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Proxima b is an Earth-like (i.e. rocky) planet, in orbit around the star Proxima Centauri, approximately 4.243 light years from Sol, our home star.

If I had a space vehicle capable of accelerating continuously at 9.81 m/s2 (i.e. 1g), how long would it take to reach Proxima b?

For the purposes of this question, assume that our vehicle departs from low Earth orbit and the goal is to achieve orbital capture of Proxima b, so our vehicle would probably need to turn around and decelerate with respect to the target at some point during its journey to achieve capture velocity.

Also, for the purpose of the question, I'm asking about the apparent travelling time from the frame of reference of a passenger on our vehicle. That said, it would also be interesting to know the apparent travel time and characteristics for observers on Earth and Proxima b.

Thanks.

MichaelK
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Mark Micallef
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    Given the continuous acceleration and the (I'm assuming) continuous deceleration, one would have to spin the ship at the halfway mark and start decelerating. from about half way. – Edlothiad Feb 09 '18 at 07:21
  • That's what I would imagine, but my understanding of the physics involved in such a long journey, and the effects of approaching light speed, are limited. – Mark Micallef Feb 09 '18 at 07:27
  • Well, that was just a gross over-simplification, but effectively it's like that (I haven't answered because I don't know, I like your question and just wanted to add my $0.02) – Edlothiad Feb 09 '18 at 07:35
  • Math is only part of the problem. You haven't dealt with time dilation or speed of light. – Mark Micallef Feb 09 '18 at 09:39
  • @uhoh It is not so easy, the simple Newtonian method would give an average speed of around c. So there will be huge SR corrections. – peterh Feb 09 '18 at 10:10
  • However, it could conceivably be rewritten into a question about space exploration. – uhoh Feb 09 '18 at 12:06
  • @uhoh How long would a travel last to the Proxyima b with constant 1g acceleration? It is not about space exploration? Huh? – peterh Feb 09 '18 at 12:07
  • @uhoh It is a book example of a space exploration question. I understood that it was a hard week. – peterh Feb 09 '18 at 12:26
  • @peterh I don't think so. It's a math/physics question which is lightly decorated with the name of a star. Let's see how it goes. – uhoh Feb 09 '18 at 12:30
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    @uhoh: It seems a bit odd to say "it's just math/physics with decorations", since of course all space exploration equations are straightforward math/physics problems with decorations. The important point is that they arise in situations that are very unusual outside of space. And, as it happens, acceleration with relativistic compensation is very unusual outside of space. The only other field where that is of any relevance would be particle physics. – Nathan Tuggy Feb 09 '18 at 15:27
  • @NathanTuggy It's a "What is the equation for X?" question and the answer is the one equation. Nothing in this question about SX, only about special relativity If I asked how long to Alpha Ceti at 0.7 g and then asked to Proxima Centauri at 3.2 g, it would be instantly marked as a duplicate since it's the same question and the answer is always one equation. If I asked a questions about two different rockets, or two different spacecraft, the answers would be different and possibly very different, because the questions are about the rocket or the spacecraft, not about a single physics equation. – uhoh Feb 09 '18 at 15:47
  • @NathanTuggy normally I am a softie and will take the time to re-write the question in a way that makes it more about SX, and give it a title that will bring in the largest audience. See Overzealous insta-closing of new users' first question with less than 5 site Q's per day here. Now see this comment. But as this comment suggests, it's been a hard week. – uhoh Feb 09 '18 at 15:52
  • @uhoh: What is the equation for burn time for GEO insertion for GovSat-1? Oops I accidentally closed that as a dupe of a question that gives the equation for GEO insertion burns. (OK, technically there are several approximations of varying accuracy for different situations, such as the "instantaneous impulse" version, but still.) – Nathan Tuggy Feb 09 '18 at 16:29
  • @NathanTuggy this is a gray area, subjects overlap, there are tails, asymptotes, I've explained my thinking, it's after midnight, so... But here's some physics/math I can't finish... care to have a look? – uhoh Feb 09 '18 at 16:36

2 Answers2

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5.8 years

According to this page, distance travelled under constant force acceleration, even up to relativistic speeds, is calculated by:

$s(t) = c(\frac{m_0c}{F})(\sqrt{1 + (\frac{F}{m_0c})^2t^2} - 1)$

...and since...

$\frac{F}{m_0} = a = g \approx 10 m/s^2$

...and since we accelerate half-way and decelerate halfway we calculate the time to travel half the distance and then double that. Half the distance is...

$s=300\cdot10^6m/s\cdot60s/min\cdot60min/h\cdot24h/day\cdot365day/y\cdot2.12y \approx 20\cdot10^{15}m$

...so from this we get...

$20\cdot10^{15} = 3\cdot10^8\cdot\frac{3\cdot10^8}{10}\cdot(\sqrt{1 + \frac{10\cdot10}{3\cdot10^8\cdot3\cdot10^8}t^2} - 1)$

...which gives us...

$t = 92\cdot10^6s = 2.9y$

Double this and you get a total traveltime of 5.8 years.

MichaelK
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  • 4.2 light years in 5.6 years? Thats one hell of a low estimate – Alex Robinson Feb 09 '18 at 13:36
  • @Cursed Not really because that 1 g builds up speed quick. After only 35(ish) days you have passed 0.1c. In slightly less than a year you have pretty much maxed out and are near 1c. – MichaelK Feb 09 '18 at 13:38
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    Do we know if those ~6 years are Earth or ship time? – Diego Sánchez Feb 09 '18 at 14:09
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    This is correct and the 6 years are Earth time. Ship time would be quite a bit less, probably around 3 years. This mission profile does have a few issues though -- apart from needing a ridiculously efficient engine (you might do it with an antimatter powered radiation drive and a high mass ratio) every dust grain that hit you would go off like a nuclear explosion. – Steve Linton Feb 09 '18 at 14:12
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    Now all we need to do is develop a propulsion system that can thrust continuously for ~6 years with 1 g acceleration. Simple. – John Bode Feb 09 '18 at 15:26
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    @JohnBode Even a 0.01g propulsion system would be super interesting, as long as it can keep that up continuously, as it would let us explore our own neighbourhood. And maybe... just maybe... something could. – MichaelK Feb 09 '18 at 15:33
  • OP asks for time in two different frames: "Also, for the purpose of the question, I'm asking about the apparent travelling time from the frame of reference of a passenger on our vehicle. That said, it would also be interesting to know the apparent travel time and characteristics for observers on Earth and Proxima b." Which one is missing? Is it the "frame of reference of a passenger on our vehicle"? – uhoh Feb 09 '18 at 15:57
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3.6 years, in traveler's time

5.8 years, in Earth observer's time

according to the Space travel calculator and this article

fjch1997
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