Evacuate the Planet!
In case of a global catastrophe, could we evacuate the entire planet?
The answer is perhaps hardly surprising: no, not even close. It is impossible to evacuate all human beings safely off Earth. We have to make do with our home world. There is no Plan(et) B!
Let’s do a back-of-the-envelope calculation to figure out how much energy it requires to send a single kilogram into space and from there to Mars. With it, we can estimate the energy required for all of humanity.
Theoretical minimum
To escape Earth’s gravitational pull and fly off to another planet (e.g. Mars), a spacecraft must achieve what is known as escape velocity at the surface of the planet: \(v_{\mathrm{\star}} = \sqrt{2g_{0}r}\), where standard gravity \(g_{0}=9.80665\,\mathrm{m}\cdot\mathrm{s}^{-2}\) and \(r=6,378\,\mathrm{km}\) is the radius of Earth. With these figures, the escape velocity is \(v_{\mathrm{\star}} = 11,185\,\mathrm{m}\cdot\mathrm{s}^{-1}\).
The theoretical minimum energy required is the kinetic energy at escape velocity: \(E_{\mathrm{k}} = \frac{1}{2} mv_{\mathrm{\star}}^{2}\). That amounts to \(6.3\cdot 10^{7}\,\mathrm{J}\cdot\mathrm{kg}^{-1}\).
But of course that ignores a few gnarly bits.
Rocket equation
Earth is not a perfect sphere. A rocket is not exactly massless. Its fuel is definitely not massless. Atmospheric drag (i.e. friction with the air, especially near the surface of the planet) matters. And there’s gravity loss, because the escape velocity is for an instantaneous surface-to-space manoeuvre, which is not how rockets reach space. They take a few minutes to escape the atmosphere.
The rocket equation accounts for the rocket’s mass: its own structure and fuel. With it we can calculate the so-called \(\Delta v\) (delta-v) or change of velocity:
\[\Delta v = g_{0}I_\mathrm{sp}\ln{\frac{m_{\mathrm{i}}}{m_\mathrm{f}}}.\]The \(\Delta v\) needed must not only include the escape velocity \(v_{\star}\) but also drag from the atmosphere and gravity. That amounts to an extra \(2\,\mathrm{km}\cdot\mathrm{s}^{-1}\).
The initial mass \(m_{\mathrm{i}}=m_{\mathrm{dry}}+m_{\mathrm{fuel}}\) includes the rocket’s dry mass and the propellant. This is also known as the wet mass. The final mass \(m_{\mathrm{f}}=m_{\mathrm{dry}}\) consists of the rocket’s dry mass only. The dry mass is what is left after all the fuel is expended. It includes the structural mass of the rocket \(m_{\mathrm{rocket}}\), the engines, and of course the payload itself. The dry mass is sometimes also referred to as the empty mass.
The specific impulse of the engine is denoted by \(I_\mathrm{sp}\). For hydrogen-oxygen engines, \(I_{\mathrm{sp}}\approx 450\,\mathrm{s}\). In these LOX/LH2 engines, liquid hydrogen LH2 is the fuel and liquid oxygen (LOX) is the oxidizer.
The propellant mass fraction \(p = m_{\mathrm{fuel}}/\left(m_{\mathrm{dry}}+m_{\mathrm{fuel}}\right)\). If we substitute that into the rocket equation and solve for \(p\), we find that
\[p=1-\exp\left[-\frac{\Delta v}{g_{0}I_{\mathrm{sp}}}\right],\]or \(p\approx 0.95\) to reach escape velocity. So, about 95% of the initial mass is fuel. That’s high, but close enough to the value for a single-stage-to-orbit (SSTO) launch vehicle. The rest is divided between the structural mass of the rocket and the payload. The payload is included in the dry mass and typically accounts for a few percent. Let’s pick 4%.
With a payload of only \(1\,\mathrm{kg}\), the rocket’s initial mass is only \(25\,\mathrm{kg}\) with almost \(24\,\mathrm{kg}\) of fuel. Note that such a launch vehicle is not feasible, because the structural mass is so small it would fall apart on the launch pad. For our purposes it suffices, though.
What we then need is the energy contained within a single unit of rocket fuel. The specific energy of LOX/LH2 is \(13.5\,\mathrm{MJ}\cdot\mathrm{kg}^{-1}\), which is the chemical energy contained in each kilogram of fuel. On the launch pad, we therefore need \(3.2\cdot 10^{8}\,\mathrm{J}\cdot\mathrm{kg}^{-1}\) to escape our planet’s gravity.
Multistage rockets
The idea of multistage rockets is that the structural mass of the initial stage(s) can be jettisoned, so that overall the launch vehicle has less mass to propel in the upper stage(s). This reduces the mass ratio to more reasonable figures (0.8–0.9) that can actually be built with modern materials, and it allows each stage to be designed for the specific conditions in the atmosphere or beyond.
The structural mass that is shed at each stage is only a small fraction of the initial mass. Two or three stages are common, where the lower stages provide less \(\Delta v\) and with lower specific impulses than the upper stages. The upper stages tend to be smaller.
With a two-stage (three-stage) rocket, we can save up to 50% (60%) of the fuel mass at the launch pad for a payload of 1 kg. The details of the calculation are in a Maple worksheet.
To Mars!
Of course, with that \(\Delta v\) the spacecraft is not yet on Mars; it is floating in space, which is a pretty lousy evacuation plan. To reach Mars, we need at least \(\Delta v = 18,536\,\mathrm{m}\cdot\mathrm{s}^{-1}\). In practice, the \(\Delta v\) budget also includes safety margins of at least 10%.
The energy required to reach Mars with such a 10% safety margin using a two-stage launch vehicle is \(5.7\cdot 10^{8}\,\mathrm{J}\cdot\mathrm{kg}^{-1}\). For a three-stage rocket, that number goes down to \(4.0\cdot 10^{8}\,\mathrm{J}\cdot\mathrm{kg}^{-1}\). This is with reasonable figures for the structural coefficients and specific impulses in each stage based on specifications of super heavy-lift launch vehicles.
What if we use LOX/RP-1 instead of LOX/LH2? It is more common and safer to store. Its specific impulse is lower than that of liquid hydrogen, which increases the fuel required. RP-1 is a highly refined kerosene, which means we need a lot of oil. Assuming all kerosene can be refined into RP-1, manufacturing enough would require more oil than all known reserves left on Earth. More than twice that amount, in fact.
Let’s now think about humans. Each person needs life support. Per day in space, that amounts to about 1–5 kg extra. Most food can be in dehydrated form, so a few hundred grams per meal. Most water can be recycled from urine, sweat, and condensation, but an initial supply is required. And there is also the need for medical supplies, hygiene products, oxygen, and so on. Don’t forget the electronics, seats, seat belts, space suits, and what have you. So, 1 kg is a very optimistic estimate.
Falcon 9 offers a 4,020 kg payload to Mars. With a mission duration of at least 6 months, that means an additional 180 kg per person! So, for each person of 75 kg, the payload needs to be 255 kg. That means a Falcon 9 rocket can carry at most 15 people to the red planet. We’re going to need a bigger vessel!
Starship with its Super Heavy rocket has a payload of up to 150,000 kg to low-Earth orbits (LEO). If the same payload ratio can be assumed as with Falcon 9 (from LEO to Mars), that leads to a payload of 26,000 kg to Mars. With that, a bit over 100 people and supplies for a six-month flight can be accommodated.
With 8 billion people to evacuate, we’d need at least 80 million launches. Finding that many launch windows any time soon is infeasible, especially since we have only had 5,000 launches since 1957 with fewer than 300 launches in any single year. With only 35 space ports globally, each would have to support more than two million launches. Ignoring weather conditions and orbital alignment for fuel efficiency, even at a completely unrealistic ten launches per day per launch site, we would need more than 80 years to accomplish such an evacuation, at which point there would be even more people to get off the planet.
Can Mars even accommodate 8 billion people? Earth's land surface area is roughly the same as Mars' entire surface, so the population density would be the same. The problem is that there is virtually no atmosphere and absolutely no infrastructure. We have yet to establish a permanent (research) colony on the Moon, let alone on Mars, or for that many people. We have never even terraformed any extraterrestrial surface, not even a tiny square patch. There is really no credible plan for such an interplanetary re-settlement.
Evacuation plans
The average person weighs about 75 kg on the surface of our planet, so we need \(2.4\cdot 10^{20}\,\mathrm{J}\) of energy to send them all to Mars in a three-stage launch vehicle. That is on the order of the annual global energy supply, which was \(6.2\cdot 10^{20}\,\mathrm{J}\) in 2022.
It costs a lot of energy to manufacture such an amount of (reusable) launch vehicles, fuel, and spacecraft to send all humans to our red neighbour, during which the people (or robots) that manufacture all these components need to eat food (or electricity), which requires more energy. We have also ignored social implications (e.g. people leaving behind beloved pets or possessions), the economic impact, the political ramifications, the international coordination required, and the logistical nightmare of millions of launches with suitable launch windows that can navigate safely through all the space debris.
All in all, it is impossible to evacuate all human beings with current and foreseeable technology.