STANDARD TIME DILATION TO AC

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CONTENT
1) G-FORCE FOR A SEC
2) G-FORCE FOR A DAY
3) FOR A YEAR
4) CRUISE FOR YEARS
5) FINAL YEAR, REVERSE G-FORCE
6) CRUISE TIME DILATION
7) ACCEL. TIME DILATION
a. Overall Average.
b. Average 2 Halves.
c. Average 4 Quarters.
d. Average 8 Parts.
e. Average 18 Parts.
f. Average 360 Days.
8) TOTAL TIME DILATION
CONSIDER 500 DAYS
VESSEL OBSERVERS

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tr = t ×  √(1 – dc2) let dc=VShip/c Time Shrinks at relativistic speeds.  Observers on very high speed vessels will measure time at a slower rate than "at rest" observers back on Earth. EXAMPLE: After 1 year of g-force propulsion, vessel velocity is 65% light speed (.65c). Then, let an Earth bound observer spend 1 day watching this vessel, observer on the vessel would measure a duration of only .76 days. Normal Time Vehicle Velocity Dilated Time t (days) Spot Vel (AU/dy) Ave Vel (AU/dy) Ave dc (dec. c) Incr. (days) Cum. (days) 0 days 0 0 0 c 0 days 0 days 1 days 0.489 0.245 0.0014 c 1.00000 days 1.0000 days 2 days 0.977 0.733 0.0042 c 0.99999 days 2.0000 days 3 days 1.464 1.221 0.0070 c 0.99998 days 3.0000 days 4 days 1.949 1.706 0.0099 c 0.99995 days 3.9999 days 5 days 2.433 2.191 0.0127 c 0.99992 days 4.9998 days 6 days 2.915 2.674 0.0154 c 0.99988 days 5.9997 days 7 days 3.396 3.156  0.0182 c 0.99983 days 6.9996 days 8 days 3.876 3.636 0.0210 c 0.99978 days 7.9993 days 9 days 4.354 4.11 0.0238 c 0.99972 days 8.9990 days 10 days 4.831 4.593 0.0265 c 0.99965 days 9.9987 days 11 days 5.307 5.069 0.0293 c 0.99957 days 10.9983 days 12 days 5.781 5.544 0.0320 c 0.99949 days 11.9978 days 13 days 6.254 6.018 0.0348 c 0.99940 days 12.9972 days 14 days 6.726 6.490 0.0375 c 0.99930 days 13.9964 days 15 days 7.196 6.961 0.0402 c 0.99919 days 14.9956 days t c[1 - Rct] (Vt-1+Vt) 2 VAve 173.145  √(1 - dc2) ΣIncr. 16 days 7.665 7.431 0.0429 c 0.99908 days 15.9947 days 17 days 8.133 7.899 0.0456 c 0.99896 days 16.9937 days 18 days 8.599 8.366 0.0483 c 0.99883 days 17.9925 days 19 days 9.064 8.832 0.0510 c 0.99870 days 18.9912 days 20 days 9.528 9.296 0.0537 c 0.99856 days 19.9898 days 21 days 9.990 9.759 0.0564 c 0.99841 days 20.9882 days 22 days 10.451 10.221 0.0590 c 0.99826 days 21.9864 days 23 days 10.911 10.681 0.0617 c 0.99810 days 22.9845 days 24 days 11.370 11.140 0.0643 c 0.99793 days 23.9825 days 25 days 11.827 11.598 0.0670 c 0.99775 days 24.9802 days 26 days 12.283 12.055 0.0696 c 0.99757 days 25.9778 days 27 days 12.737 12.510 0.0723 c 0.99739 days 26.9752 days 28 days 13.191 12.964 0.0749 c 0.99719 days 27.9724 days 29 days 13.643 13.417 0.0775 c 0.99699 days 28.9694 days 30 days 14.093 13.868 0.0801 c 0.99679 days 29.9661 days t (days) Spot Vel (AU/dy) Ave Vel (AU/dy) Ave dc (dec. c) Incr. (days) Cum. (days) 31 days 14.543 14.318 0.0827 c 0.99657 days 30.9627 days 32 days 14.991 14.767 0.0853 c 0.99636 days 31.9591 days 33 days 15.438 15.214 0.0879 c 0.99613 days 32.9552 days 34 days 15.884 15.661 0.0904 c 0.99590 days 33.9511 days 35 days 16.328 16.106 0.0930 c 0.99566 days 34.9468 days 36 days 16.771 16.550 0.0956 c 0.99542 days 35.9422 days 37 days 17.213 16.992 0.0981 c 0.99517 days 36.9374 days 38 days 17.654 17.433 0.1007 c 0.99492 days 37.9323 days 39 days 18.093 17.873 0.1032 c 0.99466 days 38.9269 days 40 days 18.531 18.312 0.1058 c 0.99439 days 39.9213 days 41 days 18.968 18.750 0.1083 c 0.99412 days 40.9155 days 42 days 19.404 19.186 0.1108 c 0.99384 days 41.9093 days 43 days 19.838 19.621 0.1133 c 0.99356 days 42.9029 days 44 days 20.272 20.055 0.1158 c 0.99327 days 43.8961 days 45 days 20.704 20.488 0.1183 c 0.99297 days 44.8891 days t c[1 - Rct] (Vt-1+Vt) 2 VAve 173.145  √(1 - dc2) ΣIncr. 46 days 21.135 20.919 0.1208 c 0.99267 days 45.8818 days 47 days 21.564 21.349 0.1233 c 0.99237 days 46.8741 days 48 days 21.992 21.778 0.1258 c 0.99206 days 47.8662 days 49 days 22.420 22.206 0.1283 c 0.99174 days 48.8579 days 50 days 22.846 22.633 0.1307 c 0.99142 days 49.8494 days 51 days 23.270 23.058 0.1332 c 0.99109 days 50.8405 days 52 days 23.694  23.482 0.1356 c 0.99076 days 51.8312 days 53 days 24.116 23.905 0.1381 c 0.99042 days 52.8216 days 54 days 24.537 24.327 0.1405 c 0.99008 days 53.8117 days 55 days 24.957 24.747 0.1429 c 0.98973 days 54.8015 days 56 days 25.376 25.167 0.1454 c 0.98938 days 55.7908 days 57 days 25.794 25.585 0.1478 c 0.98902 days 56.7799 days 58 days 26.210 26.002 0.1502 c 0.98866 days 57.7685 days 59 days 26.625 26.418 0.1526 c 0.98829 days 58.7568 days 60 days 27.039 26.832 0.1550 c 0.98792 days 59.7447 days t (days) Spot Vel (AU/dy) Ave Vel (AU/dy) Ave dc (dec. c) Incr. (days) Cum. (days) 61 days 27.452 27.246 0.1574 c 0.98754 days 60.7323 days 62 days 27.864 27.658 0.1597 c 0.98716 days 61.7194 days 63 days 28.275 28.069 0.1621 c 0.98677 days 62.7062 days 64 days 28.684 28.479 0.1645 c 0.98638 days 63.6926 days 65 days 29.092 28.888 0.1668 c 0.98598 days 64.6786 days 66 days 29.499 29.296 0.1692 c 0.98558 days 65.6641 days 67 days 29.905 29.702 0.1715 c 0.98518 days 66.6493 days 68 days 30.310 30.108 0.1739 c 0.98477 days 67.6341 days 69 days 30.714 30.512 0.1762 c 0.98435 days 68.6184 days 70 days 31.116 30.915 0.1785 c 0.98393 days 69.6024 days 71 days 31.518 31.317 0.1809 c 0.98351 days 70.5859 days 72 days 31.918 31.718 0.1832 c 0.98308 days 71.5689 days 73 days 32.317 32.117 0.1855 c 0.98265 days 72.5516 days 74 days 32.715 32.516 0.1878 c 0.98221 days 73.5338 days 75 days 33.112 32.913 0.1901 c 0.98177 days 74.5156 days t c[1 - Rct] (Vt-1+Vt) 2 VAve 173.145  √(1 - dc2) ΣIncr. 76 days 33.508 AU/day 33.310 AU/day 0.1924 c 0.98132 days 75.4969 days 77 days 33.902 AU/day 33.705 AU/day 0.1947 c 0.98087 days 76.4778 days 78 days 34.296 AU/day 34.099 AU/day 0.1969 c 0.98042 days 77.4582 days 79 days 34.688 AU/day 34.492 AU/day 0.1992 c 0.97996 days 78.4381 days 80 days 35.079 AU/day 34.884 AU/day 0.2015 c 0.97949 days 79.4176 days 81 days 35.470 AU/day 35.274 AU/day 0.2037 c 0.97903 days 80.3967 days 82 days 35.859 AU/day 35.664 AU/day 0.2060 c 0.97856 days 81.3752 days 83 days 36.247 AU/day 36.053 AU/day 0.2082 c 0.97808 days 82.3533 days 84 days 36.633 AU/day 36.440 AU/day 0.2105 c 0.97760 days 83.3309 days 85 days 37.019 AU/day 36.826 AU/day 0.2127 c 0.97712 days 84.3080 days 86 days 37.404 AU/day 37.212 AU/day 0.2149 c 0.97663 days 85.2846 days 87 days 37.788 AU/day 37.596 AU/day 0.2171 c 0.97614 days 86.2608 days 88 days 38.170 AU/day 37.979 AU/day 0.2193 c 0.97565 days 87.2364 days 89 days 38.551 AU/day 38.361 AU/day 0.2216 c 0.97515 days 88.2116 days 90 days 38.932 AU/day 38.742 AU/day 0.2238 c 0.97465 days 89.1862 days t (days) Spot Vel (AU/dy) Ave Vel (AU/dy) Ave dc (dec. c) Incr. (days) Cum. (days) 91 days 39.311 39.121 0.2259 c 0.97414 days 90.1604 days 92 days 39.689 39.500 0.2281 c 0.97363 days 91.1340 days 93 days 40.066 39.878 0.2303 c 0.97312 days 92.1071 days 94 days 40.443 40.255 0.2325 c 0.97260 days 93.0797 days 95 days 40.818 40.630 0.2347 c 0.97208 days 94.0518 days 96 days 41.192 41.005 0.2368 c 0.97155 days 95.0233 days 97 days 41.564 41.378 0.2390 c 0.97102 days 95.9944 days 98 days 41.936 41.750 0.2411 c 0.97049 days 96.9649 days 99 days 42.307 42.122 0.2433 c 0.96996 days 97.9348 days 100 days 42.677 42.492 0.2454 c 0.96942 days 98.9042 days 101 days 43.046 42.861 0.2475 c 0.96888 days 99.8731 days 102 days 43.413 43.229 0.2497 c 0.96833 days 100.8414 days 103 days 43.780 43.597 0.2518 c 0.96778 days 101.8092 days 104 days 44.145 43.96 0.2539 c 0.96723 days 102.7765 days 105 days 44.510 44.328 0.2560 c 0.96667 days 103.7431 days t c[1 - Rct] (Vt-1+Vt) 2 VAve 173.145  √(1 - dc2) ΣIncr. 106 days 44.873 44.692 0.2581 c 0.96611 days 104.7092 days 107 days 45.236 45.055 0.2602 c 0.96555 days 105.6748 days 108 days 45.597 45.417 0.2623 c 0.96499 days 106.6398 days 109 days 45.958 45.778 0.2644 c 0.96442 days 107.6042 days 110 days 46.317 46.138 0.2665 c 0.96384 days 108.5680 days 111 days 46.676 46.497 0.2685 c 0.96327 days 109.5313 days 112 days 47.033 46.854 0.2706 c 0.96269 days 110.4940 days 113 days 47.390 47.211 0.2727 c 0.96211 days 111.4561 days 114 days 47.745 47.567 0.2747 c 0.96152 days 112.4176 days 115 days 48.099 47.922 0.2768 c 0.96093 days 113.3786 days 116 days 48.453 48.276 0.2788 c 0.96034 days 114.3389 days 117 days 48.805 48.629 0.2809 c 0.95975 days 115.2987 days 118 days 49.156 48.981 0.2829 c 0.95915 days 116.2578 days 119 days 49.507 49.332 0.2849 c 0.95855 days 117.2164 days 120 days 49.856 49.682 0.2869 c 0.95795 days 118.1743 days t (days) Spot Vel (AU/dy) Ave Vel (AU/dy) Ave dc (dec. c) Incr. (days) Cum. (days) 121 days 50.205 50.030 0.2890 c 0.95734 days 119.1317 days 122 days 50.552 50.378 0.2910 c 0.95674 days 120.0884 days 123 days 50.899 50.725 0.2930 c 0.95612 days 121.0445 days 124 days 51.244 51.071 0.2950 c 0.95551 days 122.0000 days 125 days 51.588 51.416 0.2970 c 0.95489 days 122.9549 days 126 days 51.932 51.760 0.2989 c 0.95427 days 123.9092 days 127 days 52.275 52.103 0.3009 c 0.95365 days 124.8628 days 128 days 52.616 52.445 0.3029 c 0.95302 days 125.8159 days 129 days 52.957 52.786 0.3049 c 0.95239 days 126.7683 days 130 days 53.296 53.127 0.3068 c 0.95176 days 127.7200 days 131 days 53.635 53.466 0.3088 c 0.95113 days 128.6711 days 132 days 53.973 53.804 0.3107 c 0.95049 days 129.6216 days 133 days 54.310 54.141 0.3127 c 0.94985 days 130.5715 days 134 days 54.645 54.478 0.3146 c 0.94921 days 131.5207 days 135 days 54.980 54.813 0.3166 c 0.94857 days 132.4693 days t c[1 - Rct] (Vt-1+Vt) 2 VAve 173.145  √(1 - dc2) ΣIncr. 136 days 55.314 55.147 0.3185 c 0.94792 days 133.4172 days 137 days 55.647 55.481 0.3204 c 0.94727 days 134.3645 days 138 days 55.979 55.813 0.3223 c 0.94662 days 135.3111 days 139 days 56.310 56.145 0.3243 c 0.94597 days 136.2571 days 140 days 56.641 56.475 0.3262 c 0.94531 days 137.2024 days 141 days 56.970 56.805 0.3281 c 0.94465 days 138.1470 days 142 days 57.298 57.134 0.3300 c 0.94399 days 139.0910 days 143 days 57.626 57.462 0.3319 c 0.94332 days 140.0343 days 144 days 57.952 57.789 0.3338 c 0.94266 days 140.9770 days 145 days 58.277 58.115 0.3356 c 0.94199 days 141.9190 days 146 days 58.602 58.440 0.3375 c 0.94132 days 142.8603 days 147 days 58.926 58.764 0.3394 c 0.94065 days 143.8009 days 148 days 59.249 59.087 0.3413 c 0.93997 days 144.7409 days 149 days 59.570 59.410 0.3431 c 0.93929 days 145.6802 days 150 days 59.891 59.731 0.3450 c 0.93861 days 146.6188 days t (days) Spot Vel (AU/dy) Ave Vel (AU/dy) Ave dc (dec. c) Incr. (days) Cum. (days) 151 days 60.211 60.051 0.3468 c 0.93793 days 147.5567 days 152 days 60.531 60.371 0.3487 c 0.93724 days 148.4940 days 153 days 60.849 60.690 0.3505 c 0.93656 days 149.4305 days 154 days 61.166 61.008 0.3523 c 0.93587 days 150.3664 days 155 days 61.483 61.324 0.3542 c 0.93518 days 151.3016 days 156 days 61.798 61.640 0.3560 c 0.93448 days 152.2361 days 157 days 62.113 61.956 0.3578 c 0.93379 days 153.1699 days 158 days 62.427 62.270 0.3596 c 0.93309 days 154.1030 days 159 days 62.740 62.583 0.3614 c 0.93239 days 155.0353 days 160 days 63.052 62.896 0.3633 c 0.93169 days 155.9670 days 161 days 63.363 63.207 0.3651 c 0.93099 days 156.8980 days 162 days 63.673 63.518 0.3668 c 0.93028 days 157.8283 days 163 days 63.982 63.828 0.3686 c 0.92957 days 158.7579 days 164 days 64.291 64.137 0.3704 c 0.92886 days 159.6867 days 165 days 64.598 64.445 0.3722 c 0.92815 days 160.6149 days t c[1 - Rct] (Vt-1+Vt) 2 VAve 173.145  √(1 - dc2) ΣIncr. 166 days 64.905 64.752 0.3740 c 0.92744 days 161.5423 days 167 days 65.211 65.058 0.3757 c 0.92672 days 162.4691 days 168 days 65.516 65.364 0.3775 c 0.92601 days 163.3951 days 169 days 65.820 65.668 0.3793 c 0.92529 days 164.3203 days 170 days 66.124 65.972 0.3810 c 0.92457 days 165.2449 days 171 days 66.426 66.275 0.3828 c 0.92384 days 166.1688 days 172 days 66.728 66.577 0.3845 c 0.92312 days 167.0919 days 173 days 67.028 66.878 0.3863 c 0.92239 days 168.0143 days 174 days 67.328 67.178 0.3880 c 0.92166 days 168.9359 days 175 days 67.627 67.478 0.3897 c 0.92093 days 169.8569 days 176 days 67.925 67.776 0.3914 c 0.92020 days 170.7771 days 177 days 68.223 68.074 0.3932 c 0.91947 days 171.6965 days 178 days 68.519 68.371 0.3949 c 0.91873 days 172.6153 days 179 days 68.815 68.667 0.3966 c 0.91800 days 173.5333 days 180 days 69.110 68.962 0.3983 c 0.91726 days 174.4505 days t (days) Spot Vel (AU/dy) Ave Vel (AU/dy) Ave dc (dec. c) Incr. (days) Cum. (days) 181 days 69.404 69.257 0.4000 c 0.91652 days 175.3670 days 182 days 69.697 69.550 0.4017 c 0.91578 days 176.2828 days 183 days 69.989 69.843 0.4034 c 0.91503 days 177.1979 days 184 days 70.281 70.135 0.4051 c 0.91429 days 178.1121 days 185 days 70.572 70.426 0.4067 c 0.91354 days 179.0257 days 186 days 70.861 70.716 0.4084 c 0.91279 days 179.9385 days 187 days 71.150 71.006 0.4101 c 0.91204 days 180.8505 days 188 days 71.439 71.295 0.4118 c 0.91129 days 181.7618 days 189 days 71.726 71.582 0.4134 c 0.91054 days 182.6724 days 190 days 72.013 71.869 0.4151 c 0.90978 days 183.5821 days 191 days 72.299 72.156 0.4167 c 0.90903 days 184.4912 days 192 days 72.584 72.441 0.4184 c 0.90827 days 185.3994 days 193 days 72.868 72.726 0.4200 c 0.90751 days 186.3069 days 194 days 73.151 73.009 0.4217 c 0.90675 days 187.2137 days 195 days 73.434 73.292 0.4233 c 0.90599 days 188.1197 days t c[1 - Rct] (Vt-1+Vt) 2 VAve 173.145  √(1 - dc2) ΣIncr. 196 days 73.715 73.575 0.4249 c 0.90523 days 189.0249 days 197 days 73.996 73.856 0.4266 c 0.90446 days 189.9294 days 198 days 74.277 74.137 0.4282 c 0.90370 days 190.8331 days 199 days 74.556 74.416 0.4298 c 0.90293 days 191.7360 days 200 days 74.835 74.695 0.4314 c 0.90216 days 192.6382 days 201 days 75.112 74.974 0.4330 c 0.90139 days 193.5395 days 202 days 75.390 75.251 0.4346 c 0.90062 days 194.4402 days 203 days 75.666 75.528 0.4362 c 0.89984 days 195.3400 days 204 days 75.941 75.804 0.4378 c 0.89907 days 196.2391 days 205 days 76.216 76.079 0.4394 c 0.89830 days 197.1374 days 206 days 76.490 76.353 0.4410 c 0.89752 days 198.0349 days 207 days 76.763 76.626 0.4426 c 0.89674 days 198.9316 days 208 days 77.035 76.899 0.4441 c 0.89596 days 199.8276 days 209 days 77.307 77.171 0.4457 c 0.89518 days 200.7228 days 210 days 77.578 77.442 0.4473 c 0.89440 days 201.6172 days t (days) Spot Vel (AU/dy) Ave Vel (AU/dy) Ave dc (dec. c) Incr. (days) Cum. (days) 211 days 77.848 77.713 0.4488 c 0.89362 days 202.5108 days 212 days 78.117 77.983 0.4504 c 0.89283 days 203.4036 days 213 days 78.386 78.251 0.4519 c 0.89205 days 204.2957 days 214 days 78.654 78.520 0.4535 c 0.89126 days 205.1869 days 215 days 78.921 78.787 0.4550 c 0.89047 days 206.0774 days 216 days 79.187 79.054 0.4566 c 0.88968 days 206.9671 days 217 days 79.452 79.320 0.4581 c 0.88889 days 207.8560 days 218 days 79.717 79.585 0.4596 c 0.88810 days 208.7441 days 219 days 79.981 79.849 0.4612 c 0.88731 days 209.6314 days 220 days 80.244 80.113 0.4627 c 0.88652 days 210.5179 days 221 days 80.507 80.376 0.4642 c 0.88572 days 211.4036 days 222 days 80.769 80.638 0.4657 c 0.88493 days 212.2886 days 223 days 81.030 80.899 0.4672 c 0.88413 days 213.1727 days 224 days 81.290 81.160 0.4687 c 0.88334 days 214.0560 days 225 days 81.550 81.420 0.4702 c 0.88254 days 214.9386 days t c[1 - Rct] (Vt-1+Vt) 2 VAve 173.145  √(1 - dc2) ΣIncr. 226 days 81.809 81.679 0.4717 c 0.88174 days 215.8203 days 227 days 82.067 81.938 0.4732 c 0.88094 days 216.7013 days 228 days 82.324 82.195 0.4747 c 0.88014 days 217.5814 days 229 days 82.581 82.452 0.4762 c 0.87933 days 218.4607 days 230 days 82.837 82.709 0.4777 c 0.87853 days 219.3393 days 231 days 83.092 82.964 0.4792 c 0.87773 days 220.2170 days 232 days 83.346 83.219 0.4806 c 0.87692 days 221.0939 days 233 days 83.600 83.473 0.4821 c 0.87612 days 221.9700 days 234 days 83.853 83.727 0.4836 c 0.87531 days 222.8453 days 235 days 84.106 83.979 0.4850 c 0.87450 days 223.7198 days 236 days 84.357 84.231 0.4865 c 0.87369 days 224.5935 days 237 days 84.608 84.483 0.4879 c 0.87288 days 225.4664 days 238 days 84.858 84.733 0.4894 c 0.87207 days 226.3385 days 239 days 85.108 84.983 0.4908 c 0.87126 days 227.2097 days 240 days 85.357 85.232 0.4923 c 0.87045 days 228.0802 days t (days) Spot Vel (AU/dy) Ave Vel (AU/dy) Ave dc (dec. c) Incr. (days) Cum. (days) 242 days 85.852 85.728 0.4951 c 0.86882 days 229.8186 days 243 days 86.099 85.975 0.4966 c 0.86801 days 230.6867 days 244 days 86.345 86.222 0.4980 c 0.86719 days 231.5538 days 245 days 86.590 86.467 0.4994 c 0.86638 days 232.4202 days 246 days 86.835 86.712 0.5008 c 0.86556 days 233.2858 days 247 days 87.079 86.957 0.5022 c 0.86474 days 234.1505 days 248 days 87.322 87.200 0.5036 c 0.86392 days 235.0144 days 249 days 87.564 87.443 0.5050 c 0.86310 days 235.8775 days 250 days 87.806 87.685 0.5064 c 0.86228 days 236.7398 days 251 days 88.047 87.927 0.5078 c 0.86146 days 237.6013 days 252 days 88.288 88.168 0.5092 c 0.86064 days 238.4619 days 253 days 88.528 88.408 0.5106 c 0.85982 days 239.3218 days 254 days 88.767 88.647 0.5120 c 0.85900 days 240.1807 days 255 days 89.005 88.886 0.5134 c 0.85817 days 241.0389 days t c[1 - Rct] (Vt-1+Vt) 2 VAve 173.145  √(1 - dc2) ΣIncr. 256 days 89.243 89.124 0.5147 c 0.85735 days 241.8963 days 257 days 89.480 89.362 0.5161 c 0.85652 days 242.7528 days 258 days 89.717 89.598 0.5175 c 0.85570 days 243.6085 days 259 days 89.952 89.834 0.5188 c 0.85487 days 244.4634 days 260 days 90.187 90.070 0.5202 c 0.85404 days 245.3174 days 261 days 90.422 90.305 0.5216 c 0.85322 days 246.1706 days 262 days 90.656 90.539 0.5229 c 0.85239 days 247.0230 days 263 days 90.889 90.772 0.5243 c 0.85156 days 247.8746 days 264 days 91.121 91.005 0.5256 c 0.85073 days 248.7253 days 265 days 91.353 91.237 0.5269 c 0.84990 days 249.5752 days 266 days 91.584 91.469 0.5283 c 0.84907 days 250.4243 days 267 days 91.815 91.699 0.5296 c 0.84824 days 251.2725 days 268 days 92.044 91.930 0.5309 c 0.84741 days 252.1199 days 269 days 92.274 92.159 0.5323 c 0.84658 days 252.9665 days 270 days 92.502 92.388 0.5336 c 0.84575 days 253.8123 days t (days) Spot Vel (AU/dy) Ave Vel (AU/dy) Ave dc (dec. c) Incr. (days) Cum. (days) 272 days 92.957 92.844 0.5362 c 0.84408 days 255.5012 days 273 days 93.184 93.071 0.5375 c 0.84324 days 256.3445 days 274 days 93.410 93.297 0.5388 c 0.84241 days 257.1869 days 275 days 93.635 93.523 0.5401 c 0.84157 days 258.0285 days 276 days 93.860 93.748 0.5414 c 0.84074 days 258.8692 days 277 days 94.084 93.972 0.5427 c 0.83990 days 259.7091 days 278 days 94.307 94.196 0.5440 c 0.83907 days 260.5482 days 279 days 94.530 94.419 0.5453 c 0.83823 days 261.3864 days 280 days 94.752 94.641 0.5466 c 0.83739 days 262.2238 days 281 days 94.974 94.863 0.5479 c 0.83655 days 263.0604 days 282 days 95.195 95.084 0.5492 c 0.83572 days 263.8961 days 283 days 95.415 95.305 0.5504 c 0.83488 days 264.7310 days 284 days 95.635 95.525 0.5517 c 0.83404 days 265.5650 days 285 days 95.854 95.744 0.5530 c 0.83320 days 266.3982 days t c[1 - Rct] (Vt-1+Vt) 2 VAve 173.145  √(1 - dc2) ΣIncr. 286 days 96.072 95.963 0.5542 c 0.83236 days 267.2306 days 287 days 96.290 96.181 0.5555 c 0.83152 days 268.0621 days 288 days 96.507 96.399 0.5568 c 0.83068 days 268.8928 days 289 days 96.724 96.616 0.5580 c 0.82984 days 269.7226 days 290 days 96.940 96.832 0.5593 c 0.82900 days 270.5516 days 291 days 97.155 97.048 0.5605 c 0.82816 days 271.3797 days 292 days 97.370 97.263 0.5617 c 0.82731 days 272.2071 days 293 days 97.584 97.477 0.5630 c 0.82647 days 273.0335 days 294 days 97.798 97.691 0.5642 c 0.82563 days 273.8592 days 295 days 98.011 97.904 0.5654 c 0.82479 days 274.6839 days 296 days 98.223 98.117 0.5667 c 0.82394 days 275.5079 days 297 days 98.435 98.329 0.5679 c 0.82310 days 276.3310 days 298 days 98.646 98.540 0.5691 c 0.82225 days 277.1532 days 299 days 98.856 98.751 0.5703 c 0.82141 days 277.9746 days 300 days 99.066 98.961 0.5716 c 0.82057 days 278.7952 days t (days) Spot Vel (AU/dy) Ave Vel (AU/dy) Ave dc (dec. c) Incr. (days) Cum. (days) 302 days 99.484 99.380 0.5740 c 0.81888 days 280.4338 days 303 days 99.692 99.588 0.5752 c 0.81803 days 281.2518 days 304 days 99.900 99.796 0.5764 c 0.81719 days 282.0690 days 305 days 100.107 100.004 0.5776 c 0.81634 days 282.8854 days 306 days 100.313 100.210 0.5788 c 0.81549 days 283.7009 days 307 days 100.519 100.416 0.5800 c 0.81465 days 284.5155 days 308 days 100.725 100.622 0.5811 c 0.81380 days 285.3293 days 309 days 100.929 100.827 0.5823 c 0.81296 days 286.1423 days 310 days 101.133 101.031 0.5835 c 0.81211 days 286.9544 days 311 days 101.337 101.235 0.5847 c 0.81126 days 287.7656 days 312 days 101.540 101.438 0.5859 c 0.81041 days 288.5761 days 313 days 101.742 101.641 0.5870 c 0.80957 days 289.3856 days 314 days 101.944 101.843 0.5882 c 0.80872 days 290.1943 days 315 days 102.145 102.044 0.5894 c 0.80787 days 291.0022 days t c[1 - Rct] (Vt-1+Vt) 2 VAve 173.145  √(1 - dc2) ΣIncr. 316 days 102.346 102.245 0.5905 c 0.80702 days 291.8092 days 317 days 102.546 102.446 0.5917 c 0.80618 days 292.6154 days 318 days 102.745 102.646 0.5928 c 0.80533 days 293.4207 days 319 days 102.944 102.845 0.5940 c 0.80448 days 294.2252 days 320 days 103.143 103.043 0.5951 c 0.80363 days 295.0289 days 321 days 103.340 103.242 0.5963 c 0.80278 days 295.8316 days 322 days 103.538 103.439 0.5974 c 0.80193 days 296.6336 days 323 days 103.734 103.636 0.5986 c 0.80108 days 297.4347 days 324 days 103.931 103.833 0.5997 c 0.80024 days 298.2349 days 325 days 104.126 104.028 0.6008 c 0.79939 days 299.0343 days 326 days 104.321 104.224 0.6019 c 0.79854 days 299.8328 days 327 days 104.516 104.418 0.6031 c 0.79769 days 300.6305 days 328 days 104.710 104.613 0.6042 c 0.79684 days 301.4273 days 329 days 104.903 104.806 0.6053 c 0.79599 days 302.2233 days 330 days 105.096 104.999 0.6064 c 0.79514 days 303.0185 days t (days) Spot Vel (AU/dy) Ave Vel (AU/dy) Ave dc (dec. c) Incr. (days) Cum. (days) 333 days 105.671 105.576 0.6098 c 0.79259 days 305.3988 days 334 days 105.862 105.767 0.6109 c 0.79174 days 306.1905 days 335 days 106.052 105.957 0.6120 c 0.79089 days 306.9814 days 336 days 106.242 106.147 0.6131 c 0.79004 days 307.7715 days 337 days 106.431 106.336 0.6141 c 0.78919 days 308.5607 days 338 days 106.619 106.525 0.6152 c 0.78834 days 309.3490 days 339 days 106.807 106.713 0.6163 c 0.78749 days 310.1365 days 340 days 106.995 106.901 0.6174 c 0.78664 days 310.9231 days 341 days 107.182 107.088 0.6185 c 0.78579 days 311.7089 days 342 days 107.368 107.275 0.6196 c 0.78494 days 312.4939 days 343 days 107.554 107.461 0.6206 c 0.78409 days 313.2780 days 344 days 107.739 107.647 0.6217 c 0.78324 days 314.0612 days 345 days 107.924 107.832 0.6228 c 0.78239 days 314.8436 days t c[1 - Rct] (Vt-1+Vt) 2 VAve 173.145  √(1 - dc2) ΣIncr. 346 days 108.108 108.016 0.6238 c 0.78155 days 315.6252 days 347 days 108.292 108.200 0.6249 c 0.78070 days 316.4059 days 348 days 108.476 108.384 0.6260 c 0.77985 days 317.1857 days 349 days 108.658 108.567 0.6270 c 0.77900 days 317.9647 days 350 days 108.841 108.749 0.6281 c 0.77815 days 318.7428 days 351 days 109.022 108.931 0.6291 c 0.77730 days 319.5201 days 352 days 109.203 109.113 0.6302 c 0.77645 days 320.2966 days 353 days 109.384 109.294 0.6312 c 0.77560 days 321.0722 days 354 days 109.564 109.474 0.6323 c 0.77475 days 321.8469 days 355 days 109.744 109.654 0.6333 c 0.77390 days 322.6208 days 356 days 109.923 109.834 0.6343 c 0.77305 days 323.3939 days 357 days 110.102 110.013 0.6354 c 0.77220 days 324.1661 days 358 days 110.280 110.191 0.6364 c 0.77135 days 324.9374 days 359 days 110.458 110.369 0.6374 c 0.77050 days 325.7079 days 360 days 110.635 110.546 0.6385 c 0.76965 days 326.4776 days t (days) Spot Vel (AU/dy) Ave Vel (AU/dy) Ave dc (dec. c) Incr. (days) Cum. (days) 361 days 110.811 110.723 0.6395 c 0.76881 days 327.2464 days 362 days 110.988 110.900 0.6405 c 0.76796 days 328.0144 days 363 days 111.163 111.075 0.6415 c 0.76711 days 328.7815 days 364 days 111.338 111.251 0.6425 c 0.76626 days 329.5477 days 365 days 111.513 111.426 0.6435 c 0.76541 days 330.3131 days t c[1 - Rct] (Vt-1+Vt) 2 VAve 173.145  √(1 - dc2) ΣIncr.

1) G-FORCE FOR A SECOND Near Earth's surface, let a material object free fall for one second, and it will gain a velocity of 9.8065 m/sec (about 22 miles per hour). Let it continue free fall for a 2nd second, and velocity will become about 20 m/sec. Velocity will increase an additional 9.8 m/sec for every added second of free fall. 2) G-FORCE FOR A DAY One day has 86,400 seconds. 24 hours day × 60 min hour × 60 sec min = 86,400 sec day If a vessel can g-force accelerate for an entire day, it will achieve a very high velocity. 9.8065 m sec × 86,400 sec day = 847,281.6 m sec Conveniently convert into other equivalent values. 847,281.6 m sec × km 1,000 m = 847.28 km sec (By comparison, typical asteroid travels around the Sun at 25 km/sec.) TE further converts into AUs per day. 847.28 km sec × 86,400 sec day × AU 149,597,871 km = .489 day (Since planetary distances are typically measured in Astronomical Units (AUs), approximate value of .5 AU/day is very convenient.) Finally, TE converts to percent light speed per day as shown 847.28 km sec × c 299,792.458 km/sec = 0.2826%c = Δ (After one day of g=force, vessel achieves a small portion of light speed. Thought Experiment (TE) often calls this percentage the daily difference, Δ.) 3) G-FORCE FOR A YEAR In contrast to the daily difference (Δ), consider the daily remainder (Rc): Rc = c - Δ = .997174c is the gigantic portion of light speed not yet achieved after one day of g-force. Einstein tells us in his Special Theory of Relativity that light speed (c=299,792.458 km/sec) is fixed regardless of observer's speed.  Thus, Earth observer and vessel observer both observe c as same value even though Earth observer measures continuously, increasing velocity of g-force vessel. TE reconciles this paradox via following exponential which calculates vessel velocities which forever approach c but never get there. VShip = c[1 - Rct]  To obtain velocities as AU/day, a convenient measure for interplanetary travel, restate value of c: c = 299,792.458 km sec × 86,400 sec day × AU 149,597,871 km = 173.145 AU day EXAMPLE:  After one year of g-force acceleration: VShip = 173.145 AU day × [1 - .997174365] = 111.513 AU day = 64.4%c 4) CRUISE FOR SEVERAL YEARS To conserve fuel, vessel then terminates propulsion and maintains earth like gravity via centrifugal force by spinning vessel about its longitudinal axis at carefully planned angular velocity. 5) REVERSE G-FORCE FOR FINAL YEAR During the cruise phase, the vessel makes excellent time due to its relativistic speed (64.43%c = 193,156.281 km/sec); however, the g-force vessel must slow down (to a few km/sec) to orbit and even land on objects in Alpha Centauri system. Thus, TE envisions vessel again powering the propulsion system to reverse the propulsion vector; thus, the high energy exhaust particles would flow in the direction of flight to decelerate the vessel. AXIOMATIC: Decelerating at g-force should take same time/distance as accelerating at g-force. Thus, TE assumes the voyage's final year would consist of g-force deceleration for the same distance (.38 LY) as the initial year of acceleration.   6) TIME DILATION DURING CRUISE PHASE Simply calculate time dilation during cruise portion of voyage. Cruise Time can be expressed as normal time and relativistic. Normal Time At vessel velocity of 64.43%c,  Earth bound observer measures 1.55 years  for each LY of cruise distance. tCru=dCru/VCru tCru=3.56 LY÷.6443c = 5.523 years Relativistic Time Due to relativistic effects,  ship bound chronometer measures 0.76 yrs for each year observed by Earth. tr= tCru×√(1-VCru2/c2) tr= 5.523 years×0.76 = 4.22 yr However, this is only part of the time dilation. 7) TIME DILATION DURING ACCELERATION Computing time dilation for acceleration and deceleration is much more difficult because g-force velocity is not static; it is dynamic.  Therefore, Thought Experiment (TE) decided to approximate acceleration's time dilation by averaging velocities.  Thus, following text discusses different methods to average vessel speeds. (For convenience, round one year to 360 days for exercises a through e.)      a. Overall Average. Determine g-force vessel's overall average velocity as shown below: VFin + V0 2 =  .6385c + 0 2 =  .31925c Use above velocity to grossly approximate dilated time: tr = t √(1 - dc2) = 360 days √(1 - .319252) = 323.3 days This method indicates some time dilation, but TE seeks more precision.      b. Average 2 Halves. Divide acceleration duration into 2 equal parts, Part 1 is days 0 through 180:  t0 ⇨ t180. Part 2:  t180 ⇨ t360. Average min and max velocity for each half, V180  + V0 2 = 0.3983c + 0 2 =  .19915c V180  + V360 2 = 0.3983c + .6435c 2 =  .5209c Compute time dilation for each half: tr = t √(1 - dc2) = 180 days √(1 -.199152) = 176.4 days tr = t √(1 - dc2) = 180 days √(1 - .52092) = 153.6 days Sum the results. Σtr = 176.4 dy + 153.6 dy = 330.0 days       c. Average 4 Quarters Increase accuracy by dividing duration into four equal parts. t Spot Vel Ave dc 0 days 0.000 c 0.000 90 days 0.225 c 0.112 180 days 0.399 c 0.312 270 days 0.534 c 0.467 360 days 0.639 c 0.587 t c[1 - Rct] (VMin+VMax) 2×c With only four data points, we see that a g-force vessel quickly attains significant percentage of light speed, c. Typical 90 days qtr: t90 ⇨ t180. VMin (.225c) is spot velocity for t90 ; VMax  (.399c) is spot vel for  t180 . For VAve (.312c), Thought Experiment (TE) averages min and max. t Spot Vel Ave dc Incr. 0 days 0 c 0 0 days 90 days 0.225 c 0.112 89.4 days 180 days 0.399 c 0.312 85.5 days 270 days 0.534 c 0.467 79.6 days 360 days 0.639 c 0.587 72.9 days t c[1 - Rct]  (VMin+VMax) 2×c √(1 - dc2) ×90 days To approximate time dilation, TE uses the average vessel velocity for each 90 day interval. Each 90 day period shows increased shrinkage as velocity increases. Sum these results Σtr = 89.4 dy + 85.5 dy + 79.6 dy + 72.9 dy  Σtr = 327.4 days  for a slightly more accurate value of time dilation during first year of g-force acceleration.      d. Average 8 Parts. Further increase accuracy by dividing duration into eight equal parts. t Spot Vel  Ave dc Incr. Cumm. 0 days 0 c 0 0 days 0 days 45 days  0.120 c 0.060  44.9 days 44.9 days 90 days 0.225 c 0.172 44.3 days 89.2 days 135  days 0.318 c 0.271 43.3 days 132.5 days 180  days 0.399 c 0.358 42.0 days 174.5 days 225 days 0.471 c 0.435 40.5 days 215.1 days 270 days 0.534 c 0.503  38.9 days 254.0 days 315 days 0.590 c 0.562 37.2 days  291.2 days 360 days 0.639 c 0.614 35.5 days 326.7 days t c[1 - Rct] (VMin+VMax) 2 × c √(1 - dc2) ×45 days ΣIncr. Each successive 45 day interval has a higher average vessel velocity. Each successive 45 day int;"> As each interval's average velocity increases, the corresponding time dilation increases (shorter period as measured by on-board observers). If we accumulate total dilated time as shown, we can approximate total time dilation for the entire acceleration period, 360 days as observed from Earth.      e. Average 18 Parts. Greatly increase accuracy by dividing duration into eighteen equal parts      f. Average each day to 360 and beyond!!! Greatly improve accuracy determining spot velocity for every day of g-force.  Vessel can even accelerate beyond 360 days as fuel permits. Following diagrams show daily increments for an entire year, 365 days of g-force. Convert vessel velocities from decimal light speed to/from AU/day with conversion constant:  c = 173.145 AU/day. 8) TOTAL TIME DILATION For total time dilation, also consider relativistic time shrinkage during the acceleration/deceleration phases of voyage. Total tr =  Accel:tr + Cru:tr + Decel:tr For example, consider again the g-force journey  to the Alpha Centauri system. ACCELERATION PHASE Normal Acceleration Time Earth bound observer watches  g-force vessel accelerate  for 365.25 days to attain 64.43%c and travels .38 LY. Given: tAcc = 365 days Relativistic Acceleration Time Ship bound observer  measures duration of 330 days.(0.9 years). Rc = (1-Δ) = .99717 tTtl =Σ1day ×√(1-(1- Rct)2) CRUISE PHASE Normal Cruise Time At vessel velocity of 64.43%c, Earth observer measures 5.52 yrs  for 3.56 LY of cruise distance. tCru=dCru/VCru tCru=3.56 LY÷.6443c=5.52y Relativistic Cruise Time Due to relativistic effects, ship bound chronometer  measures: 4.22 yrs during cruise phase. tC-r= tCru×√(1-VCru2/c2) tr= 5.523 yrs×0.76 = 4.22 yr DECELERATION PHASE Normal Deceleration Time Earth bound observer watches g-force vessel decelerate for 1 yr (365.25 dy). tDec = tAcc = 365 days Relativistic Deceleration Time Ship bound observer measures duration of 0.9 years (328.7 dy). tD-r= tA-r TOTAL TIME FOR TRIP Total Normal Time tTtl = tAcc + tCru + tDec tTtl = 1.0yr + 5.52yr + 1.0yr tTtl =  7.52 yr  Total Relativistic Time tT-r = tA-r + tC-r + tD-r tT-r =  0.9yr+4.22yr+0.9yr tT-r =  6.02 yr CONSIDER 500 DAYS G-FORCE ACCELERATION NOTE: LY = 63,241 AU; c = 173.145AU/day Fuel permitting, interstellar vessels can attain higher speeds to further increase the time dilation effect.  For example, consider a g-force acceleration time of 500 days. At 500 days (1.37 yr) of observation by earth bound observer, a g-force vessel attains a velocity of 131 AUs/day (≈ 0.7566 c) and a cumulative acceleration distance of 40,201 AUs (≈ 0.6357 LY). This new acceleration distance changes cruise distance to AC: dCru  =   dTtl - dAcc - dDec dCru  =   4.3 LY - .636 LY - .636 LY = 3.03 LY Subsequently, new cruise time is observed:  tCru  =   dCru / VCru tCru   =   3.03 LY / 0.7566 c = 4.004 Yr Earth Observed Trip Times are as follows: tTtl  =   tAcc + tCru + tDec tTtl  =   1.37 yr + 4.00 yr  +1.37 yr = 6.74 yr much quicker then the total time for 1 year g-force acceleration. VESSEL OBSERVERS SEE DILATED TIME To determine dilated times for 500 days of g-force acceleration, extend the table to the left to 500 days. If one did this, one would determine at .7566c, a vessel observed day shrinks 0.65373.  Furthermore, 500 days of Earth observed acceleration time shrinks to 425.97 days of vessel observed acceleration time (assume same for deceleration times). tAcc-Rel= tDec-Rel tAcc-Rel =  425.97 days = 1.166 yr = tDec-Rel Use the final dilated day factor (0.65373) to determine vessel observed cruise time. tCru-Rel = tCru × √(1-VCru²/c²) tCru-Rel =  4.0 yrs × 0.65373 = 1.166 yr = 2.62 yrs Finally, compare Earth observed times vs. Vessel observed times for 500 days of g-force acceleration. TOTAL TRAVEL TIME FOR 500 DAYS OF G-FORCE Total Normal Time tTtl =  tAcc + tCru + tDec tTtl =1.37yr+4.00yr+1.37yr tTtl =  6.74 yr  Total Relativistic Time tT-r =  tA-r + tC-r + tD-r tT-r = 1.17 yr + 2.62 yr +1.17 yr tT-r = 4.96 yr

SLIDESHOW

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

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