Sunday, March 13, 2011


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.
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
0 days0 AU/day0 AU/day0 c0 days0 days
1 days0.489 AU/day0.245 AU/day0.0014 c1.00000 days1.0000 days
2 days0.977 AU/day0.733 AU/day0.0042 c0.99999 days2.0000 days
3 days1.464 AU/day1.221 AU/day0.0070 c0.99998 days3.0000 days
4 days1.949 AU/day1.706 AU/day0.0099 c0.99995 days3.9999 days
5 days2.433 AU/day2.191 AU/day0.0127 c0.99992 days4.9998 days
6 days2.915 AU/day2.674 AU/day0.0154 c0.99988 days5.9997 days
7 days3.396 AU/day3.156 AU/day0.0182 c0.99983 days6.9996 days
8 days3.876 AU/day3.636 AU/day0.0210 c0.99978 days7.9993 days
9 days4.354 AU/day4.115 AU/day0.0238 c0.99972 days8.9990 days
10 days4.831 AU/day4.593 AU/day0.0265 c0.99965 days9.9987 days
11 days5.307 AU/day5.069 AU/day0.0293 c0.99957 days10.9983 days
12 days5.781 AU/day5.544 AU/day0.0320 c0.99949 days11.9978 days
13 days6.254 AU/day6.018 AU/day0.0348 c0.99940 days12.9972 days
14 days6.726 AU/day6.490 AU/day0.0375 c0.99930 days13.9964 days
15 days7.196 AU/day6.961 AU/day0.0402 c0.99919 days14.9956 days
tc[1 - Rct](Vt-1+Vt)


(1 - dc2)ΣIncr.
16 days7.665 AU/day7.431 AU/day0.0429 c0.99908 days15.9947 days
17 days8.133 AU/day7.899 AU/day0.0456 c0.99896 days16.9937 days
18 days8.599 AU/day8.366 AU/day0.0483 c0.99883 days17.9925 days
19 days9.064 AU/day8.832 AU/day0.0510 c0.99870 days18.9912 days
20 days9.528 AU/day9.296 AU/day0.0537 c0.99856 days19.9898 days
21 days9.990 AU/day9.759 AU/day0.0564 c0.99841 days20.9882 days
22 days10.451 AU/day10.221 AU/day0.0590 c0.99826 days21.9864 days
23 days10.911 AU/day10.681 AU/day0.0617 c0.99810 days22.9845 days
24 days11.370 AU/day11.140 AU/day0.0643 c0.99793 days23.9825 days
25 days11.827 AU/day11.598 AU/day0.0670 c0.99775 days24.9802 days
26 days12.283 AU/day12.055 AU/day0.0696 c0.99757 days25.9778 days
27 days12.737 AU/day12.510 AU/day0.0723 c0.99739 days26.9752 days
28 days13.191 AU/day12.964 AU/day0.0749 c0.99719 days27.9724 days
29 days13.643 AU/day13.417 AU/day0.0775 c0.99699 days28.9694 days
30 days14.093 AU/day13.868 AU/day0.0801 c0.99679 days29.9661 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
31 days14.543 AU/day14.318 AU/day0.0827 c0.99657 days30.9627 days
32 days14.991 AU/day14.767 AU/day0.0853 c0.99636 days31.9591 days
33 days15.438 AU/day15.214 AU/day0.0879 c0.99613 days32.9552 days
34 days15.884 AU/day15.661 AU/day0.0904 c0.99590 days33.9511 days
35 days16.328 AU/day16.106 AU/day0.0930 c0.99566 days34.9468 days
36 days16.771 AU/day16.550 AU/day0.0956 c0.99542 days35.9422 days
37 days17.213 AU/day16.992 AU/day0.0981 c0.99517 days36.9374 days
38 days17.654 AU/day17.433 AU/day0.1007 c0.99492 days37.9323 days
39 days18.093 AU/day17.873 AU/day0.1032 c0.99466 days38.9269 days
40 days18.531 AU/day18.312 AU/day0.1058 c0.99439 days39.9213 days
41 days18.968 AU/day18.750 AU/day0.1083 c0.99412 days40.9155 days
42 days19.404 AU/day19.186 AU/day0.1108 c0.99384 days41.9093 days
43 days19.838 AU/day19.621 AU/day0.1133 c0.99356 days42.9029 days
44 days20.272 AU/day20.055 AU/day0.1158 c0.99327 days43.8961 days
45 days20.704 AU/day20.488 AU/day0.1183 c0.99297 days44.8891 days
tc[1 - Rct](Vt-1+Vt)


(1 - dc2)ΣIncr.
46 days21.135 AU/day20.919 AU/day0.1208 c0.99267 days45.8818 days
47 days21.564 AU/day21.349 AU/day0.1233 c0.99237 days46.8741 days
48 days21.992 AU/day21.778 AU/day0.1258 c0.99206 days47.8662 days
49 days22.420 AU/day22.206 AU/day0.1283 c0.99174 days48.8579 days
50 days22.846 AU/day22.633 AU/day0.1307 c0.99142 days49.8494 days
51 days23.270 AU/day23.058 AU/day0.1332 c0.99109 days50.8405 days
52 days23.694 AU/day23.482 AU/day0.1356 c0.99076 days51.8312 days
53 days24.116 AU/day23.905 AU/day0.1381 c0.99042 days52.8216 days
54 days24.537 AU/day24.327 AU/day0.1405 c0.99008 days53.8117 days
55 days24.957 AU/day24.747 AU/day0.1429 c0.98973 days54.8015 days
56 days25.376 AU/day25.167 AU/day0.1454 c0.98938 days55.7908 days
57 days25.794 AU/day25.585 AU/day0.1478 c0.98902 days56.7799 days
58 days26.210 AU/day26.002 AU/day0.1502 c0.98866 days57.7685 days
59 days26.625 AU/day26.418 AU/day0.1526 c0.98829 days58.7568 days
60 days27.039 AU/day26.832 AU/day0.1550 c0.98792 days59.7447 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
61 days27.452 AU/day27.246 AU/day0.1574 c0.98754 days60.7323 days
62 days27.864 AU/day27.658 AU/day0.1597 c0.98716 days61.7194 days
63 days28.275 AU/day28.069 AU/day0.1621 c0.98677 days62.7062 days
64 days28.684 AU/day28.479 AU/day0.1645 c0.98638 days63.6926 days
65 days29.092 AU/day28.888 AU/day0.1668 c0.98598 days64.6786 days
66 days29.499 AU/day29.296 AU/day0.1692 c0.98558 days65.6641 days
67 days29.905 AU/day29.702 AU/day0.1715 c0.98518 days66.6493 days
68 days30.310 AU/day30.108 AU/day0.1739 c0.98477 days67.6341 days
69 days30.714 AU/day30.512 AU/day0.1762 c0.98435 days68.6184 days
70 days31.116 AU/day30.915 AU/day0.1785 c0.98393 days69.6024 days
71 days31.518 AU/day31.317 AU/day0.1809 c0.98351 days70.5859 days
72 days31.918 AU/day31.718 AU/day0.1832 c0.98308 days71.5689 days
73 days32.317 AU/day32.117 AU/day0.1855 c0.98265 days72.5516 days
74 days32.715 AU/day32.516 AU/day0.1878 c0.98221 days73.5338 days
75 days33.112 AU/day32.913 AU/day0.1901 c0.98177 days74.5156 days
tc[1 - Rct](Vt-1+Vt)


(1 - dc2)ΣIncr.
76 days33.508 AU/day33.310 AU/day0.1924 c0.98132 days75.4969 days
77 days33.902 AU/day33.705 AU/day0.1947 c0.98087 days76.4778 days
78 days34.296 AU/day34.099 AU/day0.1969 c0.98042 days77.4582 days
79 days34.688 AU/day34.492 AU/day0.1992 c0.97996 days78.4381 days
80 days35.079 AU/day34.884 AU/day0.2015 c0.97949 days79.4176 days
81 days35.470 AU/day35.274 AU/day0.2037 c0.97903 days80.3967 days
82 days35.859 AU/day35.664 AU/day0.2060 c0.97856 days81.3752 days
83 days36.247 AU/day36.053 AU/day0.2082 c0.97808 days82.3533 days
84 days36.633 AU/day36.440 AU/day0.2105 c0.97760 days83.3309 days
85 days37.019 AU/day36.826 AU/day0.2127 c0.97712 days84.3080 days
86 days37.404 AU/day37.212 AU/day0.2149 c0.97663 days85.2846 days
87 days37.788 AU/day37.596 AU/day0.2171 c0.97614 days86.2608 days
88 days38.170 AU/day37.979 AU/day0.2193 c0.97565 days87.2364 days
89 days38.551 AU/day38.361 AU/day0.2216 c0.97515 days88.2116 days
90 days38.932 AU/day38.742 AU/day0.2238 c0.97465 days89.1862 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
91 days39.311 AU/day39.121 AU/day0.2259 c0.97414 days90.1604 days
92 days39.689 AU/day39.500 AU/day0.2281 c0.97363 days91.1340 days
93 days40.066 AU/day39.878 AU/day0.2303 c0.97312 days92.1071 days
94 days40.443 AU/day40.255 AU/day0.2325 c0.97260 days93.0797 days
95 days40.818 AU/day40.630 AU/day0.2347 c0.97208 days94.0518 days
96 days41.192 AU/day41.005 AU/day0.2368 c0.97155 days95.0233 days
97 days41.564 AU/day41.378 AU/day0.2390 c0.97102 days95.9944 days
98 days41.936 AU/day41.750 AU/day0.2411 c0.97049 days96.9649 days
99 days42.307 AU/day42.122 AU/day0.2433 c0.96996 days97.9348 days
100 days42.677 AU/day42.492 AU/day0.2454 c0.96942 days98.9042 days
101 days43.046 AU/day42.861 AU/day0.2475 c0.96888 days99.8731 days
102 days43.413 AU/day43.229 AU/day0.2497 c0.96833 days100.8414 days
103 days43.780 AU/day43.597 AU/day0.2518 c0.96778 days101.8092 days
104 days44.145 AU/day43.963 AU/day0.2539 c0.96723 days102.7765 days
105 days44.510 AU/day44.328 AU/day0.2560 c0.96667 days103.7431 days
tc[1 - Rct](Vt-1+Vt)


(1 - dc2)ΣIncr.
106 days44.873 AU/day44.692 AU/day0.2581 c0.96611 days104.7092 days
107 days45.236 AU/day45.055 AU/day0.2602 c0.96555 days105.6748 days
108 days45.597 AU/day45.417 AU/day0.2623 c0.96499 days106.6398 days
109 days45.958 AU/day45.778 AU/day0.2644 c0.96442 days107.6042 days
110 days46.317 AU/day46.138 AU/day0.2665 c0.96384 days108.5680 days
111 days46.676 AU/day46.497 AU/day0.2685 c0.96327 days109.5313 days
112 days47.033 AU/day46.854 AU/day0.2706 c0.96269 days110.4940 days
113 days47.390 AU/day47.211 AU/day0.2727 c0.96211 days111.4561 days
114 days47.745 AU/day47.567 AU/day0.2747 c0.96152 days112.4176 days
115 days48.099 AU/day47.922 AU/day0.2768 c0.96093 days113.3786 days
116 days48.453 AU/day48.276 AU/day0.2788 c0.96034 days114.3389 days
117 days48.805 AU/day48.629 AU/day0.2809 c0.95975 days115.2987 days
118 days49.156 AU/day48.981 AU/day0.2829 c0.95915 days116.2578 days
119 days49.507 AU/day49.332 AU/day0.2849 c0.95855 days117.2164 days
120 days49.856 AU/day49.682 AU/day0.2869 c0.95795 days118.1743 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
121 days50.205 AU/day50.030 AU/day0.2890 c0.95734 days119.1317 days
122 days50.552 AU/day50.378 AU/day0.2910 c0.95674 days120.0884 days
123 days50.899 AU/day50.725 AU/day0.2930 c0.95612 days121.0445 days
124 days51.244 AU/day51.071 AU/day0.2950 c0.95551 days122.0000 days
125 days51.588 AU/day51.416 AU/day0.2970 c0.95489 days122.9549 days
126 days51.932 AU/day51.760 AU/day0.2989 c0.95427 days123.9092 days
127 days52.275 AU/day52.103 AU/day0.3009 c0.95365 days124.8628 days
128 days52.616 AU/day52.445 AU/day0.3029 c0.95302 days125.8159 days
129 days52.957 AU/day52.786 AU/day0.3049 c0.95239 days126.7683 days
130 days53.296 AU/day53.127 AU/day0.3068 c0.95176 days127.7200 days
131 days53.635 AU/day53.466 AU/day0.3088 c0.95113 days128.6711 days
132 days53.973 AU/day53.804 AU/day0.3107 c0.95049 days129.6216 days
133 days54.310 AU/day54.141 AU/day0.3127 c0.94985 days130.5715 days
134 days54.645 AU/day54.478 AU/day0.3146 c0.94921 days131.5207 days
135 days54.980 AU/day54.813 AU/day0.3166 c0.94857 days132.4693 days
tc[1 - Rct](Vt-1+Vt)


(1 - dc2)ΣIncr.
136 days55.314 AU/day55.147 AU/day0.3185 c0.94792 days133.4172 days
137 days55.647 AU/day55.481 AU/day0.3204 c0.94727 days134.3645 days
138 days55.979 AU/day55.813 AU/day0.3223 c0.94662 days135.3111 days
139 days56.310 AU/day56.145 AU/day0.3243 c0.94597 days136.2571 days
140 days56.641 AU/day56.475 AU/day0.3262 c0.94531 days137.2024 days
141 days56.970 AU/day56.805 AU/day0.3281 c0.94465 days138.1470 days
142 days57.298 AU/day57.134 AU/day0.3300 c0.94399 days139.0910 days
143 days57.626 AU/day57.462 AU/day0.3319 c0.94332 days140.0343 days
144 days57.952 AU/day57.789 AU/day0.3338 c0.94266 days140.9770 days
145 days58.277 AU/day58.115 AU/day0.3356 c0.94199 days141.9190 days
146 days58.602 AU/day58.440 AU/day0.3375 c0.94132 days142.8603 days
147 days58.926 AU/day58.764 AU/day0.3394 c0.94065 days143.8009 days
148 days59.249 AU/day59.087 AU/day0.3413 c0.93997 days144.7409 days
149 days59.570 AU/day59.410 AU/day0.3431 c0.93929 days145.6802 days
150 days59.891 AU/day59.731 AU/day0.3450 c0.93861 days146.6188 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
151 days60.211 AU/day60.051 AU/day0.3468 c0.93793 days147.5567 days
152 days60.531 AU/day60.371 AU/day0.3487 c0.93724 days148.4940 days
153 days60.849 AU/day60.690 AU/day0.3505 c0.93656 days149.4305 days
154 days61.166 AU/day61.008 AU/day0.3523 c0.93587 days150.3664 days
155 days61.483 AU/day61.324 AU/day0.3542 c0.93518 days151.3016 days
156 days61.798 AU/day61.640 AU/day0.3560 c0.93448 days152.2361 days
157 days62.113 AU/day61.956 AU/day0.3578 c0.93379 days153.1699 days
158 days62.427 AU/day62.270 AU/day0.3596 c0.93309 days154.1030 days
159 days62.740 AU/day62.583 AU/day0.3614 c0.93239 days155.0353 days
160 days63.052 AU/day62.896 AU/day0.3633 c0.93169 days155.9670 days
161 days63.363 AU/day63.207 AU/day0.3651 c0.93099 days156.8980 days
162 days63.673 AU/day63.518 AU/day0.3668 c0.93028 days157.8283 days
163 days63.982 AU/day63.828 AU/day0.3686 c0.92957 days158.7579 days
164 days64.291 AU/day64.137 AU/day0.3704 c0.92886 days159.6867 days
165 days64.598 AU/day64.445 AU/day0.3722 c0.92815 days160.6149 days
tc[1 - Rct](Vt-1+Vt)


(1 - dc2)ΣIncr.
166 days64.905 AU/day64.752 AU/day0.3740 c0.92744 days161.5423 days
167 days65.211 AU/day65.058 AU/day0.3757 c0.92672 days162.4691 days
168 days65.516 AU/day65.364 AU/day0.3775 c0.92601 days163.3951 days
169 days65.820 AU/day65.668 AU/day0.3793 c0.92529 days164.3203 days
170 days66.124 AU/day65.972 AU/day0.3810 c0.92457 days165.2449 days
171 days66.426 AU/day66.275 AU/day0.3828 c0.92384 days166.1688 days
172 days66.728 AU/day66.577 AU/day0.3845 c0.92312 days167.0919 days
173 days67.028 AU/day66.878 AU/day0.3863 c0.92239 days168.0143 days
174 days67.328 AU/day67.178 AU/day0.3880 c0.92166 days168.9359 days
175 days67.627 AU/day67.478 AU/day0.3897 c0.92093 days169.8569 days
176 days67.925 AU/day67.776 AU/day0.3914 c0.92020 days170.7771 days
177 days68.223 AU/day68.074 AU/day0.3932 c0.91947 days171.6965 days
178 days68.519 AU/day68.371 AU/day0.3949 c0.91873 days172.6153 days
179 days68.815 AU/day68.667 AU/day0.3966 c0.91800 days173.5333 days
180 days69.110 AU/day68.962 AU/day0.3983 c0.91726 days174.4505 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
181 days69.404 AU/day69.257 AU/day0.4000 c0.91652 days175.3670 days
182 days69.697 AU/day69.550 AU/day0.4017 c0.91578 days176.2828 days
183 days69.989 AU/day69.843 AU/day0.4034 c0.91503 days177.1979 days
184 days70.281 AU/day70.135 AU/day0.4051 c0.91429 days178.1121 days
185 days70.572 AU/day70.426 AU/day0.4067 c0.91354 days179.0257 days
186 days70.861 AU/day70.716 AU/day0.4084 c0.91279 days179.9385 days
187 days71.150 AU/day71.006 AU/day0.4101 c0.91204 days180.8505 days
188 days71.439 AU/day71.295 AU/day0.4118 c0.91129 days181.7618 days
189 days71.726 AU/day71.582 AU/day0.4134 c0.91054 days182.6724 days
190 days72.013 AU/day71.869 AU/day0.4151 c0.90978 days183.5821 days
191 days72.299 AU/day72.156 AU/day0.4167 c0.90903 days184.4912 days
192 days72.584 AU/day72.441 AU/day0.4184 c0.90827 days185.3994 days
193 days72.868 AU/day72.726 AU/day0.4200 c0.90751 days186.3069 days
194 days73.151 AU/day73.009 AU/day0.4217 c0.90675 days187.2137 days
195 days73.434 AU/day73.292 AU/day0.4233 c0.90599 days188.1197 days
tc[1 - Rct](Vt-1+Vt)


(1 - dc2)ΣIncr.
196 days73.715 AU/day73.575 AU/day0.4249 c0.90523 days189.0249 days
197 days73.996 AU/day73.856 AU/day0.4266 c0.90446 days189.9294 days
198 days74.277 AU/day74.137 AU/day0.4282 c0.90370 days190.8331 days
199 days74.556 AU/day74.416 AU/day0.4298 c0.90293 days191.7360 days
200 days74.835 AU/day74.695 AU/day0.4314 c0.90216 days192.6382 days
201 days75.112 AU/day74.974 AU/day0.4330 c0.90139 days193.5395 days
202 days75.390 AU/day75.251 AU/day0.4346 c0.90062 days194.4402 days
203 days75.666 AU/day75.528 AU/day0.4362 c0.89984 days195.3400 days
204 days75.941 AU/day75.804 AU/day0.4378 c0.89907 days196.2391 days
205 days76.216 AU/day76.079 AU/day0.4394 c0.89830 days197.1374 days
206 days76.490 AU/day76.353 AU/day0.4410 c0.89752 days198.0349 days
207 days76.763 AU/day76.626 AU/day0.4426 c0.89674 days198.9316 days
208 days77.035 AU/day76.899 AU/day0.4441 c0.89596 days199.8276 days
209 days77.307 AU/day77.171 AU/day0.4457 c0.89518 days200.7228 days
210 days77.578 AU/day77.442 AU/day0.4473 c0.89440 days201.6172 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
211 days77.848 AU/day77.713 AU/day0.4488 c0.89362 days202.5108 days
212 days78.117 AU/day77.983 AU/day0.4504 c0.89283 days203.4036 days
213 days78.386 AU/day78.251 AU/day0.4519 c0.89205 days204.2957 days
214 days78.654 AU/day78.520 AU/day0.4535 c0.89126 days205.1869 days
215 days78.921 AU/day78.787 AU/day0.4550 c0.89047 days206.0774 days
216 days79.187 AU/day79.054 AU/day0.4566 c0.88968 days206.9671 days
217 days79.452 AU/day79.320 AU/day0.4581 c0.88889 days207.8560 days
218 days79.717 AU/day79.585 AU/day0.4596 c0.88810 days208.7441 days
219 days79.981 AU/day79.849 AU/day0.4612 c0.88731 days209.6314 days
220 days80.244 AU/day80.113 AU/day0.4627 c0.88652 days210.5179 days
221 days80.507 AU/day80.376 AU/day0.4642 c0.88572 days211.4036 days
222 days80.769 AU/day80.638 AU/day0.4657 c0.88493 days212.2886 days
223 days81.030 AU/day80.899 AU/day0.4672 c0.88413 days213.1727 days
224 days81.290 AU/day81.160 AU/day0.4687 c0.88334 days214.0560 days
225 days81.550 AU/day81.420 AU/day0.4702 c0.88254 days214.9386 days
tc[1 - Rct](Vt-1+Vt)


(1 - dc2)ΣIncr.
226 days81.809 AU/day81.679 AU/day0.4717 c0.88174 days215.8203 days
227 days82.067 AU/day81.938 AU/day0.4732 c0.88094 days216.7013 days
228 days82.324 AU/day82.195 AU/day0.4747 c0.88014 days217.5814 days
229 days82.581 AU/day82.452 AU/day0.4762 c0.87933 days218.4607 days
230 days82.837 AU/day82.709 AU/day0.4777 c0.87853 days219.3393 days
231 days83.092 AU/day82.964 AU/day0.4792 c0.87773 days220.2170 days
232 days83.346 AU/day83.219 AU/day0.4806 c0.87692 days221.0939 days
233 days83.600 AU/day83.473 AU/day0.4821 c0.87612 days221.9700 days
234 days83.853 AU/day83.727 AU/day0.4836 c0.87531 days222.8453 days
235 days84.106 AU/day83.979 AU/day0.4850 c0.87450 days223.7198 days
236 days84.357 AU/day84.231 AU/day0.4865 c0.87369 days224.5935 days
237 days84.608 AU/day84.483 AU/day0.4879 c0.87288 days225.4664 days
238 days84.858 AU/day84.733 AU/day0.4894 c0.87207 days226.3385 days
239 days85.108 AU/day84.983 AU/day0.4908 c0.87126 days227.2097 days
240 days85.357 AU/day85.232 AU/day0.4923 c0.87045 days228.0802 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
241 days85.605 AU/day85.481 AU/day0.4937 c0.86964 days228.9498 days
242 days85.852 AU/day85.728 AU/day0.4951 c0.86882 days229.8186 days
243 days86.099 AU/day85.975 AU/day0.4966 c0.86801 days230.6867 days
244 days86.345 AU/day86.222 AU/day0.4980 c0.86719 days231.5538 days
245 days86.590 AU/day86.467 AU/day0.4994 c0.86638 days232.4202 days
246 days86.835 AU/day86.712 AU/day0.5008 c0.86556 days233.2858 days
247 days87.079 AU/day86.957 AU/day0.5022 c0.86474 days234.1505 days
248 days87.322 AU/day87.200 AU/day0.5036 c0.86392 days235.0144 days
249 days87.564 AU/day87.443 AU/day0.5050 c0.86310 days235.8775 days
250 days87.806 AU/day87.685 AU/day0.5064 c0.86228 days236.7398 days
251 days88.047 AU/day87.927 AU/day0.5078 c0.86146 days237.6013 days
252 days88.288 AU/day88.168 AU/day0.5092 c0.86064 days238.4619 days
253 days88.528 AU/day88.408 AU/day0.5106 c0.85982 days239.3218 days
254 days88.767 AU/day88.647 AU/day0.5120 c0.85900 days240.1807 days
255 days89.005 AU/day88.886 AU/day0.5134 c0.85817 days241.0389 days
tc[1 - Rct](Vt-1+Vt)


(1 - dc2)ΣIncr.
256 days89.243 AU/day89.124 AU/day0.5147 c0.85735 days241.8963 days
257 days89.480 AU/day89.362 AU/day0.5161 c0.85652 days242.7528 days
258 days89.717 AU/day89.598 AU/day0.5175 c0.85570 days243.6085 days
259 days89.952 AU/day89.834 AU/day0.5188 c0.85487 days244.4634 days
260 days90.187 AU/day90.070 AU/day0.5202 c0.85404 days245.3174 days
261 days90.422 AU/day90.305 AU/day0.5216 c0.85322 days246.1706 days
262 days90.656 AU/day90.539 AU/day0.5229 c0.85239 days247.0230 days
263 days90.889 AU/day90.772 AU/day0.5243 c0.85156 days247.8746 days
264 days91.121 AU/day91.005 AU/day0.5256 c0.85073 days248.7253 days
265 days91.353 AU/day91.237 AU/day0.5269 c0.84990 days249.5752 days
266 days91.584 AU/day91.469 AU/day0.5283 c0.84907 days250.4243 days
267 days91.815 AU/day91.699 AU/day0.5296 c0.84824 days251.2725 days
268 days92.044 AU/day91.930 AU/day0.5309 c0.84741 days252.1199 days
269 days92.274 AU/day92.159 AU/day0.5323 c0.84658 days252.9665 days
270 days92.502 AU/day92.388 AU/day0.5336 c0.84575 days253.8123 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
271 days92.730 AU/day92.616 AU/day0.5349 c0.84491 days254.6572 days
272 days92.957 AU/day92.844 AU/day0.5362 c0.84408 days255.5012 days
273 days93.184 AU/day93.071 AU/day0.5375 c0.84324 days256.3445 days
274 days93.410 AU/day93.297 AU/day0.5388 c0.84241 days257.1869 days
275 days93.635 AU/day93.523 AU/day0.5401 c0.84157 days258.0285 days
276 days93.860 AU/day93.748 AU/day0.5414 c0.84074 days258.8692 days
277 days94.084 AU/day93.972 AU/day0.5427 c0.83990 days259.7091 days
278 days94.307 AU/day94.196 AU/day0.5440 c0.83907 days260.5482 days
279 days94.530 AU/day94.419 AU/day0.5453 c0.83823 days261.3864 days
280 days94.752 AU/day94.641 AU/day0.5466 c0.83739 days262.2238 days
281 days94.974 AU/day94.863 AU/day0.5479 c0.83655 days263.0604 days
282 days95.195 AU/day95.084 AU/day0.5492 c0.83572 days263.8961 days
283 days95.415 AU/day95.305 AU/day0.5504 c0.83488 days264.7310 days
284 days95.635 AU/day95.525 AU/day0.5517 c0.83404 days265.5650 days
285 days95.854 AU/day95.744 AU/day0.5530 c0.83320 days266.3982 days
tc[1 - Rct](Vt-1+Vt)


(1 - dc2)ΣIncr.
286 days96.072 AU/day95.963 AU/day0.5542 c0.83236 days267.2306 days
287 days96.290 AU/day96.181 AU/day0.5555 c0.83152 days268.0621 days
288 days96.507 AU/day96.399 AU/day0.5568 c0.83068 days268.8928 days
289 days96.724 AU/day96.616 AU/day0.5580 c0.82984 days269.7226 days
290 days96.940 AU/day96.832 AU/day0.5593 c0.82900 days270.5516 days
291 days97.155 AU/day97.048 AU/day0.5605 c0.82816 days271.3797 days
292 days97.370 AU/day97.263 AU/day0.5617 c0.82731 days272.2071 days
293 days97.584 AU/day97.477 AU/day0.5630 c0.82647 days273.0335 days
294 days97.798 AU/day97.691 AU/day0.5642 c0.82563 days273.8592 days
295 days98.011 AU/day97.904 AU/day0.5654 c0.82479 days274.6839 days
296 days98.223 AU/day98.117 AU/day0.5667 c0.82394 days275.5079 days
297 days98.435 AU/day98.329 AU/day0.5679 c0.82310 days276.3310 days
298 days98.646 AU/day98.540 AU/day0.5691 c0.82225 days277.1532 days
299 days98.856 AU/day98.751 AU/day0.5703 c0.82141 days277.9746 days
300 days99.066 AU/day98.961 AU/day0.5716 c0.82057 days278.7952 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
301 days99.276 AU/day99.171 AU/day0.5728 c0.81972 days279.6149 days
302 days99.484 AU/day99.380 AU/day0.5740 c0.81888 days280.4338 days
303 days99.692 AU/day99.588 AU/day0.5752 c0.81803 days281.2518 days
304 days99.900 AU/day99.796 AU/day0.5764 c0.81719 days282.0690 days
305 days100.107 AU/day100.004 AU/day0.5776 c0.81634 days282.8854 days
306 days100.313 AU/day100.210 AU/day0.5788 c0.81549 days283.7009 days
307 days100.519 AU/day100.416 AU/day0.5800 c0.81465 days284.5155 days
308 days100.725 AU/day100.622 AU/day0.5811 c0.81380 days285.3293 days
309 days100.929 AU/day100.827 AU/day0.5823 c0.81296 days286.1423 days
310 days101.133 AU/day101.031 AU/day0.5835 c0.81211 days286.9544 days
311 days101.337 AU/day101.235 AU/day0.5847 c0.81126 days287.7656 days
312 days101.540 AU/day101.438 AU/day0.5859 c0.81041 days288.5761 days
313 days101.742 AU/day101.641 AU/day0.5870 c0.80957 days289.3856 days
314 days101.944 AU/day101.843 AU/day0.5882 c0.80872 days290.1943 days
315 days102.145 AU/day102.044 AU/day0.5894 c0.80787 days291.0022 days
tc[1 - Rct](Vt-1+Vt)


(1 - dc2)ΣIncr.
316 days102.346 AU/day102.245 AU/day0.5905 c0.80702 days291.8092 days
317 days102.546 AU/day102.446 AU/day0.5917 c0.80618 days292.6154 days
318 days102.745 AU/day102.646 AU/day0.5928 c0.80533 days293.4207 days
319 days102.944 AU/day102.845 AU/day0.5940 c0.80448 days294.2252 days
320 days103.143 AU/day103.043 AU/day0.5951 c0.80363 days295.0289 days
321 days103.340 AU/day103.242 AU/day0.5963 c0.80278 days295.8316 days
322 days103.538 AU/day103.439 AU/day0.5974 c0.80193 days296.6336 days
323 days103.734 AU/day103.636 AU/day0.5986 c0.80108 days297.4347 days
324 days103.931 AU/day103.833 AU/day0.5997 c0.80024 days298.2349 days
325 days104.126 AU/day104.028 AU/day0.6008 c0.79939 days299.0343 days
326 days104.321 AU/day104.224 AU/day0.6019 c0.79854 days299.8328 days
327 days104.516 AU/day104.418 AU/day0.6031 c0.79769 days300.6305 days
328 days104.710 AU/day104.613 AU/day0.6042 c0.79684 days301.4273 days
329 days104.903 AU/day104.806 AU/day0.6053 c0.79599 days302.2233 days
330 days105.096 AU/day104.999 AU/day0.6064 c0.79514 days303.0185 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
331 days105.288 AU/day105.192 AU/day0.6075 c0.79429 days303.8128 days
332 days105.480 AU/day105.384 AU/day0.6086 c0.79344 days304.6062 days
333 days105.671 AU/day105.576 AU/day0.6098 c0.79259 days305.3988 days
334 days105.862 AU/day105.767 AU/day0.6109 c0.79174 days306.1905 days
335 days106.052 AU/day105.957 AU/day0.6120 c0.79089 days306.9814 days
336 days106.242 AU/day106.147 AU/day0.6131 c0.79004 days307.7715 days
337 days106.431 AU/day106.336 AU/day0.6141 c0.78919 days308.5607 days
338 days106.619 AU/day106.525 AU/day0.6152 c0.78834 days309.3490 days
339 days106.807 AU/day106.713 AU/day0.6163 c0.78749 days310.1365 days
340 days106.995 AU/day106.901 AU/day0.6174 c0.78664 days310.9231 days
341 days107.182 AU/day107.088 AU/day0.6185 c0.78579 days311.7089 days
342 days107.368 AU/day107.275 AU/day0.6196 c0.78494 days312.4939 days
343 days107.554 AU/day107.461 AU/day0.6206 c0.78409 days313.2780 days
344 days107.739 AU/day107.647 AU/day0.6217 c0.78324 days314.0612 days
345 days107.924 AU/day107.832 AU/day0.6228 c0.78239 days314.8436 days
tc[1 - Rct](Vt-1+Vt)


(1 - dc2)ΣIncr.
346 days108.108 AU/day108.016 AU/day0.6238 c0.78155 days315.6252 days
347 days108.292 AU/day108.200 AU/day0.6249 c0.78070 days316.4059 days
348 days108.476 AU/day108.384 AU/day0.6260 c0.77985 days317.1857 days
349 days108.658 AU/day108.567 AU/day0.6270 c0.77900 days317.9647 days
350 days108.841 AU/day108.749 AU/day0.6281 c0.77815 days318.7428 days
351 days109.022 AU/day108.931 AU/day0.6291 c0.77730 days319.5201 days
352 days109.203 AU/day109.113 AU/day0.6302 c0.77645 days320.2966 days
353 days109.384 AU/day109.294 AU/day0.6312 c0.77560 days321.0722 days
354 days109.564 AU/day109.474 AU/day0.6323 c0.77475 days321.8469 days
355 days109.744 AU/day109.654 AU/day0.6333 c0.77390 days322.6208 days
356 days109.923 AU/day109.834 AU/day0.6343 c0.77305 days323.3939 days
357 days110.102 AU/day110.013 AU/day0.6354 c0.77220 days324.1661 days
358 days110.280 AU/day110.191 AU/day0.6364 c0.77135 days324.9374 days
359 days110.458 AU/day110.369 AU/day0.6374 c0.77050 days325.7079 days
360 days110.635 AU/day110.546 AU/day0.6385 c0.76965 days326.4776 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
361 days110.811 AU/day110.723 AU/day0.6395 c0.76881 days327.2464 days
362 days110.988 AU/day110.900 AU/day0.6405 c0.76796 days328.0144 days
363 days111.163 AU/day111.075 AU/day0.6415 c0.76711 days328.7815 days
364 days111.338 AU/day111.251 AU/day0.6425 c0.76626 days329.5477 days
365 days111.513 AU/day111.426 AU/day0.6435 c0.76541 days330.3131 days
tc[1 - Rct](Vt-1+Vt)


(1 - dc2)ΣIncr.

1) g-force for a sec. 
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/hr × 60 sec/min).
If a vessel can g-force accelerate for an entire day, it will achieve a very high velocity of 847,281.6 m/sec  (= 86,400 sec × 9.8065 m/sec).
This velocity can be conveniently converted into several 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.)

847.28 km/sec × 86,400 sec/day × AU/149,597,871 km = .489 AU/day
(Since planetary distances are typically measured in Astronomical Units (AUs),
approximate value of .5 AU/day is very convenient.)

847.28 km/sec × c/(299,792.458 km/sec) = 0.2826%c
(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, the daily remainder:
(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 k/s) is fixed regardless of observer's speed.  Thus, Earth observer and vessel observer both observe c as same value even though Earth observes 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.
c[1 - Rct] = VShip
To obtain velocities as AU/day, a convenient measure for interplanetary travel, restate value of 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/dy[1 - .997174365] = 111.513 AU/dy = 64.4%c
4) Cruise for several years
In this example of an interstellar profile, g-force vessels accelerates for one year (365.25 days) to achieve a cruise velocity of VCru  = 64.43%c.  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 approach objects in Alpha Centauri system.
Thus, TE envisions vessel again powering on the propulsion system and reversing the propulsion vector; thus, the high energy exhaust particles would flow in the direction of flight to cause a decelerating effect.  It's axiomatic that decelerating at g-force should take same time/distance as accelerating at g-force.  Since the first year of g-force acceleration took .38 LY, the voyage's final year would consist of g-force deceleration for the same distance.

6)  Partial Time Dilation
Simply calculate time dilation during cruise portion of voyage.
Cruise Time can be expressed as normal time and relativistic.
At vessel velocity of 64.43%c, 
Earth bound observer measures 1.55 years 
for each LY of cruise distance.
tCru=3.56 LY÷.6443c = 5.523 years
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.  Fortunately, it's reasonable to assume that dilated time for acceleration is likely to equal dilated time for deceleration.  Thus, following text focuses on computing dilated for g-force acceleration during first year of voyage.

(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

= .6385c + 0

= .31925c
Use above velocity to grossly approximate dilated time:
trt √(1 - dc2) = 360 days √(1 - .319252) = 323.3 days
This method indicates some time dilation, but accuracy is likely compromised.

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

=0.3983c + 0

= .19915c
V180  + V360

=0.3983c + .6435c

= .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.
tSpot VelAve dc
0 days0.000 c0.000
90 days0.225 c0.112
180 days0.399 c0.312
270 days0.534 c0.467
360 days0.639 c0.587
tc[1 - Rct](VMin+VMax)


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  t180VMin (.225c) is spot velocity for t90 ; VMax  (.399c) is spot vel for  t180 .
For VAve (.312c), Thought Experiment (TE) averages min and max.
tSpot Vel Ave dcIncr.
0 days0 c00 days
90 days0.225 c0.11289.4 days
180 days0.399 c0.31285.5 days
270 days0.534 c0.46779.6 days
360 days0.639 c0.58772.9 days
tc[1 - Rct] (VMin+VMax)

(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 = 327.4 days 
for a slightly more accurate value of time dilation during first year of g-force acceleration.

d. Average 8 Octants
Further increase accuracy by dividing duration into eight equal parts.
Ave dcIncr.Cumm.
0 days0 c00 days0 days
45 days 0.120 c0.060 44.9 days44.9 days
90 days0.225 c0.17244.3 days89.2 days
135  days0.318 c0.27143.3 days132.5 days
180  days0.399 c0.35842.0 days174.5 days
225 days0.471 c0.43540.5 days215.1 days
270 days0.534 c0.503 38.9 days254.0 days
315 days0.590 c0.56237.2 days 291.2 days
360 days0.639 c0.61435.5 days326.7 days
tc[1 - Rct](VMin+VMax)

2 × c
(1 - dc2)
×45 days

Each successive 45 day interval has a higher average vessel velocity.

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 Everyday 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 =  Acceleration:t+ Cruise:t+ Deceleration:tr
For example, consider again the g-force journey to the Alpha Centauri system.

Earth bound observer watches 
g-force vessel accelerate for 365 days (1 year).
It achieves 64.43%c and travels .38 LY.
Given: tAcc = 365 days

Ship bound observer 
measures duration
of 330 days.(0.9 years).

At vessel velocity of 64.43%c, 
Earth bound observer measures 5.52 years 
for 3.56 LY of cruise distance.
tCru=3.56 LY÷.6443c = 5.523 years

Due to relativistic effects, 
ship bound chronometer measures:4.22 yrs 
during cruise phase.
tC-r= tCru×(1-VCru2/c2)
tr= 5.523 years×0.76 = 4.22 yr

Earth bound observer 
watches g-force vessel
decelerate for 365 days (1 year).
tDec tAcc = 365 days

Ship bound observer 
measures duration
of 330 days.(0.9 years).
tTtl =  tAcc + tCru + tDec
tTtl =  1.0 yr + 5.52 yr +1.0 yr
tTtl =  7.52 yr 
tT-r =  tA-r + tC-r + tD-r
tT-r =  0.9 yr 4.22 yr +0.9 yr
tT-r =  6.02 yr
Final Note:
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
dCru =  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
which is even quicker then the earth observed trip time for 1 year g-force acceleration.
Vessel Observers would measure Dilated Times:
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). 
tA-r = 425.97 days = 1.166 yrs = tD-r
Use the final dilated day factor (0.65373) to determine vessel observed cruise time.
tC-r = tCru×(1-VCru2/c2) = 4.0 yrs × 0.65373= 2.62 yrs
Finally, compare Earth observed times vs. Vessel observed times for 500 days of g-force acceleration.
tTtl =  tAcc + tCru + tDec
tTtl =  1.37 yr + 4.00 yr +1.37 yr
tTtl =  6.74 yr 
tT-r =  tA-r + tC-r + tD-r
tT-r = 1.17 yr + 2.62 yr +1.17 yr
tT-r = 4.96 yr



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