Space Math
[include file=../includes/math1.txt]
115 Comments

These are fabulous! Thanks!

These look good. But what I’m looking for is a calculator that you can input the planets gravity & radius and determine the planets mass.

The gravity calculation is easy, as long as you settle for Earthunits. The formula is Mass/(Radius^2)=Gravity, with Mass in Earth masses, Radius in Earth radii, and Gravity in Earth gees. If you need meters/sec multiply the result by 9.8, or as accurate as you want to take it.

I like the calculator for long planetary voyages. The relativity threw me off though. How much time passes if the ship makes a round trip? If a ship goes from Earth to Tau Ceti (about 20 ly) and back, it should take more than 12 years even if it just seems like 12 years to the crew on the ship.

Keith, I meant how old is the other twin in the twin paradox? How much time passes for the observer on Earth when his brother on the ship returns.

Just wanted to say thanks. Took me 4 hrs to find these. Although being a math idiot I think I learned some stuff trying to find it.
G®îMÅü®Å <<<<<<Hates formulas 🙂

Thanks for posting! I wish some idiots would answer the Yahoo questions correctly regarding relativistic travel. The major misconceptions about it should be corrected! Glad to see somebody is publishing correct information though.

Awesome! If you travel at the highest known nonlight speed acceleration (which would land you anywhere infinitely instantaneously from the travelers perspective at least), you would accelerate at 10^12 g’s, the acceleration experienced in free fall into a neutron star. If you accelerated at a constant rate for ~2 minutes 49, then decelerated with the same force for the same amount of time, you would reach the end of the known universe, 13 billion light years from home. Of course, by then, the Earth and our solar system would have long since passed into dust (just a few seconds after launch, from the travelers perspective), but you would have lived to see the physical boundary of space!

In your formula for Long Relativistic Journeys, what percentage of speed of light (c) did you max out at? Also, theoretically, how would Length Contraction affect a living organism?

Thanks for the answer. The reason I was wondering was that I found that using the Newtonian Constant Acceleration in G’s put me at 99.99% of c at 354 days of 1g acceleration. The story I am writing involves communication between a person on Earth and another in a ship traveling to another solar system. I’m using speed of light communication and having to determine distance at various points of the journey and calculating time needed to receive transmissions.
Thanks again – this is a great site!

Great site thank you very much. My question is in regards to your first calculator; does it calculate the trip time based on accelerating at Xgs half way, turning around, and decelerating? Or is it just a straight shot past whatever destination? I am not great at math but is it safe to assume I can cut the distance in half, double the time and arrive at my destination at 0c relative to that position?

For space travel conversion I PUT IN 36,000,000 Miles ( MIN. TO MARS) at i G.
It gave a time of NaN. What is NaN? Also Is the G force additive to the human
body at a constant acceleration? Could human body survive or would a constant
velocity at a high speed with no G force be best way to travel ? 
OK , at 1 G constant acceleration it says 1.2 days to reach Mars at its closest point to Earth. Not counting deceleration to reach surface. My question still is if G force is additive at constant acceleration humans would perish at 30 Gs right? Therefore a reasonable cruising speed is need at which point constant velocity would yiled 0 G force until time to deceleration or make turns. Some one guide me on this please. the new show on Discovery Channel ” Bad Universe” seems to have got this wrong.

G force is not additive at constant acceleration. G force is constant at constant acceleration. If a ship is traveling at one G, then the occupants feel earth like gravity the whole way. If the ship is traveling at two G, then the occupants feel twice as heavy the whole way. If you have them travel at 0 g then they will be weightless and the ship will coast in the direction you sent it, but not accelerate at all.
What you really want to do is go half the distance between Earth and Mars accelerating at N G, then turn the ship around and accelerate at N G, where N is the acceleration you are traveling. (I don’t recommend above 2 G with humans on board. It makes walking around the ship difficult.) You should end up at rest in the orbit of Mars. 1 g will take 1.8 days.

These calculators are awesome!
I’d like to point out human jet fighter pilots routinely experience G forces in the 9G range. A G suit is required, and they can only handle it for a few minutes without blacking out. Humans have survived G forces over 50, for a very short time–but with permanent damage to the brain and eyes.

In the “long relativistic journeys” part, you need to add a line feed; the far right part of the calculator is not visible.

Just found this site. Excellent stuff!
I didn’t see an answer to Will’s question – whether or not the first calculator was for the trip under constant acceleration the whole way, or for the trip with deceleration at half way, so the ship arrives at 0 velocity. I’m assuming it’s constant acceleration (since that’s what it actually says).
Also, it would be great if Astronomical Units (AUs) was an option under the units of distance.

could you make a program for it in c please? x x

Keith/admin, please, please put in a carriage return/line feed at the end of the “relativistic journeys” one!!! The “earth time” is not visible; it does not appear to be visible in either Firefox or IE. Although I think it would just be the same as the distance in light years, right?
–Brian

Thanks! 🙂

Are your relativistic calculations made using fourdimensional acceleration w=c^2du/ds or are they made by small increments of velocity under Lorenz transforms?

Thanks, Keith.
This is much more accurate than my rough approximations. I’ll be making use of this in a month or two in my Gail Scott comic. I’ll be sure to put in some references on the site.

i need for a school project a program for a relativistic travel calculator, i have no idea how to do it. if someone knows how to do one could you please post or mail it.?
thanks X X 
I have a relativity qustion… Suppose planets Earth and Woozle are basically stationary with respect to one another. Suppose you have two spaceships, A and B. Suppose A leaves Earth for Woogle (to set up a base), and then six months later in earth’s frame of reference, B leaves Earth for Woogle (resupply mission). Both ships have the same acceleration schedule (say, 1g acceleration first half of trip, 1g deceleration second half of trip). Will B arrive at Woogle six months after A does, by the Woogle base’s chronometers? If not, why not?

Does the long distance relativistic calculation assume that your top speed is just short of the speed of light?

Can’t thank you enough for putting these up. Only issue is that the “conversions” calculators seem to be broken. Everything else works perfectly with FF v4.0.1 but the last two don’t seem to want to update anything.
Or, *Much More Likely*, am I doing something blatantly stupidly wrong?
¡Muchas Gracias!

Hello and thanks for the wonderful calculator!
Would it be possible to have a long relativistic journey calculator based on propulsion force instead of constant acceleration?
If i’m not mistaken the mass gets bigger when velocity goes towards c.
I think if the spacecraft could reach c, the mass would be infinite?I have no idea how hard that would be to do script or if I am even correct.
thanks anyway,
Branx 
I don’t agree with all these scientific hypothesis that a ship accelerating at 1G shrinks or grows in size/ becomes heavier or lighter. passes one year of time while 2 years pass back on earth etc etc.. Where do any of these people get one shred of evidence for all of these mathematical theory claims? Non scientifically a ship at 1G constant acceleration reaches the speed of light in 300400 days? then it turns around – slows down for 300400 days. Then it repeats the process in the opposite direction and ends up back here on Earth at the same day / same time on it’s on board clocks and Calendars. None of you have any proof that time or mass changes. When they develop the ability to travel that way they’ll look back at all your far out inaccurate undecipherable scientific double talk theories you had to that effect in this time and laugh at you.
Also the one parameter I think would be helpful on the first calculator (Which is great by the way,excellent job)Is.. What speed would you be going after say a 10 day 1G acceleration. Not how many billions of milimeters did you travel.

John – your GPS device has to correct for the theories that you claim that there is no evidence. There is empirical evidence for time dilation. http://en.wikipedia.org/wiki/Time_dilation#Experimental_confirmation

I probably shouldn’t be so condescending but I won’t believe it until the day (decade?) I actually climb on a 1G perpetually accelerating self contained spacecraft and perform that 4+ year experiment myself. I’ll arrive back here at Earth and the clocks and calendars in my spacecraft will be the same as the clocks and calendars on Earth. Time travel is science fiction. Anyways – I’ll never be around to see that day. Thanks for the way to calculate the speed at the end.

Hi, I have a question about the “1G” used in the constant acceleration calculater:
Which reference frame are you using to hang the “1G” in? I think maintaining 1G in the earth reference frame would be impossible but not necessarilly in the traveler’s reference frame (if they have a magic fuel source). Within their reference frame 1G always requires the same energy/time input no matter what so it stays manageable from their perspective. The same energy input (for them) yields the same acceleration “G” experience for them regardless.
From the earth reference though I think energy input is dropping the longer they accelerate (as they approach C) because their time is slowing down. Eventually their clock is running so slow that what they call “1G” is negligable in our reference frame. Maybe the Lorentz transform cancels it all out with the mass increase, I’m not too sure of that.
But I do think you have to use the traveler’s frame to calculate the 1G, otherwise they get smooshed as their time slows relative to earth frame and they experience relative acceleration much higher than it looks to us. If you’re not using the traveler’s reference to calculate the 1G then the travel time calculation is probably giving a shorter answer than it should.
I’m guessing you did the right thing because this all looks pretty thorough but wanted to ask the question.
To everyone who thinks relativity isn’t real – it’s already been said – throw away your GPS because if relativity isn’t real, GPS won’t work. All the calculations for GPS require precision time for every satellite in the constellation, and all of these GPS satellites experience enough relativistic time dilation (all traveling in different directions) to screw up your GPS, unless they adjust the calculations to accomodate relativity. Every time you use a GPS, you prove Einstein was right.

Beautiful! Just had to ask… Thanks for such an amazing resource!

The 1G in spaceship frame is the best idea as it helps get around all those medical problems they have on the ISS. That’s why I’m adopting this calculator for an up and coming episode of my web comic. I may get to that bit before the end of the year. 🙂

The first calculator “Newtonian Constant Acceleration in g’s.” seems to be displaying the wrong units for v in the velocity result listed as meters/hour when I use it. It should be meters/second.

Ah, I see where your calculator seems to be screwing up. I input my units and selected hours from the drop down for time. This changed the units in the v output, but did not change the actual value.

Late coming back to comment Kieth, but it is good to see you’re maintaining such a useful site and right on top of squashing little bugs. Kudos! And thanks.

Sorry about the typo misspelling of your name Keith.

this is great. There’s one thing I think you could use here… a way to calculate the earthobserved time versus traveler’s observe time between two points at a given rate of acceleration.
example: spaceman spiff leaves earth at an acceleration rate of 1 G. How long will it take him to travel from earth to, say sirius (10 ly) assuming he accelerates at that rate the entire time and decelerates at the same rate when he reaches the halfway point? What amount of elapsed time does an observer on earth see and what would the shipboard time be?

Oops, looks like a new bug.
velocity box on Newtonian acceleration. Changing from seconds to minutes or hours or days yields bad data.
Seemed to work well enough about a week ago.

It seems like the duration link to velocity is broken; I.E. velocity doesn’t update if I change from seconds to minutes or hours. Incrementing the force level changes the velocity, but not the duration.

Understood. Time eventually changes that problem — trading age for greater freedom. I think it’s terrific that you’re maintaining this resource. I’d hate to be doing the math required otherwise.

Nope…that didn’t work. 1g for 1hr = 32ft/hour
You might need some quiet time to fix this.
There’s an additional problem now where changing the time duration interval now changes whatever you had in the time box in terms of units.
And if I change the force level, that also changes the time duration interval now, which can be an irritation when trying to work backward from a desired plot result.

I’m 64. When Digital was acquired by Compaq, I got a golden parachute into the promised land (Montana). Never regretted it.

Looks like you got it. In a day or two, I’ll be back into the long haul acceleration thing and it will get a workout then.
Intuitively, I get how Newtonian acceleration blends with time dilation. If you pile on 1000g acceleration for ten hours, you end up going faster than light with the Newtonian tool. But the ship doing 1000g’s would never get to ten hours as time dilation would slow the relative (ship) passage of time as the velocity approaches .999999+ percent of C.
Using the Relativistic Journey tool, I can approximate what that dilation is and how it affects the plot. Super tools. Thanks for sharing them.

I use to do these calculations when I was a kid on my TI83. Very cool to see someone else writing these programs for people.

This is really an awesome webpage. I stumbled across it when I was randomly Googling interstellar propulsion. I have one question about constant acceleration drives. Once a ship is nearly at the speed of light, is it necessary to keep accelerating to get the same kinds of speeds indicated in the “Long Relativistic Journeys” section?
Basically once one gets to 0.999% C, can they just coast for the light speed portion of the trip to save fuel for the deceleration, or do they need to keep adding .9’s to the percentage (if that makes sense). Oh, and I realize that the answer is probably “no” to the first and “yes” to the second, but I’m curious how much difference those extra .9’s make in journey time (for both traveler and Earther).

It takes a lot of 9’s behind the decimal point to really dilate time for the travelers. Essentially as their time dilates, the rate of acceleration drops toward zero, never getting there of course, just like they never get to the speed of light.
But while pouring on the power, time continues to dilate, making a very long trip, like 500 light years, pass in just days for the travelers, while 500 years go by in realtime.
In my current plot, one of the characters constructed a virtual chronometer which compares ship’s time and real time, giving a sense of the passing years in realtime as the 9’s mount up behind the decimal point. At the point of flipover, three months pass in about 150 seconds. This takes at least nine more nines behind 99% and a prodigious impulse engine – like a beam core antimatter source where the fuel has two orders of magnitude more energy than fusion.
It is an awesome set of tools however. Much better to be composing the prose instead of doing the math long hand.

Catastrophic for the nonC object, perhaps. Not so much for the ship traveling at .9999999 C, with its huge kinetic load all the way down to the subatomic level. Seems to me it would be doing the vaporizing instead of being the vaporizee.
Imagine trying to take a fix with sensors. Nothing but blueshifted garbage coming in from up front, where 18 hours of data might arrive in a few seconds, and from astern, data redshifted to…well nothing. Can’t detect the incoming photon because it takes too long to be detectable. Might be a 360 deg ring around midship, but somehow that concept seems to elude my imagination.

This is a great tool. I usually just do these calculations in my head or on a regular calculator if I’m lazy. It’s nice to have a quick cheat to see if I have a mistake. Keep up the good work.

Hey, I’m not getting any results out of the time dilation section. I’m curious to know the dilation that a speed of 0.5c will spit out.
Can you help?Cheers
🙂 
I love this page, thank you. It would be cool to have fuel/payload ratio for long distances assuming some theoretically perfect engine.

Thanks, Keith! I’m fairly conversant with several sciences on verbal, graphical and statistical (as they are concrete) levels, but not the math. I only passed basic calculus via memorization & a generous curve (managed a B, but I remember nothing but the graphical concepts and that there was a lot of trig having nothing to do with measuring the height of flagpoles). I couldn’t do the relativistic calculations on my own, and was excited to find this site to generate plausible numbers for my scifi stories. Your other sections are thoughtprovoking as well. At the risk of offending your Weasel Word Detector: thank you very much for your time & generosity in providing this tool.

I stumbled across this resource today and have already shared it with the writers who follow me on Twitter.
Shoot me an email if you want some help with those acsh calculations for the relativistic travel tool.

Hey, Keith. Just wanted to thank you again for the calculators and for the fix, some months back, on the units gadgets.
If you ever have the time/inclination to do the velocity/mass>KE calculator you’ll be an even bigger hero in my book. After much frustration I’ve got my TI calculator to give mostly believable numbers but I’d be happier if I could compare them to your proven QA.
Many thanks!
Tom

Hi Keith, this is a nice calculator. I have a suggestion for Long Relativistic Journeys, can you add the Average and Peak speeds in terms of c (and maybe other units)? Thanks.

Just a courtesy notice that I linked your site in my piece “Star Wars, Einstein and When Lucas Got It Right” ( http://members.cox.net/maggieameanderings/2011Oct16.htm ). And thanks……this page perfectly illustrated, in an easily understood way, the point I was making.

Hi 🙂
This is a impressive effort. One thing I would love to see though is the math behind your calculations. Yeah, I’m sort of curious.

32000 ly journey at 1G will take only 20 yrs for the ship ?
are you sure that is correct? 
Let me put in a vote for relativistic energies. Especially after reading the piece of prose pollution _Last Train From Hiroshima_. It’d be nice to have a onestop source for the energy of a nucleus traveling at relativistic speeds, to say nothing of rocks, ships, planets, and whathaveyou.
I really appreciate the fixes you did a few months back after I bugged you and I hope that there’s enough support and you have enough time and interest to add the energy calculators to the site.
Once again, humble and unworthy thanks from a frequent visitor.
TK

THANK YOU! THANK YOU! THANK YOU! As a scifi writer this is PERFECT! Never let this site go 😀

Thanks for this Keith. I’ve wondered about the math since I read A.C. Clarke’s “Rendevous with Rama” series in high school. Using your site for 10 or 15 minutes helped me get a much better mental model of the implications of relativity.
A ship sets out from Earth to visit Sirius (about 8.6 LY distant) with an acceleration/decel of 1G for each half of the trip. It takes about 4.6 years ship time, and about 10.3 years earth time. A second ship heads out to visit Kepler 22B (~600 LY distant) at the same rates. The trip takes 12.5 years ship time, but over 602 earth years pass.
I never had a firm mental picture of time dilation until I played with your calculator. STL speeds increase asymptotically, so the decimal places have incredible impact. I never visualized how much impact. Travel at .999C for 30 days ship time and 670 days pass on Earth. But travel at .999995C for 30 days and 9,486 days pass on earth. Observers on Earth see ship time slow; onship observers see Earth time speed up. I knew that but I couldn’t visualize it as clearly as I can now. Cool, thank you!

Amazing!
I love the relativistic journey calculator. I am planning on going into engineering and studying the building of spacecraft and propulsion systems. Playing around with the calculator provokes some interesting thoughts how about a time capsule send out a time capsule and it could return to earth thousands of years later. You could also do a twoinone mission; send a heavily shielded time capsule and a robot to take observations of a star of your choice; imagine what people would think hundreds of years later. Of course, it also raises questions about the practical application of space travel and the real difficulties in maintaining a type II civilization. 
I argued with my later 4time coauthor Sir Arthur C. Clarke about this, and in 1969 he agreed that I was right. The relativistic multistage rocket equation, not within the reach of your calculators, allows for “Practical Robotic Interstellar Flight” even with fission power (which converts roughly 0.1% of mass to energy), and more easily with fusion power (roughly 1% conversion).
See my refereed paper (republished by the British Interplanetary Society):
INTERSTELLAR
http://magicdragon.com/ComputerFutures/SpacePublications/STAR.htmlHydrogen Ice Spacecraft for
Robotic Interstellar Flight
by
Jonathan Vos Post, F.B.I.S.ABSTRACT
Spacecraft constructed from cryogenic hydrogen (or deuterium
and tritium) ice can use the same material for structure,
shielding, coolant, and fuel. This type of “autophage”
(selfconsuming) spacecraft achieves an extremely low deadweight
fraction, which is a critical parameter for optimizing the
performance of interstellar spacecraft.To reduce the volatility of hydrogen ice, a particular
selfrefrigerating structure is presented. Hydrogen ice by itself
is imperfect as a structural element; various methods of
stiffening by the admixture of carbon or boron fibers are
discussed. Other cryogens relevant to specific fusion reactions
are considered, including deuterium, tritium, boron11, and
saturated solutions of lithium in anhydrous ammonia.A quantitative analysis is presented of the relativistic
kinematics of multistaged interstellar iceships. In the limiting
case of a 5stage deuterium ice fusion spacecraft on a oneway
mission with no deceleration at the destination, a deadweight
fraction of 101 for each stage, and a total payload fraction of
105, then the final burnout velocity of the 5th stage is 0.64c,
which at constant 0.0485 g acceleration would reach Alpha Centauri
in 12.81 years, and at 1g acceleration would get a probe zipping
through the Alpha Centauri system in 6.7 years.Appears in: the Proceedings of “Practical Robotic Interstellar Flight: Are We Ready?”, New York University, New York City, 29 Aug1 Sep 1994, and in Journal of the British Interplanetary Society, April 1996

What a wonderful tool for laymen writing speculative fiction! Thank you for making this superb resource available to the world.

Great page, thanks! Could you add light years as optional distance units for the first calculator of distance acceleration?

The first calculator (Newtonian Constant Acceleration) is broken. It works fine if you enter time to get the distance but it doesn’t work the other way around. I tried entering 124,000 miles and got 0. Clearing and not clearing the time field before entering the distance makes no difference.

nice work !
a little buggy & can be improved a bit….all calculations should be completely reversible in relation to other fields & all input fields magnitudes should be expandable @ least to the ability of the script.
but way cool overall….Thanks ! 
Just found thiis website during some research. Just in case anyone is interested on board velocity can be calculated by:
([v/c]x[yg]x[time in seconds]/c in meters. y = number of g’s.
e.g. ([0.999]x[100×9.8]x[86400])/c = approx 0.2822c or after 10 days approx 2.822c. So in this way the crew of the ship consider they are moving faster than light because the on board guage is cummulalitive. 
The “Newtonian Constant Acceleration in G’s” Calculator only allows you to change the first two variables.
Also, despite that I must say I love this and its going to be bookmarked forever because it makes things so much easier.

Good to know. 🙂

this tool helped me to get aware that ‘a constant 1 g acceleration would permit humans to travel through the entire known Universe in one human lifetime’ and that is great idea !

This is an excellent resource. You really should consider creating a smart phone app with this. I’ve made this a favorite on my computer,

@Larryd: the crew wouldn’t consider themselves to be travelling faster than c, nothing can travel faster than c when viewed from any reference frame. You’re using Newtonian logic in a relativistic setting.

sean, I didn’t write the comment correctly. I intended that ‘IF’ the on board computer was cummulalitive. As you point out on board would probably be using a continuous form of the SR v addition equation and show the fraction of c obtained.

Keith although there various ideas around at present with regard attaining light speed (warping space for example) they remain ideas only, so we don’t have the technology to test these ideas…yet.
Disregarding Tachyons, a photon travels a c and present theory says that a photon does not realise the passage of time and could, theoretically, travel the known universe in zero time. We, of course would record X billions of years for that photon to travel. But this suggests even deeper questions for if the photon experiences no lapse of time then…
TOE theories have a variety of points to make but they have problems of their. Also remember that the components of photon propagation exceed c. 
Would you check the escape velocity conversion calculation? For Mars it shows MPH = 2251 but KMph = 1399 ?

This page is dynamite, thank you for helping those of us who love math but haven’t done it in years!

I’m trying to find a gravitational time dilation calculator. Just a simple nonmoving planet with (Y)mass and the distance from it would calculate the time difference during an (x) amount of time period. Would you be willing to provide that too? Looks like many more people are asking for it too.
I’d be forever grateful. 
I need to know how long it would take to travel from the asteroid belt to earth at a constant 1 g acceleration without needing to consider fuel or mass.

I need to know the time it would take for a trip from the asteroid belt to earth at a constant 1 g acceleration without having to consider fuel or mass.

specifically (2611) Boyce. However, all I need is the average time for a zerozero at earth orbit with an acceleration/deceleration of 1 g constant. Once in earth orbit, the pilot can maneuver to either catch up to the earth or let the earth catch up to him then go into an orbit of the earth that avoids space habitats and satellites.
would have been nice to go the other way with some of these like i know the 2 lengths of time but don’t know the velocity