Milky Way Galaxy

The Sinking of Young Stars Towards Sagittarius A^*

Far from the central Galactic black hole, beyond 1 parsec, star formation occurs as it does in the Galactic disk: dense, cool molecular clouds become gravitationally unstable, collapsing and fragmenting into new stars of all sizes. This is the likely birth place of the massive B main-sequence stars that we now find orbiting inside 0.1 parsecs of the central black hole, because star formation within 0.1 parsecs of the black hole is implausible; star formation from molecular clouds close to the massive black hole should be suppressed by the tidal forces exerted by the black hole, and star formation from an accretion disk orbiting the black hole would create stars with orbits that have preferential orientations, which would contradict the observed random orientations. For this reason astronomers hypothesize that the B main-sequence stars are born far from the central black hole. But how does one transport these stars into close orbit around the central black hole? This has proven to be a difficult theoretical problem, because this transport must occur on a timescale much shorter than the lifetimes of B main-sequence stars, which can be as short as 10 million years.

Thermodynamics

Transporting massive stars from a distant birth place to close orbit around Sgr A^* is fundamentally a problem of timescale; the mass segregation we see?heavy stars moving deeper into the central black hole?s gravitational potential?is what one expects from thermodynamics. If we think of the stars at the galactic center as particles, like the atoms in a gas, then thermodynamics tells us that they redistribute energy among themselves until they are in thermal equilibrium.

The idea of something as big as a star acting like a microscopic particle may seem peculiar, but keep in mind that the physical size of a star is quite small compared to the average distance between the stars at the Galactic center. With over 1 million solar masses distributed in stars within a sphere of 1 parsec radius, the average distance between stars, assuming 1 solar mass per star, is 0.016 parsecs, or 510¹⁶ cm, compared to the solar radius of 710¹⁰ cm, which gives a radius to distance ratio of order 10^?6. As the stars move past each other, there is little chance that one star will physically collide with another. More importantly, the gravitational interactions between these stars are effectively elastic collisions, where no energy is lost as they pass by one another through their tidal distortions of each other. Stars change direction as they interact with each other, but the total kinetic energy is conserved, and the stars remain intact. These interactions are similar to the scatterings between ions and electrons in a plasma, and like a plasma, the evolution of a cluster of stars is constrained by the principles of thermodynamics. In particular, like particles in a plasma, stars at the center of the Galaxy evolve towards a thermal equilibrium.

The stars we see at the Galactic center are far from thermal equilibrium. This is inevitable, given that stars are still being born there. When stars are born from a molecular cloud, they are born with a distribution of masses, a distribution that is dominated in number by low-mass stars. Stars that are born together follow similar orbits around the Galactic center. This means that these stars have similar velocities, but because they have different masses, they carry different energies: the high-mass stars carry more energy than the low-mass stars. The high-mass stars are therefore out of thermal equilibrium with the low-mass stars.

For the stars of the Galactic center to evolve towards thermal equilibrium, the high-mass stars must lose kinetic energy to the nearby low-mass stars. But losing kinetic energy means losing orbital velocity at periapsis (the minimum distance from the Galactic center), so the massive stars drop deeper into the gravitational potential well. Conversely, the low-mass stars acquire energy, which forces them into larger orbits. This exchange in energy eventually segregates the stars, with the massive stars deep in the gravitational potential well, and the low-mass stars farther out of the potential well. But does this segregation happen over a reasonable timescale? More precisely, can a B main-sequence star sink into a close orbit around the central black hole over its hydrogen-burning lifetime?

Dynamical Friction

The simplest theory for the sinking of massive stars towards the central black hole assumes that massive stars lose energy to the surrounding stars through many brief binary interactions. This theory resembles the theory of energy exchange between ions and electrons in a plasma; like electrons and ions, stars in the Galactic center deflect each other as they pass, exchanging some kinetic energy and momentum. Each scattering is treated as a single event between two stars, independent of all other scattering that take place over time. In such a theory, two types of interaction between pairs of stars are possible: strong and infrequent deflections that exchange a large fraction of the total kinetic energy of the pair of stars in a single scattering, and weak and frequent deflections that exchange a modest fraction of the total kinetic energy in each scattering.

The amount of energy transferred in a binary scattering depends on the distance between the two stars, the relative velocity of the stars, and the relative masses of the stars. The smaller the minimum distance between two stars as they pass each other, the larger the deflection angle, and the larger the change of kinetic energy and momentum for each star. The slower the relative velocities of two widely-separated stars, the smaller the minimum distance between the stars, and the larger the change in kinetic energy and momentum. So the most direct way to change the orbit of a star is for a second star of nearly equal mass to pass close by a low velocity.

But how close must one star be from another to cause a large deflection? If the two stars, each of 1 solar mass, have a relative velocity of 500 km s^?1, they would need to pass within 0.001 AU of one another to be deflected by a large angle. Given that the average distance between the stars at the Galactic center is of order 0.01 parsecs, or 2,000 AU, the chances of a large-angle scattering compared to a scattering with a star at the average distance is of order 1 in 10¹². In fact, despite removing only a tiny fraction of a star?s kinetic energy, the weak scatterings occur at such a high rate that they usually sap a star of its kinetic energy long before the star experiences a single strong scattering. This weak-scattering drag on a star is called ?dynamical friction.?

Dynamical friction in the Galactic Center acts slowly despite the high density of stars. A star 1 parsec from the central black hole loses all of its kinetic energy in 1 billion years. This value scales proportionally with V³R^1.7/M, where M is the star?s mass, V is the star?s velocity, and R is the star?s distance to Sgr A^*.^[1] Compared to the age of the Galaxy, which is of order the age of the universe (around 16 billion years), a 1 billion year timescale is quite short, certainly short enough to redistribute energy among the low-mass stars and the compact stars. The 1 billion timescale is much too long, however, for dynamical friction to transport B main-sequence stars from a distant birth place into close orbit around Sgr A^*. We need to find a better mechanism for transporting the B main-sequence stars if the theory that these stars are born far from the central black hole is to survive.