Existence of a small planet (theoretically)

[alpha372]

I was trying to go about deciding whether a planet such as the following could exist:

It has a radius of only 2m.
It has an acceleration rate of 9.8 m/s^2 (gravity)

This way, you can walk on it, and at the same time, you could see whether you were "upside down", although, physically you probably couldn't tell the difference.

So I've done the work that proves this thing isn't a black hole. However, I've come to an unproved conclusion that this planet is actually a star because it's density is between that of a white dwarf and a neutron star.

The trouble is, I'd like to be able to say that solids won't exist at this high of a density (5.877X10^11 (standard units))

because that kind of density requires pressures that exceeds what solids can stand, and that along w/ those high pressures come high temperatures, melting or boiling anything that would have been a solid -- thus, I couldn't walk on the "planet" anyway.

However, I feel this only works if high density implies high pressure which implies high temperature, meaning in the end, that high density implies high temperature, which fails.

As shown in this equation: density = pressure(mass)/(RT), where T is temperature

It would be cool if someone could fix my logic! Or show me something that proves this thing has to be a star.

 

[Jonathan Scott]

I was trying to go about deciding whether a planet such as the following could exist:

It has a radius of only 2m.
It has an acceleration rate of 9.8 m/s^2 (gravity)

This way, you can walk on it, and at the same time, you could see whether you were "upside down", although, physically you probably couldn't tell the difference.

So I've done the work that proves this thing isn't a black hole. However, I've come to an unproved conclusion that this planet is actually a star because it's density is between that of a white dwarf and a neutron star.

The trouble is, I'd like to be able to say that solids won't exist at this high of a density (5.877X10^11 (standard units))

because that kind of density requires pressures that exceeds what solids can stand, and that along w/ those high pressures come high temperatures, melting or boiling anything that would have been a solid -- thus, I couldn't walk on the "planet" anyway.

However, I feel this only works if high density implies high pressure which implies high temperature, meaning in the end, that high density implies high temperature, which fails.

As shown in this equation: density = pressure(mass)/(RT), where T is temperature

It would be cool if someone could fix my logic! Or show me something that proves this thing has to be a star.

I don't think it could exist at all. The temperature doesn't have to be a problem, in that in theory you could extract the heat produced by compressing the material. The problem I see is the high pressure; there isn't a material you could use which would hold something confined at anywhere near that pressure without the help of huge gravitational forces, which you obviously can't have in this case.

 

[marcus]

I was trying to go about deciding whether a planet such as the following could exist:

It has a radius of only 2 m.
It has an acceleration rate of 9.8 m/s^2 (gravity)
...

I don't think it could exist at all...

Perhaps not as a naturally formed object. But could such a thing could be constructed by positioning a black hole of the correct mass at the center of a hollow metal sphere with the correct radius of 2 meters?
Besides whether it would be physically possible there's the question, I guess, of whether alpha wants to consider artificially constructed planets, or insists on having them formed by natural processes.

The Schwarzschild radius of the black hole would presumably be about 9 x 10^-16 meters
Let's see what the Hawking temperature would be (could be a problem!)

hbar*c^3/(8 pi k* (2m)^2*9.8 m/s^2)

such a small black hole would have a very high Hawking temperature. Unless I'm mistaken it would be over 200 billion kelvin!
Would anyone care to check that?

What a dreadful small planet I've proposed! Unless I've made a mistake the power output of a black hole that size would be a billion watts! It would instantly vaporize the metal shell and give anyone standing on the planet a hot-foot deluxe.

Very uncool.

So my idea of putting a small black hole at the center of a hollow sphere doesn't work. Can anyone think of another way to construct such a planet? Or show that my estimate of the power output of the appropriate mass black hole is wrong? Otherwise I have to agree that Jonathan is right. The thing couldn't exist (either as artifact or naturally formed).

 

[marcus]

If we generalize your problem slightly, it turns into a rather nice exercise in basic black holes.
You specified a 2 meter radius. That led to the black hole being too hot. Suppose we ask how big we have to make the sphere so that the power output is only, say, one megawatt.

Perhaps we think we can cope with a megawatt. And we still want a spherical planet that you can walk on---9.8 m/s^2 gravity.
This is readymade for Wikipedia and the google calculator. Ideal sample problem:
http://en.wikipedia.org/wiki/Hawking_radiation
You see there the power P is given by
P=hbar*c^6/(15360 pi (GM)^2)
so the mass M is given by
(GM)^2 = hbar*c^6/(15360 pi megawatts)
GM = sqrt (hbar*c^6/(15360 pi megawatts))
and all we need to do is divide GM by 9.8 m/s^2 and we will have the radius squared.
So we put this into google window
"(sqrt (hbar*c^6/(15360 pi megawatts)))/(9.8 m/s^2)"

It comes to about 128 or 129 square meters! So we are talking about a planet with radius of about 11 meters! That is not so bad.
To get it more precisely
"sqrt((sqrt (hbar*c^6/(15360 pi megawatts)))/(9.8 m/s^2))"
And the calculator says 11.3 meters for the planet radius.

A megawatt is still a lot of radiant power to have to cope with, so let's take it another step and reduce the black hole power output by a factor of ten. Now it will be 0.1 megawatt. I will paste the same thing into google but with a zero knocked off.
"sqrt((sqrt (hbar*c^6/(1536 pi megawatts)))/(9.8 m/s^2))"
Now it tells me the radius is 20.1 meters.

OK, so it is physically possible to have a planet which is 20 meter radius, and has normal Earth surface gravity of 9.8 m/s^2.

What you initially asked about is almost possible. You just said 2 meters instead of 20 meters.

The planet can consist of a light hollow metal sphere with 20 meter radius, at the center of which is positioned a small black hole. You have to figure out a servomechanism to keep the black hole centered. You could put a small amount of electric charge on the hole to make it possible to exert force on it. Jonathan and I will assume that you can take care of that detail, alpha

It occurs to me you could cover the metal shell with anything you like. White rug, artificial leopard skin, leather. Something with texture will make it more pleasant to walk on.

Please let me know if I miscalculated anything. I think the calculation is right but one never knows. ;-)

 

[alpha372]

Yes, I want a "small planet" or thing, to be naturally occurring; I want the universe to be able to construct it; not humans or "divine" intervention.

20 meters isn't a bad size though... The basic idea is that it be small so that you could tell by walking that you were going around something curved, and that it would be cool enough, like the moon, to walk on. This way, unlike the earth, you could actually experience going "upside down", meaning in opposite directions vertically, relative to someone on the opposite side of the planet. Here on earth, you can't tell that you are 'upside-down' because we really can't even tell that the Earth is curved because it is so big.

The whole point being, if such a small planet could exist, it would be cool if we could see ourselves going "upside-down" although, physically, we probably wouldn't be able to tell-- just like on the Earth because the force of gravity acts the same no matter where you are.

However, it has been suggested that walking on such a planet would be quite different from walking on the Earth - perhaps it would pull our feet more than our heads, I don't really know why it was suggested. It would be interesting to know why.

 

[marcus]

... it has been suggested that walking on such a planet would be quite different from walking on the Earth - perhaps it would pull our feet more than our heads, I don't really know why it was suggested. It would be interesting to know why.

certainly, if you are on a 20 meter radius planet, and your height is 2 meters, the pull at your head level is only about (20/22)^2 of the pull at ground level. That's about 82 percent. You might notice the difference.

 

[alpha372]

certainly, if you are on a 20 meter radius planet, and your height is 2 meters, the pull at your head level is only about (20/22)^2 of the pull at ground level. That's about 82 percent. You might notice the difference.

Thanks! I was too lazy to do the calculation myself!

 

[Janus]

You'd better be prepared to wear a pressure suit also. I calculate that the escape velocity would be a mere 20 meters/sec. The average velocity of air molecules at 0°C and standard pressure is about 460 meters/sec. Such a planet would have a hard time maintaining a breathable atmosphere.

Orbital velocity would be about 50.4 kph or 31.5 mph

Running at 10 mph would reduce your weight by 10%

An Olympic class sprinter could reduce their weight by 50%

 

[alpha372]

You'd better be prepared to wear a pressure suit also. I calculate that the escape velocity would be a mere 20 meters/sec. The average velocity of air molecules at 0°C and standard pressure is about 460 meters/sec. Such a planet would have a hard time maintaining a breathable atmosphere.

Orbital velocity would be about 50.4 kph or 31.5 mph

Running at 10 mph would reduce your weight by 10%

An Olympic class sprinter could reduce their weight by 50%

Thanks, this is all interesting stuff (next time show your work! hehe). I also assumed that being in a space suit was a given... I didn't even think about the possibility that it might actually have an atmosphere (the thing is so small).