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Neutron Stars Disappointment

  1. Apr 20, 2014 #1
    So, I was disappointed to find out that the surface of a neutron star is at 1 million kelvin, not 100 billion kelvin. I did some calculations a while back using the 100 billion K as a temperature (the core temperature of a neutron star), and found that it would be emitting more radiation than the sun (even 1000 l-y away), but mostly at energies that would likely not interact with particles except on a nuclear basis (though still looking at this one).

    So, imagine my disappointment when I found my error... Goodbye dreams of renewable nuclear energy (at least using neutron stars). On the plus side, glad we're not being vaporized by the relatively "lower" energy photons that such a star would put out... (I just took a look at the visible spectrum for such a star, and let me just say, could spell problems...)

    So, on a separate note, is there any other way of collecting nuclear renewables that doesn't involve hauling asteroids back to earth, or pressures beyond what are sustainable on earth (ie most nuclear fusion processes)? I figure, if we ever want to really start to move around the local cluster (or even our own solar system for that matter), we'll need to use nuclear energy to power that travel. Maybe this last part is better left for another forum, but it's an interesting line of thinking anyway.

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    Last edited: Apr 20, 2014
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  3. Apr 20, 2014 #2


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    When you say "the surface of a neutron star is at 1 million Kelvin", it really depends on the neutron star. Neutron stars do not produce energy, so they are just radiating their thermal energy away over time. The Neutron star starts at temperatures of billions of kelvin and rapidly cools to the millions of kelvin range due to the ridiculous amount of energy they are giving off, and then take much longer to cool down from there (since their rate of radiation is much much lower then).

    With regards to your statement "core temperature of neutron star" being 100 billion kelvin. From my intuition, it seems to me that a neutron star should be very well thermalized considering how dense it is and so should have (roughly at least) but 1 temperature throughout. As there is no energy generation in a neutron star, it's surface temperature should be roughly also its internal temperature. Certainly there should not be a 5 order of magnitude difference between the two.
  4. Apr 20, 2014 #3
    Possibly, I'm just going off Wikipedia on this one. It seems legit. When I said 1 million degrees, I meant order of magnitude. As far as I understand, there is a process going on in neutron stars that produces energy. While typical stars like our sun are fueled by fusion (a nuclear process that releases energy stored in the form of potential energy due to strong nuclear force fields), a neutron star undergoes a different reaction as I understand, unlocking potential energy found in changing the bonds of weak nuclear forces. This is a rudimentary understanding at best, but you can look it up for yourself in the wikipedia article.

    As for the temperature gradients, I doubt it has anything to do with the conductivity of the star. An increase in density does not necessarily mean better thermal conductivity. It probably has more to do with an increased rate of the weak force nuclear reaction I was talking about at the higher densities, and a lower rate of reaction at the lower densities found at the surface.

  5. Apr 20, 2014 #4


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    Quote from Wikipedia article: "The temperature inside a newly formed neutron star is from around 10^11 to 10^12 kelvin.[7] However, the huge number of neutrinos it emits carry away so much energy that the temperature falls within a few years to around 10^6 kelvin.[7] Even at 1 million kelvin, most of the light generated by a neutron star is in X-rays. In visible light, neutron stars probably radiate approximately the same energy in all parts of visible spectrum, and therefore appear white."

    This appears to validate my first point.

    In regards to my second point: 1. I can't find mention of energy generation inside a neutron star in that article (I only skimmed it), can you point me to the relevant quote? 2. At nuclear densities (~neutron star densities), they sound-crossing time of a neutron star is going to be on the order of milliseconds or lower (if I recall correctly), and this would correspond roughly to the thermalization time (provided no new energy is being generated, so I have to see your quote for that), so the neutron star should maintain roughly the same temperature throughout. If you have a source for your claim that the core of a neutron star is at 100 billion Kelvin, I'd very much like to see it.
  6. Apr 20, 2014 #5
    Interesting, yeah, I didn't read that part carefully enough (about the cooling of the neutron star). So, that's interesting.

    Also, seeing how neutronium (neutron star material?) is unlike anything testable on earth, I feel it is perhaps reasonable to assume it's highly conductive thermally? (Though, density is less important than electron bonds on earth when determining both thermal and electrical conductivity, am I right?)

    As for the energy generation, I'm still a little bit skeptical. Based on my back of the envelope calculations, on the order of 7x10^25 Watts of power are leaving the neutron star (equivalent to 800 billion kg/s), it has to be coming from somewhere. Is it also reasonable to assume that neutronium (is this the correct term or just made up?) is also supercapacitative in terms of thermal energy storage? I was under the impression, I think from a documentary I saw, that some sort of weak force interaction (maybe beta decay, is that exothermic?) was responsible for the energy released by neutron stars. That's the impression I was under. Am I confusing this with something else perhaps?

    (Oh, and the temperature INSIDE a neutron star implied to me that it was a core temperature, it's not clear if that's what they meant in the context, since they mention it in regards to cooling and then surface temperature. Just to clarify where my confusion came from, it says "inside." Also, 10^11 = 100 billion?)
    Last edited: Apr 20, 2014
  7. Apr 20, 2014 #6


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    Neutron star material is basically just a bunch of degenerate neutrons. The energy being radiated is their inherent thermal energy (gotten from gravitational collapse of the stellar core), which, for totally degenerate matter is ##\frac{3}{5}N\epsilon_F##, where ##\epsilon_F## is the Fermi energy and N is the total number of neutrons inside the neutron star. This is why neutron stars cool over time, they are losing their internal heat! I have not read any sources that say neutron stars have significant energy production inside them.

    What may be confusing you is that why do neutron stars stay at ~1 million Kelvin for a long period of time if they are constantly losing this much heat? (Your estimate for the energy loss is roughly correct) Well, the answer is that neutron stars contain a LOT of heat!

    Let's do an order of magnitude estimate:

    We can define the Fermi temperature (not the actual temperature of the neutron star!) to be:
    $$T_F=\frac{\epsilon_F}{k}$$ Where k is the Boltzmann constant. For degenerate matter the actual temperature is much much less than the Fermi temperature. So, let's just plug in the actual temperature for the Fermi temperature and get a lower bound (which will be many orders of magnitude lower) on the thermal energy inside a neutron star. Neutron masses are comparable to solar masses, so they are of order ##M\approx10^{30}kg##. Given the mass of a neutron is ~1 amu, then the number of neutrons inside a neutron star is ##N\approx\frac{10^{30}kg}{1 amu}\approx10^{57}##. The lower bound on the energy is then:


    So we know that the actual thermal energy contained inside the neutron star is many orders of magnitude greater than this: ##E>>E_{lb}##. As you can see then your radiative power is puny compared to this energy, and so neutron stars can stay at very high temperatures for very long periods of time.
  8. Apr 20, 2014 #7
    That explains it, essentially, they are supercapacitative (correct term?). And this whole time I thought they had a separate energy generation process. It sounds like this is analogous to the sun. Once I did a calculation of how much mass was leaving the sun as energy compared to the total mass of the sun. I don't remember the exact numbers, but it was losing a very small fraction of its total mass over time (but it was still a very astounding number). In this case, using your energy stored figure and my energy released figure, it would take 1.4 x 10^14 seconds (a lot) for it to lose all its energy radiating at a constant temperature (which it wouldn't as it lost energy since its temperature would drop, making it take even longer!)

    Thanks for the help and explanations.
  9. Apr 20, 2014 #8


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    Yes, and also remember that the energy I gave is a lower bound, probably lower by several orders of magnitude from the real thermal energy stored inside a neutron star.

    As an aside, I'd like to note that what we have neglected in our analysis is cooling due to neutrinos.In fact, hot neutron stars lose MOST of their energy through neutrino emissions and not through light emissions. The neutrinos are produced through possibly the URCA reactions, the modified URCA reaction, Nucleon pair bremsstrahlung, Neutrino pair bremsstrahlung, pionic reactions, or even quark beta decay, at low temperatures, though, photon losses will always take over (all of those interactions are strongly dependent on temperature), and our analysis is found to be quite valid for the temperatures of ~1 million Kelvin that we care about. In fact, the rate of neutron star cooling gives us a good bound on the types of reactions that CAN occur inside a neutron star. I should note that these reaction all take AWAY heat from the neutron star by producing energetic neutrinos which leave the star.

    When you were doing your calculation with the Sun, you were looking at the entire mass of the Sun as a potential energy source. This amount of energy is MASSIVE! Note; however, that the Sun will not convert a significant portion of its rest mass into energy. Fusion, though extremely efficient compared with chemical energy, is still extremely inefficient at converting rest mass into energy (<1%). In addition, the Sun will not burn through all of its hydrogen, as only its core is hot enough to accomplish fusion. In the end, the Sun will not lose an appreciable portion of its rest mass to energy. Order of magnitude estimate suggests the total mass that the Sun will lose to radiative energy over its lifetime is:

    $$\Delta M\approx\frac{(4*10^{26}W)*(10,000,000,000 yrs)}{c^2}\approx 10^{27}kg<<M_{sun}$$

    This is <.1% of the Sun's mass. A much more efficient way of converting rest mass into energy is achieved during a core-collapse supernova explosion, during which 5-10% (don't quote me on this number) of the rest mass of a stellar core will leave as neutrinos.
  10. Apr 24, 2014 #9
    When a neutron star is created it has about a trillion degree K. This decreases within a few minutes to about a billion K via exotic processes. After this it loses heat from radiation from the surface and also from neutrinos created while the core is becoming superfluid and superconductive. The later process begins after the temperature has decreased to about half a billion K. After a thousand to maybe ten thousand years the core is as superfluid and superconductive as it is going to get and the neutrinos cease. The core is maybe several hundred million K. The star cools fairly slowly after that because its surface area is so small.

    There is heat produced in the star due to viscosity and such, but not much. Neutron stars are from 1.4 to 2 solar masses and are much hotter, so they contain a great deal of heat. The rate of cooling can be observed via their radiation.

    The superfluid core is extremely conductive of heat so it is essentially at one temperature. The surface is at a million K or so. It does seem surprising that there is that much of a differential. I'm sure there are papers about this. The crust is made of heavy metal nuclei that are polymerized by the superstrong magnetic field. It is quite an exotic material so intuition is not of all that much use in predicting its properties.

    Neutronium is a science fiction thing. It's called degenerate matter.
    Last edited: Apr 24, 2014
  11. Apr 24, 2014 #10


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    How would you sustain a 100 million Kelvin temperature differential between the core and surface? How do you not get ridiculous heat flow to the surface with such a temperature differential?
  12. Apr 24, 2014 #11
    The only reason I even used the term neutronium is because I saw it elsewhere on the physics forums here. I wasn't sure whether it was an actual proper term or not. By the way, I also did some heat transfer analysis to see how quickly they would cool. They cool off extremely fast, as it turns out, even after the initial cool down period. I did a quick calculation of a neutron star cool down, assuming black body radiation (and some of matterwave's analysis shown above). As it turns out, the cooling scale for this thing to go from 1e6 K degrees to 1e5 degrees is in the hundreds of millions of years (a shorter time scale for anything astronomical). And, that's not including energy lost to neutrino emissions. Just goes to show that neutron stars are an enigma of their own.

    Also, is the core really that high of a temperature difference from the surface? Especially if the center material is superconductive? Anyway, we've already gone far beyond the scope of my science training, so I'll let you guys duke that one out.
  13. Apr 26, 2014 #12

    I don't know and I'm too busy to look it up.
  14. Apr 26, 2014 #13


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    Neutronium is right next to Unobtainium on the secret extended Periodic Table.
  15. Apr 27, 2014 #14


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    My guess is thermalization of degenerate matter is an open question. Our very own sun requires thousands of years to permit photons to emerge from core processes. My guess is the degenerate nature of neutron star mass inhibits such a process even more than that of our sun.
  16. Apr 29, 2014 #15


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    It might inhibit photons from diffusing, but how does it inhibit conduction? The sound-crossing time in a neutron star is of order microseconds. I would intuitively expect a neutron star to thermalize in same order-of-magnitude time scales.

    Of course, I'm not saying a large temperature differential is impossible, I'm just saying it seems very weird, and I don't know how to make it happen.
  17. Apr 30, 2014 #16
    I've given it a look. When the core is 10^8K then the surface is at 10^6K.

    The heat is conducted by electrons. The thermal conductance is inversely proportional to the "Fermi energy" and effective frequency of electron collisions. Since the density and energy of free electrons is extremely high, the frequency of collision is extremely high and the conductance is very low. Amazing! The electrons are so dense and energetic that they serve as an insulator.

    I got this from http://www.jinaweb.org/docs/thesis/thesis_Ouellette.pdf. Like I said, when it comes to neutron stars intuition usually does more harm than good.
  18. Apr 28, 2015 #17
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