I have written elsewhere in these pages, observers like to joke that theorists approach all problems by assuming an object is a sphere. In the observer’s mind, a theorist would dot a farm with round buildings and populate its fields with round cows. But theorists have a much more expansive view of the world than this; while the cows, farm house, and barn might be round (to first approximation), our farm would also have columns and disks—silos and fields. The point is that there are several simple shapes that, to first approximation, give a reasonable description to the structure of most astronomical objects. These shapes arise from several simple physical principles.
The spherical structure that is so common in the universe occurs when the dominant force countering gravity in a system is the random motion of the objects in the system. In a gas, this random motion of the atoms provides pressure that counteracts gravity; in a stellar system, the random motion of the stars counteract their mutual gravitational attraction. The result is a universe filled with spherical objects: planets, stars, globular star cluster, elliptical galaxies, and rich galaxy clusters.
But random motion is not the only process that can counter gravity. In a system with sufficiently-high angular momentum, gravity is counteracted predominately by centrifugal force. These systems are disks, and like spheres, they appear on all scales in astronomy. On the smallest level, disks appear around planets and around compact stars such as neutron stars. They are found around black hole candidates (objects that should be black holes if general relativity is the correct theory of gravity). They are seen as the disks of spiral galaxies. The Kuiper belt is the remnant of the disk that rotated around the Sun, the disk that gave rise to the planets of our Solar System.
The basic structure of a disk is the same in all of these systems. A disk rotates around a central object. Usually the central object provides all of the gravitational force on the disk, but in some cases there are additional sources of gravity, such as the self-gravity of the disk itself. The orbit of the material in the disk is close to circular.
The rotation of a disk is generally differential, so that the rotation velocity and the rotation period change with distance from the center. If all of the gravitational force is provided by the central object, the disk is a Keplerian disk, and the rotation period depends on radius according to the Keplerian laws of orbital motion: P ∝ R3/2, where P is the orbital period and R is the distance from the center of the disk. Keplerian disks are the most common disks that are encountered. The disk of Saturn, the ancient disk that surrounded the Sun, the accretion disks around neutron stars in binary systems are all examples of Keplerian accretion disks. Disks that are not Keplerian are the disks in spiral galaxies, which are embedded in an invisible halo of gravitating material, and the massives self-gravitating disks that are found around some black hole candidates at the cores of galaxies.
While centrifugal force is the dominant force that balances gravity within an accretion disk, the random motion of the particles in the disk provide a second force that balances gravity. The random motion balances the gravitational tidal force perpendicular to the plane of the disk. This balance determines the thickness of the disk. For disks such as the ring around Saturn, the random motion is very small, and the rings are very thin. In gas disks around black-hole candidates, the temperatures are high, and the disks are relatively thick. Which regime a disk is in depends on the efficiency of converting the energy associated with differential rotation into random motion versus the efficiency of converting the energy of random motion into electromagnetic radiation. In other words, the temperature of the disk depends on the efficiency of heating the disk versus the efficiency of cooling the disk. For gas disks, this temperature is the temperature of the gas, but in a disks of objects, such as the disk of ice blocks around Saturn, the temperature is the average kinetic energy of the ice blocks’ random motion, and not the temperature of the ice itself. A gas disk loses energy by the direct creation of infrared, optical, ultraviolet, and x-ray light. A disk of objects, such as ice chunks, loses energy through inelastic collisions, which convert the random kinetic energy into thermal energy, which is radiated away.
The energy lost by the disk usually does not extract angular momentum from the disk; instead, the energy loss causes the disk to transports angular momentum outward. Angular momentum from the inner portions of a disk is carried outward to the outer portions, causing the inner edge of the disk to move inward towards the central source, and the outer edge to move outward. The energy loss within a disk therefore causes it to spread out.
The flow of energy from differential rotation into random motion and then into electromagnetic radiation drives the broad range of phenomena seen in disks. For the rings of Saturn, the flow of energy creates its complex ringlet structure. For the plane of a spiral galaxy, the flow of energy drives the spiral waves. For gas disks around neutron stars and black-hole candidates, the flow of energy produces a bright source of x-rays. For gas disks around the massive black-hole candidates at the centers of active galaxies, the flow of energy drives jets of gas moving at nearly the speed of light and extending hundreds of parsecs way from their source. The energy flow through astrophysical jets therefore drives some of the most interesting phenomena in astrophysics.
The accretion disk is a gas disk found around numerous types of object, ranging from newborn stars to massive black hole candidates at the centers of galaxies. As implied by their name, these disks transport gas to the object at their centers; depending on the system, the gas is pulled into the accretion disk from the interstellar medium or from another star. Accretion disks exist because gas falling onto a gravitating object inevitably has some angular momentum that forces it into orbit around the object. Once in the disk, gas slowly spirals to the disk’s center, becoming hotter as its gravitational potential energy is converted into thermal energy. This energy is radiated away as infrared, visible, ultraviolet, and x-ray light. Accretion disks around white dwarfs, neutron stars, and black hole candidates that are in binary systems can radiate massive amounts of energy at the ultraviolet and x-ray frequencies; some of these binary systems are among the brightest x-ray sources in the sky.
The rotation of a disk is differential, with the inner portions completing an orbit faster than the outer portions. The basic idea behind the accretion disk is that viscosity in the gas disk converts the free energy of differential rotation into thermal energy, which is then radiated away. As the potential energy is released, the gas slowly spirals inward, completing many revolutions around the central object before significantly changing its distance from the central source. The amount of gravitational potential energy release by the gas in the disk increases as the gas draws closer to the central object. This means that most of the energy released by an accretion disk comes from the disk’s inner edge. In its spiral to the inner edge of the disk, the gas liberates half of its gravitational potential energy as radiation; the remaining half is the kinetic energy of a circular orbit around the central source. This remaining energy can be released if the gas falls onto the central object, and that object is a star; if the object is a black hole, then the energy will be lost into the black hole.
This conversion of potential energy into radiation conserves angular momentum, so as the gas spirals inward, angular momentum must be transported outward. The fate of the angular momentum depends on the details of the disk physics. One outcome is that magnetic fields generated within the disk dump the angular momentum into a wind blowing away from the disk. A second outcome is that the angular momentum is dumped into the outer edges of the accretion disk, forcing the outer portions of the disk to expand in radius.
While some basic characteristics can be derived for Keplerian accretion disks from Newtonain mechanics, a complete understanding of disks requires study with sophisticated computer codes. The physics of creating and transporting light through a disk is similar to that encountered in stars, so this part of the problem is well understood. The big problem encountered by theorists is understanding the source of the viscosity in the accretion disk. It is the viscosity that converts the energy associated with the differential rotation of the disk into thermal energy, and it is the viscosity that transports the angular momentum to the outer portions of the accretion disk. For many decades, theorists had no firm explanation for the viscosity in accretion disks. Current studies suggest that turbulence and the generation of magnetic fields produces the required viscosity; the nature of accretion disk viscosity is an ongoing topic of research.
Structure of Keplerian Accretion Disks
The accretion disks we encounter around degenerate dwarfs, neutron stars, and the smaller (several solar masses) black hole candidates that are in compact binary systems are thin steady-state Keplerian disks, as are the accretion disks around newly-formed stars. These disks do not have sufficient mass to be self-gravitating; a Keplerian disk spins in a gravitational field that is set by the object at its center. The discussion that follows concentrates on compact objects in close binary systems.
Many of the characteristics of a Keplerian disk are set by the simple Newtonian mechanics of a low-mass object orbiting in a circle around a star. These characteristics are independent of how the gravitational potential energy is converted into thermal energy. The rotation of a Keplerian accretion disk is given from Kepler’s law of orbital motion. The gas in the disk orbits in a circle. The time to complete one orbit is proportional to the radius to the 3/2 power. The structure of the disk is determined by the disk temperature, which can be related to the accretion luminosity, which is the maximum power that can be generated from material falling onto the central object.
The free-fall velocities at the surfaces of degenerate dwarfs, neutron stars, and black holes are extremely high. The lowest velocities are associated with degenerate dwarfs, which are stellar remnants held up by electron degeneracy pressure that are about the mass of the Sun the radius of the Earth; the free-fall velocity for these stars is about 2% of the speed of light, which releases 1.5×10-4 of the rest mass energy of material falling onto the star’s surface. For a neutron star, which has a mass somewhat larger than the Sun and a radius of order 106 cm, the free-fall velocity is around half the speed of light, and the amount of energy released is over 10% of the rest mass energy. The energy released around a black hole can be greater than this, although the amount will remain below the rest-mass energy of the matter.
The maximum amount of energy liberated through free-fall onto a compact object can be expressed as a temperature. This can be expressed as
where M is the mass of the central source, m is the average mass of the particles in the gas, Rc is the radius of the central source (assumed not to be a black-hole), and G is the gravitational constant. For a degenerate dwarf, the temperature for hydrogen in free-fall is of order 100 keV (109°K), which is in the hard x-ray/soft gamma-ray frequency range, and for a neutron star, it is of order 100 MeV (109°K), which is in the high-energy gamma-ray range. These temperatures are the maximum temperatures possible for the inner edges of accretion disks—radiative cooling prevents an accretion disk from ever realizing them.
While this exercise of deriving free-fall temperatures may appear academic, it actually plays a role in defining the structure of an accretion disk. The thickness of an accretion disk is set by the balancing of pressure within the disk against the gravitational tidal force. As in the case of a star, this structure depends on how the temperature of the gas varies with altitude above the plane of the disk. This in turn depends on the viscosity mechanisms that heat the disk and the radiative processes and the convection that transport the energy to the surface of the disk.
We can easily derive the structure of a disk if we make the temperature independent of altitude above the disk plane and if we only let the gas provide the pressure. This gives us a minimum thickness for the disk, since it ignores an important source of pressure, radiative pressure, that would assist in counteracting the gravitational tidal force. The gas density varies with altitude as
ρ = ρ0 e-G M m z2/2 k T R3
where ρ0 is the mass density at the center of the disk, z is the altitude above the disk plane, R is the disk radius at this point, T is the gas temperature, and k is the Boltzmann constant. The remaining variables are as defined earlier. For a constant-temperature stellar atmosphere, the exponent would have the factor z/R rather than the factor z2/2 R2, so the atmosphere of the accretion disk falls much faster than the atmosphere of a star.
The disk is thin if z/R ≪ 1 when the exponent in the density equation is of order unity. This means that the disk is thin when the characteristic temperature of the disk is much less than the free-fall temperature of the material in the disk. The altitude confining most of the matter is given by
z/r = (kT R/G M m)1/2
The disk is therefore only thin if the temperature in the disk is much smaller that the magnitude gravitational potential energy. In other words, the disk is thin when the temperature of the gas is much less that the free-fall temperature.
One thing that changes this picture is the effects of radiation pressure. Light carries momentum, so when it strikes an object, it exerts a pressure on that object. Normally this pressure is insignificant, but in bright astronomical sources, this pressure can blow away the outer layers of a star. The Eddington luminosity is a measure of when this effect is important. It is derived by calculating when the radiative pressure exerted on a fully ionized hydrogen gas equals the force of gravity. The Eddington limit, which is independent of the radius of the star, is given by
LE = 1.25×1038(M/Msun) ergs/s,
which is 32,000 (M/Msun) times the luminosity of the Sun. This is the maximum luminosity that can occur before gas is driven off; generally a wind is radiatively driven off of a star at lower luminosities than this because radiation exerts a greater pressure on an unionized gas than on ionized gas.
The binary systems with accretion disks that we see in astronomy are very bright, with luminosities that can hover around the Eddington limit. This means that the structure of an accretion disk at times is set not by the gas pressure, but by the radiative pressure. In some cases, the disk is thick, with a thickness of order the radius, and with a wind driven off of its surface by the radiation.
For an accretion disk, the comparison is between the radiative force and the tidal force on a fully-ionized gas. Radiative pressure becomes important when the accretion luminosity satisfies the limit
L/LE > z/Rc,
where L is the accretion luminosity and Rc is the radius of the central object. This limit shows that an accretion disk can radiate at well below the Eddington limit and still have an impact on the structure of the accretion disk. If this inequality is satisfied, the radiation will puff the disk up until the ratio of the thickness to the radius equals the ratio of the luminosity to the Eddington luminosity.
The radiative pressure can play an important role in the structure of an accretion disk around a neutron star or black hole candidate, particularly if the temperature in the disk is very low, so that z/R is very small. The role of radiative pressure is less important in an accretion disk surrounding a degenerate dwarf, which generally have accretion luminosities of 1034 ergs s-1 or less.
Energetics of Keplerian Accretion Disks
For a steady-state thin accretion disk, it is easy to place a lower limit on the surface temperature of the disk at each radius. This lower limit is set solely by the rate at which mass flows through the disk. Because the disk is assumed to be in a steady state, the rate at which mass flows towards the central source is constant across the disk. The lower limit on temperature arises because there is an upper limit on how much light a body of a given temperature can radiate. This limit, set by the black body spectrum, is proportional to T4. A body can release less energy than this if it is sufficiently thin, but not more.
In a steady state accretion disk, the gravitational potential energy released as mass flows inward must be counterbalanced by the energy carried away as radiation. For a Keplerian disk, the combined gravitational potential energy and orbital kinetic energy of a particle (electron, hydrogen atom, etc.) orbiting the central object is given by
where G is the gravitational constant, M is the mass of the central object, m is the mass of the particle, and R is the distance from the accretion disk center. As a particle drifts inward, it loses energy. The amount of potential and orbital kinetic energy lost per unit distance traveled inward is given by dE/dR = GMm/2R2. Multiplying this by the thickness of the ring dR and replacing the particle mass with the rate at which mass flows through the disk, dM/dt, and we have the rate at which potential and orbital kinetic energy is convert into thermal energy in a ring of thickness dR. Expressed in terms of the free-fall luminosity, the energy loss rate per unit radius is
In this equation, L is the free-fall luminosity, which desribes the maximum amount of power that can be released from the gas falling onto the central object, R is the accretion disk radius at the point of interest, and Rc is the radius of the central source.
This equation shows that most of the energy of an accretion disk is released at its inner edge. For an accretion disk with inner radius R0 and an outer radius so large as to be considered infinite, the fraction of energy released by the portion of the disk outside of radius R is
From this equation, we see that half of the energy liberated in an accretion disk is release before the matter is at twice the disk’s inner radius.
This rate of energy release can give us a lower limit on the temperature of the accretion disk’s photosphere if we relate it to the flux of black-body radiation. This temperature limit is
T≥1.4 [ (L/1038ergs s-1) (Rc/106cm) (106 cm/R3)]1/4 keV.
In this equation, Rc is the radius of the central object. The various parameters in this equation are normalized to characteristic values encountered with an accreting neutron star. The equation shows that a large change in the accretion luminosity has a small effect on the minimum possible surface temperature. The surface temperature, however, drops significantly as one moves farther out in the accretion disk. The minimum surface temperature at the inner edge of the disk is 68% higher than the minimum surface temperature at twice the inner-edge radius.
For a neutron star, we see that accretion at close to the Eddington limit produces a disk with a surface temperature at its inner edge of over 1 keV, which places its radiation in the soft x-ray band. Dropping the luminosity by a factor of 100, which is a very common accretion luminosity in neutron star close binary systems, and we see that the surface temperature drops to 0.5 keV. Since the free-fall temperature is in the 100 keV range, a disk supported by gas pressure would have a thickness that is 10-3 times the radius, assuming that the surface temperature is equal to the temperature at the center of the accretion disk. But at 10-2 of the Eddington luminosity, radiative pressure balances the tidal force at z ∼ 10-2 Rc. This gives a thin disk, but not as thin as gas pressure alone would give. radiative pressure becomes less important, so the gas pressure becomes dominant in the outer regions of the accretion disk.
Degenerate dwarfs, which have about the mass of the Sun confined to a volume of Earth, do not have accretion disks that are bright in the x-ray. Bright cataclysmic variables, which are compact binary systems that contain a degenerate dwarf star, have accretion luminosities of 1034 ergs s-1, or about 10-4 times the Eddington limit. Combined with a radius of about 109cm, and we see that the characteristic temperature is above 5 eV, which places this disk in the ultraviolet. Gas pressure gives a disk thickness of around 10-2 times the inner disk radius, which is larger than the ratio of the accretion luminosity to the Eddinton luminosity. This means that radiation pressure is never important in the accretion disks of these objects.
One complication ignored in all of this is the effects of outside heating of the disk. There are instances when the radiation from the central source or from the companion star heat portions of the accretion disk. This additional energy source will increase the temperature of the disk, which will made the disk thicker and more luminous than it would otherwise be.
source http://www.astrophysicsspectator.com/ May 2005.