A Thought Experiment: 7G TABLE: Decelerate for 48 ¼ Days

VOLUME 0: ELEVATIONAL
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 (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% 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)+	V_t ln(1-Δ)

d_t-1 - d_t

d_Δ + Σd_Δ

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 table describes daily progress throughout the 48¼ days of 7G deceleration.

Interstellar 1G Vessel

starts 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).

Decel.
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% 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)+	V_t ln(1-Δ)

d_t-1 - d_t

d_Δ + Σd_Δ

1-(1-ε∇)^100+t

Points ⓐ thru ⓕ map out following events:

ⓐ After 360 days, resupply vessel ends 100 days of 7G propulsion and starts cruise at 149.9 AU/day.

ⓑ After 365¼ days of 1G acceleration, primary vessel cruises at 111.5 AU/day.

ⓓ 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 even more 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.

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% 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)+	V_t ln(1-Δ)

d_t-1 - d_t

d_Δ + Σd_Δ

1-(1-ε∇)^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% 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)+	V_t ln(1-Δ)

d_t-1 - d_t

d_Δ + Σd_Δ

1-(1-ε∇)^100+t

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.

Determine Intersection Point

Assume the 1G pax vessel accelerates at 1G for one year; then, cruises at constant velocity. Thus, determine distance traveled per following linear equation:

d_1G = -.267 LY + .644 c × t

Assume resupply vessel starts 7G accelerates at 265.25 days after mission vessels begins. 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 of the two linear equations, set them equal as shown:

d1G = -.267 LY + .644c × t = -.711 LY + .866c × t = d7G

For convenience, substitute following terms: 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 enable us to determine that the two cruising vessels will meet at time equals 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. A following section proposes a deceleration method to synchronize velocity in addition to time and distance.

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% 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)+	V_t ln(1-Δ)

d_t-1 - d_t

d_Δ + Σd_Δ

1-(1-ε∇)^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% 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)+	V_t ln(1-Δ)

d_t-1 - d_t

d_Δ + Σd_Δ

1-(1-ε∇)^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% 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)+	V_t ln(1-Δ)

d_t-1 - d_t

d_Δ + Σd_Δ

1-(1-ε∇)^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%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

16,228.0AU

38.470%GW₀

Resupply Vessel Matches Cruise Velocity of Baseline "Pax" Vessel (1.0 yr of 1G Accel.)
at exactly the time/distance of the intercept.

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)+	V_t ln(1-Δ)

d_t-1 - d_t

d_Δ + Σd_Δ

1-(1-ε∇)^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%GW₀

53 days

105.64AU/dy

61.01%c

2,860.5AU

0.2652LY

106.0AU

16,773.7AU

39.471%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 days

105.10AU/dy

60.70%c

2,818.4AU

0.2659LY

42.0 AU

16,815.7AU

39.551%GW₀

54 days

104.27AU/dy

60.22%c

2,755.7AU

0.2662LY

62.6 AU

16,836.3AU

39.669%GW₀

Given

(1 - (1-Δ)^100-t) × c

c×(100-t)+	V_t ln(1-Δ)

d_t-1 - d_t

d_Δ + Σd_Δ

1-(1-ε∇)^100+t

VOLUME 0: ELEVATIONAL

VOLUME I: ASTEROIDAL

VOLUME II: INTERPLANETARY

VOLUME III: INTERSTELLAR

SLIDESHOW

A Thought Experiment

Sunday, July 20, 2014

7G TABLE: Decelerate for 48 ¼ Days

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