GRAVITATIONAL COLLAPSE, FRAGMENTATION AND HALO FORMATION
Copyright 1988-2015 John Bruce Davies
This research combines the 1988 and 1999 awarded essays in the annual Gravity Research Foundation competition.
We show that a simple model of instability of a self-gravitating spheroid leading to fragmentation and subsequent pressure-free collapse subject to angular momentum conservation can explain a variety of astrophysical phenomena. That oscillations and jets are a consequence of such collapse is intuitively satisfying. Evolution of galaxies based on this model is also readily comprehended and fits well with a variety of observational data. This includes the distribution of globular clusters and angular velocity in our galaxy and the oscillations and jets observed in young galaxies. Halo formation is predicted with the mass ratio to the luminous central region being close to the observed. The age of our galaxy is obtained by utilizing the present intergalactic density relative to that at the time of collapse in the early Universe.
There is not a consensus among astrophysicists as to whether the larger elliptical galaxies, observed to be an older generation, are virgin or the result of a merger between smaller disk galaxies. Probably both! From our research on the Evolutionary Universe, awarded an Honorable Mention in the 1992 Gravity Research Foundation competition, we see that self-gravitating systems formed at temperature appropriate to their critical mass for rotating spheroids. Elliptical galaxies are slower rotating than disk galaxies so we can expect less alteration of shape from the original spheroid.
Our approach is to examine the most general and simplest case that can encompass phenomena from galaxy formation to solar systems and star formation. Such phenomena include jets, central bulges (stars), disks, globular clusters, halos and the age of galaxies. We shall see that the products of collapse and fragmentation continue in time from Universe scale to planetary scale.
The evolution of an unstable gas spheroid is controlled by its initial fragmentation that results in the subsequent pressure-free collapse subject to conservation of angular momentum. Various solution approaches for material flow show that contracting outer disks and jets may be present. For galaxies, collapse evolution is shown to be from proto-system through possible quasar-like systems to normal disc and spheroidal systems. The present distribution of globular clusters and disc angular velocity is shown to be a consequence of the fragmentation and collapse. Jets are observed from collapsing systems with halos remaining where Dark Matter is considered to be concentrated. The age of our galaxy is estimated using the calculated external density and radius at birth. Both spiral galaxy birth and newly-forming stars with gas-dust disks and planets are shown to be formed in such collapse and fragmentation processes.
We have shown in the Evolutionary Universe essay that collapse of increasingly-smaller gas spheroids is inevitable as our Universe cools and evolves. Collapse processes and their resultant structures are thus expected to be important in the formation of disk galaxies and in star and solar system development.
Young galaxies with the expected high rates of star formation have been found at redshifts up to 3.4, Lilly (1988). That the hydrogen systems seen in the spectra of many quasars are the progenitors of present day disk galaxies has been argued by Wolfe et al. (1986). Recent observations have discovered a blue compact dwarf galaxy with extremely high rates of star formation and a massive H I halo, Bergvall and Jorsater (1988). It has been also argued that nonevolving galaxies may be an important component of the Universe, such systems being large and diffuse with large fractions of mass in neutral gas that is difficult to observe, Impey and Bothun (1989).
Recent radio and infrared observations of young stars reveal complicated patterns of gas flow, stellar winds, jets and mobile blobs, together with excretion disks. Reipurth (1989) has argued that such jets are transient phenomena related to eruptive accretion events in the formation of the central star. Observations of T Tauri stars have been interpreted successfully by assuming the presence of circumstellar disks, Adams et al. (1987). That disk formation and collimated outflows are related has been argued by Pudritz, (1988), and Shu et al. (1988).
Such varied phenomena should be explainable by an appropriate model of collapse. By assuming a homogeneous uniformly rotating ellipsoid, Thuan and Ostriker (1974) reduced the pressure-free collapse problems to ordinary differential equations. This model was extended by Shapiro (1977) to encompass internal pressure and time-symmetric collapse and rebound. Most other models have been based on hydrodynamic considerations which have been analysed from self-similar arguments, Goldreich and Weber (1980). Growth of assymmetry in such pressure-dependent collapse is expected after several bounces with production of quadropole gravitational radiation, Saenz and Shapiro (1981). Relativistic collapse models are highly non-linear though Glass (1989) has determined exact solutions with shear and radial heat flow.
The structure of an isothermal gas spheroid is controlled by its total mass, angular velocity distribution and the external pressure. At instability, the spheroid will initially fragment with each fragment tending to coalesce through its own self-gravitation. This fragmentation and coalescing will result in the subsequent pressure-free collapse of the total system of fragments. The flow of fragments will initially be toward the spheroid centre and will be subject to the conservation of angular momentum in these cylindrically symmetric systems.
We examine these collapse equations in detail and show that the material flow is dependent on the fragments' initial location and velocity. Analysis of special exact solutions on the equatorial plane shows that oscillations will be present. Near the central rotation axis, it is shown that outward-flowing jets may be created. The fragments formed near the outer spheroid surface will collapse inward a distance proportionally much less than those near the centre producing a "halo" effect. Such phenomena is observed in a wide variety of astrophysical objects over a large scale range.
For galaxies, such a collapse evolution is shown to map in time and structure from protosystem through possible quasar-like systems to normal disc and spheroidal systems. The present distribution of globular clusters is shown to be a consequence of this fragmentation and collapse. Reasonable values for the distribution of angular velocity with the radius of the galactic disc are derived. Importantly, the size and mass of the non-luminous halo is consistent with the observed distribution of the so-called ‘Dark Matter’ around the luminous galaxies.
We assume initially an isothermal spheroid, with a small amount of axial angular momentum, embedded in an external medium at finite pressure. The equilibrium density structure, outside the core of this almost spherical body, has a self-similar structure, such that:
(1) ρ/ρe = A2/a2
where ρe is the external density at radius A. This implies that the mass m(a) of the spheroid scales approximately proportional to radius a and external density, i.e.:
(2) m(a) = 4πρe.A2.a
By examining the virial conditions for steady-state stability of these spheroids, we showed in Evolutionary Universe that:
1. For a given mass of gas at a fixed surface pressure, there is a minimum temperature below which no equilibrium exists.
2. For a given external pressure and temperature, there is a critical mass above which no stable spheroids exist.
A small amount of kinetic energy does not alter this general conclusion.
When the system is in unstable equilibrium, small perturbations in density will grow. The wavelength of such perturbations is obtained from the applicable dispersion relation and for non-rotating systems is given by Jean's relation, where c is the wave velocity:
(3) Λ = (4πGρ/c2)-1/2
This result is not altered appreciably if small amounts of angular momentum are included.
At instability, the individual fragment now starts to self-coalesce due to self-gravitation with the result that the interaction between fragments becomes essentially pressure-free. The subsequent collapse differs thus from the usual hydrodynamic model. Such assumptions have been found valid previously. We will also assume cylindrically symmetric rotation as an initial approximation.
A fragment initially formed at spheroid radius "a" will collapse in toward the centre and its subsequent location is governed by the following equations:
(4) d2r/dt2 = -Gm(a).r/(r2+z2)3/2 + r.Ω2(r,a)
(5) d2z/dt2 = -Gm(a).z/(r2+z2)3/2
where r and z are cylindrical coordinates, G is the gravitational constant, Ω(r,a) is angular velocity and m(a) is spheroid mass inside radius a. Conservation of angular momentum through this collapse requires:
(6) r2.Ω(r,a) = C(a)
where C(a) is a constant for that initial radius. Thus the radial equation becomes:
(7) d2r/dt2 = -Gm(a).r/(r2+z2)3/2 + C2(a)/r3
We can combine these equations (4) and (5) into a single one relating the r and z components of motion, namely:
(8) 1/r.d2r/dt2 = 1/z.d2z/dt2 + Ω2(r,a)
In the special case of spherical collapse with no rotation, it is inevitable that all the matter ends up toward the spheroid center. The conservation of angular momentum in this more realistic approach that allows for cylindrically-symmetric rotation produces far more interesting and powerful results.
Equatorial Plane Solution
The solution on the equatorial plane may be determined exactly by setting z=0 in Eqn. (7). Multiplying by dr/dt and integrating once, where K(a) is the integration constant, we get:
(9) 1/2.[dr/dt]2 = Gm(a)/r - C2(a)/2r2 + K(a)
In order to determine the integration constant, K(a), initial boundary conditions must be used. At t=0, when instability and collapse are initiated, the radial inflow velocity of the fragment is zero. Thus at t=0, r=a:
(10) K(a) = -Gm(a)/a + C2(a)/2a2
so that K(a) is fully determined from knowledge of the initial mass inside and angular velocity at location r=a.
In order to solve for the time dependence of the radial position of the fragment on the equatorial plane, we integrate (9); this is a standard integral which has the exact logarithmic solution:
(11) t + L(a) = 1/2K.(2Kr2+2Gmr–C2)1/2 +
where L(a) is the integration constant. This is a complex function with the radial location in the first term on the r.h.s having a roughly linear dependence on time and the second term being oscillatory. Thus, we see from this solution that the fragments radial location along the equatorial plane may approach its final position asymptotically. The period or rate of oscillation of its location is given approximately by:
(12) Gm/(2K)3/2 = Gm(a)/(-2Gm(a)/a + C2(a)/a2)3/2
and we shall investigate the implications and evidence for such behaviour.
For the equatorial-plane solution, the radius R at which the fragment finally comes to rest can be calculated. Putting dr/dt=0 in equation (9), we obtain:
(13) Gm(a)/R - C2(a)/2R2 + K(a) = 0
which is readily solved to yield, after substitution, the trivial result, the initial radius, a, and the final radius R(a) given by:
(14) R(a) = a/(2Gma/C2-1)
Expressing the mass in terms of the self-similar isothermal relation discussed above and incorporating the angular velocity, allows this expression to be reduced to:
(15) R(a)/A = (a/A)/[(8πGρe/Ω2).(A/a)2-1]
where the final radius on the equatorial plane is shown to be dependent on the the ratio of total gravitational potential energy and the rotational kinetic energy at the originating radius a.
For a complete solution of the coupled equations (4,5) in the r and z dimensions, it is necessary to perform numerical calculations. However, important information can be obtained from analysis of the single equation (8). If the distance from the centre to a fragment at an arbitrary point is l where, as usual, l2 = r2 + z2 then differentiating with time, we have:
(16) l.dl/dt = r.dr/dt + z.dz/dt
An important locus is the line where, at some instant, the fragment has this velocity dl/dt equal to zero. From (16), this line corresponds to where:
(17) r.dr/dt = -z.dz/dt
On the line where dl/dt=o, at some particular time, the radial velocity will be negative, the radius positive yet equation(17) demands for Z>0 the z velocity will be positive. At the same time, for Z<0, the z velocity is negative. This can be seen in the sketch, Fig.1. Inward of this zero-velocity line and close to the central axis, we see that material may be flowing away from the centre while nearer the equatorial plane matter is collapsing toward the centre. This is a primitive but illuminating model of Jets which have been observed close to the rotation axis in a wide variety of astrophysical bodies. The above simple argument also explains why jets are usually observed emanating in equal and opposite directions from the poles. We can thus assume that where jets are observed, that the body is probably in a state of collapse.
Final Rotational Velocities
Equations (15) shows the dependence of the final radial location of a fragment on its starting location relative to the external radius A. From this relation we can see that near the outer rim of the spheroid the final normalized radius R/A will be proportional to the initial radius a/A. From this simple relation and conservation of angular momentum, the final angular velocity on the equatorial plane is Ωf = Ωi.(a/R)2. Thus near the outer edge, where the radius r -> A, the angular velocity will approach a constant and vary little from its initial velocity.
Close in toward the central axis, the normalized final radius will tend to be proportional to the cube of the normalized initial radius as the (A/a) term dominates in the denominator. In between these two radii extremes, and thus through much of the disk, we can expect R to be approximately proportional to the square of the initial radius. Using the above relation where the rotational final velocity is Ωf.R = Ωi.(a)2/R, we can expect the rotational velocity through most of the disk on the equatorial plane to be approximately constant as observed in most disk galaxies. In the central core it is expected that the angular velocity will approach a constant at the centre, as depicted in Fig. 2.
From the Jean's criterion we can expect the length of a fragment to be proportional to the inverse square root of the isothermal density at instability which is thus proportional to a, i.e. the original radius. The volume of the fragment is thus proportional to a3 and the number per unit volume is proportional to a-3. Subsequently, we expect these fragments, as they collapse toward the galactic centre, to form star-forming clouds and globular clusters.
The central portion of the protogalaxy will collapse much further toward the centre producing a disc galaxy if the angular momentum is sufficient. Examination of the present distribution of globular clusters in our galaxy lends credence to our model. In the outer galaxy, where final radius is proportional to initial radius, the number of globular clusters per unit volume should thus approach the R-3 slope while toward the centre the relation will approach an R-1 slope. This behaviour is readily seen in the globular cluster distribution in our galaxy, Fig. 3.
Most normal halo clusters are usually within 100,000 lt.yr of the galactic center with a maximum distance of about 300,000 lt.yr. At this distance a few dwarf spheroidal galaxies are also found and are considered to be in gravitationally bound orbits around the galaxy. These objects, as the most distant in the galaxy, have been used to measure the mass of the galaxy through their orbital velocities, which assuming virial equilibrium, have yielded an estimate of 5 x 1011 solar masses. This is far larger than the mass of visible stars and galaxies and has been used as evidence for Dark Matter in the galaxy.
We have seen that near the outer boundary of the spheroid the final normalized radius R/A will be proportional to the initial radius a/A. Further in toward the center, the normalized final radius will tend to be proportional to the square of the normalized initial radius and finally, close to the center, proportional to the cube of the initial distance to the center. This shows why the central part of the spheroid collapses far more than the outer part thereby resulting in a "halo" effect.
Numerous observational methods and analyses using rotation curves, motion of satellite galaxies and other bound systems, have shown the existence of a halo of matter around this and other galaxies. Caldwell and Kamionkowski ( Nature v458/2 p587 2009) claim that rotational velocities of gas and stars indicate that the Milky Way is embedded in a halo of about 600,000 lt-yrs in radius. This compares to the radius of luminous matter in the disk observed to be 50,000 light-years, Allen (1973).
We can test our halo model against these numbers; the ratio of the present radius of luminous matter to the larger estimates for the galaxy's radius is thus 1/12. For an isothermal spheroid, the mass within an outermost radius is approximately proportional to the radius, equation (2). Thus, this implies that the dark halo of the galaxy may have approximately 12 times the mass of the luminous matter, close to agreement with the observational data.
Age of the Galaxy
We can use the above formulations to determine the age of the galaxy, assuming that the present density of the Universe is at its critical value. We assume, as usual, that mass is conserved during expansion of the Universe. This means that the ratio of the mean density ρb at the birth of the Galaxy to the present mean density ρp of the Universe is given by:
(18) ρb/ρp = (Rp/Rb)3
where Rp is the value of the present Cosmic Scale Factor and Rb is its value at the time of the birth of the Galaxy. The redshift Z of the Galaxy if now observed at its birth would be given by Z = Rp/Rb-1, so that:
(19) ρb/ρp = (Z+1)3
We take the present radius of the galaxy as 300,000 lt-yr and assume, as per the above collapse scenario, that this is close to the original radius at the onset of collapse. Then, using the calculated mass from globular cluster motions as 5 x 1011 solar masses, the external density for an isothermal spheroid being approximately 1/3 the average density, gives a value of the external density at birth as 0.3 x 10-26 gm/cc. However, our galaxy is part of a cluster and assuming that this too was an isothermal spheroid surrounded by the mean density of the Universe at that time, then the value of ρb is just 1/3 again and equal to approximately 10-27 gm/cc. The present day critical density that separates a closed from open Universe is approximately 10-29 gm/cc. Substituting into (19) and solving for the redshift Z at birth of the Milky Way proto-galactic collapse gives a value of Z = 3.6. This corresponds to an age of about 12 billion years assuming the standard age of the Universe as 13.7 byrs.
Quasars, postulated to be galaxies under formation, have redshifts out to at least 4 and show jets and disc-like structures with probably a blackhole forming at their centers. The Milky Way appears to be a normal spiral galaxy of intermediate mass and has a relatively quiet blackhole at its center. Thus, this value 3.6 of redshift is reasonable given the probable inexactness of the external radius and density determined above for the time of birth of the Galaxy.
Much evidence exists for the presence of an unknown component in galaxies and clusters termed Dark Matter. The original premise was based on assuming that these structures were in a steady-state and that the virial relation governing gravitational and kinetic energies was valid. However, we have seen above that galaxies may still not have reached such a steady-state and that the constant rotational velocity observed in the disk of spiral galaxies may only be a result of collapse. But, other measurement techniques such as gravitational lensing and the separation of baryonic matter and dark matter in the Bullet Cluster have convinced astrophysicists of its real effects.
There are situations where dark matter is unexpectedly not present. Globular clusters appear to contain no such component though their interaction with the galaxy indicates that the galaxy contains dark matter. There are also a small number of galaxies whose gas cloud orbital velocities indicate the absence of any dark matter. Whether any dark matter exists in the plane of the Milky Way disk is doubtful but not yet proven.
Our collapse and stability arguments for the evolution of self-gravitating systems are dependent on the cooling of the Universe’s temperature due to expansion. This top-down scenario for formation of galactic clusters and subsequently galaxies depends on dark matter behaving similarly to baryonic matter with respect to its temperature. Until we know otherwise concerning dark matter and its properties, this scenario is more compelling than the competing bottom-up arguments.
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Figure 1. Diagram showing line of zero velocity in collapse.
Figure 2. Diagram showing distribution of angular velocity on equatorial plane.
Figure 3. Distribution of globular clusters in our Galaxy, Allen (1973).