Illustration by Callie_Wugange
This article on gravity is part of the Science in Sci-fi, Fact in Fantasy blog series. Each week, we tackle one of the scientific or technological concepts pervasive in sci-fi (space travel, genetic engineering, artificial intelligence, etc.) with input from an expert.
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The Expert: Dan Allen
Dan Allen is a physicist and cell phone sensor designer in Silicon Valley, in the San Francisco Bay Area. His first novel for publication is in preparation at Jolly Fish Press. He has designed lasers for the government that see through envelopes and (eek!) clothing, lit a three-story electron accelerator on fire, and created nanoparticles in a radioactive hot lab (sorry, no spiders allowed).
He is dad to his five children and husband to his drummer artist wife. You should check out his blog and follow on Twitter.
Gravity Basics for Sci-fi Authors
To understand gravity, the first thing to do is get a big trampoline. Or one of those coin-eating toys 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.
Key Principles of Gravity
- You can’t just fall into the sun or back to earth 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). You have to slow down somehow, for instance air-braking in an atmosphere or firing retro rockets.
- To orbit you need a certain speed. More angular velocity will give you a bigger orbit.
- An impulse of energy in a particular direction will change the shape of an orbit from circular to elliptical or vice-versa. That’s how you change from one orbit to another.
Gravity calculations for sci-fi authors
Don’t look up formulas on Wikipedia and do calculations from scratch. Just scale everything from earth gravity. It’s easier and you’ll be less likely to make a mistake. It’s just simple multiplying and dividing on your smart phone calculator app.
- Escape velocity is 1.4 (√2) times orbital velocity. Once you are in orbit, you are more than 2/3 of the way to freedom. Escape velocity on earth is 11.2 km/s, so minimum orbital speed is 11.2/1.4 = 8 km/s. Earth’s diameter is 40,000 km, so it takes 40,000/8/3600 sec/hr = 1.4 hours to get around earth in low orbit or about 85 minutes (90 minutes if you are out of the atmosphere).
- Orbital time near the surface doesn’t depend on the size of the object. If two planets or moons, or even asteroids, have density similar to earth, it will take about 90 minutes to get around in low orbit. Size doesn’t matter! (Only density and distance from the surface.)
- Gravity drops on linearly inside a planet, going to zero at the center. Anything dropping through a cored planet or asteroid will bounce up and down the shaft just like a kid on a swing or a pendulum, until they fry, slow down from air resistance and get stuck in the middle, get crushed by pressure, or hit the walls and die a gruesome, but exhilarating death.
- By the same token, the force of gravity scales linearly with a planet’s diameter. Half the diameter of a planet and you get half the gravitational force at the surface. Earth has a diameter of about 8000 miles, so an eight mile wide asteroid has 1/1000th earth’s gravity. But even if you weigh in at a slim 0.35 pounds, you should probably go easy on the tribble jerky.
- Since I already told you that the orbital times are the same for objects of similar density, you immediately know that orbital and escape velocities also scale with diameter. Earth’s escape velocity is 11.2 kilometer/sec. So to get off this asteroid you need to be moving at 11.2 meters/sec or about 25 miles per hour. You also could have calculated that from earth’s orbital time being about 85 minutes, the circumference being about 3.14*8,000 miles, so orbital velocity is 3.14*8,000 miles/1.4 hr = 18 mph. Escape velocity is 1.4x that, or about 25 mph.
Making the jump to light speed
Now prepare to go one step deeper. The experiment you just did with soccer balls or coins also works with light.
Einstein’s special theory of relativity (the one that says you can’t go faster than the speed of light and time slows down the faster you go) is based on a simple idea: physics can’t tell whether you are not moving or moving at a constant speed. Einstein’s general theory of relativity goes one step further: physics can’t tell whether you are accelerating or in a gravitational field.
The implication of the general theory of relativity is that gravity warps space itself. Imagine the lines on a ruler getting further 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 recent blockbuster film Interstellar.
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. And the closer you get to a black hole the more the light bends, until you can see your own backside around the black hole—I’ve always wanted to do that.
“Honey, do these pants make my butt look big?”
“You know dear, you can jump into a black hole and see for yourself.”
Uh…nevermind.
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.
Congratulations—you understand gravity.
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Earth’s diameter is closer to 13000 km than 40000 km.
In escape velocity calc, you left off the thousands.
Should be 18,000 and 25,000 mph.
You might want to revisit this statement:
“More angular velocity will give you a bigger orbit.” For low earth orbit, the orbit time is around 90 minutes, which gives an angular velocity of 2π radians/90 minutes (around .07 radians/min). Geosynchronous orbits are 24 hours or 2π radians/1440 minutes (around .004 radians/min).
I wish JJ Abrams had a physicist on his staff for when he decides to do something like have the Enterprise “fall” to earth from near the moon’s orbit.
There are pretty good codes available that allow people to plot how objects move in orbits (powered and non-powered). I would suggest any science fiction writer who wants to talk about orbital dynamics seek one out.