Science and Doctor Who: The End of Time

I have decided it would be very fun to do a series exploring specific instances of science on Doctor Who — mostly where they get it wrong in interesting ways.  😉  To be fair, it’s usually wrong for reasons of either plot or budget, but sometimes it betrays widespread misconceptions about how the universe works.  I will start with “The End of Time”, and more specifically, just how catastrophic the arrival of Gallifrey would really have been.  Suffice it to say, it was a bit more than the government having to explain the big weird thing that suddenly appears in the sky.

As a reminder, in “The End of Time: Part Two”, the Master is able to use the entire human race, converted into duplicates of himself, to amplify a signal enough to penetrate the time lock around the final days of the Time War.  This allows Gallifrey to ride that signal out of the time lock.  Since the Master is on Earth, Gallifrey ends up materializing very close to Earth, and still approaching.  This is depicted as calamitous mostly becuase it would bring the horrors of the Time War with it (and, as we soon discover, also the Final Sanction that Rassilon has persuaded the High Council to accept — the destruction of time itself).  But honestly, the Time War is actually not something Earth is going to have to worry about.  It has much more immediate problems.

GallifreyAndEarth

First off, we see a lot of shaking; Gallifrey is triggering earthquakes.  Would it cause earthquakes?  Oh, yes indeed.  Gravity may be the weakest of the fundamental forces, but get enough mass/energy in one place and it adds up in a hurry.  Tidal forces would definitely cause earthquakes.  But would it stop there?

To determine how much tidal strain the newly arrived Gallifrey would impart on the Earth (and also how much strain the Earth would put on Gallifrey — gravity does pull both ways, after all), we first need to figure out how massive Gallifrey is.  The series has never given concrete information on this point, but we do have some clues.

First, the surface gravity must be pretty Earthlike, given all the people running around in various episodes set on Gallifrey, including people who aren’t native to Gallifrey and who haven’t had any time to acclimate.  Leela, for instance, is able to accurately throw her knife just as well on the world of the Sevateem as on Earth in “The Talons of Weng-Chiang” and as on Gallifrey in “The Invasion of Time”, so it must be pretty close to 1 G.  We also notice that it appears to have an atmosphere breathable by ordinary humans, so it must have oxygen with a partial pressure of around 160 mmHg or 21.33kPa or the Doctor’s companions would’ve struggled to remain conscious.  It’s hard to estimate what the rest of that atmosphere is like, but it’s probably not a great deal denser or thinner, since otherwise the human visitors to Gallifrey on the series could have experienced aeroembolism if they had not first taken the time to purge dissolved nitrogen from their blood, as spacewalkers do prior to EVA.  The temperature, meanwhile, is clearly comparable to what we experience here on Earth, so this is a Goldilocks Zone planet — able to sustain liquid water on the surface over long periods of time due to sufficient atmosphere to keep the water liquid, and sufficient temperature to melt water ice.  This in turn means it has to be a silicate world, like Earth, and the geology we see on the series appears to support this.

WarDoctoronGallifrey

So the composition is probably fairly similar to Earth.  But to tell how massive it is, next we must look at another clue: in “The End of Time”, we get an unprecedented look at its size.  The perspective is tricky; we don’t know how far apart the worlds are or the angle at which we’re seeing them, but Gallifrey definitely appears to be significantly larger in diameter, perhaps as much bigger than Earth as Earth is than Mars, and it subtends a terrifying amount of the sky.  So Gallifrey is clearly what astronomers call a “super-Earth”, a rocky planet bigger than Earth.  This means it is likely more massive.

The force of gravity on the surface of a perfectly spherical object depends on two things: the object’s mass and its radius.  This is because while the mass determines the strength of its gravitational field, gravitational strength decreases following the inverse square rule.  So the further you are from the center, the less force you’ll experience.  Since Gallifrey is much larger than Earth, to have 1G on its larger surface means it must have considerably greater mass.  We need to figure out approximately how much mass.

What does 1G equate to?  Well, a G is actually not a measure of force but of acceleration.  So in the classic F = ma equation, the a (acceleration) is 1G if you’re on Earth.  And we’re going to assume also on Gallifrey.  Weight is a measure of the force you exert due to that downward acceleration while you’re standing on a fixed surface.  This is why astronauts are weightless even though they are very much still being accelerated by the Earth’s gravity — they’re being pulled towards the Earth, but the container they’re in is not fixed and is being pulled just as much.

Gravitational force is defined in Newton’s gravity formula as
GravityEquation

(No, I don’t know how to do formulas neatly on the Internet.  So I’m using Gimp.  You must suffer through my penmanship.)

So, force equals the gravitational constant times the mass of the first body and the mass of the second body, divided by the square of the distance between them.  That “square of the distance” is why gravitational force drops off, and why there is a measurable difference in weight between Death Valley and the top of Mount Everest.  (Heck, there’s a measurable difference between the archipelago of Svalbard and the archipelago of the Galapagos — so if you’re looking for quick weightloss, travel to the equator, where you’re a bit further from the center of the Earth, or to the top of a mountain to be even further still.  Can we get Jenny Craig to fund construction of a space elevator, or is this weight loss method considered cheating?)

So, back to our equations.  Let’s start with calculating the acceleration due to gravity on Earth’s surface, to reassure us that this all works.  Object 1 will be an item of negligible mass.  So we can use the same object on both Earth and Gallifrey, let’s use our favorite Time Lord.  We’ll pick a mass of 75 kg for the Doctor, because it’s convenient and not unusual for a 6′ tall human male, which seems like a decent approximation.

G = 6.67 x 10^-11, m_1 = 75 kg, m_2 = 5.98 X 10^24 kg, d = 6.38 x 10^6 m

drnewtons

So the Doctor exerts 735 Newtons of force on the ground while he’s standing on Earth at sea level.  For a little fun with conversion factors, this works out to about 75 kg of force, or 165 lbs of force.  This doesn’t actually help us figure how big Gallifrey is, but it’s kind of fun to see the equation spit back out one of the values we started with.

Since we are assuming Gallifrey to have the same surface gravity as Earth, the Doctor must weigh the same there.  If we put him on a force gauge on a location whose elevation matched the average radius of Gallifrey, it should read 735 Newtons.  Or pretty close, but we’re going to assume it’s the same to give us somewhere to go with our math.  For fun, let’s pretend that’s the Panopticon.  Maybe it’s important to have that be at the average surface elevation of Gallifrey, so that the Eye of Harmony works right, since that thing’s all about gravity.  (ref. “The Deadly Assassin”)

F = 735 Newtons, G = 6.67 x 10^-11, m_1 = 75 kg, m_2 = ? kg, d = ? m

We need to solve for m_2, the mass of Gallifrey, and to do that, we’re going to need to estimate its radius, which is what we’ll be using for d (distance).  They do give us several different views of it alongside the Earth, not enough to be sure of the relative size, but enough to at least gives us some constraints.  Eyeballing extremely roughly, in the foreground it subtends about 2 1/3 as much as Earth does.  In the background, it subtends about 2 1/5.  So it is definitely larger than 2 Earth diameters.  Let’s call it 2 1/4 Earth diameters, which works out to a radius of 1 1/8 Earth radii, or 7,180 kilometers.  So, now we have:

F = 735 Newtons, G = 6.67, m_1 = 75 kg, m_2 = ? kg, d = 7.18 x 10^6 m

We can now solve for the mass of Gallifrey.

gallifreymass

That’s about 1.3 Earth masses.  To produce 1G at its surface and be so huge, Gallifrey must have 1.3 Earth masses, which means it must be less dense than the Earth.  Gallifrey is depicted as a terrestrial planet, with rocky terrain, mountains, a decent atmosphere, and a Goldilocks temperature, etc, which rules out an ice world or a gas world.  Still, it must be richer in light elements like carbon, oxygen, and hydrogen, and poorer in metals such as iron that are so abundant on Earth.  Perhaps this forced Gallifrey, early on, to develop a space program merely so they could mine other worlds for the elements theirs lacked.  They may have been heavily dependent on meteorites for raw material.

So next we need to work out how far apart the two planets are in this encounter.  Based on the images in the show, I’m gonna just take a generous stab and call it 2 Earth diameters, which honestly looks generous from the pictures — but as we’re about to see, isn’t really generous at all, at least in terms of how well this turns out for the people involved.  The gravitational force between these two bodies, at 2 Earth diameters or 12,460 km, would be:

G = 6.67 x 10^-1, m_e = 5.98 x 10^24 kg, m_g = 7.57 x 10^24 kg, d = 1.240 x 10^7

earth-gallifrey-attraction

That’s kind of a lot.  Honestly, these two bodies are close enough together that we should probably start talking about the Roche Limit.

The Roche Limit is the limit in a two-body system where the tidal forces of the larger body are no longer sufficient to overcome gravitational self-attraction in the smaller body.  Broadly speaking, objects orbiting outside a planet’s Roche limit will be fine, but objects within it will be torn apart and/or prevented from coalescing.  The disrupted smaller body will form a ring around the larger body; this is the process believed to have created Saturn’s spectacular rings.  It certainly prevents that material from reforming into a large moon. In practice, it depends on what the secondary object is made of as well as it size, and objects *at* the Roche limit will be unable to hold loose material to their surfaces.  Very small bodies won’t experience enough gravity differential across their bodies to be disrupted (which is why the ISS doesn’t get torn apart — although tidal forces *are* enough to try to pull it into a particular orientation), and liquid bodies will be disrupted more easily than solid ones.

The Roche Limit for a rigid secondary is the distance from the primary where tidal forces exerted by the primary match the gravitational pull the secondary has on objects placed on its surface.  That is, the point where the primary will lift them off the surface just as much as the secondary tries to hold them down.  Here’s the formula:

rocheequation

D is the distance, R_M is the radius of the primary, P_M is the density of the primary, and P_m is the density of the secondary.  Earth’s Roche Limit for a rigid spherical rocky satellite in a circular orbit with a density like the Moon is at 9,492 km.  But in this case, because Gallifrey is the more massive object, we must consider it as the primary.  To use Roche’s approximation above, we need to calculate Gallifrey’s density.  (Earth’s we can just look up.)

volume of Gallifrey = 4/3 * pi * (7.18 x 10^6)^3 = 2.16 x 10^14 kg/m^3
density = 7.57 x 10^24 / 2.16 x 10^14 = 4,890 kg/m^3

This puts Gallifrey between the Earth (5,513 kg/m^3) and Moon (3,346 kg/m^3) in density, and gives us enough information to fill out the formula:

gallifreyroche
Uh oh.

Gallifrey’s Roche limit for a rigid object as dense as the Earth is 8,692 km from its center.  We’re estimating that Gallifrey’s center is 12,400 km from Earth’s center.  Since Earth’s radius is 6,380 km, that means the nearest point on the Earth’s surface is 6,020 km from the center of Gallifrey.  (Closer if the nearest point happens to be, say, Mount Everest.)

This is bad.

Let’s assume Gallifrey is coming through directly above Great Britain.  Objects in Great Britain (and a substantial portion of the hemisphere around it) which are held down purely by gravity will depart the Earth’s surface.  This includes the oceans, at least on this side of the planet.  That’s right — Gallifrey isn’t just close enough to cause a tsunami, it’s close enough to lift the water right off.    The water won’t be pulled down to Gallifrey, though; it will find an equilibrium between the gravitational pull of Earth and Gallifrey, and likely be gradually smeared out into a ring around Gallifrey.  Also joining that ring will be loose objects — rocks, sand, loose soil, animals, people, cars, airplanes, ships, gravel….  And then we’ll start to see the stuff that *is* screwed down start to pull free.  There won’t just be earthquakes.  It’ll crack the crust.  Volcanoes will erupt, and chunks of Earth will be wrenched into space.  Oh, and the atmosphere will definitely not be hanging around anymore either.

Meanwhile, things aren’t going so well for Gallifrey either.  The part of Gallifrey closest to Earth will be having pretty much the same problem, as it’s within Earth’s Roche Limit.  Any surface water is being lifted into space.  The atmosphere is pouring away as well.  Loose rocks and boulders and presumably lots of Time War rubble is being swept up.  If we assume the Capitol is on the Earth-facing side of Gallifrey, we can also enjoy the hilarious mental image of Timothy Dalton as Rassilon, levitating helplessly within the Panopticon along with all the rest of the High Council.  And it will be going badly for the crust of Gallifrey too.  If the Doctor thought Gallifrey was burning before, that’s nothing to what it’s doing now.  It’s being ripped apart.

Ultimately, both Gallifrey and Earth will be devastated.  They may not be entirely destroyed, but if they are allowed to remain alongside they will probably eventually merge into a new, larger planet.  All life on both worlds will likely be extinct.

Oops.  😛

The bottom line, of course, isn’t that the Time Lords made a huge blunder in the physics department.  It’s that people tend to drastically underestimate scale when it comes to astronomy, and the producers of Doctor Who are no exception to that.  THey made Gallifrey much bigger and much closer than they needed to inorder to get the sense of urgency.

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