There is a quadrillion light years

Travel times with an interstellar spaceship

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Mathematics in everyday life, amazing math puzzles, stochastics and polyhedra, terrestrial and extraterrestrial life


According to the laws of special relativity, it would actually be possible for a spaceman to reach even the most distant galaxies during his life. To do this, his spaceship would have to be able to accelerate at a constant acceleration of 9.81 m / sec during the first half of the journey2 to accelerate and to brake again in the same way on the second half of the journey. The acceleration and deceleration would then create an artificial gravity in the spaceship that would correspond to the conditions on the earth's surface. The spaceman would have the same weight in the spaceship as on earth.

The technical difficulties e.g. for the drive, the energy supply and for the protective shield against interplanetary and interstellar dust are so great that one will probably never be able to travel beyond our solar system in this way (see also: Possibilities and limits of manned space travel). For these and other reasons, I also consider it practically impossible that intelligent extraterrestrial beings can ever reach us with spaceships (see also: extraterrestrial life in our Milky Way).

Nevertheless, it is interesting to realize at what times and at what speeds the spaceman could cover which distances. According to Einstein's special theory of relativity, the longer the spaceship is accelerated, the slower the time in the spaceship is compared to the earth. As a result, a spaceman could actually achieve very distant goals in his lifetime. However, on his return, so much time would have passed under certain circumstances that he would not find any person he knew, and possibly not even meet people there. It could even be that even the earth wouldn't exist anymore.

The following table contains the travel times of the spaceman, the times past on earth and the maximum speeds of the spaceship for various travel distances (formulas at the end of the page). One light year is the distance that light travels at a speed of c = 299,792.458 km / s in one year. The length of this route is approximately 9.46 trillion kilometers.


distanceTravel timepast timeTop speed
from Earthfor space travelersfor earth dwellersof the spaceship
 
1 billion km 7.39 days7.39 days3132 km / s
10 billion km23.4 days23.4 days9900 km / s
100 billion km73.9 days74.0 days31 193 km / s
1 trillion km 0.64 years0.65 years95 193 km / s
1 light year 1.89 years2.21 years225 357 km / s
10 light years 4.85 years11.8 years295 820 km / s
100 light years 9.02 years102 years299 738 km / s
1000 light years13.4 years1002 years299,791.9 km / s
10,000 light years17.9 years10 002 years299,792.452 km / s
100,000 light years22.3 years100 002 years299 792.458 km / s (almost)
1 million light years26.8 years1 000 002 years299 792.458 km / s (almost)
10 million light years31.3 years10 000 002 years299 792.458 km / s (almost)
100 million light years35.7 years100,000,002 years299 792.458 km / s (almost)
1 billion light years40.2 years1 000 000 002 years299 792.458 km / s (almost)
10 billion light years44.7 years10 000 000 002 years299 792.458 km / s (almost)


The second table again contains travel distances, travel times and maximum speeds, but this time for a number of known cosmic destinations:


Trip fromdistanceTravel timepast timeTop speed
the earth tofrom Earthfor space travelersfor earth dwellersof the spaceship
 
Earth moon 384,000 km3.48 hours3.48 hours61.4 km / s
Venus 39 million km (minimal)1.46 days1.46 days619 km / s
Mars 55 million km (minimal)1.73 days1.73 days735 km / s
Mercury 92 million km (minimal)2.24 days2.24 days950 km / s
Jupiter 590 million km (minimal)5.68 days5.68 days2406 km / s
Saturn 1.20 billion km (minimal)8.10 days8.10 days3431 km / s
Uranus 2.59 billion km (minimal)11.9 days11.9 days5040 km / s
Neptune 4.31 billion km (minimal)15.3 days15.3 days6501 km / s
Proxima Centauri (constellation Centaur) 4.24 light years3.54 years5.87 years284 673 km / s
Alpha Centauri (constellation Centaur) 4.36 light years3.58 years5.99 years285 259 km / s
Sirius (constellation Great Dog) 8.6 light years4.61 years10.4 years294 684 km / s
Epsilon Eridani * (constellation Eridanus) 10.5 light years4.93 years12.3 years296 135 km / s
Tau Ceti * (constellation whale) 11.9 light years5.14 years13.7 years296 841 km / s
40 Eridani A * (constellation Eridanus) 16.5 light years5.70 years18.3 years298 134 km / s
Wega (constellation lyre) 25 light years6.44 years26.9 years299 017 km / s
Arctur (constellation Bootes) 37 light years7.15 years38.9 years299 421 km / s
Aldebaran (constellation Taurus) 67 light years8.26 years68.9 years299 674 km / s
Dubhe (in the big dipper) 124 light years9.43 years126 years299 757 km / s
Pleiades (seven stars) 370 light years11.5 years372 years299 788 km / s
Polarstern (in the little wagon) 430 light years11.8 years432 years299 789 km / s
Betelgeuse (Orion constellation) 640 light years12.6 years602 years299 791 km / s
Rigel (Orion constellation) 800 light years13.0 years802 years299,791.6 km / s
Deneb (constellation swan) 1500 light years14.2 years1502 years299,792.2 km / s
Black hole in the center of the Milky Way26,000 light years19.8 years26 002 years299,792.457 km / s
Large Magellanic Cloud 160,000 light years23.3 years160 002 years299 792.458 km / s (almost)
Andromeda Galaxy (M31) 2.6 million light years28.7 years2 600 002 years299 792.458 km / s (almost)
Firewheel Galaxy (M101) 27 million light years33.2 years27 000 002 years299 792.458 km / s (almost)
Spiral galaxy NGC 1232 85 million light years35.4 years85 000 002 years299 792.458 km / s (almost)
Edge of the observable universe 46 billion light years **47.6 years46 000 000 002 years299 792.458 km / s (almost)
* Sun-like star
** The distance is given by
the extent of space, where
the rate of expansion
of the room not through the
Speed ​​of light limited
becomes, as opposed to movement
of objects in space.


From the tables one can see that the spaceman could theoretically reach any known object in the universe during his life. However, the travel times only apply to a universe that is not expanding. However, as our universe is expanding, the results for travel distances of several billion light years are no longer exact. In addition, the travel destinations may no longer exist on arrival because we currently see these destinations as they looked in the past when the corresponding light was emitted there.



The trips just described are at least theoretically possible according to the level of knowledge of physics. However, this would require a photon drive, in which the photons would be generated by the fusion of matter and antimatter, whereby the antimatter would first have to be produced with a gigantic expenditure of energy.

On the other hand, travel through wormholes or travel with warp drive are not possible according to the state of knowledge of physics. For this one would need a hypothetical form of matter (matter with negative mass) that has not yet been discovered and that probably does not even exist. With positive and negative mass one would then have to bend space-time in such a way that either wormholes or warp bubbles arise. With the wormholes you would then have to create a shortcut to the travel destination and transport spaceships to the destination with warp bubbles that are faster than light. In contrast to matter in space, there is no maximum velocity for space itself.

In order to show that even a photon rocket with matter and antimatter as fuel has nothing to do with reality, I choose the above-mentioned trip to the fixed star Vega, only 25 light years away, as an example. At an acceleration equal to the acceleration of gravity, the rocket would reach 99.74% of the speed of light. The actual spaceship, i.e. the payload, is said to have a mass of only 100,000 tons (among other things for oxygen, water and food, for machines for supplying energy, for shielding against cosmic radiation, and for machines that ensure long-term survival at the destination enable). Even if the mass for the photon engine (engine jet speed = c) and the fuel tanks are not taken into account, braking according to the relativistic rocket formula requires about 2.77 million tons of fuel (50% matter and 50% antimatter):

Final mass / initial mass = ((1 - final speed / c) / (1 + final speed / c))c / (2 engine jet speed)

For the acceleration phase, this amount of pulp has to be added to the payload of 0.1 million tons. To accelerate these 2.87 million tons, you need a further amount of pulp of 79.5 million tons. In total, that's around 82.3 million tons of fuel. This fuel contains an energy of about 7.4 quadrillion joules. According to this, only this one spaceship needs an amount of energy with which mankind could be supplied for more than 12 million years with the current energy consumption (600 trillion joules per year). And of course this spaceship does not allow a return to earth.

If you then realize that there are no deposits for antimatter as there are for coal or uranium, but that antimatter must first be produced artificially with the help of a huge additional amount of energy, and that no one knows how to transport antimatter safely in a rocket can, and that ultimately no one has an idea what a corresponding engine might look like, one gets an inkling that manned interstellar space travel will probably remain a fiction forever.

What would be realistic in the very distant future? An unmanned reconnaissance mission to the nearest exoplanet (Proxima Centauri b) at a distance of 4.24 light years? Even if you only assume a payload of 20 tons for the landing capsules with the exploration robots, you have to realistically assume a total mass of the spaceship of 200,000 tons in order to be able to reach the exoplanet with a nuclear fusion drive in about 50 years. Only heavy hydrogen (deuterium) and helium-3 can be used as fuel. About 100,000 tons of the total mass are likely to be made up of helium-3. Since there are no significant deposits of helium-3 on earth, one would have to fall back on the deposits of the moon, which are estimated at a maximum of 1 million tons. A single flight would therefore consume at least 10% of the total lunar supplies. Let the reader judge for himself whether this is realistic. It remains to be mentioned that such a mission would also devour many times the current annual energy consumption of mankind.



For those interested in mathematics, there are now the formulas that can be used to calculate the time passed on earth and the time passed in the spaceship depending on the distance to the travel destination:

tr = c / br Arccosh (xe · Br / c2 + 1)

te = c / br · √ ((xe · Br / c2 + 1)2 – 1)

In the formulas, x ise the distance of the spaceship from the earth, te those on earth and tr the time elapsed in the spaceship. br denotes the constant acceleration with which the spaceship increases its speed, and c = 299 792 458 m / s is the speed of light. For the above tables, br the acceleration due to gravity assumed:

br = 9.81 m / sec2

For the calculation it is also important for xe to use half the distance to the travel destination, because the journey consists of an acceleration phase and a symmetrical braking phase. The calculated te or tr must then of course be doubled in order to get the desired travel times.

Is xe small, the relativistic formulas merge into the classical formulas:

tr = c / br Ln (xe · Br / c2 + 1 + √ ((xe · Br / c2 + 1)2 – 1))
= c / br Ln (1 + √ (2 xe · Br / c2)) = c / br · √ (2 · xe · Br / c2) = √ (2 xe / br)

te = c / br √ ((2 xe · Br / c2 + 1) - 1) = c / br · √ (2 · xe · Br / c2) = √ (2 xe / br)

To calculate the speed v at the end of the acceleration phase, the following formula applies:

v = br · Te / √ ((br · Te / c)2 + 1)

Is the expression br · Te small compared to c, we get the classic formula:

v = br · Te




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