Acceleration/Deceleration equation for travel within a solar system

Basics: There is a wormhole in a fixed position within a solar system. It is roughly 40 million miles from the orbit of the habitable planet, which kind of mimics Earth (roughly 12 month solar orbit at a distance of 90 million miles, etc).

Ships typically travel from our Solar system through said wormhole during a specific window and arrive at a time where the planet is approaching its closest point to the wormhole as this is the most cost efficient journey.

What I can't seem to get my head round is the speed, acceleration and deceleration. So how fast would the ships need to accelerate, to what optimum speed, before slowing down and the rate of deceleration required to avoid turning the humans inside the ships into jelly.

Would it make more sense to continue acceleration to an optimum speed and then immediately switch to deceleration, or accelerate at a faster initial rate to an effective "cruising speed" and then hit the brakes closer to the target?

Ideally, I'd like them to make the 40 million mile journey in about two weeks.

• No, the principle reason for them basically waiting till its close is the same as it ever was... cost. Supporting crews and engines for an 11 week 220 million mile journey would be many times more expensive than waiting till the planet is at its closest point, and going through to coincide. Jun 17 '19 at 19:22
• "in a fixed position within a solar system" this complicates things. Letting it orbit the star dimpligies stuff. Jun 17 '19 at 21:06
• @Renan but would it simplify the spelling of "simplifies"? Jun 17 '19 at 21:16
• Having the wormhole in a fixed position is also weird. How does it stay there? It seems slightly out of keeping with your desire to have plausible rocketry. The wormhole metrics I've read about all have an actual mass, and would orbit the star like any other mass would. Jun 17 '19 at 21:21
• You should just look up the same information for getting to Mars. Your wormhole is at 130 million miles from the sun, Mars is 141 million miles from the sun and about 48 million from Earth (at least at it's closest). We have been doing that equation for 40 years, probably longer. Jun 17 '19 at 22:00

TL;DR: For your journey, a rocket that does a short boost, coasts for two weeks, and does another short boost is much more efficient than one that runs its engine for the whole two weeks. It can make do with a much less exotic and power-hungry engine, or vastly less fuel. It also makes provision of artificial gravity simpler, and relaxes debris shielding requirements. Its higher thrust also makes orbital manoevers more straightforward.

In neither case do you need to worry about mashing your passengers.

Ideally, I'd like them to make the 40 million mile journey in about two weeks.

There are (very, very loosely speaking) three things you can do here (there are other options, but they'll take far longer than you want, so I'm ignoring them).

1. Hohmann transfer (as puppetsock mentioned). This involves injecting yourself into an elliptical solar orbit with an aphelion that juuuuust reaches your target's orbit such that when you reach the aphelion your target reaches you. You boost twice, once to inject yourself into your transfer, and once to inject yourself into an orbit around your target (or, if your target isn't really massive enough to be orbitable, you inject yourself into a solar orbit that matches it).
2. Use a Moderately Powerful Rocket Engine to push yourself into a faster intercept trajectory, at the cost of using somewhat more fuel and/or requiring a more powerful and efficient rocket. The aphelion of your transfer orbits reaches much further out, and you're travelling much faster when you approach your target, so you need a beefier engine to slow yourself down.
3. Using an Outrageously Powerful Rocket Engine, blast yourself into a hyperbolic solar escape trajectory which happens to intersect your targets orbit as above. Then you blast yourself into an appropriate rendezvous orbit. Optionally run your engine up as far as the half way point, then flip over and run it again to slow you down to the intercept point, depending on how much excess fuel and power you have.

You've already kinda ruled out 1, because given your earthlike system and your 2 week requirement you'll never have the patience. A Hohmann transfer to Mars (a bit closer than your target, as its around 35 million miles away at a closest approach) takes over 8 months. You've already basically ruled out 2 as well, as you can squeeze transfer times down to about 3 weeks that way.

That basically leaves 3, which obviously requires an OPRE.

What's left is to establish exactly how outrageous your rocket is.

Lets break out good old Project Rho, and fire up the Brachistochrone Equations. These describe a continuous acceleration trajectory, which is a nice easy thing to work with. I'll use sensible units, so your flight distance will be a user-friendly 64373760km.

$$T = 2 \sqrt{ D/A }$$

Where $$T$$ is time, $$D$$ is distance and $$A$$ is acceleration. If your rocket can manage a whole 1G of acceleration (and this is no mean feat) you could do the trip in 2 days. So that's a bit excessive.

Rearranging to get $$A = D/(\frac{T}{2})^2$$ where $$T$$ is two weeks gets you a nice, relaxing $$0.18m/s^2$$, or a mere 1.8% of earth's gravity, as the required constant acceleration for your trip.

As an alternative, if you use your 1G rocket just to boost you up to a cruising speed (and to slow you down at the end) and then coast for the rest of the time in microgravity, you can use a slightly different version of the equation: $$T = \frac{D - A t^2}{A t} + 2t$$ where $$t$$ is the time spent boosting (or braking). You can rearrange this into a nice quadratic, $$Tt-t^2 - \frac{D}{A}= 0$$ which you can then solve to get $$t \approx 5500s$$ (or a little over an hour and a half of comfy acceleration to start and the same to finish, bracketing nearly 2 weeks of microgravity).

Why would you prefer one over the other?

Lets think about $$\Delta V$$ (that is, the totaly change in velocity required during your trip). For both trajectories, there's a constant factor... you've got to inject yourself into a solar escape, and then inject yourself back into a regular solar orbit. To escape Earth's gravity from low earth orbit, you'll need at most $$~3km/s$$. To escape the Sun's gravity from Earth's orbital radius is a further $$16.6km/s$$ and to inject yourself into a Mars-like orbit you'll need another $$11.2km/s$$ for a total of $$~30.8km/s$$. Note: for your "fixed" wormhole, that final manoever will be rather different, but as I don't know how something could possibly be "fixed" I can't tell you what the total value would be. Everything is simpler if it just orbits like a sensible mass.

The continuous 0.018G trip needs $$\Delta V = 2\sqrt{DA}$$ which is about $$213km/s$$ on top of that constant amount. The boost'n'coast trajectory on the other hand needs $$\Delta V = 2tA$$ or about $$107km/s$$. That's a pretty serious difference! This means that the continuous thrust option either needs quite a bit more fuel, or a much more powerful rocket. Given that $$\Delta V = V_e \log_e(R)$$ (where $$V_e$$ is the exhaust velocity of your rocket and $$R$$ is the ratio of the fully fuelled mass of your ship to its dry mass), using the same exhaust velocity the continuous boost trajectory needs 10 times the mass ratio. That means either tens times the fuel, or a much smaller payload (just one tenth, and probably smaller because you still need the same engine...). Alternatively you could use a rocket with twice the exhaust velocity, but that's easier said than done... doubling your thruster power whilst maintaining the same fuel expenditure means that your cooling requirements have now gone up (because you can only dump so much excess heat into your reaction mass) and so you'll need bigger, heavier heat radiators. You need bigger reactors (which are heavier, obviously) or more energetic and efficient nuclear engines (probably also heavier). All that extra weight eats into your useful payload, and that's clearly undesirable.

Next, shielding (note: I'm again ignoring solar escape velocity here as a constant, but you might want to take it into account). Your constant boost ship reaches a top speed of a little over $$106km/s$$ , but the boost'n'coast ship only reaches $$~54km/s$$ (but of course reaches that speed much quicker and sustains it for a long time). By reaching only half the speed, the kinetic energy of impacts with the same mass of debris are reduced to a quarter. That means you don't need as much shielding, saving weight to use on useful payloads.

Additionally, your continuous boost trajectory means your ship must stay pointing in the appropriate direction for the entire flight. If you want artificial gravity, you need a centrifuge. This are heavy and inconvenient and you need to make sure they're shielded from any radiation your drive is producing. Your boost'n'coast ship, on the other hand, can point whichever way it feels like, and this lets you use a "tumbling pigeon" artificial gravity technique. Not only is this mechanically simpler, but you can also use a much longer diameter on your "centrifuge" letting you reduce the unpleasant coriolis effects and keeping your crew happier as well as healthier.

Finally, there's the small matter of manoeuvres. Really, you want your orbit-changing manoeuvres to be done instantaneously, but generating infinite accelerations is awkward. If your engine can't generate sufficient thrust, you end up doing a slow spiral-out (or in) rather than just blasting yourself into a new orbit. The 1G engine will take you from LEO orbit to Earth escape in a little over 5 minutes, which is much less than a single orbit. This is nice and prompt and efficient (and lets you make use of tricks like the Oberth effect). The 0.018G engine takes nearly 5 hours to reach escape velocity, which may require multiple circuits around the planet (and possibly through the radiation belts, if you're not careful). These are all important issues for real world spaceflight, and maybe you might care about them too.

Edit: just remembered that I said we were going to find out just how outrageous your OPRE would need to be. Well! Lets just concentrate on the sensible boost'n'coast rocket. Again, I'll lazily elide the effort required to break solar orbit and park at your destination.

If you wanted a mass ratio of about $$e$$ (eg. your fully fuelled mass is about 2.72 times heavier than the dry mass), $$V_e = \Delta V \approx 107km/s$$. That's high, but not crazy high. This is in the realms of nuclear rocketry... a refined Orion Drive could manage it, certainly. It would have enough thrust to do the job, which is the most difficult requirement, so congratulations: your OPRE is almost achievable with modern-day technology! A nuclear electric plasma rocket might be able to do it, but getting a beefy enough lightweight and efficient nuclear reactor (might have to be fusion with direct energy conversion ) and a powerful enough plasma engine might be tricky... but y'know, you've got wormholes so maybe you'll be fine, tech-wise. Remember that you'll have to refuel when you reach the wormhole, but flying out extra fuel and reaction mass via slow, efficient bu human-unfriendly orbits will be straightforward.

If the dry mass of your ship (engine, hull, life support, crew, cargo, all the rest...) was about 1000 tonnes, it'd be about 2720 tonnes fully fuelled. For the orion drive, that's quite a substantial number of nuclear weapons, which will give you a good way of visualising Niven's Kzinti Lesson: "a reaction drive's efficiency as a weapon is in direct proportion to its efficiency as a drive."

The engine would need to develop a thrust of $$F = MA = 26MN$$. That means the rocket power is $$F_p = (FV_e)/2 \approx 1.4*10^{12}W$$ or a 1.4 terawatts... if it is 100% efficient which clearly the gods of thermodynamics are never going to allow. So 1.4TW is a minimum power.

And that is why I called it an Outrageously Powerful Rocket Engine.

• And if the OPRE doesn't seem that big 'a deal to you, queue up Doc Brown from Back to the Future, tape your eyes open with duct tape, have your significant other use the rest of the duct tape to strap you to the chair so you can't use the bathroom, then watch the clip 1,157 times. If by the end of all that you don't get the magnitude of 1.4 terawatts, we at Worldbuilding can't help you. 😜 Jun 17 '19 at 22:25
• Starfish, that is fantastic, thank you. Jun 18 '19 at 6:35
• Part of the back story for the wormhole plot is based on several centuries prior, research students stumbling on the means to identify various forms of "Exotic Matter" (with some VERY loose RW science attached...) isolating, containing and working with said EM is what allows the wormhole to remain both locked in place, and stable. It also allows space vehicles to carry much more fuel in a hugely reduced area. Said fuel is also very efficient. Just wait till I start asking questions about Antimatter containment and transfer theories... :) Jun 18 '19 at 6:54
• @Tommy I'd suggest you continue this discussion in the main question above (because other people have asked about it), but locked in place relative to what? Using electromagnetic forces to lock it to the sun (which is quite a long way away, and is rotating pretty quickly, and has quite a changeable magnetic field, and produces flares and CMEs, and and and...) seems a) really hard, b) unneccessary c) a way to complicate your trajectory planning and d) to prevent the use of most low-energy efficient trajectories to take cargo there. Jun 18 '19 at 7:01
• @Tommy Fuel density is usually not the issue, antimatter and fusion, it is the weight of the fuel that usually becomes the issue, and at terawatt power levels, which you won't get around any which way you go, the hard part is to keep your engine from disintegrating. I would really suggest you a read of Atomic Rockets: projectrho.com/public_html/rocket/enginelist.php Jun 18 '19 at 7:49