A Thought Experiment: 7G TABLE: Accelerate 100 Days

CONTENT
IV TIME(t)
VELOCITY AS AU/DAY
VELOCITY AS %c
DISTANCE AS AUs
DISTANCE AS LYs
7G ACCEL DAYS 0-15
7G ACCEL DAYS 16-30
7G ACCEL DAYS 31-45
7G ACCEL DAYS 46-60
7G ACCEL DAYS 61-75
7G ACCEL DAYS 76-90
7G ACCEL DAYS 91-100

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. EXAMPLE: the 7G vessel needs only 100 days to greatly exceed the 1G vessel cruise velocity. This table describes daily progress throughout a typical 7G acceleration duration planned to stop near the end of the 1G vessel's 1 year acceleration.

INDEPENDENT VARIABLE: time (t)
Each row is numbered sequentially
from 0 days to 100 days.

time

Velocity

Incr. Dist.

Total Distance

Total Fuel

0 days

0.00 AU/dy

0.00% c

0.0 AU

0.0000 LY

0.000% GW₀

1 days

3.43 AU/dy

1.98% c

1.7 AU

0.0000 LY

0.328% GW₀

2 days

6.80 AU/dy

3.92% c

5.1 AU

6.8 AU

0.0001 LY

0.654% GW₀

3 days

10.09 AU/dy

5.83% c

8.4 AU

15.3 AU

0.0002 LY

0.980% GW₀

4 days

13.32 AU/dy

7.70% c

11.7 AU

27.0 AU

0.0004 LY

1.304% GW₀

5 days

16.49 AU/dy

9.53% c

14.9 AU

41.9 AU

0.0007 LY

1.627% GW₀

6 days

19.60 AU/dy

11.32% c

18.0 AU

60.0 AU

0.0009 LY

1.950% GW₀

7 days

22.64 AU/dy

13.08% c

21.1 AU

81.1 AU

0.0013 LY

2.271% GW₀

8 days

25.62 AU/dy

14.80% c

24.1 AU

105.2 AU

0.0017 LY

2.591% GW₀

9 days

28.55 AU/dy

16.49% c

27.1 AU

132.3 AU

0.0021 LY

2.910% GW₀

10 days

31.42 AU/dy

18.14% c

30.0 AU

162.3 AU

0.0026 LY

3.228% GW₀

11 days

34.23 AU/dy

19.77% c

32.8 AU

195.1 AU

0.0031 LY

3.545% GW₀

12 days

36.98 AU/dy

21.36% c

35.6 AU

230.7 AU

0.0036 LY

3.861% GW₀

13 days

39.68 AU/dy

22.92% c

38.3 AU

269.1 AU

0.0043 LY

4.176% GW₀

14 days

42.33 AU/dy

24.45% c

41.0 AU

310.1 AU

0.0049 LY

4.490% GW₀

15 days

44.92 AU/dy

25.94% c

43.6 AU

353.7 AU

0.0056 LY

4.803% GW₀

Velocity (V_AU) computed as AU/day.
AU = 149,597,871 km & c = 173.145 AU/day
EXAMPLE: t = 16 days, compute as follows:
V_AU= (1 - (1-Δ)^t) × c
V_AU=(1 - (0.98019)¹⁶)×173.145 AU/day
V_AU= 47.46 AU/day
NOTE: Δ, daily difference, is the 7G vessel's daily velocity increase, 1.981% of c, light speed. Thus, 1-Δ is .98019.

Given

(1 - (1-Δ)^t) ×	c

d_t - d_t-1

c × t +	V_t ln(1-Δ)

f_t = 1 - (1-ε∇)^t

16 days

47.46 AU/dy

27.41% c

46.2 AU

399.9 AU

0.0063 LY

5.115% GW₀

17 days

49.96 AU/dy

28.85% c

48.7 AU

448.6 AU

0.0071 LY

5.426% GW₀

18 days

52.40 AU/dy

30.26% c

51.2 AU

499.8 AU

0.0079 LY

5.735% GW₀

19 days

54.79 AU/dy

31.65% c

53.6 AU

553.4 AU

0.0088 LY

6.044% GW₀

20 days

57.14 AU/dy

33.00% c

56.0 AU

609.4 AU

0.0096 LY

6.352% GW₀

21 days

59.44 AU/dy

34.33% c

58.3 AU

667.7 AU

0.0106 LY

6.659% GW₀

22 days

61.70 AU/dy

35.63% c

60.6 AU

728.2 AU

0.0115 LY

6.965% GW₀

23 days

63.91 AU/dy

36.91% c

62.8 AU

791.1 AU

0.0125 LY

7.269% GW₀

24 days

66.08 AU/dy

38.16% c

65.0 AU

856.1 AU

0.0135 LY

7.573% GW₀

25 days

68.20 AU/dy

39.39% c

67.1 AU

923.2 AU

0.0146 LY

7.876% GW₀

26 days

70.28 AU/dy

40.59% c

69.2 AU

992.4 AU

0.0157 LY

8.178% GW₀

27 days

72.32 AU/dy

41.77% c

71.3 AU

1,063.7 AU

0.0168 LY

8.479% GW₀

28 days

74.32 AU/dy

42.93% c

73.3 AU

1,137.1 AU

0.0180 LY

8.778% GW₀

29 days

76.29 AU/dy

44.06% c

75.3 AU

1,212.4 AU

0.0192 LY

9.077% GW₀

30 days

78.21 AU/dy

45.17% c

77.3 AU

1,289.6 AU

0.0204 LY

9.375% GW₀

Velocity (V_c) computed as %c.
V_c= V_AU/ c
EXAMPLE: 31 days of 7G acceleration.
V_c= 80.09 AU/dy/ 173.145 AU/day
V_c= 46.26% c

time

Velocity

Incr. Dist.

Total Distance

Total Fuel

31 days

80.09 AU/dy

46.26% c

79.2 AU

1,368.8 AU

0.0216 LY

9.672% GW₀

32 days

81.94 AU/dy

47.32% c

81.0 AU

1,449.8 AU

0.0229 LY

9.968% GW₀

33 days

83.75 AU/dy

48.37% c

82.8 AU

1,532.7 AU

0.0242 LY

10.263% GW₀

34 days

85.52 AU/dy

49.39% c

84.6 AU

1,617.3 AU

0.0256 LY

10.557% GW₀

35 days

87.26 AU/dy

50.40% c

86.4 AU

1,703.7 AU

0.0269 LY

10.850% GW₀

36 days

88.97 AU/dy

51.38% c

88.1 AU

1,791.8 AU

0.0283 LY

11.142% GW₀

37 days

90.64 AU/dy

52.35% c

89.8 AU

1,881.6 AU

0.0298 LY

11.433% GW₀

38 days

92.28 AU/dy

53.30% c

91.5 AU

1,973.1 AU

0.0312 LY

11.723% GW₀

39 days

93.88 AU/dy

54.22% c

93.1 AU

2,066.2 AU

0.0327 LY

12.012% GW₀

40 days

95.46 AU/dy

55.13% c

94.7 AU

2,160.8 AU

0.0342 LY

12.301% GW₀

41 days

97.00 AU/dy

56.02% c

96.2 AU

2,257.1 AU

0.0357 LY

12.588% GW₀

42 days

98.51 AU/dy

56.90% c

97.8 AU

2,354.8 AU

0.0372 LY

12.874% GW₀

43 days

100.00 AU/dy

57.75% c

99.3 AU

2,454.1 AU

0.0388 LY

13.160% GW₀

44 days

101.45 AU/dy

58.59% c

100.7 AU

2,554.8 AU

0.0404 LY

13.444% GW₀

45 days

102.87 AU/dy

59.42% c

102.2 AU

2,657.0 AU

0.0420 LY

13.728% GW₀

Total Distance (D_AU) computed as AUs.

D_AU = c × t +	V_t ln(1-Δ)	EXAMPLE: t = 46 days

D_AU = 173.145 AU/dy×46 dy +	104.27 AU/dy ln(.98019)	= 2,760.6 AU

Given

(1 - (1-Δ)^t) ×	c

d_t - d_t-1

c × t +	V_t ln(1-Δ)

f_t = 1 - (1-ε∇)^t

46 days

104.27 AU/dy

60.22% c

103.6 AU

2,760.6 AU

0.0437 LY

14.010% GW₀

47 days

105.64 AU/dy

61.01% c

105.0 AU

2,865.5 AU

0.0453 LY

14.292% GW₀

48 days

106.98 AU/dy

61.79% c

106.3 AU

2,971.8 AU

0.0470 LY

14.573% GW₀

49 days

108.30 AU/dy

62.55% c

107.6 AU

3,079.5 AU

0.0487 LY

14.853% GW₀

50 days

109.58 AU/dy

63.29% c

108.9 AU

3,188.4 AU

0.0504 LY

15.132% GW₀

51 days

110.85 AU/dy

64.02% c

110.2 AU

3,298.6 AU

0.0522 LY

15.410% GW₀

52 days

112.09 AU/dy

64.74% c

111.5 AU

3,410.1 AU

0.0539 LY

15.687% GW₀

53 days

113.30 AU/dy

65.44% c

112.7 AU

3,522.8 AU

0.0557 LY

15.963% GW₀

54 days

114.49 AU/dy

66.12% c

113.9 AU

3,636.7 AU

0.0575 LY

16.238% GW₀

55 days

115.66 AU/dy

66.80% c

115.1 AU

3,751.8 AU

0.0593 LY

16.513% GW₀

56 days

116.80 AU/dy

67.46% c

116.2 AU

3,868.0 AU

0.0612 LY

16.786% GW₀

57 days

117.92 AU/dy

68.11% c

117.4 AU

3,985.4 AU

0.0630 LY

17.059% GW₀

58 days

119.02 AU/dy

68.74% c

118.5 AU

4,103.8 AU

0.0649 LY

17.331% GW₀

59 days

120.10 AU/dy

69.36% c

119.6 AU

4,223.4 AU

0.0668 LY

17.601% GW₀

60 days

121.15 AU/dy

69.97% c

120.6 AU

4,344.0 AU

0.0687 LY

17.871% GW₀

Total Distance (D_LY) as decimal LY.
D_LY= D_AU/LY_AUNOTE: Light Year = 63,420 AUs = LY_AU
EXAMPLE: t = 61 days
D_LY= 4,465.7 AU/63,420 AU = 0.0706 LY

time

Velocity

Incr. Dist.

Total Distance

Total Fuel

61 days

122.19 AU/dy

70.57% c

121.7 AU

4,465.7 AU

0.0706 LY

18.140% GW₀

62 days

123.20 AU/dy

71.15% c

122.7 AU

4,588.4 AU

0.0726 LY

18.409% GW₀

63 days

124.20 AU/dy

71.73% c

123.7 AU

4,712.1 AU

0.0745 LY

18.676% GW₀

64 days

125.17 AU/dy

72.29% c

124.7 AU

4,836.8 AU

0.0765 LY

18.942% GW₀

65 days

126.13 AU/dy

72.84% c

125.6 AU

4,962.4 AU

0.0785 LY

19.208% GW₀

66 days

127.06 AU/dy

73.38% c

126.6 AU

5,089.0 AU

0.0805 LY

19.472% GW₀

67 days

127.98 AU/dy

73.91% c

127.5 AU

5,216.5 AU

0.0825 LY

19.736% GW₀

68 days

128.88 AU/dy

74.43% c

128.4 AU

5,345.0 AU

0.0845 LY

19.999% GW₀

69 days

129.76 AU/dy

74.94% c

129.3 AU

5,474.3 AU

0.0866 LY

20.261% GW₀

70 days

130.62 AU/dy

75.44% c

130.2 AU

5,604.5 AU

0.0886 LY

20.523% GW₀

71 days

131.47 AU/dy

75.93% c

131.0 AU

5,735.5 AU

0.0907 LY

20.783% GW₀

72 days

132.30 AU/dy

76.41% c

131.9 AU

5,867.4 AU

0.0928 LY

21.042% GW₀

73 days

133.12 AU/dy

76.88% c

132.7 AU

6,000.1 AU

0.0949 LY

21.301% GW₀

74 days

133.91 AU/dy

77.34% c

133.5 AU

6,133.6 AU

0.0970 LY

21.559% GW₀

75 days

134.70 AU/dy

77.79% c

134.3 AU

6,268.0 AU

0.0991 LY

21.816% GW₀

Given

(1 - (1-Δ)^t) ×	c

d_t - d_t-1

c × t +	V_t ln(1-Δ)

f_t = 1 - (1-ε∇)^t

76 days

135.46 AU/dy

78.24% c

135.1 AU

6,403.0 AU

0.1012 LY

22.072% GW₀

77 days

136.21 AU/dy

78.67% c

135.8 AU

6,538.9 AU

0.1034 LY

22.327% GW₀

78 days

136.95 AU/dy

79.10% c

136.6 AU

6,675.5 AU

0.1056 LY

22.582% GW₀

79 days

137.67 AU/dy

79.51% c

137.3 AU

6,812.8 AU

0.1077 LY

22.835% GW₀

80 days

138.38 AU/dy

79.92% c

138.0 AU

6,950.8 AU

0.1099 LY

23.088% GW₀

81 days

139.08 AU/dy

80.32% c

138.7 AU

7,089.5 AU

0.1121 LY

23.340% GW₀

82 days

139.76 AU/dy

80.72% c

139.4 AU

7,229.0 AU

0.1143 LY

23.591% GW₀

83 days

140.42 AU/dy

81.10% c

140.1 AU

7,369.0 AU

0.1165 LY

23.842% GW₀

84 days

141.08 AU/dy

81.48% c

140.7 AU

7,509.8 AU

0.1188 LY

24.091% GW₀

85 days

141.72 AU/dy

81.85% c

141.4 AU

7,651.2 AU

0.1210 LY

24.340% GW₀

86 days

142.34 AU/dy

82.21% c

142.0 AU

7,793.2 AU

0.1232 LY

24.588% GW₀

87 days

142.96 AU/dy

82.57% c

142.7 AU

7,935.9 AU

0.1255 LY

24.835% GW₀

88 days

143.56 AU/dy

82.91% c

143.3 AU

8,079.1 AU

0.1278 LY

25.081% GW₀

89 days

144.15 AU/dy

83.26% c

143.9 AU

8,223.0 AU

0.1300 LY

25.326% GW₀

90 days

144.73 AU/dy

83.59% c

144.4 AU

8,367.4 AU

0.1323 LY

25.571% GW₀

1G Acceleration for 1 year.
7G Acceleration for 100 days.
7G Deceleration for 48¼ days.
1G Deceleration for 1 year.

time

Velocity

Incr. Dist.

Total Distance

Total Fuel

91 days

145.30 AU/dy

83.92% c

145.1 AU

8,512.5 AU

0.1346 LY

25.815% GW₀

92 days

145.86 AU/dy

84.24% c

145.6 AU

8,658.0 AU

0.1369 LY

26.058% GW₀

93 days

146.40 AU/dy

84.56% c

146.1 AU

8,804.2 AU

0.1392 LY

26.300% GW₀

94 days

146.94 AU/dy

84.86% c

146.7 AU

8,950.8 AU

0.1415 LY

26.541% GW₀

95 days

147.46 AU/dy

85.17% c

147.2 AU

9,098.0 AU

0.1439 LY

26.782% GW₀

96 days

147.98 AU/dy

85.46% c

147.7 AU

9,245.8 AU

0.1462 LY

27.022% GW₀

97 days

148.48 AU/dy

85.76% c

148.2 AU

9,394.0 AU

0.1485 LY

27.261% GW₀

98 days

148.98 AU/dy

86.04% c

148.7 AU

9,542.7 AU

0.1509 LY

27.499% GW₀

99 days

149.46 AU/dy

86.32% c

149.2 AU

9,691.9 AU

0.1533 LY

27.737% GW₀

100 days

149.93 AU/dy

86.59% c

149.7 AU

9,841.6 AU

0.1556 LY

27.974% GW₀

Given

(1 - (1-Δ)^t) ×	c

d_t - d_t-1

c × t +	V_t ln(1-Δ)

f_t = 1 - (1-ε∇)^t

1) FUEL FACTOR: Exhaust flow, ∇

Assume exhaust particle speed is 99% light speed (.99c).

Thus, ship needs exhaust flow, ∇= 0.2492% ship’s GW

for each day of powered flight (%GW/day),

for 7G acceleration.

2) FUEL FACTOR: Efficiency Factor, ε

Compensate for inevitable inefficiencies by increasing ∇ by factor, ε.

Compute daily fuel consumption rate:

ε∇ =0.003394 = .3394% GW/day.

1-ε∇ = 0.9966 = daily Remainder of ship's GW.

SLIDESHOW

VOLUME 0: ELEVATIONAL
VOLUME I: ASTEROIDAL
VOLUME II: INTERPLANETARY
VOLUME III: INTERSTELLAR

A Thought Experiment

Monday, January 29, 2007

7G TABLE: Accelerate 100 Days

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