Saturday, February 24, 2007

ENHANCED TIME DILATION TO AC

Previous chapters have discussed G-force accelerations of various durations.EXAMPLE-1:  100 days of G-force enables a cruise velocity of about 25%c.EXAMPLE-2:  G-force propulsion for a year (365.25 days) achieves a cruise velocity of about 65%c.  (NOTE: Further discussed in "Standard" Time Dilation to AC.) Thought Experiment (TE) assumes that inherent fuel limitations would not enable a vessel to G-force propel for much longer than one year.EXAMPLE-3:  HOWEVER, Interstellar Snowballs (huge spheres of ice sent to refuel star bound ships) might enable an interstellar vessel to G-force accelerate for as long as 712 days to enable a cruise velocity of 86.7%c.  At such a speed, vessel observers would measure time at half the rate as an Earth bound observer (due to relativistic time dilation).
Normal
Time 		Vehicle
Velocity 		Dilated
Time
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
361 days 110.811 AU/day 110.723 AU/day 0.6394 c 0.7688 days 327.2464 days
362 days 110.988 AU/day 110.900 AU/day 0.6405 c 0.7680 days 328.0144 days
363 days 111.163 AU/day 111.075 AU/day 0.6415 c 0.7671 days 328.7815 days
364 days 111.338 AU/day 111.251 AU/day 0.6425 c 0.7663 days 329.5477 days
365 days 111.513 AU/day 111.426 AU/day 0.6435 c 0.7654 days 330.3131 days
366 days 111.687 AU/day 111.600 AU/day 0.6445 c 0.7646 days 331.0777 days
367 days 111.861 AU/day 111.774 AU/day 0.6455 c 0.7637 days 331.8414 days
368 days 112.034 AU/day 111.948 AU/day 0.6465 c 0.7629 days 332.6043 days
369 days 112.207 AU/day 112.121 AU/day 0.6475 c 0.7620 days 333.3663 days
370 days 112.379 AU/day 112.293 AU/day 0.6485 c 0.7612 days 334.1275 days
371 days 112.551 AU/day 112.465 AU/day 0.6495 c 0.7603 days 334.8878 days
372 days 112.722 AU/day 112.636 AU/day 0.6505 c 0.7595 days 335.6473 days
373 days 112.893 AU/day 112.807 AU/day 0.6515 c 0.7586 days 336.4059 days
374 days 113.063 AU/day 112.978 AU/day 0.6525 c 0.7578 days 337.1637 days
375 days 113.233 AU/day 113.148 AU/day 0.6534 c 0.7569 days 337.9206 days
tc[1 - Rct](Vt-1+Vt)
2		VAve
173.145 		√(1-dc2)ΣIncr.
376 days 113.402 AU/day 113.318 AU/day 0.6545 c 0.7561 days 338.6767 days
377 days 113.571 AU/day 113.487 AU/day 0.6554 c 0.7552 days 339.4320 days
378 days 113.739 AU/day 113.655 AU/day 0.6564 c 0.7544 days 340.1864 days
379 days 113.907 AU/day 113.823 AU/day 0.6574 c 0.7536 days 340.9399 days
380 days 114.075 AU/day 113.991 AU/day 0.6584 c 0.7527 days 341.6926 days
381 days 114.242 AU/day 114.158 AU/day 0.6593 c 0.7519 days 342.4445 days
382 days 114.408 AU/day 114.325 AU/day 0.6603 c 0.7510 days 343.1955 days
383 days 114.574 AU/day 114.491 AU/day 0.6612 c 0.7502 days 343.9457 days
384 days 114.740 AU/day 114.657 AU/day 0.6622 c 0.7493 days 344.6950 days
385 days 114.905 AU/day 114.822 AU/day 0.6632 c 0.7485 days 345.4435 days
386 days 115.069 AU/day 114.987 AU/day 0.6641 c 0.7476 days 346.1911 days
387 days 115.233 AU/day 115.151 AU/day 0.6651 c 0.7468 days 346.9379 days
388 days 115.397 AU/day 115.315 AU/day 0.6660 c 0.7459 days 347.6839 days
389 days 115.560 AU/day 115.479 AU/day 0.6669 c 0.7451 days 348.4290 days
390 days 115.723 AU/day 115.642 AU/day 0.6679 c 0.7443 days 349.1732 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
391 days 115.885 AU/day 115.804 AU/day 0.6688 c 0.7434 days 349.9167 days
392 days 116.047 AU/day 115.966 AU/day 0.6698 c 0.7426 days 350.6592 days
393 days 116.208 AU/day 116.128 AU/day 0.6707 c 0.7417 days 351.4010 days
394 days 116.369 AU/day 116.289 AU/day 0.6716 c 0.7409 days 352.1419 days
395 days 116.530 AU/day 116.449 AU/day 0.6726 c 0.7400 days 352.8819 days
396 days 116.690 AU/day 116.610 AU/day 0.6735 c 0.7392 days 353.6211 days
397 days 116.849 AU/day 116.769 AU/day 0.6744 c 0.7384 days 354.3595 days
398 days 117.008 AU/day 116.929 AU/day 0.6753 c 0.7375 days 355.0970 days
399 days 117.167 AU/day 117.088 AU/day 0.6762 c 0.7367 days 355.8337 days
400 days 117.325 AU/day 117.246 AU/day 0.6772 c 0.7358 days 356.5695 days
401 days 117.483 AU/day 117.404 AU/day 0.6781 c 0.7350 days 357.3045 days
402 days 117.640 AU/day 117.562 AU/day 0.6790 c 0.7342 days 358.0387 days
403 days 117.797 AU/day 117.719 AU/day 0.6799 c 0.7333 days 358.7720 days
404 days 117.953 AU/day 117.875 AU/day 0.6808 c 0.7325 days 359.5045 days
405 days 118.109 AU/day 118.031 AU/day 0.6817 c 0.7316 days 360.2361 days
tc[1 - Rct](Vt-1+Vt)
2		VAve
173.145 		√(1-dc2)ΣIncr.
406 days 118.265 AU/day 118.187 AU/day 0.6826 c 0.7308 days 360.9669 days
407 days 118.420 AU/day 118.343 AU/day 0.6835 c 0.7300 days 361.6969 days
408 days 118.575 AU/day 118.497 AU/day 0.6844 c 0.7291 days 362.4260 days
409 days 118.729 AU/day 118.652 AU/day 0.6853 c 0.7283 days 363.1543 days
410 days 118.883 AU/day 118.806 AU/day 0.6862 c 0.7274 days 363.8817 days
411 days 119.036 AU/day 118.959 AU/day 0.6871 c 0.7266 days 364.6083 days
412 days 119.189 AU/day 119.113 AU/day 0.6879 c 0.7258 days 365.3341 days
413 days 119.341 AU/day 119.265 AU/day 0.6888 c 0.7249 days 366.0590 days
414 days 119.494 AU/day 119.417 AU/day 0.6897 c 0.7241 days 366.7831 days
415 days 119.645 AU/day 119.569 AU/day 0.6906 c 0.7233 days 367.5064 days
416 days 119.796 AU/day 119.721 AU/day 0.6914 c 0.7224 days 368.2288 days
417 days 119.947 AU/day 119.872 AU/day 0.6923 c 0.7216 days 368.9504 days
418 days 120.097 AU/day 120.022 AU/day 0.6932 c 0.7208 days 369.6712 days
419 days 120.247 AU/day 120.172 AU/day 0.6941 c 0.7199 days 370.3911 days
420 days 120.397 AU/day 120.322 AU/day 0.6949 c 0.7191 days 371.1102 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
421 days 120.546 AU/day 120.471 AU/day 0.6958 c 0.7183 days 371.8284 days
422 days 120.695 AU/day 120.620 AU/day 0.6966 c 0.7174 days 372.5459 days
423 days 120.843 AU/day 120.769 AU/day 0.6975 c 0.7166 days 373.2624 days
424 days 120.991 AU/day 120.917 AU/day 0.6984 c 0.7158 days 373.9782 days
425 days 121.138 AU/day 121.064 AU/day 0.6992 c 0.7149 days 374.6931 days
426 days 121.285 AU/day 121.211 AU/day 0.7001 c 0.7141 days 375.4072 days
427 days 121.431 AU/day 121.358 AU/day 0.7009 c 0.7133 days 376.1205 days
428 days 121.578 AU/day 121.505 AU/day 0.7018 c 0.7124 days 376.8329 days
429 days 121.723 AU/day 121.650 AU/day 0.7026 c 0.7116 days 377.5445 days
430 days 121.869 AU/day 121.796 AU/day 0.7034 c 0.7108 days 378.2552 days
431 days 122.014 AU/day 121.941 AU/day 0.7043 c 0.7099 days 378.9652 days
432 days 122.158 AU/day 122.086 AU/day 0.7051 c 0.7091 days 379.6743 days
433 days 122.302 AU/day 122.230 AU/day 0.7059 c 0.7083 days 380.3825 days
434 days 122.446 AU/day 122.374 AU/day 0.7068 c 0.7074 days 381.0900 days
435 days 122.589 AU/day 122.517 AU/day 0.7076 c 0.7066 days 381.7966 days
tc[1 - Rct](Vt-1+Vt)
2		VAve
173.145 		√(1-dc2)ΣIncr.
436 days 122.732 AU/day 122.661 AU/day 0.7084 c 0.7058 days 382.5024 days
437 days 122.874 AU/day 122.803 AU/day 0.7093 c 0.7050 days 383.2073 days
438 days 123.017 AU/day 122.945 AU/day 0.7101 c 0.7041 days 383.9115 days
439 days 123.158 AU/day 123.087 AU/day 0.7109 c 0.7033 days 384.6148 days
440 days 123.299 AU/day 123.229 AU/day 0.7117 c 0.7025 days 385.3172 days
441 days 123.440 AU/day 123.370 AU/day 0.7125 c 0.7016 days 386.0189 days
442 days 123.581 AU/day 123.511 AU/day 0.7133 c 0.7008 days 386.7197 days
443 days 123.721 AU/day 123.651 AU/day 0.7141 c 0.7000 days 387.4197 days
444 days 123.861 AU/day 123.791 AU/day 0.7150 c 0.6992 days 388.1189 days
445 days 124.000 AU/day 123.930 AU/day 0.7158 c 0.6983 days 388.8172 days
446 days 124.139 AU/day 124.069 AU/day 0.7166 c 0.6975 days 389.5147 days
447 days 124.277 AU/day 124.208 AU/day 0.7174 c 0.6967 days 390.2114 days
448 days 124.415 AU/day 124.346 AU/day 0.7182 c 0.6959 days 390.9073 days
449 days 124.553 AU/day 124.484 AU/day 0.7190 c 0.6951 days 391.6024 days
450 days 124.690 AU/day 124.622 AU/day 0.7198 c 0.6942 days 392.2966 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
451 days 124.827 AU/day 124.759 AU/day 0.7205 c 0.6934 days 392.9900 days
452 days 124.964 AU/day 124.896 AU/day 0.7213 c 0.6926 days 393.6826 days
453 days 125.100 AU/day 125.032 AU/day 0.7221 c 0.6918 days 394.3744 days
454 days 125.236 AU/day 125.168 AU/day 0.7229 c 0.6909 days 395.0653 days
455 days 125.371 AU/day 125.303 AU/day 0.7237 c 0.6901 days 395.7554 days
456 days 125.506 AU/day 125.439 AU/day 0.7245 c 0.6893 days 396.4447 days
457 days 125.641 AU/day 125.573 AU/day 0.7253 c 0.6885 days 397.1332 days
458 days 125.775 AU/day 125.708 AU/day 0.7260 c 0.6877 days 397.8209 days
459 days 125.909 AU/day 125.842 AU/day 0.7268 c 0.6868 days 398.5077 days
460 days 126.042 AU/day 125.976 AU/day 0.7276 c 0.6860 days 399.1938 days
461 days 126.175 AU/day 126.109 AU/day 0.7283 c 0.6852 days 399.8790 days
462 days 126.308 AU/day 126.242 AU/day 0.7291 c 0.6844 days 400.5634 days
463 days 126.441 AU/day 126.374 AU/day 0.7299 c 0.6836 days 401.2470 days
464 days 126.573 AU/day 126.507 AU/day 0.7306 c 0.6828 days 401.9297 days
465 days 126.704 AU/day 126.638 AU/day 0.7314 c 0.6819 days 402.6117 days
tc[1 - Rct](Vt-1+Vt)
2		VAve
173.145 		√(1-dc2)ΣIncr.
466 days 126.835 AU/day 126.770 AU/day 0.7322 c 0.6811 days 403.2928 days
467 days 126.966 AU/day 126.901 AU/day 0.7329 c 0.6803 days 403.9731 days
468 days 127.097 AU/day 127.032 AU/day 0.7337 c 0.6795 days 404.6526 days
469 days 127.227 AU/day 127.162 AU/day 0.7344 c 0.6787 days 405.3313 days
470 days 127.357 AU/day 127.292 AU/day0.7352 c 0.6779 days406.0092days
471 days 127.486 AU/day 127.421 AU/day 0.7359 c 0.6771 days 406.6863 days
472 days 127.615 AU/day 127.551 AU/day 0.7367 c 0.6763 days 407.3625 days
473 days 127.744 AU/day 127.679 AU/day 0.7374 c 0.6754 days 408.0379 days
474 days 127.872 AU/day 127.808 AU/day 0.7382 c 0.6746 days 408.7126 days
475 days 128.000 AU/day 127.936 AU/day 0.7389 c 0.6738 days 409.3864 days
476 days 128.128 AU/day 128.064 AU/day 0.7396 c 0.6730 days 410.0594 days
477 days 128.255 AU/day 128.191 AU/day 0.7404 c 0.6722 days 410.7316 days
478 days 128.382 AU/day 128.318 AU/day 0.7411 c 0.6714 days 411.4030 days
479 days 128.508 AU/day 128.445 AU/day 0.7418 c 0.6706 days 412.0736 days
480 days 128.634 AU/day 128.571 AU/day 0.7426 c 0.6698 days 412.7434 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
481 days 128.760 AU/day 128.697 AU/day 0.7433 c 0.6690 days 413.4123 days
482 days 128.886 AU/day 128.823 AU/day 0.7440 c 0.6682 days 414.0805 days
483 days 129.011 AU/day 128.948 AU/day 0.7447 c 0.6674 days 414.7478 days
484 days 129.135 AU/day 129.073 AU/day 0.7455 c 0.6665 days 415.4144 days
485 days 129.260 AU/day 129.198 AU/day 0.7462 c 0.6657 days 416.0801 days
486 days 129.384 AU/day 129.322 AU/day 0.7469 c 0.6649 days 416.7451 days
487 days 129.507 AU/day 129.446 AU/day 0.7476 c 0.6641 days 417.4092 days
488 days 129.631 AU/day 129.569 AU/day 0.7483 c 0.6633 days 418.0725 days
489 days 129.754 AU/day 129.692 AU/day 0.7490 c 0.6625 days 418.7351 days
490 days 129.876 AU/day 129.815 AU/day 0.7497 c 0.6617 days 419.3968 days
491 days 129.999 AU/day 129.937 AU/day 0.7505 c 0.6609 days 420.0577 days
492 days 130.121 AU/day 130.060 AU/day 0.7512 c 0.6601 days 420.7178 days
493 days 130.242 AU/day 130.181 AU/day 0.7519 c 0.6593 days 421.3771 days
494 days 130.363 AU/day 130.303 AU/day 0.7526 c 0.6585 days 422.0357 days
495 days 130.484 AU/day 130.424 AU/day 0.7533 c 0.6577 days 422.6934 days
tc[1 - Rct](Vt-1+Vt)
2		VAve
173.145 		√(1-dc2)ΣIncr.
496 days 130.605 AU/day 130.545 AU/day 0.7540 c 0.6569 days 423.3503 days
497 days 130.725 AU/day 130.665 AU/day 0.7547 c 0.6561 days 424.0064days
498 days 130.845 AU/day 130.785 AU/day 0.7553 c 0.6553 days 424.6617 days
499 days 130.964 AU/day 130.905 AU/day 0.7560 c 0.6545 days 425.3163 days
500 days 131.084 AU/day 131.024 AU/day 0.7567 c 0.6537 days 425.9700 days
501 days 131.203 AU/day 131.143 AU/day 0.7574 c 0.6529 days 426.6229 days
502 days 131.321 AU/day 131.262 AU/day 0.7581 c 0.6521 days 427.2751 days
503 days 131.439 AU/day 131.380 AU/day 0.7588 c 0.6513 days 427.9264 days
504 days 131.557 AU/day 131.498 AU/day 0.7595 c 0.6505 days 428.5769 days
505 days 131.675 AU/day 131.616 AU/day 0.7601 c 0.6497 days 429.2267 days
506 days 131.792 AU/day 131.733 AU/day 0.7608 c 0.6490 days 429.8757 days
507 days 131.909 AU/day 131.850 AU/day 0.7615 c 0.6482 days 430.5238 days
508 days 132.025 AU/day 131.967 AU/day 0.7622 c 0.6474 days 431.1712 days
509 days 132.141 AU/day 132.083 AU/day 0.7628 c 0.6466 days 431.8178 days
510 days 132.257 AU/day 132.199 AU/day 0.7635 c 0.6458 days 432.4635 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
511 days 132.373 AU/day 132.315 AU/day 0.7642 c 0.6450 days 433.1085 days
512 days 132.488 AU/day 132.430 AU/day 0.7649 c 0.6442 days 433.7527 days
513 days 132.603 AU/day 132.546 AU/day 0.7655 c 0.6434 days 434.3962 days
514 days 132.718 AU/day 132.660 AU/day 0.7662 c 0.6426 days 435.0388 days
515 days 132.832 AU/day 132.775 AU/day 0.7668 c 0.6418 days 435.6806 days
516 days 132.946 AU/day 132.889 AU/day 0.7675 c 0.6410 days 436.3217 days
517 days 133.059 AU/day 133.003 AU/day 0.7682 c 0.6403 days 436.9619 days
518 days 133.173 AU/day 133.116 AU/day 0.7688 c 0.6395 days 437.6014 days
519 days 133.286 AU/day 133.229 AU/day 0.7695 c 0.6387 days 438.2401 days
520 days 133.398 AU/day 133.342 AU/day 0.7701 c 0.6379 days 438.8780 days
521 days 133.511 AU/day 133.454 AU/day 0.7708 c 0.6371 days 439.5151 days
522 days 133.623 AU/day 133.567 AU/day 0.7714 c 0.6363 days 440.1515 days
523 days 133.734 AU/day 133.678 AU/day 0.7721 c 0.6355 days 440.7870 days
524 days 133.846 AU/day 133.790 AU/day 0.7727 c 0.6348 days 441.4218 days
525 days 133.957 AU/day 133.901 AU/day 0.7733 c 0.6340 days 442.0558 days
tc[1 - Rct](Vt-1+Vt)
2		VAve
173.145 		√(1-dc2)ΣIncr.
526 days 134.067 AU/day 134.012 AU/day 0.7740 c 0.6332 days 442.6890 days
527 days 134.178 AU/day 134.123 AU/day 0.7746 c 0.6324 days 443.3214 days
528 days 134.288 AU/day 134.233 AU/day 0.7753 c 0.6316 days 443.9530 days
529 days 134.398 AU/day 134.343 AU/day 0.7759 c 0.6309 days 444.5839 days
530 days 134.507 AU/day 134.453 AU/day 0.7765 c 0.6301 days 445.2139 days
531 days 134.616 AU/day 134.562 AU/day 0.7772 c 0.6293 days 445.8432 days
532 days 134.725 AU/day 134.671 AU/day 0.7778 c 0.6285 days 446.4718 days
533 days 134.834 AU/day 134.780 AU/day 0.7784 c 0.6277 days 447.0995 days
534 days 134.942 AU/day 134.888 AU/day 0.7790 c 0.6270 days 447.7265 days
535 days 135.050 AU/day 134.996 AU/day 0.7797 c 0.6262 days 448.3527 days
536 days 135.158 AU/day 135.104 AU/day 0.7803 c 0.6254 days 448.9781 days
537 days 135.265 AU/day 135.211 AU/day 0.7809 c 0.6246 days 449.6027 days
538 days 135.372 AU/day 135.319 AU/day 0.7815 c 0.6239 days 450.2266 days
539 days 135.479 AU/day 135.426 AU/day 0.7822 c 0.6231 days 450.8497 days
540 days 135.585 AU/day 135.532 AU/day 0.7828 c 0.6223 days 451.4720 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
541 days 135.692 AU/day 135.638 AU/day 0.7834 c 0.6215 days 452.0935 days
542 days 135.797 AU/day 135.744 AU/day 0.7840 c 0.6208 days 452.7143 days
543 days 135.903 AU/day 135.850 AU/day 0.7846 c 0.6200 days 453.3343 days
544 days 136.008 AU/day 135.956 AU/day 0.7852 c 0.6192 days 453.9535 days
545 days 136.113 AU/day 136.061 AU/day 0.7858 c 0.6185 days 454.5720 days
546 days 136.218 AU/day 136.165 AU/day 0.7864 c 0.6177 days 455.1897 days
547 days 136.322 AU/day 136.270 AU/day 0.7870 c 0.6169 days 455.8066 days
548 days 136.426 AU/day 136.374 AU/day 0.7876 c 0.6161 days 456.4227 days
549 days 136.530 AU/day 136.478 AU/day 0.7882 c 0.6154 days 457.0381 days
550 days 136.633 AU/day 136.582 AU/day 0.7888 c 0.6146 days 457.6527 days
551 days 136.737 AU/day 136.685 AU/day 0.7894 c 0.6138 days 458.2666 days
552 days 136.840 AU/day 136.788 AU/day 0.7900 c 0.6131 days 458.8797 days
553 days 136.942 AU/day 136.891 AU/day 0.7906 c 0.6123 days 459.4920 days
554 days 137.044 AU/day 136.993 AU/day 0.7912 c 0.6116 days 460.1035 days
555 days 137.146 AU/day 137.095 AU/day 0.7918 c 0.6108 days 460.7143 days
tc[1 - Rct](Vt-1+Vt)
2		VAve
173.145 		√(1-dc2)ΣIncr.
556 days 137.248 AU/day 137.197 AU/day 0.7924 c 0.6100 days 461.3243 days
557 days 137.350 AU/day 137.299 AU/day 0.7930 c 0.6093 days 461.9336 days
558 days 137.451 AU/day 137.400 AU/day 0.7936 c 0.6085 days 462.5421 days
559 days 137.552 AU/day 137.501 AU/day 0.7941 c 0.6077 days 463.1498 days
560 days 137.652 AU/day 137.602 AU/day 0.7947 c 0.6070 days 463.7568 days
561 days 137.753 AU/day 137.702 AU/day 0.7953 c 0.6062 days 464.3630 days
562 days 137.853 AU/day 137.803 AU/day 0.7959 c 0.6055 days 464.9685 days
563 days 137.952 AU/day 137.902 AU/day 0.7965 c 0.6047 days 465.5732 days
564 days 138.052 AU/day 138.002 AU/day 0.7970 c 0.6039 days 466.1771 days
565 days 138.151 AU/day 138.101 AU/day 0.7976 c 0.6032 days 466.7803 days
566 days 138.250 AU/day 138.200 AU/day 0.7982 c 0.6024 days 467.3827 days
567 days 138.348 AU/day 138.299 AU/day 0.7987 c 0.6017 days 467.9844 days
568 days 138.447 AU/day 138.398 AU/day 0.7993 c 0.6009 days 468.5853 days
569 days 138.545 AU/day 138.496 AU/day 0.7999 c 0.6002 days 469.1854 days
570 days 138.643 AU/day 138.594 AU/day 0.8004 c 0.5994 days 469.7848 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
571 days 138.740 AU/day 138.691 AU/day 0.8010 c 0.5986 days 470.3835 days
572 days 138.837 AU/day 138.789 AU/day 0.8016 c 0.5979 days 470.9814 days
573 days 138.934 AU/day 138.886 AU/day 0.8021 c 0.5971 days 471.5785 days
574 days 139.031 AU/day 138.983 AU/day 0.8027 c 0.5964 days 472.1749 days
575 days 139.127 AU/day 139.079 AU/day 0.8033 c 0.5956 days 472.7705 days
576 days 139.224 AU/day 139.175 AU/day 0.8038 c 0.5949 days 473.3654 days
577 days 139.319 AU/day 139.271 AU/day 0.8044 c 0.5941 days 473.9596 days
578 days 139.415 AU/day 139.367 AU/day 0.8049 c 0.5934 days 474.5530 days
579 days 139.510 AU/day 139.463 AU/day 0.8055 c 0.5926 days 475.1456 days
580 days 139.605 AU/day 139.558 AU/day 0.8060 c 0.5919 days 475.7375 days
581 days 139.700 AU/day 139.653 AU/day 0.8066 c 0.5911 days 476.3286 days
582 days 139.795 AU/day 139.747 AU/day 0.8071 c 0.5904 days 476.9190 days
583 days 139.889 AU/day 139.842 AU/day 0.8077 c 0.5897 days 477.5087 days
584 days 139.983 AU/day 139.936 AU/day 0.8082 c 0.5889 days 478.0976 days
585 days 140.077 AU/day 140.030 AU/day 0.8087 c 0.5882 days 478.6858 days
tc[1 - Rct](Vt-1+Vt)
2		VAve
173.145 		√(1-dc2)ΣIncr.
586 days 140.170 AU/day 140.123 AU/day 0.8093 c 0.5874 days 479.2732 days
587 days 140.263 AU/day 140.217 AU/day 0.8098 c 0.5867 days 479.8599 days
588 days 140.356 AU/day 140.310 AU/day 0.8104 c 0.5859 days 480.4458 days
589 days 140.449 AU/day 140.402 AU/day 0.8109 c 0.5852 days 481.0310 days
590 days 140.541 AU/day 140.495 AU/day 0.8114 c 0.5845 days 481.6154 days
591 days 140.633 AU/day 140.587 AU/day 0.8120 c 0.5837 days 482.1991 days
592 days 140.725 AU/day 140.679 AU/day 0.8125 c 0.5830 days 482.7821 days
593 days 140.817 AU/day 140.771 AU/day 0.8130 c 0.5822 days 483.3643 days
594 days 140.908 AU/day 140.863 AU/day 0.8136 c 0.5815 days 483.9458 days
595 days 140.999 AU/day 140.954 AU/day 0.8141 c 0.5808 days 484.5266 days
596 days 141.090 AU/day 141.045 AU/day 0.8146 c 0.5800 days 485.1066 days
597 days 141.181 AU/day 141.135 AU/day 0.8151 c 0.5793 days 485.6859 days
598 days 141.271 AU/day 141.226 AU/day 0.8157 c 0.5785 days 486.2644 days
599 days 141.361 AU/day 141.316 AU/day 0.8162 c 0.5778 days 486.8422 days
600 days 141.451 AU/day 141.406 AU/day 0.8167 c 0.5771 days 487.4193 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
601 days 141.541 AU/day 141.496 AU/day 0.8172 c 0.5763 days 487.9957 days
602 days 141.630 AU/day 141.585 AU/day 0.8177 c 0.5756 days 488.5713 days
603 days 141.719 AU/day 141.674 AU/day 0.8182 c 0.5749 days 489.1461 days
604 days 141.808 AU/day 141.763 AU/day 0.8188 c 0.5741 days 489.7203 days
605 days 141.896 AU/day 141.852 AU/day 0.8193 c 0.5734 days 490.2937 days
606 days 141.985 AU/day 141.940 AU/day 0.8198 c 0.5727 days 490.8664 days
607 days 142.073 AU/day 142.029 AU/day 0.8203 c 0.5720 days 491.4383 days
608 days 142.160 AU/day 142.117 AU/day 0.8208 c 0.5712 days 492.0095 days
609 days 142.248 AU/day 142.204 AU/day 0.8213 c 0.5705 days 492.5800 days
610 days 142.335 AU/day 142.292 AU/day 0.8218 c 0.5698 days 493.1498 days
611 days 142.422 AU/day 142.379 AU/day 0.8223 c 0.5690 days 493.7188 days
612 days 142.509 AU/day 142.466 AU/day 0.8228 c 0.5683 days 494.2872 days
613 days 142.596 AU/day 142.552 AU/day 0.8233 c 0.5676 days 494.8548 days
614 days 142.682 AU/day 142.639 AU/day 0.8238 c 0.5669 days 495.4216 days
615 days 142.768 AU/day 142.725 AU/day 0.8243 c 0.5661 days 495.9878 days
tc[1 - Rct](Vt-1+Vt)
2		VAve
173.145 		√(1-dc2)ΣIncr.
616 days 142.854 AU/day 142.811 AU/day 0.8248 c 0.5654 days 496.5532 days
617 days 142.940 AU/day 142.897 AU/day 0.8253 c 0.5647 days 497.1179 days
618 days 143.025 AU/day 142.982 AU/day 0.8258 c 0.5640 days 497.6818 days
619 days 143.110 AU/day 143.068 AU/day 0.8263 c 0.5632 days 498.2451 days
620 days 143.195 AU/day 143.153 AU/day 0.8268 c 0.5625 days 498.8076 days
621 days 143.280 AU/day 143.237 AU/day 0.8273 c 0.5618 days 499.3694 days
622 days 143.364 AU/day 143.322 AU/day 0.8278 c 0.5611 days 499.9305 days
623 days 143.448 AU/day 143.406 AU/day 0.8282 c 0.5604 days 500.4909days
624 days 143.532 AU/day 143.490 AU/day 0.8287 c 0.5597 days 501.0505 days
625 days 143.616 AU/day 143.574 AU/day 0.8292 c 0.5589 days 501.6095 days
626 days 143.699 AU/day 143.658 AU/day 0.8297 c 0.5582 days 502.1677 days
627 days 143.782 AU/day 143.741 AU/day 0.8302 c 0.5575 days 502.7252 days
628 days 143.865 AU/day 143.824 AU/day 0.8307 c 0.5568 days 503.2820 days
629 days 143.948 AU/day 143.907 AU/day 0.8311 c 0.5561 days 503.8380days
630 days 144.031 AU/day 143.989 AU/day 0.8316 c 0.5554 days 504.3934 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
631 days 144.113 AU/day 144.072 AU/day 0.8321 c 0.5546 days 504.9480days
632 days 144.195 AU/day 144.154 AU/day 0.8326 c 0.5539 days 505.5020 days
633 days 144.277 AU/day 144.236 AU/day 0.8330 c 0.5532 days 506.0552 days
634 days 144.358 AU/day 144.318 AU/day 0.8335 c 0.5525 days 506.6077 days
635 days 144.440 AU/day 144.399 AU/day 0.8340 c 0.5518 days 507.1595 days
636 days 144.521 AU/day 144.480 AU/day 0.8344 c 0.5511 days 507.7106 days
637 days 144.602 AU/day 144.561 AU/day 0.8349 c 0.5504 days 508.2610 days
638 days 144.682 AU/day 144.642 AU/day 0.8354 c 0.5497 days 508.8106 days
639 days 144.763 AU/day 144.723 AU/day 0.8358 c 0.5490 days 509.3596 days
640 days 144.843 AU/day 144.803 AU/day 0.8363 c 0.5483 days 509.9078 days
641 days 144.923 AU/day 144.883 AU/day 0.8368 c 0.5475 days 510.4554 days
642 days 145.003 AU/day 144.963 AU/day 0.8372 c 0.5468 days 511.0022 days
643 days 145.082 AU/day 145.043 AU/day 0.8377 c 0.5461 days 511.5484 days
644 days 145.162 AU/day 145.122 AU/day 0.8382 c 0.5454 days 512.0938 days
645 days 145.241 AU/day 145.201 AU/day 0.8386 c 0.5447 days 512.6385 days
tc[1 - Rct](Vt-1+Vt)
2		VAve
173.145 		√(1-dc2)ΣIncr.
646 days 145.320 AU/day 145.280 AU/day 0.8391 c 0.5440 days 513.1826 days
647 days 145.398 AU/day 145.359 AU/day 0.8395 c 0.5433 days 513.7259 days
648 days 145.477 AU/day 145.437 AU/day 0.8400 c 0.5426 days 514.2685 days
649 days 145.555 AU/day 145.516 AU/day 0.8404 c 0.5419 days 514.8104 days
650 days 145.633 AU/day 145.594 AU/day 0.8409 c 0.5412 days 515.3517 days
651 days 145.711 AU/day 145.672 AU/day 0.8413 c 0.5405 days 515.8922 days
652 days 145.788 AU/day 145.749 AU/day 0.8418 c 0.5398 days 516.4320 days
653 days 145.865 AU/day 145.827 AU/day 0.8422 c 0.5391 days 516.9712 days
654 days 145.942 AU/day 145.904 AU/day 0.8427 c 0.5384 days 517.5096 days
655 days 146.019 AU/day 145.981 AU/day 0.8431 c 0.5377 days 518.0473 days
656 days 146.096 AU/day 146.058 AU/day 0.8436 c 0.5370 days 518.5844 days
657 days 146.172 AU/day 146.134 AU/day 0.8440 c 0.5363 days 519.1207 days
658 days 146.249 AU/day 146.211 AU/day 0.8444 c 0.5356 days 519.6564 days
659 days 146.325 AU/day 146.287 AU/day 0.8449 c 0.5350 days 520.1913 days
660 days 146.400 AU/day 146.363 AU/day 0.8453 c 0.5343 days 520.7256 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
661 days 146.476 AU/day 146.438 AU/day 0.8458 c 0.5336 days 521.2591 days
662 days 146.551 AU/day 146.514 AU/day 0.8462 c 0.5329 days 521.7920 days
663 days 146.627 AU/day 146.589 AU/day 0.8466 c 0.5322 days 522.3242 days
664 days 146.702 AU/day 146.664 AU/day 0.8471 c 0.5315 days 522.8557 days
665 days 146.776 AU/day 146.739 AU/day 0.8475 c 0.5308 days 523.3865 days
666 days 146.851 AU/day 146.814 AU/day 0.8479 c 0.5301 days 523.9166 days
667 days 146.925 AU/day 146.888 AU/day 0.8484 c 0.5294 days 524.4461 days
668 days 146.999 AU/day 146.962 AU/day 0.8488 c 0.5287 days 524.9748 days
669 days 147.073 AU/day 147.036 AU/day 0.8492 c 0.5281 days 525.5029 days
670 days 147.147 AU/day 147.110 AU/day 0.8496 c 0.5274 days 526.0302 days
671 days 147.220 AU/day 147.183 AU/day 0.8501 c 0.5267 days 526.5569 days
672 days 147.293 AU/day 147.257 AU/day 0.8505 c 0.5260 days 527.0829 days
673 days 147.367 AU/day 147.330 AU/day 0.8509 c 0.5253 days 527.6083 days
674 days 147.439 AU/day 147.403 AU/day 0.8513 c 0.5246 days 528.1329 days
675 days 147.512 AU/day 147.476 AU/day 0.8517 c 0.5240 days 528.6568 days
tc[1 - Rct](Vt-1+Vt)
2		VAve
173.145 		√(1-dc2)ΣIncr.
676 days 147.584 AU/day 147.548 AU/day 0.8522 c 0.5233 days 529.1801 days
677 days 147.657 AU/day 147.621 AU/day 0.8526 c 0.5226 days 529.7027 days
678 days 147.729 AU/day 147.693 AU/day 0.8530 c 0.5219 days 530.2246 days
679 days 147.801 AU/day 147.765 AU/day 0.8534 c 0.5212 days 530.7459 days
680 days 147.872 AU/day 147.836 AU/day 0.8538 c 0.5206 days 531.2664 days
681 days 147.944 AU/day 147.908 AU/day 0.8542 c 0.5199 days 531.7863 days
682 days 148.015 AU/day 147.979 AU/day 0.8547 c 0.5192 days 532.3055 days
683 days 148.086 AU/day 148.050 AU/day 0.8551 c 0.5185 days 532.8240 days
684 days 148.157 AU/day 148.121 AU/day 0.8555 c 0.5178 days 533.3418 days
685 days 148.227 AU/day 148.192 AU/day 0.8559 c 0.5172 days 533.8590 days
686 days 148.298 AU/day 148.262 AU/day 0.8563 c 0.5165 days 534.3755 days
687 days 148.368 AU/day 148.333 AU/day 0.8567 c 0.5158 days 534.8913 days
688 days 148.438 AU/day 148.403 AU/day 0.8571 c 0.5151 days 535.4065 days
689 days 148.508 AU/day 148.473 AU/day 0.8575 c 0.5145 days 535.9209 days
690 days 148.577 AU/day 148.543 AU/day 0.8579 c 0.5138 days 536.4347 days
t Spot Vel Ave AU/dy Ave dc Incr. Cum.
691 days 148.647 AU/day 148.612 AU/day 0.8583 c 0.5131 days 536.9479 days
692 days 148.716 AU/day 148.681 AU/day 0.8587 c 0.5125 days 537.4603 days
693 days 148.785 AU/day 148.751 AU/day 0.8591 c 0.5118 days 537.9721 days
694 days 148.854 AU/day 148.820 AU/day 0.8595 c 0.5111 days 538.4832 days
695 days 148.923 AU/day 148.888 AU/day 0.8599 c 0.5105 days 538.9937 days
696 days 148.991 AU/day 148.957 AU/day 0.8603 c 0.5098 days 539.5035 days
697 days 149.059 AU/day 149.025 AU/day 0.8607 c 0.5091 days 540.0126 days
698 days 149.127 AU/day 149.093 AU/day 0.8611 c 0.5085 days 540.5211 days
699 days 149.195 AU/day 149.161 AU/day 0.8615 c 0.5078 days 541.0289 days
700 days 149.263 AU/day 149.229 AU/day 0.8619 c 0.5071 days 541.5360 days
701 days 149.330 AU/day 149.297 AU/day 0.8623 c 0.5065 days 542.0424 days
702 days 149.398 AU/day 149.364 AU/day 0.8627 c 0.5058 days 542.5482 days
703 days 149.465 AU/day 149.431 AU/day 0.8630 c 0.5051 days 543.0534 days
704 days 149.532 AU/day 149.498 AU/day 0.8634 c 0.5045 days 543.5578 days
705 days 149.598 AU/day 149.565 AU/day 0.8638 c 0.5038 days 544.0616 days
706 days 149.665 AU/day 149.632 AU/day 0.8642 c 0.5032 days 544.5648 days
707 days 149.731 AU/day 149.698 AU/day 0.8646 c 0.5025 days 545.0673 days
708 days 149.798 AU/day 149.764 AU/day 0.8650 c 0.5018 days 545.5691 days
709 days 149.864 AU/day 149.831 AU/day 0.8653 c 0.5012 days 546.0703 days
710 days 149.929 AU/day 149.896 AU/day 0.8657 c 0.5005 days 546.5708 days
711 days 149.995 AU/day 149.962 AU/day 0.8661 c 0.4999 days 547.0707 days
712 days 150.060 AU/day 150.028 AU/day 0.8665 c 0.4992 days 547.5699 days
tc[1 - Rct](Vt-1+Vt)
2		VAve
173.145 		 √(1-dc2) ΣIncr.












1) G-force for a sec. 
Let an object free fall for one second near Earth's surface, and it will gain a velocity of about 10 m/sec (actually 9.8065 m/s). Let it continue free fall for a 2nd and 3rd second, and velocity will become about 20 m/s and 30 m/s respectively.  SUMMARY:  Velocity will increase about 10 m/s for every 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		=86,400 sec
day
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		=847,281.6 m
sec
This velocity can be conveniently converted into other equivalent values.
847,281.6 m
sec		×km
1,000 m		=847.28 km
sec
847.281.6 km
sec		×c
299,792.458 km		 =0.2826%c
Daily Difference (Δ). After one day of g-force propulsion, vessel achieves a small portion of light speed. Thought Experiment (TE) calls this percentage the daily difference.
Δ= 0.2826%c
3) G-force for 712 days
Phase 1. Accelerate for 712 days at g-force. This achieves a high cruise speed to shorten flight duration and also simulates gravity for contents and occupants. Since g-force comes from particle flow leaving aft of vessel, occupants feel a gravity like force likely from decks (axial planes intersecting the longitudinal axis). Thus, "up" is toward forward end of vessel which is pointing toward AC, our destination.
Remainder. In contrast to the daily difference, TE designates the daily remainder (Rc):
Rc=c - Δ = .997174c
as 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 to ever approach c but never attain it.
VShip=c[1 - Rct] 
To obtain velocities as Astronomical Units per day (AU/dy), a convenient measure for g-force 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 712 days of g-force acceleration:
VShip=173.145 AU
day		×[1 - .997174365]=150.060 AU
day		=86.7%c
4) Cruise for several years
In this example of an interstellar profile, g-force vessel accelerates for 712 days to achieve a cruise velocity of VCru  = 86.7%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.
Phase 2. Cruise for 909.5 days (about 2.5 years) at .866c. TE assumes that even with augmentation from snowballs, vessel must still conserve fuel.  Thus, vessel must "cruise" at constant speed for middle portion of flight; without G-force acceleration, flight planners must determine another way to simulate Earth gravity. This is done by spinning craft in such a manner to produce centrifugal g-force against inside of outer hull. Thus, from Phase I to Phase II,, ship must transform from living area on decks (floors perpendicular to exhaust flow of particles from aft end of vessel) to living area on hull. Thus, occupants must quickly adjust from decks being floors and hull being wall to decks becoming walls and hull becoming floor. Thus, "up" is toward the central, longitudinal axis of the cylindrical vessel. (While Phase I's up would be parallel for all occupants, Phase II's up would differ for every occupant.)
5) Reverse g-force for final 712 days
During the cruise phase, the vessel makes excellent time due to its relativistic speed (86.67%c = 259,818.131 km/sec); however, the g-force vessel must slow down (to a few km/sec) to approach and orbit objects in Alpha Centauri system.
Thus, TE envisions vessel again powering the G-force propulsion system to reverse 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 1.11 LY, the voyage's final year would consist of g-force deceleration for the same distance.
Phase 3. Decelerate for 712 day at g-force; ship must slowdown from enormous velocity (86.6% of light speed) to operational velocity required to maneuver around AC system. Thus, hull becomes a wall again, and decks once more becomes a floor. Of course, exhaust flow must point toward destination because the exhaust force must now counteract motion of vessel. (Of course, "up" once again points forward, but the forward end of vessel is now pointing toward Sol, our place of departure.)
6)  Cruise Time Dilation
Since we assume an interstellar vessel travels at constant relativistic speed during the cruise portion of flight, it is a straight forward procedure to simply calculate time dilation during cruise portion of voyage.
Previous chapters provide procedures for determining distance during G-force accelerations.
For 712 days of G-force acceleration, TE determines distance to be 1.11 LY. For 712 days of G-force deceleration, TE assumes equal distance.  Since observed distance to AC is 4.38 LY, we assume cruise distance:
dCru = dTtl - dAcc - dDec 
dCru  = 4.38 LY -1.11 LY - 1.11 LY = 2.16 LY
Cruise Time can be easily expressed as both normal time as well as relativistic.
Normal
Time
At vessel velocity of 86.66%c, 
Earth bound observer measures 1.155 years 
for each LY of cruise distance.
tCru=dCru/VCru
tCru=2.16 LY÷.8666c = 2.495 yrs
Relativistic
Time
Due to relativistic effects, 
ship bound chronometer measures:0.5 yrs 
for each year observed by Earth.
tr= tCru×√(1-VCru2/c2)
tr= 2.495 yrs×0.5=1.25yr
However, cruise is only part of the interstellar flight, and time dilation also happens during acceleration and deceleration phases.
7) Time Dilation During Acceleration
Computing time dilation for g-force acceleration more difficult because such velocity is ever increasing.  Thus, TE uses following technique to approximate time dilation for each day of acceleration.
Determine Spot Velocities. To determine spot velocities for end of each travel day (which TE also assumes as the initial spot velocity for next travel day), TE uses following formula as previously described.
VShip = c[1 - Rct] 
Determine Daily Average. To determine average velocity for each acceleration day, TE determines mid velocity.  Next, TE converts this average from AU/day to decimal light speed (c), designated as dc.
Convert vessel velocities from decimal light speed to/from AU/day with conversion constant:  c = 173.145 AU/day.


8) Time Dilation During Deceleration  Since TE assumes that acceleration duration equals deceleration duration, TE further assumes that corresponding time dilations are also equal.  
Determine Daily Dilation. Following diagram shows daily dilations after 365 days of g-force.  Of course, we assume that interstellar vessel can even accelerate well beyond 365 days as fuel permits.


 9) Total Time Dilation
For total time dilation (tr-Ttl), consider relativistic time shrinkage during all three phases of interstellar voyage: g-force acceleration, cruise, and g-force deceleration.
 1) ACCELERATION PHASE
Normal
Acceleration
Time
Earth observer watches 
g-force vessel accelerate 
for 712 days 
for 86.6%c for 1.11 LY.
Given: tAcc = 712 days
Relativistic
Acceleration
Time
Ship bound observer 
measures duration
of 547.6 days.(1.5 years).
tr-Acc= Σ√(1-dc2)
tr-Acc= 547.6 dy = 1.5 yr
2) CRUISE PHASE
Normal
Cruise
Time
Vessel velocity = 86.66%c, 
Earth bound observer 
measures duration of 2.5 yr 
for 2.16 LY cruise distance.
tCru=dCru/VCru
tCru=2.16LY÷.866c=2.5yr
Relativistic
Cruise
Time
Due to relativistic effects, 
ship bound chronometer 
measures:1.25 yrs 
during cruise phase.
tr-Cru= tCru×√(1-VCru2/c2)
tr-Cru= 2.5 yr×0.5 = 1.25 yr
3) DECELERATION PHASE
Normal
Deceleration
Time
Earth bound observer 
watches g-force vessel
decelerate for 712 days.
tDec = tAcc = 712 days
Relativistic
Deceleration
Time
Ship bound observer 
measures duration
of 547.6 days.(1.5 years).
tDec = tAcc
TOTAL TIME FOR TRIP
Total
Normal
Time
tTtl =  tAcc + tCru + tDec
tTtl = 1.95 yr + 2.5 yr +1.95 yr
tTtl =  6.4 yr 
Total
Relativistic
Time
tr-Ttl = tr-Acc+tr-Cru+tr-Dec
tT-r =  1.5 yr + 1.25 yr +1.5 yr
tT-r =  4.25 yr

NOTE: LY = 63,241 AU; c = 173.145AU/day


SUMMARY
Total tr =  tr-Acc + tr-Cru + tr-Dec
EXAMPLE:  G-force journey to the Alpha Centauri system.
tr-Ttl =  547.6 days + 454.75 days  + 547.6 days = 1,550 days
tr-Ttl =  1.5 yr + 1.25 yr + 1.5 yr = 4.25 yr
VOLUME 0: ELEVATIONAL
VOLUME I: ASTEROIDAL
VOLUME II: INTERPLANETARY
VOLUME III: INTERSTELLAR

posted by JimODell at 4:11 AM

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