Astronautics

A person could travel to alpha centauri using an ordinary chemical rocket. We know that we can construct rockets which travel at 17,000 miles per hour, because that is the escape velocity required to leave the earth's gravity. We have done that before going to the moon and to Mars. The only problem with traveling to alpha centauri at this speed, which is, by the way, faster than a speeding bullet, is that the journey would take 171,929 years!

Actually, you would have to travel a little faster than this to ever get to alpha because the escape velocity required to leave the solar system is approximately 67,052 miles per hour. This is approximately Mach 89. Presumably with a little more engineering, we could build a chemical rocket, or an ion drive rocket, which would be able to travel this fast through space.

If you were to try to travel to alpha at this speed, and held your speed constant, you would arrive in 43,590 years. This might not be very satisfying to the average mortal who has a life span of 77 years. On the other hand it is conceivable that a multi-generation voyage could be made, or that some form of "suspended animation" might be found which would assure that some form of intelligent life arrived someday in the alpha system. Note that even if the average lifespan were 100 years, the number of generations needed to accomplish a possible multi-generation scheme would be 435. This number starts to approximate the total number of human generations extant on the earth since the days of the Neanderthals.

What we have been talking about above is travelling at a constant speed from one star system to another. An alternative scenario presents itself in which we would accelerate at some constant rate throughout the first part of the journey, constantly gaining speed as we do so, and then, midway through the voyage, we would decelerate and slowdown at a constant rate so that when we arrived at the alpha system we'd stop. The advantage of this idea is that we would have something in the nature of an artificial gravity on board ship during the journey to make life on the ship a little easier, and prevent the wasting away of our joints and limbs.

An acceleration of 1 gee naturally presents itself as the right acceleration to use since this is the gravity we experience on earth, and would be comfortable for us over the long term. Such an acceleration implies a gain in velocity of 9.8 meters per second every second.

There is an equation in physics which tells us how far an accelerating object will travel in a given amount of time. The distance is equal to one-half the acceleration times the time squared. We know the distance to alpha centauri is 25.603 trillion miles. So we can certainly compute the distance to the half-way point which is just one half that figure. And we know the acceleration we have chosen. So if we solve for the time, we find that it will take 2.064 years to get to the half-way point at a constant 1 gee acceleration, which means that it will take 2 x 2.064 = 4.128 years to get to alpha if we then decelerate for the second half of the trip. But wait, something is wrong here. If it takes light 4.365 years to get to alpha, how can we make it in less time, i.e., 4.128 years?

The trouble is that we have ignored an important relativistic effect. As we accelerate faster and faster, the inertial mass of the ship begins to increase according to Einstein's Special Theory of Relativity. (Einstein's original paper was entitled "Does the inertia of a moving body depend on its energy content?" The answer, as he discovered, is yes.) We know the ship is gaining energy as it begins to move faster and faster, so we know its inertia is also increasing. Thus it takes greater and greater force to make it move just a little bit faster, as it gets faster and faster. (This is necessary to prevent objects which have mass from ever traveling as fast as light.)

There is a formula which allows us to calculate just how much more massive an object feels traveling at a given velocity. The formula is a simple one which states that the apparent mass is the rest mass times a factor known as the Gamma Factor, which is the reciprocal of the square root of 1 minus beta squared. 'Beta' is the ratio of the speed of the object to the speed of light. Thus an object moving at the speed of light will have beta = 1, and a zero in the denominator of the equation, and therefore an infinite apparent mass. Lesser speeds imply lesser apparent masses. And a speed of zero implies the rest mass. So we can compute our way out of this difficulty if we try, by trial and error.

A priori, there are two possible solutions. First of all, we could accelerate at 1 gee for a long time, get to a final speed of our choosing, say 98% of the speed of light, and then stop accelerating, coast for a while, say a year or two (or twenty!), and then start to decelerate at one gee. Or we could accelerate for exactly half the voyage at an acceleration of less than 1 gee (say .25 gees) and then decelerate for the second half of the voyage at the same deceleration rate. This type of trajectory planning is called mission planning, or voyage planning and varies depending on the type of rocket technology employed, amount of cryosleep available to the crew, etc..

It turns out that if we want to get to alpha really really fast, then we need to burn more energy, in total, than if we accept the idea of a slower voyage which will take longer, because of the need to propel the ship forward with a greater kinetic energy and also to overcome the relativistic mass increase with speed.

The total mass-energy used to get to a final speed, v, on a relativistically correct basis, is expressed by the following equation which takes into consideration the concept that all of the energy used is used with 100% efficiency to propel the ship forward (which obviously is an idealization, but useful for a first approximation):

                                       v/c = (1-a^2)/(1+a^2)

where a = mass fraction of ship left over after acceleration (decelleration) and c=velocity of light

If one uses this equation and then asks the question, how long will it take to get to alpha centauri under this or that scenario, one ends up with the following type of table: 

Voyage Planning Scenario         A                             B                                      C

Velocity of Ship (v/c):              .98                          .88                                    .60

Mass Fraction Remaining            10%                        25%                                  50%

Gamma Factor                         5.03                        2.11                                 1.25

Time (Earth) -years                 4.45                         4.96                                7.28

Time (Ship)  -years                  .89                         2.36                                5.82

The above assumes a constant velocity trip.