Don't wake me up

The questions

The journey to Adrian and back makes perfect sense to me:

I got us going a good head of steam and we’re on course for Tau Ceti e. I’ll do the orbital-insertion burn in about eleven days. While we wait, we may as well have gravity. So we’re back to centrifuge mode. Eleven days. Truly astonishing. The total distance we’ll be traveling to get there is over 150 million kilometers. That’s about the same as the distance from Earth to the sun. And we’re doing it in eleven days. How? By having an absurd velocity. I did three hours of thrust to get us going, and I’ll do another three when we get to Tau Ceti e to slow down. Right now, we’re cruising along at 162 kilometers per second. It’s just ridiculous. If you left Earth at that speed, you’d get to the moon in forty minutes. This entire maneuver, including the burn I’ll have to do to slow down at the end, will consume 130 kilograms of fuel. (Chapter 16)

I’m shooting for another eleven-day transfer. It takes 130 kilograms of fuel to make that happen—about a quarter of what the beetles have aboard (if you include George, who is sitting on the lab table full of Astrophage). That should give us enough left over to correct whatever idiotic mistakes I made in my trajectory math. We’ll get up to cruising speed in three hours, then we’ll coast for most of eleven days. I don’t want to deal with spinning up or spinning down the centrifuge. Yes, it can be done—Rocky proved it when he zeroed us out before. But it was a delicate process with lots of guessing and opportunities for spinning out of control. Or worse—getting the cables tangled up. So, for the next three hours I have 1.5 g’s to work with. After that it’ll be zero g for a while. (Chapter 23)

They thrust to accelerate to the desired speed; turn off the engines and coast for most of the journey, can use the centrifuge and the Petrovascope; turn the ship around and thrust to decelerate when they're nearing their destination. Sounds simple and fuel-efficient.

The beetles seem to travel the same way too:

They accelerate at five hundred g’s until they reach a cruising speed of 0.93 c. (Chapter 18)

But here's what confuses me. When Rocky describes his journey, it sounds different:

“Then more strange: Reach halfway point earlier than should. Much earlier. I turn ship around. Thrust to slow down. But Tau get farther away. How? Still moving toward Tau but Tau moving away. Much confusion.” (Chapter 18)

Unless his explanation skipped a few steps, the Blip-A accelerates for 50% of the journey, then decelerates for the other 50%. Why would you do that?! Just burn fuel flying backwards for a year or more?! That sounds insanely wasteful and pointless to me. Even if Eridians had plenty of Astrophage.

I don't think it's specified anywhere whether the Hail Mary's main journey used the three-step flight plan or the 50-50 one. But when Grace looks for Rocky at the end of the book, he does the 50-50:

Whatever. I set course for where I saw the blip and fire up the engines. I don’t even need to account for relativity on this one. Just high-school physics. I’ll accelerate half the way, then decelerate the other half. (Chapter 29)

I can explain it to myself by guessing that he did this just because the course would be easier to plot, and the fuel difference is negligible at that distance. Maybe specifying that he'll "accelerate half the way, then decelerate the other half" implies that it's not the normal way. On the other hand, he specifies a lot of things for the reader.

So, here are my questions. Is there any indication which of these strategies was used for the Hail Mary's main journey from Earth to Tau Ceti, or for Grace's journey back? If it didn't use a coast phase, why? And why didn't the Blip-A? Is there some math beyond my understanding that means accelerating and decelerating a lot leads to an overall speed so much higher than coasting that it's worth the fuel cost?

The answers

As usual, the first place I searched was the subreddit. This thread:

mentioned a "relativistic travel calculator", and the first result for this term led me to this website that had all the data for PHM as the default.

linked to this calculator for a three phase trip

The the first website has not only the numbers for PHM's journey already plugged in, but a couple of notes, the first of which is:

Andy Weir has acknowledged the mass ratio error, noting that the Hail Mary includes a cruise phase — coasting at peak velocity for part of the trip — which dramatically reduces the required fuel. Try the coast phase slider above: at ~85% coast, the brachistochrone mass ratio drops to roughly 21:1, matching the book's fuel load, while adding less than a year to the Earth-frame journey.

And here are some of the numbers that the calculator gives (for 1.5 g, as described in the book):

No coast phase:

ship time: 3 years, 11 months (roughly matches the book/movie)

Earth time: 13 years, 2 months

mass ratio: 416 : 1

fuel required: 41.49 thousand tonnes

Coast phase: 50% of distance:

ship time: 4 years, 2 months

Earth time: 13 years, 2 months

mass ratio: 124 : 1

fuel required: 12.29 thousand tonnes

Coast phase: 85% of distance:

ship time: 6 years, 8 months

Earth time: 13 years, 11 months

mass ratio: 20.67 : 1

fuel required: 1.97 thousand tonnes (roughly matches the book/movie)

Also, turns out, the Youtube video about time dilation in PHM which I saved into my watchlist a while ago was made by the same person as the website, and he showcases these exact numbers in the video. Which of course I watched after playing with the sliders and writing down the numbers. So, uh, good job me awkwardly stumbling onto the facts I could have learned from a video with 100k likes and 3.3m views at the time of writing...

The video also mentioned that the Blip-A's journey was different, which made me reread the scene again more carefully, and this time I got it:

“Science Eridians do much math. Calculate trip. More fuel mean faster trip. So we make much much much Astrophage.” [...] “Science Eridians design ship and fuel requirements. Journey to take 6.64 years.” That trips me up for a moment. 40 Eridani is ten light-years away from Tau Ceti, so you can’t get from one to the other in less than ten years from Erid’s point of view. He must mean 6.64 years of time experienced by his ship thanks to time dilation. [...] “Your engines failed?” “No. Not fail. Thrust normal. But speed…not increase. No can explain.” [...] “Then more strange: Reach halfway point earlier than should. Much earlier. I turn ship around. Thrust to slow down. But Tau get farther away. How? Still moving toward Tau but Tau moving away. Much confusion.” [...] “I speed up. Slow down. Much confuse. But get here. Even with all mistakes and confusion, I get here in three years. Half of time science Eridian say should be. So much confuse.” [...] “Much much much fuel remain. Much more than should have. No complain. But confuse.” Eridians don’t know about relativistic physics. They calculated their entire journey with Newtonian physics. They worked it all out by assuming they could just accelerate faster and faster and the speed of light wasn’t an issue. [...] They thought they’d need a whole lot more fuel. (Chapter 18)

So, the Eridians didn't know that the speed of light is a hard limit, and expected to accelerate past it and without diminishing returns. They expected that if they burned enough fuel on speeding up as much as possible, the journey would take less than 7 years, both on Erid and on the ship.

I checked the numbers with the calculator, and they can actually match up with the book description... as long as you keep the ship weight the same as the Hail Mary and, more importantly, ignore the fuel required for the return journey:

distance: 10.0 ly (IDK what the exact number is, but the book says 10 so close enough)

acceleration: 2 g

coast phase: 0% of distance

ship dry mass: 100 tonnes

ship time: 3 years

fuel required: 50.98 thousand tonnes

Let's compare the calculator's fuel estimate with what we have in canon. Rocky says when Grace rescues him at the end: “I have twenty-two million kilograms of Taumoeba in fuel bays” (Chapter 29). The mass of taumoeba may or not may match the mass of astrophage they ate, which in turn should be slightly less than half of what the Blip-A started with.

The book doesn't give us a specific number for the Blip-A's mass, but does give a number for its length, which is three times the size of the Hail Mary. If I triple the mass too, the amount of fuel required for a one-way trip also triples to 153 thousand tonnes. Maybe xenonite is super duper light, or maybe most of that length is just the fuel tanks, so perhaps we don't need to triple the number. But in any case, whatever amount of fuel is required for a one-way trip from Erid to Tau Ceti must be doubled way more than doubled, because the Blip-A was meant to come back. (Edit: I made a mistake that's probably very similar to the one that kicked off this whole train of thought: if a ship is going on a two-way trip, then on the way to the destination it will have to also carry all the fuel that is required for the trip back, which means that the total amount of fuel is not just double but much more. The calculator is not designed for a two-way trip and I obviously can't figure that out myself, but simply changing the above calculation from 10ly to 20ly makes the fuel requirement go up to 187.11 thousand tonnes.)

The Eridian worldbuilding document specifies that the Blip-A “set out with 31 million kg of Astrophage fuel”. Which is much lower than even the lowest estimate I could get from the calculator. I guess it makes sense if there was an error in the formula that was used for both ships.

Now, just for fun, I'll try to use the calculator and see if there's a realistic flight plan for the Blip-A's return journey based on the numbers we have in canon. I don't think we know the exact acceleration the Blip-A is capable of, nor its exact mass. For the return journey, we have somewhere around “22 million kilograms” of fuel (see above), and the more definitive number of “You’ll reach Erid in under three Earth years” (Chapter 25). By keeping these numbers and stretching all the others as much as possible, I was able to get this from the calculator:

distance: 10.0 ly

acceleration: 3 g

coast phase: 73% of distance

ship dry mass: 200 tonnes

ship time: 2 years, 11 months

peak velocity: 98.12% c

fuel required: 20.87 thousand tonnes

Close enough, I'll say!

There's another section in the document that complicates the situation from a different angle:

The only issues they face are when excreting and eating. Both require gravity to work correctly. Most critically, the incision and healing of the fissure in the abdomen that accompanies digestion. That process does not work correctly in zero-g. So the ship has a centrifuge aboard specifically for sitting in while eating and excreting. If that centrifuge were to break, they would simply use the ship’s engines to provide thrust during the critical three hours or so of eating. If that also weren’t an option, they would end up with poorly-healed abdominal wounds and would be susceptible to infection—a very serious and potentially fatal problem.

On the one hand, this need for gravity would have been an additional argument for having the engines on all the time. In a crew of 23, at least one of them would be eating often enough that even if the gravity weren't needed 100% of the time, turning the engines on and off back and fourth just for that would have been too chaotic. On the other hand, if they bothered to equip the ship with a specially made centrifuge just for eating, that implies they did expect to spend a lot of time in zero-g. But maybe it was meant to be used only after they arrived at Tau Ceti and turned off the engines.

Back to the Hail Mary — I found some additional canonical info about the main trip after leafing through the book again.

In chapter 8, Grace in the flashback confirms that constant acceleration was the plan:

I mean, we already have to design the ship to withstand four years of constant acceleration at one and a half g’s or so.

Chapter 25 confirms constant acceleration again and gives a precise number for travel duration:

The trip from Earth to Tau Ceti took three years and nine months. And it was done by accelerating constantly at 1.5 g’s the entire time—which is what Dr. Lamai decided was the maximum sustained g-force a human should be exposed to for almost four years.

The following two quotes are useful for understanding whether it would be an option to turn off the engine and use the centrifuge mode during transit.

Chapter 22:

The Hail Mary has always looked like something out of a Heinlein novel. Shiny silver, smooth hull, sharp nose cone. Why do all that for a ship that’ll never have to deal with an atmosphere? Because of the interstellar medium. There’s a teeny, tiny amount of hydrogen and helium wandering around out there in space. It’s on the order of one atom per cubic centimeter, but when you’re traveling near the speed of light, that adds up. Not only because you’re hitting a whole bunch of atoms but also because those atoms, from your inertial reference frame, weigh more than normal. Relativistic physics is weird. Long story short: I need the nose intact.

If the ship needs a reinforced nose while traveling at near light speed, then it probably shouldn't rotate. I guess this doesn't apply to the movie version of the ship, which doesn't seem to have any reinforcement, and overall looks more like the ISS, complex and delicate; good for envoking familiarity in the viewer, bad for looking like the sturdiest vessel ever built.

Chapter 8:

Stratt peered at the diagram. “You put all the fuel on the same side. That’s two million kilograms.” “No.” I shook my head. “The fuel would be gone.” They both looked at me. “It’s a suicide mission,” I said. “The fuel will be gone when they get to Tau Ceti. Lokken picked a split point where the back of the ship will weigh three times as much as the front. It’s a good mass ratio for a centrifuge. It could work.”

The centrifuge mode is designed for working with the empty or almost empty fuel tanks. I don't know enough physics to guess what would happen if you tried to activate the centrifuge on the fully fueled ship, but it's probably dangerous and probably wouldn't work properly.

There's another very important argument against the coast phase, and that's the health of the astronauts. Several years at 1.5 g are bad enough, but add another couple of years at zero g in the middle? While they're in an induced coma that hasn't been tested under these conditions? That sounds like a huge risk. My first guess was that the ship's autopilot should be able to activate centrifuge mode while the engines are off — but even if that weren't risky enough by itself, the two quotes above provide evidence that it isn't feasible. My other guess is that the medical robot might provide some kind of not-yet-invented-irl medical care that would compensate for the harmful effects of weightlessness, but that's just a bandaid on a big problem that the author clearly didn't have in mind.

You know what would have killed both birds with one stone? If the ship just travelled at a constant acceleration of 1 g. Now, obviously, this is not just "not canon" but actively against canon; but, just for fun, I plugged the numbers into the calculator and got this:

acceleration: 1 g

coast phase: 0% of distance

ship time: 5 years, 2 months

Earth time: 13 years, 9 months

mass ratio: 202 : 1

fuel required: 20.13 thousand tonnes

Frankly, that doesn't sound too bad! The fuel requirement is still crazy, of course, but it's always crazy unless you coast most of the time or accelerate so slowly that the ship time grows to 8+ years.

And last but not least! I almost forgot to address the matter that is probably of the most interest to us in the fandom: what would Grace and Rocky's journey to Erid on the Hail Mary look like? Here are the numbers from the calculator based on the canonical capacity of the fuel tanks:

distance: 10.0 ly

acceleration: 1.5 g

coast phase: 81% of distance

ship dry mass: 100 tonnes

ship time: 5 years, 7 months

peak velocity: 91.45% c

mass ratio: 22.38 : 1

fuel required: 2.14 thousand tonnes

So, if we go by these numbers, Grace and Rocky spent five and a half years on the ship, and 4/5 of that was in zero g. Have fun incorporating this into fanworks, because it sounds like absolute hell! Or have mercy on the two of them and give the ship a more generous fuel capacity, because the narrative comes first. But they'll still have to deal with zero g, unless you ignore the fuel problem completely. The easiest way out is just to ignore the canon fuel tank capacity and insist that using the centrifuge mid-transit is actually totally fine. This is a plausible enough handwave and takes away most of the discomfort. For example:

coast phase: 60% of distance

ship time: 4 years, 2 months

peak velocity: 96.98% c

fuel required: 6.41 thousand tonnes

The takeaway, aka tl;dr

There is a travel calculator specifically designed to work with PHM's situation, and I'm relying on it for all the numbers.

According to the calculator, the book massively underestimated the amount of fuel required for the interstellar journeys. To spend only that much fuel, the Hail Mary would need to travel the vast majority of the way with the engine off, and in that case the amount of time experienced by the crew almost doubles.

In other words, the book describes both the Hail Mary and the Blip-A as traveling under constant acceleration, but the canon numbers for fuel requirements only add up if there is a long coast phase in the middle.

Since the author didn't take the coast phase into account, the canonical description of the Hail Mary doesn't provide it with any ways to mitigate weightlessness while traveling at near light speed.

Realistically, Grace and Rocky would probably spend some part of their journey to Erid with the engines off, and would have to deal with weightlessness one way or another.