7G TABLE: Decelerate for 48 ¼ Days

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CONTENT
INTRODUCTION
MUST SLOWDOWN!!!
7G ACCEL FOR 100 DAYS
DECELERATE
INTERSTELLAR VESSELS
CRUISE AT CONSTANT VELOCITY
CRUCIAL EVENTS
DAILY DIFFERENCE
INTERSECTION POINT
SPOT VELOCITY
MISSING 800 AU
SPOT DISTANCE
7G DISTANCES
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SUMMARY
7G DECEL DAYS 0-6
7G DECEL DAYS 7-12
7G DECEL DAYS 13-18
7G DECEL DAYS 19-24
7G DECEL DAYS 25-30
7G DECEL DAYS 31-36
7G DECEL DAYS 37-42
7G DECEL DAYS 43-48¼

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INTRODUCTION

To effectively accomplish an interstellar resupply mission, an Artificial Intelligent (AI) vessel could conceptually use 7G propulsion to quickly intercept the primary 1G vessel. After 100 days of 7G propulsion, the resupply vessel reaches an enormous velocity of 149.9 AU/day (.866 c).  THEREFORE,  the superfast, resupply vessel must slow down to match cruise velocity of the pax vessel. This brainstorm chapter proposes daily progress throughout a conceptual, deceleration duration of 48¼ days.

RESUPPLY VESSEL MUST SLOWDOWN!!!

7-G PROPULSION FOR 100 DAYS

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 (0.866 c=86.6% light speed). Assume 7G vessel cruises at this velocity until the optimal time/distance to start decelerating at 7g.
DECELERATE FOR 48¼ DAYS
Decel. Time(t) Spot Velocity (Vt) Spot Distance (dt) Daily Dist. (dΔ) 7G Cum Dist.(d7G) Consumed 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% GW0 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% GW0 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% GW0 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% GW0 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% GW0 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% GW0 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% 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 For success, the 7G resupply vessel must slow down to match the 1G vessel's cruise velocity (.644c = 111.5 AU/day). This chapter describes daily progress throughout the 48¼ days of 7G deceleration.
INTERSTELLAR VESSELS: 1G PAX/CARGO & 7-G RESUPPLY (SNOWBALL) Interstellar 1G Vesselstarts its one year of acceleration at Day 0. After this year, vessel stops acceleration and cruises at 111.5 Astronomical Units per day  (.644 c). One year of 1G acceleration has taken vessel to 23,841 AUs,  .377 Light Year (LY). Interstellar 7G Snowball In this example, 7G snowball (traditional habitat encased in large volume of ice) 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).
AFTER 1G ACCELERATION, PAX VESSEL CRUISES AT CONSTANT VELOCITY
Decel. Time(t) Spot Velocity (Vt) Spot Distance (dt) Daily Dist. (dΔ) 7G Cum Dist.(d7G) Consumed 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% GW0 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% GW0 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% GW0 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% GW0 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% GW0 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% 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
CRUCIAL EVENTS FOR SUCCESSFUL RENDEZVOUS
Points ⓐ thru ⓕ map out following events: ⓐ After 360 days, resupply vessel ends 100 days of 7G prop; starts cruise at 149.9 AU/day. ⓑ After 365¼ days of 1G acceleration, primary vessel cruises at 111.5 AU/day. ⓒ After 400 days (about 35 days of constant velocity), resupply vessel is about 12,000 AU behind primary vessel. ⓓ After 500 days (about 135 days of constant velocity), resupply vessel closes the gap to slightly over 8,000 AU. ⓔ  600 days (100 more days of constant velocity), gap closes to just over 4,000 AU. ⓕ  684 days, gap shrinks to about 1,000 AU in prep for 7G vessel’s deceleration duration of 48¼ days needed to match speed of primary vessel. Describe primary vessel’s cruise distance by linear equation: d = 111.5 t - 16,884 AU Resupply vessel’s cruise distance: d= 149.9 t - 44,152 AU By inspection, we expect a pending intercept of these two cruise tracks about one LY and two years from start of primary vessel’s track. HOWEVER, the resupply vessel track must become nonlinear at about 684 days for 48¼ day deceleration maneuver.
DAILY DIFFERENCE IN DAILY VELOCITY
Decel. Time(t) Spot Velocity (Vt) Spot Distance (dt) Daily Dist. (dΔ) 7G Cum Dist.(d7G) 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% GW0 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% GW0 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% GW0 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% GW0 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% GW0 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% 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 DAILY DIFF: Δ= 0.0198 Thot Exp. brainstorm proposes  persistent 7G deceleration exponentially  decreases vessel velocity by about 2% c/day.
7G DECELERATION DURING DAYS 19 THRU 24
Decel. Time(t) Spot Velocity (Vt) Spot Distance (dt) Daily Dist. (dΔ) 7G Cum Dist.(d7G) Consumed 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% GW0 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% GW0 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% GW0 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% GW0 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% GW0 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% 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
DETERMINE INTERSECTION POINT
Assume 1G pax vessel accelerates at 1G for one year; then, cruises at constant velocity. Thus, determine distance traveled per following linear equation: d1G = -.267 LY + .644 c × t Assume resupply vessel starts 7G acceleration at 265.25 days after mission begins (1G vessel starts).  100 days later (365.25 days into mission), 7G vessels stops acceleration and starts cruise. Thus, determine total distance traveled via following linear equation:    d7G = -.711 LY + .866 c × t To determine intersection point, set equations equal as shown:      d1G = -.267 LY + .644c × t = -.711 LY + .866c × t  = d7G For convenience, substitute: LY = 63,241 AU and c = 173.15 AU/day:      -16,885 AU + 111.5 AU/day × t = -44,964 AU + 149.9 AU/day × t thus, solve for: tInt = 731.2 days and dInt= 64,644 AUs.   These linear equations show two cruising vessels will meet at time of 731.2 days (2 years) since day zero and distance equals 64,644 AU (about 1 LY) from 1G pax vessel's starting point. However, the two vessels cannot gracefully rendezvous because they still have a huge velocity differential of about .22 c.  Following section proposes a graceful deceleration method.
INITIATE 7G DECELERATION After 684 days of mission time, resupply vessel flight profile used 7G acceleration and high speed cruise to reach a position 58,340 AU from Sol. At this point, 7G vessel is only 1,076 AU behind the 1G vessel. The 7G vessel must start the 48¼ days of  7G deceleration required to slow down from 149.9 AU/day to 111.5 AU/day, cruise velocity of the primary vessel. For functional rendezvous, both the primary mission vessel and the resupply vessel must match velocities  at the same position at the same time. In this example, both vessels simultaneously reach 64,796 AU (about one LY) along the track from Sol to neighboring star.
SNOWBALL's SPOT VELOCITY FOR EACH DAY ON ITS WAY TO RENDEZVOUS
For Spot Velocity (Vt), see 7G Accel for 100 Dy. Countdown in reverse order; i.e., 100th day in 7G Accel Table is 0 day in 7G Decel Table. 75th day in 7G Accel has 134.70 AU/dy & 77.79% c; same values as 25th day in 7G Deceleration Table. Decel. Time(t) Spot Velocity (Vt) Spot Distance (dt) Daily Dist. (dΔ) 7G Cum Dist.(d7G) Consumed 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% GW0 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% GW0 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% GW0 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% GW0 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% GW0 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% 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
7G DECELERATION FOR 48¼ DAYS LEAVES SNOWBALL 800 AUs SHORT OF GOAL
LINEAR EQUATIONS show cruise interception after 2 years (730.5 days) of mission time at 64,567 AUs (slightly more than one LY).  PROBLEM: 7G vessel ("snowball") is cruising at 150 AU/day; much faster than 1G vessel ("pax") at 111 AU/day. SLOWDOWN: Inspect appropriate tables to see snowball needs 7G deceleration of 48¼ days and 6,445 AUs to match velocity of pax vessel. ADJUSTMENT: 48¼ day cruise distance (48¼ x 150 AU/day = 7,237.5 AU) is 792.6 AU further than the 48¼ day 7G deceleration distance. CONCLUDE: If we back off 48¼ days of cruise time to start the required deceleration of 48¼ days, we will fall short of desired intersection by almost 800 AUs.  Thus, we must stop 7G deceleration at slightly above pax's cruise speed and slowly adjust as snowball approaches pax.
SNOWBALL's SPOT DISTANCE FOR EACH DECELERATION DAY
Decel. Time(t) Spot Velocity (Vt) Spot Distance (dt) Daily Dist. (dΔ) 7G Cum Dist.(d7G) 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% GW0 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% GW0 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% GW0 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% GW0 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% GW0 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% 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 For Spot Distance (dt), see 7G Accel: 100 Dy. Look for close correspondence in reverse order. I.e., Day 69 in 7G Accel has 5,474.3 AU & 0.0866 LY; very near values in Day 31 in 7G Decel Table. NOTE: Just like 7G Accel values count up 100 days to 9841AUs Spot Dist, 7G Deceleration values count down similar values from 9,811 AUs, but stop at 48¼ days.
7G DAILY DISTANCE AND CUMULATIVE
Decel. Time(t) Spot Velocity (Vt) Spot Distance (dt) Daily Dist. (dΔ) 7G Cum Dist.(d7G) Consumed 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% GW0 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% GW0 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% GW0 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% GW0 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% GW0 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% 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 Cum Dist.(d7G) 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%GW0 44 days 116.80 AU/dy 67.46% c 3,850.9 AU 0.0609LY 117.1 AU 15,771.4 AU 37.657%GW0 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%GW0 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%GW0 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%GW0 48 days 112.09 AU/dy 64.74% c 3,394.2 AU 0.0537 LY 112.4 AU 16,228.0 AU 38.470%GW0 Resupply Vessel Matches Cruise Velocity of Baseline "Pax" Vessel (1.0 yr of 1G Accel.) at exact 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%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
SUMMARY
SLIDESHOW 1G Acceleration for 1 year. 7G Acceleration for 100 days. 7G Deceleration for 48¼ days. 1G Deceleration for 1 year. VOLUME 0: ELEVATIONAL VOLUME I: ASTEROIDAL VOLUME II: INTERPLANETARY VOLUME III: INTERSTELLAR   Oort Snowball contains water ice and many other goodies to resupply an interstellar vessel spinning in deep space. Interstellar vessel spins to simulate Earth gravity for its human passengers and a widely diverse set of cargo. Snowball cruises at 149.9 AU/day, 86.6% light speed (c) after accelerating at 7G for 100 days. Pax vessel cruises at 111.5 AU/day (.644c) after accelerating at 1G for 365.25 days. Successful rendezvous requires snowball to 7G decelerate for 48¼ days.