Interstellar Rockets

Some time ago I commented on how it is really impossible to get a "feeling" for the distance to even nearby stars.  Where our imagination becomes unreliable, we have to use mathematics. 

One way to bring these distances into a form we really can imagine is to perform some basic calculations for the relationship between the size of a rocket and how long it would take to get there.  By "rocket" I do not necessarily mean a chemical rocket of the type we're familiar with, just something in which mass is pushed out the back at some specified exhaust velocity, so that conservation of momentum pushes the rocket forward.  Conservation of momentum is a fundamental fact of all physics -- one that is routinely ignored in science fiction.

 
So let's assume you want to send a probe to Alpha Centauri, the nearest star (other than the sun) aside from Proxima Centauri, which is about the same distance but much less likely to be interesting.  Alpha Centauri is 4.27 light-years away, so under the assumption that you don't want to wait longer than 42.7 years for the probe to arrive (not counting the time spent accelerating), the probe must reach a speed of 10% the speed of light.  Perhaps surprisingly, this is still not a relativistic speed, so we can continue to use standard Newtonian physics.

We can take 1000 kg for the mass of the probe.  That's still rather light, given that we would at least want readings or pictures to be transmitted back to Earth, but perhaps with a more advanced technology it's not unreasonable.  

The most important factor determining whether or not the rocket will be feasible is the effective exhaust velocity, v_E.  For now let us assume that the exhaust velocity is given by the thermal velocity of hydrogen at the temperature of the center of the sun, which is perhaps a reasonable choice because

1. the energy source would probably be fusion, and
2. at any fixed temperature, the thermal velocity of a light particle is smaller than the thermal velocity of a heavy particle.

The thermal velocity can be calculated from the Equipartition Theorem, which states that the (average) energy in each degree of freedom is (1/2) k_BT, where T is the absolute temperature (Kelvin scale, where 0 K is absolute zero) and k_B is Boltzmann's constant. The temperature at the center of the sun is somewhere between 15 million and 20 million K; let's say 20,000,000 K. The contribution to the kinetic energy of a proton from motion out the nozzle is (1/2) m_p v_E², yielding an exhaust velocity of v_E = 4.06310 x 10⁵ m/s.  Please note that this is still much slower than the speed of light, so we are justified in using classical, Newtonian physics.

Not only is it slower than light, it is slower than the desired final velocity -- and that is a big problem.  Starting from rest, the rocket equation indicates that the final velocity is given by

v_f = v_E ln (m_i / m_f),

where m_i is the initial mass and m_f is the final mass (1000 kg in our case).  This means that 

m_i = m_f exp (v_f / v_E) = 1.10673 x 10³⁵ kg = 55.64 x m_sun.

That is, to put it mildly, a discouraging result, yet it makes the assumption that everything except the final payload mass is used as propellant. It gets worse, though. What if you you want the probe to slow back down to a stop when it reaches its destination? Then the initial mass you need is

m_i = m_f exp (2v_f / v_E) = 1.2249 x 10⁶⁷ kg 
= 6.158 x 10³⁶m_sun = 1 x 10²⁵ m_galaxy

To put this in perspective, the mass of the known universe, including dark matter, is estimated to be about 10⁵³ kg, so the initial mass would have to be about one hundred trillion times the estimated mass of the universe.

There are only two known ways to get around this.  One is to let the probe take a much longer time to get to the star.  Say you were willing to let this take 427 years to get to Alpha Centauri -- then the initial mass would only need to be 1600 metric tons with no slowing down and 2.56 million tons (about 3 times the mass of the Golden Gate Bridge) to stop when it gets there.  These are numbers are not outrageous, but they do put travel to another star beyond the limits of a single (human) lifetime.

The other way is to increase the exhaust velocity.  This would be difficult to justify if we are using thermal velocities, but maybe we could use a particle accelerator to accelerate the reaction mass to near the speed of light.  Such ion drives have already been used to limited extent.  In order for an ion drive to actually improve on our generous previous estimates, it would be necessary to significantly increase the speed of the ions, but a multi-century probe using ion propulsion would be a good objective for later in this century. 

What about warp drives?  It must be understood that there is no evidence for anything in nature moving faster than the speed of light in a vacuum, let alone anything we have been able to build.  Even when we had no airplanes or balloons, we still saw birds and insects fly, and before we could pilot an airplane past the speed of sound, rifle bullets were breaking the sound barrier.  There have been some theoretical attempts to develop a warp drive that is consistent with existing theory (even if it requires "exotic matter"), but these attempts have yet to overcome some serious problems.  For the time being, warp drives seem less likely to work than outright magic.