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A star journey

In science fiction it is quite common to portray our galaxy as a "big Earth", where planets fulfill the role of countries and they are linked by "wormholes", or quickly reached by mounting starships with faster-than-light "warp drives". The physics underlying those phenomena is very little science and very much fiction. Also, the model of planets acting as "new worlds" to be colonized by humans, like the Americas were from Europe and Asia, is not very imaginative. Below I offer an alternative which has a stronger scientific basis, though still largely fictional. Maybe in the future humanity will achieve something similar.
There a quite a lot of numbers and a few formulas in the text below, which quantify the vastness of space and the problems it poses for space travel. The metric system is used throughout and numbers are frequently listed in scientific notation, to prevent their notation running off the lines in the text. Remember that XeY means X times 10 to the power of Y, so for example 5e6 = 5 times 1000000 = 5000000 = 5 million.

Why?

The target of our example journey is a star at a distance of 30 light years from our sun. The first question is: Why would we want to board a spaceship and such a distance through cold, empty and hostile space, when we already have a beautiful planet for a home?
Some people would argue that colonizing space will become necessary to relieve the pressure of overpopulation. But as we will see in the next section, our spaceship can only hold a limited number of people. Earth could probably breed more than that number in the time that is needed to build the ship, making the attempt futile. If Earth's population is to be kept in check, it has to be done through other means.
But we know that moving to another planet will become necessary in the far future. In about 1 billion years our Sun will become so hot that our planet will be almost literally fried dry. So the colonization argument is not entirely empty. The spaceship envisioned in this article could be built in centuries rather than hundreds of millions of years, so this not an urgent problem.
Instead, the purpose of our spacecraft is primarily exploration.

The ship and life aboard it

Our ship is built in space, so that its frame does not have to bothered with the trouble of lifting off from a planetary surface and conquering the planet's gravity. It is shaped like a simple cylinder, 5 kilometers long and 1 in diameter. It is mostly hollow, not just to increase the room inside, but mainly to keep its mass low. The outer shell is a 20 meter thick layer of basalt stone, which seems a lot, but is only 4% of the cylinder radius. It forms the bedrock for the living compartments, workshops, farms and parks that make up the interior. The inner landscape is at least as varied as on Earth varied because that is how we humans like it. It is here that the crew lives.
The surface area of the inner space in the cylinder is about 1.5e7 m². If packed densely together, the ship could maybe house and feed a million people, but the spacefarers see no need for such a number. The crew size is only 5000 people, making average population density as low as that of an Earth wilderness area. The population size is kept in check by strict birth control. 5000 people is enough to maintain genetic diversity with conventional breeding, though our descendants probably make heavy use of genetic engineering.
From a spot inside the cylinder one can see its entire length and circumference; there is not horizon like on a planet. The cylinder is lighted by very strong lamps on the central axis, wrapped around the propulsion systems that are also situated there. The light is brightened and dimmed to create day and night to create some variation. The day and night rhythm is a tradition that is inherited from life on planet Earth.
Because the mass of the ship is much too low to create any substantial gravity, the cylinder rotates along its axis, making one full rotation every 6.93 seconds, which means that the outside rotates at 453 meters / second. This creates a centrifugal force of 10 m/s², slightly more than Earth surface gravity.

The maximum speed of the ship is only 3e5 m/s, 1‰ of the speed of light. If the speed were higher, any debris in the form of interstellar dust, however scarce, would strike the ship with dangerous force as it flies into it. Also, higher speeds require extraordinary amounts of energy, as will be explained later. At the target speed Einsteinian relativistic effects are marginal. (Relative) mass increases and (relative) time slows by a factor of 1.000005, low enough to ignore in our calculations, which are not that precise.
But even at this high speed, the entire journey takes about 30000 years. This is several orders of magnitude longer than the maximum age of a human being, even taken into account the increased lifespan of our descendants.
One way to cope with this problem is to deep-freeze the crew, fly the ship on computers and to awaken them on arrival. But our spacefarers have opted for a different strategy. The ship is manned by people who remain awake and active. They are a breed of humans who have been born, raised and have lived on spaceships all their life. Many of them have never even set foot on a planet. For them, the curved interior of a spaceship is home; a planet's wide open ranges just makes them uncomfortable. They are just as content to live out their life aboard an artificial construct as you and I are to live on Earth. These people will live and die and pass through many generations while en route. This does not bother them, because the ship is their world.

Speed, propulsion and energy consumption

The main problem in reaching the stars is that they are so very far away. In this example, our spaceship aims to travel to a star that is 30 light years away. As long as there is no believable theory that can topple Einstein's physics, we stick to the German's point of view. That means that the spaceship and its crew an cargo, being composed of matter, cannot exceed the speed of light, which is 3e8 meters / second. Traveling at that speed, the journey would take 30 years.
In reality even the speed of light is beyond our reach. To reach high speeds, the spacecraft must accelerate and by doing so, convert energy from some kind of energy source into kinetic energy. Newtonian physics state that the kinetic energy of an object is ½mv², i.e. half of the mass multiplied by the square of the speed - excluding Einsteinian relativistic effects. Even for relatively low speeds, like our intended 1‰ light speed velocity, this comes down to a tremendous amount of energy.
Preferably, propulsion energy is picked up along the way, but space is very empty and there is little energy to be found in the interstellar region, so our ship must carry its "fuel" with it, burning it up as it goes. The fuel adds to the weight of the spaceship and thus increases the amount of kinetic energy that is needed. This in turn increases the need for fuel, increasing mass, etc. etc. The result is a "snowball effect" that can lead to very high fuel demands.
Apollo 4 launch With chemical fuel, this puts severe limits on the maximum speed that can be reached. The Saturn V rockets that carried American astronauts to the Moon transported much more fuel than actual crew and cargo. For an interstellar journey, chemical fuels simply do not suffice. We need something that produces more energy per unit of mass. Radioactive fuels like uranium do a better job, but still fall short of our target, so we switch to the best fuel available: anti-matter.
Anti-matter is a rare occurrence in our world and that is a good thing, because if it comes into contact with normal matter, both react violently with each other, releasing an amount of energy that is 9e16 times greater than their combined mass: E = mc², where m is the mass and c is the speed of light. We will suppose that our descendants will have mastered the technique to keep the two kinds of matter separated and bring them together in a controlled way.

As said, the intended speed of the spaceship for most of the journey is 3e5 m/s. The mass of the basalt outer shell is 9e12 kg. The earth and buildings on the inside, structures along the central axis and reinforcing spokes increase this to a total of 2e13 kg. This means that the maximum kinetic energy of the ship is 9e23 J and the total energy required for propulsion is 1.8e24 J, because the ship needs to accelerate to full speed and decelerate again when it arrives at its destination. Stored in matter and antimatter, the fuel required weighs only 2e7 kg, which is negligible compared to the mass of the ship itself.
The ship uses a plasma thruster for propulsion, which expels plasma at a speed of 1.2e8 m/s, 40% of the speed of light. According to Newtonian physics, the impulse of ship and reaction mass must be equal. This means that 2.5e10 kg of reaction mass is needed for the acceleration and the same amount for deceleration. This is much more than the mass of the fuel, but still insignificant compared to the total ship mass. The actual requirements are slightly less, due to the Einsteinian relativistic effect which makes the reaction mass 1.09 times heavier at 40% times the speed of light than at zero speed.

While propulsion is the main energy consumer, it is not the only one. To keep the crew comfortable, the ship is kept at an average temperature of 293 K, or 20° Celcius. This is significantly warmer than the surrounding space, so the ship loses warmth to it constantly by radiation.
The rate of energy loss = AεσT⁴, or outer surface times emissivity times Stefan-Boltzmann constant times absolute temperature to the fourth power. The outer shell of the ship is made of basalt which has an emissivity of 0.72, but to decrease energy loss the engineers have applied a metallic coating that has an emissivity of only 0.02. With an outer surface area of 1.885e7 m², the rate of energy loss is 1.58e8 W.
During its 30000 year long voyage, the ship will radiate out 1.5e20 J. Fortunately this is insignificant when compared to the energy required for propulsion. The activity inside the ship also requires energy, but that is even less than the heat loss through the hull.

Energy source

It is important to realize that the matter + antimatter fuel of the spaceship is only a way of storing energy, not to create it. Anti-matter is not readily available and cannot be harvested. It can be created, though. Physicists today are able to create tiny amounts of anti-matter, using massive particle accelerators, wasting massive amounts of energy in the process. We will suppose that our descendants can do this with near-100% efficiency. But they still need a source of energy.
Sun Fortunately space is dotted with many abundant sources of that stuff. They are called stars. Our own sun emits energy in the form of radiation at a rate of 3.9e26 W, of which the Earth receives only 1.74e17 W, which comes down to an energy density of 1.4e3 W/m². Other planets also pick up pieces of it, but most is shone off into deep space. Our descendants have rightfully recognized the sun as the solution for all their energy needs.
As the radiation intensity decreases with the square of the distance to its source, it pays to huddle close to it. In our hypothetical future, the sun is ringed by solar panels that orbit at a radius of 1e9 m, closer than any planet. Here, they receive 3.1e7 W/m². Each panel has a surface of 1e6 m² and acts like a "black body", absorbing all radiation that falls on it, converting everything to matter and anti-matter on the fly. This 100% efficiency is remarkable feat of engineering that not only prompts envy, but also prevents that panels from overheating, because if they only spilled a small fraction of the energy they received, that energy could heat them up to dangerously high temperatures.
One hundred of these super-guzzlers generate 3.1e15 W. Though still but a fraction of the total solar power, they could "fill" our ships' fuel tanks in just 18.4 years - quite short compared to the 30000 years of the journey.
Once the ship arrives at its destination its fuel will be spent, but there our descendants will build new solar panels around the target star to power up the ship for another journey.

Star system colonization

When after, 30000 years, the spaceship finally arrives at its destination, people can disembark and start colonizing a new planet. It may be that their telescopes were inaccurate and that the target planet is not inhabitable because it lacks certain elements; is too light; too close to its star and thus too hot or too far away and too cold.
But with solar power at full capacity at their disposal, our descendants could work miracles. They can harvest asteroids and comets from the star's outer regions to bring in minerals; build planetary engines to steer the planet into the right orbit; pump energy into its core to rekindle plate tectonics, etcetera. That kind of work may take millennia or even millions of years, but the spacefarers have already shown that they no longer plan on the scale of the lifetime of individuals, but for the benefit of the entire race.
We can only wander why they would bother to terraform a planet. Our spacefarers are perfectly content to live their lives in spaceships; why change that habit? Maybe they will make planets habitable, not because they need to, but just to give other forms of life a chance.