1342 LY145.3 AU11,137.6 AU30.070% GW0
10 days144.73 AU/dy83.59% c8,340.0 AU0.1319 LY144.7 AU11,282.3 AU30.299% GW0
11 days144.15 AU/dy83.26% c8,195.8 AU0.1296 LY144.1 AU11,426.4 AU30.527% GW0
12 days143.56 AU/dy82.91% c8,052.3 AU0.1273 LY143.6 AU11,570.0 AU30.755% GW0
Given
(1 - (1-Δ)100-t) × c
c×(100-t)+ Vt

ln(1-Δ)
dt-1 - dt dΔ + ΣdΔ 1-(1-ε∇)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 (Vt)
Spot
Distance (dt)
Daily
Dist. (dΔ)
7G
Dist.(d7G)
Total
Fuel (F)
13 days142.96 AU/dy82.57% c7,909.3 AU0.1251 LY143.0 AU11,712.9 AU30.982% GW0
14 days142.34 AU/dy82.21% c7,767.0 AU0.1228 LY142.3 AU11,855.3 AU31.208% GW0
15 days141.72 AU/dy81.85% c7,625.3 AU0.1206 LY141.7 AU11,997.0 AU31.433% GW0
16 days141.08 AU/dy81.48% c7,484.2 AU0.1183 LY141.1 AU12,138.1 AU31.658% GW0
17 days140.42 AU/dy81.10% c7,343.7 AU0.1161 LY140.4 AU12,278.6 AU31.881% GW0
18 days139.76 AU/dy80.72% c7,203.9 AU0.1139 LY139.8 AU12,418.3 AU32.105% GW0
Given
(1 - (1-Δ)100-t) × c
c×(100-t)+ Vt

ln(1-Δ)
dt-1 - dt dΔ + ΣdΔ 1-(1-ε∇)100+t
Decel.
Time(t)
Spot
Velocity (Vt)
Spot
Distance (dt)
Daily
Dist. (dΔ)
7G
Dist.(d7G)
Total
Fuel (F)
19 days139.08 AU/dy80.32% c7,064.8 AU0.1117 LY139.1 AU12,557.5 AU32.327% GW0
20 days138.38 AU/dy79.92% c6,926.4 AU0.1095 LY138.4 AU12,695.9 AU32.549% GW0
21 days137.67 AU/dy79.51% c6,788.7 AU0.1073 LY137.7 AU12,833.6 AU32.770% GW0
22 days136.95 AU/dy79.10% c6,651.7 AU0.1052 LY137.0 AU12,970.6 AU32.990% GW0
23 days136.21 AU/dy78.67% c6,515.4 AU0.1030 LY136.3 AU13,106.9 AU33.210% GW0
24 days135.46 AU/dy78.24% c6,379.9 AU0.1009 LY135.5 AU13,242.4 AU33.428% GW0
Given
(1 - (1-Δ)100-t) × c
c×(100-t)+ Vt

ln(1-Δ)
dt-1 - dt dΔ + ΣdΔ 1-(1-ε∇)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 1-g 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 (Vt)
Spot
Distance (dt)
Daily
Dist. (dΔ)
7G
Dist.(d7G)
Total
Fuel (F)
25 days134.70 AU/dy77.79% c6,245.1 AU0.0988 LY134.8 AU13,377.2 AU33.646% GW0
26 days133.91 AU/dy77.34% c6,111.1 AU0.0966 LY134.0 AU13,511.2 AU33.864% GW0
27 days133.12 AU/dy76.88% c5,977.9 AU0.0945 LY133.2 AU13,644.4 AU34.080% GW0
28 days132.30 AU/dy76.41% c5,845.5 AU0.0924 LY132.4 AU13,776.8 AU34.296% GW0
29 days131.47 AU/dy75.93% c5,713.9 AU0.0904 LY131.6 AU13,908.4 AU34.512% GW0
30 days130.62 AU/dy75.44% c5,583.1 AU0.0883 LY130.7 AU14,039.1 AU34.726% GW0
Given
(1 - (1-Δ)100-t) × c
c×(100-t)+ Vt

ln(1-Δ)
dt-1 - dt dΔ + ΣdΔ 1-(1-ε∇)100+t
Decel.
Time(t)
Spot
Velocity (Vt)
Spot
Distance (dt)
Daily
Dist. (dΔ)
7G
Dist.(d7G)
Total
Fuel (F)
31 days129.76 AU/dy74.94% c5,453.2 AU0.0862 LY129.9 AU14,169.0 AU34.940% GW0
32 days128.88 AU/dy74.43% c5,324.2 AU0.0842 LY129.0 AU14,298.1 AU35.153% GW0
33 days127.98 AU/dy73.91% c5,196.1 AU0.0822 LY128.1 AU14,426.2 AU35.366% GW0
34 days127.06 AU/dy73.38% c5,068.9 AU0.0802 LY127.2 AU14,553.4 AU35.577% GW0
35 days126.13 AU/dy72.84% c4,942.6 AU0.0782 LY126.3 AU14,679.7 AU35.788% GW0
36 days125.17 AU/dy72.29% c4,817.3 AU0.0762 LY125.3 AU14,805.0 AU35.999% GW0
Given
(1 - (1-Δ)100-t) × c
c×(100-t)+ Vt

ln(1-Δ)
dt-1 - dt dΔ + ΣdΔ 1-(1-ε∇)100+t

Decel.
Time(t)
Spot
Velocity (Vt)
Spot
Distance (dt)
Daily
Dist. (dΔ)
7G
Dist.(d7G)
Total
Fuel (F)
37 days124.20 AU/dy71.73% c4,692.9 AU0.0742 LY124.4 AU14,929.4 AU36.208% GW0
38 days123.20 AU/dy71.15% c4,569.5 AU0.0723 LY123.4 AU15,052.8 AU36.417% GW0
39 days122.19 AU/dy70.57% c4,447.1 AU0.0703 LY122.4 AU15,175.2 AU36.626% GW0
40 days121.15 AU/dy69.97% c4,325.7 AU0.0684 LY121.4 AU15,296.6 AU36.833% GW0
41 days120.10 AU/dy69.36% c4,205.4 AU0.0665 LY120.3 AU15,416.9 AU37.040% GW0
42 days119.02 AU/dy68.74% c4,086.1 AU0.0646 LY119.3 AU15,536.1 AU37.246% GW0
Given
(1 - (1-Δ)100-t) × c
c×(100-t)+ Vt

ln(1-Δ)
dt-1 - dt dΔ + ΣdΔ 1-(1-ε∇)100+t
Decel.
Time(t)
Spot
Velocity (Vt)
Spot
Distance (dt)
Daily
Dist. (dΔ)
7G
Dist.(d7G)
Total
Fuel (F)
43 days117.92 AU/dy68.11% c3,968.0 AU0.0627 LY118.2 AU15,654.3 AU37.452%GW0
44 days116.80 AU/dy67.46% c3,850.9 AU0.0609LY117.1 AU15,771.4 AU37.657%GW0
45 days115.66 AU/dy66.80%c3,735.0 AU0.0591 LY115.9 AU15,887.3 AU37.861%GW0
46 days114.49 AU/dy66.12% c3,620.2 AU0.0572 LY114.8 AU16,002.1 AU38.065%GW0
47 days113.30 AU/dy65.44% c3,506.6 AU0.0554 LY113.6 AU16,115.7 AU38.268%GW0
48 days112.09 AU/dy64.74% c3,394.2 AU0.0537 LY112.4 AU16,228.0AU38.470%GW0
Resupply Vessel Matches Cruise Velocity of Baseline "Pax" Vessel (1.0 yr of 1G Accel.)
at exactly the time/distance of the intercept.
48¼days111.78 AU/dy64.56% c3,366.3 AU0.0532 LY27.9 AU16,256.0 AU38.520%GW0
Given
(1 - (1-Δ)100-t) × c
c×(100-t)+ Vt

ln(1-Δ)
dt-1 - dt dΔ + ΣdΔ 1-(1-ε∇)100+t
Decel.
Time(t)
Spot
Velocity (Vt)
Spot
Distance (dt)
Daily
Dist. (dΔ)
7G
Dist.(d7G)
Total
Fuel (F)
52 days106.98AU/dy61.79%c2,966.6 AU 0.2636LY107.3AU16,667.7AU39.272%GW0
53 days105.64AU/dy61.01%c2,860.5AU 0.2652LY106.0AU16,773.7AU39.471%GW0
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 days105.10AU/dy60.70%c2,818.4AU 0.2659LY42.0 AU16,815.7AU39.551%GW0
54 days104.27AU/dy60.22%c2,755.7AU 0.2662LY62.6 AU16,836.3AU 39.669%GW0
Given
(1 - (1-Δ)100-t) × c
c×(100-t)+ Vt

ln(1-Δ)
dt-1 - dt dΔ + ΣdΔ 1-(1-ε∇)100+t

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