Mag-sails acting as interstellar brakes follow the equation:
V = Vo/(1 + k.Vo^(1/3).t)^3
…and as we discovered the distance the mag-sail travels while deccelerating from Vo to V during a fixed time, t, is:
s = Vo.t/2 * [ (V/Vo)^(1/3).(1 + (V/Vo)^(1/3)) ]
But the behaviour of a starship isn’t obvious from an equation alone. So here’s a table. What you’re seeing is the velocity broken down into short steps of 0.05 c, except the last, and the time taken for each step, plus the cumulative total. As you can see the decceleration is high – 5.6 gee at the start – and rapidly declining. The final skid from 0.05 c to just 0.0054 c takes a whopping 522 days, while the whole 0.9 c delta-vee prior takes a mere 297 days.
| V1 | V2 | factor | time (days) | time (cumulative) |
| 0.95 | 0.9 | 0.0184994 | 3.239800003 | 3.239800003 |
| 0.9 | 0.85 | 0.019923023 | 3.489119075 | 6.728919078 |
| 0.85 | 0.8 | 0.021550153 | 3.774078302 | 10.50299738 |
| 0.8 | 0.75 | 0.023425071 | 4.102432734 | 14.60543011 |
| 0.75 | 0.7 | 0.025605464 | 4.484284936 | 19.08971505 |
| 0.7 | 0.65 | 0.028167793 | 4.933025557 | 24.02274061 |
| 0.65 | 0.6 | 0.031215428 | 5.466757946 | 29.48949855 |
| 0.6 | 0.55 | 0.034891343 | 6.110520895 | 35.60001945 |
| 0.55 | 0.5 | 0.039398605 | 6.899877768 | 42.49989721 |
| 0.5 | 0.45 | 0.04503483 | 7.886949865 | 50.38684708 |
| 0.45 | 0.4 | 0.052252928 | 9.151055264 | 59.53790234 |
| 0.4 | 0.35 | 0.061774604 | 10.81858634 | 70.35648869 |
| 0.35 | 0.3 | 0.07481817 | 13.10290615 | 83.45939483 |
| 0.3 | 0.25 | 0.09359947 | 16.39207514 | 99.85146998 |
| 0.25 | 0.2 | 0.122574895 | 21.46654131 | 121.3180113 |
| 0.2 | 0.15 | 0.172096111 | 30.13919193 | 151.4572032 |
| 0.15 | 0.1 | 0.272362632 | 47.69886779 | 199.156071 |
| 0.1 | 0.05 | 0.559982927 | 98.06980993 | 297.2258809 |
| 0.05 | 0.0054 | 2.985502206 | 522.8510013 | 820.0768822 |
If you switched to rockets at 0.1 c then the mag-sail braking would last a mere 200 days. But on interstellar flights what matters is fuel expenditure and so with multiyear trip-times, a few hundred days isn’t a big reason for wasting reaction mass. But the above assumes that the mass density of the Inter-Stellar Medium (ISM) is a constant for the whole trip. This isn’t necessarily so. Larry Niven’s classic description of a ramjet in flight (in A Gift From Earth) uses the increased density of the ISM around the UV Ceti star system to provide extra braking for a rapid flight to Tau Ceti. Maps of the ISM will prove vital to future starship captains, for many reasons.