Kinetic Theory for Dense Planetary Rings · 2018. 2. 18. · I would like to thank Prof. Ishan Sharma for introducing me to this ﬁeld and helping me learn kinetic theory of granular

Kinetic Theory for Dense Planetary Rings

Akshit Seth

December 2016

Birla Institute of Technology and Science- Pilani

Department of Chemical Engineering

Kinetic Theory for Dense Planetary Rings

Akshit Seth

Supervisor Prof. Ishan SharmaDepartment of Mechanical EngineeringIIT-Kanpur

December 2016

Submitted in partial fulfillment of B.E.(Hons.) Chemical Engineering Degree at BITS-Pilanion 7th December 2016.

Abstract

A kinetic theory is applied to study the structure of a planetary ring normal toits mid-plane. The ring is assumed to be composed of identical, smooth, slightlyinelastic spheres. Axially symmetric steady-states about a spherical central body arenumerically investigated by phrasing a one-dimensional boundary-value problemwith an unknown boundary point corresponding to ring thickness and an additionalpressure boundary condition. The boundary-value problem is solved and the pressureboundary condition is satisfied using a bisection method to determine the ringthickness. This is done for several values of the optical depth and self-gravity.The results are then extended for first-order corrections in the potential for anoblate-spheroidal central body.

As an aside, a simple numerical algorithm is developed for computing the shapes ofan axially symmetric liquid drop resting on a membrane.

v

Acknowledgement

I would like to thank Prof. Ishan Sharma for introducing me to this field and helpingme learn kinetic theory of granular gases. I would also like to thank the groupmembers Akash and Bharath for their suggestions and criticism, and Dr. MangalKothari and Dr. Sharvari Nadkarni-Ghosh for their guidance. Aside from this, I’d liketo thank Dr. Suresh Gupta for allowing me to work on this project as a part of mycourse-work at BITS-Pilani.

A sincere thanks is extended to Dr. Sai Jagan Mohan for shaping the way I look atproblems in Engineering and Applied Mathematics.

I would also like to express great admiration and gratitude towards the sagesof ancient India who have figured out almost everything that a human needs toreconcile himself with the external world.

vii

Contents

1 Background 11.1 Constitution and Geometry of Planetary Rings . . . . . . . . . . . . . 11.2 Planetary Rings as a Continuum: Kinetic Theory . . . . . . . . . . . . 3

2 Kinetic Theory of Granular Gases: Chapman-Enskog Perturbation Ap-proach 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Inelastic Collision Model . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Statistical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Evolution of f1: Boltzmann Equation . . . . . . . . . . . . . . . . . . 82.5 Maxwell Transport Equation . . . . . . . . . . . . . . . . . . . . . . . 102.6 Chapman-Enskog Perturbation Approach . . . . . . . . . . . . . . . . 11

3 Planetary Rings around a Spherical Central Body 153.1 Kinetic Theory Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Steady Axisymmetric Flow and Constitutive Relations . . . . . . . . . 153.3 Numerical Method and Results . . . . . . . . . . . . . . . . . . . . . 21

4 An aside: Equilibrium Shape of Axially Symmetric Liquid Drop on aMembrane 254.1 Problem Statement and Dimensionless Groups . . . . . . . . . . . . . 254.2 Governing Equations and Scaling . . . . . . . . . . . . . . . . . . . . 264.3 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Rings about Oblate Bodies 315.1 First-Order Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Numerical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Bibliography 35

ix

1Background

„Mathematical science is in my opinion anindivisible whole, an organism whose vitality isconditioned upon the connection of its parts.

— David Hilbert(Mathematician)

1.1 Constitution and Geometry of Planetary Rings

Giant planets are known to possess complex ring-satellite systems. Although theSaturnian system of rings is the most prominent, rings have also been discoveredaround the other three gas giants( Jupiter, Uranus and Neptune). Recently, ringshave been discovered around minor planet Chariklo as well.

Planetary rings are made up of countless particles with sizes from specks of dust tosmall moons. All rings lie predominantly within the planet’s Roche limit, where tidalforces would destroy a self-gravitating body composed of ideal fluid. Above is anartist’s depiction of Saturn’s rings as a thin axially symmetric disk.

After centuries of observations and recent explorations by spacecrafts such as Cassiniand Voyager, the following features of rings have come to light:

1

Fig. 1.1: Rings as flat cosmic structures: A Cassini image of the Saturnian system and Titan

• Flatness of Rings: Most planetary rings are flat cosmic structures. Compared totheir huge lateral extent of 80,000 kilometres, the vertical thickness of Saturn’srings is just 100 metres. Thus, Saturn’s rings are about 1000 times thinner thana razor blade.[SS06] Such a vast scale separation in the vertical and lateralextent justifies the approximation of planetary rings as thin annular disks.

Heuristically, frequent inelastic collisions and self-gravity between ring particlesprovide a mechanism for the stability of such a flat structure. Any ring particledislodged into an inclined orbit would be attracted to the ring mid-plane as aresult of self-gravity. Collisional damping of the velocity component normalto the ring mid-plane stabilises the nearly planar orbit of the ring particle.Heuristic insight is commensurate with the general observation of a ring whosethickness h is O(dp), where dp is the average particle diameter.

• Ring Composition and Coefficient of Restitution: The ring composition is well-understood only for Saturn, in which case the ring particles are made almostentirely of water ice. For other ring systems, particles contain significantamounts of silicate and, in the case of Uranus and Neptune, carbonaceousmaterial. Direct laboratory measurements of collisions between icy bodiesshow a declining coefficient of restitution with an increasing impact velocity.Researchers have argued that this is a necessary condition for the thermalstability of planetary rings as any other kind of dependence might increase

2 Chapter 1 Background

the temperature of rings, by way of collisional dissipation, to cause evapo-ration. Estimates based on astronomical observations are also available inliterature.[Esp14] A ring with an optical depth τ = 1.2 is expected to have anequilibrium coefficient of restitution ep = 0.9, as per these estimates.

• Size Distribution of Ring Particles: The current understanding of the size distri-bution of ring particles is that the particles have a very broad size distribution,characterised by power-law probability distributions. However, modellingthe ring as a bi-disperse granular gas might also lead to good qualitativeunderstanding of its structure. [Esp14]

• Oblateness of the Central Body: Gas giants like Saturn and Jupiter are known topossess figures that can be approximated as oblate spheroids, and the extent oftheir oblateness can be characterised by a flattening parameter f = a−b

a whichincreases with the oblateness. f takes on the values 0.1 and 0.06 for Saturnand Jupiter respectively. Thus far, no systematic study has been undertakento understand the effect of central-body oblateness on the structure of theplanetary ring. Rings around highly flattened central bodies like Chariklo, withf = 0.31, can be expected to possess a very different structure.[BR14]

• Other interesting phenomena associated with rings: Planetary rings exhibit otherinteresting phenomena such as dark lanes, gaps, and other density variations;eccentric and inclined rings; wakes and waves. Many of these effects havebeen attributed to ring-satellite interactions. The interplay between frequentcollisions between ring particles and self-gravity establish an equilibrium ofrubble-pile aggregates, continuously undergoing collision-induced disruptionand accretion due to self-gravity. [Esp14]

1.2 Planetary Rings as a Continuum: KineticTheory

The kinetic theory of granular gases finds a natural application in the study ofplanetary rings. In most studies, rings are modelled as thin axially symmetricdisks composed of identical, spherical particles that interact via binary, inelasticcollisions.

Kinetic theories average microscopic balance laws over a large number of particles toprovide the study of granular gases an elegant continuum framework, commensuratewith the simple microscopic understanding of inelastic collisions. The frameworkleads to governing equations which are similar to those of a compressible and

1.2 Planetary Rings as a Continuum: Kinetic Theory 3

heat-conducting Newtonian fluid with an additional collisional dissipation term.Analytical approximations for the transport coefficients like viscosity and thermalconductivity can be derived in terms of the microscopic parameters like dp, mp

andep, which gives the study a precision that cannot be arrived at by heuristichydrodynamical theories.[Esp14] However, the kinetic theory approach neitheradmits phenomena like accretion and disruption of particles within its fold nor doesit take into account the non-trivial size distribution of ring particles.

Collisional dissipation acts as a sink for the fluctuational energy, which leads tocondensation of the granular gas unless the mean motion is driven by an externalinfluence such as gravity or imposed shear.

The flow within the ring assumes a steady Keplerian velocity distribution as a resultof balance between centrifugal force and the central body’s gravity . Self-gravitybetween ring particles provides a cohesive influence that makes the rings denser andthinner. The energy equation is interpreted as a balance between the rates of viscouswork and collisional dissipation and conduction of fluctuational energy.[Jen12]

4 Chapter 1 Background

2Kinetic Theory of Granular Gases:Chapman-Enskog PerturbationApproach

2.1 Introduction

The term ’granular gas’ is used to refer to the bulk motion of a collection of grainsin a state of constant agitation, interacting via binary, inelastic collisions. Thispicture differs from molecular gases due to the fact that kinetic energy is lost duringcollisions.

Inelastic collisions lead to a sink in the fluctuational energy. This means that, unlikemolecular gases, that can have fluctuational motion even in static equilibrium,granular gases undergo condensation unless driven by an external influence.

Planetary rings are a good example of a granular gas in a state of equilibrium, withthe sink in fluctuational energy balanced by the constant supply of energy to themean motion by the planet’s gravitational attraction.

Another application of the kinetic theory described below is the motion and defor-mation of a self-gravitating cloud of a gas composed of smooth, slightly inelasticspheres. Such a study within the purview of homogeneous dynamics of deformablebodies could lead to interesting insights into structure formation and evolution ofinter-galactic and proto-planetary dust clouds.

The following assumptions are imposed to simplify the theory and make analyticalprogress:

• Particles interact only through binary collisions. ep = 1 corresponds to pefectlyelastic collisions.

• The particles are assumed to be identical spheres with mass mp and diameterdp.

5

Fig. 2.1: Inelastic Collision Model -The pre-collision velocities are ~c1 and ~c respectively.Post-collision velocities are ~c1

′ and ~c′ respectively.

• The collisions between particles are inelastic and the coefficient of restitutionep is constant.

• No tangential impulse is exerted at the point of contact during a collision i.e.the particles are smooth.

2.2 Inelastic Collision Model

The simple inelastic collision model relates the post-collisional velocities to thepre-collisional velocities. The normal component of the pre-collisional approachvelocity ~g = ~c1 − ~c gets damped by a factor of ep in a collision. Here, ~k is the unitvector along the line joining the centers of the two particles.

~g′ · ~k = −ep(~g · ~k) (2.1)

Momentum conservation implies that both particles experience the same impulseduring the collision

~c1′ = ~c1 − ~J,~c′ = ~c+ ~J (2.2)

6 Chapter 2 Kinetic Theory of Granular Gases: Chapman-Enskog Perturbation Approach

Further, the constraint of frictionless collisions imply that the impulse has zerotangential component i.e. (~k × ~J)× ~k = 0. Thus,

~J = η1(~g · ~k)~k (2.3)

where η1 = (1 + ep)/2 as determined from (2.1) and (2.2).

Consequently, the loss in kinetic energy in a single collision is calculated as

∆K = −mp

4 (1− e2p)(~g · ~k)2 (2.4)

The loss in kinetic energy in a single collision is commensurate with how we definethe perturbation parameter ε = 1 − e2

p quantifying the degree of inelasticity of agranular gas.

2.3 Statistical Preliminaries

Frictionless collisions mean that, during a collision, there is no torque exerted oneither particle. This means that the angular momentum remains fixed for eachparticle in the flow. The 6-dimensional phase space of position and velocity (~r,~c) issufficient to describe the state of a given particle. Further, we can define a smoothdistribution function f1(t, ~r,~c) that characterises the spatial and velocity distributionof particles. The number of particles in the infinitesimal phase volume d~rd~c centredabout the phase point (~r,~c) is f1(t, ~r,~c)d~rd~c . The number density of particles is nowrecovered as

n(t, ~r) =∫f1(t, ~r,~c)d~rd~c (2.5)

Now, we can define the continuum variables as averages of corresponding micro-scopic properties ψ(~c). The bulk density is recovered as ρ(t, ~r) = n(t, ~r)mp. Similarly,we define the mean velocity as

n(t, ~r)~v(t, ~r) =∫~cf1(t, ~r,~c)d~c (2.6)

2.3 Statistical Preliminaries 7

The fluctuational velocity is defined as ~C = ~c − ~v. And the granular temperatureis then defined as the two-thirds of the mean fluctuational kinetic energy per unitmass,

32n(t, ~r)T (t, ~r) =

∫C2f1(t, ~r,~c)d~c (2.7)

The particles interact through frequent binary collisions. To quantify the probabilityof such collisions, it is important to define the distribution function for a pair ofparticlesf2(t, ~r1, ~c1, ~r2, ~c2) wherein the number of pairs with one particle located ind~r1d~c1 and another located in d~r2d~c2 is f2(t, ~r1, ~c1, ~r2, ~c2)d~r1d~c1d~r2d~c2.

The mechanical state of a granular gas is completely determined by the distributionfunction f1. Determining the distribution function for non-equilibrium systems,however, is a difficult mathematical problem, as we shall see. Therefore, we workwith analytical approximations about the base state of a molecular gas at equilibrium.For the equilibrium state of a gas of elastic particles, the distribution function isnormal in the particle speed and is known as the Maxwell-Boltzmann velocitydistribution.

2.4 Evolution of f 1: Boltzmann Equation

In the absence of collisions, the particles situated in the phase volume d~rd~c move tod~rd~c where ~r = ~r + ~cdt and ~c = ~c+~bdt. This implies

f1(t+ dt, ~r, ~c)d~rd~c = f1(t, ~r, ~c)d~rd~c (2.8)

The Jacobian of the transformation acting on the phase volume J = 1 +O(dt2), andthus the volume remains conserved. Taylor expanding f1(t+ dt, ~r, ~c) about (t, ~r,~c)and dividing the equation by dt, we get the Boltzmann equation for the evolution off1

∂f1

∂t+ ~c · ∇f1 +~b · ∇cf1 = ˙fcoll (2.9)

where ˙fcoll is the rate at which collisions change the distribution function.


To determine ˙fcoll, we must determine the net rate of influx of particles in theinfinitesimal phase volume d~rd~c. We consider two kinds of collisions, direct andinverse collisions. Direct collisions are those collisions that push particles outsidethe differential phase volume d~rd~c centred at (~r,~c) and inverse collisions are thosecollisions that bring a particle inside the same differential phase volume. Integratingover all possible velocities of particle 1, ~c1 and all possible orientations of collisions ~kwithin the constraint ~g · ~k > 0 that admits only impending collisions, the Boltzmannequation reduces to the form

∂f

∂t+ ~c · ∇f +~b · ∇cf = d2

p

∫~g·k>0

[1e2p

f ′′1 (t, ~r + dpk)f ′′(t, ~r)g(t, ~r + dpk, ~r)

− f1(t, ~r − dpk)f(t, ~r)g(t, ~r − dpk, ~r)]~g · kd~kd~c1

(2.10)

Here, g(t, ~r1, ~r) = n2(t, ~r1|~r)/n is the pair distribution function characterising theconditional probability of finding a particle in the infinitesimal volume d~r1, given thatthere is a particle present at ~r. This provides a correction to Boltzmann’s assumptionof molecular chaos1, which breaks down in the case of dense molecular gases andgranular gases which are typically dense.

The Boltzmann equation for hard, smooth, inelastic spheres is a non-linear integro-differential equation and yields only to analytical approximations. The Chapman-Enskog perturbation approach expands the non-equilibrium distribution functionin an asymptotic expansion about the Maxwell-Boltzmann distribution for a gas ofelastic spheres, that is

f(~c) = n

(2πT )32exp(−C

2

2T ) (2.11)

where ~C = ~c− ~v is the fluctuational velocity.

For calculating collisional fluxes, the knowledge of pair distribution function atcontact g(t, ~r+dpk, ~r) = g0(ν(~r+ dp

2 k)) under the assumption of isotropy is sufficient.

1Boltzmann proposed that the two-particle distribution function is the product of the individual singleparticle distribution functions in dilute molecular gases, which means that the particle velocitiesjust prior to a collision are uncorrelated.

2.4 Evolution of f1: Boltzmann Equation 9

Typically used is the Carnahan-Starling formula for the isotropic radial distributionfunction

g0(ν) =1− ν

21− ν3 (2.12)

2.5 Maxwell Transport Equation

Any microscopic property of a granular gas ψ(~c) has a corresponding bulk property〈ψ〉 whose transport equation can be derived by multiplying ψ(~c) with the Boltzmannequation and integrating over the velocity space. This is known as the Maxwelltransport equation

∂(n〈ψ〉)∂t

+∇ · (n〈~cψ〉) = n〈~b · ∇cψ〉+ ψcoll (2.13)

where ψcoll is the rate of change per unit volume due to collisions. After symmetriz-ing, the expression for the collisional rate of change becomes

ψcoll = −∇ · θ + χ (2.14)

where θ is the collisional flux and χ is the collisional source. The expressions forthese are not included for the sake of brevity.

The Maxwell Transport Equation assumes the form of mass, momentum and energybalance under suitable choices of the microscopic property ψ(~c).

For ψ = mp, we get the equation of continuity

Dρ

Dt= −ρ∇ · ~v (2.15)

For ψ = mp~c, we get linear momentum balance for a deformable continuum

ρD~v

Dt= −∇ · σ + ρ~b (2.16)


For the balance of fluctuational energy, we subtract the mechanical energy balancefrom the energy equation, which leads to

32ρDT

Dt= −∇ · ~q − σT : ∇~v − Γ (2.17)

Now, for closure of these equations, we need constitutive relations for the fluxes σand ~q. The sink in fluctuational energy Γ is the collisional dissipation term presentdue to the inelasticity of collisions. But we require the knowledge of non-equilibriumdistribution function in order to close these hydrodynamic equations.

2.6 Chapman-Enskog Perturbation Approach

The Chapman-Enskog perturbation approach exploits the presence of two smallparameters. These are ε = 1 − e2

p, the degree of inelasticity, and K = πdp

6νsH, the

inverse Knudsen number. The perturbation approach expands the distributionfunction in an ansatz about the local Maxwellian f = f0(1 + Φ) where Φ is givenby

Φ = KΦK + εΦε +O(K2; ε2) (2.18)

The O(K) and O(ε) corrections to the distribution functions are derived by solvingthe Boltzmann equation at O(K) and O(ε). The resultant corrections are used tocorrect the leading-order constitutive relations.

O(1) Constitutive Relations:The leading order constitutive relations are obtained by evaluating the expressionsfor σ, ~q and Γ by taking the distribution function to be the local Maxwellian f0. Weget

σ0 = (1 + 4νg0)νT ; ~q0 = 0; Γ = 0 (2.19)

These constitutive relations describe a dense molecular gas which behaves as a com-pressible, inviscid fluid without heat conduction or collisional dissipation. Kinetictheory shows that viscous behaviour is only seen at O(K), known as Navier-Stokes

2.6 Chapman-Enskog Perturbation Approach 11

order.

O(K) Constitutive Relations:The expression for the first-order correction in K, ΦK , is a linear function of thelogarithm of the mean-field temperature gradient∇lnT and the symmetrised velocitygradient tensor D. These mean-field gradients exist as the source terms in thelinearised Boltzmann equation at O(K) as O(K) behaviour describes the behaviourof a gas of elastic particles with spatial inhomogeneity. The O(K) corrections to theconstitutive relations are recovered as

σK = −6νsπ

[2µD + µb(∇ · ~v)I

](2.20)

~qK = −6νsπ

κ∇T (2.21)

ΓK = 0 (2.22)

where µ is the dimensionless shear viscosity, µb is the dimensionless bulk viscosityand κ is the dimensionless pseudo-thermal conductivity. Expressions for thesetransport coefficients may be found in [NR06]. We notice that O(K) behaviour iscommensurate with a compressible, heat-conducting Newtonian fluid. CombiningO(1) and O(K) behaviour completes the constitutive description of a dense gas ofelastic particles with externally imposed mean-field gradients.

O(ε) Constitutive Relations:The first-order correction in ε results in an isotropic contribution to the Φ. The O(ε)corrections to the constitutive relations are obtained as

σε = −ν2g0T I (2.23)

qε = 0 (2.24)

Γε = 12√πν2g0T

32 (2.25)


O(ε) corrections lead to the inclusion of a dissipation term in the energy equation,and a correction to the pressure.

Constitutive Relations to the First Order in K and ε:

Combining the aforementioned results leads to the complete description of a fluidcomposed of smooth, inelastic spheres. The stress tensor in dimensional form is

σ = [p− µb∇ · ~v]I− 2µD (2.26)

where p = ρb(1 + 4η1νg0)νT is the pressure.

The flux of fluctuational energy attains the well-known form

~q = −κ∇T (2.27)

Finally, the rate of energy dissipation attains the dimensional form

Γ = 12√π

(1− e2p)ρpT

32

dpν2g0 (2.28)

The expressions for transport coefficients are given in [NR06] One can now workwith these constitutive relations to study the structure of planetary rings, which isdone in the next two chapters.

2.6 Chapman-Enskog Perturbation Approach 13

3Planetary Rings around aSpherical Central Body

3.1 Kinetic Theory Model

Balance equations:The planetary ring is assumed to be composed of identical, smooth, inelastic, spheresof density ρ and diameter d, interacting via binary collisions with given coefficient ofrestitution ep. In [Jen12], e = e2

p is specified instead. The balance equations for sucha continuum are

Dρ

Dt= −ρ∇ · ~u (3.1)

ρD~u

Dt= ∇ · t + ρ~f (3.2)

32ρDT

Dt= tr(tD)−∇ · ~q − Γ (3.3)

where D is the symmetric part of the velocity gradient, ~q is the flux of fluctuationalenergy, and Γ is the rate of collisional dissipation.

The constitutive relations used for closing this set of balance equations are describedin the previous chapter.

3.2 Steady Axisymmetric Flow and ConstitutiveRelations

Assumptions related to geometry:

15

The ring is described as a thin, axially symmetric disk in the cylindrical co-ordinatesystem with the spherical central body placed at the origin and z = 0 as the mid-plane of the ring. The mean velocity field is assumed to be purely circumferentiali.e. ~u = u(r)θ as an approximation. The ring is also assumed to possess reflectionsymmetry with respect to its mid-plane. As an approximation, the derivatives in rare considered to be much smaller than the derivatives in z.

Constitutive relations:

The constitutive relations provide expressions for the pressure p = −tr(t), the shearstress S = trθ, the flux of fluctuational energy Q = qz and the rate of collisionaldissipation Γ.

S = µ∂u

∂r(3.4)

p = ρ(1 + 4νg0)T (3.5)

Q = −κ∂T∂z

(3.6)

Γ = 24√π

ρνg0d

(1− e)T32 (3.7)

µ = 8J5√πρdνg0T

12 (3.8)

J = 1 + π

12

(1 + 5

8νg0

)2

(3.9)

κ = K√πρdνg0T

12 (3.10)

K = 1 + 9π32

(1 + 5

12νg0

)2

(3.11)

16 Chapter 3 Planetary Rings around a Spherical Central Body

κ and µ are the transport coefficients whose expressions are known from kinetictheory.[NR06] The Carnahan-Starling formula for the radial distribution functionis employed here g0 = 1−ν/2

(1−ν)3 . This kinetic theory works for the volume fractionν = nπd3/6 less than .49.

The body force for a spherical planet is given by the standard expressions:

fr = −GM r

(r2 + z2)32

and fz = −GM z

(r2 + z2)32

(3.12)

where G is the gravitational constant and M is the mass of the central body. We ob-serve that the z-component is much weaker than the r-component as a consequenceof z/r << 1

The r-momentum balance leads to a simple balance between the inward radialbody force and the outward centrifugal force. As an approximation, we ignore theself-gravity in the radial direction, supposing it to be much weaker than the radialcomponent of the body force due to the planet.

u2

r= GM

r

(r2 + z2)32

(3.13)

whence forth the expression for u(r) is derived in the approximation of zr << 1

u = Ωr where Ω =

√GM

r3 (3.14)

Thus, the mean flow inside a planetary ring is a radial shearing flow where the innerlayers rotate faster than the outer layers.

We can write the z-component of the body force at O( zr )

fz = −GMz

r3 = −Ω2z (3.15)

3.2 Steady Axisymmetric Flow and Constitutive Relations 17

The component of self-gravity acting in the z-direction is

fself = −2πG∫ z

−zρ(ξ)dξ (3.16)

The z-momentum balance leads to a balance between the z-component of the bodyforce ,due to the planet’s gravity, the z-component of self gravity and the pressuregradient. This equation is analogous to the vertical component of the hydrostaticcondition in the fluid. Self-gravity can thus be interpreted as a cohesive influence.

∂p

∂z= −ρzΩ2 − 2πGρ

∫ z

−zρ(ξ)dξ (3.17)

The energy equation at O( zr ) in the steady state assumption attains the followingform

∂Q

∂z= S

∂u

∂r− Γ (3.18)

Optical Depth Specification: Optical depth is a measure of the opaqueness of theplanetary ring. Occultation of the incident radiation provides an observationaldefinition of the optical depth.

Im = I0e−τ (3.19)

In the context of kinetic theory, optical depth is defined as the average number ofparticles encountered in the path of incident light multiplied with the projectedsurface area of each particle. It can be redefined in terms of the volume fraction νafter re-scaling z by the particle diameter d.

τ = πd2

4

∫ ∞−∞

n(z)dz = 3∫ ∞

0ν(ξ)ξ (3.20)

ODE BVP Formulation:Scaling the lengths by d, velocities by Ωd, stresses by ρp(Ωd)2 and energy flux(Q) by


ρp(Ωd)32 , the dimensionless pressure and shear stress can be written in terms of the

dimensionless measure w = T12 of the mean fluctuational velocity.

p = 4ν2g0Fw2 where F = 1 + 1

4νg0(3.21)

S = − J

5√π

p

Fw(3.22)

The dimensionless form of the energy equation becomes

dQ

dz= −6√

π

[(1− e)− 5π

12J

(FS

p

)2]pw

F(3.23)

The dimensionless form of the constitutive relation for thermal energy flux be-comes

dw

dz= −√π

2FQ

Kp(3.24)

We define I as a dummy variable to specify the optical depth as a boundary conditionand to convert (3.17) from an integral equation to a differential equation.

I(z) = 3∫ z

0ν(ξ)dξ (3.25)

Consequently, (3.17) takes on the following form

dν

dz= −

[νz −

√πFQ

Kw+AνI(z)

]1

4w2 d(ν2Fg0)dν

(3.26)

Here, A is a dimensionless parameter specifying the strength of the self-gravity.

A = r3

(rs)3ρp

ρs(3.27)

3.2 Steady Axisymmetric Flow and Constitutive Relations 19

where rs and ρs are the radius and the density of the planet respectively.

Additionally, we write a differential equation in terms of I as well

dI

dz= 3ν (3.28)

Equations (3.26), (3.24), (3.23) and (3.28) is the resultant one-dimensional ODEBVP system that we need to solve in order to study the vertical structure of aplanetary ring.

Boundary Conditions:

Due to equatorial symmetry, the granular temperature field possesses a reflectionsymmetry about the plane. Also, above the point z = H that demarcates thecollisional flow from the the collision-less flow, there is no energy flux due tocollisions. Thus,

Q(0) = Q(H) = 0 (3.29)

Also, by the definition of I, we obtain more Drichlet boundary conditions.

I(0) = 0 andI(H) = τ − τ b (3.30)

where τ b is the contribution of the collision-less flow to the optical depth.

τ b = 3√π

2 ν(H)w(H) (3.31)

Now, the half-thickness of the ring H is an unknown parameter of the problem.Thus, the boundary value problem in question is an inverse problem. To solve it, wemake an a priori guess for H and and iterate until the pressure boundary conditionis satisfied.

p(H) = 4ν2g0Fw2 = .039 (3.32)


Fig. 3.1: No Self-Gravity: Volume Fraction Profiles

3.3 Numerical Method and Results

Numerical Method:

To solve the above Boundary-Value Problem, MATLAB’s two point BVP-solver wasused in conjunction with the bisection method to determine H such that the pressureboundary condition is satisfied.

The problem is solved for different values of the optical depth τ and the self-gravityparameter A. The volume fraction and granular temperature profiles are obtained.The numerical method could not capture the dilute branch of solutions reported in[Jen12].

Results:

The problem is solved in the absence of self-gravity for τ = 1.0, 1.1, 1.2. Thecorresponding volume fraction and mean fluctuational velocity profiles are obtainedand represented in the following plots. We notice that the rings become denser andthe granular temperature decreases as the optical depth is increased.

3.3 Numerical Method and Results 21

Fig. 3.2: No Self-Gravity: Granular Temperature Profiles

Another set of solutions is obtained by keeping the self-gravity fixed at A = 8 andvarying τ . Again, we observe that the rings become denser with increasing opticaldepth. (Figure 3.3)

The last set of results study how the ring behaves as the self-gravity is increased atthe same value of the optical depth τ = 1.0. The corresponding volume fraction andmean fluctuational velocity plots are in Figure 3.4 and Figure 3.5 respectively.

We notice that the rings become thinner and denser and the granular temperaturedecreases upon increasing the strength of self-gravity.


Fig. 3.3: A= 8, Volume Fraction Profiles

Fig. 3.4: Volume Fraction Profiles for A =0,2,4,6,8

3.3 Numerical Method and Results 23

Fig. 3.5: Granular Temperature Profiles for A=0,2,4,6,8


4An aside: Equilibrium Shape ofAxially Symmetric Liquid Drop ona Membrane

4.1 Problem Statement and DimensionlessGroups

The problem can be simply formulated as follows:

Given a liquid drop of volume V0, density ρl and surface tension σ kept on an circularelastic membrane with tension TII and radius l such that it makes a contact angle ofθe, find the equilibrium interface of the drop h(r) and the membrane deformation η1(r)and η2(r) corresponding to the loaded and the unloaded part of the membrane.

Inverse Approach:

Taking the inverse approach to solve the problem, we specify the droplet spreadb instead of the volume V and the tension in the loaded membrane TI instead ofthe tension in the unloaded membrane TII . We shall subsequently invert using abisection method to satisfy the volume constraint.

The dimensionless numbers that define the inverse formulation are as follows:

Bo = ρlgb2

σ R = σTI

θe

In the following sections, the problem is formulated and solved for the restrictedcase of R → 0 and θe <

π2 . This translates to determining equilibrium shapes of

sessile drops kept on nearly elastic hydrophilic membranes.

25

4.2 Governing Equations and Scaling

The governing equations for the region bounded by the drop interface and the loadedmembrane are given by the hydrostatic condition.

∂p

∂r= 0 and

∂p

∂y= −ρlg when 0 6 r 6 b and η1(r) 6 y 6 h(r) (4.1)

Thus, the hydrostatic pressure field can be calculated as

p = K − ρlgy (4.2)

The drop boundary condition is the Young-Laplace equation.

p = σ∇ · n at y = h(r) (4.3)

At O(R), the membrane is loaded vertically with constant Tension. Thus, the bound-ary condition corresponding to the membrane becomes

TI∆η1 = p at y = η1(r) (4.4)

We scale r, y and h by b and η1 by d = Rb. The dimensionless boundary conditionsthen attain the following simple form, in the loaded part of the domain.

∆η1 +RBoη1 = K = hBo+∇ · n when 0 6 r 6 1 (4.5)

We call the loaded part of the membrane Domain I and the unloaded part ofthe membrane Domain II, and a triple point boundary condition is derived as aconsequence of force balance at r = 1.

Domain I (0 6 r 6 1):

26 Chapter 4 An aside: Equilibrium Shape of Axially Symmetric Liquid Drop on a Membrane

The drop interface is determined by the following non-linear differential equation.

hBo+∇ · n = K when 0 6 r 6 1 (4.6)

The following boundary conditions are imposed on h(r).

hr(0) = 0 = h(1) (4.7)

The membrane deformation is given by the following linear differential equation.

∆η1 +RBoη1 = K when 0 6 r 6 1 (4.8)

The following homogeneous boundary conditions are imposed on η1(r).

η1r(0) = 0 = η1(1) (4.9)

The contact angle condition can be satisfied by choosing K such that

tan−1(Rη1r(1))− tan−1(hr(1)) = θe (4.10)

The triple point force balance condition leads to the following constraints relatingthe slopes and tensions of the loaded and unloaded membranes.

TII = TI√

1 +R2 + 2R cos θe (4.11)

tanψ = sinφ+R sin ξcosφ+R cos ξ (4.12)

Domain II (1 6 r 6 l/b):

4.2 Governing Equations and Scaling 27

The unloaded membrane’s deformation in this domain is described by the Laplaceequation.

∆η2 = 0 (4.13)

with the following boundary conditions

η2r(1) = tanψ and η2(1) = 0 (4.14)

The solution to this is a frustum of a cone given by

η2 = tanψ(r − 1) (4.15)

4.3 Numerical Solution

The membrane in Domain I possesses the following analytical solution

η1(r) = K

RBo

[1− J0(r

√RBo)√

RBo

](4.16)

We can thus calculate the slope of the membrane at the triple point as

tanφ = RK√RBo

J1(√RBo)

J0(RBo) (4.17)

The drop equation can be split into a first-order system.

hr = p and pr = (hBo−K)(1 + p2)32 − p/r (4.18)

We notice that there is a singularity at r =0. We use Matlab’s bvp4c to solve thiscoupled problem and fix K by a bisection-based iteration so that the contact anglecondition is satisfied. Some sample equilibrium shapes are represented by thefollowing plots.


Fig. 4.1: Low Bond Number- Nearly Spherical Drop

Fig. 4.2: Moderate Bo Number

Fig. 4.3: High Bo Number- Highly Flattened Drop

4.3 Numerical Solution 29

Fig. 4.4: Low Surface Tension to Tension Ratio

Fig. 4.5: High Surface Tension to Tension Ratio


5Rings about Oblate Bodies

5.1 First-Order Corrections

Formulation: The gravitational potential for an oblate spheroid linearised in theellipticity parameter assumes the following expression in the spherical polar co-ordinates.

φ(R, θ) = −GMR

+ 2GMa2ε

5R3 P2(sin θ) (5.1)

We re-write this in the cylindrical polar co-ordinates.

φ(r, z) = − GM

(r2 + z2)12

+ GMa2ε

5(r2 + z2)32

( 3z2

r2 + z2 − 1)

(5.2)

Thus, components of body force can be calculated as ~f = −∇φ.

fr = − GMr

(r2 + z2)32

+ GMra2ε

5(r2 + z2)52

( 15z2

r2 + z2 − 3)

(5.3)

fz = − GMz

(r2 + z2)32

+ GMza2ε

5(r2 + z2)52

( 15z2

r2 + z2 − 9)

(5.4)

Now, we reduce the expressions to O(z/r) because the lateral extent of the ring ismuch greater than its vertical extent, which yields

fr = −GMr2

(1 + 3ε

5a2

r2

)and fz = −GMz

r3

(1 + 9ε

5a2

r2

)(5.5)

31

The r-momentum balance, while ignoring the self-gravity in r-direction leads to

u2

r= GM

r2

(1 + 3ε

5a2

r2

)(5.6)

Thus, the θ-component of the velocity is given by

u = G1Ωr where G1 =

√1 + 3ε

5a2

r2 and Ω =

√GM

r3 (5.7)

To evaluate the shear stress S = µdudr , we take the derivative of the the θ-componentof velocity.

du

dr= −Ω

2

(G1 + 3ε

5G1

a2

r2

)(5.8)

The z-components of the central body’s force field and the self-gravity at O(z/r)corrected to the first order are

fz = −G22Ω2z where G2 =

√1 + 9ε

5a2

r2 (5.9)

fself = −2πG∫ z

−zρ(ξ)dξ (5.10)

Thus, the z-momentum balance assumes the following form, which indicates thatthe self-gravity has a cohesive influence. The first-order correction in ellipticity addsa factor G2 to the z-momentum balance.

∂p

∂z= −ρG2

2Ω2z − 2πρG∫ z

−zρ(ξ)dξ (5.11)

Similarly, the energy equation at O(z/r) can be written as

∂Q

∂z= S

∂u

∂r− Γ (5.12)

32 Chapter 5 Rings about Oblate Bodies

ODE BVP: Lengths are scaled by d, velocities by Ωd, stresses by ρp(Ωd)2 and energyflux by ρp(Ωd)

32 .

Thus, the expressions for pressure and shear stress are given by

p = 4ν2g0Fw2 where F = 1 + 1

4νg0(5.13)

S = − J

5√π

p

Fw

(G1 + 3ε

5G1

a2

r2

)(5.14)

The dimensionless form of the energy equation becomes

dQ

dz= − 6√

π

[(1− e)− 5π

12J

(FS

p

)2]pwF

(5.15)

The constitutive relation for the conduction of fluctuational energy becomes

dw

dz= −√π

2FQ

Kp(5.16)

The equation in terms of the dummy variable I remains the same.

dI

dz= 3ν (5.17)

Finally, the equation corresponding to the z-momentum balance becomes

dν

dz= −

[G2

2νz −√πFQ

Kw+AνI(z)

] 14w2 d(ν2g0F )

dν

(5.18)

The boundary conditions remain the same as the previous calculations.

5.1 First-Order Corrections 33

5.2 Numerical Calculations

The boundary-value problem is solved with the help of MATLAB’s bvp4c functionand the value of ring half-thickness is iterated until the pressure condition is satisfiedat the boundary.

p(H) = .04 (5.19)

The value of M = a2/r2 is kept at .4, .6 and .8 and the value of the flatteningparameter is kept at ε = 0, .15 and .3. The self-gravity is varied as A = 0 and 8. Thecoefficient of restitution and optical depth are kept constant at e = .8 and τ = 1.2.

No Self-Gravity:

The ring thickness increases as we increase the oblateness of the figure. Correspond-ing to this is a decrease in the volume fraction. The grain temperature also increasesas we increase the oblateness of the figure.Also, as we go closer to the central body, the ring thickness increases.

Ring half-thickness

ε = 0 ε = 0.15 ε = 0.3M=.4 1.3211 1.3461 1.3711M = .6 1.3211 1.3586 1.3945M = .8 1.3211 1.3711 1.4164

Volume Fraction

ε = 0 ε = 0.15 ε = 0.3M=.4 .2389 .2353 .2348M = .6 .2389 .2336 .2319M = .8 .2389 .2348 .2118

Grain Temperature

ε = 0 ε = 0.15 ε = 0.3M=.4 .4347 .4608 .4862M = .6 .4347 .4738 .5112M = .8 .4347 .4862 .5367

34 Chapter 5 Rings about Oblate Bodies

Bibliography

[BR14] Braga-Ribas. „A ring system detected around the Centaur (10199) Chariklo“. In:Nature 508 (2014), pp. 72–75 (cit. on p. 3).

[Esp14] Larry W. Esposito. Planetary Rings: A Post-Equinox View. Cambridge, U.K.: CambridgeUniversity Press, 2014 (cit. on pp. 3, 4).

[Jen12] James Jenkins. „Simple Kinetic Theory for a Dense Planetary Ring“. In: Progress ofTheoretical Physics 195 (2012) (cit. on pp. 4, 15, 21).

[NR06] Prabhu R. Nott and K. Kesava Rao. An Introduction to Granular Flow. Cambridge,U.K.: Cambridge University Press, 2006 (cit. on pp. 12, 13, 17).

[SS06] Frank Spahn and Jurgen Schmidt. „Hydrodynamic Description of Planetary Rings“.In: GAMM-Mitt 29 (2006), pp. 116–143 (cit. on p. 2).

35

List of Figures

1.1 Rings as flat cosmic structures: A Cassini image of the Saturnian systemand Titan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Inelastic Collision Model -The pre-collision velocities are ~c1 and ~c re-spectively. Post-collision velocities are ~c1

′ and ~c′ respectively. . . . . . . 6

3.1 No Self-Gravity: Volume Fraction Profiles . . . . . . . . . . . . . . . . . 213.2 No Self-Gravity: Granular Temperature Profiles . . . . . . . . . . . . . 223.3 A= 8, Volume Fraction Profiles . . . . . . . . . . . . . . . . . . . . . . 233.4 Volume Fraction Profiles for A =0,2,4,6,8 . . . . . . . . . . . . . . . . 233.5 Granular Temperature Profiles for A=0,2,4,6,8 . . . . . . . . . . . . . 24

4.1 Low Bond Number- Nearly Spherical Drop . . . . . . . . . . . . . . . . 294.2 Moderate Bo Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3 High Bo Number- Highly Flattened Drop . . . . . . . . . . . . . . . . . 294.4 Low Surface Tension to Tension Ratio . . . . . . . . . . . . . . . . . . . 304.5 High Surface Tension to Tension Ratio . . . . . . . . . . . . . . . . . . 30

37

Documents

Kinetic Theory for Dense Planetary Rings · 2018. 2. 18. · I would like to thank Prof. Ishan Sharma for introducing me to this ﬁeld and helping me learn kinetic theory of granular