The effect of gravity at large scales Dimensional analysis

As shown in the preceding section, the propagation of an acoustic wave with speed csis essentially a consequence of a balance between the fluid inertia and an elastic restoring force due to the fluid compressibility. Gravity plays a role only at length scales above c_s²/g, which is about 10 km for the Earth’s atmosphere and 200 km for the ocean. Thus the role of gravity is essential at the scale of astrophysical phenom-ena. In particular, it gives rise to an instability of an interstellar gas cloud which causes the cloud to collapse on itself, leading to star formation. This instability can be analyzed qualitatively as follows.

First we recall that a mass m produces a gravitational field G= Gm/r² at a distance r , where G= 6.67 × 10⁻¹¹m³/kg/s² is the gravitational constant, and

2 The isentropic compressibility of an ideal gas isκs= γ /P, where γ = cp/cvis the adiabatic index (the ratio of the heat capacity at constant pressure to the heat capacity at constant volume). Taking into account the Boyle–

Mariotte law P/ρ = rT , where r = R/M is the ratio of the ideal gas constant R = 8.314 J K⁻¹mol⁻¹and the molar mass M, one obtains c_s²= γ rT0. For air,γ = 1.4 and M = 29.0 g mol⁻¹.

that this gravitational field exerts on a mass m^ an attractive force m^G. Now let us take a large mass of gas at rest, for example, an interstellar gas cloud, and perturb its equilibrium by density and pressure nonuniformitiesδρ and δP on a spatial scale L. This leads to a nonuniformity of the gravitational field of order δG ∼ G δρL³/L²= G δρL. A fluid particle of typical size a is then subject to two competing forces, an elastic force and a gravitational force. The elastic force, which is related to the compressibility of the fluid, points from compressed regions to neighboring rarefied regions, and its order of magnitude is

Fa∼ (δP/L)a³∼ δρ c_s²a³/L.

The net gravitational force is directed toward the compressed region containing a mass excess of orderδρL³, and its order of magnitude is

Fg∼ ρa³δG ∼ ρa³Gδρ L.

The ratio of these two forces is then Fg/Fa=ρGL²/c²s. Thus, if the spatial scale of the nonuniformities is such that L cs/√

ρG, the elastic force dominates and the perturbation propagates as an acoustic wave, without amplification. Conversely, if L cs/√

ρG, the gravitational force dominates and the gas moves from rarefied regions to compressed ones, which enhances the initial density difference. Thus, the effect amplifies the cause, and an instability develops which destroys the initial uniformity of the cloud and causes it to collapse.³

The dimensional analysis carried out above therefore indicates a length scale Lc= cs

√ρG (2.10)

above which a mass of gas is susceptible to collapsing on itself. This length cor-responds (up to a numerical factor) to the length discovered by Jeans in 1902.

For the Earth’s atmosphere this length is about 36,000 km, which is much larger than the Earth’s radius: gravity does not enter into the propagation of sound waves, and so the Earth’s atmosphere is not at risk of collapse! For a low-density cloud of interstellar gas, where the parameters are typically cs= 200 m s⁻¹ and ρ =2×10⁻¹⁷kg m⁻³, the length Lcis about 5×10¹²km, that is, about a thousand times the size of the Solar System, or half a light-year.⁴

3The analogy with a liquid–vapor phase transition should be noted: the liquid results from “collapse” of the gas.

However, in this case it is not the gravitational interaction which is responsible for the instability, but rather interactions of the van der Waals type.

4This analysis is incorrect for astrophysical objects which are very dense such as white dwarfs, whose density is of order 5×10⁶g/cm. For such objects it is a pressure of quantum rather than thermal origin which opposes the gravitational attraction and ensures equilibrium.

The base state, perturbations, and linearization

We now make the above dimensional analysis quantitative by considering solutions to the linearized equations. Consider a large cloud of interstellar gas of densityρ, pressure P, and temperature T , assumed to be an ideal gas (Chandrasekhar, 1961,

§119). The stability of this cloud can be studied from the thermodynamic point of view by seeking the extrema of a suitable thermodynamic potential (the entropy for an isolated system described by the microcanonical formalism, or the free energy for a system held at constant temperature and described by the canonical formalism (Chavanis, 2002)). Equivalently, this stability can be studied using the mechanics of continuous media. It is this latter approach that we follow here. The conserva-tion equaconserva-tions are then identical to Eqs(2.2) governing acoustic waves, with the gravitational acceleration g replaced by the gravitational fieldG=−grad, where

 is the gravitational potential:

∂tρ +divρU = 0, (2.11a)

ρ(∂tU+(U·grad)U) = −gradP −ρ grad. (2.11b) The gravitational potential is related to the density distribution by the Poisson equation:

 = 4πGρ. (2.12)

Finally, as before, we assume that the perturbations evolve isentropically, with the entropy given by the equation of state (Callen, 1985):

s−s0= cvln P

P₀−cpln ρ ρ0

, (2.13)

where c_v and c_p are the heat capacities at constant volume and pressure, and the subscript 0 refers to a reference equilibrium state.

We note first of all that equations (2.11)–(2.12) do not admit a solution with uniform P andρ: for such a solution, the potential  would have to be uniform according to (2.11b), and the Poisson equation (2.12) would then implyρ = 0. A nonuniform hydrostatic equilibrium can be found for a spherical mass distribu-tion, corresponding to a uniform temperature and a density which varies as 1/r², where r is the distance to the center of the distribution (Binney and Tremaine, 1988). However, this distribution corresponds to infinite total mass (the mass ρ(r)4πr²dr in a sphere of radius R diverges as R). This difficulty can be over-come by artificially confining the gas inside a sphere of radius R (Chavanis, 2002).

A full treatment of this problem lies outside the scope of the present book, and so here we shall consider the simplified version studied by Jeans in 1902. The sim-plification is to study the stability of a uniform base state, although this base state

does not satisfy the Poisson equation (2.12). In spite of this simplification, known as the “Jeans swindle” (Binney and Tremaine, 1988), the calculation merits repro-duction owing to its historical importance and the fact that in the end it gives a stability condition close to that obtained in the solution of the full problem.⁵

Let us consider a stationary and uniform base state defined as

U= 0, ρ = ρ0, s= s0,  = 0, P= P0, (2.14) and a small isentropic perturbation of this base state, u,ρ^,φ, and p. The equations governing the evolution of these perturbations are then, according to (2.11)–(2.13),

∂t(ρ0+ρ^)+div

(ρ0+ρ^)u

= 0, (2.15a)

(ρ0+ρ^)(∂tu+(u·grad)u) = −grad(P0+ p)−(ρ0+ρ^)grad(0+φ), (2.15b)

φ = 4πGρ^, (2.15c)

ln(1+ p

P0)−γ ln(1+ρ^

ρ0) = 0. (2.15d)

We note that, although the base state does not satisfy the Poisson equation, the perturbations do. For small perturbations we can linearize the equations about the base state neglecting products of perturbations. The linearized equations are then

∂tρ^+ρ0div u= 0, (2.16a)

ρ0∂tu= −gradp −ρ0gradφ, (2.16b)

φ = 4πGρ^, (2.16c)

p= cs²ρ^. (2.16d)

Normal modes and the dispersion relation

Like the problem of acoustic waves treated above, the problem (2.16) is linear with constant coefficients, again a reflection of its translation invariance. It therefore has solutions which are exponentials in time and space and can be written as plane waves of wave vector k and frequencyω:

u=1

2ˆu e^{i(k·r−ωt)}+c.c., (2.17)

with similar expressions for the other velocity components v and w, as well as for ρ^, p, andφ.

5The “Jeans swindle” in fact becomes legitimate in cosmology when the expansion of the Universe is taken into account, in which case the Poisson equation is involved only for the perturbations.

Taking the divergence of (2.16b), by substitution we obtain an equation involving only the pressure perturbation p:

∂t²p−c²sp −4πρ0G p= 0. (2.18) For a normal mode of the form (2.17) this equation becomes

ω²ˆp = c_s²k²ˆp −4πGρ0ˆp, (2.19) where k= |k|. This equation has a nonzero solution only when the coefficient of ˆp vanishes, from which we find the dispersion relation:

ω²= c²sk²−4πGρ0. (2.20)

Note that this is the dispersion relation(2.8) for acoustic waves with the corrected gravitational term 4πGρ0.

2.2.3 Discussion

Let us now study the dispersion relation (2.20) by considering a plane wave with real wave vector k whose temporal stability is to be determined. Since the dispersion relation is quadratic inω, there exist two eigenmodes with frequency

ω±(k) = ±

c²_sk²−4πGρ0. This indicates the existence of a critical wave number

kJ=

√4πGρ0

cs

(2.21) for which the expression in the square root vanishes. Any perturbation with wave number above kJ has real frequency, and therefore corresponds to a wave which propagates without growth or decay with velocity

c_±= ω±/k = ±cs



1−k_J²/k² (k > kJ),

which is close to csfor short wavelengths (k kJ) and vanishes at k= kJ. On the other hand, any perturbation with wave number below kJ has purely imaginary frequency. The mode corresponding toω+= +iωigrows exponentially at a rate

ωi= cs



k_J²−k² (k < kJ),

while the other mode of frequencyω−= −iωidecreases exponentially. The phase velocity of these modes, c= ωr/k, is zero.

We conclude that in a gas cloud of size smaller than the Jeans length L_J=2π/kJ, any perturbation propagates without amplification, and so the cloud is linearly stable. On the other hand, if the cloud is larger than the Jeans length, it is unstable with respect to perturbations of large wavelength. We note that LJ corresponds, up to a numerical factor√

π, to the critical length(2.10)obtained in our earlier dimensional analysis.

Subrahmanyan Chandrasekhar (1910–1995)

Subrahmanyan Chandrasekhar was born in Lahore, India (now Pakistan) into a well-off, cultivated Brahmin family. One of his uncles, Sir Chandrasekhara Venkata Raman, was awarded the Nobel Prize in Physics in 1930 for his discovery of the optical effect bearing his name. He did his secondary and undergraduate studies in Madras, at which time he published his first article in the Pro-ceedings of the Royal Society. Following the example of his uncle, he left India in 1930 to study at Trinity College, Cambridge, England, where he obtained his doctorate in 1933. He got married in India in 1936, and then moved to the University of Chicago where he remained for the rest of his life. During World War II he worked on shock waves and detonation. He was elected a Fellow of the Royal Society of London in 1944. He and his wife became American citizens in 1953. He con-tributed to many fields in physics, including white dwarfs, stellar dynamics, radiation, hydrodynamic and hydromagnetic stability, general relativity, black holes, and gravitational waves. In particular, he showed that a star of mass less than 1.44 solar masses evolves at the end of its life into a white dwarf, while a star of mass greater than this limit collapses violently into an object of enor-mous density, a neutron star or a black hole. He won the Nobel Prize in 1983 for his work on stellar structure and evolution. He published about 400 papers and many important books, in particular, Hydrodynamic and Hydromagnetic Stability (1961). He received many honors, including the Royal Medal (1962) and the Copley Medal (1984) of the Royal Society of London. He was edi-tor in chief of the Astrophysical Journal for 19 years, and raised its stature enormously. He was interested in the connections between art and science, and later in life wrote the books Truth and Beauty: Aesthetics and Motivations in Science (1987) and Newton’s Principia for the Common Reader (1995), and

gave thought-provoking lectures such as Shakespeare, Newton and Beethoven or patterns of creativity and one on comparison of Newton’s Principia and Michelangelo’s frescoes in the Sistine Chapel.

Let us now return to the “Jeans swindle” for the case of a uniform base state. This swindle can be considered acceptable on a length scale over which the gravitational potential is nearly constant, i.e. the scale given by the Poisson equation. Actually, since the equilibrium condition imposes ρ/L ∼ P/L with

P/ρ = cs²/γ ≈ c²s, the Poisson equation gives c_s²

L²∼ 

L²∼ 4πGρ.

The length scale L that appears is again none other than the critical length (2.10) (up to a numerical factor of order unity)! Therefore, the conclusion that a cloud of uniform density and size smaller than LJis stable appears completely reason-able for a realistic cloud of nonuniform density satisfying the Poisson equation.

The appearance of an instability for a cloud of size of order LJ suggested by the dimensional analysis also seems quite plausible. On the other hand, the result of maximal growth rate for k= 0 is not reasonable. Recent studies on this subject confirm these conclusions (Chavanis, 2002). Interestingly, a study of the nonlinear evolution of the instability suggests a fractal organization of the mass distribution at the time of its collapse (de Vega et al., 1996).

2.3 The Rayleigh–Taylor interface instability