1) The sun's luminosity is decreasing. Specifically, we have the numbers 0.01% for the start of the novel, then 1% over 9y and 5% in 20y.

I shrugged. “So? Where are we in the solar cycle?”

She [Marissa] shook her head. “It’s not the eleven-year cycle. It’s something else. JAXA accounted for the cycle. There’s still a downward trend. They say the sun is 0.01 percent less bright than it should be.”

(...)

“It’s right,” she said. “The sun’s output will drop a full percent over the next nine years. In twenty years that figure will be five percent. This is bad. It’s really bad.”

- Chapter 2, Project Hail Mary

p.s. love marissa. wish we could've seen her in the movie 💔

2) A year after this, the ArcLight probes are launched. These are noted to be outside the typical preferred launch window for a Venus-Earth alignment, hence the extra time.

Browne cleared his throat. “We received confirmation about ninety minutes ago that ArcLight successfully inserted into orbit around Venus. Now we’re just waiting for that first batch of data.”

It had been a heck of a year since the JAXA announcement about the Petrova problem.

- Chapter 2, Project Hail Mary

3) However! All stars affected by Astrophage will only lose up to 10% luminosity per the text.

"They only get to about ten percent dimmer before they stop dimming. We don't know why. It's not obvious to the naked eye, but"

"But if our sun dims by ten percent, we're all dead,"I said.

"Pretty much."

- Chapter 5, Project Hail Mary

4) The Sahara Paving - 2 trillion m² of paved blackpanels

“But it would be ridiculously slow,” I said. “If you had a one-square-meter box and ideal weather conditions…say, one thousand watts per square meter of solar energy…”

“It’s about half a microgram per day,” he said. “Give or take.”

“That’s a far cry from ‘a thousand kilograms’ per day.”

He smiled. “It’s just a matter of how many square meters you make of it.”

“You’d need two trillion square meters to get a thousand kilograms per day.”

“The Sahara Desert is nine trillion square meters.”

My jaw dropped open.

“That went by fast,” said Stratt. “Explain.”

“Well,” I said. “He wants to pave a chunk of the Sahara Desert with blackpanels. Like…a quarter of the entire Sahara Desert!”

“It’d be the biggest thing ever made by humanity,” he said. “It’d be starkly visible from space.”

I glared at him. “And it would destroy the ecology of Africa and probably Europe.”

“Not as much as the coming ice age will.”

- Chapter 13, Project Hail Mary

5) Bombing Antarctica extends Dr. LeClerc's halved population prediction from 19y to 27y. LeClerc is using a 1500 calorie model to determine his population numbers here. Some effects are discussed.

Stratt pinched her chin. “Nineteen years isn’t enough time. It’ll take thirteen years for the Hail Mary to get to Tau Ceti, and another thirteen for any results or data to come back. We need at least twenty-six years. Twenty-seven would be better.”

(...)

He looked out to sea. “And I’m ordering a nuclear strike on Antarctica. Two hundred and forty-one nuclear weapons, courtesy of the United States, buried fifty meters deep along a fissure at three-kilometer intervals. All going off at the same time.”

(...)

He checked his tablet. “The shelf will cleave at the line of explosions and slowly work its way into the sea and melt. Sea levels will rise about a centimeter over the next month, the ocean temperature will drop a degree—which is a disaster of its own but never mind that for now. Enormous quantities of methane will be released into the atmosphere. And now, methane is our friend. Methane is our best friend. And not just because it’ll keep us warm for a while.”

“Oh?”

“Methane breaks down in the atmosphere after ten years. We can knock chunks of Antarctica into the sea every few years to moderate the methane levels. And if Hail Mary finds a solution, we just have to wait ten years for the methane to go away. You can’t do that with carbon dioxide.”

- Chapter 14, Project Hail Mary

+1) Temperatures could drop 10 to 15 degrees <30y.

I stood and paced slowly in front of the class. "We don't know. But if it breeds like algae does, at about that same speed, climatologists are saying Earth's temperature could drop ten to fifteen degrees."

(...)

"Climatologists think it'|l happen within the next thirty years," I said.

Chapter 4, Project Hail Mary

The Challenge

So! This last piece of lore established by the text is the one we will be challenging with these models. Will the average temperature actually drop ten to fifteen degrees? Or will they go even lower? If this theory is proven incorrect, we'll also model out how we could've potentially ended up with 10-15 + how Earth will fare.

Science Time, baby!

Let's start with the central focus of the story: the sun's luminosity! Referencing (1), if we take (9,0.01) & (20,0.05) as points, x-axis as our time in years and y-axis as luminosity percent loss, we can figure out a rough exponential to determine when a luminosity percent loss will achieve that 10% limit.

y = 0.00267988e^(0.146313*x)

y = lumosity percent loss

x = time in years

So plugging in our 10% for the y, we can figure out it will take 24.75 years for the sun to reach maximum luminosity loss. We also have a general model for what the year over year loss will be, which will be helpful later.

(Edit: Hi! Just realized I forgot to account for (0,0.001) in my equation. This doesn't really fit the current exponential I'm using, so we're going to use a separate formula for 0-9 then 9-20. Formula for 0-9 is y = 0.001*e^(0.25585*x). Graph updated!)

Okay, next point (5): methane!! Everyone loves methane in this 'verse, and for good reason: it has a relatively powerful global warming potential of 81.2 (GWP) in comparison to its residence time. GWP as a measurement, btw, is comparing the warming ability of greenhouse gasses to the standard of carbon dioxide so in 20 years a tonne of methane warms the Earth as much as 81.2 tonnes of carbon dioxide. Residence time can vary a lot depending on what source you're looking at, because over the course of this time, the carbon in methane is oxidized (producing carbon dioxide and water vapor) and leaves the atmosphere through various other sinks. This can be best modelled through exponential decay, which can get very complicated very fast.

We're going to say that half of methane will be gone in say 6 years and all of it by 12 years, simplifying this vastly.

Now moving on: how much methane are we adding, exactly? Great question. We don't know :) Even with the specificity in the text, unfortunately we just don't really know how much methane is contained in glaciers and can only utilize what little work has been done on it so far. But in this particular instance, we can figure it out by utilizing some back of the napkin math and the text. So Stratt is asking for 27 years at a minimum (and confirming that we're in the launch year all at once. Thanks Stratt!), when LeClerc previously placed the halved population at 19 years. This means that in a standard model without methane, we will see the average temperature that will halve the population at 19 years post-launch. From there, we'll rewind time and add methane until that temperature doesn't occur until year 27. Now, I'll have to figure out if this will require more than one methane addition (probably), but my general theory here is that the goal will be the minimum emissions possible to achieve this goal.

Amazing! We will then have methane and luminosity... Wait. I'm forgetting something. Oh right. The Sahara!

This is a massive project. Redell rightfully notes that it'll be visible from space, and that affects albedo. What's albedo? Great question! Albedo is a measure of the percentage of sunlight that a surface reflects. You've experienced this if you've ever stuck your hand on a white car versus a black one in hot weather. The white car has a higher albedo percentage, so it absorbs less sunlight thus ends up less hot. In the Astrophage Polar Age, however, we really like having less albedo, not more. So how do we work this out to figure how much paving two trillion square meters would affect the Earth? Back to the napkin we go!

So first we get two trillion into a more usable form. Let's convert to kilometers squared instead, which works out to 2 million km². Excellent. Earth has a surface of 510 million km² of general surface area, divided between 361.13 million km² of ocean and 148.94 million km² of land. So this project would alter 1.34% of all terrestrial surface area and 0.39% of Earth's surface in total. Great job, Bob. Now, just for note, the equator is obviously where the majority of sunlight directly impacts the Earth, so it would likely have an outsized effect in comparison to more extreme latitudes. We can't really account for that without literally losing the plot, but be aware that I'm aware. So we can presume they utilize unoccupied regions of the Sahara for this. So we'll say that the original material present was sand, which has an albedo of around 0.5. Now we don't have an exact equivalent for blackpanels, but we'll use asphalt in a pinch for this exercise, here is a simple module exploring this topic. So that would be an albedo of 0.1 - incredibly low.

Okay so, let's get this show on the road. How is planetary albedo calculated? It is essentially an inventory of the reflectivity of the globe, which we figure out through use of special satellites and a lot of math. A common figure used for planetary albedo is 0.31. Now this one was fairly tricky to work out, but here's how I ended up calculating the potential change in albedo. I utilized this data set from NASA Earth Observations in 720x360, then I altered the "black/9999" cells until we ended up with an average of about 0.31 for the whole set. Yay! Now I need to change 0.39% of that, so 1012 of 259560 cells. My goal is to find cells similar to the sand and replace them with asphalt, so in the 0.5 region. This turned up 957 cells for me which I quickly replaced, and did the additional 55 manually (hahaha, suffering).

So what's the end result? Well. It makes a difference of about 0.002063869 (🥲). I will be utilizing that in the final model, because I'm incredibly petty, but it's negligible in this scenario. In case you were wondering.

Let's Get Gassy: The Methane Model

I nicknamed this model "Ethan"

Now we have our number to work towards for 2049-50. This requires the introduction of methane - a lot of it. As explored in Part 1, methane has some unique characteristics that make it excellent for the Project's purposes. They also add more manual input factors, as they exit the atmosphere through various sinks.

There were nine Ethans overall. I misjudged two problems when setting up my premise: the afterlife of methane and the habitability zone.

Let's start with the solvable one: the afterlife of methane is what happens when methane breaks down. The majority of this happens in the atmosphere, and results in carbon dioxide - another greenhouse gas, but one with considerably less immediate impact but a very long residence time. In Ethan 1-3, I ran the model with various "Nuking" events spread out to consider the relevant outcomes. Even when fully matching the concentration of nitrogen in the atmosphere (almost certainly a death sentence for life on Earth), the numbers were awful. Methane simply ages out fast relative to this timeline, and I wasn't accounting for the carbon dioxide that would be left behind.

For Ethan 3-6, I was able to narrow down a specific range for our methane, and utilized a 1:1 ratio with carbon dioxide as it aged out on the 6/12 cycle outlined in Part 1. This is a vast oversimplification of hydroxyl oxidation, but it works for our purposes in this model, and is close enough to reality to give us relatively good numbers.

Our Final Girl: The Survivor Model

So what happened in Ethan 7-9? Habitability. Let's refer back to our luminosity figures, derived in Part 1.

There is a turning point in luminosity, where there simply isn't enough solar radiation reaching Earth to support life long term. A ten percent loss in luminosity puts us directly on the knife's edge of habitability. This is exacerbated by the albedo situation, which is rather exponential - once ice develops, it creates more ice, and that reflects the light right back into space. The amount of gasses you would need to get to that 5.45°C figure is incompatible with life, in every way I tried to work this out.

So instead, I determined a best outcome model that worked to utilize the least amount of emissions for the outcome. The goal for this outcome would be to have the last of the methane emissions leaving the atmosphere around five years post-Beetle.

After working out my spacing, i determined that in this timeline, we would see Arctic Ice Nuking events in 2029, 2035, and 2041. This is used to inform the arc of my methane emission cycle.

This is our final Ethan.

Let's Draw Some Conclusions:

Referring back to our original question, we can find that we did end up in Weir's projected loss of 10-15°C range, at 13.38°C change over the 30 year period. Pretty neat, right? There can be a world of difference depending on how you model out your methane, though, so be aware that this is just one conclusion to be drawn of many.

If I were to revisit this part in the process, I would run the simulations at a slower speed to better model temperature changes over time. In addition, I would try to track other elements, such as greenhouse effect and albedo across time elapsed. I think these factors could be very informative.

Now, I will be working on the final part of this series, where I will be focusing on life on such an altered planet. In this, I plan to address the primary points of the likely climate shift - from clothing to food to governmental response.

Additional Notes:

I utilized the 2011 game "Universe Sandbox" for this exercise. In order to do justice to the original text, I needed to have a program that could model the Sun's luminosity with relevant climate effects on Earth, with an active atmosphere that could be altered, and finally, albedo. I also needed it to not explode my laptop upon loading. "Universe Sandbox" ticked the boxes. There are some inherent limitations to this: the game runs on Newtonian physics, has to optimize some aspects of the simulation to run successfully, etc.

These numbers are derived running the simulator for a full year and utilizing the chart feature to record the temperature data. Each year required manual input on luminosity change / gas changes. From there, I organized the data into an understandable format, and summarized annual change.

Parameters (Start of Simulation):

So in order to derive a specific date to start from, I'll be breaking my own rules to steal a date from movie canon. Namely, the watch in the post-explosion scene puts the date that Grace is originally detained at February 14th, 2023. Valentine's Day: cute! Not so great for the happy couple, rip Drs. Dubois and Shapiro. Real ones. Grace is then held in a cell for three days and the launch of the Hail Mary occurs on February 19th, 2023 (though the crew is ferried up two days earlier, February 17th). This is derived from the book - Dimitri lays out the timeline during the tertiary scientist conversation.

"The Hail Mary will leave on schedule" said Stratt. "Five days from now. The crew will have two days of preflight checks, so that means the Soyuz has to launch in three days."

- Chapter 23, Project Hail Mary

Okay, so as previously established in Part 1, we know that the Antarctica Nuking occurred within a year of this launch. Now, for the sake of easier travel, let's presume they went when the ice was at minimum, thus establishing the Nuking of Antarctica in Late February to Early March 2022.

Overall, the timeline of the Petrova Taskforce is intentionally vague. We are working with bread crumbs, but thankfully, those breadcrumbs have numbers. So let's turn to everyone's favorite negligent gambler, Dr. Robert Redell. One of the early primary constraints on the Mary's launch is fuel, and Redell provides a solution in Chapter 14 with his blackpanels - he notes that the fuel will be enough for the Hail Mary in a few years, and that the ship won't be done by then anyway. If you do the math based on that 2000 kg/day figure, btw, it works out to 1000 days for the 2mil kg of fuel for the Hail Mary. Roughly. There are a couple of other factors we would throw in - construction time, scaling of astrophage, restructuring of entire industries to support this kind of mass construction etc. For the sake of mitigation of incoming headaches, let's say that 1200 days is our number from Redell's meeting (October 8th, 2019) to Launch, and that any difference may be assigned to the fact that Stratt definitely never had all her eggs in one basket on the energy front, and likely was still utilizing nuclear energy to produce astrophage in excess of the Sahara farm.

“We know that,” I said. “I’ve been using a nuclear reactor’s heat to breed up Astrophage for the last several months.”

“What’d you get? Maybe a few grams? My idea can get you a thousand kilograms per day. In a few years you’ll have enough for the whole Hail Mary mission. It’ll take you longer than that to build the ship anyway.”

- Chapter 13, Project Hail Mary

So now we turn to Venus to figure out our ArcLight intercept in 2019. Now, time to admit that I'm not a trained astrophysicist, and so when I chose a launch, I'm just choosing based on two factors - the least amount of distance between the two planets, and the lack of obstructing planetary bodies (the Sun & Mercury). There are two periods in 2019 where the odds aren't entirely against us, and it's at the very beginning and end of the year. Now, these still aren't great in terms of proximity, it would've been better in November/December 2018, but considering the text specifically references that this wasn't a great time to go to Venus, I think we're on the right track. Let's place that intercept at roughly January/February 2019, which puts a launch at late 2018, and has the probe coming down roughly mid 2019.

“Tell us about going to Venus,” the reporter said. “What makes it so hard?”

“The main problem is fuel,” said Browne. “There are specific transfer windows when interplanetary travel takes the minimum amount of fuel, but we were nowhere near an Earth-Venus window. So we had to put a lot more fuel in orbit just to get ArcLight there in the first place.”

“So it’s a case of bad timing, then?” the reporter asked.

“I don’t think there’s ever a good time for the sun to get dimmer.”

- Chapter 2, Project Hail Mary

Now, take a year off of that per Grace's inner monologue about the ArcLight, and here we are in 2018. This is where everyone first hears about the Petrova Problem, and Grace has his conversation with Marissa. Now we have a general scale for those luminosity figures she gave us: loss of 0.01% in 2018, 1% in 2027, and 5% by 2038. Yay! Let's check back in with those numbers later on, but for now, we're assuming the events of Project Hail Mary (Earth-side!) take place over roughly five years. I know that seems super compressed and accelerated, but in canon Grace spends a lot of time talking about how fast everything is moving - Stratt is utilizing her power to the fullest extent to command resources across the globe, and entirely ignoring bureaucratic red tape. It's breathtaking.

Project Hail Mary Timeline

2065 - Eridani 40 sees that Sol is restored

2049 - Astrophage is cured

2048 - Beetles Arrive at Earth

2036 - Arrival of Hail Mary in Tau Ceti

February 19th, 2023 - Hail Mary Launch

February 17th, 2023 - Soyuz to Hail Mary

March 31st, 2022 - 1st Nuking of Antarctica

October 8th, 2019 - Redell's Meeting / Paving of the Sahara

June/July 23rd, 2019 - ArcLight Return / Lab!

January/February 2019 - ArcLight Intercept

September/October 2018 - ArcLight Launch