Small Body Gravity: How much is enough?

Okay, back-of-the-envelope type calculations here:

The "escape velocity" for a planet is the speed you need to achieve lift-off, i.e. to completely escape the gravity of a planet (or other body) and escape into space.

Obviously, if a human jumps up--using their usual body strength--and winds up shooting off into space forever, they're in trouble, so let's get a handle on how small an asteroid has to be before that can happen.

Say a person is strong enough, when they're standing on Earth, to jump a half metre into the air before they fall back down. We want to find out how fast they're moving when they leave the ground at the start of their jump, because that's a good benchmark for how hard they can jump (i.e. how fast they can accelerate their body mass) on an asteroid too.

The kinematics equation we'll need is:

(vf)^2 = (vi)^2 + 2ad

where:
vf is their final velocity,
vi is their initial velocity,
a is the acceleration due to gravity near the surface of the Earth, and
d is how far they have travelled.

So we have:
d=0.5 m (for jumping up half a metre),
a = -9.81 m/s^2 (acceleration due to gravity near the surface of Earth)
vf=0 m/s (because they stop moving upward once they reach the height of 0.5m)

After substituting in those values and making a cancellation that gives us:

vi^2 = 9.81 (m/s)^2

It'll be easier for us later to just leave vi squared like that. Sorry if it makes the notation confusing.

So the next thing we want to do is figure out how small an asteroid has to be before this rather-modest amount of human strength is enough to launch a person off the asteroid permanently.

The formula for the escape velocity of a body is:

ve^2 = 2GM/r

where:
ve = escape velocity, i.e. the velocity needed to achieve lift-off,
G = the universal gravitational constant for our universe, which is 6.67 x 10^(-11) m^3/kg/s^2
M = the mass of the asteroid, and
r = the radius of the asteroid

Now before we stick anything into this equation, let's first simplify things a little by getting rid of the mass M.

The mass of a planet is equal to its density times its volume, and the volume of a sphere is (4/3)*(Pi)*(r^3).

So M/r can be re-written as (density)*(4/3)*(Pi)*(r^2).

That gives us:

ve^2 = 2G*(density)*(4/3)*(Pi)*(r^2)

Old Mother Google tells me a typical asteroid has a density of 2 g/cm^3, which converts to 2000 kg/m^3. So now we have all the numbers we need to figure out how big an asteroid is when a human, who has the strength to jump 0.5 m into the air on Earth, is capable of launching themselves into space using that same amount of strength.

We want to set vi^2 in our first equation equal to ve^2 in the escape velocity equation. That gives:

(9.81) = (2) * (6.67x10^[-11]) * (2000) * (4/3) * (Pi) * (r^2)

Solving for r, which is the radius of the asteroid, we get:

r = 2962 m = 2.962 km

So any asteroid with a diameter of about (2 x 2.962km) ~ 6 km across is one that a normal human can launch themselves into space off of just by jumping really hard.

If you want to calculate how high up they can jump on LARGER asteroids, that's something we can work out too.

First, we figure out what the acceleration due to gravity is on that asteroid using the following expression:

a = GM/r^2 = G * (density) * (4/3) * (Pi) * r

Substituting in the values:
G = 6.67x10^(-11) m^3/kg/s^2
density = 2000 kg/m^3

we get:

a = 5.59x10^(-7) * r

where r is still the radius of the asteroid. The units for r will be metres, and the units for 'a' will be metres per second-squared.

To then figure out how high up the person can jump on the larger asteroid, you substitute that 'a' value into the equation we used above:

(vf)^2 = (vi)^2 + 2ad

where
a = the acceleration we just figured out, 5.59x10^(-7) * r,
(vf)^2 = 0, because they stop moving upward at the top of their trajectory,
(vi)^2 is the value we calculated above for a typical human, (vi)^2 = 9.81 (m/s)^2, and
d = the height they manage to jump to, which is the quantity we're trying to find.

This gives a simplified expression of:

9.81 (m/s)^2 = 2 * (5.59x10^[-7]) * r * d

Or, solving for d, the distance the person can jump upward:

d = 8,775,000 * (1/r)

So, for example, on a 10km wide asteroid, r=5km=5000m, and that means a person can jump:

d = 8,775,000 / 5000 = 1755m = 1.755 km high

Which is still really freaking high, and they will take ages to float back down again.

In other words, jumping around on asteroids is bad news for us big, strong, Earth-adapted humans.

So walking would be a delicate, tip-toe-don't-sneeze problem, the cards on the card table need to have weights on them, or every little breeze would knock them around, and drinks would probably work but everyone would be prone to splashing them everywhere, i.e. anyone who slams their glass down on the table will paint the ceiling with whatever was still in it.

That said, asteroids can be up to 500km across, so you can figure out whether that would be enough to keep people stuck to the ground:

diameter=500km => radius = 250km = 250,000m

So:

d = 8,755,000 / 250,000 = 35 metres

People can still bound around like grasshoppers, but they also can stay on the ground a bit more easily.

You're most welcome!

Ceres is the largest of the asteroids, so we can take that as a best-case example of a place to live on - or more likely IN since vacuum includes micro- and mini-meteorites and extremes of heat and cold. It's gravity is a bit less than 3% of that of Earth. So even there you will need ways to handle near-zero-g conditions.

So walking will need some assists. Likely solutions include a Velcro patch on the heel and forefoot plus floors with Velcro. There may be better solutions such as magnetic or electrostatic patches on shoes. Plus it is likely that inhabitants will choose a preferred direction as "down" to simplify traffic rules for moving about.

Playing cards, eating utensils, etc. will likewise have some kind of attractive tech so that they can be held "down" on working surfaces.

Drinking will need to be in squeeze bulbs. I don't think you can get around that.

You need to think about such issues but be subtle in introducing them. The less said the better. Not only does lots of detail give you more room to hang yourself, it's hard to write about them without being boring and slowing down narrative pace too much.

Space.com has an interesting article on this topic.

https://www.space.com/28640-living-on-ceres-asteroid-belt.html

Playing cards brings to mind Microsoft's old solitaire game. Chances are most things like card games are going to be on electronic devices.

Games with moving pieces can be magnetic. I even have magnetic cards and a metal board for playing outside in a breeze.

MaeZe said:

Playing cards brings to mind Microsoft's old solitaire game. Chances are most things like card games are going to be on electronic devices.

Games with moving pieces can be magnetic. I even have magnetic cards and a metal board for playing outside in a breeze.

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You never know. Old school games you can touch and see could enjoy a renaissance at some point in the future. I imagine turning poker into an entirely virtual game--played with devices between people who don't even have to be in the same room--takes away a lot of what makes the game so complex and interesting.

I kind of wonder what the future of devices like phones, computers, and tablets will be as well. At some point we will probably be able to plug them into our nervous systems directly. Unless this causes problems humans can't cope with, we might be living more and more in our own little isolated-but-connected entertainments realities in the future. People walking around, glassy eyed and unaware of the people in their vicinity as they chat with ones who aren't there (or chatting electronically with someone who is right next to them).

Our species might stop having sex entirely.

Still, I'm a sucker for older SF settings where people are (however improbably) living more or less the way we do and doing the things that feel normal to me, like interacting face to face with people in various ways outside of an electronic reality.

Roxxsmom said:

I kind of wonder what the future of devices like phones, computers, and tablets will be as well. At some point we will probably be able to plug them into our nervous systems directly...

Click to expand...

We already have a preview of this situation in the prevalence of smartphones. This "infection" we can see not only in our teenagers, each of whom insists they MUST have one, but in older people.

I live in an elder apartment, meaning here anyone over 55 (!). It's moderately luxurious but not terribly expensive. Smartphones have gotten easy enough to use that even the most technophobic can use the most basic functions easily, and do so often. It's not unusual to see phones in use in the lobby which overflows into a game room where small groups can sit at tables chatting & maybe sipping something brought from their rooms a few minutes away. There's also an interior patio which is also used most of the year. This is Southern California, after all, and the weather is mild most of the year. Even in the hottest days parts of the patio are shaded and cool.

So I've sometimes seen an entire four-person table of elders with their smartphones in hand arguing/discussing various subjects. Sometime waving a phone which has just settled (or started!) and argument. At least in this small sample phones have not distanced people from each other. And may have brought us together. So I don't buy the doomsday scenarios of tech ruining the "natural" way of life.

Facebook is often censured for its various malign influences, and some of those accusations are certainly true. But it's the main way for me to keep up with what my children and grandchildren are doing (and vice versa). Especially the one who's moved to New Zealand. She's been posting humorous videos about her experiences there. For us Fb has helped us stay in each other's lives despite the differences in distance and time zones.

It's wise to be wary of every new advance in tech or anything else - especially political "advances" - but eventually we discover the both the uses and abuses of them. Which is where science fiction writers comes in. WE are part of the vanguard of the human race, giving the rest of us early warning about possible disasters to come.

We can only hope "the rest of us" heed our warnings!