The waiting has an end: In Chaps.1 and 2 we learned how charged bodies move in homogeneous constant magnetic fields. After that we are going into the endless vastness of the universe.

In this section we are going to investigate the acceleration phase: At first, we will estimate the acceleration of our spacecraft corresponding to the laser power we assume (at the begin of the acceleration). Second, we will calculate the speed of the craft at the end of the acceleration phase, and third, the length of the acceleration distance corresponding to the duration of the acceleration.

The starship has a big light sail and can be accelerated by beams emitted from lasers in the solar system (Sol), see Fig.4. The lasers are stationed in an orbit around Mercury where they transform collected sunlight and send it to a relais-satellite. This satellite generates one single bundle of beams. At last this bundle is sent to a (rotating) Fresnel lens which can focus the light over the whole distance (10 ly).

As you will see, this isn't as complicated as it sounds. Our model is as simple as possible: The light sail could be understood as a huge plane mirror ( 1000 Km!) which reflects the incoming photons perfectly. Here we will concentrate on the acceleration stage (stage 1) only.

In the following we thought the total luminosity L of the incident laser radiation as a given quantity. We ask for the force and the acceleration taking effect on the light sail-ship. To get force and acceleration, we need three fundamental equations:

1. The momentum conservation law
The whole momentum of a system before an event (e.g. an impact) is the same after:

2. The momentum of a single photon
According to special relativity a photon has the energy

Thus we have

3. Newton's equation of motion (see also (1c.03))
Force = differentiation of the momentum with respect to time:

With this three equations we are able to derive - first of all - the force resulting from the reflection of the luminosity L:

The factor "2" arises because of the 100% reflective light sail; each photon is given a momentum with the same amount - but in opposite direction. While having the force, it's not difficult to get the ship's acceleration. Again we refer to Newton's Lex Secunda and divide Force and ship mass:

Once we have the acceleration, we have the velocity and the position of the starship too, provided that the speed is not too high (see also (1c.01) (1c.02)):
(3a.01/02)
In the following we want to calculate with concrete values to get some feeling for the physics involved.

As we will see later, the starship is being accelerated over a period of 1.6 years to relativistic speeds. Therefore we can not use our classical formulas (3a.01/02); we must distinguish between the earth- ( ) and the ship's frame ( ).
According to Robert L. Forward (who has had a series of contracts from NASA) and Joel Davis we can proceed from the assumption of a (initial) laser luminosity of L = 43000 TW [Forward and Davis (1986)]. In the following, we assume a constant luminosity of L' = 43000 TW.
For an extraterrestrial mission Forward and Davis also estimated the ship's mass to m' = 7.58 ^. 10⁷ Kg [Forward and Davis (1986)].
With these two quantities (L', m') we get a tremendous force in the ship's frame,

which causes an acceleration of

The relativistic relationship between velocity v and constant acceleration a' is given by [Marder (1982); Melcher (1978); Sexl und Schmidt (1978)]:
(3a.03)
where we introduced:

(Those of you, who are interested in the derivation, please click here!)
Inserting acceleration a' and time t (the acceleration period of 1.6 years) into equation (3a.03) we get:

This seems to be very promising - let us see, how far the ship from solar system is till yet. The equation we need can easily be derived from formula (3a.03) (try it by yourself or click here!):
(3a.04)
Thus we have:
x 4.41 ^. 10¹⁵ m 0.47 ly ;
and so, in stage one our starship has already overcome a distance, which ordinary rocket-driven craft never would reach - of course not in 1.6 years!

Surely you have ask yourself how the ship could be decelerate. Unfortunately our lasers (in the solar system) can not generate bundles of beams transporting negative energies. In the next section (Turning) you will find the answer to your question. To give you a small glimpse: The ship was not be accelerated directly to Epsilon Eridani.

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Chapter Three