An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities

Kohl Gill1 David J. Wollkind1 Bonni J. Dichone2

1Department of Mathematics, Washington State University, Pullman, WA 99164-3113, USA

2Department of Mathematics, Gonzaga University, 502 E. Boone Avenue MSC 2615, Spokane, WA 99258, USA

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Abstract

An inviscid fluid model of a self-gravitating infinite expanse of a uniformly rotating adiabatic gas cloud consisting of the continuity, Euler’s, and Poisson’s equations for that situation is considered. There exists a static homogeneous solution to this model relating that equilibrium density to the uniform rotation. A systematic linear stability analysis of this exact solution then yields a gravitational instability criterion equivalent to that developed by Sir James Jeans in the absence of rotation instead of the slightly more complicated stability behavior deduced by Subrahmanyan Chandrasekhar for this model with rotation, both of which suffered from the same deficiency in that neither of them actually examined whether their perturbation analysis was of an exact solution.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Abstract (cont.)

For the former case, it was not and, for the latter, the equilibrium density and uniform rotation were erroneously assumed to be independent instead of related to each other. Then this gravitational instability criterion is employed in the form of Jeans’ length to show that there is very good agreement between this theoretical prediction and the actual mean distance of separation of formed in the outer arms of the spiral galaxy Andromeda M31. Further the uniform rotation determined from the exact solution relation to equilibrium density and the corresponding rotational velocity for a reference radial distance are consistent with the spectroscopic measurements of Andromeda and the observational data of the spiral Milky Way galaxy.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities A Self-Gravitational Uniformly-Rotating Adiabatic Inviscid Fluid Model of Infinite Extent

The governing equations for this situation are given by [2]: Dρ Continuity Equation: + ρ∇ • v = 0; Dt Dv 1 Euler’s Equation: + 2Ω × v + Ω × (Ω × r) = − P0(ρ)∇ρ + g; Dt ρ

Poisson’s Equation: ∇ • g = −4πG0ρ. The continuity and Euler’s equations follow from the conservation of mass and momentum for an inviscid fluid with the addition of the extra second and third terms on the left-hand side of the latter which represent the Coriolis effect and centrifugal force, respectively, due to the rotation [3]. Poisson’s equation follows from the divergence theorem and Newton’s law of universal gravitation [1].

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Nomenclature

Here t ≡ time, r = (x, y, z) ≡ position vector, Ω = (0, 0, Ω0) ≡ uniform rotation vector, ρ ≡ density (mass/[unit volume]), v = (u, v, w) ≡ velocity vector with respect to the rotating frame, ∇ = (∂/∂x, ∂/∂y, ∂/∂z) ≡ gradient operator, D/Dt = ∂/∂t + v • ∇ ≡ material derivative, γ P(ρ) = p0(ρ/ρ0) 0 ≡ adiabatic , g = −∇ϕ ≡ gravitational acceleration vector with ϕ ≡ self-gravitating potential, and G0 ≡ universal gravitational constant.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities The Governing Euler’s and Poisson’s Equations

Since Ω × v = Ω0(−v, u, 0), 2 Ω × (Ω × r) = (Ω • r)Ω − (Ω • Ω)r = −Ω0(x, y, 0), ∇ • g = −∇2ϕ; the Euler’s and Poisson’s equations become:

Dv Euler’s Equation: + 2Ω (−v, u, 0) − Ω2(x, y, 0) Dt 0 0 1 = − P0(ρ)∇ρ − ∇ϕ; ρ

2 2 Poisson’s Equation: ∇ ϕ = 4πG0ρ where ∇ = ∇ • ∇.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities The Exact Static Homogeneous Density Solution

There exists an exact static homogeneous density solution of these equations of the form

v ≡ 0 = (0, 0, 0), ρ ≡ ρ0, ϕ = ϕ0

where ϕ0 satisfies

2 2 ∇ϕ0 = Ω0(x, y, 0), ∇ ϕ0 = 4πG0ρ0,

or x2 + y2 ϕ (x, y) = Ω2 with Ω2 = 2πG ρ > 0. 0 0 2 0 0 0 This equilibrium state physically corresponds to the situation of a homogeneous quiescent gas for which the centrifugal and gravitational forces balance out each other.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Linear Perturbation Analysis of that Exact Solution

Now seeking a linear perturbation solution of these basic equations of the form

2 2 v = εv1 + O(ε ), ρ = ρ0[1 + εs + O(ε )],

2 ϕ = ϕ0 + εϕ1 + O(ε ) where v1 = (u1, v1, w1);

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities ...with |ε| << 1, we deduce that the perturbation quantities to this exact solution satisfy ∂s ∂u ∂v ∂w + 1 + 1 + 1 = 0; ∂t ∂x ∂y ∂z

∂u1 2 ∂s ∂ϕ1 2 0 p0 − 2Ω0v1 + c0 + = 0 where c0 = P (ρ0) = γ0 > 0; ∂t ∂x ∂x ρ0 ∂v ∂s ∂ϕ 1 + 2Ω u + c2 + 1 = 0; ∂t 0 1 0 ∂y ∂y ∂w ∂s ∂ϕ 1 + c2 + 1 = 0; ∂t 0 ∂z ∂z 2 2 2Ω0s − ∇ ϕ1 = 0.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Normal Mode Solution of these Perturbation Equations

Assuming a normal mode solution for these perturbation quantities of the form

i(k1x+k2y+k3z)+σt [u1, v1, w1, s, ϕ1](x, y, z, t) = [A, B, C, E, F ]e ; √ where |A|2 + |B|2 + |C|2 + |E|2 + |F |2 6= 0, i = −1, and k1,2,3 ∈ R to satisfy the implicit far-field boundedness property for those quantities; we obtain the following equations for [A, B, C, E, F ]:

ik1A + ik2B + ik3C + σE = 0; 2 σA − 2Ω0B + ic0k1E + ik1F = 0; 2 2Ω0A + σB + ic0k2E + ik2F = 0; 2 σC + ic0k3E + ik3F = 0; 2 2 2 2 2 2 2Ω0E + k F = 0 where k = k1 + k2 + k3.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities The Secular Equation or Dispersion Relation

Setting the determinant of the 5 × 5 coefficient matrix for the system of constants equal to zero to satisfy their nontriviality property, we obtain the secular equation 2 4 2 2 2 2 2 2 2 2 2 k [σ + (c0k + 2Ω0)σ ] + 4Ω0(c0k − 2Ω0)k3 = 0.

Defining the wavenumber vector k = (k1, k2, k3), its dot product with Ω satisfies

k • Ω = k3Ω0 = |k| |Ω| cos(θ) = kΩ0 cos(θ), θ being the azimuthal angle between k and Ω; which implies that k3 = k cos(θ). Then, substitution of this result and cancellation of k2, yields 4 2 2 2 2 2 2 2 2 2 σ + (c0k + 2Ω0)σ + 4Ω0(c0k − 2Ω0) cos (θ) = 0.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Stability Analysis of the Secular Equation

Since this secular equation is a quadratic in σ2, we can first 2 conclude that σ ∈ R by showing that its discriminant

2 2 2 2 2 2 2 2 2 D = (c0k + 2Ω0) − 16Ω0(c0k − 2Ω0) cos (θ) ≥ 0.

2 2 2 2 2 2 Consider the two cases of c0k − 2Ω0 ≤ 0 and c0k − 2Ω0 > 0 separately. For the former case it is obvious, while for the latter one it can be deduced by noting that 2 2 2 2 2 2 2 2 2 2 2 2 D ≥ (c0k + 2Ω0) − 16Ω0(c0k − 2Ω0) = (c0k − 6Ω0) .

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities For θ = π/2, we can conclude that σ2 = 0 or 2 2 2 2 σ = −(c0k + 2Ω0) < 0, while for θ 6= π/2, the stability criteria governing such quadratics: Namely, given

ω2+aω+b = 0 with D = a2−4b ≥ 0, ω < 0 if and only if a, b > 0;

implies that σ2 < 0 if and only if

2 2 2 c0k − 2Ω0 > 0.

Making an interpretation of these results, we can deduce that there will only be σ2 > 0 and hence unstable behavior should

2 2 c0k − 4πG0ρ0 < 0,

which is usually referred to as Jeans’ gravitational instability criterion after Sir James Jeans [5] who first proposed it.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Jeans’ Instability Criterion

This instability criterion can be posed in terms of wavelength λ = 2π/k and then takes the form r π λ > λJ = c0 ≡ Jeans’ length, G0ρ0

where c0 ≡ speed of in an adiabatic gas of density ρ0. This formula is of fundamental importance in and cosmology where many significant deductions concerning the formation of galaxies and stars have been based upon it. In particular Jeans’ [5] interpretation of the criterion now bearing his name was that a gas cloud of characteristic dimension much greater than λJ would tend to form condensations with mean distance of separation comparable to λJ that then developed into those observable in the outer arms of spiral galaxies such as Andromeda M31. David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Figure: A Galaxy Evolution Explorer image of the Andromeda galaxy M31, courtesy of NASA/JPL-Caltech.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Comparisons with Jeans

Jeans [4] treated his static solution involving ρ0 and ϕ0 symbolically. Since Jeans’ original analysis was for a nonrotating system with Ω0 = 0, when he assumed in addition that ρ0 was a constant in his perturbation equations to make them constant coefficient this implicitly required ∇ϕ0 = 0 2 which implied ∇ ϕ0 = 0 = 4πG0ρ0 or ρ0 = 0 and hence is termed Jeans’ swindle by Binney and Tremaine [1]. The problem under examination illustrates the point that adding rotation to the system as Chandrasekhar did and then performing a standard linear stability analysis of its exact static solution yields Jeans’ instability criterion but in a systematic manner and such a model also has the added advantage of being more astrophysically realistic.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Sekimura et al. [10] have demonstrated that for a secular 2 2 2 2 equation similar in form to Jeans’ σ = 2Ω0 − c0k , λJ actually corresponds to the so-called critical wavelength λc of linear stability theory associated with σ = 0 while nonlinear stability analyses of physical phenomena involving related secular equations have shown that the observed wavelengths are determined to a close approximation by this λc rather than by the dominant wavelength λd at which σ achieves its maximum value from linear theory.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Jeans’ Secular Equation I

Figure: Schematic plots in the k-σ plane depicting the methodology employed by Sekimura et al. [10] applied to the Jeans’ σ secular equation q 2 2 2 σ = σ(k; c) = 2Ω0 − c k . That curve is Ω 2 0 plotted for both a general c and our specific speed c0 > c in this figure where kJ = 2π/λJ is such that σ(kJ ; c0) = 0. In a weakly nonlinear σ = σ (k;c0 ) σ = σ (k;c) stability analysis one takes the disturbance

c < c0 wavenumber k ≡ kJ and its growth rate to 2 δ 2 be equal to σJ (c) = σ(kJ ; c) = δ > 0 where c is close enough to c0 so that δ is a small parameter. Then in the limc→c0 σJ (c) = 0 which is a requirement for the application of weakly nonlinear stability theory and any k re-equilibrated pattern will exhibit a 0 2 kJ wavelength of λJ . Here c = γp0/ρ0 with γ < γ0 and hence the operation limc→c0 is equivalent to limγ→γ0 .

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Jeans’ Secular Equation II

Therefore Jeans’ interpretation, although unusual for linear stability theory (where it is often presumed that such a disturbance associated with the largest growth rate predominates), both anticipated and is consistent with these nonlinear results, since, by the time perturbations have grown enough for the effect of the maximum growth rate to be observed, the neglected nonlinearities may have rendered that linear analysis inaccurate. In this context note that for a typical value of θ 6= π/2, namely θ = 0, we can factor our secular 2 2 2 2 2 2 equation to obtain the roots σ = −4Ω0 and σ = 2Ω0 − c0k . Observe that this last condition which yields our instability is equivalent to Jeans’ secular equation.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Jeans’ Length

Using the formula r π λJ = c0 ≡ Jeans’ length G0ρ0

with the parameters c0 and ρ0 assigned the values 2 cm gm c = × 104 and ρ = 10−22 0 3 sec 0 cm3 employed by Jeans [5] for this purpose but when the polytropic index γ0 = 4/3 while taking cm3 G = 6.67 × 10−8 0 gm sec2 in cgs units, David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities ...yields 18 λJ = 4.58 × 10 cm = 1.48 pc where 1 pc ≡ 3.09 × 1018 cm, which compares quite favorably with the mean distance between actual adjacent condensations originally formed in the outer arms of Andromeda since, in those parts of M31, the averaged observed distance between protostars in such chains is about 1.4 pc or somewhat more if allowances are made for foreshortening [5].

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Jeans’ Mass

Using the definition for Jeans’ mass as that contained in a sphere of diameter λJ [1], we find that

4π λ 3 π M = J ρ = λ3 ρ = 5 × 1030 kgm, J 3 2 0 6 J 0

which represents the mass surrounding each of these condensations as compared with the mass of the sun 30 M = 2 × 10 kgm or exactly two and one-half times as much. This differs slightly from Jeans’ [5] original critical mass defined by 3 30 Mc = λJ ρ0 = 9.55 × 10 kgm, or about four and three-quarters times the mass of the sun.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Figure: A plot of size versus color for Population I stars found in the outer arms of spiral galaxies. Here the scale on the vertical axis indicates size in terms of solar masses and that on the horizontal, color from blue at the left to red at the right with yellow between them. The spiral arm stars show a simple color-size relationship: The largest ones are blue giants and the smallest, red dwarfs; while the sun-type of intermediate size is a so-called yellow dwarf.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Jeans’ Genius

Thus, Jeans got the right answer for the wrong reason. In his review of nonlinear hydrodynamic stability theory, the renowned comprehensive applied mathematical modeler Lee Segel [9] stated that “Anyone can get the right answer for the right reason. It takes a genius or a physicist to get the right answer for the wrong reason.” In this context, Sir James Jeans was both.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Comparisons with Chandrasekhar

Chandrasekhar’s [2] analysis with rotation added to Jeans’ perturbation equations also suffered from the same deficiency as the latter in that he did not develop a parameter relationship for his implicit exact solution and thus erroneously treated ρ0 and Ω0 as independent. Besides Jeans’ criterion for θ 6= π/2, this yielded an extraneous instability criterion for the case 2 2 2 2 θ = π/2; Namely: c0k < 4(πG0ρ0 − Ω0) should Ω0 < πρ0G0. Chandrasekhar [2] plotted σ2 versus k for θ = 0, π/4, π/2 and 2 2 2 Λ ≡ Ω0/(πG0ρ0) = 0.5, 1.0, 2.0. In point of fact, Λ = 0.5 is a representative value of that quantity for this extra instability condition while Λ2 = 2.0 , his upper bound, actually corresponds to its value as per our formula relating these parameters which violates that extra condition identically.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Chandrasekhar’s Extraneous Instability Criterion for θ = π/2 and Λ2 < 1.0

Figure: Plots of 2 2 2 2 2 ω = −σ = c0k + 4(Ω − πGρ0) versus k for Chandrasekhar’s dispersion relation when θ = π/2 and Λ2 < 1.0 or Λ2 > 1.0, the lower or upper curves, respectively. Note that Λ2 = 0.5 and 2.0 serve as representative values for these two cases, separated by Λ2 = 1.0.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Comparison with Rubin and Ford

Let us examine the plausibility of our formula for p Ω0 = 2πρ0G0.

In conjunction with the values for ρ0 and G0, this yields the uniform rotation

−15 Ω0 = 6.47 × 10 /sec

and the corresponding rotational velocity km V = Ω R = 200 , 0 0 0 sec

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities for the reference radial distance of

3 21 16 R0 = 1 kpc = 10 pc = 3.09 × 10 cm = 3.09 × 10 km,

both of which are consistent with the spectroscopic measurements of the Andromeda nebula and the observational data of the spiral Milky Way galaxy as reported by Rubin and Ford [7].

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Rotation Curves for M31 and the Galaxy

Figure: Comparison of rotational velocities V0 (km/sec) of Andromeda M31 and the Milky Way Galaxy as functions of radial distance to the center R0 (kpc) where the solid and dashed curves represent these plots for M31 [7] and the Galaxy [8], respectively. The filled circles are the 7 observed rotational velocities for the Galaxy adopted by Roogour and Oort [6] to make the features of the latter conform with those of Andromeda. David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Comparison with Binney and Tremaine

We close by noting that Binney and Tremaine [1] considered this gravitational instability model in a cylindrical rotating system as a problem in Chapter 5 of their book “Galactic Dynamics”. They observed that rotation allowed the Jeans’ instability to be analyzed exactly. Since the first part of their problem was to find the condition on Ω0 so that the homogeneous quiescent gas would be in equilibrium, Binney and Tremaine [1] did not examine the plausibility of this condition.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Further, the last part of their problem was to show, upon finding the resulting dispersion relation from its linear stability analysis, that waves propagating perpendicular to the rotation vector were always stable while those propagating parallel to it were unstable if and only if the usual Jeans’ criterion without rotation was satisfied. Although the latter conclusion for θ = 0 agrees with our predictions, the former does not since, when θ = π/2, we predicted σ2 = 0 as well as those σ2 < 0 which only implies a condition of neutral stability.

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Sir Arthur Conan Doyle Sherlock Holmes Quote

These results demonstrate that the best way to test the validity of a model for a natural science phenomenon is to compare its theoretical predictions with observable data of that phenomenon. Sir Arthur Conan Doyle characterized this philosophy perhaps as well as anyone by a Sherlock Holmes quote from A Scandal in Bohemia in his 1891 collection entitled The Adventures of Sherlock Holmes: “It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts.”

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities ReferencesI

[1] Binney, J., Tremaine, S.: Galactic dynamics, Princeton University Press, Princeton (1987)

[2] Chandrasekhar, S.: Hydrodynamic and hydromagnetic stability, Oxford University Press, Oxford (1961)

[3] Greenspan, H.P.: The theory of rotating fluids, Cambridge University Press, Cambridge (1968)

[4] Jeans, J.H.: The stability of a spherical nebula. Phil. Trans. Roy. Soc. of London 199, 1-53 (1902)

[5] Jeans, J.H.: and cosmogony, Cambridge University Press, Cambridge (1928)

[6] Rougoor, G.W., Oort, J.H.: Distribution and motion of interstellar hydrogen in the galactic system with particular reference to the region within 3 kiloparsecs of the center. PNAS 46, 1-13 (1960)

[7] Rubin, V. C., Ford, W. K. J.: Rotation of the Andromeda Nebula from a spectroscopic survey of emission. Astrophysical Journal. 159, 379-403 (1970)

[8] Schmidt, M.: Rotation parameters and distribution of mass in the galaxy, In Galactic structure. Blaauw, A., Schmidt, M. (eds.) University of Chicago Press, Chicago, pp. 513-530 (1965)

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities ReferencesII

[9] Segel, L.A.: Nonlinear hydrodynamic stability theory and its applications to thermal convection and curved flows, In Nonequilibrium thermodynamics, variational techniques, and stability. Donnelly, R.J., Herman, R., Prigogine, I. (eds.): University of Chicago Press, Chicago, pp. 164-197 (1966)

[10] Sekimura, T., Zhu, M., Cook, J., Maini, P. K., Murray, J. D.: Pattern formation of scale cells in Lepidoptera by differential origin-dependent cell adhesion. Bull. Math. Biol. 61, 807-827 (1999)

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities Thank you! Questions?

David J. Wollkind, et al. WSU An Exact Derivation of Jeans’ Criterion with Rotation for Gravitational Instabilities