Authors' Mistakes #3 - Lost in Space

Lost in Space is a Science Fiction film made in 1998. The script was written by Akiva Goldsman, who a few years later received an Academy Award for his adaptation to the big screen of A Beautiful Mind.

Akiva wrote scripts of many successful films like The Client, I Robot, The Da Vinci Code, I Am Legend, and Angels & Demons, but in Lost in Space, which he also produced, he made a very bad mistake.

*** Warning: spoiler! ***

At the end of the film, Prof. John Robinson (William Hurt) saves their spaceship by ordering the pilot (Major Don West, played by Matt LeBlanc) to fly through a crumbling planet.

The problem with that solution is that the ship will gain speed flying towards the planet’s centre, but it will lose it all again to emerge on the other side. If the ship was not able to reach escape velocity before passing through the planet’s centre, it will still be unable to reach it going through the planet’s core.

Perhaps, the writer thought of the so-called “slingshot” effect, used by deep-space probes to exploit the gravitational pull of one planet to gain speed. But that only works because the probe is well off-centre, not plunging towards the middle of the planet, like in the film. If the probe approaches the planet from “behind” (with respect of the direction of orbital movement of the planet), the planet pulls the probe along. The probe had from the very start enough speed not to remain trapped in the gravitational well of the planet, but this “pulling”, beside changing the direction of its motion, gives it some additional speed.

There are two other mistakes in the same scene.

The first mistake was that to cross the planet our adventurers would have needed longer than half an hour, while in the film everything happened within a few minutes.

As the planet had a gravity comparable to Earth, we can assume that it had similar mass and volume, at least as a first approximation. Now, the potential energy of the ship on the surface of the planet is Given by GMm_s/r (forget the signs), where G is the gravitational constant, M is the mass of the planet, m_s is the mass of the spaceship, and r is the radius of the planet (~6000 km, like Earth). The acceleration due to gravity on the surface of the planet is GM/r². As we know that on the surface of Earth the gravitational acceleration is about 10 m/s², we can calculate without much fuss that GM/r = 10 m/s² * r = 60 km²/s², without having to look up the values of G and M. This means that the potential energy of the ship on the surface of the planet is U = 60 km²/s² * m_s. Now, when the ship passes through the core, it will have converted all its potential energy to kinetic energy. As the kinetic energy is given by ½ m_s * v², where v is the ship’s speed, you can easily calculate that the ship, if it was at rest at the beginning, will have flown through the centre of the planet with a speed given by: v = sqrt(2 * 60) km/s = ~11 km/s (which, incidentally, is the escape velocity, which on Earth is 11.2 km/s, as I could have stated at once). To cross the whole planet, the ship would have needed 12000 km / (5.5 km/s) = 2182 s = ~ 36 minutes.

But there is still another mistake.

After emerging from the planet, West says: “The planet’s gravity field is collapsing,” which is total nonsense, as gravitational forces don’t collapse. Even in a supernova it is the matter forming the star that implodes, not its gravitational field.

Maureen, John’s wife, then observes: “We’ll be sucked in.” This is also moronic for two reasons: firstly, the planet had been “sucking them in” all the time; secondly, the gravity pull of a planet remains the same, whether it is in one piece or in a million pieces.