The Wandering Earth is a novella written by a Chinese Sci-Fi writer Liu Cixin. He is also well-known for the trilogy Remembrance of Earth's past (the original Chinese title is 地球往事 whose literal translation is Earth’s past). The English translation of the trilogy won a Hugo Award for Best Novel and it was the firsst novel by an Asian author to win a Hugo Award. The first novel of the trilogy is 三体 meaning three-body and its English title is The Three-Body Problem. There is a Chinese TV series (Title: 三体) as well as an American TV series (Title: 3 Body Problem, available on Netflix) adapted from the first novel.
The premise of the Wandering Earth 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 detonating $3.712\times 10^{12}$, 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 may be determined by the famous Tsiolkovsky rocket equation
$$
\Delta v=v_e\ln\frac{m_0}{m_f}=I_{\mathrm{sp}}g_0\ln\frac{m_0}{m_f} \tag{1}
$$
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 (1), we obtain
$$
\frac{m_0-m_f}{m_0}=1-\frac{m_f}{m_0}=1-e^{-\frac{\Delta v}{v_e}} \tag{2}
$$
(2) gives rise to the percentage of the initial total mass which has to be propellant. This tell us how efficient the rocket engine is.
I don't see why Tsiolkovsky Rocket Equation wouldn't apply beyond Earth. 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 (see [1]). We need effective exhaust velocity using the gravitational acceleration for the Sun which is $g_0=274\ \mathrm{m}/\mathrm{sec}^2$. 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, (2) 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 is 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 (2) 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.
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