# Space travel using constant acceleration drive: Earth to Europa

Long story short: I'm writing sci-fi and taking my protagonist to Europa. He's got 2 weeks to one month to get there from Earth, give or take a few days. That sounds, of course, preposterous in this day and age. However, it's the future, he's stolen an alien ship, and I need to know, using the ship's constant acceleration drive, how fast the spaceship would need to travel to get there in my literary time constraints (2 weeks-month). Any flight path between here and Jupiter is perfectly okay by me, but the ship will need to accelerate most of the way there and then decelerate before looping around Jupiter to land on Europa. Hoping I don't have to pull out the warp drive to make it happen. Any thoughts?

http://en.wikipedia.org/wiki/Space_travel_using_constant_acceleration

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Just found this: Relativistic Star Ship Calculator mysite.verizon.net/res148h4j/javascript/script_starship.html –  adamholtwrites Mar 5 at 19:27
Early Jan., 2070. My rough calculation, based on the calculator, looks like, at 5.2 AU, it's ~6.5 days, not counting an orbit around the Earth at departure and a pretty sweet fly-by of Jupiter in order to decelerate for landing on Europa. Am I missing anything? Propulsion isn't my area of expertise, to say the least. –  adamholtwrites Mar 5 at 20:34
Accurately calculating interplanetary travel time with this much attention to detail?? Nice... can I read this book/story when you're done? What's on Europa?? Gimme :) –  dvnrrs Mar 5 at 21:42
Somewhat related: How fast will 1g get you there? –  TildalWave Mar 6 at 9:49
Hopefully, the alien ship is calculating the matching trajectories for him. Because, FWIW, I wrote a simulation program of just such a thing a few years ago, and I can assure you it is virtually impossible to just eyeball this yourself without one heck of a lot of practice. I spent an awful lot of time missing planets by a wide berth. –  RBarryYoung Mar 18 at 3:13

Jupiter is 778,500,000 km away from the sun, on average. Earth is 149,600,000 km. Thus, the distance to Jupiter is always between 630-930 million km. So, let's take that range, and figure out what the time would be, given 1 g of acceleration, and ignoring for the moment relativity. Let's also ignore starting/ending velocity, as I'm feeling lazy... Okay, so 1 g of acceleration, go half way, 1 g of deceleration. What does that give?

$d=\frac{1}{2}a\cdot t^2$

For the min case:

$t=2\cdot \sqrt{\frac{\frac{2}{2}630\cdot10^9}{9.8}}=253546s$, or about 71 hours.

For the max case:

$t=2\cdot \sqrt{\frac{\frac{2}{2}930\cdot10^9}{9.8}}=308055$, or about 86 hours.

Bottom line, 2 weeks should be no problem, presuming you could accelerate that fast. Okay, so you want to stretch it 2 full weeks? Let's take the long path distance, and figure out what the acceleration would need to be:

$a=\frac{d}{(\frac{t}{2})^2}=\frac{930\cdot10^9}{(\frac{1209600}{2})^2}=2.54 m/2$, or about half of the gravitational force.

The max speed in each case:

• Short- 2484750.8 m/s, or 0.83% of the speed of light
• Long- 3018939 m/s, or 1.01% of the speed of light
• 2 week- 1536192 m/s, or 0.51% of the speed of light
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Thanks for the help. That works well. I may have the craft decelerate midstream to mimic the gravity of Europa, their destination. That, plus a lap or two around Jupiter for sightseeing purposes, should put me in on schedule. Am I missing anything? –  adamholtwrites Mar 5 at 20:46
The surface gravity of Europa is around 1.3 m/s, which isn't a high enough speed. But I suppose you could gradually go from 1g to Europa gravity, and still have time left over. –  PearsonArtPhoto Mar 5 at 20:53
I can set that scene near the midpoint when the ship moves from acceleration to deceleration. They can prepare for Europa's surface gravity at that point. –  adamholtwrites Mar 5 at 21:13
Usually acceleration is $a$, not $v$. You should divide the actual distances by $2$ because you accelerate for half the distance, then decelerate for the other half. This divides the times by $\sqrt 2$ The conclusion that you can get there easily and are well below the speed of light is correct. –  Ross Millikan Mar 6 at 1:22
I did end up dividing the distances by 2, that wasn't made clear. Essentially, the 1/2 cancels the division by two out, I did make that clearer now though. And I thought I had typed a, not v... Sigh, not sure what mental lapse happened there... Thanks for the feedback! –  PearsonArtPhoto Mar 6 at 3:37

@PearsonArtPhoto's (excellent) answer doesn't consider efficiency.

Direct travel – by accelerating directly toward the spot Jupiter will be at when you arrive, and then decelerating for arrival – is not the most efficient way to reach Jupiter. To date, no human-made spacecraft has used such a method. So taking the distance to Jupiter and calculating travel time based on simple "point A to point B" classical mechanics may not be the best answer, depending on the needs of your story.

With present-day technology, our spacecraft achieve interplanetary travel using specially calculated orbital transfer maneuvers. These require far lower quantities of energy, but also take far longer (several years would not be uncommon for a voyage to Jupiter). These methods work by using a transfer orbit which intersects both Earth's orbit and Jupiter's orbit. A small orbital change is made at the beginning of the journey to move from Earth's orbit to the transfer orbit, and then (perhaps years later), at the point where the transfer orbit intersects Jupiter's orbit, another small change is made to join Jupiter.

Physically, there is absolutely nothing wrong with your craft going directly there in the allotted 2-week time span (per the other answers) using a direct thrust method. But I wanted to point out that doing so in your story will imply, to a knowledgeable reader, that your characters/technology possess effectively "unrestricted" amounts of energy, and have reached a point where they simply don't care about energy efficiency the way modern humans do.

I don't know anything about your story so I have no idea if this is a problem at all, but it may be something to consider. You did mention this was a stolen alien craft, so if the craft happened to have a "full tank of gas" when it was stolen, and sufficiently advanced alien propulsion/energy technology, then there should be no problem.

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Great point, dvnrrs. Thank you for these thoughts. Efficiency is, for this alien race, not a concern and our thieves do have a full tank. If I was writing hard sci-fi, I would definitely be using orbital transfer maneuvers. I looked into the Hohmann transfer orbit of New Horizons. –  adamholtwrites Mar 6 at 0:16
@user2802 Well it certainly sounds like you're being quite thorough and I think that's awesome! Will we be able to read the completed work? Good luck! –  dvnrrs Mar 6 at 5:38
Hey dvnrrs, certainly! My debut novel, The Conspiracy Game, is available on Amazon right now. Print/ebook/both. I'm working on the followup, The Rathmore Chaos, which will be available late this year. –  adamholtwrites Mar 10 at 19:51