"How fast can these vehicles travel these distances?"
That is up to you, my friend. It depends on how much money (anti-money) you want to load into your engines. The bigger and badder you're willing to make your engine, the faster it can go.
I'll try to give you some equations and parameters you can use to model your antimatter engine. I'll be taking the figures for an antimatter beam core rocket at face value, found here on Atmotic Rockets:
- Exhaust velocity: 100,000,000 m/s
- Specific impulse: 10,200,000 s
- Thrust: 10,000,000 N
- mass flow: 0.10 kg/s
A beam core rocket converts small and equal amounts of matter and antimatter into gamma rays and charged particles. Charged particles are channeled by a magnetic nozzle and used directly for thrust. The dangerous levels of gamma rays are wasted and must be shielded against.
Say we want to travel distance $d$ [m] in time $t$ [s]. We can calculate our delta-v by:
$$v_{d}=2\frac{d}{t}$$
$\frac{d}{t}$ gives us the velocity we'd need to cover distance $d$ in time $t$, and our delta-v is twice this amount because we have to speed up to that amount at the start of the trip, then slow down from that amount, coming to a relative stop again.
With $v_d$, and our given exhaust velocity, $v_e$, we can calculate the rocket's mass ratio, $R$:
$$R=e^{\left(\frac{v_{d}}{v_{e}}\right)}$$
The mass ratio is the ratio of fueled spacecraft to fuel-less spacecraft. And with that, we can calculate the mass of fuel required for the acceleration-deceleration, $m_p$ [kg] (based on the parameters for beam core we take at face value):
$$m_{p}=m_{1}R-m_{1}$$
$m_1$ [kg] is the mass of the spacecraft without matter-antimatter fuel, the "dry" mass. This includes the mass of the engine (which was given to us as 10,000 kg) and the mass of the cargo, or "everything else". This includes the brain support machinery and the shielding against the deadly gamma rays released by the beam core engine. As a rough guess, I used 20,000 kg for an estimate of $m_{engine}+m_{cargo}$.
Let's throw some numbers at it. Let's say we want to travel 0.02 ly in 2 years time. I get the following values:
- $v_{d}$ = 6,000,000 m/s
- $R$ = 1.062
- $m_{1}$ = 20,000 kg
- $m_{p}$ = ~1,240 kg
Half the mass of the propellent, $m_p$, is regular matter (protons), which means our antimatter mass is half that, ~620 kg.
Additionally, if you want to calculate acceleration, $a$ [m/s^2], divide thrust (given as 10,000,000 N) by the fully-loaded "wet" spacecraft mass, $m_0$ [kg]:
$$m_{0}=m_{p}+m_{1},$$
$$a=\frac{F}{m_{0}}.$$
For the 0.02 ly in 2 yr flight, I get an acceleration during the burn-phases of ~47.1 g. For a person, that kind of gee-force would really suck, but for a brain sitting in supporting armatures, pressurized fluids, etc., I would consider that near the limit of what's tolerable before damage starts occurring. You can reduce the acceleration by increasing the cargo mass (or engine mass, if you're feeling really conservative).
You can also reduce acceleration by reducing the efficiency of the rocket, but that would in turn result in using more antimatter propellent for a similar journey. If you're going to be reducing efficiency, you might want to consider using another rocket engine; beam core is already one of the most efficient ones out there.
To calculate the burn-time for the acceleration-deceleration phases, use:
$$b_{t}=\frac{\left(m_{p}\cdot v_{e}\right)}{F}$$
This gives you the total available burn time for the engine. Divide that amount by two for the burn time of each acceleration phase. For our example, I get a "half"-burn time of ~6,400 s, or about 3 1/2 hours. That's 3.5 hours of acceleration, followed by ~2 years of coasting, then 3.5 hours of deceleration. (Note: this is a hyperbolic trajectory, which, for a set delta-v budget, is technically faster than "brachistochronic" trajectories (constant acceleration), despite the name. You can plot the graph of velocity/time to see that for yourself.)
(Edit: Part of the problem with humans and gees is getting blood to the brain and different densities in the body (cavities, bones, etc.) leading to collapsing and tearing tissues. Assuming you have a mechanism to deliver blood to the brain adequately, the oxygenation problem isn't a problem. The brain is about as dense as water. Immersing the brain in a fluid as dense as it allows the acceleration pressure to be delivered across and throughout the brain, if all the nooks and crannies are filled. I'm lacking a source atm, but the last time I looked into this ~70 gees was right up against the threshold of survivability.)
