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An Exact Derivation of Jeans' Criterion with Rotation for Gravitational 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 density 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 stars 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: + ρr • v = 0; Dt Dv 1 Euler's Equation: + 2Ω × v + Ω × (Ω × r) = − P0(ρ)rρ + g; Dt ρ Poisson's Equation: r • 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, r = (@=@x; @=@y; @=@z) ≡ gradient operator, D=Dt = @=@t + v • r ≡ material derivative, γ P(ρ) = p0(ρ/ρ0) 0 ≡ adiabatic pressure, g = −r' ≡ gravitational acceleration vector with ' ≡ 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); r • g = −∇2'; 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)rρ − r'; ρ 2 2 Poisson's Equation: r ' = 4πG0ρ where r = r • r: 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;' = '0 where '0 satisfies 2 2 r'0 = Ω0(x; y; 0); r '0 = 4πG0ρ0; or x2 + y2 ' (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 = "v1 + O(" ); ρ = ρ0[1 + "s + O(" )]; 2 ' = '0 + "'1 + O(" ) where v1 = (u1; v1; w1); David J. Wollkind, et al. WSU An Exact Derivation of Jeans' Criterion with Rotation for Gravitational Instabilities ...with j"j << 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 @'1 2 0 p0 − 2Ω0v1 + c0 + = 0 where c0 = P (ρ0) = γ0 > 0; @t @x @x ρ0 @v @s @' 1 + 2Ω u + c2 + 1 = 0; @t 0 1 0 @y @y @w @s @' 1 + c2 + 1 = 0; @t 0 @z @z 2 2 2Ω0s − r '1 = 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; '1](x; y; z; t) = [A; B; C; E; F ]e ; p where jAj2 + jBj2 + jCj2 + jEj2 + jF j2 6= 0, i = −1, and k1;2;3 2 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 = jkj jΩj 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 2 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 !2+a!+b = 0 with D = a2−4b ≥ 0; ! < 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 sound in an adiabatic gas of density ρ0. This formula is of fundamental importance in astrophysics 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 protostars 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 '0 symbolically.
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