# Why There Are Most Mass Limits for Compact Objects

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On this article, we are going to take a look at why there are most mass limits for objects which are supported towards gravity by degeneracy stress as a substitute of kinetic stress. We are going to take a look at the 2 recognized instances of this, white dwarfs and neutron stars; but it surely must be famous that related arguments will apply to *any* postulated object that meets the overall definition given above. For instance, the identical arguments would apply to “quark stars” or “quark-gluon plasma objects”, and so forth.

## The Chandrasekhar Restrict

First, we’ll take a look at the utmost mass restrict for white dwarfs, the Chandrasekhar restrict. (Observe that the primary derivation we are going to give beneath, utilizing the TOV equation, is a simplified model of the argument given in Shapiro & Teukolsky, who use numerical integration of the Lane-Emden equation. For this text we will likely be happy with a heuristic argument utilizing averages and gained’t have to go to that excessive.)

We begin with the overall relativistic equation for hydrostatic equilibrium for a static, spherically symmetric object (we is not going to take into account the rotation of neutron stars right here; that complicates the mathematics and adjustments the numerical worth of the utmost mass restrict, but it surely doesn’t take away it). That is the Tolman-Oppenheimer-Volkoff equation, which we are going to write in a type considerably completely different from the one by which it normally seems. Observe that we’re utilizing models by which ##G = c = 1##.

$$

frac{dp}{dr} = – rho frac{m}{r^2} left( 1 + frac{p}{rho} proper) left( 1 + frac{4 pi r^3 p}{m} proper) left( 1 – frac{2m}{r} proper)^{-1}

$$

This type of the equation makes it simpler to see that what we’ve right here is the Newtonian (non-relativistic) equation for hydrostatic equilibrium, with some relativistic correction components. For white dwarfs, nevertheless, it seems that we are able to ignore all of these correction components and simply take a look at the non-relativistic system for hydrostatic equilibrium. It’s because the radius of white dwarfs is far bigger than their mass in geometric models, so ##r >> 2m## and the final issue on the RHS above might be taken to be ##1##, and their stress is at all times too small to make the correction phrases within the different two components important, so these components may also be taken to be ##1##.

We now make use of the truth that, for degenerate matter, we’ve ##p = Okay rho^Gamma##, the place ##Okay## is a continuing that is dependent upon whether or not the degeneracy is non-relativistic or relativistic, so we’ll designate its two values as ##K_text{n}## and ##K_text{r}## (we are going to solely take into account the 2 extremes and won’t take a look at the transition between them), and ##Gamma## is the “adiabatic index”, which is ##5/3## within the non-relativistic restrict and ##4/3## within the relativistic restrict. This offers ##dp / dr = Okay Gamma rho^{Gamma – 1} d rho / dr##. Lastly, we make use of the truth that, for a static, spherically symmetric object, ##dm / dr = 4 pi rho r^2##, to place issues by way of derivatives of ##m##.

We plug all this into the non-relativistic hydrostatic equilibrium equation to acquire:

$$

frac{d}{dr} left( frac{1}{4 pi r^2} frac{dm}{dr} proper) = – frac{1}{Okay Gamma} left( frac{1}{4 pi r^2} frac{dm}{dr} proper)^{2 – Gamma} frac{m}{r^2}

$$

Increasing and simplifying provides:

$$

frac{d^2 m}{dr^2} – frac{2}{r} frac{dm}{dr} + frac{left( 4 pi proper)^{Gamma – 1}}{Okay Gamma} left( frac{1}{r^2} frac{dm}{dr} proper)^{2 – Gamma} m = 0

$$

Quite than attempt to clear up this nasty differential equation instantly, we will likely be happy right here with making tough order of magnitude estimates. For this goal, we outline ##M## as the full mass of the white dwarf and ##R## as its floor radius, and we approximate ##dm / dr## with its common, ##M / R##, and ##d^2 m / dr^2## with ##M / R^2##. Substituting these into the above equation provides, after simplifying:

$$

M^{2 – Gamma} = frac{Okay Gamma}{left( 4 pi proper)^{Gamma – 1}} R^{4 – 3 Gamma}

$$

Now we’re ready to have a look at our two regimes. Within the non-relativistic regime, ##Gamma = 5/3## and we’ve:

$$

M^{1/3} = frac{5}{3} frac{K_text{n}}{left( 4 pi proper)^{2/3}} frac{1}{R}

$$

Inverting this tells us that, as ##M## will increase, ##R## decreases because the dice root of ##M##. In different phrases, because the white dwarf will get extra large, it will get extra compact. And because it will get extra compact, its density and stress enhance and it turns into relativistic. So so as to assess whether or not there’s a most mass restrict, we have to take a look at the relativistic regime. Right here, ##Gamma = 4/3## and we’ve:

$$

M^{2/3} = frac{4}{3} frac{K_text{r}}{left( 4 pi proper)^{1/3}}

$$

Observe that now, ##R## *doesn’t seem in any respect* within the equation! It’s simply an equation for ##M## by way of recognized constants. In different phrases, within the ultra-relativistic restrict, ##M## approaches a continuing limiting worth *and can’t exceed it*. That worth is the Chandrasekhar restrict. (Observe that, to get the precise numerical worth for the restrict that’s utilized by astrophysicists, which is 1.4 photo voltaic plenty, the tough order of magnitude calculation we’ve achieved right here just isn’t sufficient, however we gained’t go into additional particulars about how that worth is definitely calculated right here. Our goal right here is just to see, heuristically, why there have to be a mass restrict in any respect.)

Let’s take a step again now and attempt to perceive what’s going on right here. A method of it’s to ask the query: what’s the white dwarf’s power of gravity as a operate of density? By “power of gravity” right here we imply, heuristically, the inward pull that have to be balanced by the outward drive of stress so as to preserve hydrostatic equilibrium. It seems that this will increase with density as ##rho^{4/3}##. So we should always count on that in *any* state of affairs by which ##Gamma to 4/3##, there will likely be a most mass restrict as a result of stress can now not proceed to extend quicker than gravity. And we are able to see from the above {that a} relativistically degenerate electron fuel, as in a white dwarf, is one such state of affairs. (One other seems to be a supermassive star supported by radiation stress; because the mass will increase, the efficient ##Gamma## for radiation stress turns into relativistic and we’ve the identical qualitative state of affairs as a white dwarf, although after all with completely different constants so the precise numerical worth of the mass restrict is completely different.)

This argument concerning the power of gravity can in truth be made mathematically. As Shapiro and Teukolsky be aware, the primary physicist to do that was Landau, in 1932, who got here up with an alternate method of understanding Chandrasekhar’s outcome, printed the yr earlier than, on the utmost mass of white dwarfs. Landau’s argument is easy: first, we discover an expression for the full vitality ##E## (excluding relaxation mass vitality) of an object that’s supported by degeneracy stress; then we glance to see below what situations ##E## can have a minimal, which signifies a secure equilibrium.

The whole vitality has two parts: the (constructive) vitality of the fermions as a result of degeneracy stress, and the (damaging) gravitational potential vitality as a result of fermions all pulling on one another. The vitality as a result of degeneracy stress is the Fermi vitality ##E_F## per fermion, and we are able to use the Newtonian system for the gravitational potential vitality per fermion since we noticed above that the relativistic corrections to the TOV equation, that are of the identical order of magnitude because the relativistic corrections to the gravitational potential, are negligible. The whole vitality per fermion, due to this fact, seems to be like this (I’m penning this in a barely completely different from that utilized in Shapiro & Teukolsky, for simpler comparability to the derivation given above):

$$

E = frac{hbar M^{1/3}}{mu_B^{1/3} R} – frac{M mu_B}{R}

$$

the place ##mu_B## is the baryon mass that’s related to the fermions offering the degeneracy stress; this would be the common of the proton and neutron mass in a typical white dwarf, since every electron is related to one proton and the proton-neutron ratio is roughly ##1##. (In a neutron star ##mu_B## would simply be the neutron mass.)

For there to be a secure equilibrium at a given worth of ##M##, there have to be a minimal of ##E## at a finite worth of ##R##. This may happen if we’ve ##dE / dR = 0## at a finite worth of ##R##. Since each phrases in ##E## scale as ##1 / R##, the expression for ##dE / dR## is easy:

$$

frac{dE}{dR} = – frac{1}{R^2} left( frac{hbar M^{1/3}}{mu_B^{1/3}} – M mu_B proper)

$$

Now we take a look at how ##dE / dR## varies with ##M##. If ##M## is small, the issue contained in the parentheses will likely be constructive, so ##dE / dR## will likely be damaging and ##E## will lower with growing ##R##. That can make the starless relativistic and finally nonrelativistic. As soon as the star turns into nonrelativistic, the radial dependence of the Fermi vitality will change; it’s going to scale as ##1 / R^2## as a substitute of ##1 / R##. Which means that the gravitational potential vitality will, at some worth of ##R##, turn out to be bigger than the Fermi vitality, and that can trigger the signal of ##dE / dR## to flip from damaging to constructive for the reason that gravitational potential vitality will increase with growing ##R## (to a limiting worth of ##0## as ##R to infty##). The finite worth of ##R## the place the signal flip happens will likely be a minimal of ##E## and due to this fact a secure equilibrium.

Nonetheless, if ##M## is giant, the issue contained in the parentheses will likely be damaging, so ##dE / dR## will likely be constructive. In that case, ##E## might be decreased with out certain by reducing ##R##; each phrases scale the identical method with ##R## and reducing ##R## makes the star extra relativistic so the radial dependence of the Fermi vitality is not going to change. Meaning there is no such thing as a secure equilibrium; the star will collapse.

The boundary between these two regimes will happen on the worth of ##M## at which the issue contained in the parentheses above is zero, and that would be the most doable mass, which will likely be given by:

$$

M^{2/3} = frac{hbar}{mu_B^{4/3}}

$$

Evaluating this with the heuristic system above provides not less than a tough order of magnitude estimate for the fixed ##K_r##. Observe, nevertheless, that this system will likely be formally the identical for a white dwarf and a neutron star; in truth, it is going to be the identical for *any* object that’s supported by degeneracy stress, since we made no assumptions that had been particular to a selected sort of object. The one distinction between various kinds of objects will likely be a distinct worth of ##mu_B## based mostly on chemical composition. This system itself is heuristic, and there grow to be different numerical components concerned; nevertheless, it’s going to certainly end up that the suggestion implied by the above system, that the utmost mass of a neutron star just isn’t that completely different from the utmost mass of a white dwarf, is mainly appropriate.

After all, as we famous earlier, to truly calculate the generally recognized numerical worth of the Chandrasekhar restrict for white dwarfs, the above formulation aren’t sufficient; we must do extra sophisticated numerical calculations. Chandrasekhar did these calculations when he initially printed his derivation of the restrict that got here to be named after him, in 1934; and subsequent calculations haven’t made any important adjustments to the worth he obtained. Nonetheless, the numerical worth *does* rely considerably on the chemical composition of the white dwarf. By way of the primary system above, the chemical composition can have an effect on the worth of ##K_r##; by way of the second, it will probably have an effect on the worth of ##mu_B## in keeping with the fraction of baryons which are protons. Chandrasekhar’s worth assumed that the chemical composition of the white dwarf was principally hydrogen and helium, and that’s the foundation for the generally used worth of 1.4 photo voltaic plenty for his restrict. Nonetheless, in a while within the Fifties, when Harrison, Wakano, and Wheeler had been deriving a common equation of state for chilly matter, they used a distinct chemical composition for white dwarfs, one which was considerably richer in neutrons, and obtained a price of 1.2 photo voltaic plenty. So when values within the literature for white dwarf most mass limits, one has to remember to verify the chemical composition that’s being assumed.

## The Tolman-Oppenheimer-Volkoff Restrict

In 1938, Tolman, Oppenheimer, and Volkoff investigated the query of most mass limits for neutron stars. It was in the midst of these investigations that they derived the relativistic equation for hydrostatic equilibrium that we noticed within the earlier article, and which is called after them. They went via a derivation much like the one Chandrasekhar had achieved for white dwarfs and got here up with an analogous outcome: there’s a most mass restrict for neutron stars. By way of the above formulation, the one change can be a distinct worth of ##K_r## within the first system, or ##mu_B## within the second, to account for the change in the kind of fermions, from electrons to neutrons, and the truth that the identical fermions now account for each the mass and the degeneracy stress (whereas in a white dwarf, the electrons account for the degeneracy stress whereas the baryons account for the mass).

The fascinating half was that the numerical worth of the mass restrict that they obtained for neutron stars was 0.7 photo voltaic plenty–i.e., *smaller* than the white dwarf restrict that Chandrasekhar had calculated! The explanation for this, by way of the formulation we checked out within the final article, is easy: along with the adjustments talked about above, Oppenheimer and Volkoff didn’t assume that the relativistic correction components within the TOV equation had been negligible, as we did within the earlier article. They included these components, and for neutron stars within the relativistic restrict, they don’t seem to be all negligible; the tip result’s to extend the RHS of the TOV equation by a numerical issue that finally ends up showing within the denominator of our formulation for the utmost mass and thus reduces the anticipated most mass by about half.

On the time, this was not essentially a serious difficulty, since no neutron stars had been noticed; however now we all know of many neutron stars which are considerably extra large, so we all know one thing have to be mistaken with the unique TOV calculation. However even on the time, Tolman, Oppenheimer, and Volkoff had good motive to not take that quantity at face worth. Why? As a result of, regardless that not quite a bit was recognized concerning the robust nuclear drive on the time, it was evident that, at brief sufficient distances, smaller than the scale of an atomic nucleus, that drive should turn out to be strongly repulsive; in any other case, atomic nuclei wouldn’t be secure on the measurement ranges they had been recognized to have.

This issues as a result of the derivations we went via above made an necessary assumption that we didn’t point out earlier than: that the fermions in query didn’t work together with one another in any respect, besides via the Pauli exclusion precept. If we add an interplay that’s repulsive at brief ranges, that adjustments issues. Within the first derivation within the earlier article, the impact is to extend ##Gamma##, the adiabatic index, above the traditional worth it could have as a result of degeneracy and the Pauli exclusion precept alone. Within the second derivation, the impact is so as to add one other constructive time period within the vitality as a result of repulsive interplay.

On the face of it, this would appear to point that the 2 derivations will now give us completely different solutions! Rising ##Gamma## ought to imply that the primary derivation now seems to be extra like its nonrelativistic type, which does *not* result in a most mass. Nonetheless, including a constructive vitality time period within the second derivation doesn’t change the general logic resulting in a most mass so long as that vitality scales as ##1 / R##, which we’d count on it to do. The impact will simply be to extend the numerical worth of the utmost mass that we calculate.

The decision of this obvious contradiction between the 2 derivations is that, within the neutron star case, the “important” worth of ##Gamma##, at which the star turns into unstable, is now not ##4/3##, because it was for white dwarfs; that’s solely a limiting worth within the absence of different interactions. Within the presence of different interactions, the important worth of ##Gamma## will increase, to the purpose the place even the bigger precise worth of ##Gamma## as a result of repulsive interactions continues to be lower than the important worth of ##Gamma## within the relativistic restrict. And meaning the identical logic as earlier than nonetheless goes via within the first derivation for neutron stars: within the relativistic restrict, ##Gamma## reaches a important worth at which the mass turns into unbiased of radius and there’s a most mass.

Why should the worth of ##Gamma## within the first derivation for neutron stars at all times find yourself lower than the important worth? The reply to this comes from a restrict that relativity imposes on the equation of state of any form of matter: that the pace of sound within the matter can not exceed the pace of sunshine. The pace of sound is given by ##v_s^2 = dp / drho##, and we are able to see that, if ##p = Okay rho^Gamma##, the restrict ##dp / drho le 1## will drive ##Gamma## to lower because the star turns into increasingly large and increasingly compressed and ##rho## due to this fact will increase. So there is no such thing as a method for the mass to extend indefinitely.

As we famous above, our conclusions right here, whereas they need to be common and apply to any equation of state, solely give a tough order of magnitude estimates of numerical values. Physicists have achieved extra detailed calculations utilizing varied equations of state for neutron star matter and have confirmed the existence of most mass limits for all of them, with values starting from about 1.5 to about 2.7 photo voltaic plenty. Analyses of the conduct of the important worth of ##Gamma## have additionally been achieved utilizing varied fashions; in not less than one case, the idealized case of a neutron star with uniform density, the calculations might be achieved analytically, with out requiring numerical simulation, since closed type equations for this case are recognized. For this case, the important level at which the limiting worth of ##Gamma## imposed by the situation that the pace of sound can not exceed the pace of sunshine is the same as the important worth of ##Gamma## is on the level ##p = rho / 3##, which agrees with the prediction from an ideal fluid mannequin within the ultrarelativistic restrict (for instance, this is similar worth that applies to a “fuel” of photons). It’s noteworthy that, for this case, the worth of ##Gamma## akin to this restrict could be very giant, about ##3.5##. This confirms that even an especially stiff equation of state just isn’t ample to withstand compression indefinitely within the relativistic restrict.

Trendy observations have discovered that the overwhelming majority of neutron stars we observe are pulsars, quickly rotating, and speedy rotation invalidates the calculations we’ve been making right here since we assumed a static, spherically symmetric object. We might intuitively count on that rotation would compensate considerably for elevated gravity and due to this fact would possibly enhance the utmost mass restrict, and certainly it seems to; we’ve noticed pulsars at shut to three photo voltaic plenty, and no trendy calculations for non-rotating neutron stars have indicated a restrict that enormous. Calculations for rotating neutron stars are extra sophisticated, however don’t change the fundamental conclusion: there’s nonetheless a most mass restrict, and it’s nonetheless essentially as a result of similar mechanism as above: that relativity locations final limits on the flexibility of degenerate matter to withstand compression. That could be a key motive why astrophysicists are extremely assured that darkish objects which are not directly detected by their gravitational results, and whose plenty are estimated to be a lot bigger than the utmost mass restrict for neutron stars, are black holes.

## A Closing Observe

As was talked about above, there are different, extra speculative configurations of degenerate matter proposed within the literature, comparable to “quark stars”, however all of them are topic to the identical common mechanism we’ve seen right here for max mass limits. As in comparison with neutron stars, these speculative configurations are simply adjustments in chemical composition, which might alter the equations by numerical components of order unity however can not change the fundamental conduct. So, though such speculative objects, in the event that they end up to exist, might need most mass limits considerably completely different from neutron stars, the variations will nonetheless be of order unity and won’t have an effect on the fundamental conclusion said above, that after we detect the oblique gravitational results of darkish objects with plenty a lot better than the neutron star mass restrict, these objects must be assumed to be black holes.

References:

Shapiro & Teukolsky, 1983, Sections 3.3, 3.4, 9.2, 9.3, 9.5, 9.6

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