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I'm a noob in physics & math, so I'm just looking for a scientific answer for a small fiction project.

My question is : How far can we realistically go in space, if we managed to "use" energy in the best way possible ?

I don't want to take any huge assumptions in technology / science fiction, other than the ability to provide energy to the ship.

I am not sure what's the formula to provide that, so I wouldn't know the parameters that are needed for the calculus. I will give some below, if I forget any, feel free to make a reasonable assumption :

  • energy available : let's take a year of current global supply fully dedicated to that goal, which is around 10^21 J if I'm not mistaken

  • weight of the transport : let's say a single person with its ship, for a total of 100kg

  • acceleration/deceleration rate : something bearable for a human, let's say 1G or anything else you would feel more realistic

  • time passed from the traveler's point of view: 50 years

The answer needs to take into account relativistic effects, both in terms of time contraction, and also distance contraction.

On top of the distance traveled, I'm also looking for how much time has passed from the person left on earth.

More broadly, and as a conclusion, I would also want an answer to this question : it is often said that due to the big distances in the universe, it is impossible to travel even to the nearest galaxy, due to the finite speed of light (without having highly improbable technologies such as wrap drive which uses negative energy, or cryogeny of the body). But actually it is quite possible to go far in less than a life time, from the traveler's point of view, due to relativity, and without requiring huge amounts of energy ? All we need is to be able to channel that energy ?

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    $\begingroup$ Would Space Exploration be a better home for this question? $\endgroup$
    – Qmechanic
    Feb 23 at 7:30
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    $\begingroup$ Hi Django. Welcome to Phys.SE. Did you try to do a back-of-an-envelope-calculation? $\endgroup$
    – Qmechanic
    Feb 23 at 7:40
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    $\begingroup$ Some crucial info is missing in the task,- namely ship propulsion type and/or how this input energy is supplied, because it is related with initial ship mass and so would help to verify your $100 kg$ number. For example, lets say ship is propelled by fusing matter with antimatter, then according to your supplied input energy, you have to carry $(10^{21}J)/c^2 \approx 10~ \text{metric tonnes}$ of matter and antimatter + dry transport (without fuel) weight + passengers weight. In this case covered distance would be a A LOT smaller than predicted by @gandalf. Energy doesn't come from nowhere. $\endgroup$ Feb 23 at 9:09
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    $\begingroup$ There are a lot of transportation mechanisms out there, but all of them have to cope with a fundamental issue: conservation of momentum. To make your space-craft go one direction, with some amount of momentum, you must make something else move the opposite direction with the same magnitude. For the vast majority of cases, that means having a fuel on board that is accelerated away from the craft. We have played a little with beaming. While still in the atmosphere, you can heat up air and have that go backwards to give you momentum. Once out of the atmosphere... $\endgroup$
    – Cort Ammon
    Feb 23 at 17:36
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    $\begingroup$ ... you're more limited. There are indeed experimental systems based on light-pressure, but you're not talking about 1G. You're talking far lower accelerations. A one square kilometer solar sail, powered by the sun and its extraordinary energy budgets, gives about 8N or so -- miligees of acceleration or less. $\endgroup$
    – Cort Ammon
    Feb 23 at 17:39

2 Answers 2

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Since your objective is to go as far as possible, rather than rendezvous with a particular destination, then I will assume you do not need the spaceship to decelerate. The best way to utilise the limited energy budget is to accelerate as hard as you can until all the energy is used up, then coast for the remainder of the mission.

Accelerating at a constant rate of $1$ g reaches close to light speed in less than one year. To simplify calculations I am going to assume that the acceleration period is negligible compared to the rest of the $50$ year mission. So let's assume that you coast at a constant top speed for $50$ years in the reference frame of the spaceship. What is this top speed ?

The relativistic formula for kinetic energy is

$E_k = (\gamma - 1)mc^2$

Assuming $100\%$ efficiency converting the $10^{21}$ J energy budget to kinetic energy (and not making any allowance for maintaining life support systems) then we find that the $\gamma$ value corresponding to maximum speed for a $100$ kg spaceship (did you mean $100$ kg ? that seems very small) is

$\displaystyle \gamma_{max} = \frac {10^{21}} {9 \times 10^{18}} +1 \approx 112$

This is equivalent to a top speed of about $99.996 \%$ of the speed of light. With a $\gamma$ value of $112$ then $50$ years in the reference frame of the spaceship will be equivalent to $112 \times 50 = 5600$ years in the reference frame of Earth. So, to a good approximation, the spaceship will have travelled about $5600$ light years by the end of its $50$ year mission. This is about a fifth of the distance from the Earth to the centre of our galaxy.

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  • $\begingroup$ thanks. I also just edited my post to add a conclusion question, could you please edit yours to add your input on that ? $\endgroup$
    – Django
    Feb 23 at 8:54
  • $\begingroup$ If I understood you correctly, there is a linear relation between weight of the ship and energy required. So if I have instead 100 years of global supply, I could have instead a 10T spaceship, which begins to be slightly probable for a very small 1 man ship with a closed loop system (air, water, food, cf here), and if I have 1000 years and still 10T, I get gamma=1120 so 56kly so the whole diameter of the galaxy ? $\endgroup$
    – Django
    Feb 23 at 10:11
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    $\begingroup$ @Django Yes. If you increase both your spaceship mass and your energy budget by a factor of $100$ then your gamma factor and distance travelled remain the same. If you increase you spaceship mass by a factor of $100$ and your energy budget by a factor of $1000$ then your gamma factor and distance travelled increase by a factor of $10$. However, to reach, say, the Andromeda galaxy you need to increase your distance travelled by a factor of about $500$, which means an enormous energy budget and/or a very long mission duration (in human terms). $\endgroup$
    – gandalf61
    Feb 23 at 12:10
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    $\begingroup$ @Django though remember, in reality, you will likely have much lower efficiency. You’ll have to get around the tyranny of the rocket equation somehow (causes nonlinear relation of fuel mass required to change velocity). $\endgroup$ Feb 23 at 20:09
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But actually it is quite possible to go far in less than a life time, from the traveller's point of view, due to relativity,

Your final conclusion is correct.

Working on these two specifications:

  • acceleration/deceleration rate : something bearable for a human, let's say 1G or anything else you would feel more realistic
  • time passed from the traveler's point of view: 50 year

In 50 years of the travellers time, the spaceship will travel $10^{22}$ light years in approximately $10^{22}$ years measured in Earth time. This is an enormous distance! It is more than $1,000,000,000,000,000$ times the distance of our nearest galaxy. All the equations you require to carry out your own calculations can be found in The Relativistic Rocket and an online calculator here.

The correct equation for the time dilation factor is $$t = \frac c a sinh \left( \frac{a T} {c} \right)$$ where T is the traveller's time and t is the time measured in the Earth frame. Using units such c = 1 and approximating g as 1 $ ly/ y^2$ the equation simplifies to $$t = sinh ( T )$$

Traveller Time Earth Time Distance Example Velocity
1 year $\approx$ 1.2 years $\approx$ 0.6 light years - $\approx$ 0.77c
2.3 years $\approx$ 5.2 years $\approx$ 4.3 light years Nearest star $\approx$ 0.98c
3.9 years $\approx$ 28 years $\approx$ 27 light years Vega $\approx$ 0.999c
10.7 years $\approx$ 29,680 years $\approx$ 30,000 light years Centre of our Galaxy $\approx$ 0.9999999c
14.8 years $\approx$ 2,000,000 years $\approx$ 2,000,000 light years Andromeda Galaxy $\approx$ c
50 years $\approx$ $10^{22}$ years $\approx$ $10^{22}$ light years Very very very far $\approx$ c

My answer differs from the previous one, because that one is working from your specified fuel quantity, while mine is based on your specified proper time of the traveller and you cant specify both at the same time for a given constant acceleration. That is the reason for the tens of orders of magnitude difference in our answers.

This table differs from the one given in the first link, in that it is the distance travelled when accelerating continuously without turning around and decelerating to come to a stop.

The equations for energy requirements are also in the first link, if you want to do them, but it is worth noting that once you have got to final velocity of say 0.99c you can switch off the engines and continue travelling at that speed indefinitely and still obtain the time dilation benefits. Just remember to keep a reserve of at least half your fuel to able to come to a stop again. As mentioned in the link the equation becomes less accurate for extreme distances and times where the expansion of the universe becomes more significant.

In summary, you can theoretically go as far as you like in one lifetime, providing we have sufficient energy, which is currently unrealistic. I think a more 'reasonable' goal would be to plan for a spaceship that can support several generations of travellers. To have a more realistic energy budget, we can travel for free for indefinite distances (as mentioned previously), when we switch the engine off and coast, as long as we accept it will take longer.

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  • $\begingroup$ Thanks, your answer is also very interesting. I'll just react to the last sentence, before delving into the links you mentioned. In one lifetime, and in amounts of energy that are realistic for the human race to be able to obtain ? Because it is often said that to accelerate near to c you would need nearly infinite amounts of energy. $\endgroup$
    – Django
    Feb 23 at 13:47
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    $\begingroup$ @Django nothing about this question is really feasible. A 100 kg spacecraft could not contain 10^21 J of fuel using any technology we can imagine today. A rocket providing 1 g of acceleration for 50 years is not realistic for anything short of perhaps a black hole drive. Good luck creating/finding an appropriately sized black hole + fuel for that project. And if you could assemble all this, you would not waste it on a trip for a single human. A craft with these capabilities would weigh thousands of tons, at minimum, so you may as well bring a whole city. $\endgroup$ Feb 23 at 20:33
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    $\begingroup$ @Django You are correct that the amount of energy to keep even a small rocket accelerating with a proper acceleration of 1 g for 50 years is not realistic, but if we perfect fusion power in the future the amount of energy we can produce per year may be orders of magnitude greater than we can produce currently. I will add a bit to the answer to clarify the last sentence. $\endgroup$
    – KDP
    Feb 23 at 20:54

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