Constant Acceleration: Across the Solar System and Beyond

When doing some research for my novel, which now stands at 19.557 words, the Venus chapter (12,949 words) having been completed yesterday, I did some calculations for how long spaceflight would take under constant acceleration, the method the main characters in my novel are using to travel throughout the solar system. In their case it is nuclear pulse propulsion and the desired acceleration is 1g, enough to provide full Earth gravity by way of the thrust pulling you back. With the spaceship designed so that the propulsion system is below the habitable volume’s floor, the artificial gravity pulls in the correct direction. If you speed toward your destination, your destination will appear in front of the vehicle, i.e. precisely over your head or over the ceiling of the habitable volume.

This is rather convenient for space travelers since aside from helping to get to their destinations faster it also eliminates the need for rotation to provide artificial gravity, meaning spaceships can be any size or shape and still have full gravity. Deceleration is provided in my novel by simply rotating the ship 180 degrees and applying reverse thrust, so for the second half of the trip in my characters’ case would involve the destination they’re going towards being right under them under the floor; where they came from will be visible over the ceiling. This takes a lot of fuel compared to braking using solar or magnetic sails, but they cannot provide sufficient thrust for the gravity aspect to be maintained, so nuclear pulse propulsion in the second half it is.

Nuclear Pulse with Constant Acceleration in my Novel

My novel is set in the 2020 of an alternate histoory, early-to-mid June to early September 2020 to be precise, and braking to your destination in space using solar or magnetic sails is already a common technique. It is thought that theoretically they might be able to provide enough thrust for high-speed spaceflight at 1g, but the technology is just not there yet. A key disadvantage of using fuel to decelerate, nuclear explosives in nuclear pulse propulsion’s case, is that you have to retain half of your fuel for the deceleration phase, which limits the amount of thrust that can be employed in the acceleration phase. Nuclear pulse propulsion could accelerate to perhaps 10% of the speed of light, but the use of nuclear pulse to decelerate cuts that maximum speed to just 5% of light speed.

Fortunately for our main characters and virtually everyone else traveling through space in the alternate 2020 of my world, this is not a serious concern, as their trajectories don’t demand anywhere near that much velocity.

Constant acceleration is commonplace in the 2020 of my world, at least for those hotshots that are the equivalent of private jet travelers on Earth today, and so I will lay out in the rest of this post just how long it would take to get between common destinations in the solar system. For the sake of comparison and speculation I will also explore 1g constant acceleration trajectories far beyond the solar system, as well as other level of acceleration. But for now, I’ll start with the planetary destination closest to home: the Moon.

Wandering the Inner Solar System at 1g Constant Acceleration

The Moon is the only other world we have visited to date in real life; the Apollo spacecraft somewhat famously got from Earth to the Moon in 3 days or so, which honestly is not bad. While it isn’t as fast as a transatlantic jetliner, it’s considerably shorter than many road trips people routinely take. Applying 1g of thrust to accelerate and decelerate would take us from the Earth to the Moon in 3 hours 28 minutes.

At the halfway point, acceleration and thus gravity ceases; the interior of the ship is plunged into weightlessness while the ship rotates 180 degrees and then resumes thrust to decelerate. While it is conceivable that thrust could be employed during the rotation process to maintain gravity constantly, a period of weightlessness would be slightly more fuel-efficient and probably much more fun for the passengers. The ships in my novel go through this weightlessness at the halfway mark, which is dubbed in-universe as “the turnaround.” Flying and tumbling around the cabin with a lover, keeping aloft for as long as possible before gravity resumes, is considered a thrilling and romantic activity.

The maximum velocity achieved during the trip from the Earth to the Moon is 61 kilometers per second. For comparison the maximum speed achieved by any of the Apollo missions was 11 kilometers per second, which gives you an idea of just how fast you need to go to accomplish constant acceleration. And this just from the Earth to the Moon, right next door in interplanetary terms!

As an aside, a constant acceleration trajectory with a maximum speed of 11 kilometers per second would take you only 8000 miles and take about 38 minutes. This is similar to the maximum speed and distance theoretically achievable by a gravity train through the Earth’s center, for the simple reason that such a train also travels at 1g, 11 kilometers per second is roughly the same as Earth’s escape velocity, and is also roughly the speed the Apollo missions traveled at.

It’s also worth noting that these calculations are for the distance between Earth and the Moon as of today. The Moon’s distance varies between around 225,000 and 250,000 miles, so the travel time under constant acceleration would vary too, from 3. hours 22 minutes to 3 hours 34 minutes.

The next most obvious destination is Mars. The distance from Earth to Mars varies much more than the distance to the Moon does, from roughly 0.4 to 2.6 AU. At constant acceleration of 1g, this trip would take anywhere from 43 hours (1.8 days!) to 4.6 days depending on the distance. Maximum velocity would range from 766 to 1,956 kilometers per second, much faster than anything we’ve sent out in real life so far. Still, 1,956 kilometers per second is only 0.652% of light speed, very easily attainable by nuclear pulse propulsion. Contrary to what you often hear trips to Mars need not involve weeks or months of travel time even with near future technology.

Although Mars is the most obvious interplanetary destination Venus is actually closer to Earth, the distance to that planet ranging between 0.25 and 1.75 AU. A 1g trip from Earth to Venus therefore would take between 34 hours and 91 hours, or 1.4 to 3.8 days. Maximum velocity would be between 606 and 1,602 kilometers per second.

Earth to Mercury ranges between 0.6 to 1.4 AU, implying 1g travel times between 53 and 81 hours, or 2.2 to 3.4 days. Maximum velocity ranges between 938 and 1,433 kilometers per second. As you can see at 1g constant acceleration any place in the inner solar system can be reached from Earth well within a week.

Of course in a future with space colonization Earth will not always be a destination on a journey. According to Wolfram Alpha, Mercury and Venus are currently 0.65 AU apart, so a journey at 1g between those planets would take 55 hours, or 2.3 days, and peak at 976 kilometers per second. Mercury to Mars would take 79 hours, or 3.3 days, peaking at 1,401 kilometers per second. Perhaps the longest ride through the inner solar system would be Mercury to Ceres, which at their maximum possible distance from each other of 3.45 AU would take 5.3 days and peak at 2,249 kilometers per second, or 0.075% of light speed. Between opposite parts of the asteroid belt, a distance of perhaps 6 AU, the trip would take 7 days and peak at 2,966 kilometers per second, or 0.098% of light speed.

More Distance, More Challenge: into the Outer Solar System

The distance between Earth and Jupiter is comparable, ranging from 4 to 6.5 AU. This means that rather than years-long trip times enough thrust to provide Earth gravity could get you to Jupiter in a week. Earth to Saturn is at most a distance of 11 AU, which would take 9.48 days to traverse, peaking at 4,016 kilometers per second, or 1.33% of light speed. Now we’re starting to get into “fractions of the speed of light” territory, but the maximum speed of nuclear pulse with half of the explosives saved for deceleration is still comfortably higher than this, at 5% of light speed. Saturn is starting to get deeper into the outer solar system.

Further out still, we have Uranus, which is perhaps 21 AU away at its furthest point from Earth. To traverse this at 1g would take only 13.1 days, peaking at 1.8% of light speed. Earth to Neptune, the most distant of our solar system’s gas giants, is 31 AU and at 1g would take 15.9 days and peak out at 2.2% of light speed. That’s really fast by interplanetary standards, but still well within nuclear pulse propulsion’s limits. Technically, therefore, there is no reason for travel in our solar system to take longer than a fortnight or so in the near future. In the world my novel is set in, exactly this technical dream is realized, at least for the main characters.

The Kuiper Belt, containing ice planets such as Pluto and Eris, is the next furthest region. Sedna, the most distant known planet in this region, is currently 85 AU away from Earth. Earth to Sedna would therefore at 1g take 26 days and peak out at 3.72% of light speed. Closer planets such as Pluto would of course take shorter times and slower peak speeds. This is only the beginning, however, of the outermost solar system. If a telescope cloud, as I blogged about once, is ever realized in the inner Oort cloud region just past where the sun can act as a gravitational lens, 550 AU and further out, it would take a while to get there even at 1g constant acceleration. At 1g 550 AU takes 67 days to traverse with a maximum velocity of 9.44% of light speed. This is well beyond the ability of nuclear pulse to decelerate from and is nearing the maximum speed it could accelerate to, 10% of light speed. A realistic near future implementation, therefore, would take even longer than 67 days to reach the telescope cloud or inner Oort cloud.

It is for this reason that in my novel the plot point of the main characters visiting an inner Oort cloud research station has been revised to the main characters meeting a spaceship in the Kuiper Belt region carrying the same characters on a return to the inner solar system. Taking 67 days or more, probably something like 3 or 4 months, to go on one leg of the treasure hunt would just drag the story down far too much. As it is now they still take a languid 18 days, but that’s manageable. Fortunately for me, nothing about what I planned for the story necessitates they be on the actual research station, just that they meet the characters that were on it and discuss it with them.

Of course, one way to decrease this time would be to increase the acceleration; that’s easier said than done given the technical constraints of my alternate history’s 2020, but it might be relevant in other settings or later eras. On their currently-slated 18 day trip, our main characters’ speed will peak at 7,847 kilometers per second, or 2.6% of light speed. Increasing the acceleration to 2g gets them there in 12 days, peaking at 3.6% of light speed. So far so good, but it would take 4g to get them there in just 9 days. As an example, to match the Earth to Mars 1g trip time of 1.4 days for this 40 AU distance an incredible 170g acceleration would be required, peaking out at 32% of light speed.

Clearly this is not practical, unless the passengers could be submerged in tanks of some sort of breathable liquid that is the same density as the human body, thus blunting the trauma of high acceleration. While such a thing could probably be developed eventually, it is not within the technical abilities of my novel’s version of 2020 and in any case would be an extremely uncomfortable way to save a fortnight’s travel time. Almost everyone would certainly opt for 18 days in cruise-ship or private-jet style comfort over 2 days of being immersed in some exotic liquid breathing tank.

The 550 AU trip would be helped greatly by the higher acceleration. While it takes 67 days at 1g, at 10g it would take 21 days, a saving of almost two months. At 100g it would take only 6.4 days, at least from the perspective of the ship. The peak velocity is 72% of light speed, and at that velocity time dilation becomes pronounced; from Earth’s point of view it would take them 7.4 days. Of course we’re starting to get well beyond the nuclear pulse range in terms of speeds. From now on we will leave the question of the propulsion system aside and look at cosmic distances purely in terms of acceleration, to give us some perspective.

Traveling the Universe at 1g: Possible in Principle

The nearest star of course is Proxima Centauri. Although it might feel like we’re already talking about interstellar-style propulsion systems to get to the inner Oort cloud, 550 AU is still only 0.0087 light-years, compared to 4.22 lightyears for Proxima Centauri. At this distance 1g will get you there in 3.53 years. This is much longer trip times than anywhere in the solar system, though not quite as punishing as is often portrayed. The maximum velocity needed here is 94.92% of light speed, so already to get to the nearest star under thrust-provided gravity we’re very far into the relativistic range. Accordingly, there is substantial time dilation, specifically a Lorentz factor of 3.1787 at maximum velocity; 5.85 years have passed on Earth during the ship’s 3.53 year journey.

It just gets more impressive from there. To get to the Pleiades star cluster, 444 lightyears distant, would take 11.88 years at 1g, peaking out at 99.9990562% of light speed. At these sort of speeds it makes more sense to use the Lorentz factor, which in this case is 230.17. 11.88 years have passed for the ship, but 446 years have passed on Earth.

In galactic terms, though, that’s right next door. The center of our galaxy, surely a spectacular destination for any would-be space travelers, would take 19.89 years to reach at 1g of acceleration. That’s starting to become a very long trip for the passengers; from Earth’s point of view it’s even longer, at 27,902 years. At that point it would probably be best not to ever come back, if science fiction is any indication. The maximum Lorentz factor is 14,401, corresponding to an incredible 99.99999976% of light speed. This is something of a problem, since the cosmic microwave background radiation is actually blue-shifted into visible light at a Lorentz factor of around 550, much lower than what these people are traveling at. If you go fast enough the background radiation will be blue-shifted into gamma rays, which will be hazardous to the passengers.

Nevertheless let’s assume this problem is surmounted. To reach the Andromeda Galaxy at 1g of acceleration would take 28.6 years, peaking out at a Lorentz factor of 1,300,689. This would take millions of years worth of Earth time, and would be a somewhat long trip even for the passengers. Interestingly, at still greater distances the ship travel times involved don’t exactly explode upward. For instance, a trip of 1 billion lightyears would take 40.22 years of ship time, peaking at a Lorentz factor of 516,145,976. A trip of 13.8 billion lightyears, to the edge of the observable universe, would take 45.3 years, peaking at a Lorentz factor of 7,122,814,460. We have no idea how big the universe truly is, although the observable portion (due to the expansion of space since the time light was first emitted that we see today) is currently 93 billion lightyears in diameter. The shape of the universe is to our instruments indistinguishable from being flat, which seems to imply that if the universe has a finite volume at all it must be immense compared to what we can observe.

A trip of even 100 billion lightyears at 1g would only take 49.13 years, and peak at a Lorentz factor of 51,614,597,530. A 100 year trip at 1g, the most that could be completed within a human lifetime, would cover over 10 sextillion lightyears, a trillion times as much distance as 10 billion lightyears. The Lorentz factor at maximum velocity would be an incredible 5 sextillion.

Toward the Edge of Mortal Comprehension

This doesn’t even begin to get into higher accelerations. At 10g Proxima Centauri could be reached in 8.9 months instead of 3.53 years, a substantial savings, peaking out already at 99.90365633% of light speed, or a Lorentz factor of 22.787. These savings compound over the distances we’ve explored with 1g. The Pleiades cluster could be reached in 1.63 years instead of 11.88 years, peaking at a Lorentz factor of 2,292. The center of our galaxy could be reached in 2.43 years, at a peak Lorentz factor of 144,066. Even the Andromeda Galaxy would only take 3.308 years to reach, not much longer than many sea voyages took in the Age of Discovery, at a peak Lorentz factor of 13,006,880. Even 13.8 billion lightyears would only take 4.97 years at 10g acceleration. A 100 year trip at this acceleration would cover a distance equal to 10 to the 223rd power lightyears. That’s well over a googol times more than the distance covered in 100 years at 1g.

100g acceleration, perhaps around the limit that could be achieved even with liquid breathing, could get you to Proxima Centauri in 43 days at a peak Lorentz factor of 218. To reach the Pleiades cluster would take only 76 days, speed peaking at a Lorentz factor of 22,918. The galactic center could be reached in 105 days, the Lorentz factor peaking at 1,440,048. Andromeda could be reached in 137 days, or 4 and a half months. The edge of the observable universe could be reached in only 6 and a half months at a peak Lorentz factor of 712,281,445,900. I won’t even bother trying to calculate the exact number of light-years a 100 year trip at 100g would traverse but I assure you it is many orders of magnitude greater than a googol.

Conclusion

Coming back down to earth a bit, we have seen in this post that spaceflight using constant acceleration proves that even in the absence of wormholes or other shortcuts to travel faster than light can along the usual routes to our destinations spaceflight even to distant destination is in principle perfectly possible well within a human lifetime, even at a Earthly 1g worth of acceleration. 100g of acceleration makes going to the nearest star seem like taking a leisure cruise in terms of trip time. Relativity is no obstacle; indeed, relativity actually makes spaceflight easier from the point of view of the ship.

Although it takes more and more energy to get a given unit of speed from the Earth’s point of view, from the ship’s point of view length contraction more than offsets that factor. In a Newtonian universe, to go to Proxima Centauri at 1g would take 4.04 years from everyone’s point of view (time is not relative in this scenario) and achieve a maximum velocity of 2.08 times light speed. By contrast in Einstein’s universe, the one we actually live in, a 1g trip takes only 3.53 years, substantially less time, for the ship but 5.84 years as seen from Earth.

The journey to the Andromeda Galaxy is an even starker example, taking 3,125 years in Newton’s universe, peaking at 1,612 times light speed, but only 28.62 years in Einstein’s universe from the ship’s point of view, but over 2 million years from Earth’s point of view. Interesting to think about it that way, isn’t it?

Of course the technologies it would take to reach the speeds required to constantly accelerate over such long distances are hard for us to fathom, but the examples at least within our own solar system are easily within reach of even what is at its heart 20th century technology: nuclear pulse propulsion. In the near future thrust gravity between planets will be achievable and perhaps even economical, probably providing the standard benchmark for interplanetary travel times. After all, it’s the maximum acceleration that is comfortable, and so is a natural choice based on human factors for the future of spaceflight. Worldbuilders, authors, and artists of all stripes that are interested in space travel would do well to take note of this methodology.