The Science of Project Hail Mary

Andy Weir’s latest book, Project Hail Mary, is a very good hard science fiction tale about a journey to another solar system in search of a way to save Earth from disaster. Mr. Weir always does mostly pretty good science in his books, but sadly, no author is perfect, especially as Weir is a computer engineer by trade and not an astrophysicist.

I had a few minor quibbles with the book that were mostly focused around how slow the scientists are to figure things out. Those aren’t really wrong, but I would say they felt unrealistic. However, those things could just be there so that the book doesn’t get too far ahead of the reader. More on that in my review of the book.

However, there were two aspects of the science in the book that did include rather large mistakes. Neither one is completely story-breaking, but they do add complications—one of them glaringly obvious (to me as an exoplanetary scientist), the other one much subtler.

Warning: MAJOR spoilers for Project Hail Mary below. Do not click unless you’ve read all the way to the end.

If you have read the book, you’ll know that there are aliens involved, and I don’t just mean the Astrophage.

Hey, I told you there were spoilers.

Ryland Grace’s new friend, Rocky, is a spider-like alien from the planet Erid, in the 40 Eridani system (technically 40 Eridani A, since it’s a triple star system).

Erid’s atmosphere is much denser than Earth’s and is made of almost pure ammonia (which makes little sense at that temperature, but there are a lot of acknowledged-in-story “impossibilities” in the book; my complaint is not with those). This dense atmosphere absorbs almost all of the sunlight hitting the planet, making the surface pitch black. This is why the Eridians are blind to visible light and instead can see nearly as well as we can with echolocation.

But this is wrong. First off, the surface pressure on Erid would not be 29 atmospheres. It would be 29 times the Hail Mary’s internal pressure of one third of an atmosphere—in other words, 9.67 atmospheres. But even if it were that thick, Erid’s surface would not be pitch black under 29 atmospheres of pressure. It may be very dim, maybe even dark enough for echolocation to be evolutionarily favored—but not pitch black.

People always underestimate how bright the Sun is, and to be fair, the numbers seem near-impossible at first glance. Direct sunlight is 1,000 times brighter than a 100-watt incandescent light bulb at 1.5 meters (5 feet)! We know it’s brighter, but it doesn’t seem that much brighter because our eyes are pretty good at adjusting to light and dark.

Erid gets even more sunlight than Earth does. We know the planet’s year is 42 Earth days. Crunch the numbers for 40 Eridani A, and you get about 7 times as much sunlight as Earth at the top of the atmosphere.

Now, we can do a back-of-the-envelope calculation based on light penetration in seawater (which has a broadly similar absorption spectrum to ammonia). 29 atmospheres equals 290 meters of seawater. Below about 200 meters, there’s not enough sunlight for photosynthesis to work, but there is still some. That does suggest it would be very dark, but not lightless.

Except it’s not actually 290 meters. Erid’s surface gravity is twice that of Earth. And double the surface gravity means there’s only half as much air above you for a given pressure. Erid’s 29 atmospheres of ammonia is actually equivalent to 150 meters of seawater. I don’t have an exact optical absorption spectrum for ammonia on hand, but I do know that it’s fairly close to water vapor, which absorbs modestly less light than liquid water, so this estimate is a pessimistic one. Erid’s surface may be twilight, but it would be perfectly possible to see there.

For that matter, Ryland also would have been able to see clearly when he intercepted the Blip A out past Tau Ceti’s heliopause. He’s out at the edge of the solar system, yes, but even if he’s 200 AU from Tau Ceti, he would be getting 0.001% as much sunlight as Earth—not much—little better than a full Moon—but still enough to see the Blip A instead of groping for it blindly on his tether.

The other problem, the one I didn’t realize until afterwards, is that even with a light-powered ship, even the massive fuel stores of the Hail Mary aren’t enough to get to Tau Ceti that fast. And the problem is even worse for the Blip A.

The Hail Mary was accelerating at 1.5 g for 4 subjective years, and when you do the math of special relativity, it does indeed get you 12 light-years with a turnaround halfway through. But even a light-powered ship needs reaction mass (in this case, Astrophage), especially if it’s getting up near the speed of light. To figure out how much, we need the rocket equation, and, uh-oh, that means there’s an exponential function lurking in there. The rocket equation says:

mi/mf = e^(Δv/vexhaust)

For a light-powered ship, vexhaust is the speed of light. What about Δv? That’s subject to relativity, isn’t it? Yes, but there’s a shortcut. You can replace Δv by the subjective acceleration times time of the ship. That’s how much cumulative force the engines are exerting in the ship’s reference frame, which tells you how much fuel you need.

There’s a very convenient fact in relativity that 1 g, that is, one Earth gravity, is very close to 1 light-year per year squared of acceleration (within 3%). That means acceleration in gravities multiplied by time in years gives you Δv in multiples of the speed of light. For the Hail Mary, that number is 1.5 g x 4 years = 6c. The ship doesn’t actually go 6 times the speed of light, but it does spend that much Δv . The rocket equation then says the ratio of initial mass to final (dry) mass must be mi/mf = e6 ~ 400, that is, 400 times as much fuel as dry mass.

Well, that’s impossible with off the shelf equipment. We couldn’t make the fuel tanks themselves that light, let alone the rest of the ship, even given how dense enriched Astrophage is. The Hail Mary seems to be about the size of the Space Shuttle orbiter—about 100 tons empty—and we know it carries 2000 metric tons of fuel. That’s a much more reasonable mass ratio of 20.

There’s one way to fix this; there’s a trick we can use to get an exhaust velocity of 2c instead of c: shoot the laser at the ship and reflect it off a mirror. Since the light is reversing direction, its own change in velocity is 2c, and it imparts twice the momentum. This is what real solar sails do with sunlight. Twice the exhaust velocity means taking the square root of the mass ratio, which brings us down to 20. Problem solved!

Except that no matter how well the Hail Mary’s spin drive mirrors are made, I highly doubt they could stand up to the literal nuclear bomb levels of laser light the ship puts out.

Things are much worse for the Blip A. The Blip A needed 3 subjective years to go the 11 light-years from 40 Eridani to Tau Ceti. That suggests an acceleration of 2.2 g, which is probably the same as Erid’s surface gravity. Now, that comes to 3.25 e-foldings if you use reflected light, but I’m willing to round it down to 3. That’s not the problem. The real problem is that the Blip A was built for a round-trip! It needs twice the Δv, and that means we need to square the mass ratio, bringing it back up to 400.

Now, with how great Xenonite is, the Eridians might be able to built their ship that light and use mirrors in their engines. But there’s one more problem: the Eridians didn’t know about relativity. They were trying to make the trip with Newtonian mechanics. If you apply Newtonian mechanics, the trip from 40 Eridani to Tau Ceti takes 4.5 years at 2.2 g, and that means a Δv of 10c. For a round-trip, that’s a mass ratio of e10 ~ 22,000. That’s impossible.

The bottom line is that if you make some allowances, the Hail Mary probably can work. The Blip A basically can work, too. However, there’s no way for the Blip A to be carrying as much fuel as the Eridians thought they would need without relativity.

About Alex R. Howe

I'm a full-time astrophysicist and a part-time science fiction writer.
This entry was posted in Physics, Science Fiction, Story Analysis and tagged , , , , . Bookmark the permalink.

5 Responses to The Science of Project Hail Mary

  1. i also like to read new books you can visit my blog for soem epic novels

  2. Alec Cawley says:

    Further fundamental problems:

    Astrophage breaks the Second Law of Thermodynamics. I am happy with handwaving away how it stores energy (neutrinos, waffle, waffle). But Astrophage manages to extract energy from pure heat without a cold end. In a hot environment, it extracts and stores energy without dumping a single erg to a cold place.

    And Taumoeaba violates Conservation of Energy. You have tanks of Astrophage containing gazillions of joules of energy. Then Taumoeba gets in, eats it and… Nothing except sludge.

    • Alex R. Howe says:

      My impression was that the neutrinos were supposed to be a free, nearly unlimited heat sink, i.e. the temperature of the neutrinos in a depleted Astrophage is near zero and equalizes at the “fully enriched” density. But you may be right that I’m missing some entropy considerations.

      The Taumoeba only consume the chemical energy in the astrophage. The neutrinos just escape when they die. But even just chemical metabolism on that scale should have tripped thermal alarms on the ship, so point taken.

      • Alec Cawley says:

        But even so, creating a “near absolute zero” heat sink violates the Second Law. Energy, on its own, does not do work. Energy moving from condensed (hot) to dispersed (cold) state does work. If the Astrophages had a cold end, all is well. But they do not appear to have.

        Maybe you “just” released the neutrinos. I wonder whether the unimaginable neutrino flux might have visible effects.

      • Alex R. Howe says:

        Back of the envelope says that if these were ordinary (high-energy) neutrinos, they would have died of radiation poisoning within hours. But these neutrinos have an energy of kT, where T=369 K, which is 0.03 eV. They don’t have enough energy to break chemical bonds.

Comments are closed.