Cows All the Way Down: Could Cow-Based Planetoids Support Methane Atmospheres?

William J. Roper^1,2 , Todd L. Cook¹ , Violetta Korbina¹ , Jussi K. Kuusisto¹ , Roisin O’Connor¹ , Stephen D. Riggs¹ , David J. Turner¹, Reese Wilkinson¹

¹ Astronomy Centre, University of Sussex, Falmer, Brighton, BN1 9QH, UK

² Department of Extra-Planetary Beef Expansion, Cranberry-Lemon University, Pittsburgh, PA, USA

Abstract

More often than not, a lunchtime conversation will veer off into bizarre and uncharted territories. In rare instances, these frontiers of conversation can lead to deep insights into the universe we inhabit. This paper details the fruits of one such conversation. In this paper, we will answer the question: How many cows do you need to form a planetoid entirely comprised of cows, which will support a methane atmoosphere produced by the planetary herd? We will not only present the necessary assumptions and theory underpinning the cow-culations, but also present a thorough (and suspiciously robust) discussion of the viability of, and implications for accomplishing, such a feat on all the planets, some moons, and Pluto, as well as the reasonable economic cost.

Keywords: Terra-Forming, Cow Methane, Non-Spherical Cow Models, Extra-Terrestrial Cattle Rearing, Cow Planetoids, Cow Moons, Climate Change Solutions That Could Actually Work and Let Us Keep Eating Burgers, Atmospheric Physics, Space Herding

1. Introduction

Cows are majestic beasts that have provided humanity with dairy, meat, dangerous rodeo events, and even more dangerous Spanish foot races for countless generations. As humanity has evolved, we have learned to cultivate these beasts and, in some cultures, even revere them beyond a simple source of sustenance as sacred animals. However, their environmental impact cannot be ignored. The methane they produce as waste is an extremely damaging greenhouse gas too far-reaching to be blamed on the dog. In this paper, we aim to use this damaging property and ask the following questions: How many cows do you need to form a planetoid entirely comprised of cows, which will support a methane atmosphere produced by such a planetary herd? Where would it work? How much would it cost?

2. Assumptions

For simplicity in this early work, it is necessary to make several assumptions. Although, we note that some of these can certainly be relaxed at a later date when plans are made for the Interplanetary Cattle Drive (ICD).

1. Cows are incompressible^[1].
2. The planetoid/herd is perfectly fed and hydrated. Of course, maintaining extra-terrestrial cattle is a significant challenge to this endeavor, but we leave this consideration to future work on the matter.
3. The methane atmosphere is assumed to have the same average density as Earth’s own atmosphere.
4. Contrary to what many will have you believe, cows are not spherical.
5. Cows are sufficiently intelligent to organize themselves and obtain a perfect bovine packing fraction of unity, such that the volume of the planetoid is simply V_P = N_COW V_COW
6. The cows have negligible moomentum.
7. Cows can be approximated as a blackbody with effective temperature T_COW.

3. Cow-culations

To address the proposed question, we need an expression for the fluxton of methane leaving the planetoid. There are numerous constituent parts to this calculation. Here, we will present the important steps but leave the nitty gritty as an exercise for the reader and not because we couldn’t figure it out.

The first port of call in this mathematical stroll toward bovine enlightenment is the escape velocity of methane. Let’s start with the expression for escape velocity:

Here, H is the scale height of the atmosphere, while the mass of a purely cow-based planetoid can be described as

assuming an average cow mass of M_COW = 1,390 pounds =630kg ^[2]. Using the expression above for the volume of the planet, and taking the volume of a cow to be V_COW = 1.1m^{3 [3]}, we can derive the radius of the planetoid and arrive at the final expression for escape velocity as a function of only constants and cow properties:

With the Newtonian mechanics out of the way, we must now brave the world of thermal and statistical physics. To find out how much methane is getting the flux out of the planetoid, we start with the Maxwell-Boltzmann Distribution (MBD):

We get a flux () by performing some spherical coordinate gymnastics involving lots of somersaults and backflips, multiplying by the velocity and gas density, and integrating the MBD over a hemisphere to capture the outward moving flux, yielding the following:

From this point, the cow-culation devolves into triviality for physicists. After integrating flux between the escape velocity and infinity, we arrive at the final holy fluxing expression as follows:

Here, m_g is the mass of a methane molecule and T is the atmospheric temperature measured by the Weather Channel app. is given by

is a function of the number of cows forming the planetoid, singular cow mass, and singular cow volume (N_COW, M_COW, and, V_COW):

With the beef of the work done, we are left with only two outstanding ingredients: the scale height of the atmosphere and the atmospheric temperature. For the scale height, we employ the following expression, where g is the acceleration due to gravity on the planetoid:

The atmospheric temperature is a little more involved; not only must we take into account the temperature induced by the fluxing Sun, but we must also include the contribution due to the bovine blackbody temperature. This temperature is taken to be the maximum of T_COW = 303.9K, where we have used the mean surface temperature taken from a study using infrared thermography^[4] and the temperature of the planetoid’s surface and turf induced by the Sun in a non-El Niño year,

where is the albedo of a cow, assuming the worst-case scenario of cows wearing goth clothes or just having a black hide^[5].

Combining getting fluxed over by the Sun, the temperature of the planetoid, and the greenhouse effect, we arrive at the following expression for the atmospheric temperature:

Here, is the Sun’s luminosity, is the albedo of the atmosphere (for which we assume ), D is the distance from the Sun, and f is the fraction of energy absorbed by the atmosphere (for which we assume f = 0.8).

All that is left to do now is balance the methane fluxing off the planetoid with the methane produced by the planetoid’s bovine makeup. Due to the environmental impact of cattle farming, there has been much research into the methane production of cattle and how this can be affected by their diet. By doing all of the things the papers tell us not to do, we can maximize methane production; a beans, grains, lentils, and soda only diet. We utilize this wealth of data to derive a planetoid methane production rate of

which can be converted into giving a flux by dividing by the surface area of the planetoid (A_P), yielding

In the final step, we equate the two methane flux buddies, loss and production, which leads to an analytically insoluble expression:

In the subsequent sections, we present the results from solving this expression numerically using the publicly available Python package SciPy^[6].

4. Cow planets

With all the constituent pieces in place, we can now move on to the analysis. In Figure 2.1, we present methane production as a function of the number of cows on the planetoid and the distance from the Sun.

Beyond an orbital distance of or about 831 billion bananas, the atmospheric loss is driven by the bovine blackbody temperature. At this distance, we reach a minimum of required to balance the methane production and loss. Surprisingly, this means an atmosphere can be sustained at a mass of only of the mass of the Earth with a bovine planetoid radius of of the radius of the Earth.

However, it is worth considering that, unlike traditional atmospheres, this atmosphere is significantly replenished by the planetary body it is “bound” to; because of this fact, the atmosphere is able to reach a steady state at a significantly lower mass than necessary for a celestial body without a similar atmosphere replenishment mechanism. Due to this low mass, we end up with an acceleration due to gravity of and thus a large atmospheric scale height of .

Figure 2.1: Atmospheric loss by number of cows and distance to sun

The preceding plot shows the atmospheric loss and production as a function of distance from the Sun and the number of cows comprising the planetary herd (and therefore planetoid mass). The colormap indicates the state of the atmosphere; the dark color below the line indicates a regime where and an atmosphere cannot be supported, while the upper gradient above the line indicates a regime where and the atmosphere is growing as the cow’s production outweighs the methane loss. The curve represents the hyperplane in this parameter space where methane loss is balanced by methane production and a steady-state atmosphere can be supported. The dashed vertical lines represent the distance of each planet in the solar system to aid interpretation in case you didn’t know them off the top of your head.

As we shepherd our intrepid spacefaring bovines toward the Sun for ICD day, the flying flux from the Sun becomes a far more significant factor in the temperature of the atmosphere and we begin to require a larger herd to compensate for a hot flux. Although the increase in the number of cows required is actually quite modest, with a planetoid in Mercury’s orbit only requiring an increase of . At least it is a more modest increase in the logarithmic domain. This planetoid, despite a significant increase in atmospheric temperature, is capable of balancing the atmospheric loss, due to increasing temperatures, by the sheer weight of methane production by our hardy herd.

In Figure 2.2 we present the scaling of atmospheric temperature and the number of cows for a steady-state atmosphere as a function of distance from the Sun.

Unsurprisingly, the two scale similarly with distance. As the atmospheric temperature increases, so does the methane loss, and this must be balanced by methane production (number of cows). Contrary to what we learned in Sunday school, cows evolved on Earth, which means they are not well acclimatized to temperatures beyond the ranges found on Earth, as evident by restaurant freezers and grills. Without further development of bovine survivability in extreme environments or some freakish cow-tardigrade mutation, we must limit our hopes to the habitable zone shaded in green in Figure 2.2. Luckily for us, by definition, this habitable region is very close to Earth, making the job of transporting the herd to their eventual location considerably easier. While transportation will be easy, the real estate market in this zone is a little too hot, while Mercury is a real buyer’s market.

Figure 2.2: Bovine survivability zone

The preceding plot shows the effects of distance from the Sun. The solid line shows how the atmospheric temperature scales, and the dashed line shows how the number of cows scales for comparison. The shaded region shows the “habitable zone” for our cow planetoid where the atmosphere and cows themselves are in an acceptable temperature range for the cows to exist without requiring further developments to bovine survivability or really expensive space suits with air conditioning.

5. Cow moons

Of course, we could save a significant amount of bovine mass by employing existing celestial bodies as the starting point for our bovine planetoid. We note, however, that this pollutes the purity of the bovine-only celestial body. For this demonstration, we will use the Moon as our testing ground.

The expressions presented above remain unchanged with the exception of planetary mass and radius, which must now take into account the mass and radius of the Moon, the temperature of the atmosphere, the surface temperature of the Moon, and the area of the Moon available to absorb flux from the Sun. On the plus side, the moon is made out of cheese, which will help achieve maximum methane production once devoured by the cows. Taking this into account, we have a new expression for the atmospheric temperature given by:

where:

A_abs is the area of the Moon facing the Sun not covered by cows, and A_rad is the area over which the energy is radiated.

Figure 2.3 is a reproduction of Figure 2.1 but takes into account the Moon as a starting point. Again, it shows the production and loss of methane as a function of distance from the Sun and lunar herd size and mass. We can now see a far larger variation in the number of cows in our lunar herd. In fact, rather than a bovine planetoid, in this case for most orbits we simply have a population of lunar cows.

Inside Mercury’s orbit, the number of cows is great enough to completely cover the surface of the Moon and we once again have complete shells of cows surrounding the lunar core. Possibly most surprisingly, if our herd could withstand the temperatures at Jupiter’s orbit and beyond, we would only need < 1,000 cows to sustain a methane atmosphere on the Moon!

Figure 2.3: Atmospheric loss by number of cows and distance to Sun based on Moon temperature

Much like the bovine planetoid, the fact that the atmosphere is constantly replenished and the comparatively small acceleration due to gravity leads to a larger scale height than found on Earth, varying from at Mercury’s orbit to at Jupiter’s orbit and beyond.

6. Rearing extra-terrestrial cattle

This theory is all well and good, but how possible is this agricultural marvel given our current resources? Our cows require a significant amount of rearing before we can begin to consider them for launch. We’ve considered rearing them along the way, but many experiments show reductions in fertility out of orbit and the ISS astronauts refuse to test space fertility and mating of bovines despite our many emails. The first issue we must tackle is where we could do this rearing. Cows require approximately 10m² of space according to recommendations from the Agricultural and Horticultural DataBase (AHDB)^[7]. Does this pose a problem?

In Figure 2.4, we plot the area necessary to care for our planetary herd, with some important areas highlighted for comparison. Sadly, things are looking bleak for the pure bovine planetoid. With a required number of cows, we couldn’t support this herd using the entire surface area of the Earth even under factory farming conditions. Of course, future advancements in agriculture may help us solve this issue, but for now this is but a pipe dream. More encouragingly, the 10⁶ cows necessary to sustain an atmosphere on the Moon at the Earth’s radius (conveniently where the Moon resides) could be sustained using only a fraction of the land area of the Earth. In fact, the best way forward may be to develop a method to rear our lunar herd in situ on the Moon’s surface using only a small portion of the available (currently uninhabited) surface area.

Figure 2.4: Required space herd rearing area per number of cows

The preceding plot demonstrates the area necessary to raise our spacefaring bovines prior to launch. The blue curve shows the area necessary for rearing cattle as recommended by the AHDB^[7]. The dotted line represents the area of land mass on Earth, the dashed line represents the Earth’s entire surface area, and the solid lines show the surface area of the Moon. There is no way about it; the cows will be shoulder cut to shoulder cut.

In addition to the area required, we must also consider the financial burden and nutritional needs of our herd. In Figure 2.5, we present both of these as a function of the number of cows in the herd. For the total cost of rearing a cow, we take an average cost of £190 per cow^[8]. With a Costco membership, the price may drive down to £165.99 per cow. For the required food per day, we utilize the Dry Matter Index (DMI), a measure of the dry matter consumed by the cows in tons per day. As with the area required to rear the cattle, the cost and nutritional requirement of the pure bovine planetoid is so prohibitive as to become impossible. The cows would cost at least 2 times more than the GDP of the US in 2021 and would need to consume per day tons of dry matter. For the wet matter, we plan on utilizing the subterranean moon aquifers.

We purchased the water rights from NASA at an unbelievable discount!

Figure 2.5: Cattle rearing cost without Costco card

The preceding plot demonstrates the total cost of rearing a planetary herd and the total DMI of the herd per day. For comparison, we have included the GDP of the US in 2021 as a dashed line and the total mass of hay produced in the US in 2004^[9] (a standout year) as a dotted line. Though there will be a temporary few years of starvation in food-insecure nations, we believe the space brisket and out-of-this-world milkshakes will be worth it.

For salvation, we turn to our lunar herd once more. The cows needed for a steady atmosphere at Earth’s orbit would only cost a mere dex less than the US 2021 GDP, a significantly more modest cost! However, we are still left with the prohibitive need to provide this lunar herd nutrition, a requirement that demands a significant portion of the US’s entire hay production from 2004^[9] for a single day. Given the viability of many other factors in seeing to fruition our local lunar herd, this is clearly where future efforts should be focused. It is, however, worth noting that if we could sustain bovine life at lower temperatures (and relocate the Moon), we would be able to sustain an atmosphere at Jupiter’s orbit with a significantly lower strain on rearing area, cost, and food production.

7. Conclusion

In this paper, we have investigated the viability of methane atmospheres on bovine planetoids and extended that discussion to a lunar bovine population. We find that a bovine planetoid can sustain a methane-based atmosphere at D > 0.1AU with cows but the cost, necessary rearing area, and required nutrition are impossible to meet at the current time until world leaders get their priorities straight. We also find that the temperature of the atmosphere is such that the cows can only survive the temperatures in a small orbital range around the orbit of Earth. However, when considering a lunar herd, we not only find that a methane atmosphere can be supported with cows at Earth’s orbit but also find that the cost and necessary area are much more manageable.

Future work should focus on relaxing some of the assumptions made to produce these results while endeavoring to solve some of the problems posed in this paper: How do we provide the necessary nutrition to support the planetary herd? How can we come together as a species to fund this great endeavor? Could this actually be of any use whatsoever? Could we potentially make a cow space suit to expand the available habitable planets and moons? We sketched one idea, shown in the following figure, in case any of our readers are venture capitalists and want a game-changing idea.

Figure 2.6: Space cattle suit notional concept

8. Acknowledgements

We wish to acknowledge Jeff Bezos for his comments on space-based factories, which (for reasons that are long lost to the authors) somehow led to a lunchtime conversation derailing into the concepts presented in this paper.

We acknowledge Stephen M. Wilkins for throwing fuel on the flames of this endeavor. We also acknowledge the insightful conversations we had with Christopher Brown (Sussex) and Maria Del Carmen Campos Varillas (Sussex) and their contribution to this work.

9. Disclaimer

No cows were harmed in the research presented in this paper, nor do we advocate for any unnecessary harm being brought upon our bovine friends.

References

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