MMNRAS , 1–7 (2019) Preprint 24 February 2020 Compiled using MNRAS L A TEX style file v3.0
Stability and Solution of the Time-DependentBondi-Parker Flow
Eric Keto (cid:63) Institute for Theory and Computation, Harvard College Observatory, 60 Garden St., Cambridge, MA 02138
Accepted XXX. Received YYY; in original form ZZZZ
ABSTRACT
Bondi (1952) and Parker (1958) derived a steady-state solution for Bernouilli’s equa-tion in spherical symmetry around a point mass for two cases, respectively, an inwardaccretion flow and an outward wind. Left unanswered were the stability of the steady-state solution, the solution itself of time-dependent flows, whether the time-dependentflows would evolve to the steady-state, and under what conditions a transonic flowwould develop. In a Hamiltonian description, we find that the steady state solution isequivalent to the Lagrangian implying that time-dependent flows evolve to the steadystate. We find that the second variation is definite in sign for isothermal and adiabaticflows, implying at least linear stability. We solve the partial differential equation forthe time-dependent flow as an initial-value problem and find that a transonic flowdevelops under a wide range of realistic initial conditions. We present some examplesof time-dependent solutions.
Key words: hydrodynamics; stars: winds, outflows, mass loss, formation
Spherical accretion onto a constant point mass (Bondi 1952)has found application in astronomy from stars to supermas-sive black holes and anywhere the self-gravity of the gas isinsignificant compared to the gravity of the point mass. Thesimplicity of the model combined with the non-trivial so-lution that includes a transonic critical point, has provenboth useful and interesting. The same equations but withoutward velocities (Parker 1958) has found application insolar and stellar winds and anywhere acceleration occurs asa result of a pressure-density gradient maintained by a gravi-tational field. The acceleration in the Parker wind is also thesame as occurs through a rocket nozzle with an exponentialshape.The Bondi-Parker (BP) flow is described by a combination ofthe continuity equation with Bernouilli’s equation, the lattera partial differential equation (PDE) for velocity and densityas functions of time and position. Assuming the time deriva-tive is zero results in a single ordinary differential equation(ODE) for the steady state with separable variables. Thetranscendental equation resulting from integration is easilysolved, for example with a Newton-Raphson technique or interms of the Lambert W function (Cranmer 2004). The con-stant of integration along with the branches of a quadratic (cid:63)
E-mail: [email protected] (EK) term results in a family of steady-state trajectories with ei-ther subsonic, transonic, or supersonic velocities.There have been many interesting variations of the BP flow.For example, the introduction of shock discontinuities linksdifferent trajectories in the steady-state family (McCrea1956). A non-isothermal equation of state results in multi-ple critical points (Kopp & Holzer 1976). Accounting for theself-gravity of the gas results in a similarity solution eitheras a function of time alone (Shu 1977) or time and radius(Dhang et al. 2016) depending on the equation of state andthe initial conditions.The stability of the steady-state solution has previouslybeen addressed with finite-difference methods (Balazs 1972;Stellingwerf & Buff 1978; Garlick 1979; Velli 1994; Del Zannaet al. 1998). These studies agree that the transonic flow isstable, but disagree about the stability of the subsonic andsupersonic flows based on differences in numerical methodsand choice of boundary conditions.In contrast, a Hamiltonian description of the flow deter-mines the evolution and stability independently of numericalmethods and boundary conditions. We find that the steady-state solution is equivalent to the Lagrangian for the flowand is thus the critical function for the first variation ofthe functional of the flow. This condition implies that time-dependent flows evolve to the steady-state/ We find thatthe second variation is definite in sign for isothermal and © a r X i v : . [ phy s i c s . f l u - dyn ] F e b Keto adiabatic equations of state. This implies that these time-dependent flows are at least linearly stable.The choice of trajectory in the steady-state family that re-sults from the evolution is determined by the initial veloci-ties. We use the method of characteristics to write the PDEfor the time-dependent BP flow as a pair of coupled ODEsfor velocity and position both as functions of time. Thesemay be solved numerically as an initial value problem (IVP),for example with a Runge-Kutta technique. These solutionsspecify the final trajectory for any initial conditions.Examples suggest that the time-dependent solutions evolveto a transonic flow from all initial values that lie within theregion bounded above and below by the inward and out-ward steady-state transonic trajectories. Since this regionextends asymptotically from velocities with absolute valuesfrom zero to infinity, a wide range of initial velocities resultsin a transonic flow.
Following Bondi (1952) and Parker (1958) we derive thesteady-state solution. If the gas is isothermal so that ∂ P / ∂ ρ = a for sound speed a , and the gravitational force isthat of a point of constant mass, M , then the Euler equationin spherical symmetry is, ∂ ˜ u ∂ ˜ t = − ˜ u ∂ ˜ u ∂ ˜ x − a ˜ ρ ∂ ˜ ρ∂ ˜ x − GM ˜ x (2.1)where the tilde indicates a variable with dimensional units.This can be written in non-dimensional form with the defi-nitions, ˜ x = (cid:18) GMa (cid:19) x , ˜ u = au , ˜ ρ = ˜ ρ ρ, and ˜ t = (cid:18) GMa (cid:19) t , (2.2)where ˜ ρ is an arbitrary density. With these substitutions,equation 2.1 is, ∂ u ∂ t = − u ∂ u ∂ x − ρ ∂ ρ∂ x − x . (2.3)The density may be eliminated with the help of the non-dimensional continuity equation, ρ = λ x − u − (2.4)where λ is the accretion rate. Then, ∂ u ∂ t = (cid:18) u − u (cid:19) ∂ u ∂ x + (cid:18) x − x (cid:19) . (2.5)In steady state the time derivative on the left-hand sideis zero, and the variables can be separated and integrated, L = log | u | − u + | x | + x . (2.6)In non-dimensional units, the constant of integration L = log λ is equivalent to the energy and with a simple non-linearscaling to the the mass accretion rate. This transcendentalequation can be solved either numerically, for example with Figure 1.
Representative trajectories of the steady-state BP flow(equation 2.6) in non-dimensional units. The velocities are shownas absolute values. The two transonic trajectories with L = L C are shown in pink. Green lines show two subsonic trajectories with L = L C − and L = L C − and two supersonic trajectories with L = L C + and L = L C + . The trajectories that are discontinuousacross position are shown in yellow for the same set of constants, L but represent different branches. a Newton-Raphson technique, or using the Lambert W func-tion (Cranmer 2004), u = ± (cid:26) − W (cid:20) − exp (cid:18) x + | x | + L (cid:19)(cid:21)(cid:27) / (2.7)or x = − W (cid:20) ±
12 exp (cid:18) u − ln | u | + L (cid:19)(cid:21) . (2.8)The family of solutions depends on the value of the integra-tion constant, L , and upon the branches of the quadraticterm or the Lambert W function. Representative solutionsare plotted in figure 1. Of particular interest are the twotransonic solutions, the Bondi accretion flow and the Parkerwind, both with L = L C = / − that cross atthe Bondi-Parker critical point, ( x , u ) = ( , ) or ( ˜ x , ˜ u ) = ( GM /( a ) , a ) . Also shown are subsonic flows with L < L C ,and supersonic flows with L > L C . The solutions that arediscontinuous in position derive from a different branch andare not accretion flows or winds. The region of the plot thatthey occupy is sometimes called the forbidden region. Amore complete discussion is found in Cranmer (2004) andHolzer & Axford (1970). Hamilton’s principle for a conservative system is, δ J = δ ∫ t t ( K − V ) dt = (3.1)where the difference of the kinetic and potential energies isthe Lagrangian, K − V = L . This may be generalized for afluid with internal energy, U (Herivel 1955), L = K − V − U . (3.2) MNRAS000
12 exp (cid:18) u − ln | u | + L (cid:19)(cid:21) . (2.8)The family of solutions depends on the value of the integra-tion constant, L , and upon the branches of the quadraticterm or the Lambert W function. Representative solutionsare plotted in figure 1. Of particular interest are the twotransonic solutions, the Bondi accretion flow and the Parkerwind, both with L = L C = / − that cross atthe Bondi-Parker critical point, ( x , u ) = ( , ) or ( ˜ x , ˜ u ) = ( GM /( a ) , a ) . Also shown are subsonic flows with L < L C ,and supersonic flows with L > L C . The solutions that arediscontinuous in position derive from a different branch andare not accretion flows or winds. The region of the plot thatthey occupy is sometimes called the forbidden region. Amore complete discussion is found in Cranmer (2004) andHolzer & Axford (1970). Hamilton’s principle for a conservative system is, δ J = δ ∫ t t ( K − V ) dt = (3.1)where the difference of the kinetic and potential energies isthe Lagrangian, K − V = L . This may be generalized for afluid with internal energy, U (Herivel 1955), L = K − V − U . (3.2) MNRAS000 , 1–7 (2019) ondi-Parker Flow In non-dimensional units, the Lagrangian for the BP flow isthe same as the steady-state equation, 2.6, where the non-dimensional terms, ( log | u | + | x |) , that derive from thedimensional pressure term in equation 2.1, are the internalenergy. If we define the functional gradient as the directionalderivative, ∇ J [ x ] = dd (cid:15) J [ x + (cid:15)η ] (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) = , (3.3)for any variation, η , and constant, (cid:15) , then the necessary con-dition of Euler-Lagrange for x to be a critical function of J [ x ] is that the first variation is zero (cid:104)∇ J [ x ](cid:105) = ∂∂ x L ( x , (cid:219) x , t ) − ddt (cid:18) ∂∂ (cid:219) x L ( x , (cid:219) x , t ) (cid:19) (3.4)where the velocity u = (cid:219) x .Since the right side is equivalent to the equation of motion2.5 as can be verified by substitution, then (cid:104)∇ J [ x ](cid:105) = ,and the steady-state solution, L , is the solution of sta-tionary action toward which the time-dependent solutionsevolve.This description is similar to the simple physics problem inwhich the Lagrangian determines the parabolic trajectoryof a particle in a gravitational field and the exact trajec-tory is determined from the end points as a boundary valueproblem (BVP) or alternatively from the initial velocities asan IVP. However, the Lagrangian for the BP flow includesthe internal energy of the fluid as a function of the density-pressure gradient which is constrained by the conservationequation. Therefore, it is not always possible to find an en-ergy conserving trajectory between two arbitrary velocities.Considered as a BVP, not all boundary conditions are con-sistent with a steady-state solution.As in Lagrangian mechanics, the energy serves as a Lya-punov function for the stability. The definiteness in sign ofthe second variation evaluated at the steady state implies atleast linear stability (Arnol’d, V.I. 1965, 1989). Equivalently,the integrand of the second variation, η ∂ L ∂ x + η (cid:219) η ∂ L ∂ x ∂ (cid:219) x + (cid:219) η ∂ L ∂ (cid:219) x (3.5)must be definite in sign for every nonzero variation η . Sincethe middle term with mixed partial derivatives is zero, and ∂ L ∂ x = − (cid:18) x + x (cid:19) (3.6) ∂ L ∂ (cid:219) x = − (cid:18) + (cid:219) x (cid:19) . (3.7)this is easily verified for the range x > allowed for BPflows. This implies that the steady-state solution, includingall the subsonic, transonic, and supersonic trajectories in thefamily, is at least linearly stable.The existence of a steady-state solution, a critical point inLagrangian mechanics, is required for stability. An arbitraryIVP can evolve to the steady-state, but the boundary con-ditions in a BVP must be consistent with the conservationof energy to allow a stable solution. This requirement itselfdoes not determine the steady-state solution or its stability. For example, the derivation of the Euler-Lagrange equationassumes that the functional to be maximized or minimizedhas values equal to two specified endpoints. In between, thesolution depends on the functional. To solve for the time-dependent BP flow, equation 2.5 weuse the method of characteristics to write the PDE as a pairof coupled first-order ODE’s that may be solved as an initialvalue problem. Equation 2.5 written as, d [ u ( t , x ( t ))] dt = ∂ u ∂ t + (cid:18) u − u (cid:19) ∂ u ∂ x = x − x , (4.1)and compared with the identity, dudt = ∂ u ∂ t dtdt + ∂ u ∂ x dxdt , (4.2)suggests the pair of coupled ODE’s, dxdt = u − u (4.3) dudt = x − x (4.4)to be solved with initial values, u ( t , x ) = f ( x ) at t = . Alter-natively, these may be derived from the Lagrange-Charpit(LC) equations for the PDE 2.5, dt = dxu − u − = du x − − x − . (4.5)The LC equations also yield a third ODE that is equivalentto the steady-state equation 2.6.Following the method of characteristics, we parameterize theODE’s with functions t = g ( τ, s ) and x = h ( τ, s ) to obtain thefollowing set of ODE’s with their initial conditions dtd τ = t ( , s ) = , (4.6) dxd τ = u − u with x ( , s ) = s , (4.7) dud τ = x − x with u ( , s ) = f ( s ) . (4.8)Here ( τ, s ) are the initial values for the trajectories, x ( t ) , u ( t ) .This set of coupled ODE’s may be solved numerically, forexample with a Runge-Kutta technique for u ( τ, s ) To com-plete the solution for u ( t , x ) , we need the inverse functions, t = g ( τ, s ) and x = h ( τ, s ) . From equation 4.6, τ is identi-cal to t . From a set of solutions of equations 4.7 and 4.8we determine x = h ( t , s ) to obtain the solution u ( t , h ( t , s )) = u ( t , x ) . MNRAS , 1–7 (2019)
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Example solutions suggest the proposition that a transonicflow develops if any of the initial velocities are within the for-bidden region. Because the forbidden region extends asymp-totically over < | u | < ∞ , there is a wide range of initial val-ues that will result in a transonic flow. Any constant nonzeroinitial velocities will cross a boundary of the forbidden regionif the range in position is large enough. Stated another way,an initial flow has to be constructed in a special way to avoidthe forbidden regions. The velocities of the subsonic and su-personic steady-state trajectories have this property.While generalizing from examples is short of a mathematicalproof, this proposition seems plausible on physical grounds.Consider an inflow with a subsonic velocity in the forbiddenregion with x < x C . This flow will be accelerated to approxi-mate free fall, u ∼ /√ x , by the gravitational force. Equation2.5 constrains the velocity of the steady-state solution to thesound speed at the critical point, and the flow will evolve un-til this condition is met. The critical point thus acts as anouter boundary condition for the supersonic region of the in-flow. The critical point is also an inner boundary conditionfor the subsonic region of the flow, x > x C . Here the flowis able to adjust to the transonic solution by increasing thedensity and pressure to slow the flow until its transonic pointoccurs at the critical point, and both terms on the right-handside of equation 2.5 are zero. This description also appliesto the wind with appropriate modifications.The transonic critical point effectively provides a naturalboundary condition for the inward and outward transonicflows. In the case of a stellar wind for example, if an ar-bitrary radius is chosen for the base of the wind, then thevelocity at the base is uniquely determined by the require-ment that the wind pass through the transonic critical point.In the case of an astrophysical subsonic or supersonic flow,while any single point in the ( x , u ) plane (figure 1) uniquelydetermines one steady-state trajectory, this would have to bedetermined outside of the BP flow, for example by suppos-ing that conditions in the interstellar medium set a constantvelocity at a particular radius. This velocity would need tobe outside the forbidden region, assuming the propositionabove. Figures 2 through 4 show the development of a transonicaccretion flow starting from constant initial velocities equalto -0.2, -1.0, and -2.0, respectively, for . < x < . . (So-lutions of equations 4.7 and 4.8 are not defined for initialvalues of x = or u = .) This evolution is similar to whatmight be found in a test of a numerical hydrodynamic simu-lation evolving to the steady-state starting from a constantvelocity. In figure 4 a shock develops, indicated by multi-valued velocities, as the velocities outside the critical pointdecrease to subsonic and are impacted by the supersonic ini-tial velocities at larger radii. The post-shock velocities evolveto the transonic trajectory.In the method of characteristics, the ODE for the velocity u ( t , x ( t )) is integrated along lines of x ( t ) . These lines or char-acteristics are also shown in the figures. The point where thecharacteristics, x ( t ) , first cross indicates the formation of ashock.Starting from the same but positive initial velocities, theflows evolve to the outward steady-state transonic trajec-tory, the Parker wind. Figure 5 shows the time evolutionfor initial values u ( x ) = + . and . < x < . . A shockdevelops in the outer region as the outward velocities be-come supersonic and impact the subsonic initial velocitiesfurther out. Plots of the evolution from initial velocities, u ( x ) = . and . look as expected from figure 5 and are notshown.The characteristics for the transonic solutions in figures 2through 5 have both inward and outward motion. The loca-tion of the point on the x-axis, t = , that separates theinward and outward traveling characteristics is the pointwhere the initial values cross the boundary of a forbiddenregion. For the case shown in figure 3 with initial veloci-ties equal to the sound speed, the point of separation is thecritical point. To find initial values that are everywhere outside the for-bidden regions, we start with the subsonic and supersonicsteady-state trajectories.Figure 6 shows two examples of supersonic accretion. Thefirst begins with initial values that follow the steady-statetrajectory with L = L C + inside the critical point andtransition to the steady-state trajectory with L = L C + outside the critical point by means of a Gaussian. The sec-ond example begins with initial values that transition inthe reverse sense. In both cases, these time-dependent flowsevolve asymptotically to follow the steady-state trajectoryused to set the initial velocities in the outer part of the ac-cretion flow. While difficult to imagine as an astrophysicalflow, the equations may also be solved for the same initialconditions but changing the initial direction of the flow toa wind. These solutions evolve to follow the initial steady-state trajectory in the inner part of their initial winds. Theevolution of these winds is as expected from figure 6 and isnot shown.The same calculations can be done for subsonic accretionand winds. Figure 7 shows a subsonic wind sometimes calleda Parker breeze. Both time-dependent wind solutions evolveasymptotically to follow the initial steady-state trajectoriesin the outer part of the wind. In the case of subsonic ac-cretion, figure 8, the flows evolve to the initial steady-statetrajectories in the inner region rather than the outer re-gion.As a final example, figure 9 shows the evolution of a subsonicaccretion flow with a sinusoidal perturbation of amplitude0.05 and period 0.25 imposed on the steady state between x = . and x = . . (If a finite amplitude perturbation iscontinued too close to the origin then some initial veloci-ties would be within the inner forbidden region and the flowwould be swept into approximate free fall allowing the entire MNRAS000
Example solutions suggest the proposition that a transonicflow develops if any of the initial velocities are within the for-bidden region. Because the forbidden region extends asymp-totically over < | u | < ∞ , there is a wide range of initial val-ues that will result in a transonic flow. Any constant nonzeroinitial velocities will cross a boundary of the forbidden regionif the range in position is large enough. Stated another way,an initial flow has to be constructed in a special way to avoidthe forbidden regions. The velocities of the subsonic and su-personic steady-state trajectories have this property.While generalizing from examples is short of a mathematicalproof, this proposition seems plausible on physical grounds.Consider an inflow with a subsonic velocity in the forbiddenregion with x < x C . This flow will be accelerated to approxi-mate free fall, u ∼ /√ x , by the gravitational force. Equation2.5 constrains the velocity of the steady-state solution to thesound speed at the critical point, and the flow will evolve un-til this condition is met. The critical point thus acts as anouter boundary condition for the supersonic region of the in-flow. The critical point is also an inner boundary conditionfor the subsonic region of the flow, x > x C . Here the flowis able to adjust to the transonic solution by increasing thedensity and pressure to slow the flow until its transonic pointoccurs at the critical point, and both terms on the right-handside of equation 2.5 are zero. This description also appliesto the wind with appropriate modifications.The transonic critical point effectively provides a naturalboundary condition for the inward and outward transonicflows. In the case of a stellar wind for example, if an ar-bitrary radius is chosen for the base of the wind, then thevelocity at the base is uniquely determined by the require-ment that the wind pass through the transonic critical point.In the case of an astrophysical subsonic or supersonic flow,while any single point in the ( x , u ) plane (figure 1) uniquelydetermines one steady-state trajectory, this would have to bedetermined outside of the BP flow, for example by suppos-ing that conditions in the interstellar medium set a constantvelocity at a particular radius. This velocity would need tobe outside the forbidden region, assuming the propositionabove. Figures 2 through 4 show the development of a transonicaccretion flow starting from constant initial velocities equalto -0.2, -1.0, and -2.0, respectively, for . < x < . . (So-lutions of equations 4.7 and 4.8 are not defined for initialvalues of x = or u = .) This evolution is similar to whatmight be found in a test of a numerical hydrodynamic simu-lation evolving to the steady-state starting from a constantvelocity. In figure 4 a shock develops, indicated by multi-valued velocities, as the velocities outside the critical pointdecrease to subsonic and are impacted by the supersonic ini-tial velocities at larger radii. The post-shock velocities evolveto the transonic trajectory.In the method of characteristics, the ODE for the velocity u ( t , x ( t )) is integrated along lines of x ( t ) . These lines or char-acteristics are also shown in the figures. The point where thecharacteristics, x ( t ) , first cross indicates the formation of ashock.Starting from the same but positive initial velocities, theflows evolve to the outward steady-state transonic trajec-tory, the Parker wind. Figure 5 shows the time evolutionfor initial values u ( x ) = + . and . < x < . . A shockdevelops in the outer region as the outward velocities be-come supersonic and impact the subsonic initial velocitiesfurther out. Plots of the evolution from initial velocities, u ( x ) = . and . look as expected from figure 5 and are notshown.The characteristics for the transonic solutions in figures 2through 5 have both inward and outward motion. The loca-tion of the point on the x-axis, t = , that separates theinward and outward traveling characteristics is the pointwhere the initial values cross the boundary of a forbiddenregion. For the case shown in figure 3 with initial veloci-ties equal to the sound speed, the point of separation is thecritical point. To find initial values that are everywhere outside the for-bidden regions, we start with the subsonic and supersonicsteady-state trajectories.Figure 6 shows two examples of supersonic accretion. Thefirst begins with initial values that follow the steady-statetrajectory with L = L C + inside the critical point andtransition to the steady-state trajectory with L = L C + outside the critical point by means of a Gaussian. The sec-ond example begins with initial values that transition inthe reverse sense. In both cases, these time-dependent flowsevolve asymptotically to follow the steady-state trajectoryused to set the initial velocities in the outer part of the ac-cretion flow. While difficult to imagine as an astrophysicalflow, the equations may also be solved for the same initialconditions but changing the initial direction of the flow toa wind. These solutions evolve to follow the initial steady-state trajectory in the inner part of their initial winds. Theevolution of these winds is as expected from figure 6 and isnot shown.The same calculations can be done for subsonic accretionand winds. Figure 7 shows a subsonic wind sometimes calleda Parker breeze. Both time-dependent wind solutions evolveasymptotically to follow the initial steady-state trajectoriesin the outer part of the wind. In the case of subsonic ac-cretion, figure 8, the flows evolve to the initial steady-statetrajectories in the inner region rather than the outer re-gion.As a final example, figure 9 shows the evolution of a subsonicaccretion flow with a sinusoidal perturbation of amplitude0.05 and period 0.25 imposed on the steady state between x = . and x = . . (If a finite amplitude perturbation iscontinued too close to the origin then some initial veloci-ties would be within the inner forbidden region and the flowwould be swept into approximate free fall allowing the entire MNRAS000 , 1–7 (2019) ondi-Parker Flow flow to become transonic.) The continued oscillation of thesolution shown in the figure is expected because the den-sity structure in subsonic accretion flows is approximatelyhydrostatic and the PDE has no damping. This flow is Lya-punov stable in the sense that the velocities stay within somerange of the initial flow set by the amplitude of the initialperturbations. Since the the Parker wind and subsonic accretion both haveapproximately hydrostatic density profiles inside the tran-sonic critical point, comparison with the stability of hydro-static equilibrium suggests that the BP flow should be sta-ble with an adiabatic equation of state (EOS), γ = / , butnot with a radiation dominated EOS with γ = / . Theunderstanding developed for the isothermal BP flow allowsthe study to be repeated for a flow with a more generalbarotropic EOS, P ∝ ρ γ .With a barotropic EOS, the non-dimensional Euler equationequivalent to equation 2.3 is, ∂ u ∂ t = − u ∂ u ∂ x − ρ γ − ρ ∂ ρ∂ x − x . (6.1)Following Bondi (1952), the steady-state solution, u + (cid:18) γ − (cid:19) ρ γ − − γ − − x = , (6.2)can be written in separable variables similar to equation 2.6by scaling the velocity u by the local sound speed. Substi-tuting v = u / a = u ρ −( γ − )/ into equation 6 along with thecontinuity equation, results in F ( v ) − λ − ( γ − ) γ + G ( x ) = (6.3)where F ( v ) = v γ + + γ − v − ( γ − ) γ + (6.4)and G ( x ) = γ − x ( γ − ) γ + + x γ − γ + . (6.5)The two terms of the second variation, equivalent to equa-tions 3.6 and 3.7, are then, ∂ G ( x ) ∂ x = (cid:18) γ − γ + (cid:19) (cid:18) γ − γ + (cid:19) x γ − γ − + (cid:18) γ + (cid:19) (cid:18) γ − γ + (cid:19) x γ − γ + (6.6)and ∂ F ( v ) ∂ (cid:219) x = (cid:18) γ + (cid:19) (cid:18) − γγ + (cid:19) v − ( γ − ) γ + + (cid:18) γ − γ + (cid:19) (cid:18) γ + γ + (cid:19) v − γγ + . (6.7)The second variation is definite in sign for γ = / , owingto the factors of ( γ − ) , implying that an adiabatic BPflow is stable. Since the factor containing the parameter, λ ,that determines whether the flow is subsonic, transonic, orsupersonic, is also multiplied by this zero, all the trajectories of the adiabatic BP flow are stable. For γ = / and mostother values, the second variation is indefinite in sign. Thisdoes necessarily imply that these flows are unstable, onlythat the stability cannot be determined by this Hamiltoniandescription.The method of characteristics can also be used to solve fora time-dependent barotropic BP flow. From the PDE 6.1,the two ODEs equivalent to equations 4.3 and 4.4 can bewritten for example as, dxdt = u − ρ γ − u (6.8)and dudt = ρ γ − x − x . (6.9)where ρ ( x , u ) is given by the continuity equation 2.4. Sim-ilar to the isothermal case, these equations may be solvedparametrically for u ( t , x ( t )) . Examples are best left to specificapplications. The steady-state BP flow is particularly useful as a contextto study local phenomena in accretion or winds that involvegas pressure, shocks, or fronts. In the transonic flows, thecritical point along with the conservation of energy com-pletely specifies the steady-state solution without the needfor initial values or boundary conditions. The subsonic andsupersonic steady-state flows, lacking this constraint, requireat least one boundary condition, depending on the methodof solution, and both the inner and outer boundaries areproblematic. At the inner boundary, spherical convergenceleads to unrealistically high densities. At the outer bound-ary, these flows require an asymptotic approach to zero orinfinite velocity. These are all artificial conditions. Since anyother boundary conditions lead to a transonic trajectory,this is the most likely astrophysical application.
The Hamiltonian description of the Bondi-Parker flow pro-vides a simple and definitive method for determining theevolution of time-dependent flows and the stability of thesteady state. The method of characteristics allows a sim-ple solution for the partial differential equation describingthe time-dependent flows as an initial value problem. Thesemethods provide answers to several questions about thestability and evolution of the flows that were unexplainedin Bondi (1954) and Parker (1958). In particular, time-dependent flows evolve to the steady state; the steady-statesolution for isothermal and adiabatic equations of state, in-cluding all subsonic, transonic, and supersonic trajectories isat least linearly stable; and a transonic flow develops undera wide range of realistic initial conditions.
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Figure 2.
Left: Time evolution of an accretion flow with initial values u ( x ) = − . and . < x < . plotted on top of the steady statetrajectories from figure 1. The time-dependent solution is shown in blue for a sequence of times with the initial values in red. Velocitiesare plotted as their absolute values. The flow evolves asymptotically to the steady-state transonic trajectory, the Bondi accretion flow.Right: Characteristics for the solution. Only the characteristics originating in the range (0.005 < x < Figure 3.
Time evolution of an accretion flow with initial values u ( x ) = − . and . < x < . in the same format as figure 2. Figure 4.
Time evolution of an accretion flow with initial values u ( x ) = − . and . < x < . in the same format as figure 2. REFERENCES
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Arnol’d, V.I. 1965, J. Appl. Math. Mech., 29, 1002Arnol’d, V.I. 1989, Mathematical Methods of Classical Mechan-ics, Second edn. Springer, New York, doi:10.1007/978-1-4757-2063-1Balazs N. L., 1972, MNRAS, 160, 79Bondi H., 1952, MNRAS, 112, 195 Cranmer S. R., 2004, American Journal of Physics, 72, 1397Del Zanna L., Velli M., Londrillo P., 1998, A&A, 330, L13Dhang P., Sharma P., Mukhopadhyay B., 2016, MNRAS, 461,2426Garlick A. R., 1979, A&A, 73, 171Herivel J. W., 1955, Proceedings of the Cambridge PhilosophicalSociety, 51, 344 MNRAS000 , 1–7 (2019) ondi-Parker Flow Figure 5.
Left: Time evolution of a wind with initial values u ( x ) = + . and . < x < . plotted in the same format as figure 2. Figure 6.
Time evolution of a supersonic accretion flow in the same format as figure 2. The initial values are derived as a transitionbetween two steady-state solutions as explained in the text. The trajectories evolve asymptotically to the steady-state solution in theouter region.
Figure 7.
Time evolution of a subsonic wind (Parker breeze) in the same format as figure 2. The initial values are derived as a transitionbetween two steady-state solutions as explained in the text. The trajectories evolve asymptotically to the steady-state solution in theouter region.Holzer T. E., Axford W. I., 1970, ARA&A, 8, 31Kopp R. A., Holzer T. E., 1976, Sol. Phys., 49, 43McCrea W. H., 1956, ApJ, 124, 461Parker E. N., 1958, ApJ, 128, 664Shu F. H., 1977, ApJ, 214, 488Stellingwerf R. F., Buff J., 1978, ApJ, 221, 661 Velli M., 1994, ApJ, 432, L55This paper has been typeset from a TEX/L A TEX file prepared bythe author.MNRAS , 1–7 (2019)
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Figure 8.
Time evolution of a subsonic accretion flow in the same format as figure 2. The initial values are derived as a transitionbetween two steady-state solutions as explained in the text. The trajectories evolve asymptotically to the steady-state solution in theinner region.
Figure 9.
Left: Time evolution of a subsonic accretion flow in the same format as figure 2. The initial values follow a steady-statesolution with a sinusoidal perturbation as described in the text. Right: characteristics for the solution. These indicate that the flow willnot evolve out of the subsonic region. MNRAS000