Gravity Defied (from potato asteroids to magnetised neutron stars) II : The failed stars

TL;DR

This paper explores objects that nearly became stars, focusing on the balance of gravity and other forces that prevent their collapse into true stars, highlighting the delicate conditions needed for stellar formation.

Contribution

It provides a detailed analysis of near-star objects, emphasizing the physical forces that inhibit their full transition into stars, expanding understanding of stellar formation thresholds.

Findings

Identifies key physical conditions preventing star formation

Analyzes the role of magnetic and other forces in stabilizing near-star objects

Highlights the continuum between non-stellar and stellar objects

Abstract

Gravitation, the universal attractive force, acts upon all matter (and radiation) relentlessly. Stable extended structures can exist only when gravity is held off by other forces of nature. This series of articles explores this interplay, looking at objects that just missed being stars in this particular instalment.

Equations

$$F_{\mathrm{G}}=-\frac{GM(r)dm}{r^2}=-\frac{GM(r)\rho (r)drdA}{r^2},$$

$$dP=-\frac{GM(r)\rho (r)dr}{r^2}\Rightarrow \frac{dP}{dr}=-\frac{GM(r)\rho (r)}{r^2}.$$

$$\Delta E_{\mathrm{G}}=-GM\left(\frac{1}{R_1}-\frac{1}{R_2}\right),$$

$${\lambda }_{\mathrm{deBroglie}}\simeq \frac{h}{\sqrt{2\pi m{k}_{B}T}},$$

$$f(E)=\frac{1}{e^{\frac{E-E_{\mathrm{F}}}{k_{\mathrm{B}}T}}+1}$$

$$n(p)dp$$

$$P=\frac{m}{3}{\int }_0^{\infty }n(v)v^{2}dv={\int }_0^{p_{\mathrm{F}}}\frac{p^2}{3m}n(p)dp={\left(\frac{3}{8\pi }\right)}^{\frac{2}{3}}\frac{h^2}{5m}{N}^{\frac{5}{3}},$$

$$N={\int }_0^{p_{\mathrm{F}}}n(p)dp=\frac{8\pi }{3h^3}{p}_{\mathrm{F}}^3.$$

$$P_{\mathrm{G}}\simeq \frac{G{M}^2}{R^4}.$$

$$\frac{M^2}{R^4}\propto N^{5/3}\Rightarrow MR3=\text{mbox}constant,$$

Full text

\authorIntro

Sushan Konar works at NCRA-TIFR, Pune. She tries to understand the physics of stellar compact objects (white dwarfs, neutron stars) for her livelihood and writes a blog about life in academia as a hobby.

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January 2017 \artNatureGENERAL ARTICLE

1 Self-Gravitating Fluids

The first resistance that nature puts up against gravitation is the rigidity of solids. We have seen how small rocky objects and terrestrial planets take recourse to this to retain their extended structures against gravitational pull [1]. The situation is entirely different in largely fluid (gas/liquid) Jovian planets. A fluid does not have rigidity. Instead, an element of fluid would be in hydrostatic equilibrium (at rest or at constant velocity) when external forces such as gravity are balanced by a pressure gradient. In other words, in a self-gravitating object, gravity pulling the material inwards is balanced by the internal pressure (due to heat or quantum effects) differential pushing the material outwards (Fig. 1).

1.1 Hydrostatic Equilibrium

Consider the fluid element in a cylindrical region of length $dr$, area $dA$, at a distance $r$ from the centre of a self-gravitating object having a density $\rho(r)$. Then, the volume of the cylinder is $dr\,dA$ and its mass is $dm=\rho(r)\,dr\,dA$. The force of gravity acting on the cylinder is,

[TABLE]

where $M(r)$ is the total mass contained within the radius $r$. This gravitational force is balanced by the net difference in pressure, $dP$, from above and below (Fig. 2). In equilibrium the effective pressure force should be equal to the gravitational force, giving us

[TABLE]

This equation of hydrostatic equilibrium governs the structure of all self-gravitating fluid bodies, including those of most stars. This simple form is, of course, modified when relativistic effects become important which we shall consider later.

1.2 Gas Pressure

1.2.1 Kelvin-Helmholtz Mechanism

The Kelvin-Helmholtz mechanism can be thought of as a dynamic process through which a self-gravitating body attains successive phases of hydrostatic equilibrium. To begin with, a gas cloud contracts under the mutual gravitational attraction of its constituent particles. If a spherical gas cloud of mass $M$ and initial radius $R_{1}$ contracts to a final radius of $R_{2}$ then the net change $\Delta E_{\rm G}$ in its gravitational binding energy is,

[TABLE]

where $G$ is the universal gravitational constant. Evidently, as a result of contraction an amount of energy is released. This goes into increasing the random velocities of the constituent particles raising the average temperature of the object. (This is why gravitationally bound objects are known to have ‘negative heat capacities’.) The increased temperature increases the gas pressure and the system can find a configuration where the inward gravitational pressure is balanced by the outward gas pressure, i.e. a state of hydrostatic equilibrium.

However, the increased temperature gives rise to enhanced radiative luminosity ($L\propto T^{4}$, $L$ - surface luminosity, $T$ - surface temperature) from the surface which is a process through which the object starts to lose the heat gained from gravitational contraction. Evidently, this reduces the temperature and the resultant gas pressure; eventually reaching a point when the gas pressure can no longer support the gravity. This results in an instability inducing further contraction of the object and a repeat of the whole cycle itself.

[Kelvin and Helmholtz proposed this process (late nineteenth century) to explain the energy source of the Sun. However, the total energy that can be made available through this process (E_{\rm G}^{\rm total}\sim G\mbox{{\rm M}{}{\odot}}^{2}/\mbox{{\rm R}{}{\odot}}) would last only about a few million years at the present rate of radiation from the Sun whereas the fossil records indicate that the Sun has been shining at its present rate for a few billion years at least. It was shown much later (1930s) by Hans Bethe that the source of Sun’s energy is nuclear but that is a different story altogether.]

1.2.2 Degeneracy Pressure

As a self-gravitating object continues through phases of repeated contractions its density keeps rising. When the density is such that the inter-particle spacing ($\propto n^{-1/3}$, $n$ - number density of particles) becomes comparable to the thermal de Broglie wavelength, given by

[TABLE]

(where $m$ is mass of the particles, $k_{B}$ is the Boltzmann constant and $h$ is the Planck’s constant) then quantum effects start having an increasingly important role in the behaviour of the fluid. It is then that a new agent for resisting the gravity is found in the form of quantum effects \mfnoteOnset of quantum effect : $n^{-1/3}\simeq\frac{h}{\sqrt{2\pi mk_{B}T}}$.

When quantum effects become important, Fermions (particles with half-integral spin following Fermi-Dirac statistics) follow a special energy distribution given by,

[TABLE]

where $E_{\rm F}$ is known as the Fermi energy. In particular, at absolute zero of temperature, Fermions fill up all the available energy states below $E_{\rm F}$ with one (and only one) particle in each whereas the energy levels above $E_{\rm F}$ have zero occupation in them (Fig. 3). Such a zero temperature Fermionic system is commonly known as a degenerate system.

The Fermi temperature, defined as $T_{\rm F}=E_{\rm F}/k_{B}$, provides an easy marker for the behaviour of a Fermionic system. When the physical temperature $T$ of the system is much smaller than the Fermi temperature ($T<<T_{\rm F}$) the behaviour of the system can be approximated to that of a system at (or near) absolute zero of temperature. Interestingly, quite a few astrophysical objects (Jovian planets, brown dwarfs, white dwarfs and neutron stars) fulfill this criterion and are therefore can be treated as zero temperature Fermion systems.

For a non-relativistic, zero temperature Fermion system, the density of particles (with a specific momentum $p$) is given by,

[TABLE]

where $p_{\rm F}$ is the momentum corresponding to $E_{\rm F}$. The pressure exerted by such a system is then,

[TABLE]

where $v(=p/m)$ is the velocity. And $N$ is the total number density of particles given by

[TABLE]

This rather important result for non-relativistic degenerate Fermi systems has a special significance for self gravitating objects. Consider a self-gravitating object of mass $M$ and radius $R$ composed of a degenerate Fermi gas. From simple dimensional estimates we can find that the central gravitational pressure of such an object would be,

[TABLE]

If this is to be balanced by the degeneracy pressure of the Fermi gas then it implies that,

[TABLE]

(because $N$, the number density, scales as $M/R^{3}$) giving us one of the very important relations of planetary and stellar astrophysics. [Readers are advised to look up standard Statistical Mechanics texts [2, 3] for details of Fermion systems.]

2 The Jovian Planets

Jupiter, named after the king of ancient Roman gods, is the largest planet of our solar system and is thought to consist of a dense (rocky/icy?) core with a mixture of elements, a surrounding layer of liquid metallic Hydrogen with some Helium, and an outer layer predominantly of molecular Hydrogen. Evidently, the metallic Hydrogen is largest component of this planet (Fig. 4) and therefore plays the most significant role in the structure and stability of Jupiter. Saturn, the next largest planet, is also expected to have similar interior composition as Jupiter.

Both Jupiter and Saturn indicate the presence of a positive temperature gradient from the surface to the interior. Conceivably, the higher interior temperatures allow for a transition of the molecular Hydrogen (found in the outer layers) to a metallic phase. This metallic state is expected to be quite similar to regular terrestrial metals in which the electrons form a degenerate Fermi gas. It is expected that at the end of their Kelvin-Helmholtz phase of contraction both Jupiter and Saturn would settle down into a Hydrogen-degenerate phase where the resistance to the gravitational pressure would be supplied by the degeneracy pressure of the electrons of the metallic Hydrogen (it needs to be noted that the pressure would also have a small but non-zero contribution from the Hydrogen nuclei).

However, observations indicate that both Jupiter and Saturn radiate away more energy than they receive from the Sun. This clearly suggests that both of these planets are still in their Kelvin-Helmholtz phase of contraction and heat generation. Interestingly, the Kelvin-Helmholtz process appears not to account for the entire amount of radiation from Saturn and suggests that it may have another source of power other than simple gravitational contraction.

Brown Dwarfs

A star gets its energy from nuclear burning, combining four Hydrogen (H) atoms into one Helium (He4) atom, for most of its life. The mass of He4 happens to be slightly smaller than the mass of the four Hydrogen atoms, giving rise to a small mass defect (meaning, a deficit). It is this defect that shows up as energy through the famous mass-energy equivalence relation of Einstein ($E=\Delta M\,c^{2}$) (Fig. 5).

Now the conversion of H into He4 actually happens in a number of steps, each requiring higher and higher ambient temperature for it to progress (Fig. 6). When a self-gravitating object undergoes repeated phases of Kelvin-Helmholtz contraction, the final temperature reached depends on its mass. The higher the mass, the higher is the final temperature. Though quite heavy by planetary standards, the Jovian planets fall far short of the magic temperature (about a million degree Kelvin) to begin Hydrogen fusion. Between the Jupiters and the stars there exist another class of objects, the brown dwarfs, which are really failed stars. They do achieve nuclear fusion but not of Hydrogen. Instead, brown dwarfs are known to burn Deuterium (D2, a heavier isotope of Hydrogen).

In Deuterium fusion, a Deuterium nucleus and a proton combine to form a Helium-3 (He3) nucleus. This occurs at the second stage of the $p–p$ chain of the Hydrogen fusion, but can also proceed from primordial Deuterium. This onset of Deuterium fusion separates the class of brown dwarfs from that of the Jovian planets giving the maximum mass of gas giants to be about 0.012 M*⊙* (or about 13 Jupiter masses). Whereas, the class of brown dwarfs is separated from the class of stars by the threshold for Hydrogen fusion at 0.08 M*⊙*. Once the Deuterium burning gets over a brown dwarf continues to cool and ultimately the pressure support comes from the electron degeneracy pressure of H and He3.

A particularly important characteristic of brown dwarfs is the presence of primordial Lithium in them. Stars, achieving Hydrogen fusion temperature rapidly deplete their Lithium. On the other hand, brown dwarfs (even massive brown dwarfs) never reach this temperature and therefore mostly retain their primordial Lithium. The presence of Lithium is therefore used to distinguish candidate brown dwarfs.

Acknowledgment

My thanks to G. Srinivasan for setting me on the path of Fermi-degenerate objects by explaining that the study of stellar evolution actually begins with Jupiter (among other such wonderful concepts that we’d encounter in the next installments of this series) and to D. Narasimha for introducing me to the physics of stars one exciting summer thirty years ago. I would also like to express my deep gratitude to the Resonance team for their patience and perseverance in getting this article published after I miserably failed all possible deadlines.