an exact derivation of jeans' criterion with rotation for

An Exact Derivation of Jeans’ Criterion withRotation for Gravitational Instabilities

Kohl Gill1 David J. Wollkind1 Bonni J. Dichone2

1Department of Mathematics, Washington State University, Pullman, WA99164-3113, USA

2Department of Mathematics, Gonzaga University, 502 E. Boone AvenueMSC 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.



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.



A Self-Gravitational Uniformly-Rotating AdiabaticInviscid Fluid Model of Infinite Extent

The governing equations for this situation are given by [2]:

Continuity Equation:Dρ

Dt+ ρ∇ • v = 0;

Euler’s Equation:Dv

Dt+ 2Ω× v + Ω× (Ω× r) = −1

ρP ′(ρ)∇ρ+ g;

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].



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 rotatingframe, ∇ = (∂/∂x, ∂/∂y, ∂/∂z) ≡ gradient operator,D/Dt = ∂/∂t+ v •∇ ≡ material derivative,P(ρ) = p0(ρ/ρ0)γ0 ≡ adiabatic pressure, g = −∇ϕ ≡gravitational acceleration vector with ϕ ≡ self-gravitatingpotential, and G0 ≡ universal gravitational constant.



The Governing Euler’s and Poisson’s Equations

SinceΩ× v = Ω0(−v, u, 0),

Ω× (Ω× r) = (Ω • r)Ω− (Ω •Ω)r = −Ω20(x, y, 0),

∇ • g = −∇2ϕ;

the Euler’s and Poisson’s equations become:

Euler’s Equation:Dv

Dt+ 2Ω0(−v, u, 0)− Ω2

0(x, y, 0)

= −1

ρP ′(ρ)∇ρ−∇ϕ;

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



The Exact Static Homogeneous Density Solution

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

v ≡ 0 = (0, 0, 0), ρ ≡ ρ0, ϕ = ϕ0

where ϕ0 satisfies

∇ϕ0 = Ω20(x, y, 0), ∇2ϕ0 = 4πG0ρ0,

or

ϕ0(x, y) = Ω20

x2 + y2

2with Ω2

0 = 2πG0ρ0 > 0.

This equilibrium state physically corresponds to the situation ofa homogeneous quiescent gas for which the centrifugal andgravitational forces balance out each other.



Linear Perturbation Analysis of that Exact Solution

Now seeking a linear perturbation solution of these basicequations of the form

v = εv1 + O(ε2), ρ = ρ0[1 + εs+O(ε2)],

ϕ = ϕ0 + εϕ1 +O(ε2) where v1 = (u1, v1, w1);



...with |ε| << 1, we deduce that the perturbation quantities tothis exact solution satisfy

∂s

∂t+∂u1

∂x+∂v1

∂y+∂w1

∂z= 0;

∂u1

∂t− 2Ω0v1 + c2

0

∂s

∂x+∂ϕ1

∂x= 0 where c2

0 = P ′(ρ0) = γ0p0

ρ0> 0;

∂v1

∂t+ 2Ω0u1 + c2

0

∂s

∂y+∂ϕ1

∂y= 0;

∂w1

∂t+ c2

0

∂s

∂z+∂ϕ1

∂z= 0;

2Ω20s−∇2ϕ1 = 0.



Normal Mode Solution of these Perturbation Equations

Assuming a normal mode solution for these perturbationquantities of the form

[u1, v1, w1, s, ϕ1](x, y, z, t) = [A,B,C,E, F ]ei(k1x+k2y+k3z)+σt;

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 propertyfor those quantities; we obtain the following equations for[A,B,C,E, F ]:

ik1A+ ik2B + ik3C + σE = 0;

σA− 2Ω0B + ic20k1E + ik1F = 0;

2Ω0A+ σB + ic20k2E + ik2F = 0;

σC + ic20k3E + ik3F = 0;

2Ω20E + k2F = 0 where k2 = k2

1 + k22 + k2

3.



The Secular Equation or Dispersion Relation

Setting the determinant of the 5× 5 coefficient matrix for thesystem of constants equal to zero to satisfy their nontrivialityproperty, we obtain the secular equation

k2[σ4 + (c20k

2 + 2Ω20)σ2] + 4Ω2

0(c20k

2 − 2Ω20)k2

3 = 0.

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

k •Ω = k3Ω0 = |k| |Ω| cos(θ) = kΩ0 cos(θ),

θ being the azimuthal angle between k and Ω; which impliesthat

k3 = k cos(θ).

Then, substitution of this result and cancellation of k2, yields

σ4 + (c20k

2 + 2Ω20)σ2 + 4Ω2

0(c20k

2 − 2Ω20) cos2(θ) = 0.



Stability Analysis of the Secular Equation

Since this secular equation is a quadratic in σ2, we can firstconclude that σ2 ∈ R by showing that its discriminant

D = (c20k

2 + 2Ω20)2 − 16Ω2

0(c20k

2 − 2Ω20) cos2(θ) ≥ 0.

Consider the two cases of c20k

2 − 2Ω20 ≤ 0 and c2

0k2 − 2Ω2

0 > 0separately. For the former case it is obvious, while for the latterone it can be deduced by noting thatD ≥ (c2

0k2 + 2Ω2

0)2 − 16Ω20(c2

0k2 − 2Ω2

0) = (c20k

2 − 6Ω20)2.



For θ = π/2, we can conclude that σ2 = 0 orσ2 = −(c2

0k2 + 2Ω2

0) < 0, while for θ 6= π/2, the stability criteriagoverning 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

c20k

2 − 2Ω20 > 0.

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

c20k

2 − 4πG0ρ0 < 0,

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



Jeans’ Instability Criterion

This instability criterion can be posed in terms of wavelengthλ = 2π/k and then takes the form

λ > λJ = c0

√π

G0ρ0≡ Jeans’ length,

where c0 ≡ speed of sound in an adiabatic gas of density ρ0.This formula is of fundamental importance in astrophysics andcosmology where many significant deductions concerning theformation of galaxies and stars have been based upon it. Inparticular Jeans’ [5] interpretation of the criterion now bearinghis name was that a gas cloud of characteristic dimension muchgreater than λJ would tend to form condensations with meandistance of separation comparable to λJ that then developedinto those protostars observable in the outer arms of spiralgalaxies such as Andromeda M31.



Figure: A Galaxy Evolution Explorer image of the Andromeda galaxyM31, courtesy of NASA/JPL-Caltech.



Comparisons with Jeans

Jeans [4] treated his static solution involving ρ0 and ϕ0

symbolically. Since Jeans’ original analysis was for anonrotating system with Ω0 = 0, when he assumed in additionthat ρ0 was a constant in his perturbation equations to makethem constant coefficient this implicitly required ∇ϕ0 = 0which implied ∇2ϕ0 = 0 = 4πG0ρ0 or ρ0 = 0 and hence istermed Jeans’ swindle by Binney and Tremaine [1]. Theproblem under examination illustrates the point that addingrotation to the system as Chandrasekhar did and thenperforming a standard linear stability analysis of its exact staticsolution yields Jeans’ instability criterion but in a systematicmanner and such a model also has the added advantage ofbeing more astrophysically realistic.



Sekimura et al. [10] have demonstrated that for a secularequation similar in form to Jeans’ σ2 = 2Ω2

0 − c20k

2, λJ actuallycorresponds to the so-called critical wavelength λc of linearstability theory associated with σ = 0 while nonlinear stabilityanalyses of physical phenomena involving related secularequations have shown that the observed wavelengths aredetermined to a close approximation by this λc rather than bythe dominant wavelength λd at which σ achieves its maximumvalue from linear theory.



Jeans’ Secular Equation I

σ

kJk0

( )0;k cσ σ=( );k cσ σ=

0c c<

02Ω

2δ

Figure: Schematic plots in the k-σ planedepicting the methodology employed bySekimura et al. [10] applied to the Jeans’secular equation

σ = σ(k; c) =√

2Ω20 − c2k2. That curve is

plotted for both a general speed of sound cand our specific speed c0 > c in this figurewhere kJ = 2π/λJ is such thatσ(kJ ; c0) = 0. In a weakly nonlinearstability analysis one takes the disturbancewavenumber k ≡ kJ and its growth rate to

be equal to σJ (c) = σ(kJ ; c) = δ2 > 0 wherec is close enough to c0 so that δ is a smallparameter. Then in the limc→c0

σJ (c) = 0which is a requirement for the application ofweakly nonlinear stability theory and anyre-equilibrated pattern will exhibit awavelength of λJ . Here c2 = γp0/ρ0 withγ < γ0 and hence the operation limc→c0

isequivalent to limγ→γ0

.



Jeans’ Secular Equation II

Therefore Jeans’ interpretation, although unusual for linearstability theory (where it is often presumed that such adisturbance associated with the largest growth ratepredominates), both anticipated and is consistent with thesenonlinear results, since, by the time perturbations have grownenough for the effect of the maximum growth rate to beobserved, the neglected nonlinearities may have rendered thatlinear analysis inaccurate. In this context note that for a typicalvalue of θ 6= π/2, namely θ = 0, we can factor our secularequation to obtain the roots σ2 = −4Ω2

0 and σ2 = 2Ω20 − c2

0k2.

Observe that this last condition which yields our instability isequivalent to Jeans’ secular equation.



Jeans’ Length

Using the formula

λJ = c0

√π

G0ρ0≡ Jeans’ length

with the parameters c0 and ρ0 assigned the values

c0 =2

3× 104 cm

secand ρ0 = 10−22 gm

cm3

employed by Jeans [5] for this purpose but when the polytropicindex γ0 = 4/3 while taking

G0 = 6.67× 10−8 cm3

gm sec2

in cgs units,David J. Wollkind, et al. WSU


...yieldsλJ = 4.58× 1018 cm = 1.48 pc

where 1 pc ≡ 3.09× 1018 cm, which compares quite favorablywith the mean distance between actual adjacent condensationsoriginally formed in the outer arms of Andromeda since, inthose parts of M31, the averaged observed distance betweenprotostars in such chains is about 1.4 pc or somewhat more ifallowances are made for foreshortening [5].



Jeans’ Mass

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

MJ =4π

3

(λJ2

)3

ρ0 =π

6λ3Jρ0 = 5× 1030 kgm,

which represents the mass surrounding each of thesecondensations as compared with the mass of the sunM = 2× 1030 kgm or exactly two and one-half times as much.This differs slightly from Jeans’ [5] original critical mass definedby

Mc = λ3Jρ0 = 9.55× 1030 kgm,

or about four and three-quarters times the mass of the sun.



Figure: A plot of size versus color for Population I stars found in theouter arms of spiral galaxies. Here the scale on the vertical axisindicates size in terms of solar masses and that on the horizontal,color from blue at the left to red at the right with yellow betweenthem. The spiral arm stars show a simple color-size relationship: Thelargest ones are blue giants and the smallest, red dwarfs; while thesun-type of intermediate size is a so-called yellow dwarf.



Jeans’ Genius

Thus, Jeans got the right answer for the wrong reason. In hisreview of nonlinear hydrodynamic stability theory, therenowned comprehensive applied mathematical modeler LeeSegel [9] stated that

“Anyone can get the right answer for the right reason.It takes a genius or a physicist to get the right answerfor the wrong reason.”

In this context, Sir James Jeans was both.



Comparisons with Chandrasekhar

Chandrasekhar’s [2] analysis with rotation added to Jeans’perturbation equations also suffered from the same deficiency asthe latter in that he did not develop a parameter relationshipfor 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 caseθ = π/2; Namely: c2

0k2 < 4(πG0ρ0 − Ω2

0) should Ω20 < πρ0G0.

Chandrasekhar [2] plotted σ2 versus k for θ = 0, π/4, π/2 andΛ2 ≡ Ω2

0/(πG0ρ0) = 0.5, 1.0, 2.0. In point of fact, Λ2 = 0.5 is arepresentative value of that quantity for this extra instabilitycondition while Λ2 = 2.0 , his upper bound, actuallycorresponds to its value as per our formula relating theseparameters which violates that extra condition identically.



Chandrasekhar’s Extraneous Instability Criterion forθ = π/2 and Λ2 < 1.0

Figure: Plots ofω2 = −σ2 = c20k

2 + 4(Ω2−πGρ0)versus k for Chandrasekhar’sdispersion relation when θ = π/2and Λ2 < 1.0 or Λ2 > 1.0, thelower or upper curves,respectively. Note that Λ2 = 0.5and 2.0 serve as representativevalues for these two cases,separated by Λ2 = 1.0.



Comparison with Rubin and Ford

Let us examine the plausibility of our formula for

Ω0 =√

2πρ0G0.

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

Ω0 = 6.47× 10−15/sec

and the corresponding rotational velocity

V0 = Ω0R0 = 200km

sec,



for the reference radial distance of

R0 = 1 kpc = 103 pc = 3.09× 1021 cm = 3.09× 1016 km,

both of which are consistent with the spectroscopicmeasurements of the Andromeda nebula and the observationaldata of the spiral Milky Way galaxy as reported by Rubin andFord [7].



Rotation Curves for M31 and the Galaxy

Figure: Comparison of rotational velocities V0 (km/sec) of Andromeda M31 andthe 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 theGalaxy [8], respectively. The filled circles are the 7 observed rotational velocitiesfor the Galaxy adopted by Roogour and Oort [6] to make the features of the latterconform with those of Andromeda.



Comparison with Binney and Tremaine

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



Further, the last part of their problem was to show, uponfinding the resulting dispersion relation from its linear stabilityanalysis, that waves propagating perpendicular to the rotationvector were always stable while those propagating parallel to itwere unstable if and only if the usual Jeans’ criterion withoutrotation was satisfied. Although the latter conclusion for θ = 0agrees with our predictions, the former does not since, whenθ = π/2, we predicted σ2 = 0 as well as those σ2 < 0 which onlyimplies a condition of neutral stability.



Sir Arthur Conan Doyle Sherlock Holmes Quote

These results demonstrate that the best way to test the validityof a model for a natural science phenomenon is to compare itstheoretical predictions with observable data of thatphenomenon. Sir Arthur Conan Doyle characterized thisphilosophy perhaps as well as anyone by a Sherlock Holmesquote from A Scandal in Bohemia in his 1891 collection entitledThe 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.”



References I

[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.: Astronomy and cosmogony, Cambridge University Press, Cambridge (1928)

[6] Rougoor, G.W., Oort, J.H.: Distribution and motion of interstellar hydrogen in thegalactic 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 spectroscopicsurvey of emission. Astrophysical Journal. 159, 379-403 (1970)

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



References II

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

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



Thank you!Questions?



an exact derivation of jeans' criterion with rotation for

Documents