Putting the science in fiction - Dan Koboldt, Chuck Wendig 2018
Gravity basics
Earth and other planets. Yes, Pluto counts!
By Dan Allen
Get a big trampoline or one of those coin-eating toys for donations at the mall that looks like a trumpet shape where you put a coin in the top and it spirals around and around faster and faster until it drops through the middle. If you have a trampoline, put a heavy person in the middle and roll tennis balls or soccer balls across the trampoline and watch what happens. If you play enough, you will figure out the laws of orbital mechanics and you won’t need to do any math.
These toys have surfaces the same shape as a gravitational field with a massive object in the middle. The trampoline is like Earth, with an object in the center. The penny eater is a black hole—that’s why you don’t get your money back. After some good play time on the trampoline (or a lot of lost coins), you should understand a few following key principles.
Angular momentum—when you just can’t stop
You can’t just fall into the Sun or back to Earth, a planet, moon, or even into a black hole, unless you are really far away and accidently aim straight for it, like NASA did with their failed Mars lander. To approach any gravitational object that you are moving past, you have to slow down somehow. You might try air-braking in an atmosphere or firing retro rockets. Similarly, to go from launching off the surface of an object, such as an asteroid, to orbiting the object you need to transition that upward speed into angular speed moving lateral to it. More angular velocity will give you a bigger orbit. Trajectories that trade off speed for distance from the Sun are called elliptical orbits. Trajectories with constant speed follow circular orbits.
Orbital changes
Given what we already know, if you are in a circular orbit, such as Earth’s, and you change speed, you just went into an elliptical orbit. The difference in velocity you need to change orbits is called delta-v (or Δv, meaning change in velocity; pronounced “delta-vee”). All you need is an impulse of energy in a particular direction to change the shape of an orbit from circular to elliptical and then another burst to change back to circular and match orbits with a new planet such as Mars. (You can try that on the trampoline, too.) The real trick to changing orbits is timing it so you end up in the new orbit at the same place as your destination planet. If the timing doesn’t work out given the amount of delta-v you can get with your fuel, you are “outside the launch window.”
To orbit closer to a sun, you need to lose angular velocity. To orbit farther a way, you obviously need more. Similarly, to change from one orbit to another on a minimal effort basis, you will take a mostly tangential (around the sun) path. A more direct path heading radially (toward or away from the sun) will cost you more fuel. Just remember the key concept: To leave a circular orbit, or enter one, you have to change velocity—you need delta-v. Use that word in your writing and you will definitely start to sound authentic.
Gravity calculations
When you just need enough reality to make your story plausible, don’t look up formulas on Wikipedia and do calculations from scratch. Just scale everything from Earth’s gravity. It’s easier and you’ll be less likely to make a mistake. It is as easy as multiplying or dividing with your calculator app.
Tip #1: Escape Velocity Is 1.4 Times Orbital Velocity
Once you are in orbit, you are more than two-thirds of the way to escape velocity. This is the speed at which you can get away from the “gravity well” and not get pulled back by gravity, ever. By way of example, escape velocity on earth is 11.2 km/s, so minimum orbital speed is 11.2/1.4 = 8 km/s. That’s eight kilometers every second! You really gotta move. Since Earth’s diameter is 40,000 km, at that speed it takes 40,000/8/3600 sec/hr = 1.4 hours to get around Earth in low orbit or about eighty-five minutes (ninety minutes if you are out of the atmosphere).
Tip #2: Orbital Time Near the Surface Doesn’t Depend on the Size of the Object
This is a really weird, but true, result. If two planets or moons, or even asteroids, have density similar to Earth, it will take about ninety minutes to get around in low orbit. Size doesn’t matter! (Only density and distance from the surface.)
Tip #3: Gravity Drops Linearly Inside a Planet
In fact, gravity is zero at the center. Anything dropping through a cored planet or asteroid will bounce up and down the shaft just like a yo-yo, until it fries, slows down from air resistance and gets stuck in the middle, gets crushed by pressure, or hits the walls and dies a gruesome but exhilarating death.
Tip #4: The Force of Gravity Scales Linearly With a Planet’s Diameter
Halve the diameter of a planet and you get half the gravitational force at the surface. Earth has a diameter of about eight thousand miles, so an eight-mile-wide asteroid has 1/1000 earth’s gravity. But even if you weigh in at a slim 0.35 pounds, you should probably go easy on the tribble jerky because when you get back to Earth, it all comes back.
Here is where this idea comes from. A planet’s mass grows as the cube of its radius, but the force at the surface drops the square of the distance to the center of gravity (center of planet). So there is only one factor of radius left over: Double the radius, double the gravity.
Tip #5: Orbital and Escape Velocities Also Scale With Diameter
Since I already told you that the orbital times are the same for objects of similar density, you immediately know that orbital speed scales with diameter, too. Earth’s escape velocity is 11.2 kilometer/sec. So to get off an asteroid with 1/1000th the radius you need to be moving at 11.2 meters/sec or about 25 miles per hour.
Making the jump to light speed
Now prepare to go where no accountant has gone before: general relativity.
Virtual reality theme park rides take advantage of our body’s inability to distinguish acceleration from gravity to make it feel like we are accelerating when really we are just tipping. With visual miscues, you can’t tell if you are accelerating in a car or leaning back. Einstein’s general relativity principle goes one step further. It says that not even light can tell the difference between gravity and acceleration. So the experiment you just did on the trampoline with soccer balls or coins at the mall also works with light. Yes, gravity bends light, or more accurately, gravity bends space and light travels straight through bent space. So just like the trampoline mat stretching, space stretches in the presence of gravity.
It is hard to imagine in 3D, but think of the lines on a ruler getting farther apart the closer you get to a gravitational object. Closer to the star, distances seem longer. The equivalent way to look at it is that time depends on the curvature of space (gravity). The more gravity, the slower time runs. This idea is explored in the 2015 film Interstellar.
Astronomers even use large gravitational objects like galaxies as “gravity lenses” to collect and bend light toward us from even more distant objects beyond.
Imagine a gravitational field as a sand trap or a bog. If you want to get from one side to the other you can either walk through it (slow) or go around (faster). Light always takes the shortest, quickest path from one place to the next. So light passing a star will bend. It takes a direct path through a curved space time, traveling the quickest possible route from one place to another. Gravity refracts light, just like water or glass.
The closer you get to a black hole the more the light bends, until you can see your own backside around the black hole. But it wouldn’t be pretty because the gravitational gradient would have stretched you out like a piece of taffy.
Like the coin trap’s slope, the curvature of space becomes infinite at the event horizon of a black hole. So time literally stops, just like a space traveler approaching the speed of light. The closer you get, the slower time moves. So actually you can’t reach the event horizon. The closer you get, the faster the universe behind you moves. Stars are born and die in a tick of the clock, galaxies form and collide, galactic superclusters orbit super-superclusters and before the whole universe can die the black hole glows itself out of existence (thanks to Hawking radiation). So as long as you don’t mind having your subatomic particles ripped apart in the extreme gravitation field, you can just happily wait it out on the event horizon scoping your backside as your quarks run amok.
In fiction, gravity matters, and in space, matter gravitates.