Ion thrusters trip time

Question

Suppose we had a fully equipped nuclear powered ion thruster starship.

1. If it had a constant rate of acceleration of 1g, taking into account the fact that at the half way point you would have to turn around to slow down;
   a) How long would it take to get to Proxima Centauri? 4.24ly
   b) The Trappist-1 system? 39.6ly

2. What if the approach was 2g, then a slowdown of 4g? This would shorten the trip time, but make for a less comfortable trip.
   a) 4.24ly?
   b) 39.6ly?

3. And if were talking about an uncomfortable trip,
   a) What is the maximum amount of acceleration can the human body survive?
   b) How long would both trips take if the passengers were strapped in their seats for the whole trip.

Clarification: 1g refers to the gravity of earth, 2g is 2x that. (Not 1c)

Nuclear pulse is not the type of propulsion I was thinking about, it is ion thrusters.

Answer 1

The first thing to say is that neither a nuclear powered ion rocket, nor a nuclear pulse rocket such as Orion is capable of sustaining 1g continuous acceleration over lightyears (nor indeed is any kind of rocket, except perhaps one fuelled by antimatter). To take an extreme case, consider a rocket that fuses protons to helium and manages to convert 100% of the energy released into kinetic energy of the exhaust, with no inefficiencies or losses at all. This is impractical for many reasons, but certainly no fission or fusion rocket of any kind will do better.

The four protons mass $6.69048759 \times 10^{-27} kg$ (source google), the alpha particle masses $6.64424. 10^{-27} kg$ so about about $4.6\times 10^{-29}$ is converted to energy, or a bit less than 1% of the fuel mass. That gives us $$1/2 m v^2 = 0.01 m c^2$$ where $v$ is the exhaust velocity of the rocket, so $v$ is about $c/7$. Given that we can use the Tsiolkovsky rocket equation to estimate the mass ration (fuelled rocket over empty rocket) to accelerate to say half the speed of light

$$\log(MR) = 0.5c /0.14c \sim 3$$

so the mass ration is about $e^3$ in the vicinity of 20. If we want to stop again that mass ration goes up to 400. At 1g that would be about 6 months continuous acceleration at 1g, followed (for the proxima trip) by about 8 years of coasting, followed by six months of decceleration. Nothing like continuous boot.

So continuous 1g boost over interstellar distances is not feasible using any kind of fission or fusion rocket and an even slightly realistic mass ratio.

That said, let's assume a magic engine capable of such sustained acceleration and answer your actual question. This thorough UC Riverside tutorial The Relativistic Rocket gives a lot of detail. Selecting a few of the equations from there

$$t = \sqrt{(d/c)^2 + 2 d / a}$$

where $t$ is the time to travel a distance $d$ at constant acceleration $a$ as measured by a stationary observer.

Using units of light years and years, $c = 1$ and $1g = 1.03$ so we can plug in half the distances to Proxima and Trappist (we spend the second half decclerating) and then double again to get 5.87 and 41.5 years.

However as seen from the spaceship a different formula applies:

$$T = c/a \cosh^{-1}\left({ad/c^2} + 1\right)$$

Again putting in the numbers, we get travel times of 3.54 and 7.29 years.

If you want to try other numbers (eg higher accelerations). There is a handy calculator, on line. That gives 3.5 years for the journey to Proxima and 7.3 years to Trappist (at 1g). At 2g you get 2.3 and 4.5 years, more or less.

Answer 2

The accepted answer nicely addresses the impossibility of accelerating at one g for years, regardless of fuel source / thruster type. This answer pokes deeper at the concept of a nuclear powered ion thruster starship. I'll first show that with such a vehicle it's essentially impossibly to accelerate at one g for more than a day, and then I'll show that with such a vehicle it's essentially impossibly to accelerate at one g, period.

On the impossibility of such a vehicle accelerating at one g for more than a day.

For this part of the answer I'll use the Tsiolkovsky rocket equation (the relativistic rocket equation makes matters even worse) for a vehicle whose thrusters have a specific impulse of 20000 seconds. That specific impulse (20000 seconds) is a bit higher than that of any ion thruster developed to date.

The non-relativistic rocket equation becomes nicely simple when specific impulse $I_\text{sp}$ is stated in seconds and acceleration is one g for some amount of time $\Delta t$: $$\frac{\Delta t}{I_{\text{sp}}} = \log\left({\frac{m_{\text{initial}}}{m_\text{final}}}\right)$$ The initial fuel fraction needed to be able to accelerate at one g for a time $\Delta t$ is thus $$\frac{m_\text{fuel}}{m_\text{final}+m_\text{fuel}} = 1 - \exp\left(-\frac{\Delta t}{I_{\text{sp}}}\right)$$ Substituting $\Delta t$ = 24 hours = 86400 seconds and $I_{\text{sp}}$ = 20000 seconds yields an initial fuel fraction of 98.67%. No such vehicle has ever been built. A more realistic value is that fuel initially comprises 90% of the vehicle's total mass. Such a loading would let such a vehicle accelerate for all of 12.8 hours at one g before the vehicle runs out of fuel.

On the impossibility of such a vehicle accelerating at one g, period.

Ion thrusters are extremely power-hungry. I'll be over-the-top generous and assume a 100% efficient nuclear power plant that yields one kilowatt per kilogram, coupled with a 100% efficient ion thruster that generates 20 newtons per megawatt of consumed power, coupled with a vehicle whose fuel mass, propellant mass, payload mass, structural mass, and thermal control mass are all zero. Said vehicle has an acceleration of 0.5 m/s², or about 1/20th of a g. Those assumptions are completely over the top. A more realistic result is an acceleration of a few thousandths to a few hundredths of a g.

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Ion thrusters produce very small thrusts. The key advantage of ion thrusters are their rather high specific impulse. This comes at the cost of low thrust levels and high power consumption. The problem of low thrust levels becomes moot when the thrust is applied over months to a few years. The problem of high power consumption isn't quite a problem if the power is provided by sunlight. It becomes rather problematic when the power source is onboard.