Talk:Space travel under constant acceleration

Needs wikification
I think this needs to be Wikified a bit. --88.242.208.27 (talk) 16:42, 9 May 2010 (UTC)

I'd like to know what the requirement that the spaceship flip its orientation in order to decelerate is based off. If a spaceship is equipped with opposing thrust sources, such as two rocket engines facing in opposite directions, then there's no need for it to change its orientation in order to decelerate. I think the last sentence of the "Constant-acceleration drives" paragraph can be worded better. --2603:6011:F405:442F:33F1:72D7:B60:923C (talk) 22:13, 24 September 2023 (UTC)

Interstellar Space Commerce
The article says "the benefit ... is having interstellar space commerce" At 1g acceleration, it is possible to travel to a nearby galaxy within a generation, but when you will come back it will be millions of years later. This leads to an accountancy problem.

Suppose a bank lends a company money to finance interstellar trade. When the spaceship comes back, the borrowers' accountant on the spaceship will be willing to pay only 20 years of compound intrest, while the lenders' accountant who stayed at home will want to be paid a few million years of compound intrest. — Preceding unsigned comment added by 77.228.71.21 (talk) 08:44, 6 May 2012 (UTC)
 * All in all, this statement in the article is... er... I'd put it so: meaningless. What commerce can be possible in interstellar space? It is extremely doubtful that there is something to pay for in other stellar systems that we don't have at home. Certainly what mankind seeks from interstellar travels is not commerce.


 * As for the axioms of money business for such travels, it is never a problem to change them accordingly to the situation, but the real problem is that economy changes very fast and nobody can know what happens after some many thousand years. It turns to be an investment with no revenue, like many other scientific projects. Who knows, maybe sometime in the future people will be able to afford it... - 91.122.7.53 (talk) 20:42, 19 May 2013 (UTC)


 * Yes, good point. Feel free to throw away everything related to commerce. None of it sourced. As a matter of fact, everything can be removed from this article, except the lead, which is the only thing that is sourced. When you delete something, refer to the talk page in your edit summary. That might trigger some discussion here and perhaps some contributors going after proper sources. - DVdm (talk) 21:40, 19 May 2013 (UTC)

Tagged for improvement
This article needs good sources, and a rewrite to remove speculation, techincal errors, and original research. &rarr;  Stani Stani  22:37, 21 June 2010 (UTC)

Anon: Just saying: http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html —Preceding unsigned comment added by 87.72.27.11 (talk) 11:34, 26 February 2011 (UTC)

The ship crew times in the "Interstellar Traveling Speeds -- Ship Reference Frame" are quite high by my calculations. I find that 1 G to the galactic core should take about 20 years, 2 G should take about 10.5 years, and 10 G should take 2.5 years. My calculations are derived from Landau and Lifshitz' The Classical Theory of Fields and are documented at http://ftlfactor.com/2011/06/05/special-relativity-theory-calculations-of-elapsed-time-and-distance/. I also have a relevant web-based calculator at http://ftlfactor.com/srcalc Ftlfactor (talk) 03:48, 26 August 2011 (UTC)

The times quoted are erroneous. The reference has made the mistake of equating Newtonian trip-times with the ship-frame trip-times of SR. However, as other Wikipedia entries and references note, the ship-frame elapsed time is logarithmic/hyperbolic. — Preceding unsigned comment added by 101.163.77.160 (talk) 13:12, 20 March 2012 (UTC)


 * I went ahead and removed the clearly erroneous parts and all references to that unreliable source. - DVdm (talk) 22:09, 3 December 2012 (UTC)

A destination at distance at2/4 takes time t
I reverted Patricks good faith addition per wp:unsourced and wp:original research. Also note that is wrong. That should be: "With a sufficiently small acceleration a and over a sufficiently small time t, a destination at distance at2/2 takes time t." Note that at constant proper acceleration a, the coordinate distance x as a function of coordinate time t, resp. proper time &tau; is given by
 * $$x(t) = \frac{c^2}{a} (\sqrt{1+(a t/c)^2}-1)$$
 * $$x(\tau) = \frac{c^2}{a} (\cosh(a \tau/c)-1)$$

which have series expansions
 * $$x(t) = \frac{1}{2} a t^2 - \frac{a^3}{24 c^2} t^4 ...$$
 * $$x(\tau) = \frac{1}{2} a \tau^2 + \frac{a^3}{8 c^2} \tau^4 ...$$

and thus, only for sufficiently small values of a, t and &tau; reduce to resp.
 * $$x(t) \approx \frac{1}{2} a t^2$$
 * $$x(\tau) \approx \frac{1}{2} a \tau^2$$

DVdm (talk) 09:59, 13 April 2013 (UTC)


 * It refers to "for the first half of the journey it constantly pushes the spacecraft towards its destination, and for the last half of the journey it constantly uses backthrust, so that the spaceship arrives at the destination at a standstill.", and a speed not close to the speed of light. - Patrick (talk) 12:11, 13 April 2013 (UTC)


 * Ok, that would be much better indeed, but including something along these lines here needs proper sourcing, of a better quality than just pointing to a section in another wiki-article (see wp:CIRCULAR) that has almost no inline citations. Cheers - DVdm (talk) 14:19, 13 April 2013 (UTC)


 * Is "proper time" the time measured with clocks on the ship? 92.224.97.127 (talk) 10:59, 16 December 2014 (UTC)
 * Yes, in this case t is coordinate time, measured by those who stay behind, whereas &tau; is proper time, measured by clocks on the ship. - DVdm (talk) 17:06, 16 December 2014 (UTC)

This article needs the equation
How far does one travel from the point of view of Earth as a function of the traveler's time for a given constant acceleration? It seems like the article should begin with this equation, then all the other questions can be answered with simple math. For example, if you need to turn around and slow down half way to come to a stop at a desired destination, then you calculate the time for half way and double it to get the total time. Double again to get the round trip time. Argh. Some many words in the article, but it's not answering what I think most people would want to know. Ywaz (talk) 19:12, 2 April 2016 (UTC)


 * yes, I agree. We can add something but we must make sure it is properly sourced. A search in the literature provides equation 7.12 of Don Koks' book (ref below):
 * $$x(\tau) = g^{-1} \cosh {g \tau}$$ (with initial condition $$x_0=\frac{1}{g}$$)
 * If we set g=a and choose x(0)=x0=0, and "unset" c=1, this immediately translates to the equation that I gave in the preceding thread. There are many more similar sources, but most of them forget or don't bother to specify the (important) initial condition.
 * So, how far one travels, experiencing constant acceleration, from the point of view of Earth as a function of the traveler's time, is expressed by the coordinate distance x as a function of proper time &tau; at constant proper acceleration a. It is given by:
 * $$x(\tau) = \frac{c^2}{a} \left(\cosh \frac{a \ \tau}{c} -1 \right) .$$
 * Under the same circumstances, the time elapsed on Earth (the coordinate time) as a function of the traveler's time is given by:
 * $$t(\tau) = \frac{c}{a} \sinh \frac{a \ \tau}{c}.$$


 * I have added these to the text. The reader can now also simply divide $$x(\tau)$$ by $$t(\tau)$$ to find the velocity $$v(\tau)$$. They can verify that the limit is c. - DVdm (talk) 12:35, 3 April 2016 (UTC)


 * Excellent. I had seen these equations in trying to find something, but I was not sure of the accuracy and variables. But I think you're forgetting something: Hubble's constant significantly affects the future distance of a target as $$t(\tau)$$ becomes significant, so $$x(\tau)$$ is probably not what you've stated. Hubble's constant has been observed to be slowly increasing, but I would like to see what the $$x(\tau)$$ equation is for both constant and slightly increasing $$H_0$$. Fixed cosmological distances are not physically real, so I would not state $$x(\tau)$$ as you have for large $$t$$. Maybe the correction for a target distance $$x_0$$ at $$t=0$$ and constant $$H_0$$ is
 * $$x(\tau) = x_0 \left(1+H_0 t(\tau)\right)$$
 * Like you, I have to think about things like this before I know how to begin a search for references. In expanding that, it the sinh and cosh functions remained instead of getting a tanh or something. Ywaz (talk) 17:37, 4 April 2016 (UTC)


 * I don't think that the "evolution" of Hubble's "constant" is relevant on the time scale of space travel. But even if it would be, in connection with space travel at constant acceleration—the subject of this article—I have never seen anything about it in the literature, whereas the above equations are just about eveywhere when there's talk about acceleration in the contexts of special and general relativity, but assuming "constant constant" Hubble . - DVdm (talk) 18:01, 4 April 2016 (UTC)


 * My equation is supposedly good for a constant Hubble constant. If you accelerate at 1 g and turn around half way to slow down at the same rate, Carl Sagan wrote in Cosmos it takes 21 years years to get to center of galaxy and 28 years to get to Andromeda galaxy. That's 30,000 years from Earth's view to get to center of galaxy. Using your t equation for 1/2 way (the time starts advancing for Earth in a sense only when you turn around to slow down), I get Sagan's 30,000 years. For your distance equation I get 30,000 light years, so I assume Sagan estimated distance to center of galaxy is 60,000 light years [edit: no, he estimated it to be 30,000]. So your equations seem correct. There is no difference between sinh and cosh for large values, having <0.2% error after 3 years at 1 g using
 * $$\cosh(x) = \sinh(x) =\frac{1}{2} e^x$$ for x>3. (One year times 1 g in m/s = 1.05 c) This is why the light years travelled equal to the time passage on Earth.
 * So distance travelled in light-seconds and time passage on Earth for a traveller turning around to decelerate at the half way point, I get
 * $$x(\tau) \approx t(\tau) \approx \frac{c}{2a} e^{\frac{a \tau}{c}}\left( 1+H_0 \frac{c}{2a} e^{\frac{a \tau}{c}}\right)  $$
 * for $$\tau > 3$$ years in seconds. Again, this is an underestimate since $$H$$ is increasing with $$t$$.
 * But it seems that second term remains too small unless I go to ridiculously large values of tau, about 39 years would be trillions of light years distance and time into the future. So if it is relevant, it will be because H is increasing, but I do not know how to write it as a function of time, and it would have to be a really big increase, like 100,0000 after 10 billion years.
 * So here is the equation I would like to see in the article, if it is correct:
 * $$x(\tau) \approx t(\tau) \approx \frac{c}{2a} e^{\frac{a \tau}{c}} $$
 * for distances travelled in light-seconds if you turn around at the half-way point to decelerate, and for $$\tau > 3$$ years in seconds. Round trip is two times these numbers. A lot more people can understand what an exponential is compared to cosh or sinh.
 * [edit: i previously had a calculation error where I thought H would make a difference]
 * Ywaz (talk) 18:56, 4 April 2016 (UTC)


 * (I have indented you message. Hope you don't mind. See WP:INDENT and WP:THREAD.)
 * You made a few mistakes there: some factors are wrong and the dimensions don't match. But indeed, for large X we have that cosh(X) and sinh(X) approach exp(X)/2.
 * If you can find a good source for the correct asymptotic approximation, we can take that on board, otherwise we probably should cut it short here because of our wp:talk page guidelines: we are supposed to discuss the article here, not the subject . - DVdm (talk) 21:25, 4 April 2016 (UTC)
 * I see I left out a 1/2 in the above, so I edited it to insert it. My units are not in error because I stated my x distance above was in light-seconds which is why it's c instead of c^2. Light-seconds is a time indicating distance. My 1/2 factor on x was wrong because I thought Sagan used 60,000 to center of galaxy, not 30,000. My non-H equations are not different from yours. Granted my H discussion is beyond the article, but it only occurred because I thought it was relevant.
 * Here are the equations if you turn around at the half way point. I confirmed them with Sagan's text.
 * $$x(\tau) meters = \frac{c^2}{a} \left( \cosh\left(\frac{a \tau}{2 c}\right) -1 \right)$$
 * $$t(\tau) = \frac{c}{a} \sinh\left(\frac{a \tau}{2 c}\right) $$
 * And for $$\frac{a \tau}{2 c} > 3 $$:
 * $$x(\tau) lightseconds \approx t(\tau) \approx \frac{c}{2a} e^{\frac{a \tau}{2 c}}$$
 * I'll leave it to you and others to decide if it should be included in the article.
 * [edit: I found out Sagan used 30,000 light years to center of galaxy, so x=t in light-seconds if he was not in error]. Ywaz (talk) 00:40, 5 April 2016 (UTC)

Please review this chart for inclusion in the article


Concerning the "traveler's speed" in this, there are some who argue there is a problem with "exceeding the speed of light" by a factor of 1,400 times (for example, as the chart shows) when travelling to the center of the galaxy like this. They will say "nothing can go faster than the speed of light", but this is a problematic statement because all observers from all frames of reference observe all photons as always going a speed of c, no matter how fast the observers are moving. What they mean is that no observer can observe any traveler going faster than c. All observers go zero meters/second relative to all photons.  So, the chart shows the traveler's speed as the distance he covered as measured by him before and after taking the trip divided by the time it took him according to his clock, the same as everyone does when they are driving a car. Ywaz (talk) 16:49, 5 April 2016 (UTC)


 * Note that the thing $$D/T$$ that you have plotted is not the traveller's speed but their so-called proper speed, i.e. the (observer's) coordinate distance $$x$$ divided by the (traveler's) proper time $$\tau$$. This quantity only has a physical meaning at the start and at the end of the trip, when the traveler finally has reached the destination, and is back at rest with respect to the observer. Then they can say, hey look, we have travelled a distance D in a mere time T, so we travelled much faster than light. But everybody else in the observer's frame will say that the distance was covered in a much longer time t: nah, you did not travel faster than light; your clocks and your metabolism just slowed down.
 * Anyway, I personally don't think that the chart brings much new beyond what's already there. Actually, it shows only half of the roundtrips on the two charts that are shown in the article. I also think that there's too much "straight line", so to speak. I mean, there's not much interesting going on, except perhaps in the small bottom left corner, before the lines converge beyond recognition. Mind you, apart from some minor terminology issues perhaps, everything seems correct. Have you found a similar chart in the literature somewhere?
 * What do other contributors to the article think? - DVdm (talk) 18:02, 5 April 2016 (UTC)


 * Many charts only show 1 line. This is 4 equations in 1. Three of them being similar is a bonus, not a detriment.


 * As I said, nothing with mass ever observes itself traveling faster than zero m/s compared to any photons it is looking at. I am not saying he traveled faster than light. It is impossible to travel even 1 m/s compared to any photons you want to race. But external observers do see travelers getting closer to the speed of those same photons. Which view is correct? They are equally valid, so "getting near the speed of light" is as meaningless for the traveler's viewpoint as it is valid for the observer. The D/T ratio is the only reason this article is important. Reflect on why you think the article is important and see if it is not because of D/T.  The D/T ratio is the proportion by which the traveler's fuel expenses are increased when multiplied by his 0.99999 c observed speed, and the proportion by which he does not die. The observer's will agree, so even a popularity vote ("everybody else") as to his actual speed has another consideration to make in addition to being a logical fallacy.


 * Since the chart naturally and easily follows from your equations to the points I think are the most relevant, my view is "of course there are charts like this in the literature". By my experience, if I do the work to demonstrate this, the follow-up complaint will be "but is it common enough?" which is harder to measure. Since there are 4 equations the chart is showing, asking the 2nd question first will save me effort: "is it expressing relevant and established facts?" My answer: if you point out what you think is oddball about the chart I could do a literature search on only 1 of the 4 lines or delete it. Ywaz (talk) 11:17, 6 April 2016 (UTC)


 * You certainly don't have to delete it. It can stay here on the talk page for future reference, and we don't need sources here. Let's just see what other contributors say here. PING others? - DVdm (talk) 11:24, 6 April 2016 (UTC)


 * What are the sources for the current charts in the article? Again, what are you saying in the chart needs a reference? Ywaz (talk) 11:29, 6 April 2016 (UTC)


 * Don't know about the current charts. The curves don't need sources. The conclusions and descriptions might. I have left a little message at Wikipedia talk:WikiProject Physics. - DVdm (talk) 11:34, 6 April 2016 (UTC)


 * The first chart is apparently the same equation as ours since their round-trip is double of my "one way" time for the traveler to milky way center. My line is straight because it is a log plot (a^x functions) instead of a log-log plot (for x^a functions). So it was not correct to use a log-log plot. The description of their chart is not self-contained, so I was not sure what the conditions were and what the equations were. That's why I wanted a different chart. Ywaz (talk) 17:15, 14 April 2016 (UTC)

Why is there no coverage for travel just within the solar system?
This article appears to be only concerned with interstellar travel. What about travel among the planets of the solar system? Several decades ago Robert Heinlein wrote an article on this very point and it's stuck with me ever since. He complained how everyone talked about how it would take many months just to travel to Mars and pointed out that the mistake everyone made was to think only in terms of distance and not time and not computing the time it would take by using constant boost. He then produced a chart to show the time it would take for a round trip to Mars and Pluto using different accelerations:

Aside from the fact that Mars suddenly seems not that far away, the other revelation is that although Pluto is fifty times in distance further away, it is only seven times further in time. It seems to me the article could use something on this point. Am I the only one who thinks so? __209.179.36.56 (talk) 03:45, 6 September 2016 (UTC)


 * Note that there would no significant relativistic effect here. Looking at the article figure, you'll notice that relativistic effects would only become noticable for 1g-round trips of at least a few years. For 1g, this here is just a few weeks.
 * This wouldn't off course be feasible. Imagine the amount of fuel such a rocket would need. But surely (and even more so) that could be said about interstellar travel too. So, yes, interesting, even if just theoretically.
 * As Heinlein "sort-of merely" was a science fiction author, do you know of any other author who might have taken this up? Some standard phycics textbook perhaps? If so, I wouldn't mind having a little section in the article, or perhaps even better, just a mention of what you wrote here (without the table). It should of course be noted before relativity is mentioned. Is the article available on-line somewhere? - DVdm (talk) 06:59, 6 September 2016 (UTC)


 * Thanks for the prompt reply. Did I misunderstand the point of the article? It is about space flight in general, and not just interstellar space flight? If it's the former then including it would seem like a good idea, since we've been told since forever that it would take a couple of years to do a round trip mission to Mars. It would if we coasted the whole way like we did when we went to the Moon. I personally think it would be a great thing to let readers know just how close Mars is by constant boost.


 * As for the impracticability, well, all space travel at the moment, and for the foreseeable future, is impractical. Heinliein rather confidently predicted that boosting at 1/1000 g was doable today, using the as yet unproven light-sail concept. I'll leave that proposition to greater minds than mine to contemplate.


 * I copied the chart right out of the book. It's just the straight forward results of the basic acceleration formula that anyone can do. While Heinlein is of course famous for being a fiction writer, he also wrote science articles, in fact I think he wrote a couple for Encyclopedia Brittanica. I don't know if this particular article is available online, or if others writers have tackled it. But I got this from was from "Where To?" in Expanded Universe, by Robert A. Heinlein, c. 1980, Ace Books, ISBN: 0-441-21883-0. The article is actually in two parts, as he sets up the how-long-it-takes-to-travel by laying out the basic facts, and the math involved, and then asking the reader to solve the equations themselves to see what answers they get. He then provides his answers in the second part located in a later part of the book. The page numbers in my copy are 316-353, 368-371. (The relevant portion in the first section is found on 333-338). __209.179.36.56 (talk) 22:40, 8 September 2016 (UTC)


 * Ok, no objection if you would add a little section Interplanetary travel, immediately before the section Interstellar travel. Make sure to add the source, preferably using a cite book template. Good idea. - DVdm (talk) 08:45, 9 September 2016 (UTC)


 * I would add an addendum referencing pulse jets such as the Project Orion engine. Nuclear powered engines make constant or intermittent boost interplanetary travel possible.  Freeman Dyson's group at General Atomic s decisively showed that their concept would work. There is even a clip of a chemical powered version working. ref: https://www.youtube.com/watch?v=E3Lxx2VAYi8


 * The radioactive contamination problem with launching from earth could be negated or limited by launching from space orbit to orbit the destination planet. Since payload no longer limits travel in the Freeman Dyson Group  design, it is possible to carry a chemical powered lander,and use many smaller chemical rockets to supply the vessel with food, equipment, and passengers.  Travel times as per the Robert Heinlein calculations would put travel time from Earth to Mars in the order of weeks vs. years.  Ref: Project Orion (nuclear propulsion) Wikipedia article. 23.122.255.151 (talk) 23:52, 21 January 2019 (UTC)

Original research
I removed some original research, added by user and later tweaked by user. This clearly was original research, and actually even wrong: Spacecraft strenght is not important, as acceleration can be low (typically 1g), and the powersource must not be "extremely powerful", just long-lasting. The top of the acticle ask for sources, so please let's not add even more unsourced content. I have kept The Anome's other tweaks. - DVdm (talk) 11:04, 26 October 2021 (UTC)