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Homework Help: Small theortical planet

  1. Dec 9, 2008 #1
    1. The problem statement, all variables and given/known data
    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.

    3. The attempt at a solution

    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 lead me in the right direction to prove that this thing is a star.
  2. jcsd
  3. Dec 9, 2008 #2
    Well, i could be mistaken, but given a definition of star as "ongoing hydrogen fusion", then whether or not this is a star really depends on what it is made of. If we had a big block of iron, then fusion certainly couldn't occur (iron is at the bottom of the atomic energy curve)...

    I would go about the argument of comparing the necessary density to the density supportable by electron degeneracy pressure. My guess, though I haven't done the calculation, is that is that for that density electrons would have to have merged with protons, and thus the matter would have to be composed almost entirely of a neutron/exotic particle soup.

    I'm also somewhat confused what you mean by "see if you were upside down"?

  4. Dec 9, 2008 #3


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    By "planet", do you just mean a solid sphere? Is that gravitational acceleration at the surface? Given that a person's average height is not much less than the size of your "planet", this is going to require a precise elevation at which g is defined. I'm assuming that you are just trying to figure out if there is a logical inconsistency for such a "planet".

    (Be careful with the word "prove"; you should at least state your assumptions.) I suppose you just want to do this nonrelativistically, so you probably used Newton's gravitation law to find M and then you found that escape velocity was less than c? I agree that this doesn't seem anywhere close to a black hole. BTW, what kind of object would you say is comparable in "size" (i.e. Schwarzschild radius) to a black hole with this mass?

    Well, I agree with the density range, but it just depends on what you mean by "star".

    Did you calculate the pressure? Have you ever seen a phase diagram with liquid or gas ON THE BOTTOM?! What mechanism do you think supplies the pressure (i.e. what conteracts the gravitational collapse)? Would you say that a white dwarf is not solid? What about a neutron star? What about some other kinds of material?

    Well, I would agree with your conclusion, but for a different reason. Suspend, for the moment, your disbelief about the solidity of the "planet" (because that just amounts to finding the right material). What happens when you take a step? Think center of gravity and ability to balance.


    Why? The pressure at the bottom of the ocean is immense. What do you think the temperature of the water is down there?

    Is that the ideal gas law? You should realize that the ideal gas law is an approximation whose validity is best for tenuous gases, and the approximation tends to worsen as the density increases, for various reasons.
    Last edited: Dec 9, 2008
  5. Dec 10, 2008 #4
    You're right-- I didn't carefully consider the definition of a planet. Thanks for pointing this out. I should have said something along the lines of "object in space cool enough to walk on. " And for simplicity, I'd like to first assume that this object is pretty much uniform in density. But I would also like it to be spherical in nature-- much like planets. That's probably why I called it a planet in the first place-- it reminded me of mercury or something, or the moon ( I know the moon is not a planet-- it is a satellite). So in short, let us assume that this object is a sphere, cool enough and solid enough to walk on, and is uniform in density. And let's assume that this thing is not rotating, or in any kind of orbit-- for simplicities sake.

    Yes, g= 9.8m/s^2, at the surface of the "small planet".

    Here's why I decided this thing wasn't a black hole:
    "If a spherical, non rotating body with mass M has radius less than Rs, then... the body is a black hole"

    Here is the equation associated with this:
    Rs = 2GM/(c^2)

    when I did the calculation, I found that Rs for my "small planet" was approximately 2/(c^2), which makes 2m look large! Therefore, I was able to conclude from the given definition that my "small planet" was not a black hole-- the radius of my "small planet" is too large.

    To figure out the mass of my "small planet" in the first place I used the equation:

    m = gr^2/(G)
    , where g = 9.8m/s^2, r = 2, and G = about 6.67X10^-11

    What I mean by star... Well since my "small planet" falls in the density range of a neutron star and a white dwarf, and all the "cooled" metals known on earth are about 10^7 less dense than my object, then it seems to me that if my object in fact could exist at all, it would only exist as something liquid or gaseous and certainly not a cool solid.

    That is just the logic in my head. I'm running into trouble because I can't find a way to prove the logic in my head.

    This is where I find myself "stuck." I don't even know how to even clarify my issues enough to know what I need to do to be able to calculate the pressure; All I have at my disposal that relates to pressure is the equation pv = nrt (which I know is just Ideal gas law, and may or may not be applicable to my planet at all. And even if we assume that it is, I find that the higher the temperature, the less dense the object, because of the equation d = Pm/(nRT), which goes against my logic). So I find myself in a pickle (oh, and another equation is p = df/da -- which I wouldn't even know where to begin-- I don't know anything about the force for df).

    So I've found myself in a complex situation; I want to be able to say that because of the density of my object, it would be impossible for it to be solid. I just don't know how to go about proving that because I don't even know what material my planet is. But to make things a bit more simple, we could assume that it is made of the most dense element known to man, and then some how prove that this material itself would even have to be in liquid or gaseous form on my planet to match the density of my planet.

    All I know is that platinum has a density of 21.4 x 10^3 -- higher than gold, lead, etc, so to make it match the density of my planet (something X 10^11), it would have to be compressed by pressure or something at the very least! And because of this, would most likely not be in solid form! I'm not convinced that platinum is the most dense element, but I am convinced that all metal elements only have a density around something X 10^3, more than over a million times less dense than that of my planet.

    From this statement I gather that you think such a "small planet" could exist in space, and be solid, if it were made of the right material. But keep in mind, the right material would have to be manufactured naturally by the universe, not by humans. Do you think such a material exists naturally in the universe?

    I also gather from "What happens when you take a step? Think center of gravity and ability to balance." that you believe we might need to crawl around the planet rather than walk. I'm really not sure of the implications of having the gravity of the earth on such a small planet. I draw a blank when it comes to thinking about what we might actually experience physically. I certainly wouldn't object to you saying the consequences out right!

    You know, I bet it is freezing down there. Cold air is a lot denser than hot air. I would be willing to bet that cold water compresses "nicer" than hot water (hot expands, cold shrinks). So am I thinking in opposite terms when it comes to my planet? Is the truth of the matter that my planet would have to be really really freaking cold? I would like this alternative too.
    I do believe that temperatures near absolute zero can exist in space somewhere. But the possibility of a spherical object even forming under such cold conditions is highly unlikely. Things tend not to want to move at such extremely low temperatures right? Would the particles of my planet even come together if this were the case? I assume gravity works under all conditions, so they probably would. But what kind of phenomena would cause...

    I never thought of the situation in which my planet was the result of beginning as a star, and then having been cooled down over long periods of time, having been the core of a super giant or something...

    "As you may know, a white dwarf is the cinder of a star which used to be like the Sun. At the end of its life, such a star expels much of its atmosphere, and the nuclear fusion stops. The hot core, about the size of the Earth but much denser, becomes exposed: this is the white dwarf.

    When a star has just become a white dwarf, it is hotter than 100,000 K (about 180,000 F). It then gradually cools --- after many billions of years, it can become cooler than the Sun (which is about 6,000 K). So there is no particular temperature associated with the white dwarfs. " [http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/970512e.html]

    Note however; this is still not cool enough to walk on right? It would probably still be glowing a little bit? Well, I guess we can compare this to walking on the moon. How hot does the moon get in the sun?

    "Maximum surface temperature 123°C
    Minimum surface temperature -233°C"


    I would say this is a lot cooler than 6,000 k! And we all know the moon is a lot less dense than my planet!

    I don't really care if I'm right or wrong. I just would like help proving the existence or non-existence of such a small "planet" that could exist naturally in the universe without human or "divine" intervention.

    The whole point is, could there ever be such an object that we could walk or crawl on in which we could clearly see that we were "upside-down" as opposed to the earth where we couldn't tell if we were upside down relative to someone else on the exact opposite side of the earth because the earth is too freakin' large to notice the difference.

    All I know is that I've eliminated the danger of it being a black hole. Can such a dense object exist that is cool enough to walk, crawl, be on?

    I don't know how to go about proving it one way or the other; that's what I am asking help for.

    It would be interesting to know what the physical effects of being on such a planet would feel like; would it feel like the earth, or would it be much different?
    Last edited by a moderator: Apr 24, 2017
  6. Dec 10, 2008 #5
    I guess what I meant by star was that it had to be something between a white dwarf and neutron star because of it's density. I think most white dwarfs could be considered too hot to walk on, right?

    My logic is that the density of this thing is around 10^11, and practically all elements that are "touchable" only have a density around 10^3-- more than a million times less than that of my planet, so it must be some kind of hot thing. Of course I don't have any idea on how to prove this.

    You gave me an idea, which is why I find your post the most useful. Thank you!
  7. Dec 10, 2008 #6


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    I am having trouble understanding what your goal is. It seems that perhaps you are writing some "accurate" science fiction, so you want to introduce something exotic into your story while simultaneously avoiding the problem of making your scientifically minded readers unhappy. I will just make a general subjective commentary about this; take it for whatever it is worth to you. The universe is freakin' huge and complex (to paraphrase Douglas Adams). Go far enough away from Earth, and who knows - who really has the slightest idea, with all our telescopes and whatnot - what is out there. It is even seriously considered that we are just in a matter pocket, and maybe pockets of the universe are sort of a jumbled mess of matter and anti-matter. (I wouldn't go to one of those anti-matter pockets if I were you.) It could be that there is an entirely different periodic table somewhere. It could be that the notion of space and time just doesn't extrapolate like we think, and things are a lot closer than we think, or at least way different than we think because we have ASSUMED that the distance is something else and based our measurements on that fact. The possibilities are endless. But none of that is scientific; it is just a "loophole" that can be exploited for the purposes of science fiction.

    BTW, have you ever read the story, "The Little Prince". You may get a kick out of it.

    A few specific suggestions:

    Instead of "star", just say "exotic" or something. That is, you are essentially wondering if the material can be ordinary, or must it have quite unordinary properties under the conditions. Note that the elements do not exhaust the possibilities for matter.

    For pressure, you can calculate (Newtonianly) the force of gravity on an area of "soil" due to the weight of the "soil" above. That's pretty much it, and so you know that the pressure to support the object (against gravity) has to be equal and opposite. As far as I know, the mechanisms that supply this pressure are, for example:
    - chemistry for solid planets, moons, asteroids, etc.
    - radiation pressure for suns
    - (electronic) Pauli exclusion for white dwarfs
    - ("neutronic") Pauli exclusion for neutron stars
    So, I would guess that Pauli exclusion would support your "planet" from gravitational collapse. So, you probably need to build your planet out of fermions.

    You're using the ideal gas law to imply that it has to be a gas because of the way you expect it to behave based on this law. Logically, what you're doing here is assuming it is true to prove that it is true. This is logically vacuous: P => P. Here is something to ask yourself: Is it ever possible, assuming that the material obeys the ideal gas law, to conclude that the material is not a gas? Probably the main problem is the meaning of "solid". Instead of saying "solid", I think what you really mean is that you want the surface to provide a normal force against ordinary pressure.

    Here's something to think about: water striders (insects). Even if the "planet" was a "fluid", what about surface tension? (However, if there were surface tension in this situation, that would dramatically change your pressure calculation, I think.)

    Here's something else to think about: bouyancy. Even if the planet was a "fluid", what would be the weight of such dense fluid displaced if your foot sank into it?

    BTW, the most dense element is Osmium, I believe. It is in that group of five elements or so that includes Pt, Ir, and such, but you probably won't find it in the jewelry store - too brittle, and VERY expensive I think. I actually asked to have wedding rings made of Osmium, because I thought it would be cool/funny, but they jeweler thought I was nuts. Hey, come on, people are wearing titanium now. At least my idea was for a precious metal. At any rate, as I was saying earlier, you are not limited to the known periodic table, are you?

    Think about tidal forces. Here's a question: How much different do you weigh (on Earth) if you stand straight up on the scale vs. laying down flat on it (assume a scale large enough that you can lay on)? Can you balance if you kick one leg out in front of you? What about on "your planet"? I didn't do any calculations, but I know it would definitely be "wierd". Maybe there is a simple solution. Does your "planet" really have to be so small? Can you make it, say 30 meters in radius?

    Actaully, now that I've thought about it, I'm not sure what is the temperature at the bottom of the ocean. I am assuming it is cold, because water gets denser with decreasing temperature down to 4degC. However, there are two issues that I have ignored: 1) That is for standard atmospheric pressure, and I didn't consider how that would change at higher pressures, and, continuing this issue further 2) water has "wierd" thermodynamic behavior. In fact, due to the phenomenon of regelation, I'm not sure if the water at the bottom of the ocean cannot be even "below freezing". If you squeeze ice hard enough, the crystal structure collapses and the molecules move freely. I'm not sure what is the limiting lower temperature of the ice for the phenomenon, or if the result is technically a liquid, but it makes me wonder. Anyway, this just demonstrates that, if you are thinking in terms of things like the ideal gas law, or thermal expansion, material properties may surprise you. Here's another example: what happens to rubber when you warm it up?
  8. Dec 10, 2008 #7


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    A white dwarf is only hot, AFAIK, because of the residual thermal energy from the fact that it was once undergoing fusion. This would be analogous to the fact that coals are hot even after the fire goes out. They are approximate black-bodies, and their temperature is a free parameter. The fact that the material is in coal form is completely independent of the fact that its temperature is, say 500 degrees. Come back tomorrow and you can safely touch it. For white dwarfs, I believe the issue is with the age of the universe. "come back tomorrow" would be analogous to "come back in tens of billions of years", so theoretically, no white dwarf can have cooled down enough to walk on. However, what if there was a collision that broke off a piece of white dwarf that just happened to be in the shape of a sphere with a radius of 2 meter? Eh ... eh?
  9. Dec 10, 2008 #8
    LOL. Sorry, I didn't mean to get on your defensive side, lol. Thanks for the laugh.

    About the whole, "anything can happen" in quantum mechanics world; I prefer to stick to the ordinary. And in ordinary circumstances, we would have to wait a really long long time for such a "just happened to..." incident to occur; especially with the whole, "in the shape of a sphere" phrase (I don't know any incidents of smashing something into a sphere that lead to a sphere breaking off). Who is to say that it hasn't already occurred? No one. Who is to say that probability wise, it is not likely to occur? Tons of people, including myself. If the existence of my planet is solely dependent on such a "lucky" and convenient phenomena, then I'm pretty much going to conclude that the existence of such a planet is highly unlikely, and leave it at that.

    Yes, I'd like to stick to the periodic chart we all know about. Why? Because most of those elements are naturally occurring. Most of the elements in abundance are formed in the stars themselves. Carbon, Iron, etc. If my planet has to be "exotic" and made up, I'm not interested.

    No, I'm not writing a science fiction novel. I'm studying physics and asking questions along the way, for the pure practice of learning and proving concepts. It is really boring to read a book cover to cover, linearly. I was only on page 2 of my physics book when I asked myself this question and I ended up jumping around to several different places within the book.

    It's been good practice; I learned from you and others that I need to be extra careful in how I define things. I learned that as in mathematics, I need to remember to state the assumptions of my argument.

    I really didn't intend to make the argument formal anyway; I wrote everything down in my journal for my purposes, I got stuck, and asked for help regarding which direction to go for proving that the planet would probably be too hot anyway.

    So, I'll give you that, in terms of waiting forever, there's a small chance that my planet could exist. But it seems to me, and a few others, that given the current "predictable" condition of the universe, and the current pattern of stars and what is produced in the universe naturally, that it is probably too dense.

    No, the planet doesn't just need to be 2m. It can be larger. It's just what I started with as an assumption. I wanted a planet that wouldn't take you but a few minutes to walk around it's entire circumference (...lol... that would have made a really small and uninteresting setting for a sci-fi novel anyway.). The whole idea was, you can clearly see that you are 'up-side down' relative to your initial position, at some point, but, since the gravity was the same as the earth, you probably wouldn't be able to tell the difference physically - just like here on earth, no one feels 'upside-down'. Earth is too large for anyone to realize that they are walking on a curved surface. I wanted a planet where you could at least see that you were on a sphere.
  10. Dec 10, 2008 #9


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    If you're talking about "Brian Greene" style "anything can happen in QM", then I hope you don't think that is what I was implying. If you think that you can ignore QM, I think you are mistaken. It is quite possible that you need to consider degeneracy pressure. Furthermore, I am not aware of an explanation for the stability of "ordinary matter" that does not rely on QM. In other words, without QM, E&M says that matter, as composed of electrons and charged nuclei, is inherently quite unstable. For simplicity, we can just start with the hydrogen atom and discuss. In the meantime, let's not disparage QM.

    I would also be one of those people. However, did you actually calculate the probability. Eh ... eh? (another joke ... sorry)

    What percentage of the universe, by mass, volume, or the measure of your choice, do you think that mainstream science claims to be composed of such "periodic table" matter? Is the rest of the universe "unnatural"?

    I didn't say "made up". "exotic" is a bit ambiguous, and I suppose you could argue that it does connote "made up", but then, so is the process that constently explains the why that our Sun makes energy, and so is the way our Milky Way rotates, and on and on.

    I understand probably more than you realize.

    Yuk. I just can't let this one go. There is no predictable condition of the universe. In fact, cosmology is sort of like "reverse prediction".

    Not only could you "see" it, you could do experiments to acutally quantify it on such a small planet. I'll give you two examples:

    Drop an object from height h above the surface and observe how long it takes to hit the surface. Then, drop the same object from 2h. The result should be "wierd" by Earthly standards.

    Construct a flat table 1 meter long. Measure the weight of an object at different positions on the table. The result should be "wierd" by Earthly standards.
  11. Dec 10, 2008 #10


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    Tides. I would be interested in knowing what the gravitational force is at one's head height.
  12. Dec 10, 2008 #11
    Well, at r=2 meters it would be GM/(2 meters)^2 = 9.8 m/s^2, so at GM/(4 meters)^2 = 2.45 m/s^2.... (for a 2 meter high person)

    So that's a factor of 4 in g from a person's head to their feet. I'm not sure what type of change a human can take, but that's quite a lot.

    Also, if you calculate the force due to pressure on this thing, the inward gravitational force is not anywhere near equillibrium to the pressure differential pointing out (for any pressure/density relation known to man), that's the main reason this scenario is highly unlikely.

  13. Dec 10, 2008 #12


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    Well, humans can take 9.8m/s^2 with ease and they can take 0m/s^2 with ease, so it's not outside a comfortable range.

    It's not like these are opposing forces acting to pull the body apart...

    All it means is that your feet would be planted on the ground but your head would weigh only about 2 lbs.
  14. Dec 10, 2008 #13


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    One thing that would be weird is lifting your feet. They would have a noticeable inexplicable pull toward each other. Sticking you arm out to your side would feel more like it's pointed up at an angle.

    You'd be standing in a gravitational field that's conical in shape.
  15. Dec 10, 2008 #14
    I don't think it's that easy... in both those situations, the pull on your body is equally distributed. Thinking about circulation for instance, the effort that your body has to use to get blood to your feet and back, is suddenly much different than the force you need to push blood to your head and back. Who knows what affect that would have on physiology.

  16. Dec 11, 2008 #15


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    We should put a rat in a centrifuge to find out.
  17. Dec 11, 2008 #16


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    I can see there might be minor side effects, yes, but the poster was wondering "how much a human body can take", which, to me, sounded like they were thinking of the differences as if they were forces pulling on the body in different directions.
  18. Dec 11, 2008 #17

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    Talk about light-headed!

    Well, humans can take 9.8m/s^2 with ease and they can take 0m/s^2 with ease, so it's not outside a comfortable range.[/QUOTE]
    While humans can take 1g and 0g with ease separately, whether they can take a 75% toe-to-head gradient is a different issue. A couple of studies (1 and 2) indicate artificial gravity gradient should not exceed 8% per meter ("optimal"). A later study3 gives a significantly greater gradient, 25% per meter, as tolerable. This 75% gradient over two meters exceeds the tolerable limit by quite a bit.

    1R R Gilruth, 1969, " Manned Space Stations - Gateway to our Future in Space", in Manned Laboratories in Space, ed. S F Singer, Springer-Verlag.

    2T J Gordon and R L Gervais, 1969, " Critical Engineering Problems of Space Stations", in Manned Laboratories in Space, ed. S F Singer, Springer-Verlag.

    3R W Stone, 1973, " An Overview of Artificial Gravity", in Fifth Symposium on the Role of the Vestibular Organs in Space Exploration, NASA Scientific and Technical Information Division.
  19. Dec 11, 2008 #18
    While humans can take 1g and 0g with ease separately, whether they can take a 75% toe-to-head gradient is a different issue. A couple of studies (1 and 2) indicate artificial gravity gradient should not exceed 8% per meter ("optimal"). A later study3 gives a significantly greater gradient, 25% per meter, as tolerable. This 75% gradient over two meters exceeds the tolerable limit by quite a bit.

    1R R Gilruth, 1969, " Manned Space Stations - Gateway to our Future in Space", in Manned Laboratories in Space, ed. S F Singer, Springer-Verlag.

    2T J Gordon and R L Gervais, 1969, " Critical Engineering Problems of Space Stations", in Manned Laboratories in Space, ed. S F Singer, Springer-Verlag.

    3R W Stone, 1973, " An Overview of Artificial Gravity", in Fifth Symposium on the Role of the Vestibular Organs in Space Exploration, NASA Scientific and Technical Information Division.[/QUOTE]

    Thanks! This is very interesting!
  20. Dec 11, 2008 #19


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    They may want to spend a lot of time reclining and rolling over then.

    Golf won't likely be that popular a sport as like say luging.
  21. Dec 11, 2008 #20


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    I'd have to read the articles to know for sure, but I'll bet they're talking about opposing g's, where the head is pulled in one direction and the feet in the opposite.

    Also, I assume they're creating artificial g's via rotation (afaik, there's no other way to do it), so I wonder how much the rotational factor plays into the results. I doubt the experiments had in mind a practical application with high gradient but no rotation, since it is only possible in principle (i.e. as in the OP's dense planetesimal setup).
    Last edited: Dec 11, 2008
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