There is a TV series called 3 Body Problem on Netflix. It is based on the first novel of the trilogy Remembrance of Earth’s past (the original Chinese title is 地球往事 whose literal translation is Earth’s past) by a Chinese Sci-Fi writer Liu Cixin. The first novel’s Chinese title is 三体 meaning three-body and its English title is The Three-Body Problem. The storyline of 3 Body Problem is interesting and I am planning on reading the trilogy in the near future (when I find time). I learned from my daughter that the English translation of the trilogy won a Hugo Award for Best Novel and it was the first novel by an Asian author to win a Hugo Award. The show’s storyline involves search effort of an alien civilization (like that of SETI) by the Chinese government during the time of Mao Zedong’s Cultural Revolution and a highly advanced but dying alien civilization in a triple star system due to the instability of their gravitationally bounded three stars. Is there actually a triple star system? In the Chinese TV series with the same title 三体 as that of the novel, it appears to indicate that the three star system is Alpha Centauri. Indeed, Alpha Centauri is a well-known triple star system. But in Alpha Centauri, only two of the three stars, Rigil Kentaurus (α Centuri A) and Toilman (α Centuri B) are gravitationally bound and form a binary star system. Due to their proximity to each other they look like a single star from the Earth. The other star is Proxima Centauri (α Centauri C). It is a red dwarf and is the closest star to the Sun (about 4.2 light-years). Proxima Centauri has two known planets. One of them, Proxima b is orbiting within Proxima Centrauri’s habitable zone. Also due to the closest distance from us, although its actual habitability is uncertain, Proxima b may be a good realistic candidate for Earth 2 if humans ever have to migrate themselves to another place outside of our solar system. There is also a known triple star system in which all three stars are gravitationally bound like the one described in the show 3 Body Problem or in the Chinese one 三体. It is called EZ Aquarii which is located 11.1 light-years away from the Sun. It is a part of the constellation Aquarius within the Milky Way. EZ Aquarii has two stars forming an inner binary star system and the other star orbiting around the binary system. They are red dwarfs much smaller than our Sun. The mass of each red dwarf is about only 10% of the solar mass. No planets orbiting them have been found yet.

Liu Cixin also wrote a novella The Wandering Earth. I have not read the novella but I watched a movie (available on Netflix) with the same title which is based on the novella. Its premise is that the Sun will soon become a supernova. Facing the ultimate cataclysmic extinction event, people on Earth turns their entire planet into a spaceship and attempt to relocate it to Proxima Centauri. It is going to be a long 2,500 years journey. So, at what speed the starship Earth must travel? Assuming that it makes no stops, on average, it will be 510 km/sec which is 0.19% of the speed of light. This is quite a fast speed. So far the fastest object that has ever been built is Parker Solar Probe. By 2025, it is expected to travel as fast as 191 km/sec which is 0.064% of the speed of light. Is it physically possible for the starship Earth to travel at 510 km/sec? The energy required for Earth to travel at 510 km/sec can be easily calculated using $E=\frac{1}{2}mv^2$. With $m=5.9722\times 10^{24}$ kg, the Earth’s mass and $v=510$ km/sec, the energy $E$ is calculated to be $7.767\times 10^{29}$ J=$1.856\times 10^{14}$ megatons. This is 2,000 times the energy output of the Sun per second which is $9.1\times 10^{10}$ J. The most powerful nuclear weapon that has ever been created and tested is Tsar bomb (Царь-бомба) by the Soviet Union. Interestingly, the project was overseen by the famed physicist Andrei Sakharov. Its yield was about 50 megatons. In terms of Tsar bomb, the energy is equivalent to denotating $3.712\times 10^{12}$, i.e. almost 4 trillion Tsar bombs! It seems such an enormous amount of energy is beyond our reach even in a distant future. Ultimately, the answer to the question about whether we can put giant thrusters on Earth to make it travel at 510 km/sec is determined by the famous *Tsiolkovsky rocket equation* \begin{equation}\label{eq:tre}\Delta v=v_e\ln\frac{m_0}{m_f}=I_{\mathrm{sp}}g_0\ln\frac{m_0}{m_f}\end{equation} where

- $\Delta v$ is the maximum change of velocity of the vehicle;
- $v_e=I_{\mathrm{sp}}g_0$ is the effective exhaust velocity;
- $I_{\mathrm{sp}}$ is the specific impulse in dimension of time;
- $g_0=9.8\ \mathrm{m}/\mathrm{sec}^2$ is the gravitational acceleration of an object in a vacuum near the surface of the Earth;
- $m_0$, called
*wet mass*, is the initial mass, including propellant; - $m_f$, called
*dry mass*, is the final total mass without propellant.

Tsiolkovsky rocket equation is named after the Russian rocket scientist Konstantin Eduardovich Tsiolkovsky (September 5, 1857 – September 19, 1935). He is dubbed the father of Russian rocket science. For a derivation of the rocket equation, see here. From \eqref{eq:tre}, we obtain \begin{equation}\label{eq:tre2}\frac{m_0-m_f}{m_0}=1-\frac{m_f}{m_0}=1-e^{-\frac{\Delta v}{v_e}}\end{equation} \eqref{eq:tre2} gives rise to the percentage of the initial total mass which has to be propellant. This tell us how efficient the rocket engine is. The most realistic propulsion method for the starship Earth would be a nuclear-thermal rocket. As far as I know, the best performing nuclear-thermal rocket engine that has ever been built and tested was the USSR made nuclear-thermal rocket engine RD0410. It was developed in 1965-94. Its specific impulse is $I_{\mathrm{sp}}=910$ sec [1], so its effective exhaust velocity with $g_0$ is 9 km/sec. We need effective exhaust velocity using the gravitational acceleration for the Sun which is $274\ \mathrm{m}/\mathrm{sec}^2$. The resulting effective exhaust velocity is $v_e=249$ km/sec. The orbiting speed of the Earth around the Sun can be calculated using the formula $v=\sqrt{\frac{GM}{r}}$. With $G=6.67\times 10^{-11}\ \mathrm{N}\mathrm{m}^2/\mathrm{kg}^2$, $r=1.5\times 10^{11}$ m and $M=1.99\times 10^{30}$ kg, we have $v=29.7$ km/sec. Since the escape velocity is $v_{\mathrm{escape}}=\sqrt{\frac{2GM}{r}}$, for an orbiting object its escape velocity is just $\sqrt{2}$ times its orbiting speed. So, the minimum velocity required for the Earth to break away from its orbit around the Sun is 42 km/sec. With $\Delta v=42$ km/sec and $v_e=249$ km/sec, \eqref{eq:tre2} is evaluated to be $$1-e^{\frac{-\Delta v}{v_e}}=0.155$$ This means that about 16% of the mass of Earth has to be propellant just to break away from the orbit. Since the mass of Earth is $6\times 10^{24}$ kg=$6\times 10^{21}$ tons, 16% would be $10^{21}$ tons. For a nuclear-thermal rocket, the usual propellant is liquid hydrogen. (RD0410’s propellant was also liquid hydrogen.) The basic principle is that liquid hydrogen is heated to a high temperature in a nuclear (fission) reactor and then expands through a rocket nozzle to create thrust. Earth does not even remotely have that much amount of hydrogen. While hydrogen is the most abundant element in the universe, Earth does not have it a lot. This decisively concludes that the starship Earth can’t even break away from its orbit around the Sun let alone travel at the speed of 150 km/sec. Regardless, for the sake of completion, let us calculate how much propellant the starship Earth would need just to reach the speed of 150 km/sec. Now, with $\Delta v=150$ km/sec, we calculate \eqref{eq:tre2} to be $$1-e^{\frac{-\Delta v}{v_e}}=0.453$$ That is, 45% of the mass of Earth has to be propellant!

By the way, Moving Earth is not just a science fiction. It is nothing like relocating Earth to a distant star system but scientists have been pondering how to shift Earth’s orbit farther away from the Sun in order to mitigate rising temperatures on Earth. In my opinion, such an extreme meddling in Mother Nature must not be attempted even if possible as it likely results in unintended catastrophic consequences.

*References*: