Astrophysical Gas Disks - Formation, Characteristics, and Nonlinearity

Transcription

1 Astrophysical Gas Disks - Formation, Characteristics, and Nonlinearity Introduction Giant gas clouds that formed after the Big Bang are the building blocks of the universe. The only way to understand the current formation of the universe is through the behaviour of gases; the dynamical nature of gas clouds is what leads to the formation of galaxies and their beautiful, complex structures. Gas often behaves nonlinearly whilst forming and shaping galaxies, which will be discussed in the sections below. Two models of galactic formation will be covered as well as galactic structures and merging galaxies - all in which gas dynamics play a leading role. To study the structures of galaxies however, the fundamentals of gas dynamics must first be covered. The dynamics of the gas will be governed entirely by the conservation laws in physics: conservation of momentum, conservation of energy, and conservation of mass. Conservation of Mass Consider an arbitrary volume V, with bounding surface S, in which the gas flows. V is a constant volume, so mass M in the volume will only change with time if there is a net flux of mass over S. If is mass density and v is the local velocity field then, dm dt = ρ V dv = ρv ds (1) S From divergence theorem, and since this must be true for any arbitrary volume, ρ + ρv = 0 (2) which is conservation of mass (Balbus and Potter, 2016). Conservation of Momentum The momentum density of the gas is ρv (this is also the maximum flux), and therefore the total momentum of the gas is the integral of ρv over the volume. In this case we must also take into consideration the forces that act on the surface of the volume. On the surface this force will be represented by -Pn. Putting all this together obtains, 1

2 ρvdv = ρvv nds P nds (3) V (ρvv n + P n)ds = (ρvv + IP ) (4) v where I is the unit tensor, and Pn has become PIn. The equation can now be written as, (ρv) + (ρvv + IP ) = 0 (5) (ρv) + (ρvv) + P = 0 (6) This represents the conservation of momentum within the volume being analysed. The quantity (ρvv) + P is the stress tensor of the fluid/gas (Dullemond, 2008). Equation (6) shows that the conservation of momentum in gas has a nonlinear term (ρvv) - this will lead to nonlinear behaviour in galactic gas disks seen later on. Conservation of Energy The properties of gas are strongly linked to thermal energy, and so in analysing the characteristics of energy in gas dynamics, the thermal and kinetic energy of the system must be considered. The (internal) specific thermal energy will be specified as e, and kinetic specific energy is e kin = v 2 2 v2. Therefore the total energy with be the volume integral the energy density, ρ(e + 2 ). The advection - or transfer of heat by the flow of the fluid - is then the surface integral of the ρ(e + v 2 2 )v n through the volume V. Finally the first law of thermodynamics, dv = TdS PdV, must be taken into consideration. This is also the surface integral of Pvn, which is due to the work done by the exterior on the binding volume. All together the equaiton becomes, Applying the divergence theorem yields, ρ(e + v2 2 )dv = ρ(ρe + ρ )v nds P V bds (7) V 2v2 ρ(e + v2 2 )dv = (ρe + ρ + P )v = 0 (8) ev 2v2 Again, it can be seen from equation (8) that conservation laws in gas have nonlinear terms. 2

3 Equation (8) must be true for all volumes, so the energy conservation equation now becomes (Dullemond, 2008): (ρe total) + (ρe total + P )v) = 0 (9) Now it can be seen that mass, momentum, and energy are conserved within the gas, and just how galaxies form and behave according to gas dynamics can be studied. Gas Dynamics of Galactic Formation There are two conflicting theories about how galaxies form, both of which rely heavily on gas dynamics given that only gas was present at the beginning of the universe. What is so interesting about these two models is that they almost entirely conflict with each other. Top-Down Theories In this first model, galaxies are formed from the collapse of a giant gas cloud in the early universe, hence top-down. This is also called Monolithic Collapse, because it proposes that the oldest stars and globular clusters must have also began forming here. First discussed by O. J. Eggen, D. Lynden-Bell, and A. R. Sandage in 1962, the velocities of 221 well-observed dwarf stars were used to compute eccentricities and angular momenta. What they found was that approximately 1010 years ago, a protogalaxy began to fall together from the galactic material (the protogalactic gas cloud). As the gas rushed in it eventually came to equilibrium with the gravitational force of the system. In some regions this equilibrium was violated, and the gas was able to condense. Here is where the formation of the halo stars - the oldest stars in the galaxy - and the globular clusters occur. The galactic collapse is halted in the radial direction due to rotation, but continues on in the Z-direction leading to a thin disk. Suddenly there is a much higher density of gas and the rate at which stars formed increased. Essentially what the data found is that the oldest stars were formed from collapsing gas falling towards the galactic centre, then collapsing onto the plane. During this process, the gas of the disk became very hot and most of the energy from the collapse radiated away. At first, the gas orbited in the path of the stars born from it. However, over time and many collisions with other gas clouds, thg gas disk lost enough energy to settle into 3

4 circular orbits. From this gas, stars with un-eccentric orbits are born (Eggen, Lynden-Bell, and Sandage, 1962). This model s real issues lie in its age; since 1962 quite a few advancements have been made. At the time of its creation, dark matter and dark energy were not being considered, and better observational technology have arisen to suggest some contradiction. interesting concept in dynamics and galactic formation. Nonetheless it remains an Bottom-Up Theories A more promising and current model is the broad category of Hierarchical galaxy formation. In this model, the smallest objects form first, while larger objects are then created from merging (i.e. galaxies form first then merge to form galaxy clusters). This contradicts the model in Monolithic collapse theory; here galaxies form from many tiny galaxies, and objects are formed from small instabilities. What is interesting is the role that gas dynamics plays in the processes of Hierarchical formation. In traditional N-body simulations of this model, the issue over merging often arises; dark matter halos that are much too large to be identified with real galaxies emerge. This problem however, can be overcome by including a gas component. When the gas is added to the same region with the same density as the dark halos in the simulation, even though the halos still merge, the gas cores remain distinct as they dissipate. This implies that there are multiple galaxies within the halo, negating the issue of having to assign each halo to a single galaxy (Katz, 1992). Galactic Mergers Despite comprising of a rather small amount of a galaxy over all mass, when it comes to merging events, gas plays a big role. The actual dynamics of the merger are not significantly affected by the gas component, but tidal forces during encounters can cause instabilities that lead to bar formation. The gas in these barred galaxies can then be forced by a gravitational torque to flow towards the center where it can create starbursts - regions of incredibly high star formations - that are kiloparsecs across. The consequences of these merging are often chaotic in nature, due to the nonlinear dynamics of the gas present in the galaxies. To examine the nonlinear gas dynamics within these mergers, a simulation model done by Barnes and Hernquist in 1996 will be examined. In this model, they show two galaxies of equal mass, where one is tilted 71 to the other s orbital plane. In Figure 2, it can be seen what happends when the twogalzies collide head on. The time is located in the upper corner of each image, in units 4

5 [H] Figure 1: This picture shows two galaxies merging and the consequences at different points in time. of 250 Myr. Before their first encounter both galaxies have developed spiral arms (their nonlinear characteristics will be discussed later). As the two galaxies approach they exert extreme tidal forces upon each other. By time t = 1, these forces have begun to distort the shape of the gas. At t = there is a significant amount of gas in the region between the galaxies. This gas would have been ripped from the disks during the collison. By t = 1.5 bridge arms have extended from the disks and meet in the middle, dumping gas and stars onto the inclined plane. The other two arms on the other hand have a high density of gas, and run along arms of star distrubtions (see fig. 2). At this time we see that the region of density propogates out along the filaments. As t grows past t = 1.6 the system is dominated by graviational braking. This causes the orbits to decay and the galaxies to fall into one another. This process is largely unaffected by the precess of the gas since it makes up such a small amount of the total mass ( 1.5%). After this point it is impossible to tell the two galaxies apart. What are left are very dense regions of gas, where the aformentioned starbursts are located. The mergers simultaneously create and destroy structures in the universe. The aggressive tidal forces often remove the spiral arms and complete shapes formed in the creation of the disk by the gas (Barnes and Hernquist, 1996). Spiral Arms Most of the galaxies observed in the universe are classified as spiral galaxies. As their name suggests, the main feature of this type of galaxy is their spiral arms. Initially, it was thought that these arms 5

6 were composed of mostly isothermal gas, the same material that made up the galactic disk itself. However, this led to a winding problem - after a few orbits, the arms become completely wound up and the galaxy would quickly lose its spiral structure (Gronwell, 2010). Galaxy arms are observed to last billions of years, beyond the time it would theoretically take an arm made of gas to wind itself (Mihos, 2016). This led to Lin and Shu proposing the density wave theory; spiral arms are not composed of gas nor are they rotating with the stars, but are instead density waves through which material (stars, gas) moves. Gravitational attraction between the stars at different radii can maintain the spiral pattern, thus solving the winding problem. The density wave theory is analogous to vehicles stuck in a traffic jam - the vehicles (comparable to the gas, stars in a spiral arm) move through the jam, but the jam itself does not move much (comparable to the density wave and spiral arms itself). This hypothesis explained the large scale stationary structures in spiral galaxies, but the spirals themselves are actually far from stationary (Binney and Tremaine, 459). In fact, galactic gas disks are very responsive to small distubances - the gravitational field of a point mass travelling on a circular orbit within the disk can induce a strong spiral-shaped wave in the stars of the disk - which led to the formation of the density wave theory, central to understanding astrophysical disks (Binney and Tremaine, 460). Density Waves and Tightly Wound Spiral Arms Behaviour of density waves in a disk is affected mainly the gravitational potential of the surface density pattern. This potential affects stellar orbits and alters the surface density in the galaxy, which can then be matched to the input surface density to obtain a self-consistent density wave (Binney and Tremaine, 486). The perturbed surface density of a disk (approximated to have zero thickness) for a tightly wound spiral can written as Σ 1 (R, φ, t) = H(R, t)e i[mφ+f(r,t)] (10) where m is an integer, f(r,t) is a shape function and H(R,t) is a slowly varying function of radius that gives the amplitude of the spiral pattern (Binney and Tremaine, 486). The parameters m and f(r,t) define the location of all m arms, where φ is the azimuthal angle. mφ + f(r, t) = constant (11) 6

7 It is also useful to define a radial wavenumber, k, as equal to f(r, t) k(r, t) = (12) R where R is the radius (distance from galactic disk centre). Equation (10) shows that the surface density variation can be approximated as sinusoidal in radius (taking the real of of e i[mφ+f(r,t)] ); complicated nonlinear surface density variations can be decomposed using a Fourier sum (Binney and Tremaine, 487). Now, the gravitational potential to the the surface density pattern must be determined. In a razor thin disk, as astrophysical gas disks are often approximated as, the potential of a plane wave is defined as Φ(R, φ, t) Φ a e ik(r 0,t)(R R 0 ) k(r 0,t)z (13) where Φ a = - 2πGΣ a. k Perturbations in galactic stellar disks can be modelled using fluid disks, as they are dynamically similar in nature. In tight winding approximation of perturbations in fluid disks, Euler s equations can be used to write in cylindrical coordinates, v R + v R v R R + v φ v R R φ v2 φ R = Φ R 1 p Σ d R (14) v φ + v v φ R R + v φ v φ R φ + v φ = 1 Φ v R R φ 1 p RΣ d φ where the perturbation is defined as (v )v, a nonlinear term. The density here is equal to surface density Σ d, since this models a two-dimensional disk. By choosing a simple equation of state, (15) p = KΣ d γ (16) and defining sound speed v s at a density Σ 0 as equations (14) and (15) can simplify to v 2 s(σ 0 ) = ( dp dσ ) Σ 0 = γkσ γ 1 0 (17) 7

8 Φ R 1 p Σ d R = Φ R Σ d γkσγ 2 d R = where h is the specific enthalpy equal to (Φ + h) (18) R yields h = The perturbation term s angular component, v φ, can be defined as v φ γ γ 1 KΣγ 1 0. (19) R dφ = RΩ(R) (20) dr where Ω(R) is the circular frequency. Substituting everything into equations (14) and (15) v R + Ω v R φ 2Ω v φ = (Φ + h) (21) R v φ + ( d(ωr) dr + Ω) + Ω v φ φ 2Ω v φ = 1 (Φ + h). (22) R R All solutions to equations (21) and (22) take the form v R1 = Re[v Ra (R)e i(mφ ωt) ] v φ1 = Re[v φa (R)e i(mφ ωt) ] Φ 1 = Re[Φ a (R)e i(mφ ωt) ] Σ d1 = Re[Σ da (R)e i(mφ ωt) ] h 1 = Re[h a (R)e i(mφ ωt) ] (23) where m 0 is an integer and the perturbation has m-fold rorational symmetry (Binney and Tremaine, 490). Solving for v Ra and v φa obtains v Ra (R) = i d [(ω mω) dr (Φ a + h a ) 2mΩ R (Φ a + h a )] 8

9 v Ra (R) = i [2B d dr (Φ a + h a ) m(ω mω) R (Φ a + h a )](24) where = κ 2 - (ω mω) 2 and κ, Ω, and are functions of radius (Binney and Tremaine, 489). The terms proportional to (Φ a + h a )/R are much smaller than the terms with d(φ a + h a )/dr, so they can be neglected when writing the final solution. Equation (24) can then be simplified to v Ra = (ω mω) k(φ a + h a ) (25) v Ra = 2iB k(φ a + h a ) If the perturbed velocity term, v (has components v R and v φ above), has only linear terms, the perturbed surface density becomes Σ d1 + (Σ d1 v 0 ) + (Σ 0 v 1 ) = 0. (26) By changing equation (24) into cylindrical coordinates and substituting into the equations (23), the following can be obtained i(ω mω)σ da + 1 R d dr (Rv RaΣ 0 ) + imσ 0 R v φa)=0.(27) Using equation (25) and the equations in (23), a continuity equation can be written in the form (ω mω)σ da + kσ 0 v Ra = 0. (28) In self-consistent density waves with a polarization equal to 1, the dispersion relation for a tightly wound fluid disk is (ω mω) 2 = κ 2 2πGΣ k + v 2 sk 2. (29) The potential of a tightly wound wave can then be written as Φ a (R) = F (R)e if(r) = F (R)e i R kdr (30) 9

10 In a cold disk, where v s = 0, equation (29) becomes ω 2 = κ 2 2πGΣ k + vsk 2 2. (31) The disk s local stability can then be determined using equation (31), although the dynamics in the equations above behave nonlinearly. For example, if ω 2 > 0, then ω is real and the disk is stable. If ω 2 < 0 however, then ω is complex (has roots of ±iω) and there is a nonlinear perturbation with a exponentially increasing amplitude - the disk is unstable (Binney and Tremaine, 495). Using the e imφ terms in the many equations above, it can also be seen from DeMoivre s formula that increasing m linearly leads to a nonlinear sinusoidal perturbation - e iφ leads to a cos(φ) term, but changing the perturbation to e 2iφ leads to a cos(2φ) term, which is significantly different from a cos(φ) term (cos(2φ) 2cos(φ)). Ultra-harmonic Resonances Galactic shocks in spiral galaxies lead to perturbations of motion in the spiral arms. The change in frequencies caused by these galactic shocks cause resonances to appear; resonance occurs when an external force drives a system to oscillate at a specific frequency with increased amplitude. In gas disks, shock waves causes a specific type of resonance to appear due to nonlinearity, called ultraharmonic resonance. The shock perturbations are governed by the density profile, pressure, and dispersion of gases in astrophysical disks described by the equations in the section above. The perturbation term usually takes the form e imφ, which is sinusoidal in nature. Even when m is increased linearly, the perturbation terms behaves nonlinearly, leading to nonlinear dynamical behaviour in the gas disk and ultraharmonic resonances. An example of an analysis of ultraharmonic resonance in a fluid disk with a dimensionless spiral perturbing force F is found below. For spiral galaxies with a pattern speed of Ω p and an angular velocity of Ω, ultraharmonic resonances of order n was found when mn(ω p - Ω) = ±κ (Artymowicz and Lubow, 1992). Here, m is the number of spiral arms in the galaxy and n is an integer - the wavenumber of a free wave matches n times the wavenumber of the spiral forcing (Artymowicz and Lubow, 1992). It was found in simulations that when n=2, ultraharmonic resonances were found at 4:1 - this ratio represents the material frequency relative to the pattern with the epicyclic frequency (Artymowicz and Lubow, 1992). 10

11 Spurs, Feathers, Branches Spiral arms in galaxies often contain substructures that are called spurs, feathers, or branches. Spurs can be described as distincally leading spirals, while feathers are produced by local gravitational instability of a transient kind (Chakrabati, Laughlin, Shu, 2001). Feathers and spurs do not form the same way and their progression is sensitive to initial conditions - the formations of feathers and spurs may be chaotic in nature and evolve nonlinearly. Branches in spiral arms form due to ultraharmonic resonances caused by galactic shock waves. The existence of these substructures cannot be explained solely due to ultra-harmonic resonances due to shock waves unless self gravity plays a significant role (Chakrabati, Laughlin, Shu, 2001) As a result, self gravity in the spiral arms must enhance the intrinsically nonlinear response of gas dynamics. The effect of gravitational potential of density waves in spiral arms is explained above. The nonlinear terms in the disk s self-gravity combined with ultra-harmonic resonances lead to Conclusion Astrophysical gas disks behave similarly to fluids, including the many nonlinear dynamical processes that happen in fluid dynamics. Nonlinearity can be seen most obviously in the behaviour of spiral arm galaxies, especially in density waves involving tightly wound spiral arms. The ultra-harmonic resonances caused by the local spiral motion, enhances by the self-gravity of the disk, lead to formation of spiral substructures such as spurs, feathers and branches. Galactic disk behaviours observed in the current universe are dominated by nonlinear gas dynamics, which heavily influences their creation, merging/collision, evolution, and structural formation. 11

12 Bibliography P. Artymowicz and S.H. Lubow, The Astrophysical Journal, 389 (1992), Dynamics of ultraharmonic resonances in spiral galaxies. S. A. Balbus and W. J. Potter, Rept. Prog. Phys (2016) Surprises in astrophysical gasdynamics. J. E. Barnes and L. Hernquist, ApJ, (1996) Transformations of Galaxies II: Gasdynamics in Merging Disk Galaxies. Binney and Tremaine. Galactic Dynamics. Princeton University Press: Print. S. Chakrabarti, G. Laughlin, F.H. Shu, The Astrophysical Journal, 596 (2003), Branch Spur, and Feather Formation in Spiral Galaxies. C. P. Dullemond, Equations of Hydrodynamics, (2008) WWW Document, ( dullemon/lectures/fluiddynamics08/chap 1h ydroeq.pdf). O. J. Eggen, D. Lynden-Bell, and Sandage, ApJ, (1962) Evidence from the motion of old stars that the galaxy collapsed. C. Gronwell. (2010). Spiral Structures [PDF slides]. Pennsyvania State University. N. Katz, Publication of the Astronomical Society of the Pacific, (1992) Aspect of Hierarchical Galaxy Formation Involving Gasdynamics. C. Mihos. (2016). Spiral Structure [PDF slides]. Case Western Reserve University. 12